Related papers: Totally positive elements with $m$ partitions exis…

Let $n$ be a positive integer, and let $k$ be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented $k$-algebra $R$ has infinitely many…

Recently, the authors showed that for every irrational number $\alpha$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||\alpha n||<n^{-(1/2-\varepsilon)}$…

We prove that (under the assumption of the generalized Riemann hypothesis) a totally real multiquadratic number field $K$ has a positive density of primes $p \in \mathbb{Z}$ for which the image of the unit group $(\mathcal{O}_K)^{\times})$…

We prove that there are at most 13 real quadratic fields that admit a ternary universal quadratic lattice, thus establishing a strong version of Kitaoka's Conjecture for quadratic fields. More generally, we obtain explicit upper bounds on…

Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ represents a finite extension of degree $n$ of the finite field ${\mathbb{F}_{q}}$. In this article, we investigate the existence of $m$ elements in arithmetic…

Semifields are semirings in which every nonzero element has a multiplicative inverse. A rough classification uses the characteristic of the semifield, that is the isomorphism type of the semifield generated by the two neutral elements. For…

We derive a general framework that connects every positive map with a corresponding witness for partial separability in multipartite quantum systems. We show that many previous approaches were intimately connected to the witnesses derived…

The aim of this work is to determine the quasi-filiform Lie algebras that are completable. We further prove that for any positive integer $m$ there exists a complete Lie algebra, the second cohomology group of which has dimension greater or…

We prove that, for any positive integer $m$, a segment may be partitioned into $m$ possibly degenerate or empty segments with equal values of a continuous function $f$ of a segment, assuming that $f$ may take positive and negative values,…

We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean…

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal…

Let $r > 0$ be an integer, let $\mathbb{F}_q$ be a finite field of $q$ elements, and let $\mathcal{A}$ be a nonempty proper subset of $\mathbb{F}_q$. Moreover, let $\mathbf{M}$ be a random $m \times n$ rank-$r$ matrix over $\mathbb{F}_q$…

We study totally positive, additively indecomposable integers in a real quadratic field $\mathbb Q(\sqrt D)$. We estimate the size of the norm of an indecomposable integer by expressing it as a power series in $u_i^{-1}$, where $\sqrt D$…

In this paper, we show that for any $m$-gonal form $F_m(\mathbf x)$ with $m \ge 12$ which represents every positive integer up to $m-4$, by putting together only unary $m$-gonal form, we may complete an universal form.

Fix a positive real number $\theta$. The natural numbers $m$ with largest square-free divisor not exceeding $m^\theta$ form a set $\mathscr{A}$, say. It is shown that whenever $\theta>1/2$ then all large natural numbers $n$ are the sum of…

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem $-\Delta…

The concept of the {\em half density matrix} is proposed. It unifies the quantum states which are described by density matrices and physical processes which are described by completely positive maps. With the help of the half-density-matrix…

In analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset $C$ of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of $C$, then it…

We show that for every integer $m > 0$, there is an ordinary abelian variety over ${\mathbb F}_2$ that has exactly $m$ rational points.

We show that every positive integer different from $3$ and $5$ can be realized as the $m$-invariant of a field.