Related papers: On the finitizability problem in algebraic logic; …
We survey some of the ideas behind the recent developments in additive number theory, combinatorics and ergodic theory leading to the proof of Hardy- Littlewood type estimates for the number of prime solutions to systems of linear equations…
We consider mappings, which are structure consisting of a single function (and possibly some number of unary relations) and address the problem of approximating a continuous mapping by a finite mapping. This problem is the inverse problem…
We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.
This paper is the last part of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends,…
We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…
We construct a De Morgan algebra-valued logic with quantifiers, where the truth values are in a finite De Morgan algebra, We show that there is a representation theorem of the cylindric algebra of this logic from which a completeness…
Muchnik's paradox says that enumerable betting strategies are not always reducible to enumerable strategies whose bets are restricted to either even rounds or odd rounds. In other words, there are outcome sequences x where an effectively…
Combinatorial optimization problems are notoriously challenging for neural networks, especially in the absence of labeled instances. This work proposes an unsupervised learning framework for CO problems on graphs that can provide integral…
MV-algebras are an algebraic semantics for Lukasiewicz logic and MV-algebras generated by a finite chain are Heyting algebras where the Godel implication can be written in terms of De Morgan and Moisil's modal operators. In our work, a…
Additive combinatorics asks for lower bounds on sumsets and restricted sumsets over finite fields. Central examples are the Cauchy-Davenport theorem and the Erd\H{o}s-Heilbronn conjecture. In this note, we develop Das's linear algebraic…
We study the representation theory of the algebraic Toeplitz algebra $R={\mathbb K}\langle x,y\rangle/\langle xy-1\rangle$, give a few new structure and homological theorems, completely determine one-sided ideals and survey and re-obtain…
In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. We generalise some basic facts on arithmetic degree and canonical height in positive characteristic. As applications, we…
An algebraic telic problem is a decision problem in $\textsf{NP}_\mathbb{R}$ formalizing finite-time reachability questions for one-dimensional dynamical systems. We prove that the existence of "natural" mapping reductions between algebraic…
We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…
We present a self-contained analysis of infinity from two mathematical perspectives: set theory and algebra. We begin with cardinal and ordinal numbers, examining deep questions such as the continuum hypothesis, along with foundational…
We study some versions of the statement of Hadwiger's conjecture for finite as well as infinite graphs.
A new tool is introduced, specially tailored for the study of order 2 bases of N. It is called the Erdos-Tuan algebra since it suits well their conjecture.
In this paper, we study a combinatorial problem originating in the following conjecture of Erdos and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n.…
In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative…
We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power…