%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.142", %%% date = "10 March 2022", %%% time = "06:44:34 MST", %%% filename = "benfords-law.bib", %%% address = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone = "+1 801 581 5254", %%% FAX = "+1 801 581 4148", %%% URL = "http://www.math.utah.edu/~beebe", %%% checksum = "07256 41802 183971 1851404", %%% email = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "bibliography; BibTeX; Benford's Law; %%% Bradford's Law; First Digit Phenomenon; %%% Gibrat's Law; Gibrat's Rule of Proportionate %%% Growth; Hall--Petch effect; Heaps' Law; %%% Herdan's Law; Law of Anomalous Numbers; %%% Lotka's Law; Pareto Distribution; Pareto Law; %%% Pareto Principle; power law; Significant %%% Digit Law; Simon--Yule model of text %%% generation; Stigler's Law; Yule process; %%% Zipf's Law", %%% license = "public domain", %%% supported = "yes", %%% docstring = "This is a bibliography about a numerical %%% curiousity variously known as Benford's Law %%% (see particularly entries Newcomb:1881:NFU, %%% Benford:1938:LAN, Pinkham:1961:DFS, %%% Logan:1978:FDP, Hill:1998:FDP, and %%% Block:2010:GEB, Nigrini:2012:BLA), the Law of %%% Anomalous Numbers, and the Significant Digit %%% Law: in a table of physical constants (and in %%% many other collections of measured numbers), %%% the leading digits do not have equal %%% probability of occurrence: the digit 1 is %%% more common than 2, which is more common than %%% 3, and so on. %%% %%% Publications recorded here appear in about %%% 480 different journals from accounting, %%% aerobiology, animal behavior, auditing, %%% astronomy, bible studies, biology, %%% biometrics, botany, business, chaos theory, %%% chemistry, colloid science, complexity, %%% computer science, conflict resolution, %%% cognitive science, criminology, critical %%% phenomena, demographics, drug design, %%% dynamics, earthquake detection, economics, %%% electoral studies, engineering, entropy, %%% expert systems, extinction, finance, %%% forensics, gambling, geography, geophysics, %%% human resources, hydrology, imaging science, %%% informetrics, language science, library %%% science, linguistics, marine systems, %%% marketing, mathematics, medicine, networking, %%% neuroscience, nuclear engineering, nuclear %%% medicine, nuclear science, operations %%% research, pathology, physics, plasticity, %%% political analysis, probability, psychology, %%% risk analysis, signal processing, simulation, %%% social sciences, sociology, statistical %%% mechanics, statistics, stock-market trading, %%% tax accounting, urban development, and urban %%% growth, reflecting the broad interest in the %%% unexpected phenomenon discovered and reported %%% first by the Canadian/American astronomer and %%% mathematician, Simon Newcomb, in a two-page %%% note in 1881, and then independently, and %%% with numerical evidence from much more %%% measured data, by American physicist Frank %%% Benford in 1938. %%% %%% In Benford's Law, the frequency of leading %%% decimal digit p is roughly log10(1 + (1 / %%% p)): that is, approximately 0.301, 0.176, %%% 0.125, 0.097, 0.079, 0.067, 0.058, 0.051, and %%% 0.046 for digits 1 through 9, instead of the %%% expected 1/9 ~= 0.111 for values taken from a %%% uniform distribution of random numbers. %%% %%% Another common statement of Benford's Law is %%% that the probability that a decimal number %%% representing a measured value begins with a %%% nonzero digit not greater than p is log10(1 + %%% p), that is, approximately 0.301, 0.477, %%% 0.602, 0.699, 0.778, 0.845, 0.903, 0.954, and %%% 1.000 for digits 1 through 9. %%% %%% A simple Benford-based rule of thumb is that %%% about half the values start with digit 1 or %%% 2, about 10 percent with 8 or 9, and 5 %%% percent with 9. %%% %%% For the second digits, the probabilities are %%% determined by the formula (see %%% Todter:2009:BLI) %%% %%% sum(q = 1:9) log10(1 + 1 / (10*q + p)), %%% %%% that is, approximately 0.120, 0.114, 0.109, %%% 0.104, 0.100, 0.097, 0.093, 0.090, 0.088, and %%% 0.085, for second decimal digits p = 0 %%% through 9. %%% %%% Probabilities for later digits are even %%% closer: see entries Newcomb:1881:NFU, %%% Feldstein:1976:CED, and Todter:2009:BLI. %%% %%% Benford's Law is not restricted to decimal %%% numbers: for base b (as long as b is not %%% huge, according to entry Schatte:1998:BLV), %%% just change the 10 values to b in the %%% previous formulas. Here are some results for %%% selected bases commonly used in computers, %%% showing the probability of the first (D1) or %%% second (D2) digit (first index) for base b %%% (second index): %%% %%% P[D1, 2]: 1.000 %%% P[D2, 2]: 0.585, 0.415 %%% %%% P[D1, 4]: 0.500, 0.292, 0.208 %%% P[D2, 4]: 0.304, 0.261, 0.230, 0.206 %%% %%% P[D1, 8]: 0.333, 0.195, 0.138, 0.107, %%% 0.088, 0.074, 0.064 %%% P[D2, 8]: 0.151, 0.141, 0.133, 0.126, 0.120, %%% 0.115, 0.110, 0.105 %%% %%% P[D1, 10]: 0.301, 0.176, 0.125, 0.097, %%% 0.079, 0.067, 0.058, 0.051, 0.046 %%% P[D2, 10]: 0.120, 0.114, 0.109, 0.104, 0.100, %%% 0.097, 0.093, 0.090, 0.088, 0.085 %%% %%% P[D1, 16]: 0.250, 0.146, 0.104, 0.080, %%% 0.066, 0.056, 0.048, 0.042, 0.038, %%% 0.034, 0.031, 0.029, 0.027, 0.025, %%% 0.023 %%% P[D2, 16]: 0.074, 0.072, 0.070, 0.068, 0.066, %%% 0.065, 0.064, 0.062, 0.061, 0.060, %%% 0.059, 0.058, 0.057, 0.056, 0.055, %%% 0.054 %%% %%% A small filter applied to the 1990 US Census %%% data in the Project Gutenberg file %%% uscen902.txt reports these frequencies of %%% first and second digits for the 5148 integer %%% values with at least 3 digits: %%% %%% F[D1]: 0.298, 0.215, 0.113, 0.082, %%% 0.098, 0.056, 0.055, 0.034, 0.049 %%% F[D2]: 0.166, 0.090, 0.096, 0.081, 0.100, %%% 0.122, 0.076, 0.073, 0.066, 0.130 %%% %%% For the 110 numbers in the atomic weights %%% section of the GNU units utility data file, %%% units.dat, we have %%% %%% F[D1]: 0.391, 0.309, 0.045, 0.036, %%% 0.055, 0.036, 0.036, 0.036, 0.055 %%% F[D2]: 0.173, 0.045, 0.109, 0.100, 0.145, %%% 0.145, 0.055, 0.055, 0.091, 0.082 %%% %%% From a snapshot of the country-area section %%% of the US CIA World Factbook %%% %%% https://www.cia.gov/library/publications/the-world-factbook/rankorder/2147rank.html %%% %%% taken at 18 November 2011, we find 1505 %%% numbers with these frequencies: %%% %%% F[D1]: 0.312, 0.275, 0.100, 0.067, %%% 0.058, 0.062, 0.046, 0.029, 0.050 %%% F[D2]: 0.167, 0.221, 0.092, 0.092, 0.062, %%% 0.067, 0.075, 0.067, 0.083, 0.075 %%% %%% From the country-population section %%% %%% https://www.cia.gov/library/publications/the-world-factbook/rankorder/2119rank.html %%% %%% of that book, including only the 163 values %%% of 1,000,000 or larger, we find %%% %%% F[D1]: 0.301, 0.202, 0.092, 0.135, %%% 0.055, 0.067, 0.055, 0.037, 0.055 %%% F[D2]: 0.147, 0.153, 0.110, 0.098, 0.098, %%% 0.123, 0.086, 0.043, 0.049, 0.092 %%% %%% From the section with the infant mortality %%% rates per 1000 births, %%% %%% https://www.cia.gov/library/publications/the-world-factbook/rankorder/2091rank.html %%% %%% there are 208 values with at least 4 digits, %%% with these frequencies: %%% %%% F[D1]: 0.361, 0.293, 0.043, 0.072, %%% 0.062, 0.087, 0.038, 0.014, 0.029 %%% F[D2]: 0.303, 0.139, 0.077, 0.058, 0.077, %%% 0.077, 0.106, 0.067, 0.043, 0.053 %%% %%% In the 2010 annual financial report for IBM %%% Corporation (its hundredth year of business) %%% %%% ftp://public.dhe.ibm.com/annualreport/2010/2010_ibm_annual.pdf %%% %%% there are 6126 numbers of two or more digits %%% that do not look like year dates (19xx and %%% 20xx), and they have these digit frequencies: %%% %%% F[D1]: 0.333, 0.160, 0.163, 0.086, %%% 0.068, 0.053, 0.047, 0.045, 0.046 %%% F[D2]: 0.172, 0.169, 0.096, 0.084, 0.085, %%% 0.095, 0.079, 0.074, 0.079, 0.068 %%% %%% For the first 100,000 primes, we have %%% %%% F[D1]: 0.311, 0.091, 0.090, 0.087, %%% 0.086, 0.085, 0.084, 0.083, 0.082 %%% F[D2]: 0.152, 0.151, 0.150, 0.078, 0.078, %%% 0.078, 0.078, 0.079, 0.078, 0.078 %%% %%% For the Fibonacci numbers, f(n), which %%% satisfy the recurrence f(n) = f(n-1) + f(n-2) %%% with f(1) = f(2) = 1, using the first 10000 %%% such numbers (the last of which has 2091 %%% digits), we find for the 9994 f(n) values %%% with at least two digits these frequencies: %%% %%% F[D1]: 0.301, 0.176, 0.125, 0.097, %%% 0.079, 0.067, 0.058, 0.051, 0.046 %%% F[D2]: 0.120, 0.114, 0.109, 0.105, 0.100, %%% 0.097, 0.094, 0.090, 0.088, 0.085 %%% %%% If we take the same set of Fibonacci numbers, %%% we find these frequencies for values output %%% in base 4: %%% %%% F[D1]: 0.500, 0.292, 0.208 %%% F[D2]: 0.303, 0.262, 0.230, 0.205 %%% %%% The Fibonacci numbers, and the members of %%% many other such sequences, can be proved %%% mathematically to obey Benford's Law exactly %%% (see entries Raimi:1976:FDP, Berger:2011:BLS, %%% and Berger:2011:BTB). %%% %%% Simon Newcomb made another significant %%% appearance in the history of science when he %%% proposed that the then-not-understood %%% perihelion advance of the planet Mercury %%% could be explained by modifying Newton's law %%% of gravitation from a $r^{-2}$ dependence to %%% $r^{-2.000\,000\,157\,4}$. His proposal was %%% later shown to contradict more accurate %%% measurements of the orbit of Earth's Moon. %%% That story is told on pages 93--95 of %%% Clifford M. Will's 1993 book, ``Was Einstein %%% Right?'' (ISBN 0-465-09086-9). %%% %%% Although many papers purport to explain %%% Benford's Law, a recent paper (entry %%% Berger:2011:BLS) argues that several of the %%% prior published explanations are flawed. An %%% earlier paper (entry Raimi:1976:FDP) examines %%% the literature on Benford's Law up to 1976, %%% and points out flaws in several previous %%% explanations of the law. That paper also %%% gives a bibliography of previous important %%% works on the problem. A recent survey (entry %%% Berger:2011:BTB) may provide stronger %%% mathematical grounds for Benford's Law. %%% %%% Benford's Law has been applied to the problem %%% of detection of fraud in data, elections, and %%% finance (see, among other entries, %%% Hill:1998:FDP, Nigrini:1992:DIE, %%% Swanson:2003:DPF, Durtschi:2004:EUB, %%% Geyer:2004:DFD, Bhattacharya:2005:CPA, %%% Cleary:2005:ADA, Lu:2005:DFH, Lu:2006:AFD, %%% Schneider:2006:RAS, Cho:2007:BBL, %%% Diekmann:2007:FDU, Hand:2007:DDD, %%% Todter:2009:BLI, Corazza:2010:CFM, and %%% Roukema:2014:FDA, and the cautionary view in %%% Diekmann:2010:BLF). %%% %%% Benford's Law also applies to floating-point %%% numbers in computer arithmetic (see entries %%% Konheim:1965:MDT, Hamming:1970:DN, %%% Tsao:1974:DSD, Feldstein:1976:CED, %%% Feldstein:1986:OUS, Schatte:1988:ASC), where %%% it has implications for the frequency of %%% significance loss in subtraction of %%% like-signed numbers. %%% %%% This bibliography also includes publications %%% about two related empirical laws: Heaps' Law %%% and Zipf's Law, because they often occur in %%% discussions of Benford's Law. %%% %%% Heaps' Law (see entry Heaps:1978:IRC) is an %%% empirical observation from linguistics that %%% the proportion of words from a vocabulary %%% grows exponentially (but with a small %%% exponent, often between 0.4 and 0.6) with the %%% number of words in the text of documents. %%% %%% Zipf's Law says that if S is some stochastic %%% (random) variable, the probability that S %%% exceeds s is proportional to 1 / s. The %%% variable S might be, for example, the %%% population of a city (small cities are more %%% numerous than large ones). Zipf's Law is %%% a special case of the Pareto distribution %%% (see remark in entry Pareto:1896:CEP). %%% %%% Zipf's law finds application in the design of %%% information retrieval systems (e.g., hash %%% tables, databases, filesystems, and Web %%% searches), as well as in astronomy, biology, %%% computer science, demography, earth science, %%% economics, and physics. See entry %%% Newman:2005:PLP for an outstanding survey %%% that deserves to be widely read by %%% scientists, and entry Baldi:2003:MIW for a %%% book-length treatment. %%% %%% There is an actively-maintained bibliography %%% of Benford's Law at %%% %%% http://www.benfordonline.net/ %%% %%% (see entry Berger:2009:BOB). It includes %%% material from entries Hurlimann:2006:BLB and %%% Nigrini:1992:DIE. Although BibTeX output is %%% mentioned, it may not yet be implemented. %%% %%% There is an online bibliography of Zipf's Law %%% at %%% %%% http://www.nslij-genetics.org/wli/zipf/ %%% %%% (see entry Li:2011:ZLB). %%% %%% There is a collection of several papers on %%% election-fraud detection, some of them using %%% Benford's Law analyses, at %%% %%% http://www-personal.umich.edu/~wmebane/ %%% %%% At version 1.141, the year coverage looked like %%% this: %%% %%% 1881 ( 1) 1929 ( 1) 1977 ( 9) %%% 1882 ( 0) 1930 ( 0) 1978 ( 11) %%% 1883 ( 0) 1931 ( 1) 1979 ( 6) %%% 1884 ( 0) 1932 ( 2) 1980 ( 8) %%% 1885 ( 0) 1933 ( 0) 1981 ( 12) %%% 1886 ( 0) 1934 ( 1) 1982 ( 10) %%% 1887 ( 0) 1935 ( 1) 1983 ( 6) %%% 1888 ( 0) 1936 ( 1) 1984 ( 16) %%% 1889 ( 0) 1937 ( 1) 1985 ( 10) %%% 1890 ( 0) 1938 ( 2) 1986 ( 16) %%% 1891 ( 0) 1939 ( 0) 1987 ( 18) %%% 1892 ( 0) 1940 ( 0) 1988 ( 12) %%% 1893 ( 0) 1941 ( 0) 1989 ( 14) %%% 1894 ( 0) 1942 ( 0) 1990 ( 9) %%% 1895 ( 0) 1943 ( 1) 1991 ( 11) %%% 1896 ( 1) 1944 ( 1) 1992 ( 15) %%% 1897 ( 0) 1945 ( 2) 1993 ( 11) %%% 1898 ( 0) 1946 ( 1) 1994 ( 12) %%% 1899 ( 0) 1947 ( 0) 1995 ( 19) %%% 1900 ( 0) 1948 ( 2) 1996 ( 23) %%% 1901 ( 0) 1949 ( 2) 1997 ( 17) %%% 1902 ( 0) 1950 ( 3) 1998 ( 21) %%% 1903 ( 0) 1951 ( 0) 1999 ( 28) %%% 1904 ( 0) 1952 ( 1) 2000 ( 22) %%% 1905 ( 0) 1953 ( 1) 2001 ( 26) %%% 1906 ( 0) 1954 ( 1) 2002 ( 34) %%% 1907 ( 0) 1955 ( 1) 2003 ( 42) %%% 1908 ( 0) 1956 ( 0) 2004 ( 46) %%% 1909 ( 0) 1957 ( 5) 2005 ( 68) %%% 1910 ( 1) 1958 ( 1) 2006 ( 60) %%% 1911 ( 0) 1959 ( 1) 2007 ( 49) %%% 1912 ( 1) 1960 ( 1) 2008 ( 54) %%% 1913 ( 1) 1961 ( 2) 2009 ( 78) %%% 1914 ( 1) 1962 ( 2) 2010 ( 96) %%% 1915 ( 1) 1963 ( 1) 2011 ( 88) %%% 1916 ( 1) 1964 ( 1) 2012 ( 59) %%% 1917 ( 1) 1965 ( 4) 2013 ( 46) %%% 1918 ( 0) 1966 ( 4) 2014 ( 51) %%% 1919 ( 0) 1967 ( 3) 2015 ( 78) %%% 1920 ( 1) 1968 ( 4) 2016 ( 23) %%% 1921 ( 0) 1969 ( 12) 2017 ( 12) %%% 1922 ( 1) 1970 ( 8) 2018 ( 13) %%% 1923 ( 0) 1971 ( 2) 2019 ( 7) %%% 1924 ( 0) 1972 ( 7) 2020 ( 15) %%% 1925 ( 1) 1973 ( 10) 2021 ( 10) %%% 1926 ( 1) 1974 ( 12) 2022 ( 1) %%% 1927 ( 1) 1975 ( 13) %%% 1928 ( 2) 1976 ( 13) %%% 20xx ( 1) %%% %%% Article: 1137 %%% Book: 93 %%% InCollection: 51 %%% InProceedings: 64 %%% Manual: 1 %%% MastersThesis: 6 %%% Misc: 20 %%% PhdThesis: 8 %%% Proceedings: 8 %%% TechReport: 50 %%% Unpublished: 3 %%% %%% Total entries: 1441 %%% %%% Data for this bibliography have been %%% collected from %%% %%% * the University of Utah Mathematics %%% Department bibliography archives %%% %%% * the TeX User Group bibliography %%% archives %%% %%% * the Karlsruhe Computer Science %%% bibliography archives %%% %%% * the Karlsruhe virtual catalog at %%% http://www.ubka.uni-karlsruhe.de/kvk_en.html %%% %%% * the US Library of Congress catalog at %%% http://catalog.loc.gov/ %%% %%% * the author's cattobib utility, which %%% provides Z39.50 interfaces to many %%% large libraries around the world, at %%% http://www.math.utah.edu/pub/cattobib/ %%% %%% * the American Institute of Physics %%% Scitation database at %%% http://scitation.aip.org/search_scitation %%% %%% * the American Physical Society PROLA %%% database at %%% http://publish.aps.org/search %%% %%% * the American Mathematical Society %%% MathSciNet database at %%% http://ams.rice.edu/mathscinet/search.html %%% %%% * the Bulletin of the Atomic Scientists %%% (archives available, albeit inconveniently, %%% via http://books.google.com/ and %%% http://bos.sagepub.com/search) %%% %%% * the Canadian Journal of Physics %%% database at %%% http://www.nrcresearchpress.com/search/advanced %%% %%% * the European Mathematical Society %%% Zentralblatt fuer Mathematik database %%% at http://zb.msri.org/ZMATH/zmath/en/ %%% %%% * the Europhysics journal archive at %%% http://www.europhysicsnews.org/ %%% %%% * the JSTOR database at %%% http://www.jstor.org/ %%% %%% * the Nature journal archive at %%% http://www.nature.com/search/ %%% %%% * the journal archives of the National %%% Academy of Science at %%% http://www.pnas.org/ %%% %%% * the Project Euclid archives at %%% http://projecteuclid.org/ %%% %%% * the SAO/NASA Astrophysics Data System at %%% http://adsabs.harvard.edu/ %%% %%% * the Science journal archive at %%% http://www.sciencemag.org/search %%% %%% * the SPIRES high-energy physics %%% literature database at the Stanford %%% Linear Accelerator at Stanford %%% University at %%% http://www.slac.stanford.edu/spires/hep/search/ %%% %%% * the Springer journal database at %%% http://www.springer.com/?SGWID=0-102-13-0-0 %%% %%% * the Wiley journal database at %%% http://onlinelibrary.wiley.com/ %%% %%% * many online library catalogs, including %%% those of the British Library, the %%% Karlsruhe Virtual Library catalog, the %%% Oxford University Library, the University of %%% California library system, and the US %%% Library of Congress. %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ====================================================================

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%%% ==================================================================== %%% Acknowledgement abbreviations:

@String{ack-mjn= "Mark J. Nigrini"} @String{ack-nhfb= "Nelson H. F. Beebe, University of Utah, Department of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1 801 581 4148, e-mail: \path|beebe@math.utah.edu|, \path|beebe@acm.org|, \path|beebe@computer.org| (Internet), URL: \path|http://www.math.utah.edu/~beebe/|"}

%%% ==================================================================== %%% Journal abbreviations:

@String{j-ACCOUNT-HIST-J= "Accounting Historians Journal"} @String{j-ACCOUNT-RES= "Accountability in Research"} @String{j-ACCOUNT-REV= "The Accounting Review"} @String{j-ACCOUNTANCY= "Accountancy: The journal of the {Institute of Chartered Accountants in England and Wales}"} @String{j-ACTA-ARITHMETICA= "Acta Arithmetica"} @String{j-ACTA-ASTRONAUT= "Acta Astronautica"} @String{j-ACTA-PHYS-POL-B= "Acta Physica Polonica B"} @String{j-ADV-APPL-STAT= "Advances and Applications in Statistics"} @String{j-ADV-COMPLEX-SYST= "Advances in Complex Systems"} @String{j-ADV-MATH= "Advances in Mathematics"} @String{j-AEQUATIONES-MATHEMATICAE= "Aequationes Mathematicae"} @String{j-AEROBIOLOGIA= "Aerobiologia"} @String{j-AM-DOC= "American Documentation"} @String{j-AM-J-APPL-SCI= "American Journal of Applied Sciences"} @String{j-AM-J-ECON-SOCIOL= "The American Journal of Economics and Sociology"} @String{j-AM-J-MATH= "American Journal of Mathematics"} @String{j-AM-J-MATH-MANAGE-SCI= "American Journal of Mathematical and Management Sciences"} @String{j-AM-SCI= "American Scientist"} @String{j-AMER-J-PHYSICS= "American Journal of Physics"} @String{j-AMER-J-PSYCHOLOGY= "The American Journal of Psychology"} @String{j-AMER-MATH-MONTHLY= "American Mathematical Monthly"} @String{j-AMER-STAT= "The American Statistician"} @String{j-ANALYST= "The Analyst"} @String{j-ANIM-BEHAV= "Animal Behaviour"} @String{j-ANN-APPL-PROBAB= "Annals of Applied Probability"} @String{j-ANN-INST-STAT-MATH-TOKYO= "Annals of the Institute of Statistical Mathematics (Tokyo)"} @String{j-ANN-MATH-STAT= "Annals of Mathematical Statistics"} @String{j-ANN-PHYS= "Annals of Physics"} @String{j-ANN-PROBAB= "Annals of Probability"} @String{j-ANN-STAT= "Annals of Statistics"} @String{j-APPL-ECON-LETT= "Applied Economics Letters"} @String{j-APPL-FINANC-ECON= "Applied Financial Economics"} @String{j-APPL-MATH-COMP= "Applied Mathematics and Computation"} @String{j-APPL-STAT= "Applied Statistics"} @String{j-AQUAT-BOT= "Aquatic Botany"} @String{j-ARKHIMEDES= "Arkhimedes"} @String{j-ASIA-PAC-MATH-NEWSL= "Asia Pacific Mathematics Newsletter"} @String{j-ASLIB-PROC= "ASLIB Proceedings"} @String{j-ASTROPART-PHYS= "Astroparticle Physics"} @String{j-ATMOS-ENV= "Atmospheric Environment"} @String{j-AUDITING= "Auditing: A Journal of Practice \& Theory"} @String{j-AUSTRALIAN-COMP-J= "Australian Computer Journal"} @String{j-AUSTRALIAN-MATH-SOC-GAZ= "Australian Mathematical Society Gazette"} @String{j-BELL-SYST-TECH-J= "The Bell System Technical Journal"} @String{j-BIOINFORMATICS= "Bioinformatics"} @String{j-BIOMETRIKA= "Biometrika"} @String{j-BIOSYSTEMS= "Biosystems (A6E)"} @String{j-BIT-NUM-MATH= "BIT Numerical Mathematics"} @String{j-BMC-BIOINFORMATICS= "BMC Bioinformatics"} @String{j-BOLL-UNIONE-MAT-ITAL= "Bollettino della Unione matematica italiana"} @String{j-BRAIN-COGN= "Brain and Cognition"} @String{j-BULL-INST-MATH-STAT= "Bulletin --- Institute of Mathematical Statistics"} @String{j-C-R-ACAD-SCI-I= "Comptes rendus de l'Acad{\'e}mie des sciences. S{\'e}rie I, Math{\'e}matique"} @String{j-C-R-ACAD-SCI-PARIS-SER-AB= "Comptes Rendus Hebdomadaires des S{\'e}ances de l'Acad{\'e}mie des Sciences. 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%%% ==================================================================== %%% Publisher abbreviations:

@String{pub-ACADEMIC= "Academic Press"} @String{pub-ACADEMIC:adr= "New York, USA"} @String{pub-ACM= "ACM Press"} @String{pub-ACM:adr= "New York, NY 10036, USA"} @String{pub-ADAM-HILGER= "Adam Hilger Ltd."} @String{pub-ADAM-HILGER:adr= "Bristol, UK"} @String{pub-AMS= "American Mathematical Society"} @String{pub-AMS:adr= "Providence, RI, USA"} @String{pub-AW= "Ad{\-d}i{\-s}on-Wes{\-l}ey"} @String{pub-AW:adr= "Reading, MA, USA"} @String{pub-BASIC-BOOKS= "Basic Books"} @String{pub-BASIC-BOOKS:adr= "New York, NY, USA"} @String{pub-BUTTERWORTHS= "Butterworths"} @String{pub-BUTTERWORTHS:adr= "London, UK"} @String{pub-CAMBRIDGE= "Cambridge University Press"} @String{pub-CAMBRIDGE:adr= "Cambridge, UK"} @String{pub-CLARENDON= "Clarendon Press"} @String{pub-CLARENDON:adr= "Oxford, UK"} @String{pub-CRC= "CRC Press"} @String{pub-CRC:adr= "2000 N.W. Corporate Blvd., Boca Raton, FL 33431-9868, USA"} @String{pub-DOVER= "Dover Publications, Inc."} @String{pub-DOVER:adr= "New York, NY, USA"} @String{pub-ELSEVIER= "Elsevier"} @String{pub-ELSEVIER:adr= "Amsterdam, The Netherlands"} @String{pub-FARRAR= "Farrar, Strauss, and Giroux"} @String{pub-FARRAR:adr= "New York, NY, USA"} @String{pub-HARVARD= "Harvard University Press"} @String{pub-HARVARD:adr= "Cambridge, MA, USA"} @String{pub-HOUGHTON-MIFFLIN= "Houghton-Mifflin"} @String{pub-HOUGHTON-MIFFLIN:adr= "Boston, MA, USA"} @String{pub-IEEE= "IEEE Computer Society Press"} @String{pub-IEEE:adr= "1109 Spring Street, Suite 300, Silver Spring, MD 20910, USA"} @String{pub-MATH-ASSOC-AMER= "Mathematical Association of America"} @String{pub-MATH-ASSOC-AMER:adr= "Washington, DC, USA"} @String{pub-MCGRAW-HILL= "Mc{\-}Graw-Hill"} @String{pub-MCGRAW-HILL:adr= "New York, NY, USA"} @String{pub-MIT= "MIT Press"} @String{pub-MIT:adr= "Cambridge, MA, USA"} @String{pub-MORGAN-KAUFMANN= "Morgan Kaufmann Publishers"} @String{pub-MORGAN-KAUFMANN:adr= "Los Altos, CA 94022, USA"} @String{pub-NORTH-HOLLAND= "North-Holland Publishing Co."} @String{pub-NORTH-HOLLAND:adr= "Amsterdam, The Netherlands"} @String{pub-OXFORD= "Oxford University Press"} @String{pub-OXFORD:adr= "Walton Street, Oxford OX2 6DP, UK"} @String{pub-PH= "Pren{\-}tice-Hall"} @String{pub-PH:adr= "Upper Saddle River, NJ, USA"} @String{pub-PRINCETON= "Princeton University Press"} @String{pub-PRINCETON:adr= "Princeton, NJ, USA"} @String{pub-SPIE= "SPIE Optical Engineering Press"} @String{pub-SPIE:adr= "Bellingham, WA, USA"} @String{pub-SV= "Spring{\-}er-Ver{\-}lag"} @String{pub-SV:adr= "Berlin, Germany~/ Heidelberg, Germany~/ London, UK~/ etc."} @String{pub-U-MICHIGAN= "University of Michigan Press"} @String{pub-U-MICHIGAN:adr= "Ann Arbor, MI, USA"} @String{pub-W-H-FREEMAN= "W. H. {Freeman and Company}"} @String{pub-W-H-FREEMAN:adr= "New York, NY, USA"} @String{pub-WILEY= "Wiley"} @String{pub-WILEY:adr= "New York, NY, USA"} @String{pub-WORLD-SCI= "World Scientific Publishing Co. Pte. Ltd."} @String{pub-WORLD-SCI:adr= "P. O. Box 128, Farrer Road, Singapore 9128"} @String{pub-YALE= "Yale University Press"} @String{pub-YALE:adr= "New Haven, CT, USA"}

%%% ==================================================================== %%% Series abbreviations:

@String{ser-LNAI= "Lecture Notes in Artificial Intelligence"} @String{ser-LNCS= "Lecture Notes in Computer Science"} @String{ser-LNM= "Lecture Notes in Mathematics"}

%%% ==================================================================== %%% Bibliography entries, sorted by year, and then by citation label:

@Article{Newcomb:1881:NFU, author = "Simon Newcomb", title = "Note on the frequency of use of the different digits in natural numbers", journal = j-AM-J-MATH, volume = "4", number = "1--4", pages = "39--40", year = "1881", CODEN = "AJMAAN", ISSN = "0002-9327 (print), 1080-6377 (electronic)", ISSN-L = "0002-9327", bibdate = "Thu Feb 15 16:35:24 2007", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.jstor.org/stable/2369148", abstract = "That the ten digits do not occur with equal frequency must be evident to any one making much use of logarithmic tables, and noticing how much faster the first pages wear out than the last ones. The first significant figure is oftener 1 than any other digit, and the frequency diminishes up to 9.", acknowledgement = ack-nhfb, author-dates = "March 12, 1835--July 11, 1909", fjournal = "American Journal of Mathematics", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", remark = "From p. 40: ``The law of probability of the occurrence of numbers is such that all mantissas of their logarithms are equally probable.''. The article contains no references to earlier work.", remark-2 = "The papers \cite{Boring:1920:LNL,Raimi:1976:FDP} are the earliest citations of Newcomb's work that I have yet found in connection with Benford's Law.", xxURL = "http://en.wikipedia.org/wiki/Simon_Newcomb", } @Book{Pareto:1896:CEP, author = "Vilfredo Pareto", title = "Cours d'{\'e}conomie politique profess{\'e} a l'universit{\'e} de {Lausanne}. (French) [{Course} on political economy given at the {University of Lausanne}]", publisher = "F. Rouge", address = "Lausanne, Switzerland", pages = "????", year = "1896--1897", LCCN = "HB173 .P22", bibdate = "Wed Nov 16 08:41:03 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, language = "French", remark = "Two volumes issued in successive years. The Italian economist and mathematician, Vilfredo Federico (born in Paris, France, and initially named Wilfried Fritz) Pareto (15 July 1848--19 August 1923), introduced the 80--20 rule in economics (80\% of the wealth is owned by 20\% of the people, which was true at the time in Italy, and found to be similar in other countries). He developed the Pareto distribution, in which a random variable $X$ has the property that the probability that it is greater than some number $x$ is given by $ {\rm Pr}(X > x) = (x_m / x)^\alpha $ for $ x > x_m $, and $ {\rm Pr}(X > x) = 1 $ otherwise. The positive value $ x_m $ is a cutoff, and as $ \alpha \to \infty $, the Pareto distribution approaches a Dirac delta function, $ \delta (x - x_m) $. When this models the distribution of wealth, the exponent $ \alpha $ is called the {\em Pareto index}.", } @Article{Doolittle:1910:SNF, author = "C. L. Doolittle", title = "{Simon Newcomb, F.R.S., LL.D., D.C.L.}", journal = j-PROC-AMER-PHIL-SOC, volume = "49", number = "197", pages = "iii--xviii", month = oct # "\slash " # dec, year = "1910", CODEN = "PAPCAA", ISSN = "0003-049X (print), 2326-9243 (electronic)", ISSN-L = "0003-049X", bibdate = "Sat Aug 24 19:37:51 MDT 2019", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/procamerphilsoc1900.bib", URL = "https://www.jstor.org/stable/984092", acknowledgement = ack-nhfb, ajournal = "Proc. Amer. Philos. Soc.", fjournal = "Proceedings of the American Philosophical Society", journal-URL = "http://www.jstor.org/journal/procamerphilsoci", remark = "This obituary does not mention Newcomb's short paper on the leading-digit phenomenon now known as Benford's Law, but it does give an overview of Newcomb's humble origins, education, life, and scientific influence.", } @Book{Poincare:1912:CPL, author = "H. Poincar{\'e}", title = "Calcul des Probabilit{\'e}s: Le{\c{c}}ons profess{\'e}es pendant le deuxi{\`e} me semestre 1893--1894. ({French}) [{Calculation} of Probabilities: Lectures from the second semester 1893--1894]", publisher = "Gauthier-Villars", address = "Paris, France", pages = "ii + 274", year = "1912", LCCN = "QA273 .P75", bibdate = "Mon Nov 14 10:08:18 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, author-dates = "1854--1912", language = "French", remark = "See pp. 313--320.", } @Article{Auerbach:1913:GBG, author = "Felix Auerbach", title = "{Das Gesetz der Bev{\"o}lkerungskonzentration}. ({German}) [{The} law of population concentration]", journal = "{Petermann's Geographische Mitteilungen}", volume = "59", number = "1", pages = "74--76", month = "????", year = "1913", CODEN = "PGGMA3", ISSN = "0031-6229", ISSN-L = "0031-6229", bibdate = "Thu Nov 24 10:49:06 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "This paper is reported in \cite{Newman:2005:PLP} to be the first discovery of {\em Zipf's Law\/} in connection with city populations (see \cite{Zipf:1932:SSP}). See also \cite{Estoup:1916:GSM}.", URL = "http://hdl.handle.net/2027/mdp.39015035583528?urlappend=%3Bseq=122", acknowledgement = ack-nhfb, ajournal = "Petermanns Geogr. Mitteilung", ajournal-2 = "Petermanns Geogr. Mitt.", language = "German", } @Article{Stamp:1914:NIP, author = "J. C. Stamp", title = "A New Illustration of {Pareto}'s Law", journal = j-J-R-STAT-SOC, volume = "77", number = "2", pages = "200--204", month = jan, year = "1914", DOI = "https://doi.org/10.2307/2339802", ISSN = "0952-8385", ISSN-L = "0952-8385", bibdate = "Sat Jan 24 11:18:04 MST 2015", bibsource = "http://www.jstor.org/stable/i315805; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jrss-a-1910.bib", URL = "http://www.jstor.org/stable/2339802", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society", journal-URL = "http://www.jstor.org/journals/09528385.html", } @Article{Weyl:1915:GZM, author = "Hermann Weyl", title = "{{\"U}ber die Gleichverteilung von Zahlen mod. Eins}. ({German}) [{On} the uniform distribution of numbers mod. one]", journal = j-MATH-ANN, volume = "77", number = "3", pages = "313--352", month = "????", year = "1915", CODEN = "MAANA3", DOI = "https://doi.org/10.1007/BF01475864", ISSN = "0025-5831 (print), 1432-1807 (electronic)", ISSN-L = "0025-5831", bibdate = "Sun Nov 13 10:54:49 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Math. Ann.", fjournal = "Mathematische Annalen", journal-URL = "http://link.springer.com/journal/208", language = "German", xxyear = "1916", } @Book{Estoup:1916:GSM, author = "J. B. Estoup", title = "Gammes St{\'e}nographiques: m{\'e}thode \& exercices pour l'acquisition de la vitesse ({French}) [Stenographic ranges: method and exercises for achieving speed]", publisher = "Institut St{\'e}nographique de France", address = "Paris, France", edition = "Fourth", pages = "151", year = "1916", LCCN = "????", bibdate = "Thu Nov 24 14:24:28 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "This on shorthand writing is the earliest known publication of the power-law distribution of word frequencies, generally credited to \cite{Zipf:1932:SSP}.", acknowledgement = ack-nhfb, keywords = "Zipf's Law", language = "French", } @Article{Franel:1917:PTL, author = "J. Franel", title = "{{\`A}} propos des tables de logarithmes. ({French}) [{On} tables of logarithms]", journal = "Festschrift Naturforschenden der Gesellschaft in Z{\"u}rich Vierteljahrsschrift", volume = "62", number = "??", pages = "286--295", month = "????", year = "1917", CODEN = "????", ISSN = "????", bibdate = "Mon Nov 14 10:09:40 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, language = "French", } @Article{Boring:1920:LNL, author = "Edwin G. Boring", title = "The Logic of the Normal Law of Error in Mental Measurement", journal = j-AMER-J-PSYCHOLOGY, volume = "31", number = "1", pages = "1--33", month = jan, year = "1920", CODEN = "AJPCAA", ISSN = "0002-9556 (print), 1939-8298 (electronic)", ISSN-L = "0002-9556", bibdate = "Thu Dec 01 21:35:43 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/1413989", acknowledgement = ack-nhfb, fjournal = "The American journal of psychology", remark = "A citation search in Google Scholar on 1-Dec-2011 found this article as the only one citing Newcomb's work \cite{Newcomb:1881:NFU} before it was brought to wider view by Raimi's frequently-cited paper \cite{Raimi:1976:FDP}. Boring's article has only brief mention of Newcomb's work, and then solely in connection with the believed randomness of digits of transcendental numbers.", } @InCollection{Macaulay:1922:PLG, author = "F. R. Macaulay", editor = "W. C. Mitchell", booktitle = "Income in the {United States}: Its Amount and Distribution: 1909--1919", title = "{Pareto}'s law and the general problem of mathematical describing the frequency of distribution of income", publisher = "National Bureau of Economic Research", address = "New York, NY, USA", pages = "??--??", year = "1922", LCCN = "????", bibdate = "Fri Nov 25 09:06:43 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Yule:1925:MTE, author = "G. Udny Yule", title = "A Mathematical Theory of Evolution, Based on the Conclusions of {Dr. J. C. Willis, F.R.S.}", journal = j-PHILOS-TRANS-R-SOC-LOND-SER-B-BIO-SCI, volume = "213", number = "??", pages = "21--87", day = "14", month = may, year = "1925", CODEN = "PTRBAE", DOI = "https://doi.org/10.1098/rstb.1925.0002", ISSN = "0962-8436 (print), 1471-2970 (electronic)", ISSN-L = "0962-8436", bibdate = "Tue Nov 15 10:45:15 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://rstb.royalsocietypublishing.org/content/213/402-410/21", acknowledgement = ack-nhfb, fjournal = "Philosophical transactions of the Royal Society of London Series B, Biological sciences", } @Article{Lotka:1926:FDS, author = "Alfred J. Lotka", title = "The Frequency Distribution of Scientific Productivity", journal = j-J-WASH-ACAD-SCI, volume = "16", number = "12", pages = "317--323", day = "19", month = jun, year = "1926", CODEN = "JWASA3", ISSN = "0043-0439", ISSN-L = "0043-0439", bibdate = "Thu Nov 24 14:36:11 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "This is the earliest known publication on the phenomenon known as Zipf's Law, here applied to publication counts of chemists and physicists. Lotka's Law (a term possibly first used in \cite{Zipf:1949:HBP}) says that the number of authors producing $n$ publications is about $ 1 / n^2 $ of the number producing only one. This `law' seems to have been misunderstood and misapplied in other fields; see \cite{Potter:1981:LLR}.", URL = "http://www.jstor.org/stable/24529203", acknowledgement = ack-nhfb, fjournal = "Journal of the Washington Academy of Sciences", journal-URL = "http://www.jstor.org/journal/jwashacadscie", keywords = "Lotka's Law", } @Article{Yule:1927:RS, author = "G. Udny Yule", title = "On Reading a Scale", journal = j-J-R-STAT-SOC, volume = "90", number = "3", pages = "570--587", month = "????", year = "1927", ISSN = "0952-8385", ISSN-L = "0952-8385", bibdate = "Wed Nov 23 12:29:25 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See related later work \cite{Preece:1981:DFD}.", URL = "http://www.jstor.org/stable/2341205", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society", journal-URL = "http://www.jstor.org/journals/09528385.html", remark = "This paper reports on human bias in the recording of last digits in measurements, or from roundings, or from later scale conversions.", } @Article{Condon:1928:SV, author = "E. U. Condon", title = "Statistics of vocabulary", journal = j-SCIENCE, volume = "67", number = "1733", pages = "300--??", day = "16", month = mar, year = "1928", CODEN = "SCIEAS", DOI = "https://doi.org/10.1126/science.67.1733.300", ISSN = "0036-8075 (print), 1095-9203 (electronic)", ISSN-L = "0036-8075", bibdate = "Fri Nov 25 09:04:50 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Science", journal-URL = "http://www.sciencemag.org/archive/", } @Article{Sun:1928:EFD, author = "C. P. Sun", title = "On the Examination of Final Digits by Experiments in Artificial Sampling", journal = j-BIOMETRIKA, volume = "20A", number = "1/2", pages = "64--68", month = jul, year = "1928", CODEN = "BIOKAX", DOI = "https://doi.org/10.2307/2331941", ISSN = "0006-3444 (print), 1464-3510 (electronic)", ISSN-L = "0006-3444", bibdate = "Sat Jun 21 13:38:07 MDT 2014", bibsource = "http://www.jstor.org/journals/00063444.html; http://www.jstor.org/stable/i315380; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/biometrika1920.bib", URL = "http://www.jstor.org/stable/2331941", acknowledgement = ack-nhfb, fjournal = "Biometrika", journal-URL = "http://www.jstor.org/journals/00063444.html", } @Article{Zipf:1929:RFD, author = "George Kingsley Zipf", title = "Relative frequency as a determinant of phonetic change", journal = "Harvard Studies in Classical Philology", volume = "40", number = "??", pages = "1--95", month = "????", year = "1929", CODEN = "????", ISSN = "0073-0688", ISSN-L = "0073-0688", bibdate = "Fri Nov 25 09:02:14 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/journals/00730688.html; http://www.jstor.org/stable/310585", acknowledgement = ack-nhfb, } @Book{Gibrat:1931:IE, author = "Robert Gibrat", title = "Les In{\'e}galit{\'e}s {\'e}conomiques", publisher = "Librairie du Recueil Sirey", address = "Paris, France", pages = "286", year = "1931", LCCN = "HB251.GIB", bibdate = "Mon Feb 18 14:43:11 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; library.ox.ac.uk:210/ADVANCE", note = "Applications: aux in{\'e}galit{\'e}s des richesses, {\`a} la concentration des entreprises, aux populations des villes, aux statistiques des familles, etc. d'une loi nouvelle. La loi de l'effet proportionnel.", URL = "http://en.wikipedia.org/wiki/Gibrat%27s_law", acknowledgement = ack-nhfb, author-dates = "1904--1980", remark-1 = "According to at least one library catalog (SWB), this book may be the author's doctoral dissertation from l'Universit{\'e} de Paris in 1931.", remark-2 = "This book is cited as the origin of Gibrat's Law, or Gibrat's rule of proportionate growth, that ``the size of a firm and its growth rate are independent''.", } @Misc{Anonymous:1932:FL, author = "Anonymous", title = "Fishpole Lamp", howpublished = "Web site", day = "30", month = apr, year = "1932", bibdate = "Sat Feb 23 14:52:20 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "This Web site reports that General Electric researcher Frank Benford invented the electric light pointer, the forerunner of the modern laser pointer.", URL = "http://scienceservice.si.edu/pages/012020.htm", acknowledgement = ack-nhfb, remark = "CD 2478048, E\&MP12.020, Electric Appliances \ Apparatus.", } @Book{Zipf:1932:SSP, author = "George Kingsley Zipf", title = "Selected studies of the principle of relative frequency in language", publisher = pub-HARVARD, address = pub-HARVARD:adr, pages = "51", year = "1932", LCCN = "P123 .Z5", bibdate = "Wed Nov 16 09:03:46 MST 2011", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "This book is the origin of {\em Zipf's Law}, but the phenomenon was apparently first reported two decades earlier in \cite{Auerbach:1913:GBG} for city populations, in \cite{Estoup:1916:GSM} for word frequencies, and in \cite{Lotka:1926:FDS} for researcher publication counts.", acknowledgement = ack-nhfb, author-dates = "1902--1950", subject = "Plautus, Titus Maccius; language and languages; semantics; comparative linguistics; Chinese language; phonology; Latin language; glossaries, vocabularies, etc; dialects; China; Beijing", } @Article{Bradford:1934:SIS, author = "Samuel C. Bradford", title = "Sources of Information on Specific Subjects", journal = "Engineering: An Illustrated Weekly Journal (London)", volume = "137", number = "??", pages = "85--86", day = "26", month = jan, year = "1934", CODEN = "????", ISSN = "????", bibdate = "Fri Nov 25 11:47:10 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "This paper (reprinted in \cite{Bradford:1985:SIS}) introduces an observation later known as Bradford's Law: there are exponentially-diminishing returns of extending a search for references in journals.", URL = "http://en.wikipedia.org/wiki/Bradford%27s_law", acknowledgement = ack-nhfb, } @Book{Zipf:1935:PBL, author = "George Kingsley Zipf", title = "The psycho-biology of language: an introduction to dynamic philology", publisher = pub-HOUGHTON-MIFFLIN, address = pub-HOUGHTON-MIFFLIN:adr, pages = "ix + 336", year = "1935", LCCN = "P105 .Z5", bibdate = "Wed Nov 16 09:07:09 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; prodorbis.library.yale.edu:7090/voyager", acknowledgement = ack-nhfb, author-dates = "1902--1950", subject = "language and languages; semantics; comparative linguistics", } @Book{Moon:1936:SBI, author = "Parry Moon", title = "The Scientific Basis of Illuminating Engineering", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "xi + 608", year = "1936", LCCN = "TH7703 .M65", bibdate = "Mon Nov 14 10:11:54 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, keywords = "Benford's Law", remark = "See pages 420ff.", } @Article{Bresciani-Turroni:1937:PL, author = "C. Bresciani-Turroni", title = "On {Pareto}'s Law", journal = j-J-R-STAT-SOC, volume = "100", number = "3", pages = "421--432", month = "????", year = "1937", DOI = "https://doi.org/10.2307/2980528", ISSN = "0952-8385", ISSN-L = "0952-8385", bibdate = "Sat Jan 24 11:18:08 MST 2015", bibsource = "http://www.jstor.org/stable/i349533; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jrss-a-1930.bib", URL = "http://www.jstor.org/stable/2980528", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society", journal-URL = "http://www.jstor.org/journals/09528385.html", } @Article{Benford:1938:LAN, author = "Frank Benford", title = "The Law of Anomalous Numbers", journal = j-PROC-AMER-PHIL-SOC, volume = "78", number = "4", pages = "551--572", month = mar, year = "1938", CODEN = "PAPCAA", ISSN = "0003-049X (print), 2326-9243 (electronic)", ISSN-L = "0003-049X", bibdate = "Thu Feb 15 16:28:28 2007", bibsource = "http://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib; http://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", note = "See comments about Benford's biased rounding practices \cite{Diaconis:1979:RP}.", URL = "http://links.jstor.org/sici?sici=0003-049X%2819380331%2978%3A4%3C551%3ATLOAN%3E2.0.CO%3B2-G", ZMnumber = "JFM 64.0555.03; Zbl 0018.26502", abstract = "It has been observed that the first pages of a table of common logarithms show more wear than do the last pages, indicating that more used numbers begin with the digit 1 than with the digit 9. A compilation of some 20,000 first digits taken from widely divergent sources shows that there is a logarithmic distribution of first digits when the numbers are composed of four or more digits. An analysis of the numbers from different sources shows that the numbers taken from unrelated subjects, such as a group of newspaper items, show a much better agreement with a logarithmic distribution than do numbers from mathematical tabulations or other formal data. There is here the peculiar fact that numbers that individually are without relationship are, when considered in large groups, in good agreement with a distribution law---hence the name ``Anomalous Numbers.''\par A further analysis of the data shows a strong tendency for bodies of numerical data to fall into geometric series. If the series is made up of numbers containing three or more digits the first digits form a logarithmic series. If the numbers contain only single digits the geometric relation still holds but the simple logarithmic relation no longer applies.\par An equation is given showing the frequencies of first digits in the different orders of numbers 1 to 10, 10 to 100, etc.\par The equation also gives the frequency of digits in the second, third + place of a multi-digit number, and it is shown that the same law applies to reciprocals.\par There are many instances showing that the geometric series, or the logarithmic law, has long been recognized as a common phenomenon in factual literature and in the ordinary affairs of life. The wire gauge and drill gauge of the mechanic, the magnitude scale of the astronomer and the sensory response curves of the psychologist are all particular examples of a relationship that seems to extend to all human affairs. The Law of Anomalous Numbers is thus a general probability law of widespread application.", acknowledgement = ack-nhfb, author-dates = "1883--December 4, 1948", fjournal = "Proceedings of the {American Philosophical Society} held at {Philadelphia} for promoting useful knowledge", journal-URL = "http://www.jstor.org/journal/procamerphilsoci", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", remark = "From \cite{Logan:1978:FDP}: ``Benford's paper was published in 1938 in a journal of rather limited circulation and not usually read by mathematicians. It so happened that it was immediately followed in the same issue by a physics paper which became of some importance for secret nuclear work during World War II [H. A. Bethe, M. E. Rose, and L. P. Smith, `The Multiple Scattering of Electrons', Proc. Amer. Phil. Soc. 78(4), 573--585 (1938)]. That is why Benford's paper caught the attention of physicists in the early 1940's and was much discussed. This led to the notes in Nature by Goudsmit and Furry [3] and Furry and Hurwitz [4] containing an effort to explain Benford's law. We considered it at that time merely a welcome diversion and did not expect that over thirty papers would be devoted to this subject in subsequent years.'' The 2006 bibliography \cite{Hurlimann:2006:BLB} cites 325 publications about Benford's Law.", xxURL = "http://en.wikipedia.org/wiki/Frank_Benford", } @Article{Bethe:1938:MSE, author = "H. A. Bethe and M. E. Rose and L. P. Smith", title = "The Multiple Scattering of Electrons", journal = j-PROC-AMER-PHIL-SOC, volume = "78", number = "4", pages = "573--585", month = mar, year = "1938", CODEN = "PAPCAA", ISSN = "0003-049X (print), 2326-9243 (electronic)", ISSN-L = "0003-049X", bibdate = "Wed Nov 23 14:27:52 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/b/bethe-hans.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "This paper, which immediately follows Benford's \cite{Benford:1938:LAN} in this journal issue, is reported in \cite[p. 197]{Logan:1978:FDP} to have been of considerable interest to scientists involved in secret nuclear physics work in World War II. That is how Benford's paper ``in a journal of rather limited circulation and not usually read by mathematicians'' came to be noticed by physicists.", URL = "http://www.jstor.org/stable/984803", acknowledgement = ack-nhfb, author-dates = "Hans Albrecht Bethe (1906--2005)", fjournal = "Proceedings of the {American Philosophical Society} held at {Philadelphia} for promoting useful knowledge", journal-URL = "http://www.jstor.org/journal/procamerphilsoci", remark = "Biography supplement in \cite{Lee:2007:HAB} has this as Proc. Am. Philos. Soc., 89, 373--383 (1938), but that seems to be wrong: check??", } @Article{Benford:1943:PAG, author = "Frank Benford", title = "The Probable Accuracy of the General Physical Constants", journal = j-PHYS-REV, volume = "63", number = "5--6", pages = "212--212", month = mar, year = "1943", CODEN = "PHRVAO", DOI = "https://doi.org/10.1103/PhysRev.63.212", ISSN = "0031-899X (print), 1536-6065 (electronic)", ISSN-L = "0031-899X", bibdate = "Mon Nov 21 21:52:58 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRev.63.212", acknowledgement = ack-nhfb, fjournal = "Physical Review", journal-URL = "http://prola.aps.org/browse/PR", } @Article{Goudsmit:1944:SFN, author = "S. A. Goudsmit and W. H. Furry", title = "Significant Figures of Numbers in Statistical Tables", journal = j-NATURE, volume = "154", number = "3921", pages = "800--801", day = "23", month = dec, year = "1944", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/154800a0", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Sun Sep 18 11:57:19 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See later work \cite{Logan:1978:FDP}, and severe criticism \cite{Raimi:1976:FDP,Raimi:1985:FDP}.", URL = "http://www.nature.com/nature/journal/v154/n3921/pdf/154800a0.pdf", abstract = "A rough qualitative explanation of this fact can easily be given. If we consider tables in which the entries become rarer the larger they are, we can draw the obvious conclusion that in any interval, say, between 10 and 99, or 10,000 and 99,999, there are more entries on the small side than on the large side.", acknowledgement = ack-nhfb, author-dates = "Samuel A. Goudsmit (July 11, 1902--December 4, 1978)", fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", remark = "In 1925, Samuel Goudsmit and George Uhlenbeck proposed the concept of electron spin, a critical step forward in the quantum mechanics of many-electron systems. Surprisingly, that work did not get them the Nobel Prize, but it did spur Wolfgang Pauli to present his Exclusion Principle in 1925, for which he received the 1945 Nobel Prize in Physics. After World War II, Goudsmit was the scientific head of the Alsos mission to capture German nuclear scientists. In the 1950s, he founded the journal Physical Review Letters. During the 1960s and 1970s, he was Editor-in-Chief of the main American physics journal, the Physical Review.", } @Article{Furry:1945:DND, author = "W. H. Furry and Henry Hurwitz", title = "Distribution of Numbers and Distribution of Significant Figures", journal = j-NATURE, volume = "155", number = "3924", pages = "52--53", day = "13", month = jan, year = "1945", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/155052a0", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Sun Sep 18 11:51:37 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.nature.com/nature/journal/v155/n3924/pdf/155052a0.pdf", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Unpublished{Stigler:1945:DLD, author = "George J. Stigler", title = "The distribution of leading digits in statistical tables", year = "1945", bibdate = "Sat Nov 12 16:07:40 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Unpublished, but written about 1945--1946, and presented in an 1975 address at Haskell Hall, University of Chicago. Stigler's distribution is more complex than that of Newcomb and Benford. Stigler has $ F_d = (d \ln (d) - (d + 1) \ln (d + 1) + (1 + (10 / 9) \ln (10))) / 9 $, which gives leading-digit frequencies of 0.2413, 0.1832, 0.1455, 0.1174, 0.0950, 0.0764, 0.0605, 0.0465, and 0.0342. See \cite{Lee:2010:SAR} for a comparison of the Benford and Stigler distributions, and their relations to Zipf and Pareto distributions. For derivations of Stigler's distribution, see \cite{Logan:1978:FDP,Raimi:1985:FDP}.", URL = "http://www.nobelprize.org/nobel_prizes/economics/laureates/1982/", acknowledgement = ack-nhfb, remark = "Stigler won the 1982 Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (often called the Nobel Prize in Economics, although that field was not among the original ones funded by Nobel's will).", } @Book{Furlan:1946:HSU, author = "Luigi Vladimir Furlan", title = "{Das Harmoniegesetz der Statistik: Eine Untersuchung {\"u}ber die metrische Interdependenz der sozialen Erscheinungen}. ({German}) [{The} {Law of Harmony} in statistics: an investigation of the metrical interdependence of social phenomena]", volume = "2", publisher = "Verlag f{\"u}r Recht und Gesellschaft A.-G.", address = "Basel, Switzerland", pages = "xii + 504", year = "1946", LCCN = "HA29 .F85", bibdate = "Mon Nov 14 09:40:24 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", series = "{Beihefte zum Assekuranzjahrbuch}", acknowledgement = ack-nhfb, author-dates = "1886--", keywords = "Benford--Furlan Law", language = "German", } @Article{Geiringer:1948:RVF, author = "Hilda Geiringer", title = "Review of {L. V. Furlan}'s book: {Das Harmoniegesetz der Statistik: Eine Untersuchung {\"u}ber die metrische Interdependenz der sozialen Erscheinungen}. ({German}) [{The} {Law of Harmony} in statistics: an investigation of the metrical interdependence of social phenomena]", journal = j-J-AM-STAT-ASSOC, volume = "43", number = "242", pages = "325--328", month = jun, year = "1948", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Mon Nov 14 09:42:45 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", note = "See \cite{Furlan:1946:HSU}.", URL = "http://www.jstor.org/stable/2280379", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Hsu:1948:ESM, author = "E. H. Hs{\"u}", title = "An Experimental Study on ``Mental Numbers'' and a New Application", journal = j-J-GEN-PSYCH, volume = "38", number = "1", pages = "57--67", month = "????", year = "1948", CODEN = "JGPSAY", DOI = "https://doi.org/10.1080/00221309.1948.9711768", ISSN = "0022-1309 (print), 1940-0888 (electronic)", ISSN-L = "0022-1309", bibdate = "Sat Jan 07 09:29:53 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/00221309.1948.9711768", acknowledgement = ack-nhfb, fjournal = "The Journal of General Psychology", keywords = "Benford's Law", onlinedate = "06 Jul 2010", } @Book{Hardy:1949:DS, author = "G. H. (Godfrey Harold) Hardy", title = "Divergent series", publisher = pub-CLARENDON, address = pub-CLARENDON:adr, pages = "xvi + 396", year = "1949", LCCN = "QA295 .H29", bibdate = "Wed Nov 16 08:14:50 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1877--1947", } @Book{Zipf:1949:HBP, author = "George Kingsley Zipf", title = "Human behaviour and the principle of least effort", publisher = "Hafner Pub. Co.", address = "New York, NY, USA", pages = "xi + 573", year = "1949", LCCN = "H91 .Z5 1949a", bibdate = "Tue Nov 15 09:35:14 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1902--1950", remark = "Reprinted 1965.", subject = "Human ecology", } @Article{Anonymous:1950:BRZ, author = "Anonymous", title = "Book review: {Zipf, George K., \booktitle{Human behavior and the principle of least effort}. Cambridge, (Mass.): Addison-Wesley, 1949, pp. 573}", journal = j-J-CLIN-PSYCHOL, volume = "6", number = "3", pages = "306--306", year = "1950", CODEN = "JCPYAO", DOI = "https://doi.org/10.1002/1097-4679(195007)6:3<306::AID-JCLP2270060331>3.0.CO%3B2-7", ISSN = "0021-9762 (print), 1097-4679 (electronic)", ISSN-L = "0021-9762", bibdate = "Mon Feb 25 06:13:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Clinical Psychology", } @Book{Feller:1950:IPT, author = "William Feller", title = "Introduction to Probability Theory", volume = "2", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "????", year = "1950", LCCN = "QA273 .F37", bibdate = "Mon Nov 14 09:57:35 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", series = "Wiley series in probability and mathematical statistics. Probability and mathematical statistics", acknowledgement = ack-nhfb, author-dates = "William Feller (1906--1970)", } @Article{Macon:1950:DFD, author = "N. Macon and L. Moser", title = "On the distribution of first digits of powers", journal = "Scripta Mathematica", volume = "16", number = "??", pages = "290--291", month = "????", year = "1950", CODEN = "????", ISSN = "0036-9713", ISSN-L = "0036-9713", bibdate = "Sat Nov 12 16:28:13 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Scripta Math.", fjournal = "Scripta Mathematica", } @Article{Tsuji:1952:UDN, author = "M. Tsuji", title = "On the uniform distribution of numbers $ \bmod 1 $", journal = j-J-MATH-SOC-JPN, volume = "4", number = "3--4", pages = "313--322", month = "????", year = "1952", CODEN = "NISUBC", DOI = "https://doi.org/10.2969/jmsj/00430313", ISSN = "0025-5645", ISSN-L = "0025-5645", bibdate = "Sun Feb 17 06:33:32 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://projecteuclid.org/euclid.jmsj/1261415780", acknowledgement = ack-nhfb, fjournal = "Journal of the {Mathematical Society of Japan}", } @InProceedings{Mandelbrot:1953:ITS, author = "Beno{\^\i}t B. Mandelbrot", editor = "Willis Jackson", booktitle = "{Communication theory: papers read at a Symposium on ``Applications of Communication Theory'' held at the Institution of Electrical Engineers, London, September 22nd--26th 1952}", title = "An Informational Theory of the Statistical Structure of Languages", publisher = pub-BUTTERWORTHS, address = pub-BUTTERWORTHS:adr, bookpages = "xii + 532", pages = "486--502", year = "1953", LCCN = "Q350 .S92 1952", bibdate = "Wed Nov 16 09:37:30 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://lccn.loc.gov/53004215; http://www.nytimes.com/2010/10/17/us/17mandelbrot.html", acknowledgement = ack-nhfb, remark = "The paper is incorrectly referenced as a book in \cite{Baek:2011:ZLU}. Mandelbrot got his Master of Science in Aeronautics at CalTech in 1948, and his Docteur d'{\'E}tat {\`e}s Sciences Math{\'e}matiques in 1952 in Paris, so perhaps this is a translation of his thesis work.", } @Book{Blackwell:1954:TGS, author = "David Blackwell and Meyer A. Girshick", title = "Theory of Games and Statistical Decisions", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xi + 355", year = "1954", LCCN = "QA269 .B5", bibdate = "Mon Nov 14 10:04:24 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://catalog.hathitrust.org/Record/000577721", acknowledgement = ack-nhfb, author-dates = "David Blackwell (1919--2010)", keywords = "Benford's Law", remark = "Brief mention of Benford's Law on page 74, problem 2.7.2 (??).", } @Article{Simon:1955:CSD, author = "Herbert A. Simon", title = "On a Class of Skew Distribution Functions", journal = j-BIOMETRIKA, volume = "42", number = "3/4", pages = "425--440", month = dec, year = "1955", CODEN = "BIOKAX", DOI = "https://doi.org/10.2307/2333389", ISSN = "0006-3444 (print), 1464-3510 (electronic)", ISSN-L = "0006-3444", MRclass = "62.0X", MRnumber = "0073085 (17,380e)", MRreviewer = "H. A. David", bibdate = "Sat Jun 21 14:32:48 MDT 2014", bibsource = "http://www.jstor.org/journals/00063444.html; http://www.jstor.org/stable/i315425; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/biometrika1950.bib", URL = "http://www.jstor.org/stable/2333389", acknowledgement = ack-nhfb, fjournal = "Biometrika", journal-URL = "http://biomet.oxfordjournals.org/content/by/year; http://www.jstor.org/journals/00063444.html", keywords = "distribution of income; distribution of species among genera; distributions of city populations; distributions of document word frequencies; distributions of scientific publication counts; Zipf's Law", remark = "This work is an extension of Yule's work \cite{Yule:1925:MTE}. See discussion \cite{Mandelbrot:1959:NCS} and comments \cite{Mitzenmacher:2003:BHG}.", } @Article{Devooght:1957:LZM, author = "J. Devooght", title = "Sur la loi de {Zipf--Mandelbrot}. ({French}) [On the {Zipf--Mandelbrot} law]", journal = "Acad. Roy. Belg. Bull. Cl. Sci. (5)", volume = "43", pages = "244--251", year = "1957", CODEN = "????", ISSN = "0001-4141", ISSN-L = "0001-4141", MRclass = "94.0X", MRnumber = "0088412 (19,516a)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Acad{\'e}mie Royale de Belgique. Bulletin de la Classe des Sciences. Koninklijke Belgische Academie. Mededelingen van de Klasse der Wetenschappen. 