Related papers: On the shape of a tridiagonal pair

We investigate the lattice L(V) of subspaces of an m-dimensional vector space V over a finite field GF(q) with q being the n-th power of a prime p. It is well-known that this lattice is modular and that orthogonality is an antitone…

Let $w=(w_1,\dots,w_d)$ be a $d$-tuple of positive real numbers such that $\sum_{i}w_i =1$ and $w_1\geq \cdots \geq w_d$. A $d$-dimensional vector $x=(x_1,\dots,x_d)\in\mathbb{R}^d$ is said to be $w$-singular if for every $\epsilon>0$ there…

Let $\mathbb{F}$ be a field of characteristic different from $2$ and $3$, and let $V$ be a vector space of dimension $2$ over $\mathbb{F}$. The generic classification of homogeneous quadratic maps $f\colon V\to V$ under the action of the…

We show first that a generic hypersurface $V$ of degree $d\geq 3$ in the complex projective space $ \mathbb{P}^n$ of dimension $n \geq 3$ has at least one hyperplane section $V \cap H$ containing exactly $n$ ordinary double points, alias…

We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are $F$-pure when the size of matrices is equal to 3.…

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…

This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in the three-dimensional space, it must be a parallelohedron. In other words, it must be a parallelotope, a hexagonal prism, a…

A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power…

In this paper, as part of a project initiated by A. Mallios consisting of exploring new horizons for \textit{Abstract Differential Geometry} ($\grave{a}$ la Mallios), \cite{mallios1997, mallios, malliosvolume2, modern}, such as those…

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…

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…

We consider versal deformations of 0|3-dimensional L-infinity algebras, which correspond precisely to ordinary (non-graded) three dimensional Lie algebras. The classification of such algebras over C is well known, although we shall give a…

Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or…

For an arbitrary field $\mathbb{K}$ and a family of inner products in a $\mathbb{K}$-vector space $V$ of arbitrary dimension, we study necessary and sufficient conditions in order to have an orthogonal basis relative to all the inner…

The maximal dimension of commutative subspaces of $M_n(\mathbb{C})$ is known. So is the structure of such a subspace when the maximal dimension is achieved. We consider extensions of these results and ask the following natural questions: If…

Let V be an n-dimensional vector space and let On be the orthogonal group. Motivated by a question of B. Szegedy (B. Szegedy, Edge coloring models and reflection positivity, Journal of the American Mathematical Society Volume 20, Number 4,…

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…

A Formal Orthogonal Pair is a pair $(A,B)$ of symbolic rectangular matrices such that $AB^T=0$. It can be applied for the construction of Hadamard and Weighing matrices. In this paper we introduce a systematic way for constructing such…

We present a new look at description of real finite-dimensional Lie algebras. The basic element turns out to be a pair $(F,v)$ consisting of a linear mapping $F\in End(V)$ and its eigenvector $v$. This pair allows to build a Lie bracket on…

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…