Related papers: The $k$-resultant modulus set problem on algebraic…

For a field $K$, and a root $\alpha$ of an irreducible polynomial over $K$ (in some algebraic closure) the number of roots of $f(x)$ lying in $K(\alpha)$ is studied here. Given such an $f(x)$ of degree $n$ for which $r$ of the roots are i n…

Let $V,W$ be representations of a cyclic group $G$ of prime order $p$ over a field $k$ of characteristic $p$. The module of covariants $k[V,W]^G$ is the set of $G$-equivariant polynomial maps $V \rightarrow W$, and is a module over…

We prove the following two results. \begin{enumerate} \item Let $\mathcal{A}$ be a unital commutative C*-algebra and $\mathcal{A}^d$ be the standard Hilbert C*-module over $\mathcal{A}$. Let $n\geq d$. If $\{\tau_j\}_{j=1}^n$ is any…

We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime.…

Choose $q\in {\mathbb C}$ with 0<|q|<1. The main theme of this paper is the study of linear q-difference equations over the field K of germs of meromorphic functions at 0. It turns out that a difference module M over K induces in a…

The $k$-subset sum problem over finite fields is a classical NP-complete problem.Motivated by coding theory applications, a more complex problem is the higher $m$-th moment $k$-subset sum problem over finite fields. We show that there is a…

When is a $q$-series modular? This is an interesting open question in mathematics that has deep connections to conformal field theory. In this paper we define a particular $r$-fold $q$-hypergeometric series $f_{A,B,C}$, with data given by a…

Let $\mathbb{k}$ be a field, and let $\Lambda$ be a (not necessarily finite dimensional) $\mathbb{k}$-algebra. Let $V$ be a left $\Lambda$-module such that is finite dimensional over $\mathbb{k}$. Assume further that $V$ has a weak…

Let F_q be a finite field with odd q elements. In this article, we prove that if E \subseteq \mathbb F_q^d, d\ge 2, and |E|\ge q, then there exists a set Y \subseteq \mathbb F_q^d with |Y|\sim q^d$ such that for all y\in Y, the number of…

The $k$-center problem is to choose a subset of size $k$ from a set of $n$ points such that the maximum distance from each point to its nearest center is minimized. Let $Q=\{Q_1,\ldots,Q_n\}$ be a set of polygons or segments in the…

This paper discusses various aspects of the Hecke algebra combinatorics that are related to conditions appearing in K{\aa}hrstr{\"o}m's conjecture that addresses Kostant's problem for simple highest weight modules in the…

The input to the \emph{sets-$k$-means} problem is an integer $k\geq 1$ and a set $\mathcal{P}=\{P_1,\cdots,P_n\}$ of sets in $\mathbb{R}^d$. The goal is to compute a set $C$ of $k$ centers (points) in $\mathbb{R}^d$ that minimizes the sum…

Let $\mathbb{F}_q$ be the finite field of $q$ elements and $a_1,a_2, \ldots, a_k, b\in \mathbb{F}_q$. We investigate $N_{\mathbb{F}_q}(a_1, a_2, \ldots,a_k;b)$, the number of ordered solutions $(x_1, x_2, \ldots,x_k)\in\mathbb{F}_q^k$ of…

Specializing properly the parameters contained in the maximal cyclic representation of the non-restricted A-type quantum algebra at roots of unity, we find the unique primitive vector in it. We show that the submodule generated by the…

Let $\Lambda$ be a commutative local uniserial ring of length at least seven with radical factor ring $k$. We consider the category $S(\Lambda)$ of all possible embeddings of submodules of finitely generated $\Lambda$-modules and show that…

Let $(Q,\mathfrak{n})$ be a regular local ring of dimension $c \geq 2$ with algebraically closed residue field $k = Q/\mathfrak{n}$. Let $f_1, f_2, \ldots f_{c-1}, g$ be a regular sequence in $Q$ such that $ f_i \in \mathfrak{n}^2$ for all…

The delta set of a numerical semigroup $S$, denoted $\Delta(S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set…

We study the number of the vectors determined by two sets in d-dimensional vector spaces over finite fields. We observe that the lower bound of cardinality for the set of vectors can be given in view of an additive energy or the decay of…

We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure…

In a graph $G = (V,E)$, a k-ruling set $S$ is one in which all vertices $V$ \ $S$ are at most $k$ distance from $S$. Finding a minimum k-ruling set is intrinsically linked to the minimum dominating set problem and maximal independent set…