5e S{\'e}rie", language = "French", } @Book{Feller:1957:IPT, author = "William Feller", title = "Introduction to Probability Theory", volume = "2", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "461 (vol. 1), 626 (vol. 2)", year = "1957", LCCN = "QA273 .F3712", bibdate = "Mon Nov 14 09:57:35 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://catdir.loc.gov/catdir/toc/onix01/57010805.html", acknowledgement = ack-nhfb, author-dates = "William Feller (1906--1970)", } @Article{Gini:1957:SFD, author = "Corrado Gini", title = "Sulla frequenza delle cifre iniziali dei numeri osservati. ({Italian}) [{On} the frequency of initial digits of observed numbers]", journal = "{Bulletin de l'Institut International de Statistique}", volume = "35", number = "2", pages = "57--76", year = "1957", CODEN = "BISQA3", ISSN = "????", MRclass = "62.00", MRnumber = "0117807 (22 \#8581)", bibdate = "Sun Sep 18 11:54:25 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "29th session, 2nd delivery, Rio de Janerio.", acknowledgement = ack-nhfb, ajournal = "Bull. Inst. Internat. Stat.", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", language = "Italian", xxvolume = "34 (wrong!)", } @Article{Herzel:1957:SDD, author = "Amato Herzel", title = "Sulla distribuzione della cifre iniziali dei numeri statistici. ({Italian}) [{On} the frequency of initial digits of statistical numbers]", journal = "{Atti dell XV e XVII Riunione, Societa Italiana di Statistica}", volume = "??", number = "??", pages = "205--228", year = "1957", CODEN = "????", ISSN = "????", bibdate = "Sun Sep 18 12:00:07 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", language = "Italian", } @Article{Miller:1957:SEI, author = "George A. Miller", title = "Some Effects of Intermittent Silence", journal = j-AMER-J-PSYCHOLOGY, volume = "70", number = "2", pages = "311--314", month = jun, year = "1957", CODEN = "AJPCAA", ISSN = "0002-9556 (print), 1939-8298 (electronic)", ISSN-L = "0002-9556", bibdate = "Tue Nov 15 09:40:05 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/1419346", acknowledgement = ack-nhfb, fjournal = "The American Journal of Psychology", keywords = "Zipf's Law", } @Book{Zeller:1958:TLG, author = "Karl Zeller", title = "{Theorie der Limitierungsverfahren}. (German) [{Theory} of limiting processes]", volume = "15", publisher = pub-SV, address = pub-SV:adr, pages = "242", year = "1958", LCCN = "QA295 .Z4", bibdate = "Wed Nov 16 06:43:43 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", series = "{Ergebnisse der Mathematik und ihrer Grenzgebiete}", acknowledgement = ack-nhfb, language = "German", subject = "series; convergence", } @Article{Mandelbrot:1959:NCS, author = "Beno{\^\i}t Mandelbrot", title = "A note on a class of skew distribution functions: analysis and critique of a paper by {H. A. Simon}", journal = j-INF-CONTROL, volume = "2", number = "??", pages = "90--99", month = apr, year = "1959", CODEN = "IFCNA4", DOI = "https://doi.org/10.1016/S0019-9958(59)90098-1", ISSN = "0019-9958 (print), 1878-2981 (electronic)", ISSN-L = "0019-9958", bibdate = "Wed Nov 16 09:45:49 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See \cite{Simon:1955:CSD}.", abstract = "This note is a discussion of H. A. Simon's model (1955) concerning the class of frequency distributions generally associated with the name of G. K. Zipf. The main purpose is to show that Simon's model is analytically circular in the case of the linguistic laws of Estoup--Zipf and Willis--Yule. Insofar as the economic law of Pareto is concerned, Simon has himself noted that his model is a particular case of that of Champernowne; this is correct, with some reservation. A simplified version of Simon's model is included.", acknowledgement = ack-nhfb, fjournal = "Information and Control", journal-URL = "http://www.sciencedirect.com/science/journal/00199958", } @Article{Mandelbrot:1960:PLL, author = "Beno{\^\i}t Mandelbrot", title = "The {Pareto--Levy} law and the distribution of income", journal = j-INT-ECON-REV, volume = "1", number = "??", pages = "79--106", month = "????", year = "1960", CODEN = "INERAE", ISSN = "0020-6598 (print), 1468-2354 (electronic)", ISSN-L = "0020-6598", bibdate = "Wed Nov 23 11:07:01 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.ingenta.com/journals/browse/bpl/iere", acknowledgement = ack-nhfb, fjournal = "International Economic Review", journal-URL = "http://www.jstor.org/journal/inteeconrevi", } @Article{Cigler:1961:NET, author = "J. Cigler and G. Helmberg", title = "{Neuere Entwicklungen der Theorie der Gleichverteilung}. ({German}) [{Recent} developments in the theory of uniform distribution]", journal = j-DEUTSCH-MATH-V, volume = "64", number = "??", pages = "1--50", month = "????", year = "1961", CODEN = "????", ISSN = "????", bibdate = "Sat Nov 12 16:33:37 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Jber. Deutsch. Math.--Verein", language = "German", } @Article{Pinkham:1961:DFS, author = "Roger S. Pinkham", title = "On the Distribution of First Significant Digits", journal = j-ANN-MATH-STAT, volume = "32", number = "4", pages = "1223--1230", month = dec, year = "1961", CODEN = "AASTAD", ISSN = "0003-4851", ISSN-L = "0003-4851", bibdate = "Sat Sep 17 10:19:00 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/2237922", abstract = "It has been noticed by astute observers that well used tables of logarithms are invariably dirtier at the front than at the back. Upon reflection one is led to inquire whether there are more physical constants with low order first significant digits than high. Actual counts by Benford [2] show that not only is this the case but that it seems to be an empirical truth that whenever one has a large body of physical data, Farmer's Almanac, Census Reports, Chemical Rubber Handbook, etc., the proportion of these data with first significant digit $n$ or less is approximately $ \log_{10}(n + 1) $. Any reader formerly unaware of this ``peculiarity'' will find an actual sampling experiment wondrously tantalizing. Thus, for example, approximately $ 0.7 $ of the physical constants in the Chemical Rubber Handbook begin with $4$ or less ($ \log_{10}(4 + 1) = 0.699 $ ). This is to be contrasted with the widespread intuitive evaluation $ 4 / 9 $ ths.", acknowledgement = ack-nhfb, fjournal = "Annals of Mathematical Statistics", journal-URL = "http://projecteuclid.org/all/euclid.aoms/", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", remark = "From p. 1223: ``\ldots{} the proportion of these data with first significant digit $n$ or less is approximately $ \log_{10}(n + 1) $.''\par \ldots{} ``The only distribution for first significant digits which is invariant under scale change of the underlying distribution is $ \log_{10}(n + 1) $. Contrary to suspicion this is a non-trivial mathematical result, for the variable $n$ is discrete.''.\par From p. 1230: ``The foregoing results bear on questions of round-off in computing machines. Since $ d(u v) = u d v + v d u $, the error resulting from multiplying two rounded numbers will be governed primarily by the first significant digits of the two numbers being multiplied. Now the distribution of first significant digits, favoring as it does low order digits, tends to produce less error than would be the case if first significant digits were uniform as has sometimes been assumed.''\par Cited in \cite{Sterbenz:1974:FPC}.", } @Book{Hamming:1962:NMS, author = "R. W. (Richard Wesley) Hamming", title = "Numerical methods for scientists and engineers", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "xvii + 411", year = "1962", LCCN = "QA297 .H28", MRclass = "65.00", MRnumber = "0137279", MRreviewer = "G. E. Forsythe", bibdate = "Mon Aug 6 07:24:38 2018", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/h/hamming-richard-w.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; z3950.loc.gov:7090/Voyager", series = "International Series in Pure and Applied Mathematics", acknowledgement = ack-nhfb, HDnumber = "51", remark = "Cited in \cite{Sterbenz:1974:FPC}.", subject = "Numerical analysis; Electronic digital computers; random numbers", tableofcontents = "1. The difference calculus \\ 2. Roundoff noise \\ 3. The summation calculus \\ 4. Evaluation of infinite series \\ 5. Finite difference equations \\ 6. The finite Fourier series \\ 7--9. Polynomial approximation introduction \\ 10. A uniform method for finding formulas \\ 11. On finding the error term of a formula \\ 12. Formulas for definite integrals \\ 13. Indefinite integrals \\ 14. Introduction to differential equations \\ 15. A general theory of predictor-corrector methods \\ 16. Special methods of integrating ordinary differential equations \\ 17--18. Least squares theory and practice \\ 19. Chebyshev polynomials \\ 20. Rational functions \\ 21--22. Periodic functions --- Fourier series \\ 23. Nonperiodic functions --- the Fourier integral \\ 24. Linear filters --- smoothing and differentiating \\ 25. Integrals and differential equations \\ 26. Exponential approximation \\ 27. Singularities \\ 28. On finding zeros \\ 29. Simultaneous linear algebraic equations \\ 30. Inversion of matrices and eigenvalues \\ 31. Some examples of the simulation of situations and processes \\ 32. Random numbers and Monte Carlo methods \\ $N+1$: The art of computing for scientists and engineers", } @TechReport{Mandelbrot:1962:SNR, author = "Beno{\^\i}t Mandelbrot", title = "Statistics of natural resources and the law of {Pareto}", type = "Research Note", number = "NC-146", institution = "IBM", address = "Yorktown Heights, NY, USA", day = "29", month = jun, year = "1962", bibdate = "Mon Sep 10 17:35:59 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Book{Weaver:1963:LLT, author = "Warren Weaver", title = "{Lady Luck}: The Theory of Probability", publisher = "Doubleday Anchor Series", address = "New York, NY, USA", pages = "392", year = "1963", LCCN = "QA273 .W4", bibdate = "Mon Nov 14 09:54:47 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Benford's Law is discussed on pp. 270--277.", acknowledgement = ack-nhfb, author-dates = "Warren, Weaver (1894--1978)", remark = "Reprinted in \cite{Weaver:1982:LLT}.", } @Article{Cigler:1964:MSU, author = "J. Cigler", title = "Methods of summability and uniform distribution $ \bmod 1 $", journal = j-COMPOS-MATH, volume = "16", number = "??", pages = "44--51", year = "1964", CODEN = "CMPMAF", ISSN = "0010-437X (print), 1570-5846 (electronic)", ISSN-L = "0010-437X", bibdate = "Mon Nov 14 10:15:31 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Compositio Mathematica", } @Article{Good:1965:LE, author = "I. J. Good", title = "Letter to the Editor", journal = j-AMER-STAT, volume = "19", number = "3", pages = "43--43", month = jun, year = "1965", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Mon Nov 14 09:56:33 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/2681423", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", keywords = "Benford's Law", remark = "Refers to \cite{Pinkham:1961:DFS}, and cites a review of that article published by Good in Mathematical Reviews (1962), page 214, which I have not yet been able to locate.", } @Article{Hamming:1965:NLB, author = "R. W. Hamming and W. L. Mammel", title = "A Note on the Location of the Binary Point in a Computing Machine", journal = j-IEEE-TRANS-ELEC-COMPUT, volume = "EC-14", number = "2", pages = "260--261", month = apr, year = "1965", CODEN = "IEECA8", DOI = "https://doi.org/10.1109/PGEC.1965.264258", ISSN = "0367-7508", ISSN-L = "0367-7508", bibdate = "Thu Jul 14 06:26:22 MDT 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib; http://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib", URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4038414", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Electronic Computers", remark = "The authors consider the low-level multiplication circuit efficiency of placing the binary point before or after the first bit. If the leading bit is equally likely to be a 0 or a 1, then their analysis shows that it is better to place the point before the leading bit. However, they report that a more likely distribution is logarithmic (as predicted by Benford's Law, although the Newcomb / Benford work is neither mentioned nor cited), in which case there is no advantage for either choice of placement of the binary point. They conclude that it would be humane to place it after the leading digit, by analogy with how people learn decimal arithmetic.", } @Article{Konheim:1965:MDT, author = "Alan G. Konheim", title = "Mantissa Distributions (in {Technical Notes and Short Papers})", journal = j-MATH-COMPUT, volume = "19", number = "89", pages = "143--144", month = apr, year = "1965", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2004111", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/mathcomp1960.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Book{Zipf:1965:PBL, author = "George Kingsley Zipf", title = "The psycho-biology of language: an introduction to dynamic philology", publisher = pub-MIT, address = pub-MIT:adr, pages = "xv + 336", year = "1965", LCCN = "P105 .Z5 1965", bibdate = "Wed Nov 16 09:07:09 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; prodorbis.library.yale.edu:7090/voyager", acknowledgement = ack-nhfb, author-dates = "1902--1950", remark = "Reprint of \cite{Zipf:1935:PBL}.", subject = "language; philology; semantics; psycholinguistics; comparative linguistics", } @Article{Ball:1966:BRB, author = "R. J. Ball", title = "Book Review: {{\booktitle{Random Processes and the Growth of Firms: A Study of the Pareto Law}}, by Josef Steindl}", journal = j-J-R-STAT-SOC-SER-A-GENERAL, volume = "129", number = "4", pages = "600--601", month = "????", year = "1966", CODEN = "JSSAEF", DOI = "https://doi.org/10.2307/2982275", ISSN = "0035-9238", ISSN-L = "0035-9238", bibdate = "Sat Jan 24 11:18:12 MST 2015", bibsource = "http://www.jstor.org/stable/i349635; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jrss-a-1960.bib", URL = "http://www.jstor.org/stable/2982275", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society. Series A (General)", journal-URL = "http://www.jstor.org/journals/00359238.html", } @Book{Feller:1966:IPT, author = "William Feller", title = "Introduction to Probability Theory", volume = "2", publisher = pub-WILEY, address = pub-WILEY:adr, edition = "Second", pages = "xviii + 626", year = "1966", ISSN = "0271-6232", LCCN = "QA273 .F3727 1966", bibdate = "Mon Nov 14 09:57:35 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", series = "Wiley series in probability and mathematical statistics. Probability and mathematical statistics", acknowledgement = ack-nhfb, author-dates = "William Feller (1906--1970)", remark = "Benford's Law is discussed on p. 62.", } @Article{Flehinger:1966:PRI, author = "B. J. (Betty J.) Flehinger", title = "On the probability that a random integer has initial digit {$A$}", journal = j-AMER-MATH-MONTHLY, volume = "73", number = "10", pages = "1056--1061", month = dec, year = "1966", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2314636", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Sat Nov 12 15:33:14 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/2314636", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Book{Petersen:1966:RMT, author = "Gordon Marshall Petersen", title = "Regular Matrix Transformations", publisher = pub-MCGRAW-HILL, address = pub-MCGRAW-HILL:adr, pages = "viii + 142", year = "1966", LCCN = "QA263 .P415 1966", bibdate = "Mon Nov 14 10:18:15 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Duncan:1967:AUD, author = "R. L. Duncan", title = "An Application of Uniform Distributions to the {Fibonacci} Numbers", journal = j-FIB-QUART, volume = "5", number = "2", pages = "137--140", month = apr, year = "1967", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:04:52 MDT 2011", bibsource = "http://www.fq.math.ca/5-2.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/5-2/duncan.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Groos:1967:BCB, author = "Ole V. Groos", title = "Brief Communications: {Bradford's Law} and the {Keenan--Atherton} data", journal = j-AM-DOC, volume = "18", number = "1", pages = "46--46", month = jan, year = "1967", CODEN = "AMDOA7", DOI = "https://doi.org/10.1002/asi.5090180108", ISSN = "0096-946X (print), 1936-6108 (electronic)", ISSN-L = "0096-946X", bibdate = "Fri Sep 11 06:21:26 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/amdoc.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "American Documentation", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643/", onlinedate = "23 Apr 2007", } @Article{Mosteller:1967:DSR, author = "Frederick Mosteller and Cleo Youtz and Douglas Zahn", title = "The Distribution of Sums of Rounded Percentages", journal = j-DEMOGRAPHY, volume = "4", number = "2", pages = "850--858", month = jun, year = "1967", CODEN = "????", ISSN = "0070-3370 (print), 1533-7790 (electronic)", ISSN-L = "0070-3370", bibdate = "Thu Nov 24 08:03:16 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", note = "See \cite{Diaconis:1979:RP} for further work.", URL = "http://muse.jhu.edu/journals/dem/; http://www.biomedsearch.com/nih/Distribution-Sums-Rounded-Percentages/21318695.html; http://www.jstor.org/stable/2060324", abstract = "When percentages are computed for counts in several categories or for several positive measurements0 each taken as a fraction of their sum, the rounded percentages often fail to add to 100 percent. We investigate how frequently this failure occurs and what the distributions of sums of rounded percentages are for (1) an empirical set of data, (2) the multinomial distribution in small samples, (3) spacings between points dropped on an interval --- the broken-stick model; and (4) for simulation for several categories. The several methods produce similar distributions.We find that the probability that the sum of rounded percentages adds to exactly 100 percent is certain for two categories, about three-fourths for three categories, about two-thirds for four categories, and about [Formula: see text] for larger numbers of categories, c, on the average when categories are not improbable.", acknowledgement = ack-nhfb, fjournal = "Demography", pubmedid = "21318695", } @Article{Adhikari:1968:DMS, author = "A. K. Adhikari and B. P. Sarkar", title = "Distribution of most significant digit in certain functions whose arguments are random variables", journal = j-SANKHYA-B, volume = "30", number = "??", pages = "47--58", month = "????", year = "1968", CODEN = "SANBBV", ISSN = "0581-5738", ISSN-L = "0581-5738", bibdate = "Sat Nov 12 16:09:30 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Sankhya Ser. B (Indian J. of Statist.)", fjournal = "Sankhy{\=a} (Indian Journal of Statistics), Series B. Methodological", } @Book{Feller:1968:IPT, author = "William Feller", title = "Introduction to Probability Theory and its Applications", volume = "2", publisher = pub-WILEY, address = pub-WILEY:adr, edition = "Third", pages = "xviii + 509", year = "1968", ISBN = "0-471-25708-7", ISBN-13 = "978-0-471-25708-0", LCCN = "QA273 .F3713", bibdate = "Mon Nov 14 09:57:35 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See \cite{Berger:2010:FFFa,Berger:2010:FFFb} for a discussion of flaws in the derivation of Benford's Law in this book.", acknowledgement = ack-nhfb, author-dates = "William Feller (1906--1970)", remark = "Pages 63--64 (of xxiv + 669) in 1971 printing have only brief mention of two papers \cite{Benford:1938:LAN,Pinkham:1961:DFS}.", } @Article{Shenton:1968:PDM, author = "L. R. Shenton", title = "Periodicity and Density of Modified {Fibonacci} Sequences", journal = j-FIB-QUART, volume = "6", number = "2", pages = "109--116", month = apr, year = "1968", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:02 MDT 2011", bibsource = "http://www.fq.math.ca/6-2.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/6-2/shenton.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", keywords = "Benford's Law", } @Article{Wilk:1968:PPM, author = "M. B. Wilk and R. Gnanadesikan", title = "Probability Plotting Methods for the Analysis of Data", journal = j-BIOMETRIKA, volume = "55", number = "1", pages = "1--17", month = mar, year = "1968", CODEN = "BIOKAX", DOI = "https://doi.org/10.2307/2334448", ISSN = "0006-3444 (print), 1464-3510 (electronic)", ISSN-L = "0006-3444", bibdate = "Sat Jun 21 14:33:29 MDT 2014", bibsource = "http://www.jstor.org/journals/00063444.html; http://www.jstor.org/stable/i315458; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/biometrika1960.bib", URL = "http://www.jstor.org/stable/2334448", acknowledgement = ack-mjn # "\slash " # ack-nhfb, fjournal = "Biometrika", journal-URL = "http://www.jstor.org/journals/00063444.html", keywords = "empirical cumulative distribution function (ECDF); percent ($p$--$p$) plots; quantile ($q$--$q$) plots", remark = "This article does not mention Benford's Law, but MJN \cite{Nigrini:2012:BLA} lists this reference for analysis of data.", } @Article{Adhikari:1969:SRD, author = "A. K. Adhikari", title = "Some results on the distribution of the most significant digit", journal = j-SANKHYA-B, volume = "31", number = "??", pages = "413--420", month = "????", year = "1969", CODEN = "SANBBV", ISSN = "0581-5738", ISSN-L = "0581-5738", bibdate = "Sat Nov 12 16:12:50 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Sankhya Ser. B (Indian J. of Statist.)", fjournal = "Sankhy{\=a} (Indian Journal of Statistics), Series B. Methodological", } @Article{Bumby:1969:FAM, author = "R. Bumby and E. Ellentuck", title = "Finitely additive measures and the first digit problem", journal = j-FUNDAM-MATH, volume = "65", number = "??", pages = "33--42", month = "????", year = "1969", CODEN = "????", ISSN = "0016-2736 (print), 1730-6329 (electronic)", ISSN-L = "0016-2736", bibdate = "Sat Nov 12 16:19:19 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Fundamenta mathematicae", } @Article{Duncan:1969:NID, author = "R. L. Duncan", title = "Note on the Initial Digit Problem", journal = j-FIB-QUART, volume = "7", number = "5", pages = "474--475", month = dec, year = "1969", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:21 MDT 2011", bibsource = "http://www.fq.math.ca/7-5.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/7-5/duncan.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly", journal-URL = "http://www.fq.math.ca/", } @Article{Fairthorne:1969:PDE, author = "Robert A. Fairthorne", title = "Progress in Documentation: Empirical Hyperbolic Distributions ({Bradford--Zipf--Mandelbrot}) for Bibliometric Description and Prediction", journal = j-J-DOC, volume = "25", number = "4", pages = "313--343", month = dec, year = "1969", CODEN = "JDOCAS", DOI = "https://doi.org/10.1108/eb026481", ISSN = "0022-0418 (print), 1758-7379 (electronic)", ISSN-L = "0022-0418", bibdate = "Fri Nov 25 06:27:27 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Reprinted in \cite{Fairthorne:2005:PDE}.", acknowledgement = ack-nhfb, fjournal = "Journal of Documentation", journal-URL = "http://www.emeraldinsight.com/loi/jd", remark = "According to \cite{Potter:1981:LLR}, this paper is the first to link the distributions of Bradford, Zipf, Mandelbrot, and Lotka.", } @Unpublished{Hafner:1969:CSR, author = "Everett M. Hafner", title = "Circular slide roulette", year = "1969", bibdate = "Mon Nov 14 10:00:25 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Unpublished but dated 1969. Presented in 1975 in an address at Hampshire College, Amherst, MA, USA", acknowledgement = ack-nhfb, } @Article{Haight:1969:TPD, author = "F. A. Haight", title = "Two probability distributions connected with {Zipf}'s rank-size conjecture", journal = j-ZASTOS-MAT, volume = "10", pages = "225--228", year = "1969", CODEN = "ZAMTAK", ISSN = "0044-1899", ISSN-L = "0044-1899", MRclass = "62.31", MRnumber = "0246425 (39 \#7729)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Polska Akademia Nauk. Instytut Matematyczny. Zastosowania Matematyki", } @Article{Holewijn:1969:UDS, author = "Dr. P. J. Holewijn", title = "On the uniform distribution of sequences of random variables", journal = j-Z-WAHRSCHEINLICHKEITSTHEOR-VERW-GEB, volume = "14", number = "2", pages = "89--92", month = "????", year = "1969", CODEN = "ZWVGAA", DOI = "https://doi.org/10.1007/BF00537514", ISSN = "0044-3719", ISSN-L = "0044-3719", bibdate = "Sat Apr 26 19:23:29 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/z-wahrscheinlichkeits-theor-verw-geb.bib", URL = "http://link.springer.com/article/10.1007/BF00537514", acknowledgement = ack-nhfb, fjournal = "Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und verwandte Gebiete", journal-URL = "http://link.springer.com/journal/440", } @Book{Knuth:1969:SA, author = "Donald E. Knuth", title = "Seminumerical Algorithms", volume = "2", publisher = pub-AW, address = pub-AW:adr, pages = "xi + 624", year = "1969", ISBN = "0-201-03802-1", ISBN-13 = "978-0-201-03802-6", LCCN = "QA76.5 .K57", MRclass = "68.00 (65.00)", MRnumber = "44 \#3531", MRreviewer = "M. Muller", bibdate = "Wed Dec 15 15:47:38 1993", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/cryptography.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib; http://www.math.utah.edu/pub/tex/bib/master.bib", price = "US\$19.75", series = "The Art of Computer Programming", acknowledgement = ack-nhfb, remark = "Benford's Law is discussed on pp. 219--229.", tableofcontents = "3: Random Numbers \\ 3.1. Introduction / 1 \\ 3.2. Generating Uniform Random Numbers / 9 \\ 3.2.1. The Linear Congruential Method / 9 \\ 3.2.1.1. Choice of modulus / 11 \\ 3.2.1.2. Choice of multiplier / 15 \\ 3.2.1.3. Potency / 21 \\ 3.2.2. Other Methods / 25 \\ 3.3. Statistical Tests / 34 \\ 3.3.1. General Test Procedures for Studying Random Data / 35 \\ 3.3.2. Empirical Tests / 54 \\ *3.3.3. Theoretical Tests / 69 \\ 3.3.4. The Spectral Test / 82 \\ 3.4. Other Types of Random Quantities / 100 \\ 3.4.1. Numerical Distributions / 101 \\ 3.4.2. Random Sampling and Shuffling / 121 \\ *3.5. What is a Random Sequence? / 127 \\ 3.6. Summary / 155 \\ 4: Arithmetic \\ 4.1. Positional Number Systems / 162 \\ 4.2. Floating-Point Arithmetic / 180 \\ 4.2.1. Single-Precision Calculations / 180 \\ 4.2.2. Accuracy of Floating-Point Arithmetic / 195 \\ *4.2.3. Double-Precision Calculations / 210 \\ 4.2.4. Statistical Distribution / 218 \\ 4.3. Multiple-Precision Arithmetic / 229 \\ 4.3.1. The Classical Algorithms / 229 \\ *4.3.2. Modular Arithmetic / 248 \\ *4.3.3. How Fast Can We Multiply? / 258 \\ 4.4. Radix Conversion / 280 \\ 4.5. Rational Arithmetic / 290 \\ 4.5.1. Fractions / 290 \\ 4.5.2. The Greatest Common Divisor / 293 \\ *4.5.3. Analysis of Euclid's Algorithm / 316 \\ 4.5.4. Factoring into Primes / 339 \\ 4.6. Polynomial Arithmetic / 360 \\ 4.6.1. Division of Polynomials / 363 \\ *4.6.2. Factorization of Polynomials / 381 \\ 4.6.3. Evaluation of Powers / 398 \\ 4.6.4. Evaluation of Polynomials / 422 \\ *4.7. Manipulation of Power Series / 444 \\ Answers to Exercises / 452 \\ Appendix A: MIX / 565 \\ 1. Description of MIX / 565 \\ 2. The MIX Assembly Language / 584 \\ Appendix B: Tables of Numerical Quantities / 596 \\ 1. Fundamental Constants (decimal) / 596 \\ 2. Fundamental Constants (octal) / 597 \\ 3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers / 598 \\ Appendix C: Index to Notations / 600 \\ Index and Glossary / 605", xxyear = "{\noopsort{1968c}}1969", } @Article{Kuipers:1969:RPR, author = "L. Kuipers", title = "Remark on a Paper by {R. L. Duncan} Concerning the Uniform Distribution Mod $1$ of the Sequence of the Logarithms of the {Fibonacci} Numbers", journal = j-FIB-QUART, volume = "7", number = "5", pages = "465--466", month = dec, year = "1969", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:21 MDT 2011", bibsource = "http://www.fq.math.ca/7-5.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/7-5/kuipers-a.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Raimi:1969:DFS, author = "R. A. Raimi", title = "On the Distribution of First Significant Figures", journal = j-AMER-MATH-MONTHLY, volume = "76", number = "4", pages = "342--348", month = apr, year = "1969", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2316424", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:39:22 MDT 1999", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Article{Raimi:1969:PDF, author = "R. Raimi", title = "The peculiar distribution of first significant digits", journal = j-SCI-AMER, volume = "221", number = "6", pages = "109--120", month = dec, year = "1969", CODEN = "SCAMAC", DOI = "https://doi.org/10.1038/scientificamerican1269-109", ISSN = "0036-8733 (print), 1946-7087 (electronic)", ISSN-L = "0036-8733", bibdate = "Sat Nov 12 16:39:08 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Scientific American", journal-URL = "http://www.nature.com/scientificamerican", } @PhdThesis{Wong:1969:GDS, author = "James Teng Wong", title = "On the Generalization of the Distribution of the Significant Digits Under Computation", type = "{Ph.D.} thesis", school = "Department of Mathematics, Oregon State University", address = "Corvallis, OR, USA", pages = "52", year = "1969", bibdate = "Fri Nov 25 15:41:42 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://search.proquest.com/docview/302476508?accountid=14677", acknowledgement = ack-nhfb, } @Article{Aigner:1970:EPL, author = "Dennis J. Aigner and Arthur S. Goldberger", title = "Estimation of {Pareto's Law} from Grouped Observations", journal = j-J-AM-STAT-ASSOC, volume = "65", number = "330", pages = "712--723", month = jun, year = "1970", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Jan 25 08:05:44 MST 2012", bibsource = "http://www.jstor.org/journals/01621459.html; http://www.jstor.org/stable/i314212; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc1970.bib", URL = "http://www.jstor.org/stable/2284582", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Blum:1970:SID, author = "M. Blum", title = "On the Sums of Independently Distributed {Pareto} Variates", journal = j-SIAM-J-APPL-MATH, volume = "19", number = "1", pages = "191--198", month = jul, year = "1970", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Thu Oct 15 18:16:06 MDT 1998", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/siamjapplmath.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Applied Mathematics", journal-URL = "http://epubs.siam.org/siap", } @Article{Brown:1970:MOU, author = "J. L. {Brown, Jr.} and R. L. Duncan", title = "Modulo One Uniform Distribution of the Sequence of Logarithms of Certain Recursive Sequences", journal = j-FIB-QUART, volume = "8", number = "5", pages = "482--486", month = dec, year = "1970", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:31 MDT 2011", bibsource = "http://www.fq.math.ca/8-5.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", note = "See remark \cite{Kuipers:1973:RPD}.", URL = "http://www.fq.math.ca/Scanned/8-5/brown.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Hamming:1970:DN, author = "R. W. Hamming", title = "On the Distribution of Numbers", journal = j-BELL-SYST-TECH-J, volume = "49", number = "8", pages = "1609--1625", month = oct, year = "1970", CODEN = "BSTJAN", ISSN = "0005-8580", ISSN-L = "0005-8580", bibdate = "Tue Nov 9 11:15:55 MST 2010", bibsource = "http://bstj.bell-labs.com/oldfiles/year.1970/BSTJ.1970.4908.html; http://www.alcatel-lucent.com/bstj/vol49-1970/bstj-vol49-issue08.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://bstj.bell-labs.com/BSTJ/images/Vol49/bstj49-8-1609.pdf; http://www.alcatel-lucent.com/bstj/vol49-1970/articles/bstj49-8-1609.pdf", abstract = "This paper examines the distribution of the mantissas of floating-point numbers and shows how the arithmetic operations of a computer transform various distributions toward the limiting distribution $$ r(x) = 1 / (x \ln b) \qquad (1 / b \leq x \leq 1) $$ (where $b$ is the base of the number system). The paper also gives a number of applications to hardware, software, and general computing which show that this distribution is not merely an amusing curiosity. A brief examination of the distribution of exponents is include.", acknowledgement = ack-nhfb, fjournal = "The Bell System Technical Journal", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1538-7305/issues/", } @Article{Hill:1970:ZLP, author = "Bruce M. Hill", title = "{Zipf}'s law and prior distributions for the composition of a population", journal = j-J-AM-STAT-ASSOC, volume = "65", number = "331", pages = "1220--1232", month = sep, year = "1970", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", MRclass = "62.35", MRnumber = "0279936 (43 \#5657)", MRreviewer = "I. J. Good", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", URL = "http://www.jstor.org/stable/2284288", ZMnumber = "Zbl 0224.92011", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Orlov:1970:GZM, author = "Ju. K. Orlov", title = "A generalization of the {Zipf--Mandelbrot} law", journal = "Sakharthvelos SSR Mecnierebatha Akademiis Moambe", volume = "57", pages = "37--40", year = "1970", CODEN = "????", ISSN = "????", MRclass = "94.50", MRnumber = "0274234 (42 \#9109)", MRreviewer = "T. Nemetz", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Sakharth. SSR Mech. Akad. Moambe", } @Article{Taguchi:1970:ZLC, author = "Tokio Taguchi", title = "On {Zipf}'s law. {A} characterization of distributions in linguistics and demography", journal = j-PROC-INST-STATIST-MATH, volume = "17", pages = "83--90", year = "1970", CODEN = "????", ISSN = "0912-6112", ISSN-L = "0912-6112", bibdate = "Sat Nov 12 10:39:57 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0238.68026", acknowledgement = ack-nhfb, ajournal = "Proc. Inst. Statist. Math.", classmath = "68Q45 (Formal languages)", fjournal = "Proceedings of the Institute of Statistical Mathematics", language = "Japanese", } @Book{Zeller:1970:TLG, author = "Karl Zeller and Wolfgang Beekmann", title = "{Theorie der Limitierungsverfahren}. (German) [{Theory} of limiting processes]", volume = "15", publisher = pub-SV, address = pub-SV:adr, edition = "Second", pages = "xii + 314", year = "1970", LCCN = "QA295 .Z4 1970", bibdate = "Wed Nov 16 06:43:43 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", series = "Ergebnisse der Mathematik und ihrer Grenzgebiete", acknowledgement = ack-nhfb, language = "German", subject = "series; convergence", } @Article{Thorp:1971:PCD, author = "E. Thorp and R. Whitley", title = "{Poincar{\'e}}'s conjecture and the distribution of digits in logarithm tables", journal = j-COMPOS-MATH, volume = "21", number = "??", pages = "233--250", month = "????", year = "1971", CODEN = "CMPMAF", ISSN = "0010-437X (print), 1570-5846 (electronic)", ISSN-L = "0010-437X", bibdate = "Sat Nov 12 16:36:41 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Comp. Math.", fjournal = "Compositio Mathematica", } @Article{Wlodarski:1971:FLN, author = "J. Wlodarski", title = "{Fibonacci} and {Lucas} Numbers Tend to Obey {Benford's Law}", journal = j-FIB-QUART, volume = "9", number = "1", pages = "87--88", month = feb, year = "1971", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:05:33 MDT 2011", bibsource = "http://www.fq.math.ca/9-1.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/9-1/wlodarski2.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly", journal-URL = "http://www.fq.math.ca/", } @Article{Aigner:1972:CEP, author = "Dennis J. Aigner and Arthur S. Goldberger", title = "Corrigenda: Estimation of {Pareto's Law} from Grouped Observations", journal = j-J-AM-STAT-ASSOC, volume = "67", number = "337", pages = "252--252", month = mar, year = "1972", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Jan 25 08:05:45 MST 2012", bibsource = "http://www.jstor.org/journals/01621459.html; http://www.jstor.org/stable/i314217; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc1970.bib", URL = "http://www.jstor.org/stable/2284755", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Krevitt:1972:BCC, author = "Beth Krevitt and Belver C. Griffith", title = "Brief Communications: A comparison of several {Zipf}-type distributions in their goodness of fit to language data", journal = j-J-AM-SOC-INF-SCI, volume = "23", number = "3", pages = "220--221", month = may, year = "1972", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630230310", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:44 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Krevitt:1972:CSZ, author = "Beth Krevitt and Belver C. Griffith", title = "A comparison of several {Zipf}-type distributions in their goodness of fit to language data", journal = j-J-AM-SOC-INF-SCI, volume = "23", number = "3", pages = "220--221", year = "1972", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630230310", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Mon Feb 25 06:13:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", } @Article{Varian:1972:LEB, author = "Hal R. Varian", title = "Letter to the {Editor}: {Benford's Law}", journal = j-AMER-STAT, volume = "26", number = "3", pages = "65--66", month = jun, year = "1972", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Fri Mar 30 11:34:37 2007", bibsource = "http://www.math.utah.edu/pub/tex/bib/amstat.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://links.jstor.org/sici?sici=0003-1305%28197206%2926%3A3%3C62%3ALTTE%3E2.0.CO%3B2-Q", abstract = "Around 1938 the physicist Frank Benford observed a rather strange fact: tables of logarithms in libraries tend to be dirtier at the beginning than at the end. This indicated to Benford that people had more occasion to calculate with numbers beginning with 1 or 2 than with 8 or 9.\par Benford also found that the frequency of the digit $p$ being the first digit of a decimal number was very closely approximated by $ \log (p + 1) - \log p $ [i.e., $ \log (1 + 1 / p) $ ]. This has become known as Benford's law.", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Article{Vlachy:1972:VFS, author = "Jan Vlach{\'y}", title = "Variable Factors in Scientific Communities (Observations on {Lotka's Law})", journal = "Teorie a Metoda", volume = "4", number = "??", pages = "91--120", month = "????", year = "1972", CODEN = "????", ISSN = "????", bibdate = "Fri Nov 25 07:53:43 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, xxvolume = "6", } @Article{Whitney:1972:MNI, author = "R. E. Whitney", title = "Mathematical Notes: Initial Digits for the Sequence of Primes", journal = j-AMER-MATH-MONTHLY, volume = "79", number = "2", pages = "150--152", month = feb, year = "1972", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2316536", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:36:38 MDT 1999", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Article{Wilkinson:1972:ABL, author = "Elizabeth A. Wilkinson", title = "The Ambiguity of {Bradford's Law}", journal = j-J-DOC, volume = "28", number = "2", pages = "122--130", month = "????", year = "1972", CODEN = "JDOCAS", DOI = "https://doi.org/10.1108/eb026534", ISSN = "0022-0418 (print), 1758-7379 (electronic)", ISSN-L = "0022-0418", bibdate = "Fri Nov 25 07:20:34 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Documentation", journal-URL = "http://www.emeraldinsight.com/loi/jd", } @TechReport{Diaconis:1973:LMI, author = "Persi Diaconis", title = "Limits of measures of the integers with applications to random number generators and the distribution of leading digits", type = "Memorandum", number = "NS-211", institution = "Department of Statistics, Harvard University", address = "Cambridge, MA, USA", day = "22", month = mar, year = "1973", bibdate = "Sat Nov 12 16:23:20 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Kreifelts:1973:OBG, author = "Thomas Kreifelts", title = "{Optimale Basiswahl f{\"u}r eine Gleitkomma-Arithmetik}. ({German}) [{Optimal} Choice of Basis for a Floating-Point Arithmetic]", journal = j-COMPUTING, volume = "11", number = "4", pages = "353--363", month = dec, year = "1973", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Tue Jan 2 17:40:51 MST 2001", bibsource = "Compendex database; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/computing.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib; INSPEC Axiom database (1968--date)", note = "See correction \cite{Kreifelts:1975:OBF}.", acknowledgement = ack-nj # " and " # ack-nhfb, affiliation = "Inst. Numerische Datenverarbeitung, Bonn, West Germany", classification = "723; 921; C5230", description = "digital arithmetic", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", journalabr = "Comput (Vienna/NY)", keywords = "base; computer programming; floating point arithmetic; mathematical techniques; optimal choice; rounding errors", language = "German", } @Article{Kuipers:1973:RPD, author = "L. Kuipers and Jau-shyong Shiue", title = "Remark on a Paper by {Duncan} and {Brown} on the Sequence of Logarithms of Certain Recursive Sequences", journal = j-FIB-QUART, volume = "11", number = "3", pages = "292--293", month = oct, year = "1973", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 17:58:42 MDT 2011", bibsource = "http://www.fq.math.ca/11-3.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", note = "See \cite{Brown:1970:MOU}.", URL = "http://www.fq.math.ca/Scanned/11-3/kuipers.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Loynes:1973:SRP, author = "R. M. Loynes", title = "Some results in the probabilistic theory of asymptotic uniform distribution modulo 1", journal = j-Z-WAHRSCHEINLICHKEITSTHEOR-VERW-GEB, volume = "26", number = "1", pages = "33--41", month = "????", year = "1973", CODEN = "ZWVGAA", DOI = "https://doi.org/10.1007/BF00533958", ISSN = "0044-3719", ISSN-L = "0044-3719", bibdate = "Sat Apr 26 19:24:39 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/z-wahrscheinlichkeits-theor-verw-geb.bib", URL = "http://link.springer.com/article/10.1007/BF00533958", acknowledgement = ack-nhfb, fjournal = "Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und verwandte Gebiete", journal-URL = "http://link.springer.com/journal/440", } @Article{Murphy:1973:BCL, author = "Larry J. Murphy", title = "Brief Communications: ``{Lotka's Law} in the humanities?''", journal = j-J-AM-SOC-INF-SCI, volume = "24", number = "6", pages = "461--462", month = nov, year = "1973", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630240607", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:48 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Murphy:1973:LLH, author = "Larry J. Murphy", title = "{Lotka's Law} in the Humanities?", journal = j-J-AM-SOC-INF-SCI, volume = "24", number = "6", pages = "461--462", month = nov # "\slash " # dec, year = "1973", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630240607", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Nov 25 08:09:40 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", } @Article{Petruszewycz:1973:LLE, author = "M. Petruszewycz", title = "{L'histoire} de la loi d'{Estoup--Zipf}: documents. ({French}) [{The} history of the {Estoup--Zipf} law]", journal = "Math. Sci. Humaines", volume = "??", number = "44", pages = "41--56", year = "1973", CODEN = "????", ISSN = "0025-5815", ISSN-L = "0025-5815", MRclass = "92A25", MRnumber = "0354071 (50 \#6553)", MRreviewer = "J. S. Joel", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Centre de Math{\'e}matique Sociale. \'Ecole Pratique des Hautes \'Etudes. Math{\'e}matiques et Sciences Humaines", language = "French", } @Article{Sarkar:1973:OSD, author = "B. P. Sarkar", title = "An observation on the significant digits of the binomial coefficients and factorials", journal = j-SANKHYA-B, volume = "35", number = "??", pages = "363--364", month = "????", year = "1973", CODEN = "SANBBV", ISSN = "0581-5738", ISSN-L = "0581-5738", bibdate = "Sat Nov 12 16:29:39 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Sankhya Ser. B (Indian J. of Statist.)", fjournal = "Sankhy{\=a} (Indian Journal of Statistics), Series B. Methodological", } @Article{Schatte:1973:VMG, author = "Peter Schatte", title = "{Zur Verteilung der Mantisse in der Gleitkommadarstellung einer Zufallsgr{\"o}{\ss}e}. ({German}) [Distribution of the mantissa in the floating-point representation of a random variable]", journal = j-Z-ANGE-MATH-MECH, volume = "53", number = "??", pages = "553--565", month = "????", year = "1973", CODEN = "ZAMMAX", ISSN = "0044-2267 (print), 1521-4001 (electronic)", ISSN-L = "0044-2267", bibdate = "Mon Nov 14 16:35:28 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "{Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik}", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-4001", language = "German", } @Article{Sentance:1973:FAB, author = "W. A. Sentance", title = "A Further Analysis of {Benford's Law}", journal = j-FIB-QUART, volume = "11", number = "5", pages = "490--494", month = dec, year = "1973", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 17:58:46 MDT 2011", bibsource = "http://www.fq.math.ca/11-5.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/11-5/sentance.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly", journal-URL = "http://www.fq.math.ca/", } @Article{Banks:1974:AMN, author = "William P. Banks and David K. Hill", title = "The apparent magnitude of number scaled by random production", journal = j-J-EXP-PSYCH-GEN, volume = "102", number = "2", pages = "353--376", month = feb, year = "1974", CODEN = "JPGEDD", DOI = "https://doi.org/10.1037/h0035850", ISSN = "0096-3445 (print), 1939-2222 (electronic)", ISSN-L = "0096-3445", bibdate = "Fri Feb 15 06:32:17 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://psycnet.apa.org/journals/xge/102/2/353/; http://psycnet.apa.org/psycinfo/1974-24305-001", acknowledgement = ack-nhfb, fjournal = "Journal of Experimental Psychology: General", } @PhdThesis{Diaconis:1974:WSA, author = "Persi Diaconis", title = "Weak and Strong Averages in probability and the theory numbers", type = "{Ph.D.} thesis", school = "Department of Statistics, Harvard University", address = "Cambridge, MA, USA", pages = "xvii + 117", year = "1974", bibdate = "Mon Nov 14 10:26:03 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Duran:1974:ACS, author = "J. Peter Duran", title = "Almost convergence, summability and ergodicity", journal = j-CAN-J-MATH, volume = "26", number = "??", pages = "372--387", month = "????", year = "1974", CODEN = "CJMAAB", DOI = "https://doi.org/10.4153/CJM-1974-039-6", ISSN = "0008-414X (print), 1496-4279 (electronic)", ISSN-L = "0008-414X", bibdate = "Sat Sep 10 15:38:45 MDT 2011", bibsource = "http://cms.math.ca/cjm/v26/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/canjmath1970.bib", acknowledgement = ack-nhfb, fjournal = "Canadian Journal of Mathematics = Journal canadien de math{\'e}matiques", journal-URL = "http://cms.math.ca/cjm/", remark = "In relation to Benford's Law, see theorem 6.2.", } @Article{Hill:1974:RFF, author = "Bruce M. Hill", title = "The rank-frequency form of {Zipf}'s law", journal = j-J-AM-STAT-ASSOC, volume = "69", number = "348", pages = "1017--1026", month = dec, year = "1974", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", MRclass = "62E15", MRnumber = "0426242 (54 \#14188)", MRreviewer = "S. M. Samuels", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", ZMnumber = "Zbl 0301.60011", acknowledgement = ack-nhfb, classification = "B0240 (Probability and statistics); C1140 (Probability and statistics)", corpsource = "Dept. of Statistics, Univ. of Michigan, Ann Arbor, MI, USA", fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", keywords = "Bose--Einstein allocation; distribution; random processes; rank frequency form; statistics; Zipf's law", treatment = "T Theoretical or Mathematical", } @Book{Kuipers:1974:UDS, author = "Lauwerens Kuipers and Harald Niederreiter", title = "Uniform distribution of sequences", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xiv + 390", year = "1974", ISBN = "0-471-51045-9", ISBN-13 = "978-0-471-51045-1", LCCN = "QA292 .K84", bibdate = "Sat Nov 12 15:42:53 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", series = "Pure and applied mathematics", acknowledgement = ack-nhfb, remark = "``A Wiley-Interscience publication.''.", subject = "Sequences (Mathematics); Uniform distribution (Probability theory)", } @Article{Schorr:1974:LLL, author = "Alan E. Schorr", title = "{Lotka's Law} and Library Science", journal = "RQ", volume = "14", number = "1", pages = "32--33", month = "Fall", year = "1974", CODEN = "????", ISSN = "0033-7072", ISSN-L = "0033-7072", bibdate = "Fri Nov 25 08:12:46 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "RQ", fjournal = "Reference Quarterly", } @Unpublished{Singmaster:1974:RSB, author = "D. Singmaster", title = "Recurrent sequences and {Benford}'s law", year = "1974", bibdate = "Sat Nov 12 16:25:29 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Preprint submitted to Fibonacci Quarterly in 1974, but never published there.", acknowledgement = ack-nhfb, } @Book{Sterbenz:1974:FPC, author = "Pat H. Sterbenz", title = "Floating Point Computation", publisher = pub-PH, address = pub-PH:adr, pages = "xiv + 316", year = "1974", ISBN = "0-13-322495-3", ISBN-13 = "978-0-13-322495-5", LCCN = "QA76.8.I12 S771 1974", MRclass = "68A05 (65G05)", MRnumber = "50 1556", bibdate = "Sat May 29 08:01:36 1999", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib; http://www.math.utah.edu/pub/tex/bib/mathcw.bib", series = "Prentice-Hall Series in Automatic Computation.", acknowledgement = ack-nj # " and " # ack-nhfb, keywords = "Compiling (Electronic computers); Computation by computer systems --- Floating point representation; Floating-point arithmetic; IBM 360 (Computer) --- Programming", remark = "From p. 116: ``There is general agreement that the mantissas of floating-point numbers are not uniformly distributed. [See Hamming (1962) [\cite{Hamming:1962:NMS}], Pinkham (1961) [Pinkham:1961:DFS], or Knuth (1969) [Knuth:1969:SNM].] Instead, it is customary to assume that they are distributed logarithmically, that is, that the probability density function is $$ (3.12.9) f(m) = \frac {1}{m \ln r}, \qquad r^{-1} \leq m \leq 1. $$ This assumption is based on the following observations: First, this distribution reproduces itself under multiplication, but a uniform distribution does not. [See Hamming (1962) [\cite{Hamming:1962:NMS}] and Exercise 20.] A second justification is based on the fact that many of the numbers that arise in computation represent measurable quantities such as lengths, forces, etc., and it is reasonable to believe that the distribution of the mantissas of such quantities is independent of the units in which they are measured. Pinkham (1961) shows that this leads to the conclusion that the distribution of mantissas must be logarithmic.''", reviewer = "F. J. Murray", tableofcontents = "Preface / ix \\ 1. Floating-Point Number Systems / 1 \\ 1.1 Fixed-Point Calculation / 1 \\ 1.2 Floating-Decimal Representation of Numbers / 4 \\ 1.3 Floating-Decimal Arithmetic / 6 \\ 1.4 Floating-Point Number Systems / 9 \\ 1.5 FP(r, p, c) and FP(r, p, R) / 12 \\ 1.6 Laws of Algebra / 14 \\ 1.7 Inequalities in FP(r,p, c) / 21 \\ 1.8 FP(r, p, clq) / 22 \\ 1.9 The Solution of a * x = b in FP(r,p, c) / 29 \\ 1.10 Division / 33 \\ Exercises / 35 \\ 2. Floating-Point Overflow and Underflow / 39 \\ 2.1 Bounds for Exponents / 39 \\ 2.2 $\Omega$-Zero Fixup / 41 \\ 2.3 Interrupt / 44 \\ 2.4 Messages and Tests / 46 \\ 2.5 ON OVERFLOW and ON UNDERFLOW in PL/I / 49 \\ 2.6 Example / 50 \\ 2.7 Counting Mode / 57 \\ 2.8 Gradual Underflow / 59 \\ 2.9 Imprecise Interrupt / 61 \\ 2.10 Changing the Treatment of Spill / 62 \\ 2.11 Virtual Overflow and Underflow / 64 \\ 2.12 Division by Zero and Indeterminant Forms / 66 \\ Exercises / 67 \\ 3. Error Analysis / 71 \\ 3.1 Significant Digits / 71 \\ 3.2 Relative Error / 73 \\ 3.3 Relative Error in FP(r,p, clq) / 75 \\ 3.4 Approximate Laws of Algebra / 80 \\ 3.5 Propagation of Rounding Error / 87 \\ 3.6 X**N / 92 \\ 3.7 Condition / 98 \\ 3.8 Error Analysis of a Program / 103 \\ 3.9 Backward Error Analysis / 105 \\ 3.10 Examples / 107 \\ 3.11 Changing the Problem / 109 \\ 3.12 Statistical Error Analysis / 113 \\ Exercises / 117 \\ 4. Example / 123 \\ 4.1 Quadrature / 123 \\ 4.2 Power Series / 130 \\ 4.3 Exact Sums and Differences in FP(r, p, clq) / 137 \\ 4.4 Dismantling Floating-Point Numbers / 143 \\ Exercises / 146 \\ 5. Double-Precision Calculation / 154 \\ 5.1 Programs Using Double-Precision Arithmetic / 155 \\ 5.2 Implicit Typing of Names / 162 \\ 5.3 Routines to Perform Double-Precision Arithmetic / 163 \\ 5.4 Double-Precision Multiplication / 165 \\ 5.5 Double-Precision Addition and Subtraction / 168 \\ 5.6 Double-Precision Division / 171 \\ 5.7 Writing Double-Precision Programs Without Language Support / 178 \\ 5.8 Uses of Double-Precision / 180 \\ 5.9 Higher-Precision Arithmetic / 185 \\ Exercises / 186 \\ 6. Rounding / 189 \\ 6.1 General Considerations / 189 \\ 6.2 Uses of Rounding / 191 \\ 6.3 Implementation of Rounding / 192 \\ 6.4 Bias Removal / 194 \\ 6.5 Other ``Rounding'' Procedures / 197 \\ Exercises / 198 \\ 7. Automatic Analysis of Error / 201 \\ 7.1 Introduction / 201 \\ 7.2 Significance Arithmetic / 202 \\ 7.3 Noisy Mode / 205 \\ 7.4 Interval Arithmetic / 207 \\ 7.5 Rerunning the Program in Higher-Precision / 213 \\ Exercises / 222 \\ 8. Radix Conversion / 224 \\ 8.1 Equivalent Number of Digits / 224 \\ 8.2 Properties of Conversion Transformations / 228 \\ 8.3 Conversion Techniques / 232 \\ Exercises / 238 \\ 9/ Carefully Written Programs / 240 \\ 9.1 Introduction / 240 \\ 9.2 Average Problem / 240 \\ 9.3 Quadratic Equation / 246 \\ Exercises / 252 \\ 10. Checking and Testing / 255 \\ 10.1 Range Checking / 255 \\ 10.2 Mathematical Checks / 256 \\ 10.3 Testing / 259 \\ Exercises / 261 \\ 11. Language Features for Floating-Point Computation / 263 \\ 11.1 Introduction / 263 \\ 11.2 Predictability, Controllability, Observability / 264 \\ 11.3 Ease of Programming / 265 \\ 11.4 Machine Independence / 267 \\ Exercises / 269 \\ 12. Floating-Point Hardware / 271 \\ 12.1 Choice of Radix / 271 \\ 12.2 The Representation of Floating-Point Numbers / 273 \\ 12.3 FP(r, p, c) and FP(r, p, R) / 277 \\ 12.4 Unnormalized Numbers and Unnormalized Arithmetic / 278 \\ Exercises / 282 \\ 13. Complex Numbers / 285 \\ 13.1 Programs Using Complex Numbers / 285 \\ 13.2 Relative Error / 286 \\ 13.3 Complex Arithmetic / 287 \\ Exercises / 297 \\ Bibliography / 301 \\ Glossary of Symbols / 309 \\ Index / 311", } @Article{Tsao:1974:DSD, author = "Nai Kuan Tsao", title = "On the Distributions of Significant Digits and Roundoff Errors", journal = j-CACM, volume = "17", number = "5", pages = "269--271", month = may, year = "1974", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", MRclass = "65G05", MRnumber = "49 6595", MRreviewer = "N. N. Abdelmalek", bibdate = "Mon Jan 22 07:26:38 MST 2001", bibsource = "Compendex database; ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://dblp.uni-trier.de/db/journals/cacm/cacm17.html#Tsao74; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "Generalized logarithmic law is derived for the distribution of the first $t$ significant digits of a random digital integer. This result is then used to determine the distribution of the roundoff errors in floating-point operations, which is a mixture of uniform and reciprocal distributions.", acknowledgement = ack-nhfb, classcodes = "C5230 (Digital arithmetic methods)", classification = "921", corpsource = "Aerospace Res. Labs., Wright-Patterson AFB, OH, USA", fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", journalabr = "Commun ACM", keywords = "Benford's Law; digital arithmetic; digital integer; distributions; error analysis; floating point operations; Law of Anomalous Numbers; logarithmic law; mathematical techniques; mean value; random; reciprocal; roundoff errors; significant digits; uniform distribution; variance; Zipf's Law", oldlabel = "Tsao74", treatment = "T Theoretical or Mathematical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/Tsao74", } @Article{Voos:1974:BCL, author = "Henry Voos", title = "Brief Communications: {Lotka} and information science", journal = j-J-AM-SOC-INF-SCI, volume = "25", number = "4", pages = "270--272", month = jul, year = "1974", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630250410", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:50 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Voos:1974:LIS, author = "Henry Voos", title = "{Lotka} and Information Science", journal = j-J-AM-SOC-INF-SCI, volume = "25", number = "4", pages = "270--272", month = jul # "\slash " # aug, year = "1974", CODEN = "AISJB6", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Nov 25 08:14:01 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", keywords = "Lotka's Law", } @Article{Yavuz:1974:ZLE, author = "Davras Yavuz", title = "{Zipf}'s law and entropy", journal = j-IEEE-TRANS-INF-THEORY, volume = "20", number = "5", pages = "650--650", year = "1974", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/TIT.1974.1055269", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Sat Nov 12 10:43:30 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0295.94048", acknowledgement = ack-nhfb, classmath = "94A15 (General topics of information theory)", fjournal = "IEEE Transactions on Information Theory", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18", } @InProceedings{Bennett:1975:SDI, author = "John M. Bennett", booktitle = "Int. Comput. Symp. 1975, Proc., Antibes, (1975)", title = "Storage design for information retrieval: {Scarrott}'s conjecture and {Zipf}'s law", publisher = "????", address = "????", pages = "233--237", year = "1975", bibdate = "Sat Nov 12 10:49:15 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0322.68068", acknowledgement = ack-nhfb, classmath = "68P20 (Information storage and retrieval) 68N01 (General)", } @Book{Hardy:1975:ITN, author = "Godfrey H. Hardy and Edward M. Wright", title = "An Introduction to the Theory of Numbers", publisher = pub-CLARENDON, address = pub-CLARENDON:adr, edition = "Fourth", pages = "421", year = "1975", ISBN = "0-19-853310-1", ISBN-13 = "978-0-19-853310-8", LCCN = "????", bibdate = "Fri Jun 24 12:32:14 2005", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Hill:1975:SFZ, author = "Bruce M. Hill and Michael Woodroofe", title = "Stronger forms of {Zipf}'s law", journal = j-J-AM-STAT-ASSOC, volume = "70", number = "349", pages = "212--219", month = mar, year = "1975", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", MRclass = "62E15 (62D05)", MRnumber = "0440763 (55 \#13633)", MRreviewer = "I. J. Good", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", ZMnumber = "Zbl 0326.92014", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Kemperman:1975:BDM, author = "J. H. B. Kemperman", title = "Bounds on the discrepancy modulo $1$ of a real random variable", journal = j-BULL-INST-MATH-STAT, volume = "4", number = "1", pages = "138--138", month = "????", year = "1975", CODEN = "SMBCVA", ISSN = "0146-3942", ISSN-L = "0146-3942", bibdate = "Sat Nov 12 16:26:37 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Abstract no. 75t-47.", acknowledgement = ack-nhfb, fjournal = "Bulletin --- Institute of Mathematical Statistics", keywords = "Benford's Law", } @Article{Kreifelts:1975:OBF, author = "T. Kreifelts", title = "{Optimale Basiswahl f{\"u}r eine Gleitkomma-Arithmetik (Berichtigung)} ({German}) {Optimal Basis Choice for a Floating-Point Arithmetic (Correction)}", journal = j-COMPUTING, volume = "14", number = "3", pages = "313--314", month = sep, year = "1975", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", bibdate = "Fri Sep 16 16:30:39 1994", bibsource = "garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/computing.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "See \cite{Kreifelts:1973:OBG}.", acknowledgement = ack-nj, fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", language = "German", } @Article{Partridge:1975:DDW, author = "D. Partridge", title = "A dynamic database which automatically removes unwanted generalisation for the efficient analysis of language features that exhibit a disparate frequency distribution", journal = j-COMP-J, volume = "18", number = "1", pages = "43--48", month = feb, year = "1975", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Mar 25 13:51:56 MST 1997", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www3.oup.co.uk/computer_journal/hdb/Volume_18/Issue_01/", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_18/Issue_01/tiff/43.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_18/Issue_01/tiff/44.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_18/Issue_01/tiff/45.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_18/Issue_01/tiff/46.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_18/Issue_01/tiff/47.