Related papers: Cesaro's integral formula for the Bell numbers (co…
This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer…
Solid experimental evidence has now been obtained that confirms the violation of Bell's inequality in tests of maximally entangled qubit pairs. This violation is widely interpreted as definitive proof of the impossibility of describing…
The first part of this paper contains an introduction to Bell inequalities and Tsirelson's theorem for the non-specialist. The next part gives an explicit optimum construction for the "hard" part of Tsirelson's theorem. In the final part we…
Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for…
Let $m\ge 2$ be a fixed positive integer. Suppose that $m^j \leq n< m^{j+1}$ is a positive integer for some $j\ge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this…
A positive integer $n$ is said to be a Zumkeller number or an integer-perfect number if the set of its positive divisors can be partitioned into two subsets of equal sums. In this paper, we prove several results regarding Zumkeller numbers.…
Translated from the Latin original, "Theorema arithmeticum eiusque demonstratio", Commentationes arithmeticae collectae 2 (1849), 588-592. E794 in the Enestroem index. For m distinct numbers a,b,c,d,...,\upsilon,x this paper evaluates \[…
We derive tight quadratic inequalities for all kinds of hybrid separable-inseparable $n$-particle density operators on an arbitrary dimensional space. This methodology enables us to truly derive a tight quadratic inequality as tests for…
Fourier expansion of the integrand in the path integral formula for the partition function of quantum systems leads to a deterministic expression which, though still quite complex, is easier to process than the original functional integral.…
In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to…
For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2…
We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in…
These notes explore three amazing formulas proved by Abel in his 1826 Paris memoir on what we now call Abelian integrals. We discuss the first two formulas from the point of view of symbolic computation and explain their connection to…
Let ``Faulhaber's formula'' refer to an expression for the sum of powers of integers written with terms in n(n+1)/2. Initially, the author used Faulhaber's formula to explain why odd Bernoulli numbers are equal to zero. Next, Cereceda gave…
Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor…
Because of its relation to the distribution of prime numbers, the Riemann zeta function {\zeta} (s) is one of the most important functions in mathematics. The zeta function is defined by the following formula for any complex number s with…
The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for…
In this paper, our main focus is expressing real numbers on the non-integer bases. We denote those bases as $\beta$'s, which is also a real number and $\beta \in (1,2)$. This project has 3 main parts. The study of expansions of real numbers…
This article is a survey of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned.
A classical theorem of Baranyai states that, given integers $2\leq k < n$ such that $k$ divides $n$, one can find a family of ${n-1\choose k-1}$ partitions of $[n]$ into $k$-element subsets such that every subset appears in exactly one…