Related papers: Criterion for linear independence of functions

Call a noncommutative rational function $r$ regular if it has no singularities, i.e., $r(X)$ is defined for all tuples of self-adjoint matrices $X$. In this article regular noncommutative rational functions $r$ are characterized via the…

In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Pad\'{e} approximation polynomials. Based on the…

Let F = {f_1,...,f_r} be a family of polynomials and let the ticket of F, T(F), denote the set of integers m so that ${f_j^m}$ is linearly dependent. We show that |T(F)| \le (r-1)(r-2)/2 and present many concrete examples, including one…

The paper considers functional linear regression, where scalar responses $Y_1,\ldots,Y_n$ are modeled in dependence of i.i.d. random functions $X_1,\ldots,X_n$. We study a generalization of the classical functional linear regression model.…

In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing…

Let $f:\mathbb{Q}\to \mathbb{Q}$ be a function definable in an o-minimal expansion of $(\mathbb{Q},<,+,0)$. We show that $f$ is eventually linear. In addition, we show that this holds in every elementary equivalent structure.

Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity…

Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a…

Let $\mathbb{K}$ be a finite commutative ring, and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Let $A$ and $B$ be two $n \times n$-matrices over $\mathbb{L}$ that have the same characteristic polynomial. The main result of this…

In this note we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue-Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain…

We study the (in)dependence of additivity and homogeneity conditions in the definition of linear mappings between vector spaces over the same scalar field. Unlike other works on the subject, dealing with particular fields like real or…

Let $(X,d)$ be a metric space and $f_{1,\infty}=\{f_n\}_{i=0}^{\infty}$ be a sequence of continuous maps $f_n:X\rightarrow X$ such that $(f_n)$ converges uniformly to a continuous map $f$. We investigate which conditions ensure that the…

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of…

Let N, N' and N'' be point processes such that N' is obtained from N by homogeneous independent thinning and N''= N- N'. We give a new elementary proof that N' and N'' are independent if and only if N is a Poisson point process. We present…

Let $p$ be an odd prime number. We prove that for $m\equiv1\mod p$, $x^m$ is perfectly nonlinear over $\mathbb{F}_{p^n}$ for infinitely many $n$ if and only if $m$ is of the form $p^l+1$, $l\in\mathbb{N}$. First, we study singularities of…

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…

Very simple sufficient conditions for the irreducibility of $f(X^n)$ over an arbitrary unique factorization domain $Z$ are established via a generalization of a well known theorem of A. Capelli.

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly…

The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $\mathbb R^m$ this notion is near to the separate continuity for which it is required only…

Let $G$ be one of the classical compact, simple, centre-less, connected Lie groups or rank $n$ with a maximal torus $T$, the Lie algebra $\clg$ and let $\{ E_i, F_i, H_i, i=1, \ldots, n \}$ be the standard set of generators corresponding to…