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_18/Issue_01/tiff/48.tif", abstract = "A self-organising database was developed as part of a general language analysis system. The periodic, automatic reorganisation of the database was aimed at increasing the efficiency of analysis of a language in which the constituent features exhibit a Zipfian type rank-frequency relationship. Such a distribution means that only a small number of features account for a large proportion of the information, while a large number of possible features are seldom encountered and thus seldom accessed within the database. The mechanism described aims at reconciling two conflicting procedures: condensation by generalisation of language features to minimise the total size of the database, and the particularisation of the few commonly occurring features to minimise the average analysis time. Results are presented for the application of this mechanism to the analysis of batches of FORTRAN programs that constituted part of the `normal' workload for computers within five different environments.", acknowledgement = ack-nhfb, classcodes = "C6120 (File organisation); C7820 (Humanities computing)", classification = "723", corpsource = "Univ. Nairobi, Kenya", fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", keywords = "analysis; data bases; data processing; disparate frequency distribution; dynamic database; file organisation; language; language translation and linguistics; self organising; unwanted generalisation", treatment = "P Practical", } @Article{Pope:1975:BLP, author = "Andrew Pope", title = "{Bradford's Law} and the periodical literature of information science", journal = j-J-AM-SOC-INF-SCI, volume = "26", number = "4", pages = "207--213", month = jul, year = "1975", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630260403", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:54 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Schorr:1975:BCL, author = "Alan Edward Schorr", title = "Brief Communications: {Lotka's Law} and {Map Librarianship}", journal = j-J-AM-SOC-INF-SCI, volume = "26", number = "3", pages = "189--190", month = may, year = "1975", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630260308", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:53 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Schorr:1975:LLH, author = "Alan E. Schorr", title = "{Lotka's Law} and the History of Legal Medicine", journal = "Research in Librarianship", volume = "30", number = "??", pages = "205--209", month = sep, year = "1975", CODEN = "????", ISSN = "0034-5245", ISSN-L = "0034-5245", bibdate = "Fri Nov 25 08:17:00 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Res. Librarianship", } @Article{Schorr:1975:LLM, author = "Alan E. Schorr", title = "{Lotka's Law} and Map Librarianship", journal = j-J-AM-SOC-INF-SCI, volume = "26", number = "3", pages = "189--190", month = may # "\slash " # jun, year = "1975", CODEN = "AISJB6", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Nov 25 08:11:19 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", } @Article{Webb:1975:DFD, author = "William Webb", title = "Distribution of the First Digits of {Fibonacci} Numbers", journal = j-FIB-QUART, volume = "13", number = "4", pages = "334--336", month = dec, year = "1975", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 17:59:03 MDT 2011", bibsource = "http://www.fq.math.ca/13-4.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/13-4/webb.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly", journal-URL = "http://www.fq.math.ca/", keywords = "Benford's Law", } @Article{Woodroofe:1975:ZL, author = "Michael Woodroofe and Bruce Hill", title = "On {Zipf}'s law", journal = j-J-APPL-PROBAB, volume = "12", number = "3", pages = "425--434", month = sep, year = "1975", CODEN = "JPRBAM", ISSN = "0021-9002 (print), 1475-6072 (electronic)", ISSN-L = "0021-9002", MRclass = "62E20 (62P10)", MRnumber = "0440764 (55 \#13634)", MRreviewer = "I. J. Good", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/3212857", ZMnumber = "Zbl 0343.60012", abstract = "A Zipf's law is a probability distribution on the positive integers which decays algebraically. Such laws describe (approximately) a large class of phenomena. We formulate a model for such phenomena and, in terms of our model, give necessary and sufficient conditions for a Zipf's law to hold.", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Probability", journal-URL = "http://www.jstor.org/journals/00219002.html; http://projecteuclid.org/euclid.jap/", keywords = "classical occupancy problem; laws of large numbers: weak convergence; regularly varying functions", } @Article{Worthen:1975:ABL, author = "Dennis B. Worthen", title = "The Application of {Bradford's Law} to Monographs", journal = j-J-DOC, volume = "31", number = "1", pages = "19--25", month = "????", year = "1975", CODEN = "JDOCAS", DOI = "https://doi.org/10.1108/eb026590", ISSN = "0022-0418 (print), 1758-7379 (electronic)", ISSN-L = "0022-0418", bibdate = "Fri Nov 25 07:18:55 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Documentation", journal-URL = "http://www.emeraldinsight.com/loi/jd", } @Article{Allison:1976:LLP, author = "Paul D. Allison and Derek de Solla Price and Belver C. Griffith and Michael J. Moravcsik and John A. Stewart", title = "{Lotka's Law}: a Problem in Its Interpretation and Application", journal = j-SOC-STUD-SCI, volume = "6", number = "2", pages = "269--276", day = "1", month = may, year = "1976", CODEN = "SSSCDH", DOI = "https://doi.org/10.1177/030631277600600205", ISSN = "0306-3127 (print), 1460-3659 (electronic)", ISSN-L = "0306-3127", bibdate = "Mon Dec 3 09:12:29 MST 2018", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/socstudsci.bib", URL = "https://journals.sagepub.com/doi/pdf/10.1177/030631277600600205", acknowledgement = ack-nhfb, fjournal = "Social Studies of Science", journal-URL = "http://sss.sagepub.com/content/by/year", keywords = "Lotka's Law of scientific authorship for the relative contributions of the most prolific authors", } @Article{Billera:1976:PSC, author = "Louis J. Billera and Robert E. Bixby", title = "{Pareto} Surfaces of Complexity $1$", journal = j-SIAM-J-APPL-MATH, volume = "30", number = "1", pages = "81--89", month = jan, year = "1976", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Thu Oct 15 18:16:06 MDT 1998", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/siamjapplmath.bib; JSTOR database", acknowledgement = ack-nhfb, classification = "B0250 (Combinatorial mathematics); C1160 (Combinatorial mathematics)", corpsource = "Dept. of Math., Cornell Univ., Ithaca, NY, USA", fjournal = "SIAM Journal on Applied Mathematics", journal-URL = "http://epubs.siam.org/siap", keywords = "1-commodity representation; attainable sets; complexity 1; game theory; graph theory; Pareto surfaces; set theory", treatment = "T Theoretical or Mathematical", } @Article{Brambilla:1976:DDA, author = "Francisco Brambilla", title = "La distribuzione delle aziende secondo le dimensioni. {Verifica} empirica della legge logonormale e di quella di {Zipf} e nuovi modelli interpretativi. ({Italian}) [{The} distribution of firms by size. {Empirical} test of the log-normal law and {Zipf}'s law and new models of interpretation]", journal = "Giornale degli Economisti e Annali de Economia (Nuova Serie)", volume = "35", number = "3--4", pages = "131--153, 235", month = mar # "\slash " # apr, year = "1976", CODEN = "????", ISSN = "0017-0097", ISSN-L = "0017-0097", MRclass = "90A15", MRnumber = "0452510 (56 \#10789)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/23242834", acknowledgement = ack-nhfb, ajournal = "Giorn. Econom. Ann. Econom. (N.S.)", fjournal = "Giornale degli Economisti e Annali de Economia", language = "Italian", } @Article{Cohen:1976:EFD, author = "Daniel I. A. Cohen", title = "An explanation of the first digit phenomenon", journal = j-J-COMB-THEORY-A, volume = "20", number = "3", pages = "367--370", month = may, year = "1976", CODEN = "JCBTA7", ISSN = "0097-3165 (print), 1096-0899 (electronic)", ISSN-L = "0097-3165", bibdate = "Thu Feb 15 16:29:51 2007", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Combinatorial Theory (Series A)", journal-URL = "http://www.sciencedirect.com/science/journal/00973165", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Article{Davis:1976:SRI, author = "Basil Davis", title = "Some Remarks on Initial Digits", journal = j-FIB-QUART, volume = "14", number = "1", pages = "13--14", month = feb, year = "1976", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 17:59:05 MDT 2011", bibsource = "http://www.fq.math.ca/14-1.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/14-1/davis.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly", journal-URL = "http://www.fq.math.ca/", } @Article{Feldstein:1976:CED, author = "Alan Feldstein and Richard Goodman", title = "Convergence Estimates for the Distribution of Trailing Digits", journal = j-J-ACM, volume = "23", number = "2", pages = "287--297", month = apr, year = "1976", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321941.321948", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Wed Jan 15 18:12:53 MST 1997", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "An analysis is made of the distribution of trailing digits (tail end digits) of positive real floating-point numbers represented in arbitrary base $ \beta $ and randomly chosen from a logarithmic distribution. The analysis shows that the $n$ th digit for $ n \geq 2 $ is actually approximately uniformly distributed. The approximation depends upon both $n$ and the base beta. It becomes better as $n$ increases, and it is exact in the limit as $ n \rightarrow \infty $. A table of this distribution is presented for various $ \beta $ and $n$, along with a table of the maximum digit by digit deviation $ \Delta $ of the logarithmic distribution from the uniform distribution. Various asymptotic results for $ \Delta $ are included.", acknowledgement = ack-nhfb, classification = "723", fjournal = "Journal of the Association for Computing Machinery", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", journalabr = "J Assoc Comput Mach", keywords = "Benford's Law; computer arithmetic; computer programming; Law of Anomalous Numbers; Zipf's Law", } @Article{Haspers:1976:YFB, author = "Jan H. Haspers", title = "The yield formula and {Bradford's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "27", number = "5", pages = "281--287", month = sep, year = "1976", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630270503", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:57 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Hubert:1976:NIB, author = "John J. Hubert", title = "On the {Naranan} interpretation of {Bradford's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "27", number = "5", pages = "339--341", month = sep, year = "1976", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630270510", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:57 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Orlov:1976:CBP, author = "Ju. K. Orlov", title = "The connection between the {Pareto} distribution and the generalized {Zipf--Mandelbrot} law", journal = "Sakharthvelos SSR Mecnierebatha Akademiis Moambe", volume = "83", number = "1", pages = "57--60", year = "1976", CODEN = "????", ISSN = "????", MRclass = "90A15", MRnumber = "0434355 (55 \#7321)", MRreviewer = "S. Krcevinac", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Sakharth. SSR Mech. Akad. Moambe", } @Article{Raimi:1976:FDP, author = "Ralph A. Raimi", title = "The first digit problem", journal = j-AMER-MATH-MONTHLY, volume = "83", number = "7", pages = "521--538", month = aug # "\slash " # sep, year = "1976", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2319349", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Sat Nov 12 15:49:53 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/2319349", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", remark = "This article surveys the literature up to 1976, reports defects in several earlier mathematical `explanations' of Benford's Law, and gives a 52-entry bibliography of prior work. All of those entries are incorporated in this bibliography. It criticizes parts of \cite{Goudsmit:1944:SFN} as `dead wrong' (and the author reiterates that view in \cite{Raimi:1985:FDP}). See also later work \cite{Logan:1978:FDP}. On page 529, Raimi writes ``If the first digits of all the tables in the universe obey some fixed distribution law, Stigler's or Benford's or some other, that law must surely be independent of the system of units chosen, since God is not known to favor either the metric system or the English system. In other words, a universal first digit law, if it exists, must be scale-invariant.''", remark-2 = "This is so far the second earliest reference that I have found to \cite{Newcomb:1881:NFU}; the earliest is \cite{Boring:1920:LNL}, which has only brief mention of the question of digit frequencies. Raimi comments on p. 522: ``This assertion, whatever it may mean, will be called Benford's Law because it has been thought by many writers to have originated with the General Electric Company physicist Frank Benford [2]. Certainly Benford popularized the problem, and he may well have been unaware that the polymathic Simon Newcomb, primarily an astronomer but also sometime editor of The American Journal of Mathematics, had also formulated the same law 57 years earlier [1]. There is ample precedent for naming laws and theorems for persons other than their discoverers, else half of analysis would be named after Euler. Besides, even Newcomb implied that the observation giving rise to the Benford law was an old one in his day. One would hate to change the name of the law now only to find later that another change was called for.''", } @Article{Rouault:1976:PAE, author = "Alain Rouault", title = "Propri{\'e}t{\'e}s asymptotiques d'un {$n$}-{\'e}chantillon d'une variable al{\'e}atoire d{\'e}nombrable connues sous le nom de lois de {Zipf}. ({French}) [Asymptotic properties of an $n$-sample of a denumerable random variable known under the name of {Zipf}'s law]", journal = j-C-R-ACAD-SCI-PARIS-SER-AB, volume = "283", number = "6", pages = "Aiv, A379--A380", year = "1976", CODEN = "CHASAP", ISSN = "0151-0509", ISSN-L = "0151-0509", MRclass = "62E20", MRnumber = "0423644 (54 \#11619)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Comptes Rendus Hebdomadaires des S{\'e}ances de l'Acad{\'e}mie des Sciences. S{\'e}ries A et B", language = "French", } @Article{Rouault:1976:SML, author = "Alain Rouault", title = "Sources markoviennes et lois de {Zipf}. ({French}) [{Markovian} source and {Zipf} laws]", journal = j-C-R-ACAD-SCI-PARIS-SER-AB, volume = "283", number = "10", pages = "Aiii, A789--A790", year = "1976", CODEN = "CHASAP", ISSN = "0151-0509", ISSN-L = "0151-0509", MRclass = "62E20", MRnumber = "0423645 (54 \#11620)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Comptes Rendus Hebdomadaires des S{\'e}ances de l'Acad{\'e}mie des Sciences. S{\'e}ries A et B", language = "French", } @Article{Vlachy:1976:TFL, author = "Jan Vlach{\'y}", title = "Time Factor in {Lotka's Law}", journal = "Probleme de Informare si Documentare", volume = "10", number = "2", pages = "44--87", month = "????", year = "1976", CODEN = "PIDCA6", ISSN = "0032-924X", ISSN-L = "0032-924X", bibdate = "Fri Nov 25 07:56:00 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Prob. inf. \& doc.", } @Article{Brookes:1977:TBL, author = "Bertram C. Brookes", title = "Theory of the {Bradford Law}", journal = j-J-DOC, volume = "33", number = "3", pages = "180--209", month = "????", year = "1977", CODEN = "JDOCAS", DOI = "https://doi.org/10.1108/eb026641", ISSN = "0022-0418 (print), 1758-7379 (electronic)", ISSN-L = "0022-0418", bibdate = "Fri Nov 25 07:16:18 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Documentation", journal-URL = "http://www.emeraldinsight.com/loi/jd", } @Article{Clark:1977:ESL, author = "Douglas W. Clark and C. Cordell Green", title = "An Empirical Study of List Structure in {Lisp}", journal = j-CACM, volume = "20", number = "2", pages = "78--87", month = feb, year = "1977", CODEN = "CACMA2", ISSN = "0001-0782 (print), 1557-7317 (electronic)", ISSN-L = "0001-0782", bibdate = "Mon Jan 22 06:31:28 MST 2001", bibsource = "Compendex database; ftp://ftp.ira.uka.de/pub/bibliography/Compiler/garbage.collection.bib; http://dblp.uni-trier.de/db/journals/cacm/cacm20.html#ClarkG77; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "Static measurements of the list structure of five large Lisp programs are reported and analyzed. These measurements reveal substantial regularity, or predictability, among pointers to atoms and especially among pointers to lists. Pointers to atoms are found to obey, roughly, Zipf's law, which governs word frequencies in natural languages; pointers to lists usually point to a location physically nearby in memory. The use of such regularities in the space-efficient representation of list structure is discussed.", acknowledgement = ack-nhfb, classcodes = "C6130 (Data handling techniques); C6140D (High level languages)", classification = "723", comment = "Great paper; evidence pro CDR-coding, con fancy CONS", corpsource = "Carnegie-Mellon Univ., Pittsburgh, PA, USA", fjournal = "Communications of the ACM", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J79", journalabr = "Commun ACM", keywords = "computer programming languages; data processing --- Data Structures; empirical study; LISP; list structure; measurements; regularity", oldlabel = "ClarkG77", treatment = "P Practical", XMLdata = "ftp://ftp.informatik.uni-trier.de/pub/users/Ley/bib/records.tar.gz#journals/cacm/ClarkG77", } @Article{Coile:1977:LFD, author = "Russell C. Coile", title = "{Lotka}'s Frequency Distribution of Scientific Productivity", journal = j-J-AM-SOC-INF-SCI, volume = "28", number = "6", pages = "366--370", month = nov, year = "1977", CODEN = "AISJB6", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Nov 25 08:08:19 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", keywords = "Lotka's Law", } @Article{Diaconis:1977:DLD, author = "Persi Diaconis", title = "The distribution of leading digits and uniform distribution $ \bmod 1 $", journal = j-ANN-PROBAB, volume = "5", number = "1", pages = "72--81", month = feb, year = "1977", CODEN = "APBYAE", DOI = "https://doi.org/10.1214/aop/1176995891", ISSN = "0091-1798 (print), 2168-894X (electronic)", ISSN-L = "0091-1798", bibdate = "Sun Apr 20 10:44:17 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/annprobab1970.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://projecteuclid.org/euclid.aop/1176995891", acknowledgement = ack-nhfb, fjournal = "Annals of Probability", journal-URL = "http://projecteuclid.org/all/euclid.aop", keywords = "Benford's Law", } @Article{Goudsmit:1977:PEP, author = "Samuel Goudsmit", title = "Pitfalls in Elementary Probability", journal = j-PROC-AMER-PHIL-SOC, volume = "121", number = "2", pages = "188--189", day = "29", month = apr, year = "1977", CODEN = "PAPCAA", ISSN = "0003-049X (print), 2326-9243 (electronic)", ISSN-L = "0003-049X", bibdate = "Fri Mar 22 08:50:02 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Based on a lecture presented November 11, 1976.", URL = "http://www.jstor.org/stable/986527", acknowledgement = ack-mjn # "\slash " # ack-nhfb, fjournal = "Proceedings of the {American Philosophical Society} held at {Philadelphia} for promoting useful knowledge", journal-URL = "http://www.jstor.org/journal/procamerphilsoci", keywords = "Benford's Law; biased coin flips; biased dice throws; roulette wheels", } @Article{Hubert:1977:LEL, author = "John J. Hubert", title = "Letters to the {Editor}: {Lotka's Law} in the humanities", journal = j-J-AM-SOC-INF-SCI, volume = "28", number = "1", pages = "66--66", month = jan, year = "1977", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630280115", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:58 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Krisciunas:1977:LEL, author = "Kevin Krisciunas", title = "Letters to the {Editor}: {Lotka's Law} --- year by year", journal = j-J-AM-SOC-INF-SCI, volume = "28", number = "1", pages = "65--66", month = jan, year = "1977", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630280114", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:02:58 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Kubovy:1977:RAA, author = "M. Kubovy", title = "Response Availability and the Apparent Spontaneity of Numerical Choices", journal = "Journal of Experimental Psychology: Human Performance and Performance", volume = "3", number = "2", pages = "359--364", month = may, year = "1977", DOI = "https://doi.org/10.1037/0096-1523.3.2.359", ISSN = "0096-1523 (print), 1939-1277 (electronic)", ISSN-L = "0096-1523", bibdate = "Sat Sep 10 12:51:40 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://psycnet.apa.org/index.cfm?fa=search.displayrecord&uid=1977-30946-001", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Article{Ylvisaker:1977:TR, author = "Donald Ylvisaker", title = "Test Resistance", journal = j-J-AM-STAT-ASSOC, volume = "72", number = "359", pages = "551--556", month = sep, year = "1977", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Jan 25 08:05:48 MST 2012", bibsource = "http://www.jstor.org/journals/01621459.html; http://www.jstor.org/stable/i314242; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc1970.bib", URL = "http://www.jstor.org/stable/2286216", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Brady:1978:MBL, author = "W. G. Brady", title = "More on {Benford}'s Law", journal = j-FIB-QUART, volume = "16", number = "1", pages = "51--52", month = feb, year = "1978", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 17:59:24 MDT 2011", bibsource = "http://www.fq.math.ca/16-1.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/16-1/brady.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly", journal-URL = "http://www.fq.math.ca/", } @Article{Brookes:1978:FRD, author = "Bertram C. Brookes and Jos{\'e} M. Griffiths", title = "Frequency-rank distributions", journal = j-J-AM-SOC-INF-SCI, volume = "29", number = "1", pages = "5--13", month = jan, year = "1978", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630290104", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Mon Feb 25 10:40:13 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://onlinelibrary.wiley.com/doi/10.1002/asi.4630290104/abstract", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", keywords = "Anomalous Law of Numbers; Benford's Law", } @PhdThesis{Chen:1978:ZL, author = "Wen-Chen Chen", title = "On {Zipf}'s law", type = "Thesis ({Ph.D.})", school = "University of Michigan", address = "Ann Arbor, MI, USA", pages = "114", year = "1978", MRclass = "Thesis", MRnumber = "2627928", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:7822871", acknowledgement = ack-nhfb, remark = "Available from ProQuest LLC, Ann Arbor, MI, USA. Author given name appears as Wen-Cheng in MathSciNet database.", } @Article{Drott:1978:EEB, author = "M. Carl Drott and Belver C. Griffith", title = "An empirical examination of {Bradford's Law} and the scattering of scientific literature", journal = j-J-AM-SOC-INF-SCI, volume = "29", number = "5", pages = "238--246", month = sep, year = "1978", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630290506", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:02 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Elshoff:1978:SSC, author = "James L. Elshoff", title = "A study of the structural composition of {PL/I} programs", journal = j-SIGPLAN, volume = "13", number = "6", pages = "29--37", month = jun, year = "1978", CODEN = "SINODQ", DOI = "https://doi.org/10.1145/987515.987518", ISSN = "0362-1340 (print), 1523-2867 (print), 1558-1160 (electronic)", ISSN-L = "0362-1340", bibdate = "Wed Jun 18 16:08:52 MDT 2008", bibsource = "http://portal.acm.org/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "Some research on a theory of the structural composition of an algorithm has recently been reported. The theory embodies models of operator and operand distributions that have been proposed by Bayer, Zipf, and Zweben. This paper reports a study in which the three models of operator distributions are compared with the measured distributions found in 34 PL/I programs. Two of three models, one by Zipf and one by Zweben, are shown to correlate highly with the measured distributions. Two variations of the Zipf model are then formulated and tested with good results. Also, the relationship between the method of counting operators and the models is investigated.", acknowledgement = ack-nhfb, classification = "C4240 (Programming and algorithm theory); C6140D (High level languages)", corpsource = "Computer Sci. Dept., General Motors Res. Labs., Warren, MI, USA", fjournal = "ACM SIG{\-}PLAN Notices", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J706", keywords = "counting operators; operator distributions; PL/1; PL/1 programs; programming theory; structural composition", treatment = "T Theoretical or Mathematical", } @Book{Heaps:1978:IRC, author = "Harold Stanley Heaps", title = "Information retrieval, computational and theoretical aspects", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "xii + 344", year = "1978", ISBN = "0-12-335750-0", ISBN-13 = "978-0-12-335750-2", LCCN = "Z699 .H38", bibdate = "Sat Nov 12 11:36:39 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", series = "Library and information science", acknowledgement = ack-nhfb, remark = "Heaps's Law is given on pp. 206--208.", subject = "Information storage and retrieval systems", } @Article{Hubert:1978:BCR, author = "John J. Hubert", title = "Brief Communications: A relationship between two forms of {Bradford's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "29", number = "3", pages = "159--161", month = may, year = "1978", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630290311", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:01 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "23 Mar 2007", } @Article{Logan:1978:FDP, author = "Jonothan L. Logan and Samuel A. Goudsmit", title = "The First Digit Phenomenon", journal = j-PROC-AMER-PHIL-SOC, volume = "122", number = "4", pages = "193--197", day = "18", month = aug, year = "1978", CODEN = "PAPCAA", ISSN = "0003-049X (print), 2326-9243 (electronic)", ISSN-L = "0003-049X", bibdate = "Fri Mar 30 11:20:35 2007", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "This paper contains derivations of both Stigler's Law and Benford's Law, and receives strong criticism in \cite{Raimi:1985:FDP}. This paper contains an important historical note that is recorded in entry \cite{Benford:1938:LAN}.", URL = "http://links.jstor.org/sici?sici=0003-049X(19780818)122%3A4%3C193%3ATFDP%3E2.0.CO%3B2-C; http://www.jstor.org/stable/986530", abstract = "Forty years ago an article appeared in these Proceedings, which has since then attracted much attention of mathematicians and some physicists. It was written by Frank Benford, a physicist from the General Electric Company at Schenectady. He had examined a large number of numerical tables, the kind that are printed in almanacs, and in scientific and technical handbooks. He noticed a most unusual and unexpected property of many of these tables. Consider, for example, the numbers representing the populations of states or all the countries in the world, or their areas, their budgets, their numbers of hospital beds, etc. One would expect that on the average just as many of these entries would begin with the digit $1$ as with $2$ or $3$ or $4$, and so on. But Benford noticed that small values of the first digits occur far more often.", acknowledgement = ack-nhfb, author-dates = "Samuel A. Goudsmit (July 11, 1902--December 4, 1978)", fjournal = "Proceedings of the {American Philosophical Society} held at {Philadelphia} for promoting useful knowledge", journal-URL = "http://www.jstor.org/journal/procamerphilsoci", remark = "In 1925, Samuel Goudsmit and George Uhlenbeck proposed the concept of electron spin, a critical step forward in the quantum mechanics of many-electron systems. Surprisingly, that work did not get them the Nobel Prize, but it did spur Wolfgang Pauli to present his Exclusion Principle in 1925, for which he received the 1945 Nobel Prize in Physics. After World War II, Goudsmit was the scientific head of the Alsos mission to capture German nuclear scientists. In the 1950s, he founded the journal Physical Review Letters. During the 1960s and 1970s, he was Editor-in-Chief of the main American physics journal, the Physical Review.", } @Article{Praunlich:1978:BDN, author = "Peter Praunlich and Michael Kroll", title = "{Bradford}'s distribution: a new formulation", journal = j-J-AM-SOC-INF-SCI, volume = "29", number = "2", pages = "51--55", month = mar, year = "1978", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630290203", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:01 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Rouault:1978:LZS, author = "Alain Rouault", title = "Lois de {Zipf} et sources markoviennes. ({French}) [{Zipf}'s law and {Markovian} sources]", journal = "Ann. Inst. H. Poincar{\'e} Sect. B (N.S.)", volume = "14", number = "2", pages = "169--188", year = "1978", CODEN = "AHPBAR", ISSN = "0020-2347", ISSN-L = "0020-2347", MRclass = "62E99 (60J20 62P99)", MRnumber = "507732 (80c:62026a)", MRreviewer = "Bruce M. Hill", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Annales de l'Institut Henri Poincar{\'e}. Section B. Calcul des Probabilit{\'e}s et Statistique. Nouvelle S{\'e}rie", language = "French", } @Article{Samson:1978:STU, author = "W. B. Samson and R. H. Davis", title = "Search Times Using Hash Tables for Records with Non-Unique Keys", journal = j-COMP-J, volume = "21", number = "3", pages = "210--214", month = aug, year = "1978", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Mar 25 13:51:56 MST 1997", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/hash.bib; http://www3.oup.co.uk/computer_journal/hdb/Volume_21/Issue_03/", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_21/Issue_03/tiff/210.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_21/Issue_03/tiff/211.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_21/Issue_03/tiff/212.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_21/Issue_03/tiff/213.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_21/Issue_03/tiff/214.tif", abstract = "Recent research in hash coding has concentrated on unique keys, or uniform distributions of keys. This paper is intended to clarify the effect of non-unique keys with various distributions on search times in the hash table thus enabling recommendations to be made to those who must deal with hash tables of this type. It is found that extreme rank-order frequency distribution of keys, such as the Zipf distribution, result in much higher access times than more uniform distributions, but it is possible to reduce these to some extent by loading records with common keys on to the hash table first.", acknowledgement = ack-nhfb, classcodes = "C6120 (File organisation); C6130 (Data handling techniques)", classification = "723", corpsource = "Dept. of Math. and Computer Studies, Dundee Coll. of Technol., Dundee, UK", fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", journalabr = "Comput J", keywords = "common keys; compiler symbol; computer programming; data handling; data processing --- Data Structures; direct access files; distribution; extreme; file organisation; hash tables; higher access times; non unique keys; overflow method; rank order frequency distribution of keys; records; search times; simulation; tables; uniform distributions; Zipf", treatment = "T Theoretical or Mathematical", } @Article{Brookes:1979:LEB, author = "B. C. Brookes", title = "Letters to the {Editor}: The {Bradford Law}: a new calculus for the social sciences?", journal = j-J-AM-SOC-INF-SCI, volume = "30", number = "4", pages = "233--234", month = jul, year = "1979", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630300412", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:05 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Diaconis:1979:RP, author = "Persi Diaconis and David Freedman", title = "On Rounding Percentages", journal = j-J-AM-STAT-ASSOC, volume = "74", number = "366", pages = "359--364", month = jun, year = "1979", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Wed Nov 23 11:24:59 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/b/bethe-hans.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", note = "The authors extend prior work on correctness of sums of rounded percentages \cite{Mosteller:1967:DSR}, and criticize biased rounding practices in \cite{Benford:1938:LAN}.", URL = "http://www.jstor.org/stable/2284288", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", keywords = "Benford's Law", remark = "When percentage-column entries are rounded, their sum may differ from 100\%. The authors use a prior proof \cite{Mosteller:1967:DSR} that for $ n \gg 2 $ rounded values, the probability of a correct sum is $ \sqrt {6 / (\pi n)} \approx 1.382 / \sqrt {n} $. They show that rounding may produce large changes in computed chi-squared values, and they conclude their article with this recommendation: ``it is important to calculate with many-digit accuracy when computing $ \chi^2 $ for large sample sizes.''", } @Article{Marulli:1979:BCB, author = "Luciana Marulli and Michael E. D. Koenig", title = "Brief Communications: {Bradford} distribution of data elements", journal = j-J-AM-SOC-INF-SCI, volume = "30", number = "2", pages = "107--108", month = mar, year = "1979", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630300209", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:04 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Mohanty:1979:MMQ, author = "Siba N. Mohanty", title = "Models and Measurements for Quality Assessment of Software", journal = j-COMP-SURV, volume = "11", number = "3", pages = "251--275", month = sep, year = "1979", CODEN = "CMSVAN", DOI = "https://doi.org/10.1145/356778.356783", ISSN = "0010-4892", ISSN-L = "0360-0300", bibdate = "Thu Jun 19 09:25:12 MDT 2008", bibsource = "ftp://ftp.ira.uka.de/pub/bibliography/Distributed/QLD/1979.bib; http://www.acm.org/pubs/contents/journals/surveys/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See also \cite{Dunn:1980:SFW,Mohanty:1980:SFW}.", acknowledgement = ack-nhfb, annote = "Several software quality assessment methods which span the software life cycle are discussed. The quality of a system design can be estimated by measuring the system entropy function or the system work function. The quality improvement due to reconfiguration can be determined by calculating system entropy loading measures. Software science and Zipf's law are shown to be useful for estimating program length and implementation time.", country = "USA", date = "11/10/79", descriptors = "Software; design; software engineering; reliability; measurement; model; fundamentals in statistics; performance evaluation", enum = "2189", fjournal = "ACM Computing Surveys", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J204", references = "65", } @Article{Radhakrishnan:1979:LLC, author = "T. Radhakrishnan and R. Kernizan", title = "{Lotka's Law} and Computer Science Literature", journal = j-J-AM-SOC-INF-SCI, volume = "30", number = "1", pages = "51--54", month = jan, year = "1979", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630300109", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Nov 25 07:45:09 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "6 September 2007", remark = "This paper analyzes publication statistics from Communications of the Association for Computing Machinery (CACM) and in the Journal of the ACM (JACM).", } @Article{Vlachy:1979:FDS, author = "Jan Vlach{\'y}", title = "Frequency Distribution of Scientific Performance: a Bibliography of {Lotka's Law} and Related Phenomena", journal = j-SCIENTOMETRICS, volume = "1", number = "1", pages = "107--130", month = sep, year = "1979", CODEN = "SCNTDX", DOI = "https://doi.org/10.1007/BF02016844", ISSN = "0138-9130 (print), 1588-2861 (electronic)", ISSN-L = "0138-9130", bibdate = "Fri Nov 25 07:59:34 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.springer.com/article/10.1007/BF02016844; http://www.springerlink.com/content/0138-9130/", acknowledgement = ack-nhfb, fjournal = "Scientometrics", journal-URL = "http://link.springer.com/journal/11192", xxjournal = "Scientometrics, Bibliography Section", } @Article{Brookes:1980:FISa, author = "Bertram C. Brookes", title = "The foundations of information science. {Part I}. {Philosophical} aspects", journal = j-J-INFO-SCI-PRINCIPLES-PRACTICE, volume = "2", number = "3--4", pages = "125--133", month = jun, year = "1980", CODEN = "JISCDI", DOI = "https://doi.org/10.1177/016555158000200302", ISSN = "0165-5515 (print), 1741-6485 (electronic)", ISSN-L = "0165-5515", bibdate = "Mon Feb 25 10:36:18 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://jis.sagepub.com/content/2/3-4/125.abstract", acknowledgement = ack-nhfb, fjournal = "Journal of Information Science, Principles and Practice", journal-URL = "http://jis.sagepub.com/", keywords = "Anomalous Law of Numbers; Benford's Law", } @Article{Brookes:1980:FISb, author = "Bertram C. Brookes", title = "The foundations of information science. {Part II}. {Quantitative} aspects: classes of things and the challenge of human individuality", journal = j-J-INFO-SCI-PRINCIPLES-PRACTICE, volume = "2", number = "5", pages = "209--221", month = oct, year = "1980", CODEN = "JISCDI", DOI = "https://doi.org/10.1177/016555158000200502", ISSN = "0165-5515 (print), 1741-6485 (electronic)", ISSN-L = "0165-5515", bibdate = "Mon Feb 25 11:01:20 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://jis.sagepub.com/content/2/5/209.abstract", acknowledgement = ack-nhfb, fjournal = "Journal of Information Science, Principles and Practice", journal-URL = "http://jis.sagepub.com/", keywords = "Anomalous Law of Numbers; Benford's Law", } @Article{Brookes:1980:FISc, author = "Bertram C. Brookes", title = "The foundations of information science. {Part III}. {Quantitative} aspects: Objective maps and subjective landscapes", journal = j-J-INFO-SCI-PRINCIPLES-PRACTICE, volume = "2", number = "6", pages = "269--275", month = dec, year = "1980", CODEN = "JISCDI", DOI = "https://doi.org/10.1177/016555158000200602", ISSN = "0165-5515 (print), 1741-6485 (electronic)", ISSN-L = "0165-5515", bibdate = "Mon Feb 25 10:36:18 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://jis.sagepub.com/content/2/6/269.abstract", acknowledgement = ack-nhfb, fjournal = "Journal of Information Science, Principles and Practice", journal-URL = "http://jis.sagepub.com/", keywords = "Anomalous Law of Numbers; Benford's Law", } @Article{Chen:1980:WFZ, author = "Wen Chen Chen", title = "On the weak form of {Zipf}'s law", journal = j-J-APPL-PROBAB, volume = "17", number = "3", pages = "611--622", year = "1980", CODEN = "JPRBAM", ISSN = "0021-9002 (print), 1475-6072 (electronic)", ISSN-L = "0021-9002", MRclass = "62E10 (62F10)", MRnumber = "580021 (81m:62021)", MRreviewer = "Patrick L. Brockett", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/3212955", ZMnumber = "Zbl 0435.60017", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Probability", journal-URL = "http://www.jstor.org/journals/00219002.html; http://projecteuclid.org/euclid.jap/", } @Article{Dunn:1980:SFW, author = "Stanley Dunn", title = "Surveyor's Forum: {Working} on Interpretations", journal = j-COMP-SURV, volume = "12", number = "2", pages = "255--255", month = jun, year = "1980", CODEN = "CMSVAN", DOI = "https://doi.org/10.1145/356810.356817", ISSN = "0010-4892", ISSN-L = "0360-0300", bibdate = "Thu Jun 19 09:27:10 MDT 2008", bibsource = "http://www.acm.org/pubs/contents/journals/surveys/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See \cite{Mohanty:1979:MMQ,Mohanty:1980:SFW}.", acknowledgement = ack-nhfb, fjournal = "ACM Computing Surveys", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J204", } @Article{Leimkuhler:1980:EFB, author = "Ferdinand F. Leimkuhler", title = "An Exact Formulation of {Bradford's Law}", journal = j-J-DOC, volume = "36", number = "4", pages = "285--292", month = "????", year = "1980", CODEN = "JDOCAS", DOI = "https://doi.org/10.1108/eb026699", ISSN = "0022-0418 (print), 1758-7379 (electronic)", ISSN-L = "0022-0418", bibdate = "Fri Nov 25 07:14:04 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "An exact, discrete formulation of Bradford's law describing the distribution of articles in journals is derived by showing that Bradford's law is a special case of the Zipf--Mandelbrot `rank frequency' law. A relatively simple method is presented for fitting the model to empirical data and estimating the number of journals and articles in a subject collection. This method is demonstrated with an example application.", acknowledgement = ack-nhfb, fjournal = "Journal of Documentation", journal-URL = "http://www.emeraldinsight.com/loi/jd", } @Article{McKee:1980:POR, author = "Arnold F. McKee", title = "The {Pareto} Optimum: Reconcilable with Social Thought?", journal = "International Journal of Social Economics", volume = "7", number = "7", pages = "366--375", month = "????", year = "1980", CODEN = "ISLEBC", DOI = "https://doi.org/10.1108/eb013878", ISSN = "0306-8293 (print), 1758-6712 (electronic)", ISSN-L = "0306-8293", bibdate = "Fri Nov 25 07:01:57 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Mohanty:1980:SFW, author = "Siba Mohanty", title = "Surveyor's Forum: {Working} on Interpretations", journal = j-COMP-SURV, volume = "12", number = "2", pages = "256--256", month = jun, year = "1980", CODEN = "CMSVAN", DOI = "https://doi.org/10.1145/356810.356818", ISSN = "0010-4892", ISSN-L = "0360-0300", bibdate = "Thu Jun 19 09:27:10 MDT 2008", bibsource = "http://www.acm.org/pubs/contents/journals/surveys/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See \cite{Mohanty:1979:MMQ,Dunn:1980:SFW}.", acknowledgement = ack-nhfb, fjournal = "ACM Computing Surveys", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J204", } @Article{Asai:1981:GFB, author = "Isao Asai", title = "A general formulation of {Bradford}'s distribution: the graph-oriented approach", journal = j-J-AM-SOC-INF-SCI, volume = "32", number = "2", pages = "113--119", month = mar, year = "1981", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630320206", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:10 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Brookes:1981:FIS, author = "Bertram C. Brookes", title = "The foundations of information science. {Part IV}. {Information} science: the changing paradigm", journal = j-J-INFO-SCI-PRINCIPLES-PRACTICE, volume = "3", number = "1", pages = "3--12", month = feb, year = "1981", CODEN = "JISCDI", DOI = "https://doi.org/10.1177/016555158100300102", ISSN = "0165-5515 (print), 1741-6485 (electronic)", ISSN-L = "0165-5515", bibdate = "Mon Feb 25 10:36:18 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://jis.sagepub.com/content/3/1/3", acknowledgement = ack-nhfb, fjournal = "Journal of Information Science, Principles and Practice", journal-URL = "http://jis.sagepub.com/", keywords = "Anomalous Law of Numbers; Benford's Law", } @Book{Fleiss:1981:SMR, author = "Joseph L. Fleiss", title = "Statistical Methods for Rates and Proportions", publisher = pub-WILEY, address = pub-WILEY:adr, edition = "Second", pages = "xviii + 321", year = "1981", ISBN = "0-471-06428-9", ISBN-13 = "978-0-471-06428-2", LCCN = "QA279 .F58 1981", bibdate = "Fri Mar 22 08:46:57 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", series = "Wiley series in probability and mathematical statistics", URL = "http://www.loc.gov/catdir/description/wiley032/80026382.html; http://www.loc.gov/catdir/enhancements/fy0607/80026382-b.html; http://www.loc.gov/catdir/toc/onix02/80026382.html", acknowledgement = ack-mjn # "\slash " # ack-nhfb, keywords = "Benford's Law", subject = "Analysis of variance; Sampling (Statistics); Biometry", } @Article{Kennard:1981:MDF, author = "Robert W. Kennard and John E. Reith", title = "{M32}. {On} the distribution of first digits", journal = j-COMMUN-STAT-SIMUL-COMPUT, volume = "10", number = "1", pages = "97--98", year = "1981", CODEN = "CSSCDB", DOI = "https://doi.org/10.1080/03610918108812195", ISSN = "0361-0918", ISSN-L = "0361-0918", bibdate = "Sat Jan 30 06:32:15 MST 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/communstatsimulcomput1980.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/03610918108812195", acknowledgement = ack-nhfb, fjournal = "Communications in Statistics: Simulation and Computation", journal-URL = "http://www.tandfonline.com/loi/lssp20", keywords = "Benford's Law", onlinedate = "27 Jun 2007", } @Article{Morse:1981:IEB, author = "Philip M. Morse", title = "Implications of the exact {Bradford} distribution", journal = j-J-AM-SOC-INF-SCI, volume = "32", number = "1", pages = "43--50", month = jan, year = "1981", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630320106", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:10 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "22 Mar 2007", } @Article{Pavlov:1981:DFF, author = "A. I. Pavlov", title = "On the distribution of fractions and {F. Benford}'s law", journal = j-IZV-AKAD-NAUK-SSSR-MAT, volume = "45", number = "4", pages = "760--774, 927", year = "1981", CODEN = "IZAMAT", ISSN = "0373-2436", ISSN-L = "0373-2436", MRclass = "10K10", MRnumber = "631437 (83m:10093)", MRreviewer = "Michael Keane", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Izvestiia Akademii Nauk. USSR Seriia Matematicheskaia (Moscow)", xxtitle = "On the distribution mod one and {Benford}'s law", } @Article{Peters:1981:EFB, author = "James V. Peters", title = "An Equivalent Form of {Benford's Law}", journal = j-FIB-QUART, volume = "19", number = "1", pages = "74--76", month = feb, year = "1981", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", MRclass = "60C05 (10K10 60-01)", MRnumber = "606117 (82k:60024)", MRreviewer = "L. E. Clarke", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.fq.math.ca/19-1.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/19-1/peters.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Peters:1981:SCC, author = "James V. Peters", title = "Short Communications: a combinatoric proof of {Benford}'s law", journal = j-AEQUATIONES-MATHEMATICAE, volume = "23", number = "1", pages = "122--123", year = "1981", CODEN = "AEMABN", DOI = "https://doi.org/10.1007/BF02188018", ISSN = "0001-9054 (print), 1420-8903 (electronic)", ISSN-L = "0001-9054", MRclass = "Contributed Item", MRnumber = "1553878", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.springer.com/accesspage/article/10.1007/BF02188018", acknowledgement = ack-nhfb, fjournal = "Aequationes Mathematicae", journal-URL = "http://link.springer.com/journal/10", } @Article{Potter:1981:LLR, author = "William Gray Potter", title = "{Lotka's Law} revisited", journal = j-LIBR-TRENDS, volume = "31", number = "2", pages = "21--39", month = "Summer", year = "1981", CODEN = "LIBTA3", ISSN = "0024-2594 (print), 1559-0682 (electronic)", ISSN-L = "0024-2594", bibdate = "Thu Nov 24 14:41:53 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.172.9842", acknowledgement = ack-nhfb, fjournal = "Library Trends", } @Article{Preece:1981:DFD, author = "D. A. Preece", title = "Distributions of Final Digits in Data", journal = j-J-R-STAT-SOC-SER-D-STATISTICIAN, volume = "30", number = "1", pages = "31--60", month = mar, year = "1981", ISSN = "0039-0526 (print), 1467-9884 (electronic)", ISSN-L = "0039-0526", bibdate = "Wed Nov 23 12:19:24 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/2987702", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society. Series D (The Statistician)", journal-URL = "http://www.jstor.org/journals/00390526.html", remark = "This paper extends previous work \cite{Yule:1927:RS} on human bias in the recording of last digits in measurements, or from roundings, or from later scale conversions, and how such bias can be detected.", } @Article{Schatte:1981:RVL, author = "Peter Schatte", title = "On random variables with logarithmic mantissa distribution relative to several bases", journal = j-ELEK-INFO-KYBER, volume = "17", number = "??", pages = "293--295", month = "????", year = "1981", CODEN = "EIVKAX", ISSN = "0013-5712", ISSN-L = "0013-5712", bibdate = "Sat Nov 12 15:53:03 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Elektronische Informationsverarbeitung und Kybernetik (EIK)", } @Article{Washington:1981:BLF, author = "Lawrence C. Washington", title = "{Benford's Law} for {Fibonacci} and {Lucas} Numbers", journal = j-FIB-QUART, volume = "19", number = "2", pages = "175--177", month = apr, year = "1981", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", MRclass = "10A30 (10A35)", MRnumber = "614056 (82f:10009)", MRreviewer = "S. P. Mohanty", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.fq.math.ca/19-2.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/19-2/washington.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Fedorowicz:1982:TFZ, author = "Jane Fedorowicz", title = "The Theoretical Foundation of {Zipf's Law} and Its Application to the Bibliographic Database Environment", journal = j-J-AM-SOC-INF-SCI, volume = "33", number = "5", pages = "285--293", year = "1982", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630330507", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Mon Feb 25 06:13:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", } @Article{Fedorowicz:1982:ZMA, author = "Jane Fedorowicz", title = "A {Zipfian} Model of an Automatic Bibliographic System: An Application to {MEDLINE}", journal = j-J-AM-SOC-INF-SCI, volume = "33", number = "4", pages = "223--232", month = jul, year = "1982", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630330406", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:14 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "6 Sep 2007", } @Article{Lefort:1982:LOA, author = "B. Lefort", title = "{L}'Emploi des Outils au Cours de Taches d'Entretien et la Loi de {Zipf--Mandelbrot}. ({French}) [The Use of Tools in Maintenance Tasks and the {Zipf--Mandelbrot Law}]", journal = j-TRAV-HUM, volume = "45", number = "2", pages = "307--316", month = "????", year = "1982", CODEN = "TRHUAH", ISSN = "0041-1868 (print), 2104-3663 (electronic)", ISSN-L = "0041-1868", bibdate = "Mon Sep 10 07:17:32 MDT 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/40657805", acknowledgement = ack-nhfb, fjournal = "Le Travail Humain", language = "French", } @Article{MacRoberts:1982:REL, author = "Michael H. MacRoberts and Barbara R. MacRoberts", title = "A Re-evaluation of {Lotka}'s {Law of Scientific Productivity}", journal = j-SOC-STUD-SCI, volume = "12", number = "3", pages = "443--450", day = "1", month = aug, year = "1982", CODEN = "SSSCDH", DOI = "https://doi.org/10.1177/030631282012003005", ISSN = "0306-3127 (print), 1460-3659 (electronic)", ISSN-L = "0306-3127", bibdate = "Mon Dec 3 09:12:38 MST 2018", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/socstudsci.bib", URL = "https://journals.sagepub.com/doi/pdf/10.1177/030631282012003005", acknowledgement = ack-nhfb, fjournal = "Social Studies of Science", journal-URL = "http://sss.sagepub.com/content/by/year", keywords = "Lotka's Law of scientific authorship for the relative contributions of the most prolific authors", } @Article{Pavlov:1982:DFP, author = "A. I. Pavlov", title = "On the Distribution of Fractional Parts and {Benford's Law}", journal = j-MATH-USSR-IZV, volume = "19", number = "1", pages = "65--77", month = "????", year = "1982", CODEN = "MUSIAE", DOI = "https://doi.org/10.1070/IM1982v019n01ABEH001411", ISSN = "0025-5726 (print), 2169-5075 (electronic)", ISSN-L = "0025-5726", bibdate = "Sun Nov 13 10:46:55 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://stacks.iop.org/0025-5726/19/i=1/a=A05", abstract = "This article investigates the distribution of the first digits (from the left) in the $q$-nary expansions of numerical sequences and functions ($q$ an integer $ \geq 2 $ )", acknowledgement = ack-nhfb, fjournal = "Mathematics of the {USSR} --- Izvestiya", journal-URL = "http://iopscience.iop.org/0025-5726", } @PhdThesis{Perline:1982:EVM, author = "Richard K. Perline", title = "An extreme value model of weakly harmonic ({Pareto--Zipf} type) laws", type = "Thesis ({Ph.D.})", school = "Department of Behavioral Sciences, The University of Chicago", address = "Chicago, IL, USA", pages = "129", month = dec, year = "1982", MRnumber = "2611828", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://search.proquest.com/docview/251796701", acknowledgement = ack-nhfb, remark = "Available from ProQuest LLC, Ann Arbor, MI, USA.", } @Article{Regazzini:1982:BFL, author = "Eugenio Regazzini", title = "The {Benford--Furlan} law as a statistical law", journal = j-STATISTICA-BOLOGNA, volume = "42", number = "3", pages = "351--370", year = "1982", CODEN = "STATDJ", ISSN = "0390-590X (print), 1973-2201 (electronic)", ISSN-L = "0390-590X", MRclass = "62E99 (10K05 62A15)", MRnumber = "695467 (84h:62032)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Statistica (Bologna)", } @Article{Turner:1982:DLS, author = "Peter R. Turner", title = "The distribution of leading significant digits", journal = j-IMA-J-NUMER-ANAL, volume = "2", number = "4", pages = "407--412", month = oct, year = "1982", CODEN = "IJNADH", DOI = "https://doi.org/10.1093/imanum/2.4.407", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65G99 (60E05)", MRnumber = "84f:65038", MRreviewer = "Seppo Linnainmaa", bibdate = "Sat Dec 23 17:06:35 MST 2000", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/imajnumeranal.bib; MathSciNet database", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Book{Weaver:1982:LLT, author = "Warren Weaver", title = "{Lady Luck}: the theory of probability", publisher = pub-DOVER, address = pub-DOVER:adr, pages = "392", year = "1982", ISBN = "0-486-24342-7", ISBN-13 = "978-0-486-24342-9", LCCN = "QA273 .W4 1982", bibdate = "Mon Nov 14 11:35:17 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/description/dover031/82007396.html", acknowledgement = ack-nhfb, author-dates = "1894--1978", remark = "Reprint of \cite{Weaver:1963:LLT}.", subject = "Probabilities", } @Article{White:1982:FSS, author = "Warren H. White", title = "On the form of steady-state solutions to the coagulation equations", journal = j-J-COLLOID-INTERFACE-SCI, volume = "87", number = "1", pages = "204--208", month = may, year = "1982", CODEN = "JCISA5", DOI = "https://doi.org/10.1016/0021-9797(82)90382-4", ISSN = "0021-9797 (print), 1095-7103 (electronic)", ISSN-L = "0021-9797", bibdate = "Thu Feb 28 10:46:13 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/0021979782903824", acknowledgement = ack-nhfb, fjournal = "Journal of Colloid and Interface Science", keywords = "power law; scale invariance; Zipf's law", } @Article{Iyengar:1983:SPL, author = "S. Sitharama Iyengar and A. K. Rajagopal and V. R. R. Uppuluri", title = "String patterns of leading digits", journal = j-APPL-MATH-COMP, volume = "12", number = "4", pages = "321--337", month = jul, year = "1983", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/0096-3003(83)90045-0", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Sat Feb 16 18:02:26 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.sciencedirect.com/science/journal/00963003", URL = "http://www.sciencedirect.com/science/article/pii/0096300383900450", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Montroll:1983:NOD, author = "Elliott W. Montroll and Michael F. Shlesinger", title = "On $ 1 / f $ noise and other distributions with long tails", journal = j-PROC-NATL-ACAD-SCI-USA, volume = "79", number = "10", pages = "3380--3383", day = "15", month = may, year = "1983", CODEN = "PNASA6", DOI = "https://doi.org/10.1073/pnas.79.10.3380", ISSN = "0027-8424 (print), 1091-6490 (electronic)", ISSN-L = "0027-8424", bibdate = "Thu Feb 28 11:03:24 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.pnas.org/content/79/10/3380.abstract", acknowledgement = ack-nhfb, fjournal = "Proceedings of the National Academy of Sciences of the United States of America", journal-URL = "http://www.pnas.org/search", keywords = "power law; scale invariance; Zipf's law", } @Article{Orlov:1983:SSZ, author = "Yu. K. Orlov and R. Ya. Chitashvili", title = "The statistical significance of {Zipf}'s distribution", journal = "Soobshch. Akad. Nauk Gruzin. SSR", volume = "109", number = "3", pages = "505--508", year = "1983", CODEN = "????", ISSN = "0132-1447", ISSN-L = "0132-1447", MRclass = "62E99 (62P99)", MRnumber = "723296", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Soobshcheniya Akademii Nauk Gruzinsko{\u\i} SSR", } @Article{Schatte:1983:AUD, author = "Peter Schatte", title = "On the asymptotic uniform distribution of sums reduced $ \bmod 1 $", journal = j-MATH-NACHR, volume = "115", number = "1", pages = "257--281", month = "????", year = "1983", CODEN = "MTMNAQ", DOI = "https://doi.org/10.1002/mana.19841150121", ISSN = "0025-584X", ISSN-L = "0025-584X", bibdate = "Sat Feb 16 19:12:20 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://onlinelibrary.wiley.com/doi/10.1002/mana.19841150121/abstract", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", } @Article{Schatte:1983:SUD, author = "Peter Schatte", title = "On {$ H_\infty $}-summability and the uniform distribution of sequences", journal = j-MATH-NACHR, volume = "113", number = "1", pages = "237--243", month = "????", year = "1983", CODEN = "MTMNAQ", DOI = "https://doi.org/10.1002/mana.19831130122", ISSN = "0025-584X", ISSN-L = "0025-584X", bibdate = "Sat Feb 16 19:13:27 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://onlinelibrary.wiley.com/doi/10.1002/mana.19831130122/abstract", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", } @Article{Whittaker:1983:SID, author = "James V. Whittaker", title = "On Scale-Invariant Distributions", journal = j-SIAM-J-APPL-MATH, volume = "43", number = "2", pages = "257--267", month = apr, year = "1983", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", MRclass = "60E05 (10K10 62E10)", MRnumber = "84h:60035", MRreviewer = "K. C. Chanda", bibdate = "Thu Oct 15 18:16:06 MDT 1998", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/siamjapplmath.bib; JSTOR database", acknowledgement = ack-nhfb, affiliation = "Univ of British Columbia, Dep of Mathematics, Vancouver, BC, Can", classification = "922", fjournal = "SIAM Journal on Applied Mathematics", journal-URL = "http://epubs.siam.org/siap", journalabr = "SIAM J Appl Math", keywords = "probability", } @Article{Bennett:1984:ZLS, author = "J. M. Bennett", title = "{Zipf's Law}, Structured Programming and Creativity", journal = j-AUSTRALIAN-COMP-J, volume = "16", number = "4", pages = "122--129", year = "1984", CODEN = "ACMJB2", ISSN = "0004-8917", ISSN-L = "0004-8917", bibdate = "Fri Mar 9 17:59:26 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "As software projects increase in size, the effort required to implement them increases at a greater rate. The use of higher level languages and adherence to various rules of ``good programming practice'' appear to help only marginally in reducing the impact of this non-linear effect, and progress can only be made by calling on techniques for managing complexity which have emerged in other fields. Hierarchical structures are discussed with particular reference to software as is the effect which appears to be common to a wide range of human activities, and which is known as Zipf's law. The effects of variation in individual programming skills, the roles of creativity, invention and innovation and lessons to be learned from case studies of past failure are also examined.", acknowledgement = ack-nhfb, fjournal = "Australian Computer Journal", keywords = "creativity; innovation; invention; large software projects; project failures; structured programming; Zipf's law", } @Article{Boroda:1984:ZML, author = "M. G. Boroda and A. A. Polikarpov", title = "The {Zipf--Mandelbrot} law and units of different levels of text organization", journal = "Tartu Riikl. {\"U}l. Toimetised", volume = "??", number = "689", pages = "35--60", year = "1984", CODEN = "????", ISSN = "????", MRclass = "92A25", MRnumber = "781731 (86c:92030)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Tartu Riikliku {\"U}likooli Toimetised. Uchenye Zapiski Tartuskogo Gosudarstvennogo Universiteta. Acta et Commentationes Universitatis Tartuensis", } @Article{Brookes:1984:RTE, author = "Bertram C. Brookes", title = "Ranking techniques and the empirical log law", journal = j-INFO-PROC-MAN, volume = "20", number = "1--2", pages = "37--46", month = "????", year = "1984", CODEN = "IPMADK", DOI = "https://doi.org/10.1016/0306-4573(84)90038-4", ISSN = "0306-4573 (print), 1873-5371 (electronic)", ISSN-L = "0306-4573", bibdate = "Mon Feb 25 10:38:32 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/0306457384900384", acknowledgement = ack-nhfb, fjournal = "Information Processing and Management", journal-URL = "http://www.sciencedirect.com/science/journal/03064573", keywords = "Anomalous Law of Numbers; Benford's Law", } @Article{Bunt:1984:MPL, author = "Richard B. Bunt and Jennifer M. Murphy and Shikharesh Majumdar", title = "A measure of program locality and its application", journal = j-SIGMETRICS, volume = "12", number = "3", pages = "28--40", month = aug, year = "1984", CODEN = "????", DOI = "https://doi.org/10.1145/1031382.809311", ISSN = "0163-5999 (print), 1557-9484 (electronic)", ISSN-L = "0163-5999", bibdate = "Thu Jun 26 11:00:50 MDT 2008", bibsource = "http://portal.acm.org/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/sigmetrics.bib", abstract = "Although the phenomenon of locality has long been recognized as the single most important characteristic of program behaviour, relatively little work has been done in attempting to measure it. Recent work has led to the development of an intrinsic measure of program locality based on the Bradford--Zipf distribution. Potential applications for such a measure are many, and include the evaluation of program restructuring methods (manual and automatic), the prediction of system performance, the validation of program behaviour models, and the enhanced understanding of the phenomena that characterize program behaviour. A consideration of each of these areas is given in connection with the proposed measure, both to increase confidence in the validity of the measure and to illustrate a methodology for dealing with such problems.", acknowledgement = ack-nhfb, fjournal = "ACM SIGMETRICS Performance Evaluation Review", journal-URL = "http://portal.acm.org/toc.cfm?id=J618", } @Article{Cohen:1984:PNF, author = "D. I. A. Cohen and T. M. Katz", title = "Prime numbers and the first digit phenomenon", journal = j-J-NUMBER-THEORY, volume = "18", number = "3", pages = "261--268", month = jun, year = "1984", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/0022-314X(84)90061-1", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Fri Nov 25 13:58:58 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Number Theory", } @Article{Fedorowicz:1984:DEU, author = "Jane Fedorowicz", title = "Database evaluation using multiple regression techniques", journal = j-SIGMOD, volume = "14", number = "2", pages = "70--76", year = "1984", CODEN = "SRECD8", ISBN = "0-89791-128-8", ISBN-13 = "978-0-89791-128-3", ISSN = "0163-5808 (print), 1943-5835 (electronic)", ISSN-L = "0163-5808", bibdate = "Mon Dec 9 07:58:51 MST 1996", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "A model of the inverted file of an automated bibliographic system is constructed using the Zipf distribution of word frequency. By ascertaining the parameters of the Zipfian model of the inverted file system, one can estimate the minimum data storage requirements of the database. In addition, given a few additional system parameters, access time for a specified query can be estimated. The estimation procedures are accomplished using logarithmic transformations and multiple regression techniques. This paper introduces the Zipfian models, their regression formulation, and their results and interpretation for application to database evaluation.", acknowledgement = ack-nhfb, affiliation = "Northwestern Univ, Kellogg Graduate Sch of Management, Evanston, IL, USA", affiliationaddress = "Northwestern Univ, Kellogg Graduate Sch of Management, Evanston, IL, USA", classification = "723; 901; 922", conference = "SIGMOD '84, Proceedings of Annual Meeting (ACM Special Interest Group on Management of Data).", fjournal = "SIGMOD Record (ACM Special Interest Group on Management of Data)", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J689", keywords = "access time model; bibliographic techniques; database systems; evaluation; information storage systems; mathematical models; multiple regression techniques; Zipfian model", meetingaddress = "Boston, MA, USA", sponsor = "ACM, Special Interest Group on Management of Data, New York, NY, USA", } @Article{Friedberg:1984:DFD, author = "Stephen H. Friedberg", title = "The Distribution of First Digits", journal = j-COLLEGE-MATH-J, volume = "15", number = "2", pages = "120--125", month = mar, year = "1984", CODEN = "????", DOI = "https://doi.org/10.1080/00494925.1984.11972760", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:49:53 MST 2019", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.jstor.org/stable/2686516; http://www.tandfonline.com/doi/abs/10.1080/00494925.1984.11972760", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", keywords = "Benford's Law", onlinedate = "30 Jan 2018", } @Article{Fuchs:1984:PDP, author = "A. Fuchs and G. Letta", title = "{Sur le probl{\`e}me du premier chiffre d{\'e}cimal}. ({French}) [{On} the problem of the first decimal digit]", journal = "{Bollettino della Unione Matem{\`a}tica Italiana. Serie VI. B}", volume = "3", pages = "451--461", year = "1984", ISSN = "0392-4041", ISSN-L = "0392-4041", bibdate = "Wed Aug 12 10:09:50 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0557.60011", acknowledgement = ack-nhfb, ajournal = "Boll. Unione Mat. Ital., VI. Ser., B", keywords = "analytical density; asymptotic density; first-digit problem", language = "French", ZMclass = "60B99 (Probability theory on general structures)", ZMreviewer = "R. Scozzafava", } @Article{Karmeshu:1984:RBL, author = "Karmeshu and N. C. Lind and V. Cano", title = "Rationales for {Bradford's Law}", journal = j-SCIENTOMETRICS, volume = "6", number = "4", pages = "233--241", month = jul, year = "1984", CODEN = "SCNTDX", DOI = "https://doi.org/10.1007/BF02279358", ISSN = "0138-9130 (print), 1588-2861 (electronic)", ISSN-L = "0138-9130", bibdate = "Wed Sep 2 12:02:53 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/scientometrics1980.bib", URL = "http://link.springer.com/article/10.1007/BF02279358", acknowledgement = ack-nhfb, fjournal = "Scientometrics", journal-URL = "http://link.springer.com/journal/11192", } @Article{Kohyama:1984:TIC, author = "Tamotsu Kohyama and Y{\=o}ji Aizawa", title = "Theory of the intermittent chaos. {$ 1 / f $} spectrum and the {Pareto--Zipf} law", journal = j-PROG-THEOR-PHYS, volume = "71", number = "5", pages = "917--929", year = "1984", CODEN = "PTPKAV", DOI = "https://doi.org/10.1143/PTP.71.917", ISSN = "0033-068X (print), 1347-4081 (electronic)", ISSN-L = "0033-068X", MRclass = "58F13 (76F99)", MRnumber = "756214 (86a:58063)", MRreviewer = "Joseph Ford", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Progress of Theoretical Physics", journal-URL = "http://ptp.oxfordjournals.org/content/by/year", } @Article{Maia:1984:UBL, author = "M. J. F. Maia and M. D. Maia", title = "On the Unity of {Bradford's Law}", journal = j-J-DOC, volume = "40", number = "3", pages = "206--216", month = "????", year = "1984", CODEN = "JDOCAS", DOI = "https://doi.org/10.1108/eb026765", ISSN = "0022-0418 (print), 1758-7379 (electronic)", ISSN-L = "0022-0418", bibdate = "Fri Nov 25 07:11:46 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Documentation", journal-URL = "http://www.emeraldinsight.com/loi/jd", } @Article{McLaughlin:1984:DFI, author = "William I. McLaughlin and Sylvia A. Lundy", title = "Digit Functions of Integer Sequences", journal = j-FIB-QUART, volume = "22", number = "2", pages = "105--115", month = may, year = "1984", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:00:30 MDT 2011", bibsource = "http://www.fq.math.ca/22-2.