Research Scholars in ResidenceCopyright (c) 2019 California Polytechnic State University, San Luis Obispo All rights reserved.
https://digitalcommons.calpoly.edu/rgp_rsr
Recent documents in Research Scholars in Residenceen-usWed, 03 Apr 2019 11:03:11 PDT3600How to combine independent data sets for the same quantity
https://digitalcommons.calpoly.edu/rgp_rsr/87
https://digitalcommons.calpoly.edu/rgp_rsr/87Tue, 17 Oct 2017 12:03:20 PDT
This paper describes a new mathematical method called conflation for consolidating data from independent experiments that measure the same physical quantity. Conflation is easy to calculate and visualize and minimizes the maximum loss in Shannon information in consolidating several independent distributions into a single distribution. A formal mathematical treatment of conflation has recently been published. For the benefit of experimenters wishing to use this technique, in this paper we derive the principal basic properties of conflation in the special case of normally distributed (Gaussian) data. Examples of applications to measurements of the fundamental physical constants and in high energy physics are presented, and the conflation operation is generalized to weighted conflation for cases in which the underlying experiments are not uniformly reliable. When different experiments are designed to measure the same unknown quantity, how can their results be consolidated in an unbiased and optimal way? Given data from experiments made at different times, in different locations, with different methodologies, and perhaps differing even in underlying theory, is there a straightforward, easily applied method for combining the results from all of the experiments into a single distribution? This paper describes a new mathematical method called conflation for consolidating data from independent experiments that measure the same physical quantity.
]]>
Theodore P. Hill et al.Hubble's law implies Benford's law for distances to galaxies
https://digitalcommons.calpoly.edu/rgp_rsr/86
https://digitalcommons.calpoly.edu/rgp_rsr/86Tue, 17 Oct 2017 12:03:16 PDT
A recent article by Alexopoulos and Leontsinis presented empirical evidence that the first digits of the distances from the Earth to galaxies are a reasonably good fit to the probabilities predicted by Benford’s law, the well known logarithmic statistical distribution of significant digits. The purpose of the present article is to give a theoretical explanation, based on Hubble’s law and mathematical properties of Benford’s law, why galaxy distances might be expected to follow Benford’s law. The new galaxy-distance law derived here, which is robust with respect to change of scale and base, to additive and multiplicative computational or observational errors, and to variability of the Hubble constant in both time and space, predicts that conformity to Benford’s law will improve as more data on distances to galaxies becomes available. Conversely, with the logical derivation of this law presented here, the recent empirical observations may be viewed as independent evidence of the validity of Hubble’s law.
]]>
Theodore P. Hill et al.Gender gaps in science: The creativity factor
https://digitalcommons.calpoly.edu/rgp_rsr/85
https://digitalcommons.calpoly.edu/rgp_rsr/85Tue, 17 Oct 2017 12:03:13 PDTTheodore P. Hill et al.The kilogram cabal
https://digitalcommons.calpoly.edu/rgp_rsr/84
https://digitalcommons.calpoly.edu/rgp_rsr/84Tue, 17 Oct 2017 12:03:09 PDTTheodore P. HillA Prophet Inequality Related to the Secretary Problem
https://digitalcommons.calpoly.edu/rgp_rsr/83
https://digitalcommons.calpoly.edu/rgp_rsr/83Tue, 06 Mar 2012 14:50:29 PST
Let Z1, Z2 , .. . , Zn be independent 0-1-valued random variables. A gambler gels a. reward 1 if he stop8 a.t the time of the last success and otherwise gets no reward. A simple comparison with a Poisson process is used to show that a prophet can do at most e times as well as the gambler using an optimal stopping time. For fixed n, the best constant is (n/(n -l ))"-1.
]]>
Theodore P. Hill et al.Finite-state Markov Chains Obey Benford’s Law
https://digitalcommons.calpoly.edu/rgp_rsr/82
https://digitalcommons.calpoly.edu/rgp_rsr/82Tue, 07 Feb 2012 08:42:21 PST
A sequence of real numbers(x_{n})is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_{n}), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (P^{n} −P*)and (P ^{n+1}-P^{n}) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton’s method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (“nonresonance”) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.
]]>
Arno Berger et al.A Basic Theory of Benford’s Law
https://digitalcommons.calpoly.edu/rgp_rsr/81
https://digitalcommons.calpoly.edu/rgp_rsr/81Thu, 26 Jan 2012 13:50:53 PST
Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford’s Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.
]]>
Arno Berger et al.Conflations of Probability Distributions
https://digitalcommons.calpoly.edu/rgp_rsr/80
https://digitalcommons.calpoly.edu/rgp_rsr/80Thu, 26 Jan 2012 13:49:37 PST
The conflation of a finite number of probability distributions P_{1},...,P_{n} is a consolidation of those distributions into a single probability distribution Q = Q(P_{1},...,P_{n}), where intuitively Q is the conditional distribution of independent random variables X1,...,X_{n} with distributions P_{1},...,P_{n}, respectively, given that X_{1} = ···= X_{n}. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. Q is shown to be the unique probability distribution that minimizes the loss of Shannon information in consolidating the combined information from P_{1},...,P_{n} into a single distribution Q, and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. In that sense, conflation may be viewed as an optimal method for combining the results from several different independent experiments. When P_{1},...,P_{n} are Gaussian, Q is Gaussian with mean the classical weighted-mean-squares reciprocal of variances. A version of the classical convolution theorem holds for conflations of a large class of a.c. measures.
