相关论文: The Varchenko Determinant of an Oriented Matroid

The determinant is a main organizing tool in commutative linear algebra. In this review we present a theory of the quasideterminants defined for matrices over a division algebra. We believe that the notion of quasideterminants should be one…

It is shown how to define difference operators and equations on particular lattices $\{x_n\}$, $2n\in\mathbb{Z}$, such that the divided difference operator $(\mathcal{D}f)(x_{n+1/2})= (f(x_{n+1})-f(x_n))/(x_{n+1}-x_n)$ has the property that…

The Tutte equations are ported (or set-pointed) when the equations F(N) = g_e F(N/e) + r_e F(N\e) are omitted for elements e in a distinguished set called ports. Solutions F can distinguish different orientations of the same matroid. A…

We study orthogonal polynomial systems arising from general pre-Hilbert inner products on polynomial spaces, beyond the classical framework of measures. To each such inner product we associate a canonical Laplacian defined from an abstract…

We consider the five-vertex model on a finite square lattice with fixed boundary conditions such that the configurations of the model are in a one-to-one correspondence with the boxed plane partitions (3D Young diagrams which fit into a box…

For a given Markov chain Monte Carlo (MCMC) algorithm, we define the distance between configurations that quantifies the difficulty of transitions. This distance enables us to investigate MCMC algorithms in a geometrical way, and we…

We develop a unified construction of matrix-valued orthogonal polynomials associated with discrete weights, yielding bispectral sequences as eigenfunctions of second-order difference operators. This general framework extends the discrete…

Seventy years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For some time this paper attracted little attention,…

In this paper we study higher-order difference equations which can be written as follows: $$ \mathbf{J} (y_0,y_1,...)^T = \lambda^N (y_0,y_1,...)^T, $$ where $\mathbf{J}$ is a $(2N+1)$-diagonal bounded banded matrix…

A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic…

The higher order supersymmetric partners of the Schroedinger's Hamiltonians can be explicitly constructed by iterating a simple finite difference equation corresponding to the Baecklund transformation. The method can completely replace the…

We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We…

Let $\mathscr{M}$ be a conditional oriented matroid. We define a graded algebra $\widehat{\mathscr{VG}}_\mathscr{M}$ with vector space dimension given by the number of covectors in $\mathscr{M}$ which admits a distinguished filtration…

The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…

The discrete power function on the hexagonal lattice proposed by Bobenko et al is considered, whose defining equations consist of three cross-ratio equations and a similarity constraint. We show that the defining equations are derived from…

Given vectors $v_1,\dots,v_n\in\mathbb{R}^d$ and a matroid $M=([n],I)$, we study the problem of finding a basis $S$ of $M$ such that $\det(\sum_{i \in S}v_i v_i^\top)$ is maximized. This problem appears in a diverse set of areas such as…

We show that the number of unit distances determined by n points in R^3 is O(n^{3/2}), slightly improving the bound of Clarkson et al. established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth…

In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of…

We study the semi-discrete directed random polymer model introduced by O'Connell and Yor. We obtain a representation for the moment generating function of the polymer partition function in terms of a determinantal measure. This measure is…

Milgrom (2017) has proposed a heuristic for determining a maximum weight basis of an independence system ${\mathcal I}$ given that we want an approximation guarantee only for sets in a prescribed ${\mathcal O}\subseteq {\mathcal I}$. This…