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/22-2/mclaughlin.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Nagasaka:1984:BL, author = "Kenji Nagasaka", title = "On {Benford}'s law", journal = j-ANN-INST-STAT-MATH-TOKYO, volume = "36", number = "1", pages = "337--352", month = "????", year = "1984", CODEN = "AISXAD", DOI = "https://doi.org/10.1007/BF02481974", ISSN = "0020-3157 (print), 1572-9052 (electronic)", ISSN-L = "0020-3157", MRclass = "60F99 (11K06 11K36 62E99)", MRnumber = "758506 (86a:60052)", MRreviewer = "L. Kuipers", bibdate = "Sat Jan 31 18:19:56 MST 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/anninststatmath.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.springer.com/article/10.1007/BF02481974", acknowledgement = ack-nhfb, ajournal = "Ann. Inst. Statist. Math.", fjournal = "Annals of the Institute of Statistical Mathematics (Tokyo)", journal-URL = "http://link.springer.com/journal/10463", } @Article{Schatte:1984:AUD, author = "Peter Schatte", title = "On the asymptotic uniform distribution of sums reduced $ \bmod 1 $", journal = j-MATH-NACHR, volume = "115", number = "1", pages = "275--281", month = "????", year = "1984", CODEN = "MTMNAQ", DOI = "https://doi.org/10.1002/mana.19841150121", ISSN = "0025-584X", ISSN-L = "0025-584X", bibdate = "Sat Nov 12 15:54:14 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://onlinelibrary.wiley.com/doi/10.1002/mana.19841150121/abstract", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", } @Article{Turner:1984:FRD, author = "Peter R. Turner", title = "Further revelations on l.s.d", journal = j-IMA-J-NUMER-ANAL, volume = "4", number = "2", pages = "225--231", month = apr, year = "1984", CODEN = "IJNADH", DOI = "https://doi.org/10.1093/imanum/4.2.225", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65G10 (11K99)", MRnumber = "85m:65039", MRreviewer = "Calvin T. Long", bibdate = "Sat Dec 23 17:06:35 MST 2000", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/imajnumeranal.bib; MathSciNet database", abstract = "In a recent paper the author showed that the distribution of leading significant digits, l.s.d., resulting from successive multiplications is logarithmic. In this paper these results are extended by establishing, still without any assumptions of invariance to scaling, that this distribution remains invariant under all further arithmetic operations --- both multiplicative and additive. The fact that 30\% of decimal numbers have l.s.d. 1 persists.", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", keywords = "l. s. d. (leading significant digits)", } @Article{Yuan:1984:LOC, author = "Y. Yuan", title = "On the least {$Q$}-order of convergence of variable metric algorithms", journal = j-IMA-J-NUMER-ANAL, volume = "4", number = "2", pages = "233--239", month = apr, year = "1984", CODEN = "IJNADH", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65K05", MRnumber = "85h:65133", MRreviewer = "A. I. \c Schiop", bibdate = "Sat Dec 23 17:06:35 MST 2000", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/imajnumeranal.bib; MathSciNet database", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", } @Article{Barlow:1985:RED, author = "J. L. Barlow and E. H. Bareiss", title = "On Roundoff Error Distributions in Floating Point and Logarithmic Arithmetic", journal = j-COMPUTING, volume = "34", number = "4", pages = "325--347", month = dec, year = "1985", CODEN = "CMPTA2", ISSN = "0010-485X (print), 1436-5057 (electronic)", ISSN-L = "0010-485X", MRclass = "65G05", MRnumber = "87c:65045", MRreviewer = "N. N. Abdelmalek", bibdate = "Tue Oct 12 16:33:42 MDT 1999", bibsource = "Compendex database; http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib; MathSciNet database", acknowledgement = ack-nhfb, affiliation = "Pennsylvania State Univ, Computer Science Dep, University Park, PA, USA", affiliationaddress = "USA", classification = "723", fjournal = "Computing", journal-URL = "http://link.springer.com/journal/607", journalabr = "Computing (Vienna/New York)", keywords = "computer arithmetic; computer metatheory; floating point arithmetic; logarithmic arithmetic; roundoff error distributions", } @Article{Bradford:1985:SIS, author = "Samuel C. Bradford", title = "Sources of Information on Specific Subjects", journal = j-J-INFO-SCI-PRINCIPLES-PRACTICE, volume = "10", number = "4", pages = "173--180", month = oct, year = "1985", CODEN = "JISCDI", ISSN = "0165-5515 (print), 1741-6485 (electronic)", ISSN-L = "0165-5515", bibdate = "Fri Nov 25 11:49:58 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Reprint of \cite{Bradford:1934:SIS}, the source of Bradford's Law.", acknowledgement = ack-nhfb, fjournal = "Journal of Information Science, Principles and Practice", journal-URL = "http://jis.sagepub.com/", } @Article{Egghe:1985:CLL, author = "L. Egghe", title = "Consequences of {Lotka's Law} for the {Law of Bradford}", journal = j-J-DOC, volume = "41", number = "3", pages = "173--189", month = "????", year = "1985", CODEN = "JDOCAS", DOI = "https://doi.org/10.1108/eb026780", ISSN = "0022-0418 (print), 1758-7379 (electronic)", ISSN-L = "0022-0418", bibdate = "Fri Nov 25 07:23:22 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Documentation", journal-URL = "http://www.emeraldinsight.com/loi/jd", keywords = "Bradford's Law", } @Article{Faber:1985:AML, author = "Vance Faber and Andrew B. {White, Jr.} and G. Milton Wing", title = "Analysis of a model that leads to the {Pareto} law of wealth distribution", journal = j-J-MATH-ANAL-APPL, volume = "112", number = "2", pages = "579--594", year = "1985", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/0022-247X(85)90264-1", ISSN = "0022-247X (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", MRclass = "90A14", MRnumber = "813621 (87b:90027)", bibdate = "Mon Feb 18 17:05:26 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", } @Article{Lagarias:1985:PG, author = "Jeffrey C. Lagarias", title = "The $ 3 x + 1 $ Problem and Its Generalizations", journal = j-AMER-MATH-MONTHLY, volume = "92", number = "1", pages = "3--23", month = jan, year = "1985", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "11K55 (11K31)", MRnumber = "86i:11043", MRreviewer = "A. D. Pollington", bibdate = "Mon Jun 28 12:37:47 MDT 1999", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", keywords = "Benford's Law", } @Article{Pao:1985:LLT, author = "Miranda Lee Pao", title = "{Lotka}'s law: a testing procedure", journal = j-INFO-PROC-MAN, volume = "21", number = "4", pages = "305--320", month = "????", year = "1985", CODEN = "IPMADK", DOI = "https://doi.org/10.1016/0306-4573(85)90055-X", ISSN = "0306-4573 (print), 1873-5371 (electronic)", ISSN-L = "0306-4573", bibdate = "Fri Nov 25 08:41:41 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Information Processing and Management", journal-URL = "http://www.sciencedirect.com/science/journal/03064573", } @Article{Raimi:1985:FDP, author = "Ralph A. Raimi", title = "The First Digit Phenomenon Again", journal = j-PROC-AMER-PHIL-SOC, volume = "129", number = "2", pages = "211--219", month = jun, year = "1985", CODEN = "PAPCAA", ISSN = "0003-049X (print), 2326-9243 (electronic)", ISSN-L = "0003-049X", bibdate = "Sat Nov 12 16:56:54 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "This paper contains strong criticism of a derivation of Benford's Law \cite{Logan:1978:FDP}.", URL = "http://www.jstor.org/stable/986989", acknowledgement = ack-nhfb, fjournal = "Proceedings of the {American Philosophical Society} held at {Philadelphia} for promoting useful knowledge", journal-URL = "http://www.jstor.org/journal/procamerphilsoci", remark = "This paper contains mathematical derivations of both Stigler's Law and Benford's Law. Raimi comments on p. 217: ``the Benford law itself as a conclusion to be derived is by no means as firm as Kepler's planetary curves. Indeed, it has been pointed out by Diaconis and Freedman [D-F] [\cite{Diaconis:1979:RP}] that a statistical analysis of Benford's announced data gives a high probability to the conjecture that he did a bit of rounding-off here and there, always in the direction of his law. The license that a physicist often takes on the path from his hypotheses to verified, observable, and repeatable conclusion is not justified here.''.", } @Article{Tichy:1985:UDD, author = "Robert F. Tichy", title = "Uniform distribution and diophantine inequalities", journal = j-MONAT-MATH, volume = "99", number = "2", pages = "147--152", month = jun, year = "1985", CODEN = "MNMTA2", DOI = "https://doi.org/10.1007/BF01304194", ISSN = "0026-9255 (print), 1436-5081 (electronic)", ISSN-L = "0026-9255", bibdate = "Sun Feb 17 06:26:16 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.springer.com/article/10.1007/BF01304194", ZMnumber = "0538.10039", acknowledgement = ack-nhfb, fjournal = "Monatshefte f{\"u}r Mathematik", keywords = "diophantine inequalities; discrepancy; Uniformly distributed sequences", ZMclass = "11K06 (General theory of distribution modulo 1); 11D75 (Diophantine inequalities)", } @Article{Tuldava:1985:FST, author = "Yu. A. Tuldava", title = "The frequency structure of a text and {Zipf}'s law", journal = "Tartu Riikl. {\"U}l. Toimetised", volume = "??", number = "711", pages = "93--116", year = "1985", CODEN = "????", ISSN = "????", MRclass = "92A25 (94A15)", MRnumber = "836129 (87c:92063)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Tartu Riikliku {\"U}likooli Toimetised. Uchenye Zapiski Tartuskogo Gosudarstvennogo Universiteta. Acta et Commentationes Universitatis Tartuensis", } @Article{Uppuluri:1985:DFD, author = "V. R. R. Uppuluri and S. A. Patil", title = "The distribution of the first $j$ digits of beta related random variables", journal = j-COMMUN-STAT-SIMUL-COMPUT, volume = "14", number = "2", pages = "467--472", year = "1985", CODEN = "CSSCDB", DOI = "https://doi.org/10.1080/03610918508812450", ISSN = "0361-0918", ISSN-L = "0361-0918", bibdate = "Sat Jan 30 06:32:23 MST 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/communstatsimulcomput1980.bib", acknowledgement = ack-nhfb, fjournal = "Communications in Statistics: Simulation and Computation", journal-URL = "http://www.tandfonline.com/loi/lssp20", keywords = "Benford's Law", } @Article{Chen:1986:RBL, author = "Ye-Sho Chen and Ferdinand F. Leimkuhler", title = "A relationship between {Lotka's Law}, {Bradford's Law}, and {Zipf's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "37", number = "5", pages = "307--314", year = "1986", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(198609)37:5<307::AID-ASI5>3.0.CO%3B2-8", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Mon Feb 25 06:13:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", } @Article{Chen:1986:RRB, author = "Ye-Sho Chen and Ferdinand F. Leimkuhler", title = "Research: A relationship between {Lotka's Law}, {Bradford's Law}, and {Zipf's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "37", number = "5", pages = "307--314", month = sep, year = "1986", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(198609)37:5<307::AID-ASI5>3.0.CO%3B2-8", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:27 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "7 Jan 1999", } @Article{Egghe:1986:RDB, author = "L. Egghe", title = "Research: The dual of {Bradford's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "37", number = "4", pages = "246--255", month = jul, year = "1986", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(198607)37:4<246::AID-ASI10>3.0.CO%3B2-D", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:27 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "7 Jan 1999", } @Article{Feldstein:1986:OUS, author = "Alan Feldstein and Peter Turner", title = "Overflow, underflow, and severe loss of significance in floating-point addition and subtraction", journal = j-IMA-J-NUMER-ANAL, volume = "6", number = "2", pages = "241--251", month = apr, year = "1986", CODEN = "IJNADH", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", MRclass = "65G05", MRnumber = "89h:65065", bibdate = "Sat Dec 23 14:52:49 2000", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "In this paper it is shown that, under the assumption of the logarithmic distribution of numbers, floating-point addition and subtraction can result in overflow or underflow with alarming frequency --- a frequency that increases rapidly with machine speed unless the range of the exponent is also increased. If numbers are assumed to be distributed in accordance with Sweeney's (1965) experiments, then severe loss of significance occurs with large probability in floating point subtraction. These results have implications for computer design and lead to the suggestion of a long word format which will reduce the risks to acceptable levels.", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", keywords = "Benford's Law; Law of Anomalous Numbers; logarithmic distribution; overflow; underflow; Zipf's Law", } @Article{Green:1986:BRP, author = "M. W. Green", title = "Book Reviews: {{\em Pareto Distributions}}, by {B. C. Arnold}", journal = j-APPL-STAT, volume = "35", number = "2", pages = "215--215", year = "1986", CODEN = "APSTAG", ISSN = "0035-9254 (print), 1467-9876 (electronic)", ISSN-L = "0035-9254", bibdate = "Sat Apr 21 10:24:36 MDT 2001", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "Applied Statistics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-9876/issues", } @Article{Hoare:1986:DFS, author = "G. T. Q. Hoare and E. Wright", title = "The distribution of first significant digits", journal = j-MATH-GAZ, volume = "70", number = "451", pages = "34--37", month = mar, year = "1986", CODEN = "MAGAAS", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Sat Feb 16 17:57:52 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/3615826", acknowledgement = ack-nhfb, fjournal = "Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", } @Article{Katz:1986:FDP, author = "Talbot M. Katz and Daniel I. A. Cohen", title = "The First Digit Property for Exponential Sequences is Independent of the Underlying Distribution", journal = j-FIB-QUART, volume = "24", number = "1", pages = "2--7", month = feb, year = "1986", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:00:45 MDT 2011", bibsource = "http://www.fq.math.ca/24-1.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/24-1/katz.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Lemons:1986:NTD, author = "Don S. Lemons", title = "On the number of things and the distribution of first digits", journal = j-AMER-J-PHYSICS, volume = "54", number = "9", pages = "816--817", month = sep, year = "1986", CODEN = "AJPIAS", DOI = "https://doi.org/10.1119/1.14453", ISSN = "0002-9505 (print), 1943-2909 (electronic)", ISSN-L = "0002-9505", bibdate = "Wed Nov 23 12:03:08 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://ajp.aapt.org/", acknowledgement = ack-nhfb, fjournal = "American Journal of Physics", journal-URL = "http://scitation.aip.org/content/aapt/journal/ajp", keywords = "Benford's Law", remark = "Lemons gives a derivation of Benford's Law that does not assume scale invariance, unlike other widely-cited derivations \cite{Pinkham:1961:DFS,Raimi:1976:FDP,Raimi:1985:FDP}.", } @Book{Lines:1986:NYT, author = "Malcolm E. Lines", title = "A number for your thoughts: facts and speculations about numbers from {Euclid} to the latest computers", publisher = pub-ADAM-HILGER, address = pub-ADAM-HILGER:adr, pages = "vi + 214", year = "1986", ISBN = "0-85274-495-1 (paperback)", ISBN-13 = "978-0-85274-495-6 (paperback)", LCCN = "QA241 .L617 1986", bibdate = "Wed Nov 23 11:34:02 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.gbv.de:20011/gvk", URL = "http://www.loc.gov/catdir/enhancements/fy0668/86183277-d.html", acknowledgement = ack-nhfb, remark = "Chapter 7 (pp. 43--52) is entitled: ``The Baffling Law of Benford''.", subject = "number theory", } @Article{MacRoberts:1986:QMC, author = "Michael H. MacRoberts and Barbara R. MacRoberts", title = "Quantitative Measures of Communication in Science: a Study of the Formal Level", journal = j-SOC-STUD-SCI, volume = "16", number = "1", pages = "151--172", day = "1", month = feb, year = "1986", CODEN = "SSSCDH", DOI = "https://doi.org/10.1177/030631286016001008", ISSN = "0306-3127 (print), 1460-3659 (electronic)", ISSN-L = "0306-3127", bibdate = "Mon Dec 3 09:12:44 MST 2018", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/socstudsci.bib", note = "See \cite{Stigler:1987:PMF} and reply \cite{MacRoberts:1987:MFU}.", URL = "https://journals.sagepub.com/doi/pdf/10.1177/030631286016001008", acknowledgement = ack-nhfb, fjournal = "Social Studies of Science", journal-URL = "http://sss.sagepub.com/content/by/year", } @Article{Pao:1986:EEL, author = "Miranda Lee Pao", title = "An empirical examination of {Lotka's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "37", number = "1", pages = "26--33", month = jan, year = "1986", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/asi.4630370105", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:25 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "24 May 2007", } @Article{Pao:1986:REE, author = "Miranda Lee Pao", title = "Research: An empirical examination of {Lotka's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "37", number = "1", pages = "26--33", month = jan, year = "1986", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(198601)37:1<26::AID-ASI4>3.0.CO%3B2-Z", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:25 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "7 Jan 1999", } @Article{Patil:1986:DFD, author = "S. A. Patil and V. R. R. Uppuluri", title = "The Distribution of First $j$ Digits", journal = j-COLLEGE-MATH-J, volume = "17", number = "3", pages = "240--243", month = may, year = "1986", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.1986.11972963", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:50:20 MST 2019", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.jstor.org/stable/2686982; http://www.tandfonline.com/doi/abs/10.1080/07468342.1986.11972963", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", keywords = "Benford's Law", onlinedate = "30 Jan 2018", } @Article{Pontigo:1986:QAB, author = "J. Pontigo and F. W. Lancaster", title = "Qualitative aspects of the {Bradford} distribution", journal = j-SCIENTOMETRICS, volume = "9", number = "1--2", pages = "59--70", month = jan, year = "1986", CODEN = "SCNTDX", DOI = "https://doi.org/10.1007/BF02016608", ISSN = "0138-9130 (print), 1588-2861 (electronic)", ISSN-L = "0138-9130", bibdate = "Wed Sep 2 12:02:58 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/scientometrics1980.bib", URL = "http://link.springer.com/article/10.1007/BF02016608", acknowledgement = ack-nhfb, fjournal = "Scientometrics", journal-URL = "http://link.springer.com/journal/11192", } @Article{Schatte:1986:ALD, author = "Peter Schatte", title = "On the asymptotic logarithmic distribution of the floating-point mantissas of sums", journal = j-MATH-NACHR, volume = "127", number = "1", pages = "7--20", month = "????", year = "1986", CODEN = "MTMNAQ", DOI = "https://doi.org/10.1002/mana.19861270102", ISSN = "0025-584X", ISSN-L = "0025-584X", bibdate = "Sun Feb 17 06:04:23 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://onlinelibrary.wiley.com/doi/10.1002/mana.19861270102/abstract", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", } @Article{Tichy:1986:GZU, author = "Robert F. Tichy", title = "{Gleichverteilung und zahlentheoretische Ungleichungen II}. ({German}) [{Uniform} distribution and number-theoretic inequalities {II}]", journal = "Anzeiger der {\"O}sterreichischen Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse", volume = "122(1985)", number = "??", pages = "95--99", month = "????", year = "1986", CODEN = "OSAWA8", DOI = "????", ISSN = "0065-535X", ISSN-L = "0065-535X", bibdate = "Sun Feb 17 06:24:23 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "????", acknowledgement = ack-nhfb, ajournal = "Anz. {\"O}sterr. Akad. Wiss., Math.-Naturwiss. Kl.", classmath = "11K06 (General theory of distribution modulo 1); 11J25 (Diophantine inequalities)", keywords = "diophantine inequalities; discrepancy; H infinity; logarithmic mean; summation method; uniform distribution", language = "German", xxnote = "Despite repeated searches, I have not found this journal online, not even from links at the Austrian Academy of Sciences, \path=http://www.oeaw.ac.at=.", } @Article{Bak:1987:SOC, author = "Per Bak and Chao Tang and Kurt Wiesenfeld", title = "Self-organized criticality: An explanation of the $ 1 / f $ noise", journal = j-PHYS-REV-LET, volume = "59", number = "4", pages = "381--384", month = jul, year = "1987", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.59.381", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Thu Feb 28 10:54:26 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.59.381", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "power law; scale invariance; Zipf's law", } @Article{Chen:1987:AZL, author = "Ye-Sho Chen and Ferdinand F. Leimkuhler", title = "Analysis of {Zipf}'s law: An index approach", journal = j-INFO-PROC-MAN, volume = "23", number = "3", pages = "171--182", day = "1", month = apr, year = "1987", CODEN = "IPMADK", DOI = "https://doi.org/10.1016/0306-4573(87)90002-1", ISSN = "0306-4573 (print), 1873-5371 (electronic)", ISSN-L = "0306-4573", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/0306457387900021", ZMnumber = "0614.62150", abstract = "A rigorous analysis of Zipf's law is made using an index for the sequence of observed values of the variables in a Zipf-type relationship. Three important properties relating rank, count, and frequency are identified. Using this approach, the shape of Zipf-type curves can be described in terms of three distinct regions and two parameters of the Mandelbrot--Zipf law. This result has considerable practical significance, since it provides rigorous foundations for the application of Zipf's law.", acknowledgement = ack-nhfb, classmath = "62P99 (Appl. of statistics)", fjournal = "Information Processing and Management", journal-URL = "http://www.sciencedirect.com/science/journal/03064573", keywords = "index approach; Mandelbrot-Zipf law; Zipf's law", } @Article{Chen:1987:BLI, author = "Ye-Sho Chen and F. F. Leimkuhler", title = "{Bradford's Law}: An index approach", journal = j-SCIENTOMETRICS, volume = "11", number = "3--4", pages = "183--198", month = mar, year = "1987", CODEN = "SCNTDX", DOI = "https://doi.org/10.1007/BF02016591", ISSN = "0138-9130 (print), 1588-2861 (electronic)", ISSN-L = "0138-9130", bibdate = "Wed Sep 2 12:03:03 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/scientometrics1980.bib", URL = "http://link.springer.com/article/10.1007/BF02016591", acknowledgement = ack-nhfb, fjournal = "Scientometrics", journal-URL = "http://link.springer.com/journal/11192", } @Article{Fedorowicz:1987:DPE, author = "Jane Fedorowicz", title = "Database Performance Evaluation in an Indexed File Environment", journal = j-TODS, volume = "12", number = "1", pages = "85--110", month = mar, year = "1987", CODEN = "ATDSD3", ISSN = "0362-5915 (print), 1557-4644 (electronic)", ISSN-L = "0362-5915", bibdate = "Sat Apr 14 10:34:48 MDT 2001", bibsource = "Compendex database; Database/Graefe.bib; Database/Wiederhold.bib; http://www.acm.org/pubs/toc/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.acm.org/pubs/articles/journals/tods/1987-12-1/p85-fedorowicz/p85-fedorowicz.pdf; http://www.acm.org/pubs/citations/journals/tods/1987-12-1/p85-fedorowicz/; http://www.acm.org/pubs/toc/Abstracts/tods/13675.html", abstract = "The use of database systems for managerial decision making often incorporates information-retrieval capabilities with numeric report generation. Of great concern to the user of such a system is the response time associated with issuing a query to the database. This study presents a procedure for estimating response time for one of the most frequently encountered physical storage mechanisms, the indexed file. The model provides a fairly high degree of accuracy, but is simple enough so that the cost of applying the model is not exorbitant. The model incorporates the knowledge that the distribution of access key occurrences is known to follow Zipf's law. It first estimates the access time required to complete the query, which includes the time needed for all input and output transactions, and CPU time used in performing the search. The effects of multiple users on an individual's response time are then assessed using a simple regression estimation technique. The two-step procedure allows for the separation of access time from multiuser influences.", acknowledgement = ack-nhfb, affiliation = "Boston Univ, Boston, MA, USA", affiliationaddress = "Boston Univ, Boston, MA, USA", annote = "a procedure for estimating response time; distribution of access key occurrences follow Zipf's law. Early version with Kellogg, J. L. Model provides a fairly high degree of accuracy but is simple. The effects of multiple users are assessed using simple regression estimation.", classification = "723; 912; 922", fjournal = "ACM Transactions on Database Systems", generalterms = "Design; Performance", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J777", keywords = "data processing --- File organization; database performance; database systems; indexed file environment; MANAGEMENT --- Information Systems; multiple users, design; Performance; performance; response time; statistical methods --- regression analysis; Zipf's law", subject = "{\bf D.4.3}: Software, OPERATING SYSTEMS, File Systems Management, File organization. {\bf H.2.2}: Information Systems, DATABASE MANAGEMENT, Physical Design, Access methods. {\bf H.3.2}: Information Systems, INFORMATION STORAGE AND RETRIEVAL, Information Storage, File organization. {\bf H.3.3}: Information Systems, INFORMATION STORAGE AND RETRIEVAL, Information Search and Retrieval, Retrieval models.", } @Article{Hosking:1987:PQE, author = "J. R. M. Hosking and J. R. Wallis", title = "Parameter and Quantile Estimation for the Generalized {Pareto} Distribution", journal = j-TECHNOMETRICS, volume = "29", number = "3", pages = "339--349", month = "????", year = "1987", CODEN = "TCMTA2", ISSN = "0040-1706 (print), 1537-2723 (electronic)", ISSN-L = "0040-1706", bibdate = "Fri Nov 25 16:39:38 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Technometrics", journal-URL = "http://www.jstor.org/journals/00401706.html; http://www.tandfonline.com/loi/utch20", } @Article{Kunoff:1987:FDP, author = "Sharon Kunoff", title = "{$ N! $} has the First Digit Property", journal = j-FIB-QUART, volume = "25", number = "4", pages = "365--367", month = nov, year = "1987", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:01:00 MDT 2011", bibsource = "http://www.fq.math.ca/25-4.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/25-4/kunoff.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{MacRoberts:1987:MFU, author = "Michael H. MacRoberts and Barbara R. MacRoberts", title = "Measurement in the Face of Universal Uncertainty: a Reply to {Stigler}", journal = j-SOC-STUD-SCI, volume = "17", number = "2", pages = "334--336", day = "1", month = may, year = "1987", CODEN = "SSSCDH", DOI = "https://doi.org/10.1177/030631287017002008", ISSN = "0306-3127 (print), 1460-3659 (electronic)", ISSN-L = "0306-3127", bibdate = "Mon Dec 3 09:12:46 MST 2018", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/socstudsci.bib", note = "See \cite{MacRoberts:1986:QMC}.", URL = "https://journals.sagepub.com/doi/pdf/10.1177/030631287017002008", acknowledgement = ack-nhfb, fjournal = "Social Studies of Science", journal-URL = "http://sss.sagepub.com/content/by/year", } @Article{Nagasaka:1987:BLL, author = "Kenji Nagasaka and Jau-Shyong Shiue", title = "{Benford}'s law for linear recurrence sequences", journal = "Tsukuba J. Math.", volume = "11", number = "2", pages = "341--351", year = "1987", CODEN = "????", ISSN = "0387-4982", ISSN-L = "0387-4982", MRclass = "60E99 (11B37)", MRnumber = "926460 (88m:60047)", MRreviewer = "L. Kuipers", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Tsukuba Journal of Mathematics", } @Article{Nicholls:1987:BCE, author = "Paul Travis Nicholls", title = "Brief Communications: Estimation of {Zipf} parameters", journal = j-J-AM-SOC-INF-SCI, volume = "38", number = "6", pages = "443--445", month = nov, year = "1987", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(198711)38:6<443::AID-ASI4>3.0.CO%3B2-E", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:31 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "24 May 2007", } @Article{Nicholls:1987:EZP, author = "Paul Travis Nicholls", title = "Estimation of {Zipf} parameters", journal = j-J-AM-SOC-INF-SCI, volume = "38", number = "6", pages = "443--445", year = "1987", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(198711)38:6<443::AID-ASI4>3.0.CO%3B2-E", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Mon Feb 25 06:13:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", } @Article{Sanders:1987:PPUa, author = "Robert Sanders", title = "The {Pareto Principle}: Its Use and Abuse", journal = "Journal of Consumer Marketing", volume = "4", number = "1", pages = "47--50", month = "????", year = "1987", CODEN = "????", DOI = "https://doi.org/10.1108/eb008188", ISSN = "0736-3761", ISSN-L = "0736-3761", bibdate = "Fri Nov 25 06:56:30 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Sanders:1987:PPUb, author = "Robert Sanders", title = "The {Pareto Principle}: Its Use and Abuse", journal = "Journal of Services Marketing", volume = "1", number = "2", pages = "37--40", month = "????", year = "1987", CODEN = "????", DOI = "https://doi.org/10.1108/eb024706", ISSN = "0887-6045", ISSN-L = "0887-6045", bibdate = "Fri Nov 25 06:56:30 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Schatte:1987:ABM, author = "Peter Schatte", title = "On the asymptotic behaviour of the mantissa distribution of sums", journal = j-J-INFO-PROC-CYBERNETICS-EIK, volume = "23", number = "??", pages = "353--360", month = "????", year = "1987", CODEN = "JICYE5", ISSN = "0863-0593", ISSN-L = "0863-0593", bibdate = "Sat Nov 12 15:58:50 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "J. Inform. Process. Cybern. EIK", fjournal = "Journal of Information Processing and Cybernetics: EIK", } @Article{Schatte:1987:SEU, author = "Peter Schatte", title = "Some estimates of the {$ \mathcal {H}_\infty $}-uniform distribution", journal = j-MONAT-MATH, volume = "103", number = "3", pages = "233--249", month = sep, year = "1987", CODEN = "MNMTA2", DOI = "https://doi.org/10.1007/BF01364342", ISSN = "0026-9255 (print), 1436-5081 (electronic)", ISSN-L = "0026-9255", bibdate = "Sat Nov 12 15:57:05 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.springer.com/article/10.1007/BF01364342", acknowledgement = ack-nhfb, fjournal = "Monatshefte f{\"u}r Mathematik", } @Article{Stigler:1987:PMF, author = "Stephen M. Stigler", title = "Precise Measurement in the Face of Error: a Comment on {MacRoberts} and {MacRoberts}", journal = j-SOC-STUD-SCI, volume = "17", number = "2", pages = "332--334", day = "1", month = may, year = "1987", CODEN = "SSSCDH", DOI = "https://doi.org/10.1177/030631287017002007", ISSN = "0306-3127 (print), 1460-3659 (electronic)", ISSN-L = "0306-3127", bibdate = "Mon Dec 3 09:12:46 MST 2018", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/socstudsci.bib", note = "See \cite{MacRoberts:1986:QMC} and reply \cite{MacRoberts:1987:MFU}.", URL = "https://journals.sagepub.com/doi/pdf/10.1177/030631287017002007", acknowledgement = ack-nhfb, fjournal = "Social Studies of Science", journal-URL = "http://sss.sagepub.com/content/by/year", } @Article{Tague:1987:MVZ, author = "Jean Tague and Paul Nicholls", title = "The maximal value of a {Zipf} size variable: Sampling properties and relationship to other parameters", journal = j-INFO-PROC-MAN, volume = "23", number = "3", pages = "155--170", day = "7", month = jun, year = "1987", CODEN = "IPMADK", DOI = "https://doi.org/10.1016/0306-4573(87)90001-X", ISSN = "0306-4573 (print), 1873-5371 (electronic)", ISSN-L = "0306-4573", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/030645738790001X", acknowledgement = ack-nhfb, fjournal = "Information Processing and Management", journal-URL = "http://www.sciencedirect.com/science/journal/03064573", keywords = "Zipf's Law", } @Article{Tichy:1987:GS, author = "Robert F. Tichy", title = "{Gleichverteilung zum Summierungsverfahren {$ H_\infty $}}. ({German}) [{Uniform} distribution to the {$ H_\infty $} summation method]", journal = j-MATH-NACHR, volume = "131", number = "??", pages = "119--125", month = "????", year = "1987", CODEN = "MTMNAQ", DOI = "https://doi.org/10.1002/mana.19871310112", ISSN = "0025-584X", ISSN-L = "0025-584X", bibdate = "Mon Feb 18 06:44:11 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "????", ZMnumber = "0626.10044", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", keywords = "$H_\infty$-discrepancy; difference sequences; real sequences; uniform distribution", language = "German", ZMclass = "11K06 (General theory of distribution modulo 1); 40J05 (Summability in abstract structures)", ZMreviewer = "D. Leitmann", } @Article{Turner:1987:DDI, author = "Peter R. Turner", title = "The Distribution of l.s.d. and Its Implications for Computer Design", journal = j-MATH-GAZ, volume = "71", number = "455", pages = "26--31", month = mar, year = "1987", CODEN = "MAGAAS", DOI = "https://doi.org/10.2307/3616283", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Tue Oct 30 23:36:58 2007", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "[l.s.d. = least significant digits]. The topic is variously known as Benford's Law, the Law of Anomalous Numbers, and Zipf's Law.", acknowledgement = ack-nhfb, fjournal = "Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", keywords = "Benford's Law; Law of Anomalous Numbers; logarithmic distribution; overflow; underflow; Zipf's Law", } @Article{Belevitch:1988:PQD, author = "Vitold Belevitch", title = "{{\`A}} propos de la queue de la distribution de {Zipf}. ({French}) [{On} the tail of the {Zipf} distribution]", journal = "Ann. Soc. Sci. Bruxelles S{\'e}r. I", volume = "102", number = "3", pages = "73--85 (1989)", year = "1988", CODEN = "ASSBAH", ISSN = "0037-959X", ISSN-L = "0037-959X", MRclass = "92A90", MRnumber = "1016850 (90g:92114)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Annales de la Societ{\'e} Scientifique de Bruxelles. S{\'e}rie I. Sciences Math{\'e}matiques, Astronomiques et Physiques", language = "French", } @Article{Carslaw:1988:AIN, author = "C. A. Carslaw", title = "Anomalies in Income Numbers: Evidence of Goal-Oriented Behavior", journal = j-ACCOUNT-REV, volume = "63", number = "2", pages = "321--327", month = apr, year = "1988", CODEN = "ACRVAS", ISSN = "0001-4826 (print), 1558-7967 (electronic)", ISSN-L = "0001-4826", bibdate = "Sat Sep 10 11:00:09 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/248109", acknowledgement = ack-nhfb, fjournal = "The Accounting Review", journal-URL = "https://www.jstor.org/journal/accountingreview", keywords = "Benford's Law; New Zealand firms; second-digit frequency", remark-1 = "This paper references neither Newcomb nor Benford, but it uses their digit-frequency formula from \cite{Feller:1966:IPT} to compare with data on revenues of New Zealand companies.", remark-2 = "According to \cite{Kossovsky:2015:BLT}, this may be the first published serious use of Benford's Law for fraud detection in accounting, although the possibility had been mentioned 16 years earlier \cite{Varian:1972:LEB}, and that earlier paper is not cited in this 1988 paper.", } @Article{Hill:1988:RNG, author = "Theodore Hill", title = "Random number guessing and the first digit phenomenon", journal = j-PSYCHOL-REP, volume = "62", number = "3", pages = "967--971", month = jun, year = "1988", CODEN = "PYRTAZ", DOI = "https://doi.org/10.2466/pr0.1988.62.3.967", ISSN = "0033-2941 (print), 1558-691X (electronic)", ISSN-L = "0033-2941", bibdate = "Fri Feb 15 06:41:31 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.amsciepub.com/doi/abs/10.2466/pr0.1988.62.3.967", acknowledgement = ack-nhfb, ajournal = "Psychol. rep.", fjournal = "Psychological Reports", } @InCollection{Kanemitsu:1988:BLF, author = "Shigeru Kanemitsu and Kenji Nagasaka and G{\'e}rard Rauzy and Jau-Shyong Shiue", booktitle = "Probability theory and mathematical statistics ({Kyoto}, 1986)", title = "On {Benford}'s law: the first digit problem", volume = "1299", publisher = pub-SV, address = pub-SV:adr, pages = "158--169", year = "1988", DOI = "https://doi.org/10.1007/BFb0078471", MRclass = "11K06 (60E99)", MRnumber = "935987 (89d:11059)", MRreviewer = "Peter Schatte", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", series = "Lecture Notes in Math.", acknowledgement = ack-nhfb, } @InCollection{Nicolis:1988:PBZ, author = "John S. Nicolis", booktitle = "Synergetics, order and chaos ({Madrid}, 1987)", title = "On the parallel between {Zipf}'s law and {$ 1 / f $} processes in chaotic systems possessing coexisting attractors", publisher = "World Sci. Publ.", address = "Teaneck, NJ, USA", pages = "723--734", year = "1988", MRclass = "58F13 (00A69)", MRnumber = "1115922 (92g:58087)", MRreviewer = "Dieter H. Mayer", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Prather:1988:CET, author = "R. E. Prather", title = "Comparison and Extension of Theories of {Zipf} and {Halstead}", journal = j-COMP-J, volume = "31", number = "3", pages = "248--252", month = jun, year = "1988", CODEN = "CMPJA6", ISSN = "0010-4620 (print), 1460-2067 (electronic)", ISSN-L = "0010-4620", bibdate = "Tue Mar 25 13:51:56 MST 1997", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www3.oup.co.uk/computer_journal/hdb/Volume_31/Issue_03/", URL = "http://www3.oup.co.uk/computer_journal/hdb/Volume_31/Issue_03/tiff/248.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_31/Issue_03/tiff/249.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_31/Issue_03/tiff/250.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_31/Issue_03/tiff/251.tif; http://www3.oup.co.uk/computer_journal/hdb/Volume_31/Issue_03/tiff/252.tif", acknowledgement = ack-nhfb, affiliation = "Trinity Univ, San Antonio, TX, USA", affiliationaddress = "Trinity Univ, San Antonio, TX, USA", classcodes = "C6110B (Software engineering techniques)", classification = "723", corpsource = "Comput. and Inf. Sci., Trinity Univ., San Antonio, TX, USA", fjournal = "The Computer Journal", journal-URL = "http://comjnl.oxfordjournals.org/", keywords = "behavioural psychology; computer programming; computer software; empirical law; Halstead metrics; Halstead's hypothesis; length estimates; Measurements; probabilistic hypothesis; process; programming; software engineering; software metrication; software science; Zipf metrics; Zipf's law", treatment = "T Theoretical or Mathematical", } @Article{Sanders:1988:PPU, author = "Robert Sanders", title = "The {Pareto Principle}: Its Use and Abuse", journal = "Journal of Business \& Industrial Marketing", volume = "3", number = "2", pages = "37--40", month = "????", year = "1988", CODEN = "????", DOI = "https://doi.org/10.1108/eb006057", ISSN = "0885-8624", ISSN-L = "0885-8624", bibdate = "Fri Nov 25 07:04:53 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Schatte:1988:ASC, author = "Peter Schatte", title = "On the Almost Sure Convergence of Floating-Point Mantissas and {Benford's Law}", journal = j-MATH-NACHR, volume = "135", number = "1", pages = "79--83", month = "????", year = "1988", CODEN = "MTMNAQ", DOI = "https://doi.org/10.1002/mana.19881350108", ISSN = "0025-584X", ISSN-L = "0025-584X", MRclass = "60F15 (11K31)", MRnumber = "944219 (89g:60114)", MRreviewer = "S. A. Book", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", onlinedate = "12 Nov 2006", reviewer = "S. A. Book", } @Article{Schatte:1988:LIL, author = "Peter Schatte", title = "On a law of the iterated logarithm for sums {$ \bmod 1 $} with application to {Benford}'s law", journal = j-PROBAB-THEORY-RELAT-FIELDS, volume = "77", number = "2", pages = "167--178", month = feb, year = "1988", CODEN = "PTRFEU", DOI = "https://doi.org/10.1007/BF00334035", ISSN = "0178-8051 (print), 1432-2064 (electronic)", ISSN-L = "0178-8051", MRclass = "60F15 (11K31)", MRnumber = "927235 (89b:60081)", MRreviewer = "L. Kuipers", bibdate = "Sun Apr 27 11:18:06 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/probab-theory-relat-fields.bib", URL = "http://link.springer.com/article/10.1007/BF00334035", acknowledgement = ack-nhfb, fjournal = "Probability Theory and Related Fields", journal-URL = "http://link.springer.com/journal/440", } @Article{Schatte:1988:MDC, author = "Peter Schatte", title = "On Mantissa Distribution in Computing and {Benford}'s Law", journal = j-J-INFO-PROC-CYBERNETICS-EIK, volume = "24", number = "9", pages = "443--455", month = "????", year = "1988", CODEN = "JICYE5", ISSN = "0863-0593", ISSN-L = "0863-0593", MRclass = "60E05 (11K06 11K31 65G99)", MRnumber = "984516 (90g:60016)", MRreviewer = "F. Schweiger", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb # "\slash " # ack-nj, fjournal = "Journal of Information Processing and Cybernetics: EIK", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Article{Schatte:1988:UDC, author = "Peter Schatte", title = "On the uniform distribution of certain sequences and {Benford}'s law", journal = j-MATH-NACHR, volume = "136", number = "1", pages = "271--273", year = "1988", CODEN = "MTMNAQ", DOI = "https://doi.org/10.1002/mana.19881360119", ISSN = "0025-584X", ISSN-L = "0025-584X", MRclass = "11K06 (11K31)", MRnumber = "952478 (89j:11075)", MRreviewer = "O. P. Stackelberg", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", onlinedate = "11 Nov 2006", } @Article{Tichy:1988:SRC, author = "Robert F. Tichy", title = "{Statistische Resultate {\"u}ber computergerechte Darstellungen von Zahlen}. ({German}) [Statistical results on computer-oriented representations of numbers]", journal = "Anzeiger der {\"O}sterreichischen Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse", volume = "1987", number = "??", pages = "1--8", month = "????", year = "1988", CODEN = "OSAWA8", DOI = "????", ISSN = "0065-535X", ISSN-L = "0065-535X", bibdate = "Mon Feb 18 06:45:27 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "????", ZMnumber = "0625.10041", acknowledgement = ack-nhfb, ajournal = "Anz. {\"O}sterr. Akad. Wiss., Math.-Naturwiss. Kl.", language = "German", xxnote = "Despite repeated searches, I have not found this journal online, not even from links at the Austrian Academy of Sciences, \path=http://www.oeaw.ac.at=.", ZMclass = "11K16 (Normal numbers, etc.); 11J71 (Distribution modulo one); 11K38 (Irregularities of distribution)", } @Article{Arnold:1989:BEP, author = "Barry C. Arnold and S. James Press", title = "{Bayesian} estimation and prediction for {Pareto} data", journal = j-J-AM-STAT-ASSOC, volume = "84", number = "408", pages = "1079--1084", month = "????", year = "1989", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", MRclass = "62F15 (62F10)", MRnumber = "92f:62034", bibdate = "Mon May 5 12:36:01 MDT 1997", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Chen:1989:ZLT, author = "Ye-Sho Chen", title = "{Zipf}'s laws in text modeling", journal = "International Journal of General Systems. Methodology, Applications, Education", volume = "15", number = "3", pages = "233--252", year = "1989", CODEN = "IJGSAX", DOI = "https://doi.org/10.1080/03081078908935048", ISSN = "0308-1079, 1026-7492, 1563-5104", ISSN-L = "0308-1079", MRclass = "92A90", MRnumber = "1015844", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Internat. J. Gen. Systems", fjournal = "International Journal of General Systems. Methodology, Applications, Education", } @Article{Family:1989:KDG, author = "Fereydoon Family and Paul Meakin", title = "Kinetics of droplet growth processes: Simulations, theory, and experiments", journal = j-PHYS-REV-A, volume = "40", number = "7", pages = "3836--3854", month = oct, year = "1989", CODEN = "PLRAAN", DOI = "https://doi.org/10.1103/PhysRevA.40.3836", ISSN = "1050-2947 (print), 1094-1622, 1538-4446, 1538-4519", ISSN-L = "1050-2947", bibdate = "Thu Feb 28 10:49:20 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevA.40.3836", acknowledgement = ack-nhfb, fjournal = "Physical Review A (Atomic, Molecular, and Optical Physics)", journal-URL = "http://pra.aps.org/browse", } @InProceedings{Kitsuregawa:1989:EBS, author = "Masaru Kitsuregawa and Masaya Nakayama and Mikio Takagi", title = "The Effect of Bucket Size Tuning in the Dynamic Hybrid {GRACE} Hash Join Method", crossref = "Apers:1989:VLD", pages = "257--266", month = aug, year = "1989", CODEN = "VLDBDP", bibdate = "Sat Jan 25 17:38:12 MST 1997", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/hash.bib", abstract = "In this paper, we show detailed analysis and performance evaluation of the Dynamic Hybrid GRACE Hash Join Method (DHGH Method) when the tuple distribution in buckets is unbalanced. The conventional Hash Join Methods specify the tuple distribution in buckets statically. However it may differ from estimation since join operations are applied with selection operations. When the tuple distribution in buckets is unbalanced, the processing cost of join operation becomes more costly than the ideal case when you use Hybrid Hash Join Method (HH Method). On the other hand, when you use the DHGH Method, the destaging buckets are selected dynamically, gives the same performance as the ideal case even if the tuple distribution in buckets is unbalanced such as Zipf-like distributions.", acknowledgement = ack-nhfb, affiliation = "Univ of Tokyo", affiliationaddress = "Tokyo, Jpn", classification = "723", journalabr = "Very Large Data Bases Int Conf Very Large Data Bases", keywords = "Computer Metatheory --- Programming Theory; Computer Programming --- Algorithms; Database Systems; Query Processing; Relational; Relational Algebra; VLDB", } @Article{Lang:1989:UAB, author = "Sheau-Dong Lang and James R. Driscoll and Jiann H. Jou", title = "A Unified Analysis of Batched Searching of Sequential and Tree-Structured Files", journal = j-TODS, volume = "14", number = "4", pages = "604--618", month = dec, year = "1989", CODEN = "ATDSD3", ISSN = "0362-5915 (print), 1557-4644 (electronic)", ISSN-L = "0362-5915", MRclass = "68P10 (68P20)", MRnumber = "1 073 204", bibdate = "Sat Apr 14 10:34:48 MDT 2001", bibsource = "Database/Graefe.bib; Database/Wiederhold.bib; http://www.acm.org/pubs/toc/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.acm.org/pubs/articles/journals/tods/1989-14-4/p604-lang/p604-lang.pdf; http://www.acm.org/pubs/citations/journals/tods/1989-14-4/p604-lang/; http://www.acm.org/pubs/toc/Abstracts/tods/76908.html", abstract = "A direct and unified approach is used to analyze the efficiency of batched searching of sequential and tree-structured files. The analysis is applicable to arbitrary search distributions, and closed-form expressions are obtained for the expected batched searching cost and savings. In particular, we consider a search distribution satisfying Zipf's law for sequential files and four types of uniform (random) search distribution for sequential and tree-structured files. These results unify and extend earlier research on batched searching and estimating block accesses for database systems.", acknowledgement = ack-nhfb, affiliation = "Central Florida Univ., Orlando, FL, USA", annote = "closed-form expressions for the number of accesses needed given arbitrary search distributions.", fjournal = "ACM Transactions on Database Systems", generalterms = "Algorithms; Design; Performance; Theory", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J777", keywords = "algorithms; design; performance; theory", subject = "{\bf H.3.3}: Information Systems, INFORMATION STORAGE AND RETRIEVAL, Information Search and Retrieval, Search process. {\bf H.3.2}: Information Systems, INFORMATION STORAGE AND RETRIEVAL, Information Storage, File organization. {\bf H.2.2}: Information Systems, DATABASE MANAGEMENT, Physical Design, Access methods.", } @Article{Nicholls:1989:RBM, author = "Paul Travis Nicholls", title = "Research: Bibliometric modeling processes and the empirical validity of {Lotka's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "40", number = "6", pages = "379--385", month = nov, year = "1989", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(198911)40:6<379::AID-ASI1>3.0.CO%3B2-Q", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:37 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "27 May 2007", } @Article{Nicolis:1989:CDM, author = "G. Nicolis and C. Nicolis and John S. Nicolis", title = "Chaotic dynamics, {Markov} partitions, and {Zipf}'s law", journal = j-J-STAT-PHYS, volume = "54", number = "3--4", pages = "915--924", month = feb, year = "1989", CODEN = "JSTPSB", DOI = "https://doi.org/10.1007/BF01019781", ISSN = "0022-4715 (print), 1572-9613 (electronic)", ISSN-L = "0022-4715", MRclass = "58F13 (58F14 58F20)", MRnumber = "988565 (90b:58184)", MRreviewer = "Shi Hai Li", bibdate = "Thu Aug 28 18:40:55 MDT 2014", bibsource = "http://link.springer.com/journal/10955; http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0022-4715&volume=54&issue=3; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jstatphys1980.bib", URL = "http://link.springer.com/article/10.1007/BF01019781", acknowledgement = ack-nhfb, fjournal = "Journal of Statistical Physics", journal-URL = "http://link.springer.com/journal/10955", } @Article{Nicolis:1989:PBZ, author = "John S. Nicolis and Ichiro Tsuda", title = "On the parallel between {Zipf}'s law and {$ 1 / f $} processes in chaotic systems possessing coexisting attractors", journal = j-PROG-THEOR-PHYS, volume = "82", number = "2", pages = "254--274", year = "1989", CODEN = "PTPKAV", ISSN = "0033-068X (print), 1347-4081 (electronic)", ISSN-L = "0033-068X", MRclass = "58F13 (92K20)", MRnumber = "1023357 (91d:58165)", MRreviewer = "Nima Geffen", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Progress of Theoretical Physics", journal-URL = "http://ptp.oxfordjournals.org/content/by/year", } @Article{Schatte:1989:MUD, author = "Peter Schatte", title = "On measures of uniformly distributed sequences and {Benford}'s law", journal = j-MONAT-MATH, volume = "107", number = "3", pages = "245--256", year = "1989", CODEN = "MNMTA2", DOI = "https://doi.org/10.1007/BF01300347", ISSN = "0026-9255 (print), 1436-5081 (electronic)", ISSN-L = "0026-9255", MRclass = "11K36 (60F99)", MRnumber = "1008683 (90k:11097)", MRreviewer = "Kazuo Goto", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Monatshefte f{\"u}r Mathematik", } @Article{Shanbhag:1989:BRBa, author = "D. N. Shanbhag", title = "Book Review: {{\booktitle{Pareto Distributions}}, by B. C. Arnold}", journal = j-J-R-STAT-SOC-SER-A-STAT-SOC, volume = "152", number = "2", pages = "253--254", month = "????", year = "1989", CODEN = "JSSAEF", DOI = "https://doi.org/10.2307/2982920", ISSN = "0964-1998 (print), 1467-985X (electronic)", ISSN-L = "0964-1998", bibdate = "Sat Jan 24 15:13:38 MST 2015", bibsource = "http://www.jstor.org/stable/i349642; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jrss-a-1980.bib", URL = "http://www.jstor.org/stable/2982920", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society. Series A (Statistics in Society)", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1467-985X; http://www.jstor.org/journals/09641998.html", } @Article{Shieh:1989:BZE, author = "Yeung-Nan Shieh", title = "The {Beckmann--Zipf} effect and plant location under uniform pricing", journal = j-ECONOM-LETT, volume = "30", number = "1", pages = "89--92", day = "1", month = jan, year = "1989", CODEN = "ECLEDS", DOI = "https://doi.org/10.1016/0165-1765(89)90162-6", ISSN = "0165-1765 (print), 1873-7374 (electronic)", ISSN-L = "0165-1765", MRclass = "90A12 (90B05)", MRnumber = "1010312", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/0165176589901626", acknowledgement = ack-nhfb, fjournal = "Economics Letters", keywords = "Zipf's Law", } @Article{Thomas:1989:UPR, author = "Jacob K. Thomas", title = "Unusual patterns in reported earnings", journal = j-ACCOUNT-REV, volume = "59", number = "4", pages = "773--787", month = oct, year = "1989", CODEN = "ACRVAS", ISSN = "0001-4826 (print), 1558-7967 (electronic)", ISSN-L = "0001-4826", bibdate = "Fri Feb 15 06:49:18 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/247861", acknowledgement = ack-nhfb, fjournal = "The Accounting Review", keywords = "Benford's Law; distribution of first digits; distribution of second digits", } @Article{Xekalaki:1989:APY, author = "E. Xekalaki and J. Panaretos", title = "On the Association of the {Pareto} and the {Yule} Distribution", journal = j-THEORY-PROBAB-APPL, volume = "33", number = "1", pages = "191--195", month = "????", year = "1989", CODEN = "TPRBAU", DOI = "https://doi.org/10.1137/1133028", ISSN = "0040-585X (print), 1095-7219 (electronic)", ISSN-L = "0040-585X", bibdate = "Tue Apr 1 10:40:04 MDT 2014", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/TVP/33/1; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/theoryprobabappl.bib", acknowledgement = ack-nhfb, fjournal = "Theory of Probability and its Applications", journal-URL = "http://epubs.siam.org/tvp", onlinedate = "January 1989", } @Article{Egghe:1990:RAT, author = "L. Egghe", title = "Research: Applications of the theory of {Bradford}'s {Law} to the calculation of {Leimkuhler's Law} and to the completion of bibliographies", journal = j-J-AM-SOC-INF-SCI, volume = "41", number = "7", pages = "469--492", month = oct, year = "1990", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199010)41:7<469::AID-ASI1>3.0.CO%3B2-P", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:41 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "7 Jan 1999", } @Article{Egghe:1990:RND, author = "L. Egghe", title = "Research: A note on different {Bradford} multipliers", journal = j-J-AM-SOC-INF-SCI, volume = "41", number = "3", pages = "204--209", month = apr, year = "1990", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199004)41:3<204::AID-ASI7>3.0.CO%3B2-8", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:39 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "7 Jan 1999", } @Article{Katsikas:1990:CDG, author = "A. A. Katsikas and J. S. Nicolis", title = "Chaotic dynamics of generating {Markov} partitions and linguistic sequences mimicking {Zipf}'s law", journal = "Nuovo Cimento D (1)", volume = "12", number = "2", pages = "177--195", year = "1990", CODEN = "NCSDDN", DOI = "https://doi.org/10.1007/BF02450454", ISSN = "0392-6737 (print), 1826-9893 (electronic)", ISSN-L = "0392-6737", MRclass = "92A90 (58F13 92A08 94A15)", MRnumber = "1049338 (90m:92114)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Societ{\`a} Italiana di Fisica. Il Nuovo Cimento. D. Serie 1", } @InCollection{Nagasaka:1990:BLL, author = "Kenji Nagasaka and Shigeru Kanemitsu and Jau-Shyong Shiue", booktitle = "Number theory, {Vol}.