Related papers: Non-gaussian $r-$congruences

We consider residue structures $R/G$ where $(G,+)$ is an additive subgroup of a ring $(R,+,\cdot)$, not necessarily an ideal. Special instances include Krasner's construction of quotient hyperfields, and Pumpluen's construction of…

Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning…

We construct a generalized Lagrangian that unifies the Gross-Neveu-Yukawa, Nambu-Jona-Lasinio-Yukawa, and Wess-Zumino models, allowing for arbitrary scalar and fermion flavors in $D$-dimensional regularization. This framework clarifies how…

Let $w=w(x_1,\ldots,x_r)$ be a lower central word or a derived word. We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables, thus proving a generalized version…

We prove that every odd-order group is symmetric harmonious: there exists a permutation $g_0,g_1,\ldots, g_{\ell-1}$ of elements of $G$ such that the consecutive products $g_0g_1,g_1g_2,\ldots, g_{\ell-1}g_0$ also form a permutation of…

We present the notion of non-abelian descent type, which classifies torsors up to twisting by a Galois cocycle. This relies on the previous construction of kernels and non-abelian Galois 2-cohomology due to Springer and Borovoi. The…

In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and…

Using the notion of a gauge connection on a flat superspace, we construct a general class of noncommutative ($D=2,$ $\mathcal{N}=1$) supertranslation algebras generalizing the ordinary algebra by inclusion of some new bosonic and fermionic…

Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…

Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker…

Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that…

A new subclass of AG-groupoids, so called, cyclic associative Abel-Grassman groupoids or CA-AG-groupoid is studied. These have been enumerated up to order $6$. A test for the verification of cyclic associativity for an arbitrary AG-groupoid…

The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…

Let G denote a group and let W be an algebra over a commutative ring R. We will say that W is a G-graded twisted algebra (not necessarily commutative, neither associative) if there exists a G-grading W=\bigoplus_{g \in G}W_{g} where each…

Let $N_k(n,r,\boldsymbol{a})$ denote the number of incongruent solutions of the quadratic congruence $a_1x_1^2+\ldots+a_kx_k^2\equiv n$ (mod $r$), where $\boldsymbol{a}=(a_1,\ldots,a_k)\in {\Bbb Z}^k$, $n\in {\Bbb Z}$, $r\in {\Bbb N}$. We…

The goal of this paper is to show that fundamental concepts in higher-order Fourier analysis can be nauturally extended to the non-commutative setting. We generalize Gowers norms to arbitrary compact non-commutative groups. On the…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

We introduce quiver gauge theory associated with the non-simply-laced type fractional quiver, and define fractional quiver W-algebras by using construction of arXiv:1512.08533 and arXiv:1608.04651 with representation of fractional quivers.

It is shown that in the units of augmentation one of an integral group ring $\mathbb{Z} G$ of a finite group $G$, a noncyclic subgroup of order $p^{2}$, for some odd prime $p$, exists only if such a subgroup exists in $G$. The corresponding…

In this article, we explore a series of elementary yet insightful results involving integrals related to Gaussian sums. Using techniques rooted in classical calculus, we derive several identities and evaluate nontrivial definite integrals…