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The relevance and importance of contextualizing data analytics is described. Qualitative characteristics might form the context of quantitative analysis. Topics that are at issue include: contrast, baselining, secondary data sources,…
A family of graphs optimized as the topologies for supercomputer interconnection networks is proposed. The special needs of such network topologies, minimal diameter and mean path length, are met by special constructions of the weight…
For decades, de Casteljau's algorithm has been used as a fundamental building block in curve and surface design and has found a wide range of applications in fields such as scientific computing, and discrete geometry to name but a few. With…
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks.…
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
In this paper we discuss some physical applications of topological *-algebras of unbounded operators. Our first example is a simple system of free bosons. Then we analyze different models which are related to this one. We also discuss the…
In algebraic geometry there is the notion of a height pairing of algebraic cycles, which lies at the confluence of arithmetic, Hodge theory and topology. After explaining a motivating example situation, we introduce new directions in this…
The algebro-geometric approach for integrability of semi-Hamiltonian hydrodynamic type systems is presented. This method is significantly simplified for so-called symmetric hydrodynamic type systems. Plenty interesting and physically…
The paper consists of two parts. The first part is devoted to logic for universal algebraic geometry. The second one deals with problems and some results. It may be regarded as a brief exposition of some ideas from the book in progress:…
Algebraic K-theory has applications in a broad range of mathematical subjects, from number theory to functional analysis. It is also fiendishly hard to calculate. Presently there are two main inroads: motivic and cyclic homology. I've been…
This work proposes an algebraic model for classical information theory. We first give an algebraic model of probability theory. Information theoretic constructs are based on this model. In addition to theoretical insights provided by our…
Recently, many systems for graph analysis have been developed to address the growing needs of both industry and academia to study complex graphs. Insight into the practical uses of graph analysis will allow future developments of such…
The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on…
Data-driven methods play an increasingly important role in discovering geometric, structural, and semantic relationships between 3D shapes in collections, and applying this analysis to support intelligent modeling, editing, and…
We describe the topology of singular real algebraic curves in a smooth surface. We enumerate and bound in terms of the degree the number of topological types of singular algebraic curves in the real projective plane.
This is an attempt to look at the tropical geometry from topological point of view.
We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…
Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject as well as external…
We give a survey on eta invariants including methods of computation and applications in differential topology.