Related papers: Tridiagonal pairs of $q$-Racah type and the $\mu$-…

Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$, equipped with a symmetric or alternating non-degenerate bilinear form $b$. When the characteristic of $\mathbb{F}$ is not $2$, we characterize the endomorphisms $u$ of…

We investigate point arrangements $v_i\in\mathbb R^d,i\in \{1,...,n \}$ with certain prescribed symmetries. The arrangement space of $v$ is the column span of the matrix in which the $v_i$ are the rows. We characterize properties of $v$ in…

In this paper, we seek analytically checkable necessary and sufficient condition for copositivity of a three-dimensional symmetric tensor. We first show that for a general third order three-dimensional symmetric tensor, this means to solve…

Let $V$ be a braided vector space of diagonal type. Let $\mathfrak B(V)$, $\mathfrak L^-(V)$ and $\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial…

We consider quadrangles of perimeter $2$ in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^\bot$ defines a plane…

A detailed version of preprint "Self-linking number of a real algebraic link" by the same author, alg-geom/9410030. For a nonsingular real algebraic curve in 3-dimensional projective space or 3-sphere, a new integer-valued characteristic is…

Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu, we say that a family S of pairwise orthogonal objects in T with trivial endomorphism rings is a simple-minded system if…

Let $V$ be a finite dimensional vector space. Given a decomposition $V\otimes V=\oplus_i^n I_i$, define $n$ quadratic algebras $(V, J_m)$ where $J_m=\oplus_{i\neq m} I_i$. This decomposition defines also the quantum semigroup…

Let $V$ be a vector space with countable dimension over a field, and let $u$ be an endomorphism of it which is locally finite, i.e. $(u^k(x))_{k \geq 0}$ is linearly dependent for all $x$ in $V$. We give several necessary and sufficient…

This paper is devoted to studying $R$-diagonal and $\eta$-diagonal pairs of random variables. We generalize circular elements to the bi-free setting, defining bi-circular element pairs of random variables, which provide examples of…

Given $n$ pairwise openly disjoint triangles in 3-space, their vertical depth relation may contain cycles. We show that, for any $\varepsilon>0$, the triangles can be cut into $O(n^{3/2+\varepsilon})$ connected semi-algebraic pieces, whose…

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are…

For a hypersurface V of a conformal space, we introduce a conformal differential invariant I = h^2/g, where g and h are the first and the second fundamental forms of V connected by the apolarity condition. This invariant is called the…

Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let $K$ be an algebraically closed field of characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$ endowed with…

Let $(A,\Delta)$ be a finite-dimensional Hopf algebra. The linear dual $B$ of $A$ is again a finite-dimensional Hopf algebra. The duality is given by an element $V\in B\otimes A$, defined by $\langle V,a\otimes b\rangle=\langle a,b\rangle$…

A strong arc decomposition of a digraph $D=(V,A)$ is a decomposition of its arc set $A$ into two disjoint subsets $A_1$ and $A_2$ such that both of the spanning subdigraphs $D_1=(V,A_1)$ and $D_2=(V,A_2)$ are strong. Let $T$ be a digraph…

We consider spectra of $n$-by-$n$ irreducible tridiagonal matrices over a field and of their $n-1$-by-$n-1$ trailing principal submatrices. The real symmetric and complex Hermitian cases have been fully understood: it is necessary and…

Let (A,B) and (C,D) denote Leonard pairs on V. We say these pairs are adjacent whenever each basis for V which is standard for (A,B) (resp. (C,D)) is split for (C,D) (resp. (A,B)). Our main results are as follows: Theorem 1. There exists at…

Let $V$ be a finite-dimensional vector space over the field with $p$ elements, where $p$ is a prime number. Given arbitrary $\alpha,\beta\in \mathrm{GL}(V)$, we consider the semidirect products $V\rtimes\langle \alpha\rangle$ and…

Let $V$ be a vector space of rectangular $n\times k$ matrices annihilating the Cullis' determinant. We show that $\dim(V) \le (n-1)k$, extending Dieudonn{\'{e}}'s result on the dimension of vector spaces of square matrices annihilating the…