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Generalized unitarity cut of a Feynman diagram generates an algebraic system of polynomial equations. At high-loop levels, these equations may define a complex curve or a (hyper-)surface with complicated topology. We study the curve cases,…

A group element is called a generalized torsion if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the…

Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using…

In "A note on generalized Clifford algebras and representations" (Caenepeel, S.; Van Oystaeyen, F., Comm. Algebra 17 (1989) no. 1, 93--102.) generalized Clifford algebras were introduced via Clifford representations; these correspond to…

In this article, we give the most genaral form of the quaternions algebra depending on 3-parameters. We define 3-parameter generalized quaternions (3PGQs) and study on various properties and applications. Firstly we present the definiton,…

We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient…

We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or…

Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the trivial element, then $g$ is called a generalized torsion element. To the best of our knowledge, we have no…

We define orbifold elliptic genus for general orbifolds which generalizes the definition of Borisov and Libgober, and prove their rigidity property.

For each finite subgroup G of SL(n, C), we introduce the generalized Cartan matrix C_{G} in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices…

Double coverings of the orthogonal groups of the real and complex spaces are considered. The relation between discrete transformations of these spaces and fundamental automorphisms of Clifford algebras is established, where an isomorphism…

In this paper, a way of generalizing the tensor renormalization group(TRG) is proposed. Mathematically, the connection between patterns of tensor renormalization group and the concept of truncation sequence in polytope geometry is…

We introduce a new class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. These allow us, through the process of twisting the familiar standard quantum groups, to generate new as well as…

Formal orbifolds are defined in higher dimension. Their \'etale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to…

In the first part of this article we show for some examples of surfaces of general type in toric 3-folds how to construct minimal and canonical models by toric methods explicitly. The examples we study turn out to be surfaces of general…

In the space of marked group, we determine the structure of groups which are limit points of the set of all generalized quaternion groups.

The classification of the unitary irreducible representations of symmetry groups is a cornerstone of modern quantum physics, as it provides the fundamental building blocks for constructing the Hilbert spaces of theories admitting these…

Some results that are true in classical groups are investigated in generalized groups and are shown to be either generally true in generalized groups or true in some special types of generalized groups. Also, it is shown that a Bol groupoid…

We study the isometry groups of compact spherical orientable $3$-orbifolds $S^3/G$, where $G$ is a finite subgroup of $\mathrm{SO}(4)$, by determining their isomorphism type. Moreover, we prove that the inclusion of $\mbox{Isom}(S^3/G)$…

A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without…