Related papers: Ranks Based on Strong Amalgamation Fraisse Classes

A matrix is said to have factor width at most $k$ if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single $k \times k$ principal submatrix. We explore the ``factor-width-$k$ rank'' of a matrix,…

In this paper, we give an overview of some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics. The motivations for this work come from Computer…

Recent discoveries make it possible to compute the K-theory of certain rings from their cyclic homology and certain versions of their cdh-cohomology. We extend the work of G. Corti\~nas et al. who calculated the K-theory of, in addition to…

We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures…

Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with…

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism…

For some variants of regression models, including partial, measurement error or error-in-variables, latent effects, semi-parametric and otherwise corrupted linear models, the classical parametric tests generally do not perform well. Various…

Given a variety of universal algebras. A method is suggested for describing automorphisms of a category of free algebras of this variety. Applying this general method all automorphisms of such categories are found in two cases: 1) for the…

Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. Each class consists of the…

We study automorphisms of the free associative algebra K<x,y,z> over a field K which fix the variable z. We describe the structure of the group of z-tame automorphisms and derive algorithms which recognize z-tame automorphisms and z-tame…

In a seminal 2007 paper, Andrews introduced a class of combinatorial objects that generalize partitions called $k$-marked Durfee symbols. Multivariate rank generating functions for these objects have been shown by many to have interesting…

This manuscript describes a technique for computing partial rank-revealing factorizations, such as, e.g, a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance $\varepsilon$ and an…

We present a generalization, which we call (k,m)-rank, of Dyson's notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries…

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollob\'as-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted…

Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected (but different from) group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to…

We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…

In this article we describe the $G\times G$-equivariant $K$-ring of $X$, where $X$ is a regular compactification of a connected complex reductive algebraic group $G$. Furthermore, in the case when $G$ is a semisimple group of adjoint type,…

We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy…

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class $1$ functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory…

This article is concerned with classes of relational structures that are closed under taking substructures and isomorphism, that have the joint embedding property, and that furthermore have the Ramsey property, a strong combinatorial…