\ {I} ({Budapest}, 1987)", title = "{Benford}'s law: the logarithmic law of first digit", volume = "51", publisher = pub-NORTH-HOLLAND, address = pub-NORTH-HOLLAND:adr, pages = "361--391", year = "1990", MRclass = "11K06", MRnumber = "1058225 (92b:11048)", MRreviewer = "Peter Schatte", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", series = "Colloq. Math. Soc. J{\'a}nos Bolyai", acknowledgement = ack-nhfb, } @Article{Peritz:1990:BDB, author = "Bluma C. Peritz", title = "A {Bradford} distribution for bibliometrics", journal = j-SCIENTOMETRICS, volume = "18", number = "5--6", pages = "323--329", month = may, year = "1990", CODEN = "SCNTDX", DOI = "https://doi.org/10.1007/BF02020148", ISSN = "0138-9130 (print), 1588-2861 (electronic)", ISSN-L = "0138-9130", bibdate = "Wed Sep 2 12:03:16 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/scientometrics1990.bib", URL = "http://link.springer.com/article/10.1007/BF02020148", acknowledgement = ack-nhfb, fjournal = "Scientometrics", journal-URL = "http://link.springer.com/journal/11192", } @Article{Schatte:1990:BLC, author = "Peter Schatte", title = "On {Benford}'s law for continued fractions", journal = j-MATH-NACHR, volume = "148", pages = "137--144", year = "1990", CODEN = "MTMNAQ", ISSN = "0025-584X", ISSN-L = "0025-584X", MRclass = "11K50 (11K06)", MRnumber = "1127337 (92m:11077)", MRreviewer = "Harald Niederreiter", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", } @Book{Szekely:1990:PKN, author = "G{\'a}bor J. Sz{\'e}kely", title = "{Paradoxa: klassische und neue {\"U}berraschungen aus Wahrscheinlichkeitsrechnung und mathematischer Statistik}. (German) [{Paradoxes}: traditional and new surprises in probability and mathematical statistics]", publisher = "Deutsch", address = "Thun, Switzerland", pages = "239", year = "1990", ISBN = "3-87144-850-8", ISBN-13 = "978-3-87144-850-8", LCCN = "????", bibdate = "Thu Nov 17 19:26:05 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.gbv.de:20011/gvk", note = "Translation by Ulrike Leitner from the Hungarian original.", URL = "http://www.gbv.de/dms/hbz/toc/ht003363686.pdf; http://www.zentralblatt-math.org/zmath/en/search/?an=0743.60004", acknowledgement = ack-nhfb, language = "German", } @Article{VandenBroeck:1990:LFB, author = "C. {Van den Broeck} and R. Kawai", title = "Learning in feedforward {Boolean} networks", journal = j-PHYS-REV-A, volume = "42", number = "??", pages = "6210--??", day = "15", month = nov, year = "1990", CODEN = "PLRAAN", DOI = "https://doi.org/10.1103/PhysRevA.42.6210", ISSN = "1050-2947 (print), 1094-1622, 1538-4446, 1538-4519", ISSN-L = "1050-2947", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevA.42.6210", acknowledgement = ack-nhfb, fjournal = "Physical Review A (Atomic, Molecular, and Optical Physics)", journal-URL = "http://pra.aps.org/browse", keywords = "Zipf's Law", } @Article{Yeh:1990:ODT, author = "Hsiaw-Chan Yeh", title = "One discrete time series model for fat-tailed integer random variables: {Zipf} process", journal = "Bulletin of the Institute of Mathematics. Academia Sinica", volume = "18", number = "1", pages = "19--33", year = "1990", CODEN = "BIMSDG", ISSN = "0304-9825", ISSN-L = "0304-9825", MRclass = "60J10 (62M10)", MRnumber = "1072827 (91i:60175)", MRreviewer = "Ed McKenzie", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Bull. Inst. Math. Acad. Sinica", fjournal = "Bulletin of the Institute of Mathematics. Academia Sinica", } @Article{Antonini:1991:NUD, author = "Rita Giuliano Antonini", title = "On the Notion of Uniform Distribution Mod $1$", journal = j-FIB-QUART, volume = "29", number = "3", pages = "230--234", month = aug, year = "1991", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:01:34 MDT 2011", bibsource = "http://www.fq.math.ca/29-3.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/29-3/antonini.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", keywords = "Benford's Law", } @Article{Burke:1991:BLP, author = "John Burke and Eric Kincanon", title = "{Benford}'s law and physical constants: The distribution of initial digits", journal = j-AMER-J-PHYSICS, volume = "59", number = "10", pages = "952--952", month = oct, year = "1991", CODEN = "AJPIAS", DOI = "https://doi.org/10.1119/1.16838", ISSN = "0002-9505 (print), 1943-2909 (electronic)", ISSN-L = "0002-9505", bibdate = "Mon Nov 14 17:31:11 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://adsabs.harvard.edu/abs/1991AmJPh..59..952B; http://ajp.aapt.org/resource/1/ajpias/v59/i10/p952_s1", acknowledgement = ack-nhfb, fjournal = "American Journal of Physics", journal-URL = "http://scitation.aip.org/content/aapt/journal/ajp", } @Article{Burrell:1991:BDG, author = "Q. L. Burrell", title = "The {Bradford} distribution and the {Gini} index", journal = j-SCIENTOMETRICS, volume = "21", number = "2", pages = "181--194", month = jun, year = "1991", CODEN = "SCNTDX", DOI = "https://doi.org/10.1007/BF02017568", ISSN = "0138-9130 (print), 1588-2861 (electronic)", ISSN-L = "0138-9130", bibdate = "Wed Sep 2 12:03:22 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/scientometrics1990.bib", URL = "http://link.springer.com/article/10.1007/BF02017568", acknowledgement = ack-nhfb, fjournal = "Scientometrics", journal-URL = "http://link.springer.com/journal/11192", } @Article{Chen:1991:ZLN, author = "Ye-Sho Chen", title = "{Zipf}'s law in natural languages, programming languages, and command languages: the {Simon--Yule} approach", journal = j-INT-J-SYST-SCI, volume = "22", number = "11", pages = "2299--2312", year = "1991", CODEN = "IJSYA9", DOI = "https://doi.org/10.1080/00207729108910791", ISSN = "0020-7721 (print), 1464-5319 (electronic)", ISSN-L = "0020-7721", bibdate = "Sat Nov 12 10:50:55 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0793.68020", abstract = "Zipf's law and issues in natural languages, programming languages and command languages related to it, are reviewed. We point out the need for a unified theory to resolve the issues. Based on Herbert Simon's creative process for scientific discovery, we evaluate four leading theories of language generation. As a result, the Simon--Yule model of text generation is selected as a promising theory. The implications of the theory related to the issues in natural languages, programming languages and command languages are discussed.", acknowledgement = ack-nhfb, classmath = "68N15 (Programming languages) 68T50 (Natural language processing)", fjournal = "International Journal of Systems Science", keywords = "command languages; natural languages; programming languages; Simon-Yule model; text generation; Zipf's law", } @Book{Durrett:1991:PTE, author = "Richard Durrett", title = "Probability: theory and examples", publisher = "Wadsworth and Brooks\slash Cole Advanced Books and Software", address = "Pacific Grove, CA, USA", pages = "ix + 453", year = "1991", ISBN = "0-534-13206-5", ISBN-13 = "978-0-534-13206-4", LCCN = "QA273 .D865 1991", bibdate = "Mon Nov 21 10:06:18 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", price = "US\$49.95", series = "The Wadsworth and Brooks\slash Cole statistics\slash probability series", acknowledgement = ack-nhfb, remark = "Benford's Law is treated on pp. 300ff. Hill \cite[page 893]{Hill:1995:BII} reports that Durrett has a proof that the $k$-th digit of powers of integers follows Benford's Law.", subject = "Probabilities", } @Book{Hardy:1991:DS, author = "G. H. (Godfrey Harold) Hardy", title = "Divergent series", publisher = "Chelsea Pub. Co.", address = "New York, NY, USA", edition = "Second", pages = "xvi + 396", year = "1991", ISBN = "0-8284-0334-1", ISBN-13 = "978-0-8284-0334-4", LCCN = "QA295 .H29 1991", bibdate = "Wed Nov 16 08:17:27 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1877--1947", remark = "Unaltered reprint of \cite{Hardy:1949:DS}, despite label of second edition.", subject = "Divergent series", } @Article{Katsikas:1991:CDG, author = "A. A. Katsikas and J. S. Nicolis", title = "Chaotic Dynamics of Generating {Markov} Partitions, and Linguistic Sequences Mimicking {Zipf}'s Law", journal = j-LECT-NOTES-COMP-SCI, volume = "565", pages = "335--??", year = "1991", CODEN = "LNCSD9", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Mon May 13 08:51:55 MDT 1996", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @Article{Schatte:1991:NBL, author = "P. Schatte and K. Nagasaka", title = "A note on {Benford}'s law for second order linear recurrences with periodical coefficients", journal = j-Z-ANAL-ANWEND, volume = "10", number = "2", pages = "251--254", year = "1991", CODEN = "????", ISSN = "0232-2064 (print), 1661-4534 (electronic)", ISSN-L = "0232-2064", MRclass = "11K31", MRnumber = "1155374 (93b:11101)", MRreviewer = "F. Schweiger", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "{Zeitschrift f{\"u}r Analysis und ihre Anwendungen}", } @Article{Schatte:1991:ULI, author = "P. Schatte", title = "On a uniform law of the iterated logarithm for sums mod {$1$} and {Benford}'s law", journal = "Litovsk. Mat. Sb.", volume = "31", number = "1", pages = "205--217", year = "1991", CODEN = "????", DOI = "https://doi.org/10.1007/BF00972327", ISSN = "0132-2818", ISSN-L = "0132-2818", MRclass = "60F15 (11K99)", MRnumber = "1161363 (93g:60067)", MRreviewer = "O. P. Stackelberg", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Lietuvos TSR Moksl\polhk u Akademija. Lietuvos TSR Auk\v stosios Mokyklos. Lietuvos Matematikos Rinkinys. Akademiya Nauk Litovsko{\u\i} SSR. Vysshie Uchebnye Zavedeniya Litovsko{\u\i} SSR. Litovski\u\i\ Matematicheski{\u\i} Sbornik", xxnote = "Check pages: reprinted in two version of this journal??", } @Book{Schroeder:1991:FCP, author = "Manfred Schroeder", title = "Fractals, Chaos, Power Laws", publisher = pub-W-H-FREEMAN, address = pub-W-H-FREEMAN:adr, pages = "xviii + 429", year = "1991", ISBN = "0-671-74217-5, 0-7167-2136-8, 0-7167-2357-3", ISBN-13 = "978-0-671-74217-1, 978-0-7167-2136-9, 978-0-7167-2357-8", LCCN = "QD921 .S3 1990", bibdate = "Wed Dec 15 10:41:35 1993", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/h/heisenberg-werner.bib; http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/bibnet/authors/s/shannon-claude-elwood.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://www.math.utah.edu/pub/tex/bib/master.bib; http://www.math.utah.edu/pub/tex/bib/pi.bib", abstract = "Reveals the extraordinary dimensions of new mathematical insights about the nature of physical reality; explores the powerful applications of these symmetry concepts in physics, chemistry, music, and the visual arts. Includes such areas as deterministic chaos and strange attractors, iterated mappings, nonlinear dynamics, Cayley trees, cellular automata, random fractals and related topics.", acknowledgement = ack-nhfb, shorttableofcontents = "Introduction \\ Similarity and dissimilarity \\ Self-similarity --- discrete, continuous, strict, and otherwise \\ Power laws: endless sources of self-similarity \\ Noises: white, pink, brown, and black \\ Brownian motion, gambling losses, and intergalactic voids: random fractals par excellence \\ Cantor sets: self-similarity and arithmetic dust \\ Fractals in higher dimensions and a digital sundial \\ Multifractals: intimately intertwined fractals \\ Some practical fractals and their measurement \\ Iteration, strange mappings, and a billion digits for $\pi$ \\ A self-similar sequence, the logistic parabola, and symbolic dynamics \\ A forbidden symmetry, Fibonacci's rabbits, and a new state of matter \\ Periodic and quasiperiodic structures in space --- the route to spatial chaos \\ Percolation: from forest fires to epidemics \\ Phase transitions and renormalization \\ Cellular automata", tableofcontents = "Preface / xiii \\ 1: Introduction / 1 \\ Einstein, Pythagoras, and Simple Similarity / 3 \\ A Self-Similar Array of Self-Preserving Queens / 4 \\ A Self-Similar Snowflake / 7 \\ A New Dimension for Fractals / 9 \\ A Self-Similar Tiling and a ``Non-Euclidean'' Paradox / 13 \\ At the Gates of Cantor's Paradise / 15 \\ The Sierpinski Gasket / 17 \\ Sir Pinski's Game and Deterministic Chaos / 20 \\ Three Bodies Cause Chaos / 25 \\ Strange Attractors, Their Basins, and a Chaos Game 2 / 7 \\ Percolating Random Fractals / 30 \\ Power Laws: from Alvarez to Zipf / 33 \\ Newton's Iteration and How to Abolish Two-Nation Boundaries / 38 \\ Could Minkowski Hear the Shape of a Drum? / 40 \\ Discrete Self-Similarity: Creases and Center Folds / 45 \\ Golden and Silver Means and Hyperbolic Chaos / 49 \\ Winning at Fibonacci Nim / 53 \\ Self-Similar Sequences from Square Lattices / 55 \\ John Horton Conway's ``Death Bet'' / 57 \\ 2: Similarity and Dissimilarity / 61 \\ More Than One Scale / 61 \\ To Scale or Not to Scale: A Bit of Biology and Astrophysics / 63 \\ Similarity in Physics: Some Astounding Consequences / 66 \\ Similarity in Concert Halls, Microwaves, and Hydrodynamics / 68 \\ Scaling in Psychology / 70 \\ Acousticians, Alchemy, and Concert Halls / 72 \\ Preference and Dissimilarity: Concert Halls Revisited / 74 \\ 3: Self-Similarity --- Discrete, Continuous, Strict, and Otherwise / 81 \\ The Logarithmic Spiral, Cutting Knives, and Wideband Antennas / 89 \\ Some Simple Cases of Self-Similarity / 93 \\ Weierstrass Functions and a Musical Paradox / 96 \\ More Self-Similarity in Music: The Tempered Scales of Bach / 99 \\ The Excellent Relations between the Primes 3, 5, and 7 / 102 \\ 4: Power Laws: Endless Sources of Self-Similarity / 103 \\ The Sizes of Cities and Meteorites / 103 \\ A Fifth Force of Attraction / 105 \\ Free of Natural Scales / 107 \\ Bach Composing on All Scales / 107 \\ Birkhoff's Aesthetic Theory / 109 \\ Heisenberg's Hyperbolic Uncertainty Principle / 112 \\ Fractional Exponents / 115 \\ The Peculiar Distribution of the First Digit / 116 \\ The Diameter Exponents of Trees, Rivers, Arteries, and Lungs / 117 \\ 5: Noises: White, Pink, Brown, and Black / 121 \\ Pink Noise / 122 \\ Self-Similar Trends on the Stock Market / 126 \\ Black Noises and Nile Floods / 129 \\ Warning: World Warming / 131 \\ Fractional Integration: A Modem Tool / 131 \\ Brownian Mountains / 133 \\ Radon Transform and Computer Tomography / 134 \\ Fresh and Tired Mountains / 135 \\ 6: Brownian Motion, Gambling Losses, and Intergalactic Voids: Random Fractals Par Excellence / 139 \\ The Brownian Beast Tamed / 140 \\ Brownian Motion as a Fractal / 141 \\ How Many Molecules? / 143 \\ The Spectrum of Brownian Motion / 144 \\ The Gambler's Ruin, Random Walks, and Information Theory / 145 \\ Counterintuition Runs Rampant in Random Runs / 146 \\ More Food for Fair Thought / 147 \\ The St. Petersburg Paradox / 148 \\ Shannon's Outguessing Machine / 149 \\ The Classical Mechanics of Roulette and Shannon's Channel Capacity / 150 \\ The Clustering of Poverty and Galaxies / 152 \\ Levy Flights through the Universe / 155 \\ Paradoxes from Probabilistic Power Laws / 155 \\ Invariant Distributions: Gauss, Cauchy, and Beyond / 157 \\ 7: Cantor Sets: Self-Similarity and Arithmetic Dust / 161 \\ A Comer of Cantor's Paradise / 161 \\ Cantor Sets as Invariant Sets / 165 \\ Symbolic Dynamics and Deterministic Chaos / 166 \\ Devil's Staircases and a Pinball Machine / 167 \\ Mode Locking in Swings and Clocks / 171 \\ The Frustrated Manhattan Pedestrian / 172 \\ Arnold Tongues 17 / 4 \\ 8: Fractals in Higher Dimensions and a Digital Sundial / 177 \\ Cartesian Products of Cantor Sets / 177 \\ A Leaky Gasket, Soft Sponges, and Swiss Cheeses / 178 \\ A Cantor-Set Sundial / 181 \\ Fat Fractals / 183 \\ 9: Multifractals: Intimately Intertwined Fractals / 187 \\ The Distributions of People and Ore / 187 \\ Self-Affine Fractals without Holes / 190 \\ The Multifractal Spectrum: Turbulence and Diffusion-Limited Aggregation / 193 \\ Viscous Fingering / 199 \\ Multifractals on Fractals / 200 \\ Fractal Dimensions from Generalized Entropies / 203 \\ The Relation between the Multifractal Spectrum $f(\alpha)$ and the Mass Exponents $(q)$ / 205 \\ Strange Attractors as Multifractals / 206 \\ A Greedy Algorithm for Unfavorable Odds / 207 \\ 10: Some Practical Fractals and Their Measurement / 211 \\ Dimensions from Box Counting / 213 \\ The Mass Dimension / 215 \\ The Correlation Dimension / 220 \\ Infinitely Many Dimensions / 220 \\ The Determination of Fractal Dimensions from Time Series / 223 \\ Abstract Concrete / 224 \\ Fractal Interfaces Enforce Fractional Frequency Exponents / 225 \\ The Fractal Dimensions of Fracture Surfaces / 230 \\ The Fractal Shapes of Clouds and Rain Areas / 231 \\ Cluster Agglomeration / 232 \\ Diffraction from Fractals / 233 \\ 11: Iteration, Strange Mappings, and a Billion Digits for Pi / 237 \\ Looking for Zeros and Encountering Chaos / 239 \\ The Strange Sets of Julia / 243 \\ A Multifractal Julia Set / 245 \\ The Beauty of Broken Linear Relationships / 249 \\ The Baker's Transformation and Digital Musical Chairs / 251 \\ Arnol'd's Cat Map / 253 \\ A Billion Digits for $\pi$ / 257 \\ Bushes and Flowers from Iterations / 259 \\ 12: A Self-Similar Sequence, the Logistic Parabola, and Symbolic Dynamics / 263 \\ Self-Similarity from the Integers / 264 \\ The Logistic Parabola and Period Doubling / 268 \\ Self-Similarity in the Logistic Parabola / 272 \\ The Scaling of the Growth Parameter / 274 \\ Self-Similar Symbolic Dynamics / 277 \\ Periodic Windows Embedded in Chaos / 279 \\ The Parenting of New Orbits / 282 \\ The Calculation of the Growth Parameters for Different Orbits / 286 \\ Tangent Bifurcations, Intermittency, and I/f Noise / 289 \\ A Case of Complete Chaos / 291 \\ The Mandelbrot Set / 295 \\ The Julia Sets of the Complex Quadratic Map / 297 \\ 13: A Forbidden Symmetry, Fibonacci's Rabbits, and a New State of Matter / 301 \\ The Forbidden Fivefold Symmetry / 301 \\ Long-Range Order from Neighborly Interactions / 304 \\ Generation of the Rabbit Sequence from the Fibonacci Number System / 307 \\ The Self-Similar Spectrum of the Rabbit Sequence / 308 \\ Self-Similarity in the Rabbit Sequence / 310 \\ A One-Dimensional Quasiperiodic Lattice / 310 \\ Self-Similarity from Projections / 311 \\ More Forbidden Symmetries / 315 \\ 14: Periodic and Quasiperiodic Structures in Space --- The route to Spatial Chaos / 319 \\ Periodicity and Quasiperiodicity in Space / 320 \\ The Devil's Staircase for Ising Spins / 321 \\ Quasiperiodic Spatial Distributions / 322 \\ Beatty Sequence Spins / 325 \\ The Scaling Laws for Quasiperiodic Spins / 329 \\ Self-Similar Winding Numbers / 330 \\ Circle Maps and Arnold Tongues / 331 \\ Mediants, Farey Sequences, and the Farey Tree / 334 \\ The Golden-Mean Route to Chaos / 340 \\ 15: Percolation: From Forest Fires to Epidemics / 345 \\ Critical Conflagration on a Square Lattice / 346 \\ Universality / 350 \\ The Critical Density / 353 \\ The Fractal Perimeters of Percolation / 353 \\ Finite-Size Scaling / 354 \\ 16: Phase Transitions and Renormalization / 357 \\ A First-Order Markov Process / 357 \\ Self-Similar and Non-Self-Similar Markov Processes / 358 \\ The Scaling of Markov Output's / 360 \\ Renormalization and Hierarchical Lattices / 362 \\ The Percolation Threshold of the Bethe Lattice / 363 \\ A Simple Renormalization / 367 \\ 17: Cellular Automata / 371 \\ The Game of Life / 373 \\ Cellular Growth and Decay / 375 \\ Biological Pattern Formation / 382 \\ Self-Similarity from a Cellular Automaton / 383 \\ A Catalytic Converter as a Cellular Automaton / 386 \\ Pascal's Triangle Modulo $N$ / 387 \\ Bak's Self-Organized Critical Sandpiles / 389 \\ Appendix / 391 \\ References / 395 \\ Author Index / 411 \\ Subject Index / 417", } @Article{Trubnikov:1991:SDZ, author = "B. A. Trubnikov and I. A. Rumynski{\u\i}", title = "A simple derivation of the {Zipf--Krylov} law for words and the possibility of its ``evolution'' interpretation", journal = j-DOKL-AKAD-NAUK, volume = "321", number = "2", pages = "270--275", year = "1991", CODEN = "DANKAS", ISSN = "0002-3264", ISSN-L = "0002-3264", MRclass = "00A69 (92K20)", MRnumber = "1153554", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Doklady Akademii Nauk SSSR", journal-URL = "http://istina.msu.ru/journals/366838/", } @Article{Basu:1992:RHD, author = "Aparna Basu", title = "Research: Hierarchical distributions and {Bradford's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "43", number = "7", pages = "494--500", month = aug, year = "1992", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199208)43:7<494::AID-ASI4>3.0.CO%3B2-F", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:51 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @TechReport{Busta:1992:TRN, author = "B. Busta and R. Sundheim", title = "Tax return numbers tend to obey {Benford}'s law", type = "Working Paper", number = "W93-106-94", institution = "Center for Business Research, St. Cloud State University", address = "St. Cloud, MN, USA", year = "1992", bibdate = "Sat Feb 16 17:29:47 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Dehaene:1992:CLR, author = "Stanislas Dehaene and Jacques Mehler", title = "Cross-linguistic regularities in the frequency of number words", journal = j-COGNITION, volume = "43", number = "1", pages = "1--29", year = "1992", CODEN = "CGTNAU", DOI = "https://doi.org/10.1016/0010-0277(92)90030-L", ISSN = "0010-0277 (print), 1873-7838 (electronic)", ISSN-L = "0010-0277", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/001002779290030L", abstract = "We examine the frequency of numerals and ordinals in seven different languages and/or cultures. Many cross-cultural and cross-linguistic patterns are identified. The most striking is a decrease of frequency with numerical magnitude, with local increases for reference numerals such as 10, 12, 15, 20, 50 or 100. Four explanations are considered for this effect: sampling artifacts, notational regularities, environmental biases and psychological limitations on number representations. The psychological explanation, which appeals to a Fechnerian encoding of numerical magnitudes and to the existence of numerical points of reference, accounts for most of the data. Our finding also has practical importance since it reveals the frequent confound of two experimental variables: numerical magnitude and numeral frequency.", acknowledgement = ack-nhfb, fjournal = "Cognition", keywords = "Benford's Law", } @Article{Goto:1992:SEB, author = "Kazuo Got{\^o}", title = "Some examples of {Benford} sequences", journal = "Math. J. Okayama Univ.", volume = "34", pages = "225--232 (1994)", year = "1992", CODEN = "MJOKAP", ISSN = "0030-1566", ISSN-L = "0030-1566", MRclass = "11K36", MRnumber = "1272624 (95c:11097)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Journal of Okayama University", } @Article{Gunther:1992:PCZ, author = "R. G{\"u}nther and B. Schapiro and P. Wagner", title = "Physical complexity and {Zipf}'s law", journal = j-INT-J-THEOR-PHYS, volume = "31", number = "3", pages = "525--543", year = "1992", CODEN = "IJTPBM", DOI = "https://doi.org/10.1007/BF00740003", ISSN = "0020-7748 (print), 1572-9575 (electronic)", ISSN-L = "0020-7748", MRclass = "58F40", MRnumber = "1154668 (92k:58252)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Theoretical Physics", journal-URL = "http://link.springer.com/journal/10773", } @Article{Jech:1992:LDL, author = "Thomas Jech", title = "The logarithmic distribution of leading digits and finitely additive measures", journal = j-DISCRETE-MATH, volume = "108", number = "1--3", pages = "53--57", month = "????", year = "1992", CODEN = "DSMHA4", ISSN = "0012-365X (print), 1872-681X (electronic)", ISSN-L = "0012-365X", bibdate = "Fri Nov 18 12:12:57 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Discrete Mathematics", } @Article{Lagarias:1992:PTS, author = "J. C. Lagarias and A. Weiss", title = "The $ 3 x + 1 $ Problem: Two Stochastic Models", journal = j-ANN-APPL-PROBAB, volume = "2", number = "1", pages = "229--261", month = feb, year = "1992", CODEN = "????", ISSN = "1050-5164", ISSN-L = "1050-5164", bibdate = "Fri Jun 6 11:35:34 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/annapplprobab.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://projecteuclid.org/euclid.aoap/1177005779", acknowledgement = ack-nhfb, fjournal = "Annals of Applied Probability", journal-URL = "http://projecteuclid.org/all/euclid.aoap/; http://www.jstor.org/journals/10505164.html", } @Article{Li:1992:RTE, author = "W. Li", title = "Random texts exhibit {Zipf}'s-law-like word frequency distribution", journal = j-IEEE-TRANS-INF-THEORY, volume = "38", number = "6", pages = "1842--1845", month = "????", year = "1992", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/18.165464", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", bibdate = "Fri Nov 18 12:32:19 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Information Theory", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18", } @Article{Longfellow:1992:LER, author = "Charles A. {Longfellow, Jr.}", title = "Letters to the {Editor}: Regarding {``Lotka's Law and the Kolmogorov--Smirnov Test: An Error in Calculation''}", journal = j-J-AM-SOC-INF-SCI, volume = "43", number = "7", pages = "518--518", month = aug, year = "1992", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199208)43:7<518::AID-ASI7>3.0.CO%3B2-#", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:51 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", note = "See \cite{Loughner:1992:BCL}.", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @Article{Loughner:1992:BCL, author = "William Loughner", title = "Brief Communications: {Lotka's Law} and the {Kolmogorov--Smirnov} test: an error in calculation", journal = j-J-AM-SOC-INF-SCI, volume = "43", number = "2", pages = "149--150", month = mar, year = "1992", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199203)43:2<149::AID-ASI4>3.0.CO%3B2-Y", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:49 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", note = "See comment \cite{Longfellow:1992:LER}.", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @PhdThesis{Nigrini:1992:DIE, author = "Mark John Nigrini", title = "The detection of income tax evasion through an analysis of digital distributions", type = "{Ph.D.} Dissertation", school = "Department of Accounting\slash Business Law, University of Cincinnati", address = "Cincinnati, OH, USA", pages = "xvi + 309", day = "28", month = aug, year = "1992", bibdate = "Sat Nov 12 15:47:41 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://search.proquest.com/docview/304052348", acknowledgement = ack-nhfb, advisor = "Wallace R. Wood", remark = "There are more than 230 mentions of Frank Benford, and 8 of Simon Newcomb, but no mention of Pareto or Zipf.", } @Article{Rousseau:1992:RBR, author = "Ronald Rousseau", title = "Research: Breakdown of the robustness property of {Lotka's Law}: the case of adjusted counts for multiauthorship attribution", journal = j-J-AM-SOC-INF-SCI, volume = "43", number = "10", pages = "645--647", month = dec, year = "1992", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199212)43:10<645::AID-ASI1>3.0.CO%3B2-X", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:48 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @Article{Sanders:1992:PPU, author = "Robert Sanders", title = "The {Pareto Principle}: its Use and Abuse", journal = j-J-PROD-BRAND-MANAG, volume = "1", number = "2", pages = "37--40", month = "????", year = "1992", CODEN = "????", DOI = "https://doi.org/10.1108/10610429210036762", ISSN = "1061-0421", ISSN-L = "1061-0421", bibdate = "Fri Nov 25 06:50:32 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Product \& Brand Management", keywords = "80/20 rule", } @Article{Too:1992:UDM, author = "Yeu-Hua Too", title = "On the uniform distribution modulo one of some log-like sequences", journal = "Proceedings of the Japan Academy, Series A. Mathematical Sciences", volume = "68", number = "9", pages = "269--272", month = "????", year = "1992", DOI = "https://doi.org/10.3792/pjaa.68.269", ISSN = "0021-4280", ISSN-L = "0021-4280", bibdate = "Mon Feb 18 06:55:06 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://projecteuclid.org/euclid.pja/1195511634", acknowledgement = ack-nhfb, } @Article{Trachtenberg:1992:WFR, author = "M. Trachtenberg", title = "Why failure rates observe {Zipf}'s law in operational software", journal = j-IEEE-TRANS-RELIAB, volume = "41", number = "3", pages = "386--389", year = "1992", CODEN = "IEERAJ", DOI = "https://doi.org/10.1109/24.159803", ISSN = "0018-9529 (print), 1558-1721 (electronic)", ISSN-L = "0018-9529", bibdate = "Sat Nov 12 10:50:55 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0825.68222", acknowledgement = ack-nhfb, classmath = "68M15 (Reliability and testing of computer systems) 68N01 (General)", fjournal = "IEEE Transactions on Reliability", } @Article{Anonymous:1993:WFR, author = "Anonymous", title = "Why failure rates observe {Zipf}'s law in operational software: {M. Trachtenberg. IEEE Transactions on Reliability, {\bf 41}(3), 386 (1992)}", journal = j-MICROELECT-RELIABILITY, volume = "33", number = "13", pages = "2062--2062", month = oct, year = "1993", CODEN = "MCRLAS", DOI = "https://doi.org/10.1016/0026-2714(93)90406-O", ISSN = "0026-2714 (print), 1872-941X (electronic)", ISSN-L = "0026-2714", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/002627149390406O", acknowledgement = ack-nhfb, fjournal = "Microelectronics and Reliability", keywords = "Zipf's Law", } @Article{Buck:1993:IBF, author = "B. Buck and A. C. Merchant and S. M. Perez", title = "An illustration of {Benford}'s first digit law using alpha decay half lives", journal = j-EUR-J-PHYS, volume = "14", number = "2", pages = "59--63", month = "????", year = "1993", CODEN = "EJPHD4", DOI = "https://doi.org/10.1088/0143-0807/14/2/003", ISSN = "0143-0807 (print), 1361-6404 (electronic)", ISSN-L = "0143-0807", bibdate = "Mon Nov 14 17:32:53 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://adsabs.harvard.edu/abs/1993EJPh...14...59B", acknowledgement = ack-nhfb, fjournal = "European Journal of Physics", journal-URL = "http://iopscience.iop.org/0143-0807/", } @Article{Christian:1993:NES, author = "C. W. Christian and S. Gupta", title = "New Evidence on ``Secondary Evasion''", journal = "Journal of the {American Taxation Association}", volume = "15", number = "??", pages = "72--93", month = "????", year = "1993", ISSN = "0198-9073 (print), 1558-8017 (electronic)", ISSN-L = "0198-9073", bibdate = "Thu Oct 13 06:50:53 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "J. Amer. Tax. Assoc.", fjournal = "Journal of the {American Taxation Association}", journal-URL = "http://aaajournals.org/loi/atax", remark = "No journal issues online before 1999.", } @Article{Deakin:1993:ADB, author = "Michael A. B. Deakin", title = "Another derivation of {Benford}'s law", journal = j-AUSTRALIAN-MATH-SOC-GAZ, volume = "20", number = "5", pages = "162--163", year = "1993", CODEN = "????", ISSN = "0311-0729 (print), 1326-2297 (electronic)", ISSN-L = "0311-0729", MRclass = "11K36", MRnumber = "1268513 (95j:11072)", MRreviewer = "Jean-Loup Mauclaire", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Australian Mathematical Society Gazette", journal-URL = "http://www.austms.org.au/gazette", } @Article{Nigrini:1993:CBL, author = "Mark Nigrini", title = "Can {Benford's Law} be used in Forensic Accounting?", journal = "The Balance Sheet", volume = "??", number = "??", pages = "7--8", month = jun, year = "1993", CODEN = "????", ISSN = "????", bibdate = "Mon Nov 21 22:03:06 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Oluic-Vukovic:1993:LEW, author = "Vesna Olui{\'c}-Vukovi{\'c}", title = "Letters to the {Editor}: Why has {Bradford's Law} been an elusive phenomenon so far?", journal = j-J-AM-SOC-INF-SCI, volume = "44", number = "3", pages = "182--183", month = apr, year = "1993", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199304)44:3<182::AID-ASI9>3.0.CO%3B2-S", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:54 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @TechReport{Stewart:1993:LAN, author = "Ian Stewart", title = "The law of anomalous numbers", type = "Working paper", institution = "Mathematics Institute, University of Warwick", address = "Warwick, UK", year = "1993", bibdate = "Fri Mar 22 09:15:03 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Published in Spektrum der Wissenschaft (April 1994)", acknowledgement = ack-mjn # "\slash " # ack-nhfb, keywords = "Benford's Law", } @TechReport{Sundheim:1993:FNT, author = "R. Sundheim and B. Busta", title = "{Fibonacci} numbers tend to obey {Benford's Law}: an extension of {Wlodarski} and {Sentance}", type = "Working Paper", institution = "St. Cloud State University", address = "St. Cloud, MN, USA", month = "????", year = "1993", bibdate = "Mon Feb 18 06:37:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Sury:1993:FPD, author = "B. Sury", title = "Fractional parts of $ \log p $ and a digit function", journal = j-EXPO-MATH, volume = "11", number = "4", pages = "381--384", month = "????", year = "1993", DOI = "????", ISSN = "0723-0869 (print), 1878-0792 (electronic)", ISSN-L = "0723-0869", bibdate = "Mon Feb 18 06:39:02 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "????", ZMnumber = "0787.11003", acknowledgement = ack-nhfb, ajournal = "Expo. Math.", fjournal = "Expositiones Mathematicae", journal-URL = "http://www.sciencedirect.com/science/journal/07230869", keywords = "decimal expansion; sum of digits", ZMclass = "11A63 (Radix representation; digital problems)", ZMreviewer = "G. Larcher (Salzburg)", } @Article{Wolf:1993:PHJ, author = "Joel L. Wolf and Philip S. Yu and John Turek and Daniel M. Dias", title = "A parallel hash join algorithm for managing data skew", journal = j-IEEE-TRANS-PAR-DIST-SYS, volume = "4", number = "12", pages = "1355--1371", month = dec, year = "1993", CODEN = "ITDSEO", DOI = "https://doi.org/10.1109/71.250117", ISSN = "1045-9219 (print), 1558-2183 (electronic)", ISSN-L = "1045-9219", bibdate = "Fri Apr 11 15:20:39 MDT 1997", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/ieeetranspardistsys.bib", acknowledgement = ack-nhfb, affiliation = "IBM T. J. Watson Research Cent", affiliationaddress = "Yorktown Heights, NY, USA", classification = "721.1; 722.4; 723.1; 723.2; 723.3; 921.4; C4240P (Parallel programming and algorithm theory); C4250 (Database theory); C6160D (Relational DBMS)", corpsource = "IBM Thomas J. Watson Res. Center, Yorktown Heights, NY, USA", fjournal = "IEEE Transactions on Parallel and Distributed Systems", journal-URL = "http://www.computer.org/tpds/archives.htm", journalabr = "IEEE Trans Parallel Distrib Syst", keywords = "Algorithms; combinatorial; Combinatorial mathematics; Combinatorial optimization; Complex queries; complex queries; Computational complexity; Data handling; Data skew; data skew; database theory; hash; Hash joins; hashing; heuristic optimization; hierarchical; join column; joins; load balancing; Load balancing; optimization; parallel algorithms; parallel hash join algorithm; Parallel processing systems; Program processors; Query languages; query processing; Relational database systems; relational databases; resource allocation; scheduling; Zipf-like distribution", treatment = "T Theoretical or Mathematical", } @Article{Wolf:1993:PSM, author = "Joel L. Wolf and Daniel M. Dias and Philip S. Yu", title = "Parallel sort merge join algorithm for managing data skew", journal = j-IEEE-TRANS-PAR-DIST-SYS, volume = "4", number = "1", pages = "70--86", month = jan, year = "1993", CODEN = "ITDSEO", DOI = "https://doi.org/10.1109/71.205654", ISSN = "1045-9219 (print), 1558-2183 (electronic)", ISSN-L = "1045-9219", bibdate = "Fri Apr 11 15:20:39 MDT 1997", bibsource = "Compendex database; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/ieeetranspardistsys.bib", acknowledgement = ack-nhfb, affiliation = "IBM", affiliationaddress = "Yorktown Heights, NY, USA", classification = "723.1; 723.3; 921.5; C4240P (Parallel programming and algorithm theory); C4250 (Database theory); C6130 (Data handling techniques); C6160B (Distributed DBMS); C6160D (Relational DBMS)", corpsource = "IBM T. J. Watson Res. Center. Yorktown Heights, NY, USA", fjournal = "IEEE Transactions on Parallel and Distributed Systems", journal-URL = "http://www.computer.org/tpds/archives.htm", journalabr = "IEEE Trans Parallel Distrib Syst", keywords = "Algorithms; Computer systems programming; Data skew; data skew management; distributed databases; Divide and conquer approach; divide-and-conquer; join phases; load balancing; merging; multiple processors; Multiprocessing programs; Optimization; parallel; Parallel algorithms; parallel algorithms; Parallel sort merge join algorithm; parallelizable optimization algorithm; phase; Query processing; relational algebra; Relational database systems; relational databases; scheduling; Scheduling; sort merge join algorithm; sort phase; sorting; Sorting; transfer phase; Zipf-like distribution", treatment = "T Theoretical or Mathematical", } @Book{Arthur:1994:IRP, author = "W. Brian Arthur", title = "Increasing returns and path dependence in the economy", publisher = pub-U-MICHIGAN, address = pub-U-MICHIGAN:adr, pages = "xx + 201", year = "1994", ISBN = "0-472-09496-3, 0-472-06496-7 (paperback)", ISBN-13 = "978-0-472-09496-7, 978-0-472-06496-0 (paperback)", LCCN = "HD69.S5 A77 1994", bibdate = "Wed Feb 27 07:08:40 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", note = "With a foreword by Kenneth J. Arrow.", series = "Economics, cognition, and society", acknowledgement = ack-nhfb, keywords = "Zipf's Law", subject = "Economies of scale; Mathematical models; Economic development; Economics, Mathematical", } @Article{Basu:1994:LEE, author = "Aparna Basu", title = "Letters to the {Editor}: The elusive phenomenon of {Bradford's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "45", number = "1", pages = "59--60", month = jan, year = "1994", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199401)45:1<59::AID-ASI8>3.0.CO%3B2-F", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:03:58 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @Article{Bookstein:1994:TMD, author = "A. Bookstein", title = "Towards a multi-disciplinary {Bradford Law}", journal = j-SCIENTOMETRICS, volume = "30", number = "1", pages = "353--361", month = may, year = "1994", CODEN = "SCNTDX", DOI = "https://doi.org/10.1007/BF02017233", ISSN = "0138-9130 (print), 1588-2861 (electronic)", ISSN-L = "0138-9130", bibdate = "Thu Sep 3 07:35:17 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/scientometrics1990.bib", URL = "http://link.springer.com/article/10.1007/BF02017233", acknowledgement = ack-nhfb, fjournal = "Scientometrics", journal-URL = "http://link.springer.com/journal/11192", } @Article{Boyle:1994:AFS, author = "Jeff Boyle", title = "An application of {Fourier} series to the most significant digit problem", journal = j-AMER-MATH-MONTHLY, volume = "101", number = "9", pages = "879--886", month = nov, year = "1994", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "60F99", MRnumber = "1 300 493", bibdate = "Wed Dec 3 17:17:33 MST 1997", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Article{Egghe:1994:RSF, author = "L. Egghe", title = "Research: Special features of the author--publication relationship and a new explanation of {Lotka's Law} based on convolution theory", journal = j-J-AM-SOC-INF-SCI, volume = "45", number = "6", pages = "422--427", month = jul, year = "1994", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199407)45:6<422::AID-ASI8>3.0.CO%3B2-C", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:04:01 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @Article{Fuchs:1994:PDP, author = "Aim{\'e} Fuchs and Giorgio Letta", title = "Le probl{\`e}me du premier chiffre d{\'e}cimal pour les nombres premiers. ({French}) [The problem for the first decimal digit of prime numbers]", journal = "Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Serie V. Memorie di Matem{\`a}tica", volume = "18", number = "1", pages = "81--87", month = "????", year = "1994", bibdate = "Sat Feb 16 17:40:01 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.combinatorics.org/ojs/index.php/eljc/article/download/v3i2r25/pdf", ZMnumber = "0838.11004", acknowledgement = ack-nhfb, ajournal = "Rend. Accad. Naz. Sci. XL, V. Ser., Mem. Mat.", keywords = "asymptotic densities; logarithmic density; primes; relative densities", language = "French", xxjournal = "Rend. Acad. Naz. Sci. Detta", xxvolume = "XL", ZMclass = "11B05 (Topology etc. of sets of numbers); 11K16 (Normal numbers, etc.)", ZMreviewer = "I. Z. Ruzsa (Budapest)", } @Article{Ley:1994:TPB, author = "Eduadro Ley and Hal R. Varian", title = "Are there psychological barriers in the {Dow--Jones} index?", journal = j-APPL-FINANC-ECON, volume = "4", number = "3", pages = "217--224", month = "????", year = "1994", CODEN = "????", DOI = "https://doi.org/10.1080/758526902", ISSN = "0960-3107 (print), 1466-4305 (electronic)", ISSN-L = "0960-3107", bibdate = "Sat Feb 16 18:29:52 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/758526902", acknowledgement = ack-nhfb, fjournal = "Applied Financial Economics", } @Article{Mantegna:1994:LFN, author = "R. N. Mantegna and S. V. Buldyrev and A. L. Goldberger and S. Havlin and C. K. Peng and M. Simons and H. E. Stanley", title = "Linguistic Features of Noncoding {DNA} Sequences", journal = j-PHYS-REV-LET, volume = "73", number = "??", pages = "3169--??", day = "5", month = dec, year = "1994", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.73.3169", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See comment \cite{Voss:1996:CLF} and reply \cite{Mantegna:1996:MAR}.", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.73.3169", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "Zipf's Law", } @Article{Nigrini:1994:UDF, author = "Mark J. Nigrini", title = "Using digital frequencies to detect fraud", journal = "The White Paper", volume = "??", number = "2", pages = "3--6", month = apr # "\slash " # may, year = "1994", bibdate = "Sat Feb 16 18:53:08 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, remark = "The White Paper is the precursor to Fraud Magazine; I cannot find it in the Library of Congress catalog.", } @Article{Pedrotti:1994:ALU, author = "A. Pedrotti", title = "Analysis of a list-update strategy", journal = j-INFO-PROC-LETT, volume = "52", number = "3", pages = "115--121", day = "11", month = nov, year = "1994", CODEN = "IFPLAT", ISSN = "0020-0190 (print), 1872-6119 (electronic)", ISSN-L = "0020-0190", bibdate = "Wed Nov 11 12:16:26 MST 1998", bibsource = "Compendex database; http://www.elsevier.com:80/inca/publications/store/5/0/5/6/1/2/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, affiliation = "Scuola Normale Superiore", affiliationaddress = "Pisa, Italy", classification = "721.1; 723.2; 921.6; 922.1; 922.2; C4240C (Computational complexity); C6130 (Data handling techniques)", corpsource = "Scuola Normale Superiore, Pisa, Italy", fjournal = "Information Processing Letters", journal-URL = "http://www.sciencedirect.com/science/journal/00200190", journalabr = "Inf Process Lett", keywords = "Algorithms; analysis of algorithms; average-case complexity; BIT; competitive algorithms; Computational complexity; computational complexity; convergence; Convergence of numerical methods; Data handling; Function evaluation; Heuristic methods; list processing; list update; Markov chains; Move to front algorithm; move-to-front; Probability; Random number generation; searching; Theorem proving; transpose; Zipf's law", treatment = "T Theoretical or Mathematical", } @Article{Sheppard:1994:TNA, author = "Lee A. Sheppard", title = "{Tax Notes} audits the {Clintons}", journal = "Tax Notes Today", volume = "??", number = "??", pages = "18--23", day = "4", month = apr, year = "1994", bibdate = "Fri Mar 22 09:13:48 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.bankler.com/docs/Tax_Notes_Audits_the_Clintons.pdf", acknowledgement = ack-mjn # "\slash " # ack-nhfb, keywords = "Benford's Law", } @Article{Tritch:1994:MAC, author = "Teresa Tritch and Mary L. Sprouse", title = "{Money} audits the {Clintons}", journal = "Money Magazine", volume = "??", number = "??", pages = "84--98", day = "1", month = apr, year = "1994", bibdate = "Fri Mar 22 09:16:28 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://money.cnn.com/magazines/moneymag/moneymag_archive/1994/04/01/88775/index.htm", acknowledgement = ack-mjn # "\slash " # ack-nhfb, keywords = "Benford's Law", } @Article{Berton:1995:HGT, author = "L. Berton", title = "He's Got their Number. {Scholar} Uses Math to Foil Financial Fraud", journal = j-WALL-ST-J, volume = "??", number = "??", pages = "??--??", day = "10", month = jul, year = "1995", CODEN = "WSJOAF", ISSN = "0099-9660", ISSN-L = "0099-9660", bibdate = "Sat Dec 10 11:17:37 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Wall Street Journal", } @Book{Bologna:1995:FAF, author = "Jack Bologna and Robert J. Lindquist", title = "Fraud auditing and forensic accounting: new tools and techniques", publisher = pub-WILEY, address = pub-WILEY:adr, edition = "Second", pages = "xv + 249", year = "1995", ISBN = "0-471-10646-1 (hardcover)", ISBN-13 = "978-0-471-10646-3 (hardcover)", LCCN = "HV8079.W47 B65 1995", bibdate = "Tue Nov 22 10:31:53 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/bios/wiley042/95005510.html; http://www.loc.gov/catdir/description/wiley033/95005510.html; http://www.loc.gov/catdir/toc/onix04/95005510.html", acknowledgement = ack-nhfb, remark = "This book does not discuss Benford's Law, but may be helpful background for application of that law to the detection of fraud.", subject = "white collar crime investigation; United States; forensic accounting; fraud investigation", } @Article{Chen:1995:LED, author = "Ye-Sho Chen", title = "Letters to the {Editor}: On the dynamic behavior of {Bradford's Law} --- response", journal = j-J-AM-SOC-INF-SCI, volume = "46", number = "10", pages = "799--800", month = dec, year = "1995", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199512)46:10<799::AID-ASI19>3.0.CO%3B2-P", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:04:04 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @Article{Chen:1995:RDB, author = "Ye-Sho Chen and P. Pete Chong and Morgan Y. Tong", title = "Research: Dynamic behavior of {Bradford's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "46", number = "5", pages = "370--383", month = jun, year = "1995", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199506)46:5<370::AID-ASI7>3.0.CO%3B2-J", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:04:06 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @Article{Czirok:1995:CBS, author = "Andras Czir{\'o}k and Rosario N. Mantegna and Shlomo Havlin and H. Eugene Stanley", title = "Correlations in binary sequences and a generalized {Zipf} analysis", journal = j-PHYS-REV-E, volume = "52", number = "??", pages = "446--??", day = "1", month = jul, year = "1995", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.52.446", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.52.446", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Article{Filipponi:1995:SPA, author = "Piero Filipponi and Renato Menicocci", title = "Some Probabilistic Aspects of the Terminal Digits of {Fibonacci} Numbers", journal = j-FIB-QUART, volume = "33", number = "4", pages = "325--331", month = aug, year = "1995", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:02:19 MDT 2011", bibsource = "http://www.fq.math.ca/33-4.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/33-4/filipponi.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Havlin:1995:DBZ, author = "Shlomo Havlin", title = "The distance between {Zipf} plots", journal = j-PHYSICA-A, volume = "216", number = "1--2", pages = "148--150", day = "1", month = jun, year = "1995", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/0378-4371(95)00069-J", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/037843719500069J", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Article{Hill:1995:BII, author = "Theodore P. Hill", title = "Base-invariance implies {Benford's Law}", journal = j-PROC-AM-MATH-SOC, volume = "123", number = "3", pages = "887--895", month = mar, year = "1995", CODEN = "PAMYAR", DOI = "https://doi.org/10.2307/2160815", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "60A10 (28D05)", MRnumber = "1233974 (95d:60006)", MRreviewer = "Peter Schatte", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/2160815", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", } @Article{Hill:1995:SDP, author = "Theodore P. Hill", title = "The significant-digit phenomenon", journal = j-AMER-MATH-MONTHLY, volume = "102", number = "4", pages = "322--327", month = apr, year = "1995", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2974952", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "11K99 (60E99)", MRnumber = "96f:11101", MRreviewer = "Jean-Loup Mauclaire", bibdate = "Wed Dec 3 17:17:33 MST 1997", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/page/journal/amermathmont/about.html", } @Article{Hill:1995:SDS, author = "Theodore P. Hill", title = "A statistical derivation of the significant-digit law", journal = j-STAT-SCI, volume = "10", number = "4", pages = "354--363", year = "1995", CODEN = "STSCEP", ISSN = "0883-4237 (print), 2168-8745 (electronic)", ISSN-L = "0883-4237", MRclass = "60E99 (60A10 62E99)", MRnumber = "1421567 (98a:60021)", MRreviewer = "Peter Schatte", bibdate = "Sat Nov 12 10:09:36 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "2-P&origin=MSN; http://links.jstor.org/sici?sici=0883-4237(199511)10:4<354:ASDOTS>2.0.CO", ZMnumber = "Zbl 0955.60509", acknowledgement = ack-nhfb, fjournal = "Statistical Science. A Review Journal of the Institute of Mathematical Statistics", journal-URL = "http://projecteuclid.org/all/euclid.ss", } @Article{Kanter:1995:MPL, author = "I. Kanter and D. A. Kessler", title = "{Markov} Processes: Linguistics and {Zipf's Law}", journal = j-PHYS-REV-LET, volume = "74", number = "??", pages = "4559--??", day = "29", month = may, year = "1995", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.74.4559", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.74.4559", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "Zipf's Law", } @Article{Koenig:1995:BCL, author = "Michael Koenig and Toni Harrell", title = "Brief Communications: {Lotka's Law}, {Price}'s urn, and electronic publishing", journal = j-J-AM-SOC-INF-SCI, volume = "46", number = "5", pages = "386--388", month = jun, year = "1995", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199506)46:5<386::AID-ASI9>3.0.CO%3B2-V", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:04:06 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @Article{Makse:1995:MUG, author = "Hern{\'a}n A. Makse and Shlomo Havlin and H. Eugene Stanley", title = "Modelling urban growth patterns", journal = j-NATURE, volume = "377", number = "6550", pages = "608--612", day = "19", month = oct, year = "1995", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/377608a0", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Wed Feb 27 06:06:25 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.nature.com/nature/journal/v377/n6550/abs/377608a0.html", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", keywords = "Berlin (Germany) population; London (UK) population; Zipf's Law", } @Article{Mantegna:1995:SAC, author = "R. N. Mantegna and S. V. Buldyrev and A. L. Goldberger and S. Havlin and C.-K. Peng and M. Simons and H. E. Stanley", title = "Systematic analysis of coding and noncoding {DNA} sequences using methods of statistical linguistics", journal = j-PHYS-REV-E, volume = "52", number = "??", pages = "2939--??", day = "1", month = sep, year = "1995", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.52.2939", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.52.2939", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Article{Mosimann:1995:DFC, author = "James E. Mosimann and Claire V. Wiseman and Ruth E. Edelman", title = "Data Fabrication: Can People Generate Random Digits?", journal = j-ACCOUNT-RES, volume = "4", number = "1", pages = "31--55", month = "????", year = "1995", CODEN = "ARQAEZ", DOI = "https://doi.org/10.1080/08989629508573866", ISSN = "0898-9621 (print), 1545-5815 (electronic)", ISSN-L = "0898-9621", bibdate = "Sat Dec 10 11:56:38 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/08989629508573866", acknowledgement = ack-nhfb, fjournal = "Accountability in Research", journal-URL = "http://www.tandfonline.com/toc/gacr20", keywords = "Benford's Law", remark = "The abstract ends with: ``The difficulty that people have in creating random error digits supports the utility of examining such digits in investigations of scientific misconduct.''", } @Article{Oluic-Vukovic:1995:LED, author = "Vesna Olui{\'c}-Vukovi{\'c}", title = "Letters to the {Editor}: On the dynamic behavior of {Bradford's Law}", journal = j-J-AM-SOC-INF-SCI, volume = "46", number = "10", pages = "798--799", month = dec, year = "1995", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199512)46:10<798::AID-ASI18>3.0.CO%3B2-T", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:04:04 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "4 Jan 1999", } @Article{Stanley:1995:ZPS, author = "Michael H. R. Stanley and Sergey V. Buldyrev and Shlomo Havlin and Rosario N. Mantegna and Michael A. Salinger and H. Eugene Stanley", title = "{Zipf} plots and the size distribution of firms", journal = j-ECONOM-LETT, volume = "49", number = "4", pages = "453--457", day = "15", month = oct, year = "1995", CODEN = "ECLEDS", DOI = "https://doi.org/10.1016/0165-1765(95)00696-D", ISSN = "0165-1765 (print), 1873-7374 (electronic)", ISSN-L = "0165-1765", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/016517659500696D", acknowledgement = ack-nhfb, fjournal = "Economics Letters", keywords = "Zipf's Law", } @Article{Wagner-Dobler:1995:DLL, author = "Roland Wagner-D{\"o}bler and Jan Berg", title = "The Dependence of {Lotka's Law} on the Selection of Time Periods in the Development of Scientific Areas and Authors", journal = j-J-DOC, volume = "51", number = "1", pages = "28--43", month = "????", year = "1995", CODEN = "JDOCAS", DOI = "https://doi.org/10.1108/eb026941", ISSN = "0022-0418 (print), 1758-7379 (electronic)", ISSN-L = "0022-0418", bibdate = "Fri Nov 25 07:29:22 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Documentation", journal-URL = "http://www.emeraldinsight.com/loi/jd", } @Article{Zornig:1995:URZ, author = "Peter Z{\"o}rnig and Gabriel Altmann", title = "Unified representation of {Zipf} distributions", journal = j-COMPUT-STAT-DATA-ANAL, volume = "19", number = "4", pages = "461--473", day = "1", month = apr, year = "1995", CODEN = "CSDADW", DOI = "https://doi.org/10.1016/0167-9473(94)00009-8", ISSN = "0167-9473 (print), 1872-7352 (electronic)", ISSN-L = "0167-9473", MRclass = "62E10 (62F10)", MRnumber = "1333065 (96f:62029)", MRreviewer = "Friedrich Liese", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/0167947394000098", acknowledgement = ack-nhfb, fjournal = "Computational Statistics \& Data Analysis", journal-URL = "http://www.sciencedirect.com/science/journal/01679473", keywords = "Zipf's Law", } @Book{Bak:1996:HNW, author = "Per Bak", title = "How nature works: the science of self-organized criticality", publisher = "Copernicus", address = "New York, NY, USA", pages = "xiii + 212 + 8", year = "1996", ISBN = "0-387-94791-4 (hardcover)", ISBN-13 = "978-0-387-94791-4 (hardcover)", LCCN = "QC173.4.C74 B34 1996", bibdate = "Thu Nov 24 13:34:22 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, keywords = "Zipf's Law", subject = "Critical phenomena (Physics); Complexity (Philosophy); Physics; Philosophy", } @Article{Beirlant:1996:TIE, author = "Jan Beirlant and Petra Vynckier and Jozef L. Teugels", title = "Tail Index Estimation, {Pareto} Quantile Plots, and Regression Diagnostics", journal = j-J-AM-STAT-ASSOC, volume = "91", number = "436", pages = "1659--??", month = "????", year = "1996", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Mon May 5 08:26:17 MDT 1997", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Bonhoeffer:1996:NSH, author = "Sebastian Bonhoeffer and Andreas V. M. Herz and Maarten C. Boerlijst and Sean Nee and Martin A. Nowak and Robert M. May", title = "No Signs of Hidden Language in Noncoding {DNA}", journal = j-PHYS-REV-LET, volume = "76", number = "11", pages = "1977--1977", month = mar, year = "1996", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.76.1977", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Sat Sep 10 17:27:23 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.76.1977", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "Zipf's Law", } @Article{Burgos:1996:ZSB, author = "Javier D. Burgos and Pedro Moreno-Tovar", title = "{Zipf}-scaling behavior in the immune system", journal = j-BIOSYSTEMS, volume = "39", number = "3", pages = "227--232", day = "15", month = nov, year = "1996", CODEN = "BSYMBO", DOI = "https://doi.org/10.1016/0303-2647(96)01618-8", ISSN = "0303-2647 (print), 1872-8324 (electronic)", ISSN-L = "0303-2647", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/0303264796016188", acknowledgement = ack-nhfb, fjournal = "Biosystems (A6E)", journal-URL = "http://www.sciencedirect.com/science/journal/03032647", keywords = "Zipf's Law", } @Article{Campanino:1996:IIM, author = "Massimo Campanino and Stefano Isola", title = "Infinite invariant measures for non-uniformly expanding transformations of $ [0, 1] $: Weak law of large numbers with anomalous scaling", journal = j-FORUM-MATH, volume = "8", number = "1", pages = "71--92", year = "1996", CODEN = "FOMAEF", DOI = "https://doi.org/10.1515/form.1996.8.71", ISSN = "0933-7741 (print), 1435-5337 (electronic)", ISSN-L = "0933-7741", bibdate = "Sat Nov 12 10:33:06 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0837.60022", abstract = "We consider a class of maps of $ [0, 1] $ with an indifferent fixed point at 0 and expanding everywhere else. Using the invariant ergodic probability measure of a suitable, everywhere expanding, induced transformation we are able to study the infinite invariant measure of the original map in some detail. Given a continuous function with compact support in $]0, 1]$, we prove that its time averages satisfy a `weak law of large numbers' with anomalous scaling $ n / \log n $ and give an upper bound for the decay of correlations.", acknowledgement = ack-nhfb, ajournal = "Forum Math.", classmath = "60F05 (Weak limit theorems) 28D05 (Measure-preserving transformations) 37A99 (Ergodic theory)", fjournal = "Forum Mathematicum", keywords = "decay of correlations; invariant ergodic probability measure; weak law of large numbers", } @Article{Czirok:1996:POP, author = "Andr{\'a}s Czir{\'o}k and H. Eugene Stanley and Tam{\'a}s Vicsek", title = "Possible origin of power-law behavior in $n$-tuple {Zipf} analysis", journal = j-PHYS-REV-E, volume = "53", number = "??", pages = "6371--??", day = "1", month = jun, year = "1996", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.53.6371", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.53.6371", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Article{Fuchs:1996:PDP, author = "Aim{\'e} Fuchs and Giorgio Letta", title = "Le probl{\`e}me du premier chiffre d{\'e}cimal pour les nombres premiers. ({French}) [The problem for the first decimal digit of prime numbers]", journal = j-ELECT-J-COMB, volume = "3", number = "2", pages = "R25:1--R25:7", month = "????", year = "1996", ISSN = "1077-8926 (print), 1097-1440 (electronic)", ISSN-L = "1077-8926", bibdate = "Sat Feb 16 17:40:01 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "The Foata Festschrift volume.", URL = "http://www.combinatorics.org/ojs/index.php/eljc/article/download/v3i2r25/pdf", ZMnumber = "0853.11006", acknowledgement = ack-nhfb, fjournal = "Electronic Journal of Combinatorics", language = "French", xxpages = "601--607", } @Article{Gunther:1996:ZLE, author = "R. G{\"u}nther and L. Levitin and B. Schapiro and P. Wagner", title = "{Zipf}'s law and the effect of ranking on probability distributions", journal = j-INT-J-THEOR-PHYS, volume = "35", number = "2", pages = "395--417", year = "1996", CODEN = "IJTPBM", DOI = "https://doi.org/10.1007/BF02083823", ISSN = "0020-7748 (print), 1572-9575 (electronic)", ISSN-L = "0020-7748", bibdate = "Sat Nov 12 10:53:57 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0843.60002", abstract = "Ranking procedures are widely used in the description of many different types of complex systems. Zipf's law is one of the most remarkable frequency-rank relationships and has been observed independently in physics, linguistics, biology, demography, etc. We show that ranking plays a crucial role in making it possible to detect empirical relationships in systems that exist in one realization only, even when the statistical ensemble to which the systems belong has a very broad probability distribution. Analytical results and numerical simulations are presented which clarify the relations between the probability distributions and the behavior of expected values for unranked and ranked random variables. This analysis is performed, in particular, for the evolutionary model presented in our previous papers which leads to Zipf's law and reveals the underlying mechanism of this phenomenon in terms of a system with interdependent and interacting components as opposed to the ``ideal gas'' models suggested by previous researchers. The ranking procedure applied to this model leads to a new, unexpected phenomenon: a characteristic ``staircase'' behavior of the mean values of the ranked variables (ranked occupation numbers). This result is due to the broadness of the probability distributions for the occupation numbers and does not follow from the ``ideal gas'' model. Thus, it provides an opportunity, by comparison with empirical data, to obtain evidence as to which model relates to reality.", acknowledgement = ack-nhfb, classmath = "60A99 (Foundations of probability theory)", fjournal = "International Journal of Theoretical Physics", journal-URL = "http://link.springer.com/journal/10773", keywords = "numerical simulations; ranking procedures; Zipf's law", } @TechReport{Hill:1996:NDT, author = "Theodore P. Hill", title = "A note on distributions of true versus fabricated data", type = "Preprint", institution = "Department of Mathematics, Georgia Institute of Technology", address = "Atlanta, GA, USA", pages = "4", year = "1996", bibdate = "Sat Feb 16 17:47:27 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1046&context=rgp_rsr", acknowledgement = ack-nhfb, } @Article{Israeloff:1996:CZD, author = "N. E. Israeloff and M. Kagalenko and K. Chan", title = "Can {Zipf} Distinguish Language From Noise in Noncoding {DNA}?", journal = j-PHYS-REV-LET, volume = "76", number = "11", pages = "1976--??", day = "11", month = mar, year = "1996", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.76.1976", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.76.1976", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "Zipf's Law", } @Article{Krugman:1996:CMU, author = "Paul Krugman", title = "Confronting the Mystery of Urban Hierarchy", journal = j-J-JPN-INT-ECON, volume = "10", number = "4", pages = "399--418", month = dec, year = "1996", CODEN = "JJIEBD", DOI = "https://doi.org/10.1006/jjie.1996.0023", ISSN = "0889-1583 (print), 1095-8681 (electronic)", ISSN-L = "0889-1583", bibdate = "Wed Feb 27 06:15:55 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0889158396900234", acknowledgement = ack-nhfb, fjournal = "Journal of the {Japanese} and International Economies", keywords = "Zipf's Law", } @Book{Krugman:1996:SOE, author = "Paul R. Krugman", title = "The Self-Organizing Economy", publisher = "Blackwell Publishers", address = "Cambridge, MA, USA", pages = "vi + 122", year = "1996", ISBN = "1-55786-699-6 (paperback)", ISBN-13 = "978-1-55786-699-8 (paperback)", LCCN = "HB199 .K75 1996", bibdate = "Wed Feb 27 07:08:14 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, keywords = "Zipf's Law", subject = "Economics; Self-organizing systems", } @Article{Ley:1996:PDU, author = "Eduardo Ley", title = "On the Peculiar Distribution of the {U.S.