First digit law from Laplace transform

The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benford's law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benford's law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevalence of Benford's law. The slight variation from the Benford term leads to deviations from Benford's law for distributions which oscillate violently in the inverse Laplace space. We prove that the whole family of completely monotonic distributions can satisfy Benford's law within a small bound. Our study suggests that the origin of Benford's law is from the way that we write numbers, thus should be taken as a basic mathematical knowledge.     

The law of the leading digits and the world religions

Benford’s law states that the occurrence of significant digits in many data sets is not uniform but tends to follow a logarithmic distribution such that the smaller digits appear as first significant digits more frequently than the larger ones. We investigate here numerical data on the country-wise adherent distribution of seven major world religions i.e. Christianity, Islam, Buddhism, Hinduism, Sikhism, Judaism and Baha’ism to see if the proportion of the leading digits occurring in the distribution conforms to Benford’s law. We find that the adherent data of all the religions, except Christianity, excellently does conform to Benford’s law. Furthermore, unlike the adherent data on Christianity, the significant digit distribution of the three major Christian denominations i.e. Catholicism, Protestantism and Orthodoxy obeys the law. Thus in spite of their complexity general laws can be established for the evolution of religious groups.     

Benford’s law in non-equilibrium processes: Droplet collisions case

Benford’s law is investigated for the simulation results generated from non-equilibrium molecular dynamics. A statistic to measure how closely a set of the numbers follows Benford’s law is defined. The simulation data are from the collisions of two nano droplets with different impact velocities. When a non-equilibrium system returns to its equilibrium state, some physical quantities relevant to the non-equilibrium settings follow Benford’s law more closely. The initial settings for the non-equilibrium state can be interpreted as a data fabrication of its corresponding equilibrium state. A connection with the Shannon entropy for the first digit distribution is also discussed.     

Reliability of Financial Information from the Perspective of Benford’s Law

This study investigates the conformity to Benford’s Law of the information disclosed in financial statements. Using the first digit test of Benford’s Law, the study analyses the reliability of financial information provided by listed companies on an emerging capital market before and after the implementation of International Financial Reporting Standards (IFRS). The results of the study confirm the increase of reliability on the information disclosed in the financial statements after IFRS implementation. The study contributes to the existing literature by bringing new insights into the types of financial information that do not comply with Benford’s Law such as the amounts determined by estimates or by applying professional judgment.     

Benford’s law and half-lives of unstable nuclei

We find that the experimental data of the -decay half-lives for 627 nuclei are in good agreement with Benford’s law, which states that the frequency of the appearance of each figure, 1-9, as the first significant digit, follows a logarithmic distribution favoring the smallest value. In order to generalize the applicability of Benford’s law, we systematically investigate the data of the total half-lives for 3177 nuclides in their ground and isomeric states, where the half-lives of many nuclei are determined by -decay and spontaneous fission. We find that they are also in excellent agreement with Benford’s law, although they are determined by different interactions such as strong, weak and electromagnetic interactions. The possible physics behind them is discussed. Moreover, Benford’s law can be used to test theoretical models or methods.     

Benford’s law: A Poisson perspective

Benford’s law is a counterintuitive statistical law asserting that the distribution of leading digits, taken from a large ensemble of positive numerical values that range over many orders of scale, is logarithmic rather than uniform (as intuition suggests). In this paper we explore Benford’s law from a Poisson perspective, considering ensembles of positive numerical values governed by Poisson-process statistics. We show that this Poisson setting naturally accommodates Benford’s law and: (i) establish a Poisson characterization and a Poisson multidigit-extension of Benford’s law; (ii) study a system-invariant leading-digit distribution which generalizes Benford’s law, and establish a Poisson characterization and a Poisson multidigit-extension of this distribution; (iii) explore the universal emergence of the system-invariant leading-digit distribution, couple this universal emergence to the universal emergence of the Weibull and Fréchet extreme-value distributions, and distinguish the special role of Benford’s law in this universal emergence; (iv) study the continued-fractions counterpart of the system-invariant leading-digit distribution, and establish a Poisson characterization of this distribution; and (v) unveil the elemental connection between the system-invariant leading-digit distribution and its continued-fractions counterpart. This paper presents a panoramic Poisson approach to Benford’s law, to its system-invariant generalization, and to its continued-fractions counterpart.     

