Related papers: Simplex and Polygon Equations
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in block diagonal matrix form (resulting in the Wedderburn decomposition), a general form of polyadic…
The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair…
Computational methods are an important tool for solving the Yang-Baxter equations(in small dimensions), for classifying (unifying) structures, and for solving related problems. This paper is an account of some of the latest developments on…
We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P. We devise a combinatorial method…
We present a systematic procedure to obtain singular solutions of the constant quantum Yang-Baxter equation in arbitrary dimension. This approach, inspired in the Lie (super)algebra structure, is explicitly applied to the particular case of…
In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries (J. Phys. A: Math. Theor. Vol. 42 (2009) 395202 (20pp)), mixed…
Some idea, which leads to a non-trivial solution of the quantum four-simplex equation, is exposed in this paper. We call this idea "pentagonal algebra". Few examples of the realisation of this idea are given here, and thus few examples of…
Lehmer's code defines a bijection between the symmetric group and the set of staircase compositions. In this paper, we characterize a poset structure on these compositions that is equivalent to the strong Bruhat order on the symmetric…
In this paper, we first introduce associative-Yamaguti algebras as the associative analogue of Lie-Yamaguti algebras. Associative algebras, reductive associative algebras and associative triple systems of the first kind form subclasses of…
We define a natural notion of higher order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic varieties, up to linear error. This generalizes…
We propose a specific class of matrices which participate in factorization problems that turn to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang-Baxter maps, expressed in non-commutative variables.…
We create a framework for studying symmetric chain decompositions of families of finite posets based on the geometry of polytopes. Our framework unifies almost all known results regarding symmetric chain decompositions of the Young posets…
We investigate the class of regular-ordered word equations. In such equations, each variable occurs at most once in each side and the order of the variables occurring in both sides is the preserved (the variables can be, however, separated…
Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, double cosets of the group of label preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal…
Given a small polygon S, a big simple polygon B and a positive integer k, it is shown to be NP-hard to determine whether k copies of the small polygon (allowing translation and rotation) can be placed in the big polygon without overlap.…
Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…
We introduce a new variety of set-theoretic non-associative algebras, P{\l}onka bi-magmas, to describe and classify all solutions of the set-theoretic Yang-Baxter (YB) equation of Baaj-Long-Skandalis (BLS) type. We also study new classes of…
We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…
In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general…
We present degrees of freedom to accompany the approximation spaces already presented in a companion paper and thus complete the definition of families of high-order conforming finite elements on pyramids for the spaces of the de Rham…