Related papers: The Varchenko Determinant of an Oriented Matroid

We discuss the problem of optimizing the distance function from a given point, subject to polynomial constraints. A key algebraic invariant that governs its complexity is the Euclidean distance degree, which pertains to first-order…

Let $A \in \mathbb{R}^{n \times (n - d)}$ be a random matrix with independent uniformly anti-concentrated entries satisfying $\mathbb{E}\lvert A\rvert_{HS}^2 \leq Kn(n-d)$ and let $H$ be the subspace spanned by the columns of $A$. Let $X…

In 1931 Elie Cartan constructed a geometry which was rarely considered. Cartan proposed a way to define an infinitesimal metric $ds$ starting from a variational problem on hypersurfaces in an $n$-dimensional manifold $\mathcal{M}$. This…

If V(R) is the vertex sequence of a symmetric cycle R in the tope graph of a simple acyclic oriented matroid M on a t-element ground set, then the set min V(R) of minimal elements in the subposet V(R) of the tope poset of M, based at the…

Determining the number of pieces after cutting a cake is a classical problem. Roberts (1887) provided an exact solution by computing the number of chambers contained in a plane cut by lines. About 88 years later, Zaslavsky (1975) even…

In this paper, we define a matrix which we call Vieta matrix and calculate its determinant: $$ \left( \begin{array}{cccc} 1&1&\cdots&1\\ a_{2}+a_{3}+\cdots+a_{n}&a_{1}+a_{3}+\cdots+a_{n}&\cdots&a_{1}+a_{2}+\cdots+a_{n-1}\\…

We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite-Pad\'{e} approximation and interpolation problems. We study also…

The generalised Wronskian of differential order $k\geqslant 1$ for $N$ functions $f_1$, $\ldots$, $f_N$ in $d\geqslant 1$ independent variables $x^1$, $\ldots$, $x^d$ is the determinant of the matrix with these functions' derivatives…

The determinant can be computed by classical circuits of depth $O(\log^2 n)$, and therefore it can also be computed in classical space $O(\log^2 n)$. Recent progress by Ta-Shma [Ta13] implies a method to approximate the determinant of…

Let $p_n(x)$, $n=0,1,\dots$, be the orthogonal polynomials with respect to a given density $d\mu(x)$. Furthermore, let $d\nu(x)$ be a density which arises from $d\mu(x)$ by multiplication by a rational function in $x$. We prove a formula…

In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the…

In the matroid intersection problem, we are given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ defined on the same ground set $V$ of $n$ elements, and the objective is to find a common…

We consider a function $U=e^{-f_0}\prod_j^N f_j^{\alpha_j}$ on a real affine space, here $f_0,..,f_N$ are linear functions, $\alpha_1, ...,\alpha_N$ complex numbers. The zeros of the functions $f_1, ..., f_N$ form an arrangement of…

We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well--known difference operators $\mathscr{D}_{q}$ and $\Delta$ and, as a limit case, the derivative operator. The objective…

This paper presents the generalization of weighted distances to modules and their computation through the chamfer algorithm on general point lattices. The first part is dedicated to formalization of definitions and properties (distance,…

Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and…

We develop a formula for the diagonal values of the Hadamard coefficients associated to a normally hyperbolic operator on a globally hyperbolic spacetime in terms of the advanced and retarded Green's operators. We develop a local formula as…

Let $\D$ be a set of $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the set of their center points. A hyperplane $\Hy$ is an \emph{$m$-separator} for $\D$ if each closed halfspace bounded by $\Hy$ contains at least $m$ points from $P$.…

We present two characterizations of regular matroids among orientable matroids and use them to give a measure of "how far" an orientable matroid is from being regular.

Alternative novel measures of the distance between any two partitions of a n-set are proposed and compared, together with a main existing one, namely 'partition-distance' D(.,.). The comparison achieves by checking their restriction to…