Related papers: Voevodsky's Unachieved Project
Statistical learning theory chiefly studies restricted hypothesis classes, particularly those with finite Vapnik-Chervonenkis (VC) dimension. The fundamental quantity of interest is the sample complexity: the number of samples required to…
Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.~V.~Konyagin and V.~N.~Temlyakov, A…
Despite its popularity, it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, whereas for…
Given a raw knowledge in the form of a data table/a decision system, one is facing two possible venues. One, to treat the system as closed, i.e., its universe does not admit new objects, or, to the contrary, its universe is open on…
This paper considers stochastic monotone variational inequalities whose feasible region is the intersection of a (possibly infinite) number of convex functional level sets. A projection-based approach or direct Lagrangian-based techniques…
A research problem for undergraduates and graduates is being posed as a cap for the prior antecedent regular discrete mathematics exercises. [Here cap is not necessarily CAP=Competitive Access Provider, though nevertheless ...] The object…
The manuscript is an overview of the motivations and foundations lying behind Voevodsky's ideas of constructing categories similar to the ordinary topological homotopy categories. The objects of these categories are strictly related to…
This paper is an introduction to our ongoing more comprehensive work on a critically important period in the history of Russian mathematics education; it provides a glimpse into the socio-political environment in which the famous Soviet…
The Bayesian statistical paradigm uses the language of probability to express uncertainty about the phenomena that generate observed data. Probability distributions thus characterize Bayesian analysis, with the rules of probability used to…
The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a…
Domain theory has a long history of applications in theoretical computer science and mathematics. In this article, we explore the relation of domain theory to probability theory and stochastic processes. The goal is to establish a theory in…
A simple but useful method of reciprocal values is introduced, explained and illustrated. This method simplifies the analysis of hyperbolic distributions, which are causing serious problems in the demographic and economic research. It…
We revisit the classical analysis of Karp, Vazirani, and Vazirani (KVV, STOC~1990), which established the well-known upper bound of $1 - 1/e$ as the limiting proportion of vertices that can be matched by any online procedure in a canonical…
The information in an individual finite object (like a binary string) is commonly measured by its Kolmogorov complexity. One can divide that information into two parts: the information accounting for the useful regularity present in the…
The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is to present an exposition of the circle of…
This reform was a part of a transnational movement as 'the New Math' in the United States or the activity of the Lichnerowicz commission in France. The Russian version of these events involved the whole secondary and high school (4-10…
This thesis describes work on two applications of probabilistic programming: the learning of probabilistic program code given specifications, in particular program code of one-dimensional samplers; and the facilitation of sequential Monte…
We use differential equations based approaches to provide some {\it \textbf{physics}} insights into analyzing the dynamics of popular optimization algorithms in machine learning. In particular, we study gradient descent, proximal gradient…
This introductory text arises from a lecture given in G\"oteborg, Sweden, given by the first author and is intended for undergraduate students, as well as for any mathematically inclined reader wishing to explore a synthesis of ideas…
This paper identifies several different interconnected challenges preventing the move towards more ethical and sustainable mathematics education: the entrenched belief in mathematical neutrality, the difficulty of simultaneously reforming…