]]>
Theodore P. HillCriticisms of the proposed “new SI”
https://digitalcommons.calpoly.edu/rgp_rsr/79
https://digitalcommons.calpoly.edu/rgp_rsr/79Mon, 17 Oct 2011 16:48:36 PDTTheodore P. HillObituary for Lester Eli Dubins, 1921-2010
https://digitalcommons.calpoly.edu/rgp_rsr/78
https://digitalcommons.calpoly.edu/rgp_rsr/78Wed, 13 Jul 2011 11:34:58 PDTDavid Gilat et al.Fundamental Flaws in Feller’s Classical Derivation of Benford’s Law
https://digitalcommons.calpoly.edu/rgp_rsr/77
https://digitalcommons.calpoly.edu/rgp_rsr/77Thu, 02 Jun 2011 12:49:03 PDT
Feller’s classic text An Introduction to Probability Theory and its Applications contains a derivation of the well known significant-digit law called Benford’s law. More specifically, Fellergives a sufficient condition (“large spread”) for a random variable X to be approximately Benford distributed, that is, for log_{10}X to be approximately uniformly distributed moduloone. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstratethat larges pread (or large spread on a logarithmic scale) does not imply that a random variable is approximately Benford distributed, for any reasonable definition of “spread” or measure of dispersion.
]]>
Arno Berger et al.Counterexamples in the Theory of Fair Division
https://digitalcommons.calpoly.edu/rgp_rsr/76
https://digitalcommons.calpoly.edu/rgp_rsr/76Fri, 20 May 2011 08:41:12 PDTTheodore P. Hill et al.How to Publish Counterexamples in 1 2 3 Easy Steps
https://digitalcommons.calpoly.edu/rgp_rsr/75
https://digitalcommons.calpoly.edu/rgp_rsr/75Fri, 20 May 2011 08:41:09 PDTTheodore P. HillHoisting the Black Flag
https://digitalcommons.calpoly.edu/rgp_rsr/74
https://digitalcommons.calpoly.edu/rgp_rsr/74Fri, 20 May 2011 08:41:05 PDTTheodore P. HillCutting Cakes Carefully
https://digitalcommons.calpoly.edu/rgp_rsr/73
https://digitalcommons.calpoly.edu/rgp_rsr/73Fri, 20 May 2011 08:41:00 PDTTheodore P. Hill et al.Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem
https://digitalcommons.calpoly.edu/rgp_rsr/72
https://digitalcommons.calpoly.edu/rgp_rsr/72Fri, 20 May 2011 08:40:57 PDTArno Berger et al.Towards a Better Definition of the Kilogram
https://digitalcommons.calpoly.edu/rgp_rsr/71
https://digitalcommons.calpoly.edu/rgp_rsr/71Fri, 20 May 2011 08:40:53 PDT
It is widely accepted that improvement of the current International System of Units (SI) is necessary, and that central to this problem is redefinition of the kilogram. This paper compares the relative advantages of two main proposals for a modern scientific definition of the kilogram: an ‘electronic kilogram’ based on a fixed value of Planck’s constant, and an ‘atomic kilogram’ based on a fixed value for Avogadro’s number. A concrete and straightforward atomic definition of the kilogram is proposed. This definition is argued to be more experimentally neutral than the electronic kilogram, more realizable by school and university laboratories than the electronic kilogram, and more readily comprehensible than the electronic kilogram.
]]>
Theodore P. Hill et al.A Stronger Conclusion to the Classical Ham Sandwich Theorem
https://digitalcommons.calpoly.edu/rgp_rsr/70
https://digitalcommons.calpoly.edu/rgp_rsr/70Fri, 20 May 2011 08:40:49 PDT
The conclusion of the classical ham sandwich theorem for bounded Borel sets may be strengthened, without additional hypotheses – there always exists a common bisecting hyperplane that touches each of the sets, that is, that intersects the closure of each set. In the discrete setting, where the sets are finite (and the measures are counting measures), there always exists a bisecting hyperplane that contains at least one point in each of the sets. Both these results follow from the main theorem of this note, which says that for n compactly supported positive finite Borel measures in R^{n}, there is always an (n − 1)-dimensional hyperplane that bisects each of the measures and intersects the support of each measure. Thus, for example, at any given instant of time, there is one planet, one moon and one asteroid in our solar system and a single plane touching all three that exactly bisects the total planetary mass, the total lunar mass, and the total asteroidal mass of the solar system. In contrast to the bisection conclusion of the classical ham sandwich theorem, this bisection-and-intersection conclusion does not carry over to unbounded sets of finite measure.
]]>
John H. Elton et al.Goal Problems in Gambling Theory
https://digitalcommons.calpoly.edu/rgp_rsr/69
https://digitalcommons.calpoly.edu/rgp_rsr/69Fri, 17 Sep 2010 12:17:46 PDT
A short introduction to goal problems in abstract gambling theory is given, along with statements of some of the main theorems and a number of examples, open problems and references. Emphasis is on the finite-state, countably-additive setting with such classical objectives as reaching a goal, hitting a goal infinitely often, staying in the goal, and maximizing the average time spent at a goal.
]]>
Theodore P. HillA characterisation of Newton maps
https://digitalcommons.calpoly.edu/rgp_rsr/68
https://digitalcommons.calpoly.edu/rgp_rsr/68Mon, 23 Aug 2010 10:31:52 PDT
Conditions are given for a C^{k} map T to be a Newton map, that is, the map associated with a differentiable real-valued function via Newton’s method. For finitely differentiable maps and functions, these conditions are only necessary, but in the smooth case, i.e. for k = ∞ , they are also sufficient. The characterisation rests upon the structure of the fixed point set of T and the value of the derivative T^{1}there, and it is best possible as is demonstrated through examples.
]]>
Arno Berger et al.