Related papers: Problems in additive number theory, VI: Sizes of s…

We study a zero-sum problem dealing with minimal zero-sum sequences of maximal length over finite abelian groups. A positive answer to this problem yields a structural description of sets of lengths with maximal elasticity in transfer Krull…

In this paper we study the number of finite topologies on an $n$-element set subject to various restrictions.

It was shown that in a group of bijections of an infinite set some families of subsets, related to the cardinality of some eigenspaces, are generating. Besides, we derived a criterion for generating by sets of this kind.

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds…

The concept of graph compositions is related to several number theoretic concepts, including partitions of positive integers and the cardinality of the power set of finite sets. This paper examines graph compositions where the total number…

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers,…

We indicate a natural generalization of the concept of subgroup commutativity degree of a finite group and a list of open problems on these new concepts.

For finite subsets A_1,...,A_n of a field, their sumset is given by {a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various restricted sumsets of A_1,...,A_n with restrictions of the following forms: a_i-a_j not in S_{ij},…

We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…

We study the problem of finding a subgroup of a given order in a finite group, where the group is represented by its Cayley table. We analyze the complexity of the problem in the special case of abelian groups and present an optimal…

Experimental calculations suggest that the $h$-fold sumset sizes of 4-element sets of integers are concentrated at $h$ numbers that are differences of tetrahedral numbers. In this paper it is proved that these "popular" sumset sizes always…

This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that…

When p divides the ordering of Galois group, the distribution of the Sylow p-subgroup of Cl(K) is closely related to the problem of counting fields with certain specifications. Moreover, different orderings of number fields affect the…

We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

We introduce and study four optimization problems that generalize the well-known subset sum problem. Given a node-weighted digraph, select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints…

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

The paper focuses on some versions of connected dominating set problems: basic problems and multicriteria problems. A literature survey on basic problem formulations and solving approaches is presented. The basic connected dominating set…

This paper examines a denumerable version of the nested-set theorem and derives from it a contradiction involving the formal consistency of the actual infinity assumed by the Axiom of Infinity.

We consider the growth, order, and finiteness problems for automaton (semi)groups. We propose new implementations and compare them with the existing ones. As a result of extensive experimentations, we propose some conjectures on the order…

Let $S$ be a class of groups and let $f_S (n)$ be the number of isomorphism classes of groups in $S$ of order $n$. Let $f(n)$ count the number of groups of order $n$ up to isomorphism. The asymptotic bounds for $f(n)$ behave differently…