} Stock Indexes' Digits", journal = j-AMER-STAT, volume = "50", number = "4", pages = "311--313", month = nov, year = "1996", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Mon Nov 14 15:09:04 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/amstat.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib", URL = "http://www.jstor.org/stable/2684926", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @Article{Mantegna:1996:MAR, author = "R. N. Mantegna and S. V. Buldyrev and A. L. Goldberger and S. Havlin and C.-K. Peng and M. Simons and H. E. Stanley", title = "{Mantegna} {\em et al.} Reply", journal = j-PHYS-REV-LET, volume = "76", pages = "1979--1981", month = mar, year = "1996", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.76.1979", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Sat Sep 10 17:28:12 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See \cite{Mantegna:1994:LFN,Voss:1996:CLF}", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.76.1979", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", issue = "11", journal-URL = "http://prl.aps.org/browse", } @Article{Newman:1996:ASC, author = "M. E. J. Newman and Kim Sneppen", title = "Avalanches, scaling, and coherent noise", journal = j-PHYS-REV-E, volume = "54", number = "6", pages = "6226--6231", month = dec, year = "1996", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.54.6226", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Thu Feb 28 10:44:51 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.54.6226", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "power law; scale invariance; Zipf's law", } @InProceedings{Nigrini:1996:DAR, author = "Mark J. Nigrini", editor = "M. Ettredge", booktitle = "{Proceedings of the 1996 Deloitte \& Touche I University of Kansas Symposium on Auditing Problems}", title = "Digital Analysis and the Reduction of Auditor Litigation Risk", publisher = "University of Kansas", address = "Lawrence, KS, USA", pages = "69--81", year = "1996", bibdate = "Sat Sep 10 15:20:48 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Article{Nigrini:1996:TCA, author = "Mark J. Nigrini", title = "A Taxpayer Compliance Application of {Benford's Law}", journal = "Journal of the {American Taxation Association}", volume = "18", number = "1", pages = "72--91", month = "????", year = "1996", CODEN = "????", ISSN = "0198-9073 (print), 1558-8017 (electronic)", ISSN-L = "0198-9073", bibdate = "Mon Nov 14 16:30:42 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "J. Amer. Tax. Assoc.", fjournal = "Journal of the {American Taxation Association}", journal-URL = "http://aaajournals.org/loi/atax", remark = "No journal issues online before 1999.", } @Article{Perline:1996:ZLC, author = "Richard Perline", title = "{Zipf}'s law, the {Central Limit Theorem}, and the random division of the unit interval", journal = j-PHYS-REV-E, volume = "54", number = "1", pages = "220--223", day = "1", month = jul, year = "1996", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.54.220", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.54.220", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Mandelbrot's monkey-at-the-typewriter model of Zipf's inverse power law; Zipf's Law", } @Article{Sole:1996:ESO, author = "Ricard V. Sol{\'e} and Susanna C. Manrubia", title = "Extinction and self-organized criticality in a model of large-scale evolution", journal = j-PHYS-REV-E, volume = "54", number = "1", pages = "R42--R45", month = jul, year = "1996", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.54.R42", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Thu Feb 28 10:32:32 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.54.R42", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "power law; scale invariance; Zipf's law", } @Article{Solomon:1996:SSE, author = "Sorin Solomon and Moshe Levy", title = "Spontaneous Scaling Emergence in Generic Stochastic Systems", journal = j-INT-J-MOD-PHYS-C, volume = "7", number = "5", pages = "745--752", month = oct, year = "1996", CODEN = "IJMPEO", DOI = "https://doi.org/10.1142/S0129183196000624", ISSN = "0129-1831 (print), 1793-6586 (electronic)", bibdate = "Thu Feb 28 10:20:02 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.worldscientific.com/doi/abs/10.1142/S0129183196000624", acknowledgement = ack-nhfb, fjournal = "International Journal of Modern Physics C [Physics and Computers]", journal-URL = "http://www.worldscientific.com/loi/ijmpc", } @Article{Voss:1996:CLF, author = "Richard F. Voss", title = "Comment on {``Linguistic Features of Noncoding DNA Sequences''}", journal = j-PHYS-REV-LET, volume = "76", number = "11", pages = "1978--1978", month = mar, year = "1996", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.76.1978", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Sat Sep 10 14:35:50 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See \cite{Mantegna:1994:LFN} and reply \cite{Mantegna:1996:MAR}.", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.76.1978", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "Benford's Law", } @Article{Watanabe:1996:EZL, author = "Makoto S. Watanabe", title = "Erratum: {Zipf}'s law in percolation", journal = j-PHYS-REV-E, volume = "54", number = "??", pages = "4483--4483", day = "1", month = oct, year = "1996", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.54.4483", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See \cite {Watanabe:1996:ZLP}.", URL = "http://link.aps.org/doi/10.1103/PhysRevE.54.4483", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Article{Watanabe:1996:ZLP, author = "Makoto S. Watanabe", title = "{Zipf}'s law in percolation", journal = j-PHYS-REV-E, volume = "53", number = "??", pages = "4187--4190", day = "1", month = apr, year = "1996", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.53.4187", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See erratum \cite {Watanabe:1996:EZL}.", URL = "http://link.aps.org/doi/10.1103/PhysRevE.53.4187", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Article{Allaart:1997:ISC, author = "Pieter C. Allaart", title = "An Invariant-Sum Characterization of {Benford's Law}", journal = j-J-APPL-PROBAB, volume = "34", number = "1", pages = "288--291", month = mar, year = "1997", CODEN = "JPRBAM", DOI = "https://doi.org/10.2307/3215195", ISSN = "0021-9002 (print), 1475-6072 (electronic)", ISSN-L = "0021-9002", MRclass = "60E05", MRnumber = "1429075 (98d:60029)", MRreviewer = "Peter Schatte", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/3215195", abstract = "The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit $ d (d = 1, 2, \ldots, 9) $ is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected sum of all elements with first digits $ d_1, \ldots, d_k $ is constant for every fixed $k$.", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Probability", journal-URL = "http://www.jstor.org/journals/00219002.html; http://projecteuclid.org/euclid.jap/", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Article{Castillo:1997:FGP, author = "Enrique Castillo and Ali S. Hadi", title = "Fitting the Generalized {Pareto} Distribution to Data", journal = j-J-AM-STAT-ASSOC, volume = "92", number = "440", pages = "1609--1620", month = dec, year = "1997", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274X (electronic)", ISSN-L = "0162-1459", bibdate = "Mon Oct 30 06:57:36 MST 2000", bibsource = "http://www.amstat.org/publications/jasa/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jamstatassoc.bib", URL = "http://www.amstat.org/publications/jasa/abstracts_97/december/CASTILLO.HTM; http://www.jstor.org/stable/2965432", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Chen:1997:NHS, author = "Weishing Chen and Tai-Hsi Wu", title = "A non-homogeneous software reliability model based on {Zipf}'s law", journal = "International Journal of Quality \& Reliability Management", volume = "14", number = "4", pages = "409--431", month = "????", year = "1997", DOI = "https://doi.org/10.1108/02656719710170666", ISSN = "0265-671X (print), 1758-6682 (electronic)", ISSN-L = "0265-671X", bibdate = "Fri Nov 25 07:08:17 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Denisov:1997:FBS, author = "S. Denisov", title = "Fractal binary sequences: {Tsallis} thermodynamics and the {Zipf} law", journal = j-PHYS-LET-A, volume = "235", number = "5", pages = "447--451", day = "17", month = nov, year = "1997", CODEN = "PYLAAG", DOI = "https://doi.org/10.1016/S0375-9601(97)00688-9", ISSN = "0375-9601 (print), 1873-2429 (electronic)", ISSN-L = "0375-9601", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0375960197006889", acknowledgement = ack-nhfb, fjournal = "Physics Letters A", journal-URL = "http://www.sciencedirect.com/science/journal/03759601", keywords = "Zipf's Law", } @InCollection{Gell-Mann:1997:ETZ, author = "Murray Gell-Mann", title = "Empirical Theory --- {Zipf}'s Law", crossref = "Gell-Mann:1997:QJA", pages = "92--97", year = "1997", bibdate = "Mon Sep 01 10:20:00 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @InCollection{Hill:1997:BL, author = "Theodore Hill", editor = "Michiel Hazewinkel", booktitle = "Encyclopedia of Mathematics, Supplement {I}", title = "{Benford's Law}", volume = "1", publisher = pub-SV, address = pub-SV:adr, pages = "102--103", year = "1997", ISBN = "90-481-4896-0", ISBN-13 = "978-90-481-4896-7", LCCN = "????", bibdate = "Wed Nov 16 10:29:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Irmay:1997:RBZ, author = "Shragga Irmay", title = "The relationship between {Zipf}'s law and the distribution of first digits", journal = j-J-APPL-STAT, volume = "24", number = "4", pages = "383--393", day = "1", month = aug, year = "1997", CODEN = "????", ISSN = "0266-4763 (print), 1360-0532 (electronic)", ISSN-L = "0266-4763", MRclass = "62E15", MRnumber = "1491308 (98h:62015)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.catchword.co.uk/titles/carfax/02664763/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.tandf.co.uk/journals/routledge/02664763.html", URL = "http://www.tandfonline.com/doi/abs/10.1080/02664769723594", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Statistics", journal-URL = "http://www.tandfonline.com/loi/cjas20", keywords = "Benford's Law; Zipf's Law", xxpages = "383--394", } @Article{Jeevanand:1997:BEB, author = "E. S. Jeevanand", title = "{Bayes} Estimation of {$ P(X_2 < X_1) $} for a Bivariate {Pareto} Distribution", journal = j-J-R-STAT-SOC-SER-D-STATISTICIAN, volume = "46", number = "1", pages = "93--99", month = "????", year = "1997", CODEN = "????", DOI = "https://doi.org/10.2307/2988496", ISSN = "0039-0526 (print), 1467-9884 (electronic)", ISSN-L = "0039-0526", bibdate = "Thu Jan 22 18:10:24 MST 2015", bibsource = "http://www.jstor.org/stable/i349996; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jrss-d-1990.bib", URL = "http://www.jstor.org/stable/2988496", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society. Series D (The Statistician)", journal-URL = "http://www.jstor.org/journals/00390526.html", } @Manual{Nigrini:1997:DAT, author = "Mark J. Nigrini", title = "Digital Analysis Tests and Statistics", organization = "The Nigrini Institute, Inc.", address = "Allen, TX, USA", month = "????", year = "1997", bibdate = "Sat Feb 16 18:55:18 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.nigrini.com/data_software/Program_Details_2009.docx", acknowledgement = ack-nhfb, } @Article{Nigrini:1997:UBL, author = "Mark J. Nigrini and L. J. Mittermaier", title = "The Use of {Benford's Law} as an Aid in Analytical Procedures", journal = j-AUDITING, volume = "16", number = "2", pages = "52--67", month = "????", year = "1997", CODEN = "????", ISSN = "0278-0380 (print), 1558-7991 (electronic)", ISSN-L = "0278-0380", bibdate = "Sun Nov 13 11:37:48 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://aaapubs.org/loi/ajpt", acknowledgement = ack-nhfb, fjournal = "Auditing: A Journal of Practice \& Theory", remark = "No journal issues online before 1999.", } @Article{Oluic-Vukovic:1997:BDC, author = "Vesna Olui{\'c}-Vukovi{\'c}", title = "{Bradford}'s distribution: {From} the classical bibliometric ``law'' to the more general stochastic models", journal = j-J-AM-SOC-INF-SCI, volume = "48", number = "9", pages = "833--842", month = sep, year = "1997", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(199709)48:9<833::AID-ASI7>3.0.CO%3B2-S", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:04:21 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "7 Dec 1998", } @Book{Robertson:1997:FEM, author = "Jack C. Robertson", title = "Fraud Examination for Managers and Auditors", publisher = "Viesca Books", address = "Austin, TX, USA", edition = "1997", pages = "vii + 564", year = "1997", ISBN = "0-9656785-0-4", ISBN-13 = "978-0-9656785-0-6", LCCN = "HV8079.F7 R63 1997", bibdate = "Thu Oct 13 09:04:19 MDT 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, keywords = "Benford's Law", subject = "Fraud investigation", } @Article{Salsburg:1997:DPB, author = "David Salsburg", title = "Digit Preferences in the {Bible}", journal = j-CHANCE, volume = "10", number = "4", pages = "46--48", month = "????", year = "1997", CODEN = "CNDCE4", DOI = "https://doi.org/10.1080/09332480.1997.10542065", ISSN = "0933-2480 (print), 1867-2280 (electronic)", ISSN-L = "0933-2480", bibdate = "Sat Feb 16 19:08:02 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://amstat.tandfonline.com/doi/full/10.1080/09332480.1997.10542065", acknowledgement = ack-nhfb, fjournal = "Chance: a magazine for people interested in the analysis of data", journal-URL = "http://chance.amstat.org/; http://link.springer.com/journal/144; http://www.tandfonline.com/loi/ucha20", } @InCollection{Smith:1997:EBL, author = "Steven W. Smith", booktitle = "The Scientist and Engineer's Guide to Digital Signal Processing", title = "Explaining {Benford's Law}", chapter = "34", publisher = "California Technical Publishing", address = "San Diego, CA, USA", bookpages = "xiv + 626", pages = "701--722", year = "1997", ISBN = "0-9660176-3-3", ISBN-13 = "978-0-9660176-3-2", LCCN = "TK5102.9 .S57 1997", bibdate = "Sat Sep 10 16:54:35 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.dspguide.com/ch34.htm", acknowledgement = ack-nhfb, keywords = "Benford's Law", remark = "Chapters appear to stop at 31 in 1997 edition (according to Amazon book preview), so a later addition should have a new ISBN (the 2002 edition from Newnes is \booktitle{Digital Signal Processing: A Practical Guide for Engineers and Scientists}, with ISBN 0-7506-7444-X). However, the Web site says it is the same book. The Web edition has 34 chapters, and all are freely downloadable.", } @Article{Tsonis:1997:ZLS, author = "A. A. Tsonis and C. Schultz and P. A. Tsonis", title = "{Zipf's Law} and the structure and evolution of languages", journal = j-COMPLEXITY, volume = "2", number = "5", pages = "12--13", year = "1997", CODEN = "COMPFS", DOI = "https://doi.org/10.1002/(SICI)1099-0526(199705/06)2:5<12::AID-CPLX3>3.0.CO%3B2-C", ISSN = "1076-2787 (print), 1099-0526 (electronic)", ISSN-L = "1076-2787", bibdate = "Mon Feb 25 06:13:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Complexity", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-0526", } @Article{Zanette:1997:RIU, author = "Dami{\'a}n H. Zanette and Susanna C. Manrubia", title = "Role of Intermittency in Urban Development: A Model of Large-Scale City Formation", journal = j-PHYS-REV-LET, volume = "79", number = "3", pages = "523--526", day = "3", month = jul, year = "1997", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.79.523", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Thu Feb 28 10:40:44 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.79.523", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "power law; scale invariance; Zipf's law", } @Article{Bach:1998:EPM, author = "Eric Bach", title = "Efficient prediction of {Marsaglia--Zaman} random number generators", journal = j-IEEE-TRANS-INF-THEORY, volume = "44", number = "3", pages = "1253--1257", year = "1998", CODEN = "IETTAW", DOI = "https://doi.org/10.1109/18.669305", ISSN = "0018-9448 (print), 1557-9654 (electronic)", ISSN-L = "0018-9448", MRclass = "65C10", MRnumber = "99b:65007", bibdate = "Thu Dec 22 07:42:24 2011", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/marsaglia-george.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib", ZMnumber = "0915.65003", abstract = "This paper presents two properties of the random number generator by {\em G. Marsaglia} and {\em A. Zaman} [Ann. Appl. Probab. 1, No. 3, 462-480 (1991; Zbl 0733.65005)]. First, it is shown that its successive digits are digits of certain rational $b$-adic numbers. Then, an efficient algorithm is derived to predict an unknown pseudorandom sequence of this type. Two examples of the prediction are given.", acknowledgement = ack-nhfb, classmath = "*65C10 Random number generation 11K45 Pseudo-random numbers, etc.", fjournal = "IEEE Transactions on Information Theory", journal-URL = "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18", keywords = "$b$-adic number; algorithm; continued fraction; pseudo-random number generator", ZMreviewer = "K. Uosaki (Tottori)", } @Article{Browne:1998:FBL, author = "Malcolm W. Browne", title = "Following {Benford's Law}, or Looking Out for {No. 1}", journal = j-NY-TIMES, volume = "??", number = "??", pages = "F4--F4", day = "4", month = aug, year = "1998", CODEN = "NYTIAO", ISSN = "0362-4331 (print), 1542-667X, 1553-8095", ISSN-L = "0362-4331", bibdate = "Sat Feb 16 14:29:48 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Science section.", URL = "http://search.proquest.com/hnpnewyorktimes/docview/109868161/13C4AACC02923FE97A/1", acknowledgement = ack-nhfb, fjournal = "New York Times", journal-URL = "http://www.nytimes.com/", } @Article{Busta:1998:UBL, author = "Bruce Busta and Randy Weinberg", title = "Using {Benford's Law} and Neural Networks as a Review Procedure", journal = "Managerial Auditing Journal", volume = "13", number = "6", pages = "356--366", month = "????", year = "1998", CODEN = "????", DOI = "https://doi.org/10.1108/02686909810222375", ISSN = "0268-6902 (print), 1758-7735 (electronic)", ISSN-L = "0268-6902", bibdate = "Tue Nov 22 18:50:49 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Chongde:1998:EMB, author = "Wang Chongde and Wang Zhe", title = "Evaluation of the models for {Bradford's Law}", journal = j-SCIENTOMETRICS, volume = "42", number = "1", pages = "89--95", month = may, year = "1998", CODEN = "SCNTDX", DOI = "https://doi.org/10.1007/BF02465014", ISSN = "0138-9130 (print), 1588-2861 (electronic)", ISSN-L = "0138-9130", bibdate = "Wed Sep 2 12:03:59 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/scientometrics1990.bib", URL = "http://link.springer.com/article/10.1007/BF02465014", acknowledgement = ack-nhfb, fjournal = "Scientometrics", journal-URL = "http://link.springer.com/journal/11192", } @Article{DeCeuster:1998:HPB, author = "Marc J. K. {De Ceuster} and Geert Dhaene and Tom Schatteman", title = "On the hypothesis of psychological barriers in stock markets and {Benford's Law}", journal = j-J-EMPIR-FINANCE, volume = "5", number = "3", pages = "263--279", month = sep, year = "1998", CODEN = "JEFIEC", DOI = "https://doi.org/10.1016/S0927-5398(97)00024-8", ISSN = "0927-5398 (print), 1879-1727 (electronic)", ISSN-L = "0927-5398", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0927539897000248", abstract = "We consider the hypothesis of psychological barriers at round numbers of a stock index. This hypothesis is often examined by testing the uniformity of the distribution of the trailing digits in the stock index, a rejection being interpreted as evidencing the existence of psychological barriers. By virtue of Benford's Law, we show that the uniform distribution is not the right benchmark against which to test. As an alternative we propose a test based on the cyclical permutations of the actual returns. Applying this test to the Dow Jones 30 Industrial Average, the Financial Times Stock Exchange 100 and the Nikkei Stock Average 225, we find no convincing evidence of psychological barriers, contrary to previous findings.", acknowledgement = ack-nhfb, fjournal = "Journal of Empirical Finance", keywords = "Benford's Law; Psychological barriers", } @Article{Herman:1998:TRI, author = "Tom Herman", title = "Tax report: An {IRS} blooper startles thousands of taxpayers", journal = j-WALL-ST-J, volume = "??", number = "??", pages = "1--1", day = "18", month = feb, year = "1998", CODEN = "WSJOAF", ISSN = "0099-9660", bibdate = "Fri Mar 22 09:17:50 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://online.wsj.com/article/SB887753996209804500.html", acknowledgement = ack-mjn # "\slash " # ack-nhfb, fjournal = "Wall Street Journal", keywords = "Benford's Law", remark = "From the article: ``[a taxpayer] was expecting a \$513 tax refund. Instead, he recently received a letter from the Internal Revenue Service informing him that he owes the government \$300,000,007.57. \ldots{} about 3,000 other people around the nation got similar erroneous notices, each showing a balance due of `three hundred million dollars and change.'\,''", } @Article{Hill:1998:FDP, author = "Theodore P. Hill", title = "The First Digit Phenomenon", journal = j-AM-SCI, volume = "86", number = "4", pages = "358--363", month = jul # "\slash " # aug, year = "1998", CODEN = "AMSCAC", DOI = "https://doi.org/10.1511/1998.4.358", ISSN = "0003-0996 (print), 1545-2786 (electronic)", ISSN-L = "0003-0996", bibdate = "Sat Dec 10 11:54:49 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://people.math.gatech.edu/~hill/publications/PAPER%20PDFS/TheFirstDigitPhenomenonAmericanScientist1996.pdf; http://www.math.gatech.edu/~hill/publications/cv.dir/1st-dig.pdf", abstract = "A century ago, Simon Newcomb observed an unexpected pattern in the first digits of logarithm tables: The digit $1$ is significantly more likely to occur than $2$, $2$ than $3$, and so on. More than a half-century later, Frank Benford rediscovered the first-digit phenomenon and found that it applied to many tables of numerical data, including the stock market, census statistics and accounting figures. New mathematical insights establish the empirical law developed by Newcomb and Benford as part of modern probability theory, and recent applications include testing of mathematical models, design of computers and detection of fraud in accounting.", acknowledgement = ack-nhfb, fjournal = "American Scientist", journal-URL = "http://www.americanscientist.org/issues/past.aspx", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Book{Knuth:1998:SA, author = "Donald E. Knuth", title = "Seminumerical Algorithms", volume = "2", publisher = pub-AW, address = pub-AW:adr, edition = "Third", pages = "xiii + 762", year = "1998", ISBN = "0-201-89684-2", ISBN-13 = "978-0-201-89684-8", LCCN = "QA76.6 .K64 1997", bibdate = "Fri Jul 11 15:41:22 1997", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; http://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg-2ed.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/css.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib; http://www.math.utah.edu/pub/tex/bib/master.bib; http://www.math.utah.edu/pub/tex/bib/prng.bib; http://www.math.utah.edu/pub/tex/bib/texbook2.bib", note = "See section 4.2.4V, The Fraction Parts, pages 254--262, for a discussion of Benford's Law.", price = "US\$52.75", series = "The Art of Computer Programming", acknowledgement = ack-nhfb, remark = "Knuth comments on page 255: ``The fact that the leading digits tend to be small makes the most obvious techniques of ``average error'' estimation for floating point calculations invalid. The relative error due to rounding is usually a little more than expected.", tableofcontents = "3: Random Numbers / 1 \\ 3.1. Introduction / 1 \\ 3.2. Generating Uniform Random Numbers / 10 \\ 3.2.1. The Linear Congruential Method / 10 \\ 3.2 1.1. Choice of modulus / 12 \\ 3.2.1.2 Choice of multiplier / 16 \\ 3.2.1.3. Potency / 23 \\ 3.2.2. Other Methods / 26 \\ 3.3. Statistical Tests / 41 \\ 3.3.1. General Test Procedures for Studying Random Data / 41 \\ 3.3.2. Empirical Tests / 61 \\ *3.3.3. Theoretical Tests / 80 \\ 3.3.4. The Spectral Test / 93 \\ 3.4. Other Types of Random Quantities / 119 \\ 3.4 1. Numerical Distributions / 119 \\ 3.4.2. Random Sampling and Shuffling / 142 \\ *3.5. What Is a Random Sequence? / 149 \\ 3.6. Summary / 184 \\ 4: Arithmetic / 194 \\ 4.1. Positional Number Systems / 195 \\ 4.2. Floating Point Arithmetic / 214 \\ 4.2.1. Single-Precision Calculations / 214 \\ 4.2 2. Accuracy of Floating Point Arithmetic / 229 \\ *4.2.3. Double-Precision Calculations / 246 \\ 4.2.4. Distribution of Floating Point Numbers / 253 \\ 4.3 Multiple Precision Arithmetic / 265 \\ 4.3.1. The Classical Algorithms / 265 \\ *4.3.2. Modular Arithmetic / 284 \\ *4.3.3. How Fast Can We Multiply? / 294 \\ 4.4. Radix Conversion / 319 \\ 4.5. Rational Arithmetic / 330 \\ 4.5.1. Fractions / 330 \\ 4.5.2. The Greatest Common Divisor / 333 \\ *4.5.3. Analysis of Euclid's Algorithm / 356 \\ 4.5.4. Factoring into Primes / 379 \\ 4.6. Polynomial Arithmetic / 418 \\ 4.6.1. Division of Polynomials / 420 \\ *4.6.2. Factorization of Polynomials / 439 \\ 4.6.3. Evaluation of Powers / 461 \\ 4.6.4. Evaluation of Polynomials / 485 \\ *4.7. Manipulation of Power Series / 525 \\ Answers to Exercises / 538 \\ Appendix A: Tables of Numerical Quantities / 726 \\ 1. Fundamental Constants (decimal) / 726 \\ 2; Fundamental Constants ( octal) / 727 \\ 3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers / 728 \\ Appendix B: Index to Notations / 730 \\ Index and Glossary / 735", } @Article{Manrubia:1998:IMU, author = "Susanna C. Manrubia and Dami{\'a}n H. Zanette", title = "Intermittency model for urban development", journal = j-PHYS-REV-E, volume = "58", number = "1", pages = "295--302", month = jul, year = "1998", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.58.295", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Thu Feb 28 10:42:23 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.58.295", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "power law; scale invariance; Zipf's law", } @Article{Marsili:1998:IIL, author = "Matteo Marsili and Yi-Cheng Zhang", title = "Interacting Individuals Leading to {Zipf's Law}", journal = j-PHYS-REV-LET, volume = "80", number = "??", pages = "2741--??", day = "23", month = mar, year = "1998", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.80.2741", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.80.2741", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "Zipf's Law", } @Article{Nigrini:1998:DIB, author = "Mark J. Nigrini", title = "{Dow} Illustrates {Benford's Law}", journal = j-NY-TIMES, volume = "??", number = "??", pages = "F4--F4", day = "4", month = aug, year = "1998", CODEN = "NYTIAO", ISSN = "0362-4331 (print), 1542-667X, 1553-8095", ISSN-L = "0362-4331", bibdate = "Sat Feb 16 15:46:43 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://search.proquest.com/hnpnewyorktimes/docview/109901848/13C4AB0BB5661D397CE/1", acknowledgement = ack-nhfb, fjournal = "New York Times", journal-URL = "http://www.nytimes.com/", } @Article{NunesAmaral:1998:PLS, author = "Lu{\'\i}s A. {Nunes Amaral} and Sergey V. Buldyrev and Shlomo Havlin and Michael A. Salinger and H. Eugene Stanley", title = "Power Law Scaling for a System of Interacting Units with Complex Internal Structure", journal = j-PHYS-REV-LET, volume = "80", number = "7", pages = "1385--1388", day = "16", month = feb, year = "1998", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.80.1385", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Thu Feb 28 11:09:02 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.80.1385", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "power law; scale invariance; Zipf's law", } @Article{Pietronero:1998:UDN, author = "L. Pietronero and E. Tosatti and V. Tosatti and A. Vespignani", title = "The Uneven Distribution of Numbers in Nature", journal = "ArXiv e-prints", pages = "9", day = "27", month = aug, year = "1998", bibdate = "Mon Nov 21 06:58:45 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://arxiv.org/abs/cond-mat/9808305", abstract = "Suppose you look at today's stock prices and bet on the value of the first digit. One could guess that a fair bet should correspond to the frequency of $ 1 / 9 = 11.11 \% $ for each digit from 1 to 9. This is by no means the case, and one can easily observe a strong prevalence of the small values over the large ones. The first three integers 1, 2 and 3 alone have globally a frequency of 60\% while the other six values 4, 5, 6, 7, 8 and 9 appear only in 40\% of the cases. This situation is actually much more general than the stock market and it occurs in a variety of number catalogs related to natural phenomena. The first observation of this property traces back to S. Newcomb in 1881 but a more precise account was given by F. Benford in 1938. In this note we illustrate these observations with the enlightening specific example of the stock market. We also identify the general mechanism for the origin of this uneven distribution in the multiplicative nature of fluctuations in economics and in many natural phenomena. This provides a natural explanation for the ubiquitous presence of the Benford's law in many different phenomena with the common element that their fluctuations refer to a fraction of their values. This brings us close to the problem of the spontaneous origin of scale invariant properties in various phenomena which is a debated question at the frontier of different fields.", acknowledgement = ack-nhfb, } @Article{Pocheau:1998:SRS, author = "A. Pocheau", title = "Scale ratios, statistical symmetries and intermittency", journal = j-EUROPHYS-LETT, volume = "43", number = "4", pages = "410--415", month = aug, year = "1998", CODEN = "EULEEJ", DOI = "https://doi.org/10.1209/epl/i1998-00103-6", ISSN = "0295-5075 (print), 1286-4854 (electronic)", ISSN-L = "0295-5075", bibdate = "Sat Feb 16 16:02:17 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Europhysics Letters", journal-URL = "http://iopscience.iop.org/0295-5075", } @Article{Ramsden:1998:ZLB, author = "J. J. Ramsden and J. Vohradsk{\'y}", title = "{Zipf}-like behavior in procaryotic protein expression", journal = j-PHYS-REV-E, volume = "58", number = "??", pages = "7777--??", day = "1", month = dec, year = "1998", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.58.7777", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.58.7777", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Article{Rousset:1998:CPV, author = "Olivier G. Rousset and Yilong Ma and Alan C. Evans", title = "Correction for Partial Volume Effects in {PET}: Principle and Validation", journal = j-NUCL-MED, volume = "39", number = "5", pages = "904--911", month = may, year = "1998", CODEN = "JNMEAQ", ISSN = "0161-5505 (print), 1535-5667 (electronic)", ISSN-L = "0161-5505", bibdate = "Sat Sep 10 16:10:46 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://jnm.snmjournals.org/content/39/5/904.full.pdf+html", acknowledgement = ack-nhfb, fjournal = "Journal of Nuclear Medicine", journal-URL = "http://jnm.snmjournals.org/content/by/year", keywords = "Benford's Law", } @Article{Schatte:1998:BLV, author = "P. Schatte", title = "On {Benford's law} to variable base", journal = j-STAT-PROB-LETT, volume = "37", number = "4", pages = "391--397", day = "30", month = mar, year = "1998", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/S0167-7152(97)00142-9", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", MRclass = "60F05 (60E15)", MRnumber = "1624423 (99e:60072)", MRreviewer = "Theodore P. Hill", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.sciencedirect.com/", URL = "http://www.sciencedirect.com/science/article/pii/S0167715297001429", abstract = "Benford's law is studied in dependence on the base $ b > 1 $. It can hold to large bases $b$ only approximately. The quality of approximation is estimated in cases of products and sums of random variables, respectively, and in case of some deterministic sequences. Always the approximation by Benford's law becomes worse as $ b \to \infty $, but as a rule also as $ b \to 1 + 0 $", acknowledgement = ack-nhfb, fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", keywords = "Benford's Law; Discrepancy; First-digit problem; Mantissa distribution; Products of random variables; Sums of random variables", } @Article{Simon:1998:ADC, author = "Jonathan Simon", title = "An Analysis of the Distribution of Combinations Chosen by {UK National Lottery} Players", journal = j-J-RISK-UNCERT, volume = "17", number = "3", pages = "243--277", year = "1998", CODEN = "JRUNEN", DOI = "https://doi.org/10.1023/A:1007786901776", ISSN = "0895-5646 (print), 1573-0476 (electronic)", ISSN-L = "0895-5646", bibdate = "Sat Sep 10 14:35:50 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Risk and Uncertainty", journal-URL = "http://link.springer.com/journal/11166", keywords = "Benford's Law", } @Article{Slijepcevic:1998:NID, author = "Sini{\v{s}}a Slijep{\v{c}}evi{\'c}", title = "A Note on Initial Digits of Recurrence Sequences", journal = j-FIB-QUART, volume = "36", number = "4", pages = "305--308", month = aug, year = "1998", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:02:49 MDT 2011", bibsource = "http://www.fq.math.ca/36-4.html; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/36-4/slijepcevic.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Article{Troll:1998:ZLC, author = "G. Troll and P. beim Graben", title = "{Zipf}'s law is not a consequence of the {Central Limit Theorem}", journal = j-PHYS-REV-E, volume = "57", number = "3", pages = "1347--1355", day = "1", month = feb, year = "1998", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.57.1347", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.57.1347; http://pre.aps.org/abstract/PRE/v57/i2/p1347_1", abstract = "It has been observed that the rank statistics of string frequencies of many symbolic systems (e.g., word frequencies of natural languages) follows Zipf's law in good approximation. We show that, contrary to claims in the literature, Zipf's law cannot be realized by the central limit theorem(s). The observation that a log-normal distribution of string frequencies yields an approximately Zipf-like rank statistics is actually misleading. Indeed, Zipf's law for the rank statistics is strictly equivalent to a power law distribution of frequencies. There are two natural ways to perform the infinite size limit for the vocabulary. The first one is the method of choice in the literature; it makes the upper word length bound tend to infinity and leads in the case of a multistate Bernoulli process via a central limit theorem to a log-normal frequency distribution. An alternative and for text samples actually better realizable way is to make the lower frequency bound tend to zero. This limit procedure leads to a power law distribution and hence to Zipf's law --- at least for Bernoulli processes and to a very good approximation for natural languages where it passes the $ \chi^2 $ test. For the Bernoulli case we will give a heuristic proof.", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Book{Wirsching:1998:DSG, author = "G{\"u}nther J. Wirsching", title = "The dynamical system generated by the $ 3 n + 1 $ function", volume = "1681", publisher = pub-SV, address = pub-SV:adr, pages = "vii + 158", year = "1998", ISBN = "3-540-63970-5 (softcover)", ISBN-13 = "978-3-540-63970-1 (softcover)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", LCCN = "QA3 .L28 no. 1681", bibdate = "Mon Nov 21 08:00:54 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", series = ser-LNM, URL = "http://www.loc.gov/catdir/enhancements/fy0818/97051329-d.html; http://www.loc.gov/catdir/enhancements/fy0818/97051329-t.html", acknowledgement = ack-nhfb, keywords = "Benford's Law", subject = "Sequences (Mathematics); Combinatorial probabilities; Convergence", } @Article{Ausloos:1999:PMK, author = "M. Ausloos and K. Ivanova", title = "Precise $ (m, k) $-{Zipf} diagram analysis of mathematical and financial time series when $ m = 6 $, $ k = 2 $", journal = j-PHYSICA-A, volume = "270", number = "3--4", pages = "526--542", day = "15", month = aug, year = "1999", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(99)00178-8", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437199001788", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Article{AyllonBurguillo:1999:PDS, author = "Juan David {Ayll{\'o}n Burguillo} and Manuel {Perera Dom{\'\i}nguez}", title = "El primer d{\'\i}gito significativo. ({Spanish}) [The first significant digit]", journal = "{Epsilon}: Revista de la {Sociedad Andaluza de Educaci{\'o}n Matem{\'a}tica ``Thales''}", volume = "15", number = "3(45)", pages = "339--352", month = "????", year = "1999", ISSN = "1131-9321", ISSN-L = "1131-9321", MRnumber = "K60xx", bibdate = "Sat Feb 16 14:32:14 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "MATHEDUC.02332710", acknowledgement = ack-nhfb, ajournal = "{Rev. Epsil{\'o}n SAEM `Thales'}", journal-URL = "http://thales.cica.es/epsilon/; http://dialnet.unirioja.es/servlet/revista?codigo=504", language = "Spanish", remark = "Reported in \cite{Torres:2007:HDN} to contain an extensive bibliography on Benford's Law. The publisher does not have issues before 2003, and no electronic copy has yet been found on the Internet. Attempts to get this paper via Interlibrary Loan failed on the grounds that ``We are unable to verify any owning libraries within the United States or those that we have ILL privileges.''", xxauthor = "Manuel {Perera Dom{\'\i}nguez} and Juan David {Ayll{\'o}n Burguillo}", } @Article{Barabasi:1999:ESR, author = "Albert-L{\'a}szl{\'o} Barab{\'a}si and R{\'e}ka Albert", title = "Emergence of scaling in random networks", journal = j-SCIENCE, volume = "286", number = "5439", pages = "509--512", day = "15", month = oct, year = "1999", CODEN = "SCIEAS", DOI = "https://doi.org/10.1126/science.286.5439.509", ISSN = "0036-8075 (print), 1095-9203 (electronic)", ISSN-L = "0036-8075", bibdate = "Tue Nov 15 09:57:42 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencemag.org/content/286/5439/509", acknowledgement = ack-nhfb, fjournal = "Science", journal-URL = "http://www.sciencemag.org/archive/", keywords = "power-law distribution; Zipf's Law", } @Article{Brakman:1999:RZT, author = "Steven Brakman and Harry Garretsen and Charles {Van Marrewijk} and Marianne {Van Den Berg}", title = "The Return of {Zipf}: Towards a Further Understanding of the Rank--Size Distribution", journal = j-J-REG-SCI, volume = "39", number = "1", pages = "183--213", year = "1999", DOI = "https://doi.org/10.1111/1467-9787.00129", ISSN = "0022-4146 (print), 1467-9787 (electronic)", ISSN-L = "0022-4146", bibdate = "Mon Feb 25 06:13:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Regional Science", } @Article{Coderre:1999:FDU, author = "David Coderre", title = "Fraud Detection Using Digital Analysis", journal = j-EDPACS, volume = "27", number = "3", pages = "1--8", month = "????", year = "1999", CODEN = "EDPCDF", DOI = "https://doi.org/10.1201/1079/43249.27.3.19990901/30268.1", ISSN = "0736-6981 (print), 1936-1009 (electronic)", ISSN-L = "0736-6981", bibdate = "Sat Jan 07 09:20:49 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.tandfonline.com/doi/abs/10.1201/1079/43249.27.3.19990901/30268.1", acknowledgement = ack-nhfb, fjournal = "EDPACS", keywords = "Benford's Law", onlinedate = "21 Dec 2006", } @Article{Egghe:1999:LZM, author = "L. Egghe", title = "On the law of {Zipf--Mandelbrot} for multi-word phrases", journal = j-J-AM-SOC-INF-SCI, volume = "50", number = "3", pages = "233--241", month = "????", year = "1999", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(1999)50:3<233::AID-ASI6>3.0.CO%3B2-8", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Mon Sep 10 06:09:31 2012", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "17 Feb 1999", } @Article{Egghe:1999:RLZ, author = "L. Egghe", title = "Research: On the law of {Zipf--Mandelbrot} for multi-word phrases", journal = j-J-AM-SOC-INF-SCI, volume = "50", number = "3", pages = "233--241", month = "????", year = "1999", CODEN = "AISJB6", DOI = "https://doi.org/10.1002/(SICI)1097-4571(1999)50:3<233::AID-ASI6>3.0.CO%3B2-8", ISSN = "0002-8231 (print), 1097-4571 (electronic)", ISSN-L = "0002-8231", bibdate = "Fri Sep 11 09:04:32 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasis.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2330-1643", onlinedate = "17 Feb 1999", } @Article{Gabaix:1999:ZLC, author = "Xavier Gabaix", title = "{Zipf}'s law for cities: An explanation", journal = j-Q-J-ECON, volume = "114", number = "3", pages = "739--767", year = "1999", CODEN = "QJECAT", DOI = "https://doi.org/10.1162/003355399556133", ISSN = "0033-5533 (print), 1531-4650 (electronic)", ISSN-L = "0033-5533", bibdate = "Sat Nov 12 10:57:19 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0952.91059", abstract = "Zipf's law is a very tight constraint on the class of admissible models of local growth. It says that for most countries the size distribution of cities strikingly fits a power law: the number of cities with populations greater than $S$ is proportional to $ 1 / S $. Suppose that, at least in the upper tail, all cities follow some proportional growth process (this appears to be verified empirically). This automatically leads their distribution to converge to Zipf's law.", acknowledgement = ack-nhfb, ajournal = "Q. J. Econ.", classmath = "91B62 (Dynamic economic models etc.) 91B72 (Spatial models)", fjournal = "The Quarterly Journal of Economics", keywords = "local growth; Zipf's law", } @Book{Gigerenzer:1999:SHM, author = "Gerd Gigerenzer and Peter M. Todd", title = "Simple heuristics that make us smart", publisher = pub-OXFORD, address = pub-OXFORD:adr, pages = "xv + 416", year = "1999", ISBN = "0-19-512156-2", ISBN-13 = "978-0-19-512156-8", LCCN = "BD260 .G54 1999", bibdate = "Sat Sep 10 11:19:26 MDT 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", series = "Evolution and cognition", URL = "http://www.loc.gov/catdir/enhancements/fy0640/98051084-d.html; http://www.loc.gov/catdir/enhancements/fy0640/98051084-t.html; http://www.loc.gov/catdir/enhancements/fy0726/98051084-b.html", acknowledgement = ack-nhfb, keywords = "Benford's Law", subject = "Heuristic", tableofcontents = "The ABC Research Group \\ I. The Research Agenda \\ 1. Fast and Frugal Heuristics: The Adaptive Toolbox / Gerd Gigerenzer and Peter M. Todd \\ II. Ignorance-Based Decision Making \\ 2. The Recognition Heuristic: How Ignorance Makes Us Smart / Daniel G. Goldstein and Gerd Gigerenzer \\ 3. Can Ignorance Beat the Stock Market? / Bernhard Borges et al. \\ III. One-Reason Decision Making \\ 4. Betting on One Good Reason: The Take The Best Heuristic / Gerd Gigerenzer and Daniel G. Goldstein \\ 5. How Good Are Simple Heuristics? / Jean Czerlinski, Gerd Gigerenzer, and Daniel G. Goldstein \\ 6. Why Does One-Reason Decision Making Work? A Case Study in Ecological Rationality / Laura Martignon and Ulrich Hoffrage \\ 7. When Do People Use Simple Heuristics, and How Can We Tell? / J{\"o}rg Rieskamp and Ulrich Hoffrage \\ 8. Bayesian Benchmarks for Fast and Frugal Heuristics / Laura Martignon and Kathryn Blackmond Laskey \\ IV. Beyond Choice: Memory, Estimation, and Categorization \\ 9. Hindsight Bias: A Price Worth Paying for Fast and Frugal Memory / Ulrich Hoffrage and Ralph Hertwig \\ 10. Quick Estimation: Letting the Environment Do the Work / Ralph Hertwig, Ulrich Hoffrage, and Laura Martignon \\ 11. Categorization by Elimination: Using Few Cues to Choose / Patricia M. Berretty, Peter M. Todd, and Laura Martignon \\ V. Social Intelligence \\ 12. How Motion Reveals Intention: Categorizing Social Interactions / Philip W. Blythe, Peter M. Todd, and Geoffrey F. Miller \\ 13. From Pride and Prejudice to Persuasion: Satisficing in Mate Search / Peter M. Todd and Geoffrey F. Miller \\ 14. Parental Investment by Simple Decision Rules / Jennifer Nerissa Davis and Peter M. Todd \\ VI. A Look Around, A Look Back, A Look Ahead \\ 15. Demons versus Heuristics in Artificial Intelligence, Behavioral Ecology, and Economics / Adam S. Goodie et al. \\ 16. What We Have Learned (So Far) / Peter M. Todd and Gerd Gigerenzer \\ References \\ Name Index \\ Subject Index", } @Article{Hill:1999:DFD, author = "Theodore P. Hill", title = "The Difficulty of Faking Data", journal = j-CHANCE, volume = "12", number = "3", pages = "27--31", month = "????", year = "1999", CODEN = "CNDCE4", DOI = "https://doi.org/10.1080/09332480.1999.10542154", ISSN = "0933-2480 (print), 1867-2280 (electronic)", ISSN-L = "0933-2480", bibdate = "Tue Nov 22 18:54:44 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://amstat.tandfonline.com/doi/abs/10.1080/09332480.1999.10542154", acknowledgement = ack-nhfb, fjournal = "Chance: a magazine for people interested in the analysis of data", journal-URL = "http://chance.amstat.org/; http://link.springer.com/journal/144; http://www.tandfonline.com/loi/ucha20", keywords = "Benford's Law", xxnote = "Check: Hurlimann bibliography says v26 pp. 8--13??; that is wrong!", } @Article{Hill:1999:PCS, author = "Theodore P. Hill", title = "Le premier chiffre significatif fait sa loi. ({French}) [{The} first significant digit makes its law]", journal = "La Recherche: L'actualit{\'e} des sciences", volume = "2", number = "316", pages = "72--76", month = jan, year = "1999", CODEN = "RCCHBV", ISSN = "0029-5671 (print), 1625-9955 (electronic)", ISSN-L = "0029-5671", bibdate = "Sat Feb 16 17:51:25 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.larecherche.fr/savoirs/autre/premier-chiffre-significatif-fait-sa-loi-01-01-1999-70970", acknowledgement = ack-nhfb, language = "French", xxjournal = "La Recherche Hors S{\'e}rie", } @Article{Huberman:1999:IGD, author = "Bernardo A. Huberman and Lada A. Adamic", title = "{Internet}: Growth dynamics of the {World-Wide Web}", journal = j-NATURE, volume = "401", number = "6749", pages = "131--131", day = "9", month = sep, year = "1999", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/43604", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Tue Nov 15 09:52:36 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", keywords = "Zipf's Law", } @TechReport{Lowe:1999:FDL, author = "Tomas Lowe and Sally Murphy and Justin Hayward", title = "The First Digit Law", type = "Report", number = "??", institution = "Department of Computing, Imperial College", address = "London, UK", month = jun, year = "1999", bibdate = "Wed Nov 23 16:59:50 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://users.isr.ist.utl.pt/~jmrs/research/TopicsLinks/Benford/hayward.pdf; http://www.doc.ic.ac.uk/~jjh97/suprema/main_page.html", acknowledgement = ack-nhfb, lastaccess = "23 November 2011", } @Article{Malcai:1999:PLD, author = "Ofer Malcai and Ofer Biham and Sorin Solomon", title = "Power-law distributions and {L{\'e}vy}-stable intermittent fluctuations in stochastic systems of many autocatalytic elements", journal = j-PHYS-REV-E, volume = "60", number = "2", pages = "1299--1303", month = aug, year = "1999", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.60.1299", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Thu Feb 28 10:25:38 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.60.1299", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", } @Article{Manrubia:1999:SMP, author = "Susanna C. Manrubia and Dami{\'a}n H. Zanette", title = "Stochastic multiplicative processes with reset events", journal = j-PHYS-REV-E, volume = "59", number = "5", pages = "4945--4948", month = may, year = "1999", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.59.4945", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:25:24 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.59.4945", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", issue = "5", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's law", } @Article{Manrubia:1999:TDS, author = "Susanna C. Manrubia and Dami{\'a}n H. Zanette and Ricard V. Sol{\'e}", title = "Transient Dynamics and Scaling Phenomena in Urban Growth", journal = j-FRACTALS, volume = "7", number = "1", pages = "1--8", month = mar, year = "1999", CODEN = "FRACEG", DOI = "https://doi.org/10.1142/S0218348X99000025", ISSN = "0218-348X", ISSN-L = "0218-348X", bibdate = "Wed Feb 27 07:11:42 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.worldscientific.com/doi/abs/10.1142/S0218348X99000025", acknowledgement = ack-nhfb, fjournal = "Fractals", journal-URL = "http://www.worldscientific.com/loi/fractals", keywords = "Zipf's Law", } @Article{Matthews:1999:PO, author = "Robert Matthews", title = "The Power of One", journal = j-NEW-SCIENTIST, volume = "163", number = "2194", pages = "27--30", day = "10", month = jul, year = "1999", CODEN = "NWSCAL", ISSN = "0262-4079 (print), 1364-8500 (electronic)", ISSN-L = "0262-4079", bibdate = "Mon Nov 14 16:04:36 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.fortunecity.com/emachines/e11/86/one.html; http://www.newscientist.com/article/mg16321944.600-the-power-of-one.html", acknowledgement = ack-nhfb, fjournal = "New Scientist", journal-URL = "http://www.sciencedirect.com/science/journal/02624079", xxISSN = "0262-4079, 0028-6664", } @Article{Mercik:1999:SAI, author = "Szymon Mercik and Karina Weron and Zuzanna Siwy", title = "Statistical analysis of ionic current fluctuations in membrane channels", journal = j-PHYS-REV-E, volume = "60", number = "6", pages = "7343--7348", day = "1", month = dec, year = "1999", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.60.7343", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.60.7343", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Book{Miller:1999:JFM, author = "Irwin Miller and Marylees Miller", title = "{John E. Freund}'s mathematical statistics", publisher = pub-PH, address = pub-PH:adr, edition = "Sixth", pages = "xii + 624", year = "1999", ISBN = "0-13-123613-X", ISBN-13 = "978-0-13-123613-4", LCCN = "QA276 .M4726 1999", bibdate = "Thu Oct 13 07:00:13 MDT 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, author-dates = "1928--", remark = "Revised edition of John E. Freund, \booktitle{Mathematical statistics}, 5th edition, 1992.", subject = "Mathematical statistics", } @Article{Nigrini:1999:AVD, author = "Mark J. Nigrini", title = "Adding value with digital analysis", journal = "The Internal Auditor: Journal of the {Institute of Internal Auditors}", volume = "56", number = "1", pages = "21--23", month = "????", year = "1999", CODEN = "ITAUAB", ISSN = "0020-5745", ISSN-L = "0020-5745", bibdate = "Sat Feb 16 18:56:27 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Nigrini:1999:IGY, author = "Mark J. Nigrini", title = "{I've} Got Your Number: How a Mathematical Phenomenon Can Help {CPAs} Uncover Fraud and Other Irregularities", journal = j-J-ACCOUNTANCY, volume = "187", number = "2", pages = "79--83", month = may, year = "1999", CODEN = "JACYAD", ISSN = "0021-8448 (print), 1945-0729 (electronic)", ISSN-L = "0021-8448", bibdate = "Mon Nov 14 16:31:40 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.aicpa.org/pubs/jofa/may1999/nigrini.htm; http://www.journalofaccountancy.com/issues/1999/may/nigrini.html", acknowledgement = ack-nhfb, fjournal = "Journal of Accountancy", journal-URL = "http://www.journalofaccountancy.com/", } @Article{Nigrini:1999:PPF, author = "Mark J. Nigrini", title = "Peculiar patterns of first digits", journal = j-IEEE-POT, volume = "18", number = "2", pages = "24--27", month = apr # "\slash " # may, year = "1999", CODEN = "IEPTDF", DOI = "https://doi.org/10.1109/45.755849", ISSN = "0278-6648 (print), 1558-1772 (electronic)", ISSN-L = "0278-6648", bibdate = "Mon Nov 21 22:18:30 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "Despite seeing numbers all the time only occasionally would one say that a list of numbers seems odd. For this to occur, the numbers are typically highly rounded or implausible (too much of a coincidence). However digit pattern analysis, using a computer, can produce valuable findings not revealed at a mere glance. The most widely cited paper on digit frequencies was published by Frank Benford in 1938. With powerful, low-cost personal computers, Benford's law can help us to efficiently test the integrity of data. The first step is to assess whether Benford's law should apply to the data. If the data is expected to conform, then Benford's law becomes the expected distribution. Conformity means that the actual distribution closely approximates the expected distribution", acknowledgement = ack-nhfb, fjournal = "IEEE Potentials", } @Article{Okuyama:1999:ZLI, author = "K. Okuyama and M. Takayasu and H. Takayasu", title = "{Zipf}'s law in income distribution of companies", journal = j-PHYSICA-A, volume = "269", number = "1", pages = "125--131", day = "1", month = jul, year = "1999", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(99)00086-2", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437199000862", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Growth rate of assets; Income distribution; Zipf's Law", } @Article{Piqueira:1999:ZLO, author = "J. R. C. Piqueira and L. H. A. Monteiro and T. M. C. de Magalh{\~a}es and R. T. Ramos and R. B. Sassi and E. G. Cruz", title = "{Zipf's Law} Organizes a Psychiatric Ward", journal = j-J-THEOR-BIOL, volume = "198", number = "3", pages = "439--443", day = "7", month = jun, year = "1999", CODEN = "JTBIAP", DOI = "https://doi.org/10.1006/jtbi.1999.0923", ISSN = "0022-5193 (print), 1095-8541 (electronic)", ISSN-L = "0022-5193", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022519399909232", acknowledgement = ack-nhfb, fjournal = "Journal of Theoretical Biology", journal-URL = "http://www.sciencedirect.com/science/journal/00225193", keywords = "Zipf's Law", } @Article{Stanley:1999:SUR, author = "H. Eugene Stanley", title = "Scaling, universality, and renormalization: Three pillars of modern critical phenomena", journal = j-REV-MOD-PHYS, volume = "71", number = "2", pages = "S358--S366", month = mar, year = "1999", CODEN = "RMPHAT", DOI = "https://doi.org/10.1103/RevModPhys.71.S358", ISSN = "0034-6861 (print), 1538-4527 (electronic), 1539-0756", ISSN-L = "0034-6861", bibdate = "Wed May 23 10:16:18 MDT 2012", bibsource = "http://rmp.aps.org/toc/RMP/v71/i2; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/revmodphys1990.bib", URL = "http://link.aps.org/doi/10.1103/RevModPhys.71.S358; http://rmp.aps.org/abstract/RMP/v71/i2/pS358_1", acknowledgement = ack-nhfb, fjournal = "Reviews of Modern Physics", journal-URL = "http://rmp.aps.org/browse", } @Article{Vandewalle:1999:ZAF, author = "N. Vandewalle and M. Ausloos", title = "The $n$-{Zipf} analysis of financial data series and biased data series", journal = j-PHYSICA-A, volume = "268", number = "1--2", pages = "240--249", day = "1", month = jun, year = "1999", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(99)00031-X", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037843719900031X", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Article{Vardi:1999:PCS, author = "Ilan Vardi", title = "Premiers chiffres significatifs et nombres alg{\'e}briques. ({French}) [Significant leading digits and algebraic numbers]", journal = j-C-R-ACAD-SCI-I, volume = "328", number = "9", pages = "749--754", month = may, year = "1999", CODEN = "CASMEI", DOI = "https://doi.org/10.1016/S0764-4442(99)80265-1", ISSN = "0764-4442 (print), 1778-3577 (electronic)", ISSN-L = "0764-4442", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.sciencedirect.com/", URL = "http://www.sciencedirect.com/science/article/pii/S0764444299802651", abstract = "I show that there is a meromorphic continuation of $ \sigma \log_\beta n / a < \log_\beta a n^{-s} $ to the whole plane if and only if $ \beta $ is a Pisot number, $ \alpha \in Q(\beta) $, and either the second largest conjugate of $ \beta $ is real or the conjugate of $ \alpha $ corresponding to the second largest conjugate of $ \beta $ is positive.", acknowledgement = ack-nhfb, fjournal = "Comptes rendus de l'Acad{\'e}mie des sciences. S{\'e}rie I, Math{'e}matique", keywords = "Benford's Law", language = "French", } @Article{Walthoe:1999:LNO, author = "Jon Walthoe and Robert Hunt and Mike Pearson", title = "Looking out for number one", journal = "Plus Magazine", volume = "??", number = "??", pages = "1--9", month = sep, year = "1999", DOI = "????", bibdate = "Mon Feb 18 06:59:18 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Published by the University of Cambridge Millennium Mathematics Project.", URL = "http://plus.maths.org/issue9/features/benford/2pdf/index.html/op.pdf", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Article{Banyard:2000:NAF, author = "Peter Banyard", title = "A New Aid for Fraud Detection", journal = "Credit Management: journal of the {Institute of Credit Management}", volume = "??", number = "??", address = "Stamford College, UK", pages = "32--33", month = mar, year = "2000", DOI = "", ISSN = "0265-2099", ISSN-L = "0265-2099", bibdate = "Thu Oct 13 08:34:05 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "", abstract = "A simple mathematical curiosity, discovered about a century ago, may well prove a surefire way of sniffing out swindlers. Benford's Law was just an observation until the theory that proved it was discovered 3 years ago. It seems unlikely, but about 30\% of numbers start with the digit one. Another 18\% start with the digit 2 and only 4.6\% start with 9. It would be quite easy to build a Benford fraud buster into any analysis of figures, while it remains extremely difficult to produce an artificial set of figures that will comply with the law.", acknowledgement = ack-nhfb, keywords = "Benford's Law", remark = "I cannot find a Web site for this journal, or for its archives, despite extensive searching.", } @Article{Blank:2000:PLC, author = "Aharon Blank and Sorin Solomon", title = "Power laws in cities population, financial markets and {Internet} sites (scaling in systems with a variable number of components)", journal = j-PHYSICA-A, volume = "287", number = "1--2", pages = "279--288", day = "15", month = nov, year = "2000", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(00)00464-7", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 06:24:21 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437100004647", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @InProceedings{Burns:2000:SSR, author = "B. D. Burns", editor = "Niels A. Taatgen and Hedderik van Rijn", booktitle = "{Proceedings of the 31st Annual Conference of the Cognitive Science Society 2009 (CogSci 2009): Amsterdam, Netherlands, 29 July--1 August 2009}", title = "Sensitivity to Statistical Regularities: People (Largely) Follow {Benford's Law}", publisher = "Cognitive Science Society", address = "Austin, TX, USA", pages = "2872--2877", year = "2010", ISBN = "1-61567-407-1", ISBN-13 = "978-1-61567-407-7", LCCN = "????", bibdate = "Sat Sep 10 10:53:37 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Article{Drake:2000:CAA, author = "Philip D. Drake and Mark J. Nigrini", title = "Computer assisted analytical procedures using {Benford's Law}", journal = j-J-ACCOUNT-EDUC, volume = "18", number = "2", pages = "127--146", month = "Spring", year = "2000", CODEN = "????", DOI = "https://doi.org/10.1016/S0748-5751(00)00008-7", ISSN = "0748-5751 (print), 1873-1996 (electronic)", ISSN-L = "0748-5751", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0748575100000087; http://www.sciencedirect.com/science/journal/07485751", abstract = "This case introduces students to Benford's Law and Digital Analysis, which can be used as an analytical procedure and fraud detection tool. Digital Analysis (DA) is the analysis of digit and number patterns of a data set. Actual digit frequencies in a data set are compared to the expected frequencies according to Benford's Law [Benford, F. (1938). The law of the anomalous numbers, Proceedings of the American Philosophical Society, 78, 551 572.]