Related papers: Sign variation, the Grassmannian, and total positi…

In this article, we show a new general linear independence criterion related to values of $G$-functions, including the linear independence of values at algebraic points of contiguous hypergeometric functions, which is not known before. Let…

We study the supersymmetry breaking patterns in four-dimensional $\mathcal{N}=2$ gauged supergravity. The model contains multiple (Abelian) vector multiplets and a single hypermultiplet which parametrizes SO$(4,1)/{\rm{SO}}(4)$ coset. We…

There are numerous combinatorial objects associated to a Grassmannian permutation $w_\lambda$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations…

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of their polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost…

A linear dynamical system is called positive if its flow maps the non-negative orthant to itself. More precisely, it maps the set of vectors with zero sign variations to itself. A linear dynamical system is called $k$-positive if its flow…

Let K be a skew field, and K_0 be a subfield of the central subfield of K such that K has finite dimension q over K_0. Let V be a K_0-linear subspace of n by n nilpotent matrices with entries in K. We show that the dimension of V is bounded…

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with…

In this paper we consider the mixed tensor space of a $\mathbb Z_2$-graded vector space. We obtain a spanning set of invariants of the associated symmetric algebra under the action of the general linear supergroup as well as the queer…

The present paper contains two interrelated developments. First, are proposed new generalized Verma modules. They are called k-Verma modules, k\in N, and coincide with the usual Verma modules for k=1. As a vector space a k-Verma module is…

Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $\nu$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of…

Given two graphs $G_1$ and $G_2$ on $n$ vertices each, we define a graph $G$ on vertex set $V_1\times V_2$ and the edge set as the union of edges of $G_1\times \bar{G_2}$, $\bar{G_1}\times G_2$, $\{(v,u'),(v,u"))(|u',u"\in V_2\}$ for each…

Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a given polynomial are parametrized by the convex body of…

Let $\Gamma_k(V)$ be the Grassmann graph whose vertex set ${\mathcal G}_{k}(V)$ is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its…

Suppose that R is an ordered ring, G_n(R) is a subsemigroup of $GL_n(R)$, consisting of all matrices with nonnegative elements. A.V. Mikhalev and M.A. Shatalova described all automorphisms of G_n(R), where R is a linearly ordered skewfield…

Denote by $\mathbb G(k,n)$ the Grassmannian of linear subspaces of dimension $k$ in $\mathbb P^n$. We show that, if $\varphi:\mathbb G(l,n) \to \mathbb G(k,n)$ is a non constant morphism and $l \not=0,n-1$ then $l=k$ or $l=n-k-1$ and…

The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $[1!4! >...…

Let V be a finite-dimensional vector space over a field k and let W be a 1-dimensional k-vector space. Let < , >: V x V \to W be a symmetric bilinear form. Then < , > is called anisotropic if for all nonzero v \in V we have <v,v> \neq 0.…

Zeckendorf's Theorem says that for all $k \geq 3$, every nonnegative integer has a unique $k$-Zeckendorf representation as a sum of distinct $k$-bonacci numbers, where no $k$ consecutive $k$-bonacci numbers are present in the…

Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and more generally as witnesses for states having Schmidt number not…

While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it…