Related papers: Tridiagonal pairs of shape (1,2,1)

A permanental vector is a generalization of a vector with components that are squares of the components of a Gaussian vector, in the sense that the matrix that appears in the Laplace transform of the vector of Gaussian squares is not…

Let H_1=SL(5), H_2=SL(3), H=H_1 \times H_2. It is known that (G,V) is a prehomogeneous vector space (see [22], [26], [25], for the definition of prehomogeneous vector spaces). A non-constant polynomial \delta(x) on V is called a relative…

Let ${\mathscr A}(D)$ be an algebra of functions continuous in the disk $D=\{z\in{\mathbb C}\,|\,\,\,|z|\leqslant 1\}$ and {\it holomorphic} into $D$. The well-known fact is that the set ${\mathscr M}$ of its characters (homomorphisms…

Let $\mathbb T$ be the differential field of transseries. We establish some basic properties of the dimension of a definable subset of ${\mathbb T}^n$, also in relation to its codimension in the ambient space ${\mathbb T}^n$. The case of…

We prove the Verlinde conjecture in the following general form: Let V be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0, the homogeneous subspace of V of…

The aim of this paper is to prove duality of Triebel-Lizorkin spaces $% F_{1,q\left( \cdot \right) }^{\alpha \left( \cdot \right) }$. First, we prove the duality of associated sequence spaces. Then from the so-called $% \varphi $-transform…

Denote by $\lambda K_v$ the complete graph of order $v$ with multiplicity $\lambda$. Let $\lambda K_v-\lambda K_w-\lambda K_u$ be the graph obtained from $\lambda K_v$ by the removal of the edges of two vertex disjoint complete…

A vector space over a field $\mathbb{F}$ is a set $V$ together with two binary operations, called vector addition and scalar multiplication. It is standard practice to think of a Euclidean space $\mathbb{R}^n$ as an $n$-dimensional real…

A simple ansatz is proposed for two-color R-matrix satisfying the tetrahedron equation. It generalizes, on one hand, a particular case of the eight-vertex model to three dimensions, and on another hand - Hietarinta's permutation-type…

Two vertex-labelled polygons are \emph{compatible} if they have the same clockwise cyclic ordering of vertices. The definition extends to polygonal regions (polygons with holes) and to triangulations---for every face, the clockwise cyclic…

A linear mapping $T$ on a JB$^*$-triple is called triple derivable at orthogonal pairs if for every $a,b,c\in E$ with $a\perp b$ we have $$0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}.$$ We prove that for each bounded linear mapping $T$ on…

Versal deformation of a matrix A is a normal form to which all matrices A + E, close to A, can be reduced by similarity transformation smoothly depending on the entries of A + E. In this paper we discuss versal deformations and their use in…

We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove…

Let $A=(a_1,\ldots,a_n)$ and $B=(b_1,\ldots,b_n)$ be two sequences of nonnegative integers with $a_i \le b_i$ for $1\le i\le n$. The pair $(A;B)$ is said to be realizable by a graph if there exists a simple graph $G$ with vertices…

In this paper, we deal with the question; under what conditions the points $P_i(xi,yi)$ $(i = 1,\cdots, n)$ form a convex polygon provided $x_1 < \cdots < x_n$ holds. One of the main findings of the paper can be stated as follows: "Let…

We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero…

For a pair of positive integers (k,r) with r>1 such that k+1 and r-1 are relatively prime, we describe the space of symmetric polynomials in variables x_1,...,x_n which vanish at all diagonals of codimension k of the form…

We study how the geometry of a projective variety $X$ is reflected in the positivity properties of the diagonal $\Delta_X$ considered as a cycle on $X \times X$. We analyze when the diagonal is big, when it is nef, and when it is rigid. In…

Let $b$ be a symmetric bilinear form on a finite-dimensional vector space over a field with characteristic $2$. Here, we determine the greatest possible dimension of a linear subspace of nilpotent $b$-symmetric or $b$-alternating…

We reproduce the ${\cal N}=4$ supersymmetric mechanics on curved spaces, constructed earlier within the Hamiltonian formalism, using the superfield methods. We show that for any such mechanics, given by the metric and the third order…