. Minor differences suggest that the data have passed a reasonableness test, while major differences signal possible financial statement mis-statements. After describing Benford's Law and the basic {DA} tests, the case requires auditing students to download {DA} software and the actual accounts payable file of a software company from a designated Internet site. Students then, (1) analyse the data using three {DA} tests as an analytical procedure in an external audit context, and (2) graph and a report the results and audit-related conclusions. The teaching note includes the actual findings from the audit of that data set and guidance on using the case in an auditing course.", acknowledgement = ack-nhfb, fjournal = "Journal of Accounting Education", keywords = "Benford's Law", } @Article{Lanza:2000:UDA, author = "R. B. Lanza", title = "Using digital analysis to detect fraud: Review of the {DATAS\reg} statistical analysis tool", journal = j-J-FORENSIC-ACCOUNT, volume = "1", number = "??", pages = "291--296", month = "????", year = "2000", CODEN = "????", ISSN = "1524-5586", bibdate = "Tue Nov 22 10:47:45 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.rtedwards.com/journals/JFA/", acknowledgement = ack-nhfb, fjournal = "Journal of Forensic Accounting", } @Article{Leemis:2000:SDS, author = "Lawrence M. Leemis and Bruce W. Schmeiser and Diane L. Evans", title = "Survival Distributions Satisfying {Benford's Law}", journal = j-AMER-STAT, volume = "54", number = "4", pages = "236--241", month = nov, year = "2000", CODEN = "ASTAAJ", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", MRclass = "62E15 (62N05)", MRnumber = "1803620", bibdate = "Fri Jan 27 18:16:34 MST 2012", bibsource = "http://www.amstat.org/publications/tas/; http://www.jstor.org/journals/00031305.html; http://www.jstor.org/stable/i326510; http://www.math.utah.edu/pub/tex/bib/amstat.bib; http://www.math.utah.edu/pub/tex/bib/amstat2000.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.amstat.org/publications/tas/Leemis.htm; http://www.jstor.org/stable/2685773", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @InProceedings{Lei:2000:EIN, author = "Skylar Lei and Michael Smith and Giancarlo Succi", title = "Empirical investigation of a novel approach to check the integrity of software engineering measuring processes", crossref = "ACM:2000:PIC", pages = "773--773", year = "2000", DOI = "https://doi.org/10.1145/337180.337629", bibdate = "Sat Feb 16 18:28:12 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Article{Lowe:2000:BLF, author = "R. Lowe", title = "{Benford's Law} and fraud detection", journal = "Chartered Accountants Journal of New Zealand", volume = "79", number = "10", pages = "32--36", month = "????", year = "2000", DOI = "", ISSN = "1172-9929", ISSN-L = "1172-9929", bibdate = "Thu Oct 13 08:55:37 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "", acknowledgement = ack-nhfb, journal-URL = "http://www.nzica.com/journalarchive.aspx", remark = "No archives online before 2012. Library of Congress says that publication ceased in July 2005 at volume 84, number 6.", } @Article{Lowe:2000:WBL, author = "R. Lowe", title = "When {Benford's Law} is broken", journal = "Chartered Accountants Journal of New Zealand", volume = "79", number = "11", pages = "24--27", month = "????", year = "2000", DOI = "", ISSN = "1172-9929", ISSN-L = "1172-9929", bibdate = "Thu Oct 13 08:55:37 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "", acknowledgement = ack-nhfb, journal-URL = "http://www.nzica.com/journalarchive.aspx", remark = "No archives online before 2012. Library of Congress says that publication ceased in July 2005 at volume 84, number 6.", } @TechReport{Nigrini:2000:CA, author = "Mark J. Nigrini", title = "Continuous Auditing", type = "Preprint", institution = "Ernst \& Young Center for Auditing Research and Advanced Technology, University of Kansas", address = "Lawrence, KS, USA", pages = "23", day = "30", month = aug, year = "2000", bibdate = "Sat Feb 16 18:57:40 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "https://aaahq.org/audit/midyear/01midyear/papers/nigrini_continuous_audit.pdf", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Book{Nigrini:2000:DAT, author = "Mark J. Nigrini", title = "Digital analysis tests and statistics: using digit and number patterns and Benford's law to detect errors, biases, fraud, irregularities, and processing inefficiencies", publisher = "Nigrini Institute Inc.", address = "Allen, TX, USA", pages = "242", year = "2000", LCCN = "????", bibdate = "Sat Jan 23 07:53:08 MST 2016", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Book{Nigrini:2000:DAU, author = "Mark John Nigrini", title = "Digital analysis using {Benford}'s law: tests and statistics for auditors", publisher = "Global Audit Publications", address = "Vancouver, BC, Canada", edition = "Second", pages = "xvii + 278", year = "2000", ISBN = "189449709 (??invalid ISBN??)", ISBN-13 = "189449709 (??invalid ISBN??)", LCCN = "????", bibdate = "Tue Nov 22 20:48:29 MST 2011", bibsource = "amicus.nlc-bnc.ca:210/NL; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, subject = "Numerical analysis; Data processing; Auditing; Statistical methods; Analyse num{\'e}rique; Informatique; V{\'e}rification comptable; M{\'e}thodes statistiques", } @Article{Padmanabhan:2000:CAD, author = "Venkata N. Padmanabhan and Lili Qiu", title = "The content and access dynamics of a busy {Web} server (poster session)", journal = j-SIGMETRICS, volume = "28", number = "1", pages = "122--123", month = jun, year = "2000", CODEN = "????", DOI = "https://doi.org/10.1145/339331.339405", ISSN = "0163-5999 (print), 1557-9484 (electronic)", ISSN-L = "0163-5999", bibdate = "Thu Jun 26 11:31:11 MDT 2008", bibsource = "http://portal.acm.org/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/sigmetrics.bib", abstract = "We study the MSNBC Web site, one of the busiest in the Internet today. We analyze the dynamics of content creation and modification as well as client accesses. Our key findings are (a) files tend to change little upon modification, (b) a small set of files get modified repeatedly, (c) file popularity follows a Zipf-like distribution with an $ \alpha $ much larger than reported in previous, proxy-based studies, and (d) there is significant temporal stability in file popularity but not much stability in the domains from which popular content is accessed. We discuss implications of these findings.", acknowledgement = ack-nhfb, fjournal = "ACM SIGMETRICS Performance Evaluation Review", journal-URL = "http://portal.acm.org/toc.cfm?id=J618", } @Book{Sole:2000:SLH, author = "Ricard Vicente Sol{\'e} and Brian Goodwin", title = "Signs of Life: How Complexity Pervades Biology", publisher = pub-BASIC-BOOKS, address = pub-BASIC-BOOKS:adr, pages = "xi + 322", year = "2000", ISBN = "0-465-01927-7, 0-465-01928-5 (paperback)", ISBN-13 = "978-0-465-01927-4, 978-0-465-01928-1 (paperback)", LCCN = "QH501 .S65 2000", bibdate = "Thu Feb 28 09:59:50 MST 2013", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, keywords = "Mandelbrot; Zipf's law", tableofcontents = "Nonlinearity, chaos, and emergence \\ Order, complexity, disorder \\ Genetic networks, cell differentiation, and development \\ Physiology on the edge of chaos \\ Brain dynamics \\ Ants, brains, and chaos \\ The Baroque of nature \\ Life on the edge of catastrophe \\ Evolution and extinction \\ Fractal cities and market crashes", } @Article{Soliman:2000:BPP, author = "Ahmed A. Soliman", title = "{Bayes} Prediction in a {Pareto} Lifetime Model with Random Sample Size", journal = j-J-R-STAT-SOC-SER-D-STATISTICIAN, volume = "49", number = "1", pages = "51--62", month = "????", year = "2000", CODEN = "????", DOI = "https://doi.org/10.2307/2681255", ISSN = "0039-0526 (print), 1467-9884 (electronic)", ISSN-L = "0039-0526", bibdate = "Thu Jan 22 18:10:23 MST 2015", bibsource = "http://www.jstor.org/stable/i326257; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jrss-d-2000.bib", URL = "http://www.jstor.org/stable/2681255", acknowledgement = ack-nhfb, fjournal = "Journal of the Royal Statistical Society. Series D (The Statistician)", journal-URL = "http://www.jstor.org/journals/00390526.html", } @Article{Stanley:2000:SIU, author = "H. E. Stanley and L. A. N. Amaral and P. Gopikrishnan and P. Ch. Ivanov and T. H. Keitt and V. Plerou", title = "Scale invariance and universality: organizing principles in complex systems", journal = j-PHYSICA-A, volume = "281", number = "1--4", pages = "60--68", day = "15", month = jun, year = "2000", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(00)00195-3", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Thu Feb 28 09:52:12 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437100001953", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "power law; scale invariance; Zipf's law", } @Article{Tapp:2000:UTD, author = "D. J. Tapp and D. B. Burg", title = "Using technology to detect fraud", journal = "{Pennsylvania CPA} Journal", volume = "71", number = "4", pages = "20--23", month = "????", year = "2000", ISSN = "0746-1062", ISSN-L = "0746-1062", bibdate = "Thu Oct 13 09:05:22 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, journal-URL = "https://www.picpa.org/keep-informed/pennsylvania-cpa-journal", keywords = "Benford's Law", remark = "No archives online before 2014.", } @Article{Tolle:2000:DDS, author = "Charles R. Tolle and Joanne L. Budzien and Randall A. Laviolette", title = "Do dynamical systems follow {Benford}'s law?", journal = j-CHAOS, volume = "10", number = "2", pages = "331--336", month = "????", year = "2000", CODEN = "CHAOEH", DOI = "https://doi.org/10.1063/1.166498", ISSN = "1054-1500", ISSN-L = "1054-1500", bibdate = "Mon Nov 14 17:35:25 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://adsabs.harvard.edu/abs/2000Chaos..10..331T; http://link.aip.org/link/chaoeh/v10/i2/p331/s1; http://link.aip.org/link/doi/10.1063/1.166498", abstract = "Data compiled from a variety of sources follow Benford's law, which gives a monotonically decreasing distribution of the first digit (1 through 9). We examine the frequency of the first digit of the coordinates of the trajectories generated by some common dynamical systems. One-dimensional cellular automata fulfill the expectation that the frequency of the first digit is uniform. The molecular dynamics of fluids, on the other hand, provides trajectories that follow Benford's law. Finally, three chaotic systems are considered: Lorenz, H{\'e}non, and R{\"o}ssler. The Lorenz system generates trajectories that follow Benford's law. The H{\'e}non system generates trajectories that resemble neither the uniform distribution nor Benford's law. Finally, the R{\"o}ssler system generates trajectories that follow the uniform distribution for some parameters choices, and Benford's law for others.", acknowledgement = ack-nhfb, fjournal = "Chaos (Woodbury, NY)", journal-URL = "http://scitation.aip.org/content/aip/journal/chaos", xxpages = "331--337", } @Article{Urzua:2000:SET, author = "Carlos M. Urz{\'u}a", title = "A simple and efficient test for {Zipf}'s law", journal = j-ECONOM-LETT, volume = "66", number = "3", pages = "257--260", day = "1", month = mar, year = "2000", CODEN = "ECLEDS", DOI = "https://doi.org/10.1016/S0165-1765(99)00215-3", ISSN = "0165-1765 (print), 1873-7374 (electronic)", ISSN-L = "0165-1765", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0165176599002153", ZMnumber = "0951.91059", abstract = "This paper presents a simple and locally optimal test for Zipf's law. Its use is illustrated in the case of the largest US metropolitan areas. An objection to the general relevance of that law is also presented.", acknowledgement = ack-nhfb, classmath = "91B82 (Statistical methods in economics) 91B74 (Models of real-world systems)", fjournal = "Economics Letters", keywords = "rank-size law; Zipf's law", } @Article{Vogt:2000:BGS, author = "W. Vogt", title = "{Benford's Gesetz: Steuer- und Budgets{\"u}ndern auf der Spur --- Zahlen l{\"u}gen nichts}. ({German}) [{Benford's Law}: tax and budget sinners lying on the track --- Numbers do not lie]", journal = "Schweizer Versicherung", volume = "9", number = "??", pages = "27--29", month = "????", year = "2000", DOI = "????", bibdate = "Mon Feb 18 06:57:07 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "????", acknowledgement = ack-nhfb, journal-URL = "http://www.schweizerversicherung.ch/de/archiv_suche", language = "German", remark = "No ISSN assigned. Not found in journal archive on 18 February 2013.", } @Article{York:2000:ATB, author = "D. York", title = "Auditing Technique --- {Benford}'s law", journal = j-ACCOUNTANCY, volume = "126", number = "1283", pages = "126--??", month = "????", year = "2000", CODEN = "ACTYAD", ISSN = "0001-4664", ISSN-L = "0001-4664", bibdate = "Wed Nov 23 11:30:54 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Accountancy", journal-URL = "https://www.cchdaily.co.uk/magazine", remark = "No archives online before 2014", } @Article{Axtell:2001:ZDU, author = "R. L. Axtell", title = "{Zipf} Distribution of {U.S.} Firm Sizes", journal = j-SCIENCE, volume = "293", number = "5536", pages = "1818--1820", day = "7", month = sep, year = "2001", CODEN = "SCIEAS", DOI = "https://doi.org/10.1126/science.1062081", ISSN = "0036-8075 (print), 1095-9203 (electronic)", ISSN-L = "0036-8075", bibdate = "Wed Feb 27 06:38:34 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencemag.org/content/293/5536/1818", acknowledgement = ack-nhfb, fjournal = "Science", journal-URL = "http://www.sciencemag.org/archive/", keywords = "Zipf's Law", } @Article{Dolgikh:2001:PCS, author = "Dmitry G. Dolgikh and Andrei M. Sukhov", title = "Parameters of cache systems based on a {Zipf}-like distribution", journal = j-COMP-NET-AMSTERDAM, volume = "37", number = "6", pages = "711--716", day = "15", month = dec, year = "2001", CODEN = "????", DOI = "https://doi.org/10.1016/S1389-1286(01)00243-2", ISSN = "1389-1286 (print), 1872-7069 (electronic)", ISSN-L = "1389-1286", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.elsevier.com/locate/issn/13891286; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.elsevier.nl/gej-ng/10/15/22/67/36/31/abstract.html; http://www.sciencedirect.com/science/article/pii/S1389128601002432", acknowledgement = ack-nhfb, fjournal = "Computer Networks (Amsterdam, Netherlands: 1999)", journal-URL = "http://www.sciencedirect.com/science/journal/13891286", keywords = "Zipf's Law", } @Article{Dubinsky:2001:MFF, author = "Bruce G. Dubinsky", title = "Math formula fights fraud", journal = "Legal Times of {Washington}", volume = "XXIV", number = "9", pages = "??--??", day = "26", month = feb, year = "2001", ISSN = "0732-7536", ISSN-L = "0732-7536", bibdate = "Sat Feb 16 17:35:39 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, keywords = "Benford's Law", xxjournal = "Legal Times", } @Article{Gelbukh:2001:ZHL, author = "Alexander Gelbukh and Grigori Sidorov", title = "{Zipf} and {Heaps Laws}' Coefficients Depend on Language", journal = j-LECT-NOTES-COMP-SCI, volume = "2004", pages = "332--??", year = "2001", CODEN = "LNCSD9", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Sat Feb 2 13:03:22 MST 2002", bibsource = "http://link.springer-ny.com/link/service/series/0558/tocs/t2004.htm; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.springer-ny.com/link/service/series/0558/bibs/2004/20040332.htm; http://link.springer-ny.com/link/service/series/0558/papers/2004/20040332.pdf", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @TechReport{Gent:2001:BL, author = "Ian Gent and Toby Walsh", title = "{Benford's Law}", type = "Report", institution = "School of Computer Science, University of St. Andrews, and Department of Computer Science, University of York", address = "St. Andrews, Scotland and York, England", pages = "7", day = "2", month = feb, year = "2001", bibdate = "Fri Nov 18 08:37:38 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.dcs.st-and.ac.uk/~apes/reports/apes-25-2001.pdf", abstract = "Benford's Law predicts the frequency of the leading digit in numbers met in a wide range of naturally occurring phenomena. In data following Benford's Law, numbers start with a small leading digit more often those with a large leading digit. Here we demonstrate that Benford's Law also describes a wide range of computational phenomena. In particular, we show that a number of different statistics associated with computation like space and runtime often follow Benford's Law. We also show that search cost on input data that follows Benford's Law is often very different to that on more uniform data. These results could be used to improve algorithm performance (for example, for load balancing or disk de-fragmentation), as well as to help model algorithm performance. Benford's Law can also be used to generate data for benchmarking algorithms that may be more realistic than purely random data.", acknowledgement = ack-nhfb, remark = "PDF file no longer at the St. Andrews URL on 18 November 2011, but recovered from http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.3357.", } @Article{Harremoes:2001:MEF, author = "Peter Harremo{\"e}s and Flemming Tops{\o}e", title = "Maximum Entropy Fundamentals", journal = j-ENTROPY, volume = "3", number = "3", pages = "191--226", month = sep, year = "2001", CODEN = "ENTRFG", DOI = "https://doi.org/10.3390/e3030191", ISSN = "1099-4300", ISSN-L = "1099-4300", bibdate = "Thu Feb 28 10:55:45 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.mdpi.com/1099-4300/3/3/191", acknowledgement = ack-nhfb, fjournal = "Entropy", keywords = "continuity of entropy; entropy loss; exponential family; game theoretical equilibrium; hyperbolic distributions; information topology; maximum entropy; minimum risk; Nash equilibrium code; partition function; Zipf's law", remark = "Open access journal.", } @Article{Hobza:2001:NBL, author = "Tom{\'a}{\v{s}} Hobza and Igor Vajda", title = "On the {Newcomb--Benford} law in models of statistical data", journal = "Revista Matem{\'a}tica Complutense", volume = "14", number = "2", pages = "407--420", year = "2001", CODEN = "????", ISSN = "1139-1138, 1696-8220, 1988-2807", ISSN-L = "1139-1138", MRclass = "62E10", MRnumber = "1871305 (2002j:62015)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, ajournal = "Rev. Mat. Complut.", fjournal = "Revista Matem{\'a}tica Complutense", } @Article{Huang:2001:PLE, author = "Z. F. Huang and S. Solomon", title = "Power, {L{\'e}vy}, exponential and {Gaussian}-like regimes in autocatalytic financial systems", journal = j-EUR-PHYS-J-B, volume = "20", number = "4", pages = "601--607", month = apr, year = "2001", CODEN = "EPJBFY", DOI = "https://doi.org/10.1007/PL00011114", ISSN = "1434-6028 (print), 1434-6036 (electronic)", ISSN-L = "1434-6028", bibdate = "Thu Feb 28 10:26:53 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.springer.com/article/10.1007/PL00011114", acknowledgement = ack-nhfb, fjournal = "European Physical Journal B: Condensed Matter and Complex Systems", journal-URL = "http://link.springer.com/journal/10051", keywords = "power law; scale invariance; Zipf's law", } @MastersThesis{Jamain:2001:BL, author = "Adrien Jamain", title = "{Benford's Law}", type = "{Master}'s thesis", school = "Department of Mathematics, Imperial College of London and ENSIMAG", address = "London, UK", pages = "????", year = "2001", bibdate = "Sat Feb 16 18:06:41 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Not found in Imperial College Library or COPAC catalogs on 16 February 2013. URL link is broken too.", URL = "http://www.math.ualberta.ca/~aberger/benford_bibliography/jamain_thesis01.pdf", acknowledgement = ack-nhfb, } @Article{Jansen:2001:RNP, author = "C. J. M. Jansen and M. M. W. Pollmann", title = "On Round Numbers: Pragmatic Aspects of Numerical Expressions", journal = j-J-QUANT-LINGUISTICS, volume = "8", number = "3", pages = "187--201", month = "????", year = "2001", DOI = "https://doi.org/10.1076/jqul.8.3.187.4095", ISSN = "0929-6174 (print), 1744-5035 (electronic)", ISSN-L = "0929-6174", bibdate = "Sat Jan 07 09:51:05 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.tandfonline.com/doi/abs/10.1076/jqul.8.3.187.4095", abstract = "This paper describes and explains some regularities in the frequency of numbers in text. An analysis of number frequencies in text corpora in Dutch, English, German, and French confirms the expectation that frequency is highly dependent on two factors: magnitude and roundness. Roundness (defined as number frequency in an approximation context) proves to be related to three arithmetical properties: `10-ness', `2-ness', and `5-ness'. In predicting the frequency of numbers irrespective of their context `$ 2 1 / 2 $-ness' should be added to these factors, as is suggested in the work of Sigurd (1988). The role of the four number characteristics found in this study can be explained by the preference of the language user for using base numbers, and for doubling and halving quantities.", acknowledgement = ack-nhfb, fjournal = "Journal of Quantitative Linguistics", onlinedate = "09 Aug 2010", } @Article{Jolion:2001:IBL, author = "Jean-Michel Jolion", title = "Images and {Benford}'s law", journal = j-J-MATH-IMAG-VIS, volume = "14", number = "1", pages = "73--81", year = "2001", CODEN = "JMIVEK", DOI = "https://doi.org/10.1023/A:1008363415314", ISSN = "0924-9907 (print), 1573-7683 (electronic)", ISSN-L = "0924-9907", MRclass = "94A08 (68U10)", MRnumber = "1818436 (2001m:94007)", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "See comments and extensions in \cite[Chapter 19]{Miller:2015:BLT}.", URL = "http://link.springer.com/article/10.1023/A%3A1008363415314", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Imaging and Vision: JMIV", } @Article{Knudsen:2001:ZLC, author = "Thorbj{\o}rn Knudsen", title = "{Zipf}'s law for cities and beyond: the case of {Denmark}", journal = j-AM-J-ECON-SOCIOL, volume = "60", number = "1", pages = "123--146", month = jan, year = "2001", CODEN = "AJESA3", DOI = "https://doi.org/10.1111/1536-7150.00057", ISSN = "0002-9246 (print), 1536-7150 (electronic)", ISSN-L = "0002-9246", bibdate = "Sat Feb 16 18:14:02 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.jstor.org/stable/3487947", acknowledgement = ack-nhfb, fjournal = "The American Journal of Economics and Sociology", } @Article{Kretschmer:2001:RAI, author = "Hildrun Kretschmer and Ronald Rousseau", title = "Research: Author inflation leads to a breakdown of {Lotka's Law}", journal = j-J-AM-SOC-INF-SCI-TECHNOL, volume = "52", number = "8", pages = "610--614", month = "????", year = "2001", CODEN = "JASIEF", DOI = "https://doi.org/10.1002/asi.1118", ISSN = "1532-2882 (print), 1532-2890 (electronic)", ISSN-L = "1532-2882", bibdate = "Fri Sep 11 10:41:58 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasist.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science and Technology: JASIST", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1532-2890", onlinedate = "27 Apr 2001", } @Article{Levene:2001:ZLW, author = "Mark Levene and Jos{\'e} Borges and George Loizou", title = "{Zipf}'s law for {Web} surfers", journal = j-KNOWL-INFO-SYS, volume = "3", number = "1", pages = "120--129", year = "2001", CODEN = "KISNCR", DOI = "https://doi.org/10.1007/PL00011657", ISSN = "0219-1377", bibdate = "Sat Nov 12 11:00:30 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "0991.68622", abstract = "One of the main activities of Web users, known as `surfing', is to follow links. Lengthy navigation often leads to disorientation when users lose track of the context in which they are navigating and are unsure how to proceed in terms of the goal of their original query. Studying navigation patterns of Web users is thus important, since it can lead us to a better understanding of the problems users face when they are surfing. We derive Zipf's rank frequency law (i.e., an inverse power law) from an absorbing Markov chain model of surfers' behavior assuming that less probable navigation trails are, on average, longer than more probable ones. In our model the probability of a trail is interpreted as the relevance (or `value') of the trail. We apply our model to two scenarios: in the first the probability of a user terminating the navigation session is independent of the number of links he has followed so far, and in the second the probability of a user terminating the navigation session increases by a constant each time the user follows a link. We analyze these scenarios using two sets of experimental data sets showing that, although the first scenario is only a rough approximation of surfers' behavior, the data is consistent with the second scenario and can thus provide an explanation of surfers' behavior.", acknowledgement = ack-nhfb, classmath = "68U99 (Computing methodologies) 68P15 (Database theory) 68P20 (Information storage and retrieval)", fjournal = "Knowledge and Information Systems", keywords = "Markov chain; Web navigation; Zipf's law", } @Article{Losee:2001:TDB, author = "Robert M. Losee", title = "Term dependence: a basis for {Luhn} and {Zipf} models", journal = j-J-AM-SOC-INF-SCI-TECHNOL, volume = "52", number = "12", pages = "1019--1025", year = "2001", CODEN = "JASIEF", DOI = "https://doi.org/10.1002/asi.1155", ISSN = "1532-2882 (print), 1532-2890 (electronic)", ISSN-L = "1532-2882", bibdate = "Mon Feb 25 06:13:56 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science and Technology", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1532-2890", } @Article{Malacarne:2001:QED, author = "L. C. Malacarne and R. S. Mendes and E. K. Lenzi", title = "$q$-exponential distribution in urban agglomeration", journal = j-PHYS-REV-E, volume = "65", number = "??", pages = "017106", day = "21", month = dec, year = "2001", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.65.017106", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.65.017106", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf--Mandelbrot law", } @Article{Montemurro:2001:BZM, author = "Marcelo A. Montemurro", title = "Beyond the {Zipf--Mandelbrot} law in quantitative linguistics", journal = j-PHYSICA-A, volume = "300", number = "3--4", pages = "567--578", day = "15", month = nov, year = "2001", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(01)00355-7", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437101003557", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @PhdThesis{Morters:2001:BGV, author = "Peter M{\"o}rters", title = "{Benford's Gesetz {\"u}ber die Verteilung der Ziffern}. (German) [{Benford}'s law on the distribution of digits]", type = "{Habilitationsvorlesung}", school = "????", address = "Kaiserslauten, Germany, and Bath, UK", pages = "4", month = "????", year = "2001", bibdate = "Sat Feb 16 18:47:15 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://people.bath.ac.uk/maspm/benford.ps", acknowledgement = ack-nhfb, language = "German", } @Article{Nigrini:2001:DAU, author = "Mark Nigrini", title = "Digital Analysis Using {Benford's} Law: Tests and Statistics for Auditors", journal = j-EDPACS, volume = "28", number = "9", pages = "1--2", month = "????", year = "2001", CODEN = "EDPCDF", DOI = "https://doi.org/10.1201/1079/43266.28.9.20010301/30389.4", ISSN = "0736-6981 (print), 1936-1009 (electronic)", ISSN-L = "0736-6981", bibdate = "Sat Jan 07 09:08:09 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.tandfonline.com/doi/abs/10.1201/1079/43266.28.9.20010301/30389.4", acknowledgement = ack-nhfb, fjournal = "EDPACS", onlinedate = "21 Dec 2006", } @Article{Pietronero:2001:EUD, author = "L. Pietronero and E. Tosatti and V. Tosatti and A. Vespignani", title = "Explaining the uneven distribution of numbers in nature: the laws of {Benford} and {Zipf}", journal = j-PHYSICA-A, volume = "293", number = "1--2", pages = "297--304", day = "1", month = apr, year = "2001", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(00)00633-6", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://adsabs.harvard.edu/abs/2001PhyA..293..297P; http://www.sciencedirect.com/science/article/pii/S0378437100006336", abstract = "The distribution of first digits in numbers series obtained from very different origins shows a marked asymmetry in favor of small digits that goes under the name of Benford's law. We analyze in detail this property for different data sets and give a general explanation for the origin of the Benford's law in terms of multiplicative processes. We show that this law can be also generalized to series of numbers generated from more complex systems like the catalogs of seismic activity. Finally, we derive a relation between the generalized Benford's law and the popular Zipf's law which characterize the rank order statistics and has been extensively applied to many problems ranging from city population to linguistics.", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Article{Reed:2001:PZO, author = "William J. Reed", title = "The {Pareto}, {Zipf} and other power laws", journal = j-ECONOM-LETT, volume = "74", number = "1", pages = "15--19", day = "20", month = dec, year = "2001", CODEN = "ECLEDS", DOI = "https://doi.org/10.1016/S0165-1765(01)00524-9", ISSN = "0165-1765 (print), 1873-7374 (electronic)", ISSN-L = "0165-1765", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0165176501005249", acknowledgement = ack-nhfb, fjournal = "Economics Letters", keywords = "Zipf's Law", } @TechReport{Scott:2001:BLE, author = "P. D. Scott and M. Fasli", title = "{Benford's Law}: An Empirical Investigation and a Novel Explanation", type = "CSM Technical Report", number = "349", institution = "Department of Computer Science, University Essex", address = "Colchester, UK", day = "26", month = oct, year = "2001", bibdate = "Tue Nov 22 21:05:22 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.essex.ac.uk/csee/research/publications/technicalreports/2001/CSM-349.pdf", acknowledgement = ack-nhfb, } @Article{Sellke:2001:CVT, author = "Thomas Sellke and M. J. Bayarri and James O. Berger", title = "Calibration of $p$ Values for Testing Precise Null Hypotheses", journal = j-AMER-STAT, volume = "55", number = "1", pages = "62--71", month = feb, year = "2001", CODEN = "ASTAAJ", DOI = "https://doi.org/10.1198/000313001300339950", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", bibdate = "Fri Jan 27 18:16:34 MST 2012", bibsource = "http://www.amstat.org/publications/tas/; http://www.jstor.org/journals/00031305.html; http://www.jstor.org/stable/i326511; http://www.math.utah.edu/pub/tex/bib/amstat2000.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.amstat.org/publications/tas/Sellke.htm; http://www.jstor.org/stable/2685531; http://www.tandfonline.com/doi/abs/10.1198/000313001300339950", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @Article{Snyder:2001:SAO, author = "Mark A. Snyder and James H. Curry and Anne M. Dougherty", title = "Stochastic aspects of one-dimensional discrete dynamical systems: {Benford}'s law", journal = j-PHYS-REV-E, volume = "64", number = "2", pages = "026222:1--026222:5", month = aug, year = "2001", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.64.026222", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Mon Nov 14 17:44:15 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://adsabs.harvard.edu/abs/2001PhRvE..64b6222S; http://pre.aps.org/abstract/PRE/v64/i2/e026222", abstract = "Benford's law owes its discovery to the ``Grubby Pages Hypothesis,'' a 19th century observation made by Simon Newcomb that the beginning pages of logarithm books were grubbier than the last few pages, implying that scientists referenced the values toward the front of the books more frequently. If a data set satisfies Benford's law, then its significant digits will have a logarithmic distribution, which favors smaller significant digits. In this article we demonstrate two ways of creating discrete one-dimensional dynamical systems that satisfy Benford's law. We also develop a numerical simulation methodology that we use to study dynamical systems when analytical results are not readily available.", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", pagecount = "5", } @InCollection{Solomon:2001:SPZ, author = "Sorin Solomon and Peter Richmond", booktitle = "Economics with heterogeneous interacting agents ({Marseille}, 2000)", title = "Stability of {Pareto--Zipf} law in non-stationary economies", volume = "503", publisher = pub-SV, address = pub-SV:adr, pages = "141--159", year = "2001", MRclass = "91B28 (91B62)", MRnumber = "1838925", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", series = "Lecture Notes in Econom. and Math. Systems", acknowledgement = ack-nhfb, } @Article{Zanette:2001:VTC, author = "Dami{\'a}n H. Zanette and Susanna C. Manrubia", title = "Vertical transmission of culture and the distribution of family names", journal = j-PHYSICA-A, volume = "295", number = "1--2", pages = "1--8", day = "1", month = jun, year = "2001", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(01)00046-2", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:03:20 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437101000462", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Article{Abe:2002:STE, author = "Sumiyoshi Abe", title = "Stability of {Tsallis} entropy and instabilities of {R{\'e}nyi} and normalized {Tsallis} entropies: a basis for $q$-exponential distributions", journal = j-PHYS-REV-E, volume = "66", number = "??", pages = "046134", day = "24", month = oct, year = "2002", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.66.046134", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.66.046134", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Article{Anonymous:2002:ZLM, author = "Anonymous", title = "{Zipf}'s law and maximum sustainable growth", journal = j-STAT-PROB-LETT, volume = "56", number = "2", pages = "??--??", day = "15", month = jan, year = "2002", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/j.jedc.2013.02.004", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0165188913000341", acknowledgement = ack-nhfb, fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", keywords = "Zipf's Law", onlinedate = "19 February 2013", } @InProceedings{Bhattacharya:2002:KBT, author = "S. Bhattacharya", editor = "????", booktitle = "9th World Congress of Accounting Historians, Deakin University, Melbourne, Australia July 30--August 3, 2002", title = "From {Kautilya} to {Benford} --- Trends in Forensic and Investigative Accounting", publisher = "Accounting Association of Australia and New Zealand", address = "????", pages = "??--??", year = "2002", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Sat Feb 16 17:22:29 2013", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, journal-URL = "http://ach.sagepub.com/", remark = "I cannot find the published proceedings in online catalogs or bookstores, or in the Deakin University library catalogs. Where only selected papers published from the conference in the journal Accounting History??", review-doi = "https://doi.org/10.1177/103237320200700207", review-url = "http://dro.deakin.edu.au/view/DU:30012663; http://www.highbeam.com/doc/1G1-97484707.html", } @Article{Bolton:2002:SFD, author = "Richard J. Bolton and David J. Hand", title = "Statistical Fraud Detection: A Review", journal = j-STAT-SCI, volume = "17", number = "3", pages = "235--255", month = aug, year = "2002", CODEN = "STSCEP", ISSN = "0883-4237 (print), 2168-8745 (electronic)", ISSN-L = "0883-4237", bibdate = "Fri May 30 12:48:50 MDT 2014", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/statsci.bib", URL = "http://projecteuclid.org/euclid.ss/1042727940", acknowledgement = ack-nhfb, fjournal = "Statistical Science", journal-URL = "http://projecteuclid.org/all/euclid.ss", } @Article{Brown:2002:ICF, author = "Philip Brown and Angeline Chua and Jason Mitchell", title = "The influence of cultural factors on price clustering: Evidence from {Asia Pacific} stock markets", journal = "Pacific-Basin Finance Journal", volume = "10", number = "3", pages = "307--332", year = "2002", DOI = "https://doi.org/10.1016/S0927-538X(02)00049-5", ISSN = "0927-538X (print), 1879-0585 (electronic)", ISSN-L = "0927-538X", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "13th Annual PACAP/FMA Finance Conference", URL = "http://www.sciencedirect.com/science/article/pii/S0927538X02000495", abstract = "Price clustering is the tendency of prices to be observed more frequently at some numbers than others. It increases with haziness, or imprecision, about underlying value. Most research on price clustering has been conducted in Western financial markets, where there is manifest preference for trading at round numbers. We focus on number preferences under Chinese culture. Many Chinese believe some numbers are unlucky and to be avoided. For instance, the number 4 is inauspicious because the Cantonese pronunciation of 4 is similar to the phrase to die. We first document clustering of daily closing prices on six Asia Pacific stock markets, three with predominantly Chinese populations. Next, we fit binomial logit models within these markets to estimate the association between structural and economic factors, and culture, on price clustering. We find some support for the influence of Chinese culture and superstition on year-round number preferences of traders, but it is located solely in the Hong Kong market. Furthermore, in the Hong Kong market Chinese culture and superstition help explain the increased avoidance of the number 4 during the auspicious Chinese New Year, Dragon Boat and Mid-Autumn festivals.", acknowledgement = ack-nhfb, keywords = "Attraction; Benford's Law; Haziness; Negotiation; Price clustering", } @TechReport{Diekmann:2002:DFM, author = "Andreas Diekmann", title = "{Diagnose von Fehlerquellen und methodische Qualit{\"a}t in der sozialwissenschaftlichen Forschung}. ({German}) [{Diagnosis} of errors and methodological quality in social science research]", type = "Report", number = "ITA-02-04", institution = "Institut f{\"u}r Technikfolgen-Absch{\"a}tzung", address = "Vienna, Austria", pages = "????", year = "2002", bibdate = "Thu Nov 17 19:11:00 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, language = "German", } @Article{Egghe:2002:DRC, author = "L. Egghe and I. K. Ravichandra Rao", title = "Duality revisited: {Construction} of fractional frequency distributions based on two dual {Lotka} laws", journal = j-J-AM-SOC-INF-SCI-TECHNOL, volume = "53", number = "10", pages = "789--801", month = "????", year = "2002", CODEN = "JASIEF", DOI = "https://doi.org/10.1002/asi.10103", ISSN = "1532-2882 (print), 1532-2890 (electronic)", ISSN-L = "1532-2882", bibdate = "Fri Sep 11 10:41:59 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasist.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science and Technology: JASIST", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1532-2890", onlinedate = "11 Jun 2002", } @Article{FerreriCancho:2002:ZLR, author = "Ramon {Ferrer i Cancho} and Ricard V. Sol{\'e}", title = "{Zipf}'s law and random texts", journal = j-ADV-COMPLEX-SYST, volume = "5", number = "1", pages = "1--6", year = "2002", CODEN = "????", DOI = "https://doi.org/10.1142/S0219525902000468", ISSN = "0219-5259", bibdate = "Sat Nov 12 11:00:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", ZMnumber = "1090.91595", abstract = "Random-text models have been proposed as an explanation for the power law relationship between word frequency and rank, the so-called Zipf's law. They are generally regarded as null hypotheses rather than models in the strict sense. In this context, recent theories of language emergence and evolution assume this law as a priori information with no need of explanation. Here, random texts and real texts are compared through (a) the so-called lexical spectrum and (b) the distribution of words having the same length. It is shown that real texts fill the lexical spectrum much more efficiently and regardless of the word length, suggesting that the meaningfulness of Zipf's law is high.", acknowledgement = ack-nhfb, classmath = "91F20 (Linguistics)", fjournal = "Advances in Complex Systems", journal-URL = "http://www.worldscinet.com/acs/acs.shtml", keywords = "Human language; monkey languages; random texts; scaling; Zipf's law", } @Article{Gottwald:2002:NBL, author = "Georg A. Gottwald and Matthew Nicol", title = "On the nature of {Benford's Law}", journal = j-PHYSICA-A, volume = "303", number = "3--4", pages = "387--396", day = "15", month = jan, year = "2002", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(01)00497-6", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", MRclass = "60C05 (28D05 60E05 82B05)", MRnumber = "1917635 (2003d:60015)", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.sciencedirect.com/", URL = "http://adsabs.harvard.edu/abs/2002PhyA..303..387G; http://www.sciencedirect.com/science/article/pii/S0378437101004976", abstract = "We study multiplicative and affine sequences of real numbers defined by $ N(j + 1) = \zeta (j) N(j) + \eta (j) $, where \{\zeta(j)\} and \{\eta(j)\} are sequences of positive real numbers (in the multiplicative case $ \eta (j) = 0 $ for all $j$). We investigate the conditions under which the leading digits $k$ of \{N(j)\} have the following probability distribution, known as Benford's Law, $ P(k) = \log_{10}((k + 1) / k)$. We present two main results. First, we show that contrary to the usual assumption in the literature, \{\zeta(j)\} does not necessarily need to come from a chaotic or independent random process for Benford's Law to hold. The multiplicative driving force may be a deterministic quasiperiodic or even periodic forcing. Second, we give conditions under which the distribution of the first digits of an affine process displays Benford's Law. Our proofs use techniques from ergodic theory.", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Benford's Law; Law of Anomalous Numbers; Multiplicative process; Scaling laws; Zipf's Law", } @Article{Hasan:2002:ADA, author = "Bassam Hasan", title = "Assessing data authenticity with {Benford}'s law", journal = "Information Systems Control Journal", volume = "6", pages = "??--??", year = "2002", CODEN = "ISYJFS", ISSN = "1526-7407", bibdate = "Sat Feb 16 17:45:21 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.isaca.org/Journal/Past-Issues/2002/Volume-6/Pages/Assessing-Data-Authenticity-With-Benfords-Law.aspx", acknowledgement = ack-nhfb, journal-URL = "http://www.isaca.org/", remark = "Journal publication ceased at volume 6 in 2008, according to one library catalog, but volumes up to 2013 are available at journal Web site.", } @InProceedings{Hill:2002:RAB, author = "T. P. Hill", editor = "????", booktitle = "Conference, Leiden University", title = "Recent applications of {Benford}'s law", publisher = "????", address = "????", pages = "??--??", year = "2002", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Sat Feb 16 17:55:46 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Hoyle01042002, author = "David C. Hoyle and Magnus Rattray and Ray Jupp and Andrew Brass", title = "Making sense of microarray data distributions", journal = j-BIOINFORMATICS, volume = "18", number = "4", pages = "576--584", year = "2002", DOI = "https://doi.org/10.1093/bioinformatics/18.4.576", ISSN = "1367-4803 (print), 1367-4811 (electronic)", ISSN-L = "1367-4803", bibdate = "Sat Sep 10 11:43:16 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://bioinformatics.oxfordjournals.org/content/18/4/576.abstract", abstract = "Motivation: Typical analysis of microarray data has focused on spot by spot comparisons within a single organism. Less analysis has been done on the comparison of the entire distribution of spot intensities between experiments and between organisms.Results: Here we show that mRNA transcription data from a wide range of organisms and measured with a range of experimental platforms show close agreement with Benford's law (Benford, Proc. Am. Phil. Soc., 78, 551 572, 1938) and Zipf's law (Zipf, The Psycho-biology of Language: an Introduction to Dynamic Philology, 1936 and Human Behaviour and the Principle of Least Effort, 1949). The distribution of the bulk of microarray spot intensities is well approximated by a log-normal with the tail of the distribution being closer to power law. The variance, 2, of log spot intensity shows a positive correlation with genome size (in terms of number of genes) and is therefore relatively fixed within some range for a given organism. The measured value of 2 can be significantly smaller than the expected value if the mRNA is extracted from a sample of mixed cell types. Our research demonstrates that useful biological findings may result from analyzing microarray data at the level of entire intensity distributions. Contact: david.c.hoyle@man.ac.uk", acknowledgement = ack-nhfb, fjournal = "Bioinformatics", journal-URL = "http://bioinformatics.oxfordjournals.org/", keywords = "Zipf's Law", } @Article{Hoyle:2002:MSM, author = "David C. Hoyle and Magnus Rattray and Ray Jupp and Andrew Brass", title = "Making sense of microarray data distributions", journal = j-BIOINFORMATICS, volume = "18", number = "4", pages = "576--584", month = apr, year = "2002", ISSN = "1367-4803 (print), 1367-4811 (electronic)", ISSN-L = "1367-4803", bibdate = "Fri Nov 25 11:16:37 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Bioinformatics", journal-URL = "http://bioinformatics.oxfordjournals.org/", keywords = "Benford's Law; mRNA transcription data", } @Article{Huber:2002:NMG, author = "John C. Huber", title = "A new model that generates {Lotka's Law}", journal = j-J-AM-SOC-INF-SCI-TECHNOL, volume = "53", number = "3", pages = "209--219", month = "????", year = "2002", CODEN = "JASIEF", DOI = "https://doi.org/10.1002/asi.10025", ISSN = "1532-2882 (print), 1532-2890 (electronic)", ISSN-L = "1532-2882", bibdate = "Fri Sep 11 10:42:02 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/jasist.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science and Technology: JASIST", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1532-2890", onlinedate = "28 Dec 2001", } @Article{Jones:2002:LDR, author = "B. K. Jones", title = "Logarithmic distributions in reliability analysis", journal = j-MICROELECT-RELIABILITY, volume = "42", number = "4--5", pages = "779--786", month = apr # "\slash " # may, year = "2002", CODEN = "MCRLAS", DOI = "https://doi.org/10.1016/S0026-2714(02)00031-8", ISSN = "0026-2714 (print), 1872-941X (electronic)", ISSN-L = "0026-2714", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.sciencedirect.com/", URL = "http://www.sciencedirect.com/science/article/pii/S0026271402000318", abstract = "Real-life systems are complex, with many independent parameters which can affect the system. Their behaviour can therefore be very variable. This is especially true of the more involved processes which occur in reliability and degradation processes. However, there are some characteristics which are observed which can be understood simply because they are characteristic of complex systems. These include distributions that are very often logarithmic rather than uniform, log normal failure distributions and $ 1 g / f $ noise. A wide variety of diverse examples is given to illustrate the common occurrence of such observations together with the underlying unifying themes. There are several basic reasons for the origin of logarithmic distributions. One is that they arise from multiplicative processes. Another is that although basic science is often introduced as linear, with non-linear effects added as a correction, complex systems are often inherently non-linear. This produces multiplicative effects, such as harmonic generation and fractal behaviour.", acknowledgement = ack-nhfb, fjournal = "Microelectronics and Reliability", keywords = "Benford's Law; Pareto distribution; Zipf's Law", } @Article{Kawamura:2002:UZL, author = "Kenji Kawamura and Naomichi Hatano", title = "Universality of {Zipf}'s law", journal = j-J-PHYS-SOC-JAP, volume = "71", number = "5", pages = "1211--1213", year = "2002", CODEN = "JUPSAU", DOI = "https://doi.org/10.1143/JPSJ.71.1211", ISSN = "0031-9015 (print), 1347-4073 (electronic)", ISSN-L = "0031-9015", bibdate = "Sat Nov 12 11:00:54 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://jpsj.ipap.jp/", ZMnumber = "1063.60114", abstract = "The aim of the present paper is to propose and simulate a simple and generic model that reproduces Zipf's law. In the proposed model, the time evolution is considered to be a random walk in the logarithmic scale reflecting the company asset distribution and/or the evolution of city populations (e.g. Zipf's law for income distribution asserts that the size and number of companies are in inverse proportional relationship). The paper explains theoretically and shows by numerical simulation that the introduced model of Zipf's law has a very robust behaviour when applied to various natural and social phenomena. The authors do not investigate the famous case of Zipf's law application to the frequency of English words, but they speculate that this case might be also explained within the same time development of the proposed model.", acknowledgement = ack-nhfb, classmath = "60J60 (Diffusion processes) 60J70 (Appl. of diffusion theory) 76R50 (Diffusion) 82C24 (Interface problems (dynamic and non-equilibrium))", fjournal = "Journal of the Physical Society of Japan", keywords = "city population evolution; company asset distribution; diffusion; generic model of Zipf's law; natural and social phenomena; random walk in logarithmic scale", ZMreviewer = "Neculai Curteanu (Ia\c si)", } @TechReport{Kreiner:2002:FDL, author = "W. A. Kreiner", title = "First digit law", type = "Preprint", institution = "Abteilung Chemische Physik, Arbeitsgruppe Laseranwendungen, Universit{\"a}t Ulm", address = "Ulm, Germany", pages = "7", day = "3", month = may, year = "2002", bibdate = "Sat Feb 16 18:18:49 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://vts.uni-ulm.de/docs/2002/1441/vts_1441.pdf", acknowledgement = ack-nhfb, keywords = "Benford's Law; First Digit Law; Newcomb; Statistisches Fraktal", language = "German", } @Article{Li:2002:ZLI, author = "Wentian Li and Yaning Yang", title = "{Zipf's Law} in Importance of Genes for Cancer Classification Using Microarray Data", journal = j-J-THEOR-BIOL, volume = "219", number = "4", pages = "539--551", day = "21", month = dec, year = "2002", CODEN = "JTBIAP", DOI = "https://doi.org/10.1006/jtbi.2002.3145", ISSN = "0022-5193 (print), 1095-8541 (electronic)", ISSN-L = "0022-5193", MRclass = "92C40 (92D10)", MRnumber = "2044066", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022519302931450", acknowledgement = ack-nhfb, fjournal = "Journal of Theoretical Biology", journal-URL = "http://www.sciencedirect.com/science/journal/00225193", keywords = "Zipf's Law", } @Book{Livio:2002:GRS, author = "Mario Livio", title = "The Golden Ratio: the Story of Phi, the World's Most Astonishing Number", publisher = "Broadway Books", address = "New York, NY, USA", pages = "viii + 294", year = "2002", ISBN = "0-7679-0815-5", ISBN-13 = "978-0-7679-0815-3", LCCN = "QA466 .L58 2002", bibdate = "Tue Jul 08 12:29:44 2003", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/master.bib", note = "Benford's Law is treated on pages 231--237.", price = "US\$24.95, CAN\$37.95", abstract = "Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, the author tells the tale of a number at the heart of that mystery: phi, or 1.6180339887\ldots{}. This curious mathematical relationship, widely known as ``The Golden Ratio,'' was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona Lisa to Salvador Dali's The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been found to be connected to the behavior of the stock market! This book is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, the author reveals the world as a place where order, beauty, and eternal mystery will always coexist.", acknowledgement = ack-nhfb, tableofcontents = "Prelude to a number \\ The pitch and the pentagram \\ Under a star-Y-pointing pyramid? \\ The second treasure \\ Son of good nature \\ The divine proportion \\ Painters and poets have equal license \\ From the tiles to the heavens \\ Is God a mathematician?", } @InProceedings{Manaris:2002:PTR, author = "B. Manaris and T. Purewal and C. McCormick", editor = "{IEEE}", booktitle = "Proceedings: IEEE SoutheastCon 2002: April 5--7, 2002, Embassy Suites Hotel, Columbia, South Carolina, USA", title = "Progress Towards Recognizing and Classifying Beautiful Music with Computers: {MIDI}-Encoded Music and the {Zipf-Mandelbrot Law}", publisher = pub-IEEE, address = pub-IEEE:adr, bookpages = "xviii + 482", pages = "52--57", year = "2002", DOI = "https://doi.org/10.1109/.2002.995557", ISBN = "0-7803-7252-2", ISBN-13 = "978-0-7803-7252-8", LCCN = "TK7801 .I56 2002", bibdate = "Mon Feb 18 10:40:27 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://ieeexplore.ieee.org/servlet/opac?punumber=7814", acknowledgement = ack-nhfb, } @Article{Mochty:2002:AMI, author = "L. Mochty", title = "{Die Aufdeckung von Manipulationen im Rechnungswesen --- Was leistet das Benford's Law?