Related papers: Plus and minus logarithms and Amice transform
The paper explores properties of the {\L}ukasiewicz {\mu}-calculus, or {\L}{\mu} for short, an extension of {\L}ukasiewicz logic with scalar multiplication and least and greatest fixed-point operators (for monotone formulas). We observe…
Let k be a positive integer and let G_k denote the set of non-commutative k-variable distributions \mu such that \mu (X_1) = ... = \mu (X_k) = 1. G_k is a group under the operation of free multiplicative convolution. We identify G_k as the…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
The standard approximation of a natural logarithm in statistical analysis interprets a linear change of \(p\) in \(\ln(X)\) as a \((1+p)\) proportional change in \(X\), which is only accurate for small values of \(p\). I suggest…
We come back to the construction of p-adic L-functions attached to cusp forms of even weight k in the spirit of G. Stevens, R. Pollack [7] and M. Greenberg [3] with a new unified presentation including the non-ordinary case. This…
Let $m\ge 2$ be an integer, $K$ an algebraic number field and $\alpha\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in…
Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Pad\'e approximations of…
We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the $L^{p} $ estimate $p^{\ast} -1$, where $p^{\ast} = \max \{ p,q \}$ and $p$ and $q$ are conjugate exponents. This estimate is sharp if…
We study, with the help of a computer program, the Polish Algorithm for finite terms satisfying various algebraic laws, e.g., left distributivity a(bc) = (ab)(ac). While the termination of the algorithm for left distributivity remains open…
Amdeberhan et al. (2024) introduced the notion of a generalized overcubic partition function $\overline a_c (n)$ and proved an infinite family of congruences modulo a prime $p\ge 3$ and some Ramanujan type congruences. In this paper, we…
We generalize a previous inequality related to a sharp version of the Littlewood conjecture on the minimal $L_1$-norm of $N$-term exponential sums $f$ on the unit circle. The new result concerns replacing the expression $\log(1+t|f|^2)$…
In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic…
Recently, Ko\c{c} proposed a neat and efficient algorithm for computing \[ x = a^{-1} \pmod {p^k} \] for a prime $p$ based on the exact solution of linear equations using $p$-adic expansions. The algorithm requires only addition and right…
A collector wishes to collect $m$ complete sets of $N$ distinct coupons. The draws from the population are considered to be independent and identical distributed with replacement, and the probability that a type-$j$ coupon is drawn is noted…
An extension of the theory of the Iterated Logarithmic Algebra gives the logarithmic analog of a Sheffer or Appell sequence of polynomials. This leads to several examples including Stirling's formula and a logarithmic version of the…
The Welch map $x \rightarrow g^{x-1+c}$ is similar to the discrete exponential map $x \rightarrow g^x$, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to…
Formal transformations somehow resembling the usual derivative are surprisingly common in computer science, with two notable examples being derivatives of regular expressions and derivatives of types. A newcomer to this list is the…
We describe a new arithmetic system for the Magma computer algebra system for working with $p$-adic numbers exactly, in the sense that numbers are represented lazily to infinite $p$-adic precision. This is the first highly featured such…
Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$;…
Let m be a positive integer, and let A be the set of all positive integers that belong to a union of r distinct congruence classes modulo m. We assume that the elements of A are relatively prime, that is, gcd(A) = 1. Let p_A(n) denote the…