Related papers: Leading Digit Laws on Linear Lie Groups

According to a popular belief, the decimal digits of mathematical constants such as {\pi} behave like statistically independent random variables, each taking the values 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 with equal probability of 1/10. If…

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…

Motivated by giving a meaning to "The probability that a random integer has initial digit d", we define a URI-set as a random set E of natural integers such that each n>0 belongs to E with probability 1/n, independently of other integers.…

An important aspect of religious association is that adherents, as part of their religious duty, carry out various activities. One religious group known for keeping the elaborate records of day-to-day activities of its members is the…

The renowned proverb, Numbers do not lie, underscores the reliability and insight that lie beneath numbers, a concept of undisputed importance, especially in economics and finance etc. Despite the prosperity of Benford's Law in the first…

We consider difference equations of order four and determine the one parameter Lie group of transformations (Lie symmetries) that leave them invariant. We introduce a technique for finding their first integrals and discuss the association…

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, Feller gives a sufficient condition ("large spread")…

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $\alpha_n$, where $(\alpha_n)$ is a sequence such that $\sum 1/\alpha_n$ is finite. This set is shown to have…

The first digit (FD) phenomenon i.e., the significant digits of numbers in large data are often distributed according to a logarithmically decreasing function was first reported by S. Newcomb and then many decades later independently by F.…

Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral…

In many real life situations, it is observed that the first digits (i.e., $1,2,\ldots,9$) of a numerical data-set, which is expressed using decimal system, do not follow a random distribution. Instead, smaller numbers are favoured by nature…

The \textit{ternary digits of $2^n$} are a finite sequence of 0s, 1s, and 2s. It is a natural question to ask whether the frequency of any string of 0s, 1s, and 2s in this sequence approaches the same limit for all strings of the same…

Let $f(z)=\sum_{n=1}^\infty \lambda_f(n)e^{2\pi i n z}\in S_{k}^{new}(\Gamma_0(N))$ be a normalized Hecke eigenform of even weight $k\geq2$ on $\Gamma_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We…

A universal First-Letter Law (FLL) is derived and described. It predicts the percentages of first letters for words in novels. The FLL is akin to Benford's law (BL) of first digits, which predicts the percentages of first digits in a data…

Benford's law describes the distribution of the first digit of numbers appearing in a wide variety of numerical data, including tax records, and election outcomes, and has been used to raise "red flags" about potential anomalies in the data…

We explain Kossovsky's generalization of Benford's law which is a formula that approximates the distribution of leftmost digits in finite sequences of natural data and apply it to six sequences of data including populations of US cities and…

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on $n$ integers. Under certain…

The use of statistical methods for the description of complex quantum systems was primarily motivated by the failure of a line-by-line interpretation of atomic spectra. Such methods reveal regularities and trends in the distributions of…

Gram's Law describes a pattern that frequently occurs in the distribution of the non-trivial zeros of the Riemann zeta function along the critical line. Whenever Gram's Law holds true, it reduces the difficulty of computing the…

The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and…