Linear operators on S-graded vector spaces

AbstractThe aim of this paper is to formulate and study natural generalizations of the well-known classical classification problems of linear algebra. We first consider the problem about one linear operator which acts on a finite-dimensional vector space graded by a partially ordered set with involution S=(A,*). For a fixed S and a fixed polynomial f(t), we study the problem of classifying (up to S-similarity, which is defined in a natural way) the operators ϕ satisfying f(ϕ)=0; in particular, a complete description of tame and wild cases is obtained. Furthermore, we prove that there are no new tame cases in the “most” general situation when objects of a Krull–Schmidt subcategory of modk are considered instead of graded spaces. We consider also a “most” general natural extension of the problem on the reduction of the matrix of a linear map by means of elementary row and column transformations. Finally, we introduce the notion of “dispersing representation of a quiver”; in terms of these representations one can formulate many classification problems and, in particular, all the known and new ones encountered in this paper

Face sizes and the connectivity of the dual

For each $c\ge 1$ we prove tight lower bounds on face sizes that must be present to allow $1$- or $2$-cuts in simple duals of $c$-connected maps. Using these bounds, we determine the smallest genus on which a $c$-connected map can have a simple dual with a $2$-cut and give lower and some upper bounds for the smallest genus on which a $c$-connected map can have a simple dual with a $1$-cut.Comment: 19 pages, 11 figure

Testing Convexity and Acyclicity, and New Constructions for Dense Graph Embeddings

Property testing, especially that of geometric and graph properties, is an ongoing area of research. In this thesis, we present a result from each of the two areas. For the problem of convexity testing in high dimensions, we give nearly matching upper and lower bounds for the sample complexity of algorithms have one-sided and two-sided error, where algorithms only have access to labeled samples independently drawn from the standard multivariate Gaussian. In the realm of graph property testing, we give an improved lower bound for testing acyclicity in directed graphs of bounded degree. Central to the area of topological graph theory is the genus parameter, but the complexity of determining the genus of a graph is poorly understood when graphs become nearly complete. We summarize recent progress in understanding the space of minimum genus embeddings of such dense graphs. In particular, we classify all possible face distributions realizable by minimum genus embeddings of complete graphs, present new constructions for genus embeddings of the complete graphs, and find unified constructions for minimum triangulations of surfaces

Obituary - Richard Kenneth Guy, 1916-2020

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Bifurcation of free and forced vibrations for nonlinear wave and Kirchhoff equations via Nash-Moser theory

Nonlinear wave equations model the propagation of waves in a wide range of Nonlinear wave equations model the propagation of waves in a wide range of physical systems, from acoustics to electromagnetics, from seismic motions to vibrating string and elastic membranes, where oscillatory phenomena occur. Because of this intrinsic oscillatory physical structure, it is natural, from a mathematical point of view, to investigate the question of the existence of oscillations, namely periodic and quasi-periodic solutions, for the equations governing such physical systems. This is the central question of this Thesis