Related papers: Combinatorics of factorization systems on lattices
The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we…
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…
Motivated by questions in biological classification, we discuss some elementary combinatorial and computational properties of certain set systems that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample patchworks' and…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
We establish analogues in the context of group actions or group representations of some classical problems and results in additive combinatorics of groups. We also study the notion of left invariant submodular function defined on power sets…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer…
We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put…
We extend a few fundamental aspects of the classical theory of non-unique factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on the subject, to a non-commutative and non-cancellative setting, in the same spirit of…
We prove that the existence of finite combinatorial objects such as affine planes, mutually orthogonal Latin squares, and resolvable balanced incomplete block designs can be reformulated as the existence of certain algorithmic reductions…
In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However,…
Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and…
We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…
Canonical extension of finitary ordered structures such as lattices, posets, proximity lattices, etc., is a certain completion which entirely describes the topological dual of the ordered structure and it does so in a purely algebraic and…
We introduce a general analytic approach to the study of factorization points and factorized ground states in quantum cooperative systems. The method allows to determine rigorously existence, location, and exact form of separable ground…
This paper aims to generalize and unify classical criteria for comparisons of balanced lattice designs, including fractional factorial designs, supersaturated designs and uniform designs. We present a general majorization framework for…
We study the combinatorics of the contributions to the form factor of the group U(N) in the large $N$ limit. This relates to questions about semiclassical contributions to the form factor of quantum systems described by the unitary…
We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the…
The numerical computations of many quantities of theoretical and phenomenological interest are plagued by statistical errors which increase exponentially with the distance of the sources in the relevant correlators. Notable examples are…
This paper is, essentially, a survey related to the problem of understanding the combinatorics of the action of the monoidal category of finite dimensional modules over a simple finite dimensional Lie algebra on various categories of Lie…