The significant digit law in statistical physics

The occurrence of the nonzero leftmost digit, i.e., 1,2,…,9, of numbers from many real world sources is not uniformly distributed as one might naively expect, but instead, the nature favors smaller ones according to a logarithmic distribution, named Benford’s law. We investigate three kinds of widely used physical statistics, i.e., the Boltzmann-Gibbs (BG) distribution, the Fermi-Dirac (FD) distribution, and the Bose-Einstein (BE) distribution, and find that the BG and FD distributions both fluctuate slightly in a periodic manner around Benford’s distribution with respect to the temperature of the system, while the BE distribution conforms to it exactly whatever the temperature is. Thus Benford’s law seems to present a general pattern for physical statistics and might be even more fundamental and profound in nature. Furthermore, various elegant properties of Benford’s law, especially the mantissa distribution of data sets, are discussed.     

Quantum effect,critical dynamics and one-particle excitation in 1/n expansion

By quantizing Ma's Hamiltonian, quantum effect on η, the energy spectrum of one-particle excitation and the dynamic scaling law are studied up to O(1/n). The case just at the critical point is considered.     

Possible thermodynamic structure underlying the laws of Zipf and Benford

We show that the laws of Zipf and Benford, obeyed by scores of numerical data generated by many and diverse kinds of natural phenomena and human activity are related to the focal expression of a generalized thermodynamic structure. This structure is obtained from a deformed type of statistical mechanics that arises when configurational phase space is incompletely visited in a strict way. Specifically, the restriction is that the accessible fraction of this space has fractal properties. The focal expression is an (incomplete) Legendre transform between two entropy (or Massieu) potentials that when particularized to first digits leads to a previously existing generalization of Benford’s law. The inverse functional of this expression leads to Zipf’s law; but it naturally includes the bends or tails observed in real data for small and large rank. Remarkably, we find that the entire problem is analogous to the transition to chaos via intermittency exhibited by low-dimensional nonlinear maps. Our results also explain the generic form of the degree distribution of scale-free networks.     

Benford’s law and cross-sections of A(n,$ \alpha$)B reactions

Benford’s law, also called the first-digit law, states that in lists of numbers from many quite disparate databases, the leading digit is distributed in a non-uniform but actually logarithmic way. We have investigated the first-digit distribution of experimental cross-sections of A(n,a \alpha)B reactions. In the case of below-barrier a \alpha -particle emission from compound nucleus, it is found that the (n,a \alpha) reaction cross-sections approximately follow the first-digit distribution indicated by Benford’s law. The origin of this first-digit distribution is discussed within the framework of the statistical model. In addition, Benford’s law is used to test the evaluated cross-sections of A(n,a \alpha)B reactions.     

Benford’s law and Theil transform of financial data

Among econophysics investigations, studies of religious groups have been of interest. On the one hand, the present paper concerns the Antoinist community financial reports–a community which appeared at the end of the 19-th century in Belgium. Several growth–decay regimes have been previously found over different time spans. However, there is common suspicion about sect finances. In that spirit, the Antoinist community yearly financial reports, income and expenses, are hereby examined through the so-called Benford’s law. The latter is often used as a test about possible accounting wrongdoings. On the other hand, Benford’s law is known to be invariant under scale and base transformation. Therefore, as a further test, of both such data and the use of Benford’s law, the yearly financial reports are nonlinearly remapped through a sort of Theil transformation, i.e. based on a log-transformation. The resulting data is again analyzed along the Benford’s law scheme. Bizarre, puzzling, features are seen. However, it is emphasized that such a non-linear transformation can shift the argument toward a more objective conclusion. In an appendix, some brief discussion is made on why the original Theil mapping should not be used. In a second appendix, an imperfect Benford’s law-like form–better suited for anomalous distributions–is presented.     

On subgame perfect equilibria in quantum Stackelberg duopoly with incomplete information

The Li–Du–Massar quantum duopoly model is one of the generally accepted quantum game schemes. It has applications in a wide range of duopoly problems. Our purpose is to study Stackelberg's duopoly with incomplete information in the quantum domain. The result of Lo and Kiang has shown that the correlation of players' quantities caused by the quantum entanglement enhances the first-mover advantage in the game. Our work demonstrates that there is no first-mover advantage if the players' actions are maximally correlated. Furthermore, we proved that the second mover gains a higher equilibrium payoff than the first one.     

