Related papers: Heaps and Two Exponential Structures

This paper presents results on structures in P based on tools developed from subjects of elementary number theory. Key findings are: The arithmetical sequence H = (+-3*2; 1) is in Z the smallest superset of P \ {3, 2}. H is a semigroup. A…

We introduce a homological approach to exhibiting instances of Stembridge's q=-1 phenomenon. This approach is shown to explain two important instances of the phenomenon, namely that of partitions whose Ferrers diagrams fit in a rectangle of…

Let H[N] denote the tensor product of n finite dimensional Hilbert spaces H(r). A state |phi> of H[N] is separable if |phi> is the tensor product of states in the respective product spaces. An orthogonal unextendible product basis is a…

We define an algebra $\mathcal{U}_0$ using a simplified set of generators for the quantum toroidal algebra $U_q(sl_{n+1}, tor)$ and show that there exists an epimorphism from $\mathcal{U}_0$ to $U_q(sl_{n+1}, tor)$. We derive a closed…

Let $\varepsilon>0$ be a fixed small constant, ${\mathbb F}_p$ be the finite field of $p$ elements for prime $p$. We consider additive and multiplicative problems in ${\mathbb F}_p$ that involve intervals and arbitrary sets. Representative…

The incidence algebra of a partially ordered set (poset) supports in a natural way also a coalgebra structure, so that it becomes a m-weak bialgebra even a m-weak Hopf algebra with M\"obius function as antipode. Here m-weak means that…

Let $\mathcal{M}$ be a set with $M$ elements, let $\psi :\mathcal{M}\to\mathcal{M}$ be a bijective involution, and let~$\boldsymbol{\mathcal{X}}_{\psi}$ be the set of sequences $(x_1,\dots,x_M)\in\mathcal{M}^M$ with the property that…

A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers, when represented using continued fractions, exhibit partial quotients that grow at specific rates. For any positive function…

We propose a new class of algebraic structure named as \emph{$(m,n)$-semihyperring} which is a generalization of usual \emph{semihyperring}. We define the basic properties of $(m,n)$-semihyperring like identity elements, weak distributive…

We say that a 0-1 matrix $N$ of size $a\times b$ can be found in a collection of sets $\mathcal{H}$ if we can find sets $H_{1}, H_{2}, \dots, H_{a}$ in $\mathcal{H}$ and elements $e_1, e_2, \dots, e_b$ in $\cup_{H \in \mathcal{H}} H$ such…

We consider the poset of vector partitions of $[n]$ into $s$ components, denoted $\Pi_{n,s}$, which was first defined by Stanley in 1978. In 1986, Sagan showed that this poset is CL-shellable, and hence has the homotopy type of a wedge of…

Let $j(z)$ be the modular $j$-invariant function. Let $\tau$ be an algebraic number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider and Siegel that if $\tau$ is not a CM point, i.e.,…

One of the classical notions of group theory is the notion of the exponent of a group. The exponent of a group is the least common multiple of orders of its elements. In this paper we generalize the notion of exponent to Hopf algebras. We…

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define…

The main aim of this article is to study the quantitative structure of projective symplectic groups $PSp_{4}(q)$ with $q>2$ even. Indeed, we prove that the groups $PSp_{4}(q)$ with $q>2$ even are uniquely determined by their orders and the…

A set of integers is called sum-free if it contains no triple $(x,y,z)$ of not necessarily distinct elements with $x+y=z$. In this paper, we provide a structural characterisation of sum-free subsets of $\{1,2,\ldots,n\}$ of density at least…

For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A}\setminus \{0,N\}$ verifying $\alpha_{2}r-\alpha_{1}$ divides…

For a set of nonnegative integers $A$, denote by $R_{A}(n)$ the number of unordered representations of the integer $n$ as the sum of two different terms from $A$. In this paper we partially describe the structure of the sets, which have…

We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra U_{q,p}(\hat{sl}_2). The algebra U_{q,p}(\hat{sl}_2) has a constructive definition in terms of the…

For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^{\alpha})=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{\alpha-1}(p-r), & \mbox{if } p>r…