相关论文: Algebraically constructible functions: real algebr…
Abstract axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and…
The article is devoted to a structure of topological spaces related with topological quasigroups. Regular and complete spaces over topological quasigroups are studied. Separations and embeddings are also investigated for them. Their…
This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function…
A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the…
This article introduces a method, which starting from simple and quite general mathematical data, allows to construct linear algebras of operators which are, each of them, endowed with a bialgebra structure (coproduct and counity). Moreover…
We introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory $T$ is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model $M$ of $T$ has an expansion…
In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…
This book gives a thorough introduction to topological data analysis (TDA), the application of algebraic topology to data science. Algebraic topology is traditionally a very specialized field of math, and most mathematicians have never been…
In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…
Algebras of ultradifferentiable generalized functions are introduced. We give a microlocal analysis within these algebras related to the regularity type and the ultradifferentiable property.
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
We provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions.
Real valued homomorphisms on the algebra of smooth functions on a differential space are described. The concept of generators of this algebra is emphasized in this description.
This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on 3-dimensional topological "objects" of various types, and to classify several classes of such objects using such…
The purpose of this contribution is to give a coherent account of a particular narrative which links locales, geometric theories, sheaf semantics and constructive commutative algebra. We are hoping to convey a firm grasp of three ideas: (1)…
In a recent work we have shown how to construct an information algebra of coherent sets of gambles defined on general possibility spaces. Here we analyze the connection of such an algebra with the set algebra of subsets of the possibility…
We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.
A characterization of multiplicative (and additive) arithmetical functions is given. Using this characterization, we show that the group of multiplicative arithmetical functions is isomorphic to the group of additive arithmetical functions.
Extending the `metric spaces' of Lawvere, we study `real metrics', with values in the extended real line. Formally, this ordered set is a symmetric monoidal closed category, and our structures are enriched categories on the latter.…