An introduction to the fundamentals of the renormalization group in critical phenomena

The fundamental concepts underlying the application of the renormalization group and related techniques to critical phenomena are reviewed at an elementary level. Topics discussed include: the definition of the renormalization group as a functional integral over high momentum components of the spin field, the behaviour of the renormalization group near the fixed point and the derivation of scaling, Wilson's approximate recursion relation, trivial and non-trivial fixed points of isotropic spin systems near d = 4, Feynman graph expansions for critical exponents, ? = 4 ? d and 1/n-expansions, the derivation of exact recursion relations and co-ordinate space transformations for d = 2 Ising systems     

Critical line broadening of the EPR in the two-dimensional antiferromagnet crystals Rb2MnCl4 and Rb2Mn0,8Cd0,2Cl4

The EPR spectrum of quasi-two-dimensional antiferromagnets Rb₂Mn_xCd_1?xCl₄ (x=1,0; 0,8) has been studied in a critical temperature range. Two theoretical approaches - the scaling theory and the soliton's theory - are used to explain experimental data on temperature of the EPR linewidth. In the first interpretation critical exponents are determined. For both crystal two temperature regions with different critical exponents are found. It is shown that the soliton's theory with an anomaleous great excitement energy describes the experimental data satisfactory.     

Zeno effect in quantum Newton's cradle

We describe a chain of quantum oscillators which behaves analogously to Newton's cradle. The energy swings between the ends of the chain with very low population in its interior. Moreover, the oscillators at the ends can entangle with each other with negligible entanglement with the intermediate oscillators that mediate the process. Up to a certain number of oscillators, the system evolves in a manner similar to two coupled oscillators. The conditions for such behavior and the characteristic periods are analyzed. When that number exceeds a threshold, the dynamical regime changes to virtually freezing. In the oscillatory regime, Zeno effect can be observed. The parallelism between the Zeno dynamics in quantum Newton's cradle and in two coupled oscillators is highlighted. Promising platforms to observe such phenomena in the laboratory are cavities in photonic-band-gap material and trapped ions.     

First Digits’ Shannon Entropy

Related to the letters of an alphabet, entropy means the average number of binary digits required for the transmission of one character. Checking tables of statistical data, one finds that, in the first position of the numbers, the digits 1 to 9 occur with different frequencies. Correspondingly, from these probabilities, a value for the Shannon entropy H can be determined as well. Although in many cases, the Newcomb–Benford Law applies, distributions have been found where the 1 in the first position occurs up to more than 40 times as frequently as the 9. In this case, the probability of the occurrence of a particular first digit can be derived from a power function with a negative exponent p > 1. While the entropy of the first digits following an NB distribution amounts to H = 2.88, for other data distributions (diameters of craters on Venus or the weight of fragments of crushed minerals), entropy values of 2.76 and 2.04 bits per digit have been found.     

ε-expansion calculations for spin-glasses

The critical properties of the spin-glass transition proposed by Edwards and Anderson are studied using the minimal subtraction method. The universal ratio of the second correction to scaling amplitude to the square of the first for the order parameter susceptibility χ₀ is calculated to first order in ε(ε=6?d). Comparison is made with Fisch and Harris' high temperature series analysis which incorporated Rudnick-Nelson-type corrections to scaling. Within the same formalism the critical exponents are calculated to second order in ε. They agree with the first order ε expansion of Harris, Lubensky and Chen.     

Blind quantum computation with identity authentication

Blind quantum computation (BQC) allows a client with relatively few quantum resources or poor quantum technologies to delegate his computational problem to a quantum server such that the client's input, output, and algorithm are kept private. However, all existing BQC protocols focus on correctness verification of quantum computation but neglect authentication of participants' identity which probably leads to man-in-the-middle attacks or denial-of-service attacks. In this work, we use quantum identification to overcome such two kinds of attack for BQC, which will be called QI-BQC. We propose two QI-BQC protocols based on a typical single-server BQC protocol and a double-server BQC protocol. The two protocols can ensure both data integrity and mutual identification between participants with the help of a third trusted party (TTP). In addition, an unjammable public channel between a client and a server which is indispensable in previous BQC protocols is unnecessary, although it is required between TTP and each participant at some instant. Furthermore, the method to achieve identity verification in the presented protocols is general and it can be applied to other similar BQC protocols.     

Critical behavior of a spherical model with a free surface

Critical phenomena ind-dimensional ferromagnetic spherical models on hypercubic lattices with free surfaces are studied. The surface specific heat and surface susceptibilities are obtained. The exponents characterizing the divergence of these surface quantities at the bulk critical temperature are found to satisfy recently proposed scaling relations. The variation of the susceptibility with distance from the surface is also discussed. The author's recent scaling theory for surface properties is investigated in detail, and found to give an exact representation for the free energy of a three-dimensional spherical model of finite thickness in finite bulk and surface magnetic fields. A scaling form for the surface free energy is derived.