Related papers: Groups and Combinatorial Number Theory
Given a finite abelian group $G$ and cyclic subgroups $A$, $B$, $C$ of $G$ of the same order, we find necessary and sufficient conditions for $A$, $B$, $C$ to admit a common transversal for the cosets they afford. For an arbitrary number of…
We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
Evidences have suggested that counting representations are sometimes tractable even when the corresponding classification problem is almost impossible, or "wild" in a precise sense. Such counting problems are directly related to matrix…
Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for…
This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for…
In the paper "An Abelian Loop for Non-Composites" (arXiv:110.14716), we introduced a group-like structure consisting of odd prime numbers and 1, with properties that allowed us to prove analogous results to well known theorems in Number…
In this article, we introduce the notion of representations of polyadic groups and we investigate the connection between these representations and those of retract groups and covering groups.
We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure and category…
This paper describes the homology of various simplicial complexes associated to set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition…
We study conformally invariant boundary conditions that break part of the bulk symmetries. A general theory is developped for those boundary conditions for which the preserved subalgebra is the fixed algebra under an abelian orbifold group.…
It is well known that results on zero-sum sequences over a finitely generated abelian group can be translated to statements on generators of rings of invariants of the dual group. Here the direction of the transfer of information between…
We consider $\G$-graded commutative algebras, where $\G$ is an abelian group. Starting from a remarkable example of the classical algebra of quaternions and, more generally, an arbitrary Clifford algebra, we develop a general viewpoint on…
We establish some results about large restricted Lie algebras similar to those known in the Group Theory. As an application we use this group-theoretic approach to produce some examples of restricted as well as ordinary Lie algebras which…
Keller proposed a combinatorial conjecture on construction of an n-by-infinite matrix, which comes from showing the existence of many orbits of different sizes in certain linear group actions. He proved it for the case n=4, and we show that…
We propose various problems about Borel complexity of characterized subgroups of compact abelian groups, inspired by our forthcoming paper \cite{DI3}.
We investigate numerical semigroups generated by any quadratic sequence with initial term zero and an infinite number of terms. We find an efficient algorithm for calculating the Ap\'ery set, as well as bounds on the elements of the Ap\'ery…
New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> makes an important such contribution.
In this note, we give the explicit formula for the number of multisubsets of a finite abelian group $G$ with any given size such that the sum is equal to a given element $g\in G$. This also gives the number of partitions of $g$ into a given…
A diverse collection of fusion categories may be realized by the representation theory of quantum groups. There is substantial literature where one will find detailed constructions of quantum groups, and proofs of the…