Related papers: A local-global principle for linear dependence of …

Given finite sets $X_1,\dotsc,X_m$ in $\mathbb{R}^d$ (with $d$ fixed), we prove that there are respective subsets $Y_1,\dotsc,Y_m$ with $|Y_i|\ge \frac{1}{\operatorname{poly}(m)}|X_i|$ such that, for $y_1\in Y_1,\dotsc,y_m\in Y_m$, the…

We prove a local limit theorem the number of $r$-cliques in $G(n,p)$ for $p\in(0,1)$ and $r\ge 3$ fixed constants. Our bounds hold in both the $\ell^\infty$ and $\ell^1$ metric. The main work of the paper is an estimate for the…

We give an elementary proof of a Caratheodory-type result on the invertibility of a sum of matrices, due first to Facchini and Barioli. The proof yields a polynomial identity, expressing the determinant of a large sum of matrices in terms…

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…

We propose local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if none of its partial derivatives changes in sign on tuples which differ in less than p…

Let $ V_{n} = X_{1,n} + X_{2,n} + \cdots + X_{n,n}$ where $X_{i,n}$ are Bernoulli random variables which take the value $1$ with probability $b(i;n)$. Let $\lambda_{n} = \sum\limits_{i=1}^{n} b(i;n) $, $\lambda = \lim\limits_{n \to \infty}…

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta\colon \mathbb{C}^n\to…

A form in a polynomial ring over a field is said to be homaloidal if its polar map is a Cremona map, i.e., if the rational map defined by the partial derivatives of the form has an inverse rational map. The object of this work is the search…

Given a suitable collection of partitions of sets, there exists a connection to easy quantum groups via intertwiner maps. A sufficient condition for this correspondence to be one-to-one are particular linear independences on the level of…

It is known that a linear system with a system matrix A constitutes a Hamiltonian system with a quadratic Hamiltonian if and only if A is a Hamiltonian matrix. This provides a straightforward method to verify whether a linear system is…

We discuss local-global principles for the existence of Levi factors (i.e., complements to the unipotent radical) for linear algebraic groups over one-variable function fields. We give examples of disconnected groups that fail the…

Let $X$ be a locally compact Abelian group, $Y$ be its character group. Following A. Kagan and G. Sz\'ekely we introduce a notion of $Q$-independence for random variables with values in $X$. We prove group analogues of the Cram\'er,…

In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some multilinear polynomial is a central polynomial, or…

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek-Geiringer-Laman type. Methods from matroid theory have been used to prove other interesting results, again under the…

In this paper we prove Hasse local-global principle for polynomials with coefficients in Mordell-Weil type groups over number fields like S-units, abelian varieties with trivial ring of endomorphisms and odd algebraic K-theory groups.

In this paper, we establish a local limit theorem for linear fields of random variables constructed from independent and identically distributed innovations each with finite second moment. When the coefficients are absolutely summable we do…

Linear Hamiltonian systems with time-dependent coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. It is shown that the solution map of a linear Hamiltonian system with…

This paper concerns the verification of continuous-time polynomial spline trajectories against linear temporal logic specifications (LTL without 'next'). Each atomic proposition is assumed to represent a state space region described by a…

We prove that a linear mapping on the algebra \(\mathfrak{sl}_n\) of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple…

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the…