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We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce…
This letter derives closed-form expressions for the probability density function of the distance between two nodes located in heterogeneous concentric geometries, namely a disk or sphere and a surrounding annulus or spherical shell. Two…
Given linear spaces $E$ and $F$ over the real numbers or a field of characteristic zero, a simple argument is given to represent a symmetric multilinear map $u(x_1, x_2, \ldots, x_n)$ from $E^n$ to $F$ in terms of its restriction to the…
Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are…
The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the…
Approximating a definite integral of product of cosines to within an accuracy of n binary digits where the integrand depends on input integers x[k] given in binary radix, is equivalent to counting the number of equal-sum partitions of the…
In this notice, we revisit the recent work [1] of Jung Yoog Kang and Tai Sup about special polynomials with exponential distribution in order to state some improvements and get new proofs for results therein.
In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare…
The paper suggests a slightly more rigorous justification to Wang et al.'s work from 2007, and introduces the Slanted Line Integral.
Position probability distribution of a set of massive mutually exclusive particles in one dimension has been defined. Examples with a given two mutually exclusive particles system are considered. It is emphasized that quantum particles at…
The Stirling approximation formula for $n!$ dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and…
Exact formulas are derived for the probability density functions of the sum and difference of two independent non-central gamma distributed random variables, with both series and integral representations of the density presented. These…
To complete a previous paper, the probability density functions of the center-of-gravity as positioning algorithm are derived with classical methods. These methods, as suggested by the textbook of Probability, require the preliminary…
While Gaussian probability densities are omnipresent in applied mathematics, Gaussian cumulative probabilities are hard to calculate in any but the univariate case. We study the utility of Expectation Propagation (EP) as an approximate…
The goal of this "Habilitation \`a diriger des recherches" is to present two different applications, namely computations of certain partition functions in probability and applications to integrable systems, of the topological recursion…
We embed the somewhat unusual multiplicative function, which was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics, into two families of multiplicative functions that we…
A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex…
This paper reconsiders the problem of calculating the expected set of probabilities <p_i>, given the observed set of items {m_i}, that are distributed among n bins with an (unknown) set of probabilities {p_i} for being placed in the ith…
In earlier work the authors obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process. Here formulas are obtained for joint probabilities for several particles. In the case of a single…
In the first part, by the first author's work of 1972, an integral representation for an ultraspherical polynomial of higher index in terms of one of lower index and an infinite series was obtained. While this representation works well from…