}. ({German}) [{The} disclosure of accounting manipulations --- what does {Benford's Law} mean?]", journal = "Die Wirtschaftspr{\"u}fung", volume = "14", number = "??", pages = "725--736", month = "????", year = "2002", CODEN = "????", ISSN = "0043-6313", ISSN-L = "0043-6313", bibdate = "Thu Nov 17 19:20:24 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, language = "German", } @Article{Mosimann:2002:TDE, author = "James Mosimann and John Dahlberg and Nancy Davidian and John Krueger", title = "Terminal Digits and the Examination of Questioned Data", journal = j-ACCOUNT-RES, volume = "9", number = "2", pages = "75--92", month = "????", year = "2002", CODEN = "ARQAEZ", DOI = "https://doi.org/10.1080/08989620212969", ISSN = "0898-9621 (print), 1545-5815 (electronic)", ISSN-L = "0898-9621", bibdate = "Sat Sep 10 15:08:21 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/08989620212969", acknowledgement = ack-nhfb, fjournal = "Accountability in Research", journal-URL = "http://www.tandfonline.com/toc/gacr20", keywords = "Benford's Law", } @Book{Newman:2002:ME, author = "M. E. J. (Mark E. J.) Newman and Richard G. Palmer", title = "Modeling extinction", publisher = pub-OXFORD, address = pub-OXFORD:adr, pages = "xii + 102", year = "2002", ISBN = "0-19-515946-2 (paperback), 0-19-515945-4 (hardcover)", ISBN-13 = "978-0-19-515946-2 (paperback), 978-0-19-515945-5 (hardcover)", LCCN = "QH78 .N48 2002", bibdate = "Thu Feb 28 10:36:25 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; library.ox.ac.uk:210/ADVANCE", series = "Santa Fe Institute studies in the sciences of complexity", acknowledgement = ack-nhfb, keywords = "power law; Zipf's law", subject = "Extinction (Biology); Statistical methods; Mathematical models", } @Misc{Popescu:2002:LNZ, author = "Ioan-Iovitz Popescu", title = "On the {Lavelette}'s nonlinear {Zipf}'s law", howpublished = "Web preprint.", month = feb, year = "2002", bibdate = "Sat Feb 16 19:02:15 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://alpha2.infim.ro/~ltpd/Zipf's_Law.html", acknowledgement = ack-nhfb, } @Article{Reed:2002:GFG, author = "William J. Reed and Barry D. Hughes", title = "From gene families and genera to incomes and {Internet} file sizes: Why power laws are so common in nature", journal = j-PHYS-REV-E, volume = "66", number = "6", pages = "067103", month = dec, year = "2002", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.66.067103", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Wed Feb 27 06:58:05 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.66.067103", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Pareto's Law; rank-size property; Zipf's Law", pagecount = "4", } @Article{Sandon:2002:DPC, author = "F. Sandon", title = "Do Populations Conform to the Law of Anomalous Numbers?", journal = "Population", volume = "57", number = "4", pages = "755--761", month = "????", year = "2002", bibdate = "Sat Sep 10 16:15:28 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.persee.fr/doc/pop_1634-2941_2002_num_57_4_18419", acknowledgement = ack-nhfb, journal-URL = "http://www.persee.fr/collection/pop", keywords = "Benford's Law", } @Article{Sandron:2002:DPC, author = "Fr{\'e}d{\'e}ric Sandron", title = "Do populations conform to the {Law of Anomalous Numbers}?", journal = "Population", volume = "57", number = "4--5", pages = "755--761", month = "????", year = "2002", CODEN = "????", DOI = "????", ISSN = "1169-1018", ISSN-L = "1169-1018", bibdate = "Sat Feb 16 19:08:50 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Translation from French by S. R. Hayford.", URL = "http://www.cairn.info/article_p.php?ID_ARTICLE=POPE_204_0753", acknowledgement = ack-nhfb, journal-URL = "http://www.jstor.org/journals/11691018.html", } @Article{Sandron:2002:PSE, author = "Fr{\'e}d{\'e}ric Sandron", title = "Les populations suivent-elles la loi des nombres anomaux? ({French}). [{Do} populations conform to the {Law of Anomalous Numbers}?]", journal = "Population ({French} edition)", volume = "57", number = "4--5", pages = "761--768", month = "????", year = "2002", CODEN = "POPUAQ", DOI = "????", ISSN = "0032-4663 (print), 1957-7966 (electronic)", ISSN-L = "0032-4663", bibdate = "Sat Feb 16 19:08:50 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.persee.fr/web/revues/home/prescript/article/pop_0032-4663_2002_num_57_4_16873", acknowledgement = ack-nhfb, language = "French", } @TechReport{Seaman:2002:BLB, author = "R. S. Seaman", title = "{Benford}'s law and and background field errors in data assimilation", type = "Preprint", institution = "Bureau of Meteorology Research Centre", address = "Melbourne, Australia", month = "????", year = "2002", bibdate = "Sun Feb 17 14:01:22 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://amm.bom.gov.au/amoj/docs/2002/seaman.pdf", acknowledgement = ack-nhfb, } @Article{Seaman:2002:RBL, author = "R. S. Seaman", title = "The relevance of {Benford's Law} to background field errors in data assimilation", journal = "Australian Meteorological Magazine", volume = "51", number = "1", pages = "25--33", month = "????", year = "2002", CODEN = "AMMGAS", ISSN = "0004-9743", ISSN-L = "0004-9743", bibdate = "Mon Nov 14 16:08:25 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.bom.gov.au/amm/docs/2002/seaman.pdf; http://www.bom.gov.au/amm/docs/2002/seaman_hres.pdf", acknowledgement = ack-nhfb, ajournal = "Aust. Met. Mag.", ajournal-2 = "Aust. Meteorol. Mag.", fjournal = "Australian Meteorological Magazine", journal-URL = "http://www.bom.gov.au/amm/", } @Article{Taylor:2002:STD, author = "Rosemary N. Taylor and Damian J. McEntegart and Eleanor C. Stillman", title = "Statistical techniques to detect fraud and other data irregularities in clinical questionnaire data", journal = "Drug Information Journal", volume = "36", number = "1", pages = "115--125", month = jan, year = "2002", DOI = "https://doi.org/10.1177/009286150203600115", ISSN = "0092-8615", ISSN-L = "0092-8615", bibdate = "Mon Feb 18 06:42:24 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.diahome.org/DIAHome/resources/content.aspx?type=eopdf&file=%2Fproductfiles%2F8357%2Fdiaj_11524.pdf????", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Article{VanCaneghem:2002:EMI, author = "T. {Van Caneghem}", title = "Earnings Management Induced by Cognitive Reference Points", journal = "The British Accounting Review", volume = "34", number = "2", pages = "167--178", year = "2002", DOI = "https://doi.org/10.1006/bare.2002.0190", ISSN = "0890-8389 (print), 1095-8347 (electronic)", ISSN-L = "0890-8389", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0890838902901903", abstract = "Previous studies (Carslaw, 1988; Thomas, 1989; Niskanen \& Keloharju, 2000) have shown that companies' managers tend to round-up the first digits of reported earnings (i.e. for companies reporting profits). According to Carslaw (1988), this type of behaviour is inspired by the existence of the so-called `\$1.99' phenomenon where a price of \$1.99 is perceived as being abnormally lower than one of \$2.00. In the current study, we try to determine whether managers of UK-listed companies also engage in this type of `earnings rounding-up behaviour'. Analogous to the earlier studies, our study compares observed and expected frequencies for the second-from-the-left digit in reported earnings. Our results suggest that managers of UK-listed companies tend to round-up reported pre-tax income, in a way that increases the first digit by one, when they are faced with a nine in the second-from-the-left position for this particular earnings measure. The major contribution of the current study is that it introduces discretionary accruals in this line of research. Discretionary accruals were estimated using both the Jones model (1991) and the modified Jones model as proposed by Dechow et al. (1995). Our results clearly suggest that discretionary accruals are used in order to round-up reported earnings figures. Moreover, discretionary accruals enabled us to increase the power of the tests used in previous studies.", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Article{Wallace:2002:AQD, author = "W. A. Wallace", title = "Assessing the quality of data used for benchmarking and decision making", journal = "Journal of Government Financial Management", volume = "51", number = "3", pages = "16--22", month = "????", year = "2002", DOI = "", ISSN = "1533-1385", ISSN-L = "1533-1385", bibdate = "Thu Oct 13 09:17:04 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "", acknowledgement = ack-nhfb, journal-URL = "https://www.agacgfm.org/Resources/Journal-of-Government-Financial-Management/Archives.aspx", keywords = "Benford's Law", remark = "Online archive access, even for journal metadata, requires member account!", } @Article{Yeh:2002:SMZ, author = "Hsiaw-Chan Yeh", title = "Six multivariate {Zipf} distributions and their related properties", journal = j-STAT-PROB-LETT, volume = "56", number = "2", pages = "131--141", day = "15", month = jan, year = "2002", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/S0167-7152(01)00149-3", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", MRclass = "62H10", MRnumber = "1881166 (2002m:62084)", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0167715201001493", acknowledgement = ack-nhfb, fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", keywords = "Zipf's Law", } @Article{Abdel-All:2003:GPP, author = "N. H. Abdel-All and M. A. W. Mahmoud and H. N. Abd-Ellah", title = "Geometrical properties of {Pareto} distribution", journal = j-APPL-MATH-COMP, volume = "145", number = "2--3", pages = "321--339", day = "25", month = dec, year = "2003", CODEN = "AMHCBQ", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Fri Jan 9 08:41:12 MST 2004", bibsource = "http://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", } @Article{Alvarez-Ramirez:2003:ZMS, author = "Jose Alvarez-Ramirez and Monica Meraz and Gustavo Gallegos", title = "{Zipf--Mandelbrot} scaling law for world track records", journal = j-PHYSICA-A, volume = "328", number = "3--4", pages = "545--560", day = "15", month = oct, year = "2003", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(03)00579-X", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S037843710300579X", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Book{Anastasi:2003:NFI, author = "Joe Anastasi", title = "The new forensics: investigating corporate fraud and the theft of intellectual property", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xiv + 270", year = "2003", ISBN = "0-471-26994-8 (hardcover)", ISBN-13 = "978-0-471-26994-6 (hardcover)", LCCN = "HV8079.W47 A5 2003", bibdate = "Tue Nov 22 11:42:36 MST 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/bios/wiley044/2003001697.html; http://www.loc.gov/catdir/description/wiley037/2003001697.html; http://www.loc.gov/catdir/toc/wiley031/2003001697.html", acknowledgement = ack-nhfb, remark = "Brief mention of Benford's Law on pp. 184--185.", subject = "white collar crime investigation; data processing; case studies; fraud investigation; forensic accounting; computer security", } @Article{Ausloos:2003:SIZ, author = "M. Ausloos and Ph. Bronlet", title = "Strategy for investments from {Zipf} law(s)", journal = j-PHYSICA-A, volume = "324", number = "1--2", pages = "30--37", day = "1", month = jun, year = "2003", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(02)01845-9", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", MRclass = "91B82", MRnumber = "2030919", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "International Econophysics Conference IEC2002 (Bali)", URL = "http://www.sciencedirect.com/science/article/pii/S0378437102018459", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Book{Baldi:2003:MIW, author = "Pierre Baldi and Paolo Frasconi and Padhraic Smyth", title = "Modeling the {Internet} and the {Web}: probabilistic methods and algorithms", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xix + 285", year = "2003", ISBN = "0-470-86492-3 (e-book), 0-470-84906-1", ISBN-13 = "978-0-470-86492-0 (e-book), 978-0-470-84906-4", LCCN = "TK5105.875.I57 B35 2003eb", bibdate = "Fri Jun 3 10:03:23 MDT 2011", bibsource = "fsz3950.oclc.org:210/WorldCat; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/pagerank.bib", acknowledgement = ack-nhfb, subject = "Internet; Mathematical models; Telecommunication; Traffic; World Wide Web; Cyberspace; Probabilities; Computers; Web; General; Networking; Intranets and Extranets", tableofcontents = "Mathematical Background \\ Probability and Learning from a Bayesian Perspective \\ Parameter Estimation from Data \\ Basic principles \\ A simple die example \\ Mixture Models and the Expectation Maximization Algorithm \\ Graphical Models \\ Bayesian networks \\ Belief propagation \\ Learning directed graphical models from data \\ Classification \\ Clustering \\ Power-Law Distributions \\ Scale-free properties (80/20 rule) \\ Applications to Languages: Zipf's and Heaps' Laws \\ Origin of power-law distributions and Fermi's model \\ Basic WWW Technologies \\ Web Documents \\ SGML and HTML \\ General structure of an HTML document \\ Links \\ Resource Identifiers: URI, URL, and URN \\ Protocols \\ Reference models and TCP/IP \\ The domain name system \\ The Hypertext Transfer Protocol \\ Programming examples \\ Log Files \\ Search Engines \\ Coverage \\ Basic crawling \\ Web Graphs \\ Internet and Web Graphs \\ Power-law size \\ Power-law connectivity \\ Small-world networks \\ Power law of PageRank \\ The bow-tie structure \\ Generative Models for the Web Graph and Other Networks \\ Web page growth \\ Lattice perturbation models: between order and disorder \\ Preferential attachment models, or the rich get richer \\ Copy models \\ PageRank models \\ Applications \\ Distributed search algorithms \\ Subgraph patterns and communities \\ Robustness and vulnerability \\ Notes and Additional Technical References \\ Text Analysis \\ Indexing \\ Compression techniques \\ Lexical Processing \\ Tokenization", } @TechReport{Berger:2003:BLP, author = "Arno Berger", title = "{Benford}'s law in power-like nonautonomous dynamical systems", type = "Preprint", institution = "Vienna University of Technology", address = "Vienna, Austria", year = "2003", bibdate = "Sat Feb 16 17:18:18 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Canessa:2003:TAFa, author = "Enrique Canessa", title = "Theory of Analogous Force on Number Sets", journal = "ArXiv e-prints", year = "2003", bibdate = "Mon Nov 21 06:56:01 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://arxiv.org/abs/cond-mat/0307703", abstract = "A general statistical thermodynamic theory that considers given sequences of $x$-integers to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability distributions $ p_x $ for natural numbers, we claim that they lead to a better understanding of probabilistic laws associated with number theory. Sequences of numbers are treated as the size measure of finite sets. By considering $ p_x $ to describe complex phenomena, the theory leads to derive a distinct analogous force $ f_x $ on number sets proportional to $ (\frac {\partial p_x}{\partial x})_T $ at an analogous system temperature $T$. In particular, this yields to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits. The theory also allows to establish recursion relations to predict sequences of Fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the Benford's law in Fibonacci numbers. A possible relevance to prime numbers is also analyzed.", acknowledgement = ack-nhfb, remark = "To appear in Phys. A.", } @Article{Canessa:2003:TAFb, author = "Enrique Canessa", title = "Theory of analogous force on number sets", journal = j-PHYSICA-A, volume = "328", number = "1--2", pages = "44--52", month = oct, year = "2003", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(03)00526-0", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fibquart.bib; http://www.sciencedirect.com/", URL = "http://www.sciencedirect.com/science/article/pii/S0378437103005260", abstract = "A general statistical thermodynamic theory that considers given sequences of $x$-integers to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability distributions $ p_x $ for natural numbers, we claim that they lead to a better understanding of probabilistic laws associated with number theory. Sequences of numbers are treated as the size measure of finite sets. By considering $ p_x $ to describe complex phenomena, the theory leads to derive a distinct analogous force $ f_x $ on number sets proportional to $ (\partial p_x / \partial x)T $ at an analogous system temperature $T$. In particular, this leads to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits. The theory also allows to establish recursion relations to predict sequences of Fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the Benford's law in Fibonacci numbers. A possible relevance to prime numbers is also analyzed.", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Benford's Law; Fibonacci numbers; Number theory; prime numbers; Probability theory; Statistical physics and thermodynamics", } @Article{Das:2003:RRE, author = "Somnath Das and Huai Zhang", title = "Rounding-up in reported {EPS}, behavioral thresholds, and earnings management", journal = "Journal of Accounting and Economics", volume = "35", number = "1", pages = "31--50", year = "2003", DOI = "https://doi.org/10.1016/S0165-4101(02)00096-4", ISSN = "0165-4101 (print), 1879-1980 (electronic)", ISSN-L = "0165-4101", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0165410102000964", abstract = "Reported earnings per share (EPS) are frequently rounded to the nearest cent. This paper provides evidence that firms manipulate earnings so that they can round-up and report one more cent of EPS. Specifically, we examine the digit immediately right of the decimal in the calculated {EPS} number expressed in cents. Evidence is presented that firms are more likely to round-up when managers ex ante expect rounding-up to meet analysts forecasts, report positive profits, or sustain recent performance. Further investigation provides evidence that working capital accruals are used to round-up EPS.", acknowledgement = ack-nhfb, keywords = "Behavioral threshold; Benford's Law; Earnings management; Earnings per share (EPS); Rounding", } @Article{Engel:2003:BLE, author = "Hans-Andreas Engel and Christoph Leuenberger", title = "{Benford}'s law for exponential random variables", journal = j-STAT-PROB-LETT, volume = "63", number = "4", pages = "361--365", day = "15", month = jul, year = "2003", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/S0167-7152(03)00101-9", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", MRclass = "60E99 (60C05)", MRnumber = "1996184 (2004d:60050)", MRreviewer = "Ulrich M. Hirth", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.sciencedirect.com/", URL = "http://www.sciencedirect.com/science/article/pii/S0167715203001019", abstract = "Benford's law assigns the probability $ \log_{10}(1 + 1 / d) $ for finding a number starting with specific significant digit $d$. We show that exponentially distributed numbers obey this law approximatively, i.e., within bounds of $ 0.03$.", acknowledgement = ack-nhfb, fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", keywords = "Benford's Law; Exponential distribution; Significant digit law", } @Article{FerreriCancho:2003:LEO, author = "Ramon {Ferrer i Cancho} and Ricard V. Sol{\'e}", title = "Least effort and the origins of scaling in human language", journal = j-PROC-NATL-ACAD-SCI-USA, volume = "100", number = "3", pages = "788--791", day = "4", month = feb, year = "2003", CODEN = "PNASA6", DOI = "https://doi.org/10.1073/pnas.0335980100", ISSN = "0027-8424 (print), 1091-6490 (electronic)", ISSN-L = "0027-8424", bibdate = "Tue Nov 15 09:44:47 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the National Academy of Sciences of the United States of America", journal-URL = "http://www.pnas.org/search", keywords = "Zipf's Law", onlinedate = "January 22, 2003", } @Book{Fleiss:2003:SMR, author = "Joseph L. Fleiss and Bruce A. Levin and Myunghee Cho Paik", title = "Statistical Methods for Rates and Proportions", publisher = pub-WILEY, address = pub-WILEY:adr, edition = "Third", pages = "xxvii + 760", year = "2003", ISBN = "0-471-52629-0 (hardcover)", ISBN-13 = "978-0-471-52629-2 (hardcover)", LCCN = "QA279 .F58 2003", bibdate = "Fri Mar 22 08:46:57 MDT 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", series = "Wiley series in probability and statistics", URL = "http://www.loc.gov/catdir/bios/wiley042/2002191005.html; http://www.loc.gov/catdir/description/wiley034/2002191005.html; http://www.loc.gov/catdir/toc/wiley031/2002191005.html", acknowledgement = ack-mjn # "\slash " # ack-nhfb, keywords = "Benford's Law", subject = "Analysis of variance; Sampling (Statistics); Biometry", tableofcontents = "Preface \\ Preface to the Second Edition \\ Preface to the First Edition \\ 1. An Introduction to Applied Probability \\ 2. Statistical Inference for a Single Proportion \\ 3. Assessing Significance in a Fourfold Table \\ 4. Determining Sample Sizes Needed to Detect a Difference Between Two Proportions \\ 5. How to Randomize \\ \\ 6. Comparative Studies: Cross-Sectional, Naturalistic, or Multinomial Sampling \\ \\ 7. Comparative Studies: Prospective and Retrospective Sampling \\ 8. Randomized Controlled Trials \\ 9. The Comparison of Proportions from Several Independent Samples \\ 10. Combining Evidence from Fourfold Tables \\ 11. Logistic Regression \\ 12. Poisson Regression \\ 13. Analysis of Data from Matched Samples \\ 14. Regression Models for Matched Samples \\ 15. Analysis of Correlated Binary Data \\ 16. Missing Data \\ 17. Misclassification Errors: Effects, Control, and Adjustment \\ 18. The Measurement of Interrater Agreement \\ 19. The Standardization of Rates \\ Appendix A. Numerical Tables \\ Appendix B. The Basic Theory of Maximum Likelihood", } @Article{Furusawa:2003:ZLG, author = "Chikara Furusawa and Kunihiko Kaneko", title = "{Zipf's Law} in Gene Expression", journal = j-PHYS-REV-LET, volume = "90", number = "??", pages = "088102", day = "26", month = feb, year = "2003", CODEN = "PRLTAO", DOI = "https://doi.org/10.1103/PhysRevLett.90.088102", ISSN = "0031-9007 (print), 1079-7114 (electronic), 1092-0145", ISSN-L = "0031-9007", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevLett.90.088102", acknowledgement = ack-nhfb, fjournal = "Physical Review Letters", journal-URL = "http://prl.aps.org/browse", keywords = "Zipf's Law", } @Article{Gabaix:2003:TPL, author = "Xavier Gabaix and Parameswaran Gopikrishnan and Vasiliki Plerou and H. Eugene Stanley", title = "A theory of power-law distributions in financial market fluctuations", journal = j-NATURE, volume = "423", number = "6937", pages = "267--270", day = "15", month = may, year = "2003", CODEN = "NATUAS", DOI = "https://doi.org/10.1038/nature01624", ISSN = "0028-0836 (print), 1476-4687 (electronic)", ISSN-L = "0028-0836", bibdate = "Wed Feb 27 06:45:03 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.nature.com/nature/journal/v423/n6937/full/nature01624.html", acknowledgement = ack-nhfb, fjournal = "Nature", journal-URL = "http://www.nature.com/nature/archive/", keywords = "Zipf's Law", } @InProceedings{Grekos:2003:RSC, author = "G. Grekos and R. Giuliano-Antonini", editor = "????", booktitle = "{Journ{\'e}es Arithm{\'e}tiques XXIII, Graz, July 6--12, 2003}", title = "Regular sets and conditional density: an extension of {Benford}'s law", publisher = "????", address = "????", pages = "??--??", year = "2003", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Sat Feb 16 17:43:43 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Hassan:2003:EDA, author = "Bassam Hassan", title = "Examining data accuracy and authenticity with leading digit frequency analysis", journal = "Industrial Management + Data Systems", volume = "103", number = "2", pages = "121--125", month = "????", year = "2003", CODEN = "MDSD8F", ISSN = "0263-5577 (print), 1758-5783 (electronic)", ISSN-L = "0263-5577", bibdate = "Fri Nov 25 11:07:59 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Book{Havil:2003:GEE, author = "Julian Havil", title = "{Gamma}: Exploring {Euler}'s Constant", publisher = pub-PRINCETON, address = pub-PRINCETON:adr, pages = "xxiii + 266", year = "2003", ISBN = "0-691-09983-9", ISBN-13 = "978-0-691-09983-5", LCCN = "QA41 .H23 2003", bibdate = "Thu Sep 25 16:52:46 2003", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/master.bib", price = "US\$29.95", URL = "http://www.loc.gov/catdir/description/prin031/2002192453.html", abstract = "Among the myriad of constants that appear in mathematics, $ \pi $, $e$, and $i$ are the most familiar. Following closely behind is $ \gamma $ or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery. In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Introduced by the Swiss mathematician Leonhard Euler (1707--1783), who figures prominently in this book, gamma is defined as the limit of the sum of $ 1 + 1 / 2 + 1 / 3 + \cdots + 1 / n - \ln n$, the numerical value being $ 0.5772156 \ldots {}$. But unlike its more celebrated colleagues $ \pi $ and $e$, the exact nature of gamma remains a mystery --- we don't even know if it can be expressed as a fraction. Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today, the Riemann Hypothesis (though no proof of either is offered!). Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.", acknowledgement = ack-nhfb, tableofcontents = "Foreword / xv \\ Acknowledgements / xvii \\ Introduction / xix \\ Chapter One: The Logarithmic Cradle / 1 \\ 1.1 A Mathematical Nightmare- and an Awakening / 1 \\ 1.2 The Baron's Wonderful Canon / 4 \\ 1.3 A Touch of Kepler / 11 \\ 1.4 A Touch of Euler / 13 \\ 1.5 Napier's Other Ideas / 16 \\ Chapter Two: The Harmonic Series / 21 \\ 2.1 The Principle / 21 \\ 2.2 Generating Function for $H_n$ / 21 \\ 2.3 Three Surprising Results / 22 \\ Chapter Three: Sub-Harmonic Series / 27 \\ 3.1 A Gentle Start / 27 \\ 3.2 Harmonic Series of Primes / 28 \\ 3.3 The Kempner Series / 31 \\ 3.4 Madelung's Constants / 33 \\ Chapter Four: Zeta Functions / 37 \\ 4.1 Where $n$ Is a Positive Integer / 37 \\ 4.2 Where $x$ Is a Real Number / 42 \\ 4.3 Two Results to End With / 44 \\ Chapter Five: Gamma's Birthplace / 47 \\ 5.1 Advent / 47 \\ 5.2 Birth / 49 \\ Chapter Six: The Gamma Function / 53 \\ 6.1 Exotic Definitions / 53 \\ 6.2 Yet Reasonable Definitions / 56 \\ 6.3 Gamma Meets Gamma / 57 \\ 6.4 Complement and Beauty / 58 \\ Chapter Seven: Euler's Wonderful Identity / 61 \\ 7.1 The All-Important Formula / 61 \\ 7.2 And a Hint of Its Usefulness / 62 \\ Chapter Eight: A Promise Fulfilled / 65 \\ Chapter Nine: What Is Gamma Exactly? / 69 \\ 9.1 Gamma Exists / 69 \\ 9.2 Gamma Is What Number? 739.3 A Surprisingly Good Improvement / 75 \\ 9.4 The Germ of a Great Idea / 78 \\ Chapter Ten: Gamma as a Decimal / 81 \\ 10.1 Bernoulli Numbers / 81 \\ 10.2 Euler--Maclaurin Summation / 85 \\ 10.3 Two Examples / 86 \\ 10.4 The Implications for Gamma / 88 \\ Chapter Eleven: Gamma as a Fraction / 91 \\ 11.1 A Mystery / 91 \\ 11.2 A Challenge / 91 \\ 11.3 An Answer / 93 \\ 11.4 Three Results / 95 \\ 11.5 Irrationals / 95 \\ 11.6 Pell's Equation Solved / 97 \\ 11.7 Filling the Gaps / 98 \\ 11.8 The Harmonic Alternative / 98 \\ Chapter Twelve: Where Is Gamma? / 101 \\ 12.1 The Alternating Harmonic Series Revisited / 101 \\ 12.2 In Analysis / 105 \\ 12.3 In Number Theory / 112 \\ 12.4 In Conjecture / 116 \\ 12.5 In Generalization / 116 \\ Chapter Thirteen: It's a Harmonic World / 119 \\ 13.1 Ways of Means / 119 \\ 13.2 Geometric Harmony / 121 \\ 13.3 Musical Harmony / 123 \\ 13.4 Setting Records / 125 \\ 13.5 Testing to Destruction / 126 \\ 13.6 Crossing the Desert / 127 \\ 13.7 Shuffiing Cards / 127 \\ 13.8 Quicksort / 128 \\ 13.9 Collecting a Complete Set / 130 \\ 13.10 A Putnam Prize Question / 131 \\ 13.11 Maximum Possible Overhang / 132 \\ 13.12 Worm on a Band / 133 \\ 13.13 Optimal Choice / 134 \\ Chapter Fourteen: It's a Logarithmic World / 139 \\ 14.1 A Measure of Uncertainty / 139 \\ 14.2 Benford's Law / 145 \\ 14.3 Continued-Fraction Behaviour / 155 \\ Chapter Fifteen: Problems with Primes / 163 \\ 15.1 Some Hard Questions about Primes / 163 \\ 15.2 A Modest Start / 164 \\ 15.3 A Sort of Answer / 167 \\ 15.4 Picture the Problem / 169 \\ 15.5 The Sieve of Eratosthenes / 171 \\ 15.6 Heuristics / 172 \\ 15.7 A Letter / 174 \\ 15.8 The Harmonic Approximation / 179 \\ 15.9 Different-and Yet the Same / 180 \\ 15.10 There are Really Two Questions, Not Three / 182 \\ 15.11 Enter Chebychev with Some Good Ideas / 183 \\ 15.12 Enter Riemann, Followed by Proof(s) / 186 \\ Chapter Sixteen: The Riemann Initiative / 189 \\ 16.1 Counting Primes the Riemann Way / 189 \\ 16.2 A New Mathematical Tool / 191 \\ 16.3 Analytic Continuation / 191 \\ 16.4 Riemann's Extension of the Zeta Function / 193 \\ 16.5 Zeta's Functional Equation / 193 \\ 16.6 The Zeros of Zeta / 193 \\ 16.7 The Evaluation of $\Pi(x)$ and $\pi(x)$ / 196 \\ 16.8 Misleading Evidence / 197 \\ 16.9 The Von Mangoldt Explicit Formula --- and How It Is Used to Prove the Prime Number Theorem / 200 \\ 16.10 The Riemann Hypothesis / 202 \\ 16.11 Why Is the Riemann Hypothesis Important? / 204 \\ 16.12 Real Alternatives / 206 \\ 16.13 A Back Route to Immortality-Partly Closed / 207 \\ 16.14 Incentives, Old and New / 210 \\ 16.15 Progress / 213 \\ Appendix A: The Greek Alphabet / 217 \\ Appendix B: Big Oh Notation / 219 \\ Appendix C: Taylor Expansions / 221 \\ C.1 Degree 1 / 221 \\ C.2 Degree 2 / 221 \\ C.3 Examples / 223 \\ C.4 Convergence / 223 \\ Appendix D: Complex Function Theory / 225 \\ D.1 Complex Differentiation / 225 \\ D.2 Weierstrass Function / 230 \\ D.3 Complex Logarithms / 231 \\ D.4 Complex Integration / 232 \\ D.5 A Useful Inequality / 235 \\ D.6 The Indefinite Integral / 235 \\ D.7 The Seminal Result / 237 \\ D.8 An Astonishing Consequence / 238 \\ D.9 Taylor Expansions-and an Important Consequence / 239 \\ D.10 Laurent Expansions --- and Another Important Consequence / 242 \\ D.11 The Calculus of Residues / 245 \\ D.12 Analytic Continuation / 247 \\ Appendix E: Application to the Zeta Function / 249 \\ E.1 Zeta Analytically Continued / 249 \\ E.2 Zeta's Functional Relationship / 253", } @Article{Hurlimann:2003:GBL, author = "Werner H{\"u}rlimann", title = "A generalized {Benford} law and its application", journal = j-ADV-APPL-STAT, volume = "3", number = "3", pages = "217--228", year = "2003", CODEN = "????", ISSN = "0972-3617", ISSN-L = "0972-3617", MRclass = "11K31 (11B73)", MRnumber = "2034405 (2004k:11125)", MRreviewer = "Vydas {\v{C}}ekanavi{\v{c}}ius", bibdate = "Wed Aug 16 07:36:50 2017", bibsource = "http://www.math.utah.edu/pub/tex/bib/advapplstat.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Advances and Applications in Statistics", } @Article{Ioannides:2003:ZLC, author = "Yannis M. Ioannides and Henry G. Overman", title = "{Zipf}'s law for cities: an empirical examination", journal = j-REG-SCI-URBAN-ECON, volume = "33", number = "2", pages = "127--137", day = "1", month = mar, year = "2003", CODEN = "RSUEDM", DOI = "https://doi.org/10.1016/S0166-0462(02)00006-6", ISSN = "0166-0462 (print), 1879-2308 (electronic)", ISSN-L = "0166-0462", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0166046202000066", acknowledgement = ack-nhfb, fjournal = "Regional Science and Urban Economics", journal-URL = "http://www.sciencedirect.com/science/journal/01660462", keywords = "Zipf's Law", } @Article{Kreiner:2003:NBL, author = "W. A. Kreiner", title = "On the {Newcomb--Benford} law", journal = j-Z-NATURFORSCH, volume = "58a", number = "11", pages = "618--622", month = "????", year = "2003", CODEN = "ZNTFA2", ISSN = "0372-9516", ISSN-L = "0372-9516", bibdate = "Sat Feb 16 18:19:41 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.znaturforsch.com/aa/v58a/s58a0618.pdf", acknowledgement = ack-nhfb, fjournal = "{Zeitschrift f{\"u}r Naturforschung}", journal-URL = "http://www.degruyter.com/view/j/zna", } @InCollection{Kumar:2003:BLA, author = "Kuldeep Kumar and Sukanto Bhattacharya", editor = "Cheng-Few Lee", booktitle = "Advances in Financial Planning and Forecasting", title = "{Benford}'s law and its application in financial fraud detection", volume = "11", publisher = pub-ELSEVIER, address = pub-ELSEVIER:adr, bookpages = "ix + 177", pages = "57--70", year = "2003", ISBN = "0-7623-1016-2", ISBN-13 = "978-0-7623-1016-6", ISSN = "1046-5847", LCCN = "????", bibdate = "Wed Nov 23 16:45:49 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{Lagarias:2003:PAB, author = "Jeffrey C. Lagarias", title = "The $ 3 x + 1 $ problem: An annotated bibliography (1963--1999) (sorted by author)", journal = "arxiv.org", volume = "??", number = "??", month = sep, year = "2003", bibdate = "Sat Sep 10 12:58:10 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://arxiv.org/abs/math/0309224", abstract = "The $ 3 x + 1 $ problem concerns iteration of the map on the integers given by $ T(n) = (3 n + 1) / 2 $ if $n$ is odd; $ T(n) = n / 2$ if $n$ is even. The $ 3 x + 1$ Conjecture asserts that for every positive integer $ n > 1$ the forward orbit of $n$ under iteration by $T$ includes the integer 1. This paper is an annotated bibliography of work done on the $ 3 x + 1$ problem and related problems from 1963 through 1999. At present the $ 3 x + 1$ Conjecture remains unsolved.", acknowledgement = ack-nhfb, pagecount = "74", remark = "At least 13 updates have been made up to 11-Jan-2011.", } @Article{Lanza:2003:FSP, author = "Richard B. Lanza", title = "Fear Not the Software: Proactively Detecting Occupational Fraud Using Computer Audit Reports", journal = "Fraud Magazine", volume = "??", number = "??", pages = "??--??", month = sep # "\slash " # feb, year = "2003", ISSN = "1553-6645", ISSN-L = "1553-6645", bibdate = "Sun Feb 17 08:14:47 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.fraud-magazine.com/article.aspx?id=4294967868", acknowledgement = ack-nhfb, fjournal = "Fraud Magazine. {A} Publication of the Association of Certified Fraud Examiners", keywords = "Benford's Law", } @InProceedings{Manaris:2003:EMZ, author = "Bill Manaris and Dallas Vaughan and Christopher Wagner and Juan Romero and Robert B. Davis", title = "Evolutionary Music and the {Zipf--Mandelbrot Law}: Developing Fitness Functions for Pleasant Music", crossref = "Raidl:2003:AEC", pages = "522--534", year = "2003", bibdate = "Mon Feb 18 10:49:54 2013", bibsource = "http://www.math.utah.edu/pub/bibnet/authors/m/mandelbrot-benoit.bib; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @Article{McAllister:2003:ARE, author = "James W. McAllister", title = "Algorithmic randomness in empirical data", journal = j-STUD-HIST-PHILOS-SCI, volume = "34", number = "3", pages = "633--646", month = sep, year = "2003", CODEN = "SHPSB5", DOI = "https://doi.org/10.1016/S0039-3681(03)00047-5", ISSN = "0039-3681 (print), 1879-2510 (electronic)", ISSN-L = "0039-3681", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/shps-a.bib; http://www.sciencedirect.com/science/journal/00393681", note = "See replies \cite{Twardy:2005:EDS,McAllister:2005:ACE}.", URL = "http://www.sciencedirect.com/science/article/pii/S0039368103000475", abstract = "According to a traditional view, scientific laws and theories constitute algorithmic compressions of empirical data sets collected from observations and measurements. This article defends the thesis that, to the contrary, empirical data sets are algorithmically incompressible. The reason is that individual data points are determined partly by perturbations, or causal factors that cannot be reduced to any pattern. If empirical data sets are incompressible, then they exhibit maximal algorithmic complexity, maximal entropy and zero redundancy. They are therefore maximally efficient carriers of information about the world. Since, on algorithmic information theory, a string is algorithmically random just if it is incompressible, the thesis entails that empirical data sets consist of algorithmically random strings of digits. Rather than constituting compressions of empirical data, scientific laws and theories pick out patterns that data sets exhibit with a certain noise.", acknowledgement = ack-nhfb, fjournal = "Studies in History and Philosophy of Science Part {A}", journal-URL = "http://www.sciencedirect.com/science/journal/00393681", keywords = "Algorithmic randomness; Benford's Law; Compression; Empirical data; Information; Law; Pattern", } @Article{Mitzenmacher:2003:BHG, author = "Michael Mitzenmacher", title = "A Brief History of Generative Models for Power Law and Lognormal Distributions", journal = j-INTERNET-MATH, volume = "1", number = "2", pages = "226--251", month = dec, year = "2003", CODEN = "????", DOI = "https://doi.org/10.1080/15427951.2004.10129088", ISSN = "1542-7951 (print), 1944-9488 (electronic)", ISSN-L = "1542-7951", bibdate = "Thu Nov 24 13:38:13 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://projecteuclid.org/euclid.im/1089229510; http://www.eecs.harvard.edu/~michaelm/CS223/powerlaw.pdf; http://www.tandfonline.com/toc/uinm20/1/2", acknowledgement = ack-nhfb, ajournal = "Internet Math.", fjournal = "Internet Mathematics", journal-URL = "http://projecteuclid.org/info/euclid.im", keywords = "Fibonacci sequence; lognormal distribution; monkeys typing randomly; Pareto distribution; power-law distribution; Yule distribution; Zipf's Law", remark = "This is a survey article with a six-page bibliography of earlier work, and a clear description of the differences between lognormal and power-law distributions, which otherwise look similar on log-log plots. Lognormal distributions have finite means and moments, while power-law distributions may have infinite means and/or moments, depending on the exponent. The preprint at author's Web site (last URL) contains additional material on the acrimonious debate between Beno{\^\i}t Mandelbrot and Herbert A. Simon (see \cite{Mandelbrot:1959:NCS,Simon:1955:CSD}, and other references in Mitzenmacher's paper) on the derivation and modeling of Zipf's Law distributions.", } @PhdThesis{Muller:2003:AZP, author = "Mathias M{\"u}ller", title = "{Anwendungsm{\"o}glichkeiten der Ziffernanalyse in der Pr{\"u}fungspraxis mit Schwerpunkt auf Benford's Law}. ({German}) [Applications of the digit analysis in the audit practice with a focus on {Benford's Law}]", type = "{Diplomarbeit}", school = "Wirtschaftsuniversit{\"a}t Wien", address = "Wien, Austria", pages = "v + 75", month = "????", year = "2003", bibdate = "Sat Feb 16 18:44:50 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://permalink.obvsg.at/wuw/AC03895842", acknowledgement = ack-nhfb, language = "German", } @Article{Naldi:2003:CIZ, author = "M. Naldi", title = "Concentration indices and {Zipf}'s law", journal = j-ECONOM-LETT, volume = "78", number = "3", pages = "329--334", month = mar, year = "2003", CODEN = "ECLEDS", DOI = "https://doi.org/10.1016/S0165-1765(02)00251-3", ISSN = "0165-1765 (print), 1873-7374 (electronic)", ISSN-L = "0165-1765", MRclass = "91B82", MRnumber = "1959355", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0165176502002513", acknowledgement = ack-nhfb, fjournal = "Economics Letters", keywords = "Zipf's Law", } @Article{Newby:2003:OSS, author = "Gregory B. Newby and Jane Greenberg and Paul Jones", title = "Open source software development and {Lotka's Law}: {Bibliometric} patterns in programming", journal = j-J-AM-SOC-INF-SCI-TECHNOL, volume = "54", number = "2", pages = "169--178", day = "15", month = jan, year = "2003", CODEN = "JASIEF", DOI = "https://doi.org/10.1002/asi.10177", ISSN = "1532-2882 (print), 1532-2890 (electronic)", ISSN-L = "1532-2882", bibdate = "Fri Sep 11 10:42:09 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/gnu.bib; http://www.math.utah.edu/pub/tex/bib/jasist.bib", note = "See comments \cite{Burrell:2004:LEF}.", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science and Technology: JASIST", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1532-2890", onlinedate = "10 Dec 2002", } @Article{Nguyen:2003:DPL, author = "H. T. Nguyen and V. Kreinovich and L. Longpre", title = "Dirty pages of logarithm tables, lifetime of the universe, and subjective (fuzzy) probabilities on finite and infinite intervals", journal = j-IEEE-TRANS-FUZZY-SYSTEMS, volume = "1", pages = "67--73", year = "2003", CODEN = "IEFSEV", DOI = "https://doi.org/10.1109/FUZZ.2003.1209339", ISSN = "1063-6706 (print), 1941-0034 (electronic)", ISSN-L = "1063-6706", bibdate = "Sat Feb 16 18:51:15 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "The 12th IEE International Conference on Fuzzy Systems. FUZZ'03", acknowledgement = ack-nhfb, fjournal = "IEEE Transactions on Fuzzy Systems", xxnote = "Is this the listed journal, or a conference paper??", } @Article{Peter:2003:ADE, author = "Manfred Peter", title = "The asymptotic distribution of elements in automatic sequences", journal = j-THEOR-COMP-SCI, volume = "301", number = "1 3", pages = "285--312", year = "2003", CODEN = "TCSCDI", DOI = "https://doi.org/10.1016/S0304-3975(02)00587-X", ISSN = "0304-3975 (print), 1879-2294 (electronic)", ISSN-L = "0304-3975", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S030439750200587X", abstract = "In an automatic sequence an element need not have an asymptotic density. In this paper a necessary and sufficient criterion is proved for the existence of the asymptotic density of a given element. If it does not exist the asymptotic distribution of the element can be described in terms of a function H whose graph is self-similar. An algorithm is given to decide whether H is piecewise continuously differentiable, and in this case it can be computed effectively. Finally, it is shown that the H -density of an element in an automatic sequence always exists and equals its logarithmic density.", acknowledgement = ack-nhfb, fjournal = "Theoretical Computer Science", journal-URL = "http://www.sciencedirect.com/science/journal/03043975", keywords = "Asymptotics; Automata; Benford's Law; Density; H{\"o}lder mean; Oscillating sum; Self-similarity", } @Article{Quick:2003:BLD, author = "Reiner Quick and Matthias Wolz", title = "{Benford's Law in deutschen Rechnungslegungsdaten}. ({German}) [{Benford's Law} in {German} accounting data]", journal = "Betriebswirtschaftliche Forschung und Praxis", volume = "55", number = "2", pages = "208--224", month = "????", year = "2003", CODEN = "????", ISSN = "0340-5370", ISSN-L = "0340-5370", bibdate = "Sat Dec 10 12:01:20 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, language = "German", remark = "Betriebswirtschaftliche Forschung und Praxis == Business Research and Practice.", } @Article{Reed:2003:PLI, author = "William J. Reed", title = "The {Pareto} law of incomes---an explanation and an extension", journal = j-PHYSICA-A, volume = "319", number = "1--4", pages = "469--486", year = "2003", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/S0378-4371(02)01507-8", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", MRclass = "91B70", MRnumber = "1965596", bibdate = "Mon Feb 18 17:05:26 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", } @Article{Rose:2003:TEF, author = "Anna M. Rose and Jacob M. Rose", title = "Turn {Excel} into a financial sleuth: an easy-to-use digital analysis tool can red-flag irregularities", journal = j-J-ACCOUNTANCY, volume = "196", number = "2", pages = "58--??", month = aug, year = "2003", CODEN = "JACYAD", ISSN = "0021-8448 (print), 1945-0729 (electronic)", ISSN-L = "0021-8448", bibdate = "Sat Feb 16 19:07:02 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.journalofaccountancy.com/Issues/2003/Aug/TurnExcelIntoAFinancialSleuth", acknowledgement = ack-nhfb, fjournal = "Journal of Accountancy", journal-URL = "http://www.journalofaccountancy.com/", } @Article{Sinai:2003:SP, author = "Ya. G. Sinai", title = "Statistical $ (3 x + 1) $ problem", journal = j-COMM-PURE-APPL-MATH, volume = "56", number = "7", pages = "1016--1028", month = jul, year = "2003", CODEN = "CPAMAT, CPMAMV", DOI = "https://doi.org/10.1002/cpa.10084", ISSN = "0010-3640 (print), 1097-0312 (electronic)", ISSN-L = "0010-3640", bibdate = "Mon Nov 21 07:52:19 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", note = "Dedicated to the memory of J{\"u}rgen K. Moser.", acknowledgement = ack-nhfb, fjournal = "Communications on Pure and Applied Mathematics (New York)", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0312", } @InProceedings{Swanson:2003:DPF, author = "David Swanson and Moon Jung Cho and John Eltinge", editor = "????", booktitle = "Proceedings of the Survey Research Section", title = "Detecting possibly fraudulent or error-prone survey data using {Benford's Law}", publisher = "American Statistical Association", address = "Washington, DC, USA", pages = "4172--4177", year = "2003", ISBN = "????", ISBN-13 = "????", LCCN = "????", bibdate = "Wed Nov 16 10:21:39 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.amstat.org/sections/srms/proceedings/y2003/Files/JSM2003-000205.pdf.", acknowledgement = ack-nhfb, } @Article{Wales:2003:IWI, author = "Elspeth Wales", title = "Is it Worth Investing in Software to Combat Fraud?", journal = "Computer Fraud \& Security", volume = "2003", number = "5", pages = "6--8", year = "2003", DOI = "https://doi.org/10.1016/S1361-3723(03)05009-7", ISSN = "1361-3723 (print), 1873-7056 (electronic)", ISSN-L = "1361-3723", bibdate = "Wed Jan 28 10:08:12 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S1361372303050097", abstract = "Despite the tendency for businesses to keep fraud to themselves, the consensus is that the problem is becoming worse. On the plus side though software solutions that can detect potential fraudulent activity are evolving in their scope and ability to automatically track transactions in real-time.", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @InCollection{Warren:2003:DLD, author = "Henry S. Warren", title = "The distribution of leading digits", crossref = "Warren:2003:HD", chapter = "15.3", pages = "264--267", year = "2003", bibdate = "Fri Mar 16 08:02:56 2012", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, } @Article{Wu:2003:IPD, author = "Shuo-Jye Wu and Chun-Tao Chang", title = "Inference in the {Pareto} distribution based on progressive {Type II} censoring with random removals", journal = j-J-APPL-STAT, volume = "30", number = "2", pages = "163--172", month = "????", year = "2003", CODEN = "????", ISSN = "0266-4763 (print), 1360-0532 (electronic)", ISSN-L = "0266-4763", bibdate = "Tue Dec 16 15:40:48 MST 2003", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.tandf.co.uk/journals/routledge/02664763.html", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Statistics", journal-URL = "http://www.tandfonline.com/loi/cjas20", } @Article{Wu:2003:PDC, author = "Wei Biao Wu and Chinya V. Ravishankar", title = "The performance of difference coding for sets and relational tables", journal = j-J-ACM, volume = "50", number = "5", pages = "665--693", month = sep, year = "2003", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/876638.876641", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", bibdate = "Mon Sep 8 17:55:23 MDT 2003", bibsource = "http://portal.acm.org/; http://www.acm.org/pubs/toc/; http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "We characterize the performance of difference coding for compressing sets and database relations through an analysis of the problem of estimating the number of bits needed for storing the spacings between values in sets of integers. We provide analytical expressions for estimating the effectiveness of difference coding when the elements of the sets or the attribute fields in database tuples are drawn from the uniform and Zipf distributions. We also examine the case where a uniformly distributed domain is combined with a Zipf distribution, and with an arbitrary distribution. We present limit theorems for most cases, and probabilistic convergence results in other cases. We also examine the effects of attribute domain reordering on the compression ratio. Our simulations show excellent agreement with theory.", acknowledgement = ack-nhfb, fjournal = "Journal of the Association for Computing Machinery", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", } @Article{Aoyama:2004:KDP, author = "Hideaki Aoyama and Yoshi Fujiwara and Wataru Souma", title = "Kinematics and dynamics of {Pareto--Zipf}'s law and {Gibrat}'s law", journal = j-PHYSICA-A, volume = "344", number = "1--2", pages = "117--121", day = "1", month = dec, year = "2004", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/j.physa.2004.06.099", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437104009185", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @TechReport{Berger:2004:DDU, author = "A. Berger", title = "Dynamics and digits: on the ubiquity of {Benford's Law}", type = "Preprint", institution = "Institute of Mechanics, Vienna University of Technology", address = "Vienna, Austria", pages = "3", day = "19", month = nov, year = "2004", bibdate = "Sat Feb 16 17:20:58 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.math.leidenuniv.nl/~verduyn/EQUADIFF03/MS14/berger_eqdiff03.pdf", acknowledgement = ack-nhfb, } @Article{Burrell:2004:LEF, author = "Quentin L. Burrell", title = "Letters to the {Editor}: Fitting {Lotka's Law}: {Some} cautionary observations on a recent paper by {Newby} et al. (2003)", journal = j-J-AM-SOC-INF-SCI-TECHNOL, volume = "55", number = "13", pages = "1209--1210", month = nov, year = "2004", CODEN = "JASIEF", DOI = "https://doi.org/10.1002/asi.20086", ISSN = "1532-2882 (print), 1532-2890 (electronic)", ISSN-L = "1532-2882", bibdate = "Fri Sep 11 10:42:16 MDT 2015", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/gnu.bib; http://www.math.utah.edu/pub/tex/bib/jasist.bib", note = "See \cite{Newby:2003:OSS}.", acknowledgement = ack-nhfb, fjournal = "Journal of the American Society for Information Science and Technology: JASIST", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1532-2890", onlinedate = "17 Aug 2004", } @Article{Challet:2004:BPD, author = "Damien Challet and Andrea Lombardoni", title = "Bug propagation and debugging in asymmetric software structures", journal = j-PHYS-REV-E, volume = "70", number = "4", pages = "046109", month = oct, year = "2004", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.70.046109", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Thu Feb 28 08:15:07 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.70.046109; http://pre.aps.org/abstract/PRE/v70/i4/e046109", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", issue = "4", journal-URL = "http://pre.aps.org/browse", keywords = "Linux; Unix; Zipf's Law", pagecount = "5", } @Article{Chen:2004:MFM, author = "Yanguang Chen and Yixing Zhou", title = "Multi-fractal measures of city-size distributions based on the three-parameter {Zipf} model", journal = j-CHAOS-SOLITONS-FRACTALS, volume = "22", number = "4", pages = "793--805", day = "1", month = nov, year = "2004", CODEN = "CSFOEH", DOI = "https://doi.org/10.1016/j.chaos.2004.02.059", ISSN = "0960-0779 (print), 1873-2887 (electronic)", ISSN-L = "0960-0779", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0960077904001031", acknowledgement = ack-nhfb, fjournal = "Chaos, Solitons \& Fractals", keywords = "Zipf's Law", } @Article{Cong:2004:WTP, author = "Lin Cong and Zhipeng Li", title = "On {Wilson}'s theorem and {Polignac} conjecture", journal = "ArXiv e-prints", day = "2", month = aug, year = "2004", bibdate = "Mon Feb 18 07:03:53 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://arxiv.org/abs/math.NT/0408018", abstract = "We introduce Wilson's theorem and Clement's result and present a necessary and sufficient condition for $p$ and $ p + 2 k $ to be primes where $k$ is a positive integer. By using Simiov's Theorem, we derive an improved version of Clement's result and characterizations of Polignac twin primes which parallel previous characterizations.", acknowledgement = ack-nhfb, } @TechReport{Diekmann:2004:DEE, author = "Andreas Diekmann", title = "{Datenf{\"a}lschung. Ergebnisse aus Experimenten mit der Benford Verteilung}. ({German}) [{Data} falsification. {Results} from experiments with the {Benford} distribution]", type = "Report", institution = "ETH Z{\"u}rich", address = "Z{\"u}rich, Switzerland", year = "2004", bibdate = "Fri Nov 18 12:09:49 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, language = "German", } @TechReport{Diekmann:2004:FDU, author = "Andreas Diekmann", title = "Not the First Digit! {Using} {Benford's Law} to Detect Fraudulent Scientific Data", type = "Report", institution = "ETH Z{\"u}rich", address = "Z{\"u}rich, Switzerland", pages = "26", month = oct, year = "2004", bibdate = "Sat Dec 10 11:12:29 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://rfe.cs.oswego.edu/eps/othr/papers/0507/0507001.pdf", abstract = "Digits in statistical data produced by natural or social processes are often distributed in a manner described by ``Benford's law''. Recently, a test against this distribution was used to identify fraudulent accounting data. This test is based on the supposition that real data follow the Benford distribution while fabricated data do not. Is it possible to apply Benford tests to detect fabricated or falsified scientific data as well as fraudulent financial data? We approached this question in two ways. First, we examined the use of the Benford distribution as a standard by checking digit frequencies in published statistical estimates. Second, we conducted experiments in which subjects were asked to fabricate statistical estimates (regression coefficients). These experimental data were scrutinized for possible deviations from the Benford distribution. There were two main findings. First, the digits of the published regression coefficients were approximately Benford distributed. Second, the experimental results yielded new insights into the strengths and weaknesses of Benford tests. At least in the case of regression coefficients, there were indications that checks for digit-preference anomalies should focus less on the first and more on the second and higher-digits", acknowledgement = ack-nhfb, } @Article{Durtschi:2004:EUB, author = "C. Durtschi and W. Hillison and C. Pacini", title = "The Effective Use of {Benford's Law} to Assist in Detecting Fraud in Accounting Data", journal = j-J-FORENSIC-ACCOUNT, volume = "5", number = "1", pages = "17--34", month = "????", year = "2004", CODEN = "????", ISSN = "1524-5586", bibdate = "Sun Nov 13 11:35:18 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Forensic Accounting", } @Article{Fontanari:2004:SNM, author = "J. F. Fontanari and L. I. Perlovsky", title = "Solvable null model for the distribution of word frequencies", journal = j-PHYS-REV-E, volume = "70", number = "??", pages = "042901", day = "25", month = oct, year = "2004", CODEN = "PLEEE8", DOI = "https://doi.org/10.1103/PhysRevE.70.042901", ISSN = "1539-3755 (print), 1550-2376 (electronic)", ISSN-L = "1539-3755", bibdate = "Tue Feb 26 06:40:57 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://link.aps.org/doi/10.1103/PhysRevE.70.042901", acknowledgement = ack-nhfb, fjournal = "Physical Review E (Statistical physics, plasmas, fluids, and related interdisciplinary topics)", journal-URL = "http://pre.aps.org/browse", keywords = "Zipf's Law", } @Article{Fujiwara:2004:DPZ, author = "Yoshi Fujiwara and Corrado {Di Guilmi} and Hideaki Aoyama and Mauro Gallegati and Wataru Souma", title = "Do {Pareto--Zipf} and {Gibrat} laws hold true? {An} analysis with {European} firms", journal = j-PHYSICA-A, volume = "335", number = "1--2", pages = "197--216", day = "1", month = apr, year = "2004", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/j.physa.2003.12.015", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437103011294", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Article{Fujiwara:2004:GPZ, author = "Yoshi Fujiwara and Hideaki Aoyama and Corrado {Di Guilmi} and Wataru Souma and Mauro Gallegati", title = "{Gibrat} and {Pareto--Zipf} revisited with {European} firms", journal = j-PHYSICA-A, volume = "344", number = "1--2", pages = "112--116", day = "1", month = dec, year = "2004", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/j.physa.2004.06.098", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437104009173", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Article{Fujiwara:2004:ZLF, author = "Yoshi Fujiwara", title = "{Zipf} law in firms bankruptcy", journal = j-PHYSICA-A, volume = "337", number = "1--2", pages = "219--230", day = "1", month = jun, year = "2004", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/j.physa.2004.01.037", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", MRclass = "91B38", MRnumber = "2092316", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437104001165", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03784371", keywords = "Zipf's Law", } @Article{Garcia-Berthou:2004:IBT, author = "Emili Garc{\'i}a-Berthou and Carles Alcaraz", title = "Incongruence between test statistics and {$P$} values in medical papers", journal = "BMC Medical Research Methodology", volume = "4", number = "1", pages = "1--5", year = "2004", DOI = "https://doi.org/10.1186/1471-2288-4-13", ISSN = "1471-2288", ISSN-L = "1471-2288", bibdate = "Sat Sep 10 11:16:07 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", abstract = "Given an observed test statistic and its degrees of freedom, one may compute the observed P value with most statistical packages. It is unknown to what extent test statistics and P values are congruent in published medical papers.", acknowledgement = ack-nhfb, journal-URL = "http://bmcmedresmethodol.biomedcentral.com/", keywords = "Benford's Law; digit frequencies", } @Article{Geyer:2004:DFD, author = "Christina Lynn Geyer and Patricia Pepple Williamson", title = "Detecting fraud in data sets using {Benford's Law}", journal = j-COMMUN-STAT-SIMUL-COMPUT, volume = "B33", number = "1", pages = "229--246", month = "????", year = "2004", CODEN = "CSSCDB", DOI = "https://doi.org/10.1081/SAC-120028442", ISSN = "0361-0918", ISSN-L = "0361-0918", MRclass = "62-07", MRnumber = "2044866", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.math.utah.edu/pub/tex/bib/fparith.bib", acknowledgement = ack-nhfb, ajournal = "Comm. Statist. Simulation Comput.", fjournal = "Communications in Statistics: Simulation and Computation", journal-URL = "http://www.tandfonline.com/loi/lssp20", keywords = "Benford's Law; Law of Anomalous Numbers; Zipf's Law", } @Article{Hurlimann:2004:IPB, author = "Werner H{\"u}rlimann", title = "Integer powers and {Benford}'s law", journal = j-INT-J-PURE-APPL-MATH, volume = "11", number = "1", pages = "39--46", year = "2004", CODEN = "????", ISSN = "1311-8080 (print), 1314-3395 (electronic)", ISSN-L = "1314-3395", MRclass = "62E15 (11B83 11K31 62E20)", MRnumber = "2033394 (2005c:62030)", MRreviewer = "I. N. Volodin", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Pure and Applied Mathematics", journal-URL = "http://ijpam.eu/", } @Article{Ismail:2004:SES, author = "Sanaa Isma{\"\i}l", title = "A Simple Estimator for the Shape Parameter of the {Pareto} Distribution with Economics and Medical Applications", journal = j-J-APPL-STAT, volume = "31", number = "1", pages = "3--13", month = jan, year = "2004", CODEN = "????", ISSN = "0266-4763 (print), 1360-0532 (electronic)", ISSN-L = "0266-4763", bibdate = "Sat Dec 4 12:10:37 MST 2004", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; http://www.tandf.co.uk/journals/routledge/02664763.html", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Statistics", journal-URL = "http://www.tandfonline.com/loi/cjas20", } @Article{Janvresse:2004:UDB, author = "{\'E}lise Janvresse and Thierry de la Rue", title = "From uniform distributions to {Benford}'s law", journal = j-J-APPL-PROBAB, volume = "41", number = "4", pages = "1203--1210", year = "2004", CODEN = "JPRBAM", DOI = "https://doi.org/10.1239/jap/1101840566", ISSN = "0021-9002 (print), 1475-6072 (electronic)", ISSN-L = "0021-9002", MRclass = "60J20 (60J10)", MRnumber = "2122815 (2006b:60161)", MRreviewer = "Michael Drmota", bibdate = "Sat Nov 12 09:13:23 2011", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://projecteuclid.org/getRecord?id=euclid.jap/1101840566; http://www.jstor.org/stable/4141393; https://www.univ-rouen.fr/LMRS/Persopage/Delarue/Publis/PDF/uniform_distribution_to_Benford_law.pdf", abstract = "We provide a new, probabilistic explanation for the appearance of Benford's law in everyday-life numbers, by showing that it arises naturally when we consider mixtures of uniform distributions. Then we connect our result to a result of Flehinger, for which we provide a shorter proof, and the speed of convergence.", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Probability", journal-URL = "http://www.jstor.org/journals/00219002.html; http://projecteuclid.org/euclid.jap/", } @Book{Judson:2004:GBF, author = "Horace Freeland Judson", title = "The Great Betrayal: Fraud in Science", publisher = "Harcourt", address = "Orlando, FL, USA", pages = "xiv + 463", year = "2004", ISBN = "0-15-100877-9", ISBN-13 = "978-0-15-100877-3", LCCN = "Q175.37 .J84 2004", bibdate = "Sat Sep 10 12:29:21 MDT 2016", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib; z3950.loc.gov:7090/Voyager", URL = "http://www.loc.gov/catdir/bios/har051/2004005906.html; http://www.loc.gov/catdir/description/har051/2004005906.html; http://www.loc.gov/catdir/samples/har051/2004005906.html; http://www.loc.gov/catdir/toc/ecip0416/2004005906.html", acknowledgement = ack-nhfb, remark = "No mention of Benford's Law or first digit phenomenon, but nevertheless relevant for applications of Benford's Law to detection of fraud in data.", subject = "Fraud in science", tableofcontents = "Preface / xi \\ Prologue / 1 \\ 1: A Culture of Fraud / 9 \\ 2: What's It Like? A Typology of Scientific Fraud / 43 \\ 3: Patterns of Complicity: Recent Cases / 98 \\ 4: Hard to Measure, Hard to Define: The Incidence of Scientific Fraud and the Struggle Over Its Definition / 155 \\ 5: The Baltimore Affair / 191 \\ 6: The Problems of Peer Review / 244 \\ 7: Authorship, Ownership: The Problems of Credit, Plagiarism, and Property / 287 \\ 8: The Rise of Open Publication on the Internet / 325 \\ 9: Laboratory to Law: The Problems of Institutions When Misconduct is Charged / 369 \\ Epilogue / 404 \\ Notes / 419 \\ Index / 447", } @TechReport{Kollath-Romano:2004:DSD, author = "P. Kollath-Romano", title = "On the distribution of significant digits in numbers", type = "Preprint", institution = "Rensselaer Polytechnic Institute", address = "Troy, NY, USA", year = "2004", bibdate = "Sat Feb 16 18:15:19 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, } @InCollection{Kotz:2004:FDP, author = "Samuel Kotz and N. Balakrishnan and Campbell B. Read and Brani Vidakovic", editor = "Samuel Kotz and N. Balakrishnan and Campbell B. Read and Brani Vidakovic", booktitle = "Encyclopedia of Statistical Sciences", title = "First-Digit Problem", publisher = pub-WILEY, address = pub-WILEY:adr, edition = "Second", pages = "??--??", year = "2004", DOI = "https://doi.org/10.1002/0471667196.ess0783.pub2", ISBN = "0-471-66719-6 (e-book)", ISBN-13 = "978-0-471-66719-3 (e-book)", bibdate = "Sun Feb 24 10:49:25 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", acknowledgement = ack-nhfb, keywords = "Benford's Law", } @Article{Kristiansen:2004:BTZ, author = "Kai de Lange Kristiansen and Geir Helgesen and Arne T. Skjeltorp", title = "Braid theory and {Zipf} relation used in dynamics of magnetic microparticles", journal = j-ECONOM-LETT, volume = "30", number = "1", pages = "2387--2388", day = "1", month = may, year = "2004", CODEN = "ECLEDS", DOI = "https://doi.org/10.1016/j.jmmm.2003.12.1288", ISSN = "0165-1765 (print), 1873-7374 (electronic)", ISSN-L = "0165-1765", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0304885303019127", acknowledgement = ack-nhfb, fjournal = "Economics Letters", keywords = "Zipf's Law", } @Article{Kristiansen:2004:EOZ, author = "Kai de Lange Kristiansen and Geir Helgesen and Arne T. Skjeltorp", title = "Experimental observation of {Zipf--Mandelbrot} relation", journal = j-PHYSICA-A, volume = "335", number = "3--4", pages = "413--420", day = "15", month = apr, year = "2004", CODEN = "PHYADX", DOI = "https://doi.org/10.1016/j.physa.2003.12.024", ISSN = "0378-4371 (print), 1873-2119 (electronic)", ISSN-L = "0378-4371", MRclass = "82D10 (76T99 82-05)", MRnumber = "2044153", bibdate = "Wed Feb 27 07:32:55 MST 2013", bibsource = "http://www.math.utah.edu/pub/tex/bib/benfords-law.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0378437103011804", acknowledgement = ack-nhfb, fjournal = "Physica A. Statistical Mechanics and its Applications", journal-URL = "