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Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but…
This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…
In this review, we present some advanced algorithms and programs used in our scientific school with short description of types of astrophysical systems, which we study. However, we discuss mainly mathematical methods, which may be applied…
The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the…
Objects are a centerpiece of the mathematical realm and our interaction with and reasoning about it, just as they are of the physical one (if not more). And humans' mathematical reasoning must ultimately be grounded in our general…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
We explain the motivation for looking for a predicative analogue of the notion of a topos and propose two definitions. For both notions of a predicative topos we will present the basic results, providing the groundwork for future work in…
This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and…
Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the {\em computationally dense\/} ones) are seen to be the ones…
Since there are different ways of axiomatizing and developing a mathematical theory, knowledge about a such a theory may reside in many places and in many forms within a library of formalized mathematics. We introduce the notion of a realm…
We discuss the optimal presentations of mathematical objects under well defined symbol libraries. We shall examine what light our chosen symbol libraries and syntax shed upon the objects they represent. A major part of this work will focus…
This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher…
Many mathematicians find mathematics aesthetically beautiful and even comparable to art forms such as music or painting. On the other hand, every year a great number of school students leave mathematics with total disillusionment and…
The hypothesis concerning the off-site continuum existence is investigated from the point of view of the mathematical theory of sets. The principles and methods of the mathematical description of the physical objects from different off-site…
The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data…
We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in…
Linear topological spaces with partial ordering (linear kinematics) are studied. They are defined by a set of 8 axioms implying that topology, linear structure and ordering are compatible with each other. Most of the results are valid for…
The constructive approach to mathematics has the advantage that witnesses can be extracted from statements of existence and theorems can be unwound to give algorithms. Even better, constructive theorems can be interpreted in any topos,…
I describe the general mathematical construction and physical picture of topological black holes, which are black holes whose event horizons are surfaces of non-trivial topology. The construction is carried out in an arbitrary number of…
We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…