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We refine Schmidt's problem and a partition identity related to 2-color partitions which we will refer to as Uncu-Andrews-Paule theorem. We will approach the problem using Boulet-Stanley weights and a formula on Rogers-Szeg\H{o} polynomials…

We derive exact series solutions for the Wheeler-DeWitt equation corresponding to a spatially closed Friedmann-Robertson-Walker universe with cosmological constant for arbitrary operator ordering of the scale factor of the universe. The…

We generalize Sylvester single sums to multisets (sets with repeated elements), and show that these sums compute subresultants of two univariate polyomials as a function of their roots independently of their multiplicity structure. This is…

We show that the number of reduced Young walls of type $D_{n+1}^{(2)}$ with $m$ blocks is independent of $n$ and the same as the number of strict partitions of $m$. Thus the principally specialized character $\chi_n^{\Lambda_0}(t)$ of…

The regularity lemma of Szemeredi asserts that one can partition every graph into a bounded number of quasi-random bipartite graphs. In some applications however, one would like to have a strong control on how quasi-random these bipartite…

For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form…

We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…

We give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and non-periodic…

We characterize the restrictions of B\'ekoll\'e--Bonami weights of bounded hyperbolic oscillation, to subsets of the unit disc, thus proving an analogue of Wolff's restriction theorem for Muckenhoupt weights. Sundberg proved a discrete…

In [Phys. Rev. A 70, 062101 (2004)] Gibbons et al. defined a class of discrete Wigner functions W to represent quantum states in a finite Hilbert space dimension d. I characterize a set C_d of states having non-negative W simultaneously in…

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…

We apply some recent developments of Baldoni-Beck-Cochet-Vergne on vector partition function, to Kostant's and Steinberg's formulae, for classical Lie algebras $A\_r$, $B\_r$, $C\_r$, $D\_r$. We therefore get efficient {\tt Maple} programs…

A systematic search for superintegrable quantum Hamiltonians describing the interaction between two particles with spin 0 and 1/2, is performed. We restrict to integrals of motion that are first-order (matrix) polynomials in the components…

To resolve the non-convex optimization problem in partial wave analysis, this paper introduces a novel approach that incorporates fraction constraints into the likelihood function. This method offers significant improvements in both the…

We consider a Schroedinger operator on the axis with a bipartite potential consisting of two compactly supported complex-valued functions, whose supports are separated by a large distance. We show that this operator possesses a sequence of…

Rational wave numbers are periodic sequences ${\mathbf \omega}={\bf A}{\bf w}(f,g)$ in which amplitude ${\bf A}$ a product of powers of trigonometric sequences and ${\bf w}(f,g)=\exp({\bf {i2}\pi ( f {\mathbf \xi} \oplus g{\bf 1})})$ is a…

Given a partition $\lambda$, we write $e_j(\lambda)$ for the $j^{\textrm{th}}$ elementary symmetric polynomial $e_j$ evaluated at the parts of $\lambda$ and $e_jp_A(n)$ for the sum of $e_j(\lambda)$ as $\lambda$ ranges over the set of…

We find general solutions to the generating-function equation sum c_q^{(X)} z^q = F(z)^X, where X is a complex number and F(z) is a convergent power series with |F(0)| >0. We then use these results to derive finite expressions containing…

Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1,…

We consider partitions $p_{w}(n)$ of a positive integer $n$ arising from the generating functions \[ \sum_{n=1}^\infty p_{w}(n) z^n = \prod_{m \in \mathbb{N}} (1-z^m)^{-w(m)}, \] where the weights $w(m)$ are M\"{o}bius convolutions. We…