The Yang-Baxter equation first appeared in a paper by the Nobel laureate, C.N. Yang, and in R.J. Baxter’s work. Later, Vladimir Drinfeld, Vaughan F. R. Jones and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equat...
We introduce and study a large class of coalgebras (possibly (non)coassociative) with group-algebraic structures Hopf (non)coassociative group-algebras. Hopf (non)coassociative group-algebras provide a unifying framework for classical Hopf algebras a...
Given the Yetter–Drinfeld category over any quasigroup and a braided Hopf coquasigroup in this category, we first mainly study the Radford’s biproduct corresponding to this braided Hopf coquasigroup. Then, we investigate Sweedler’s...
Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we intr...
For a group π with unit e, we introduce and study the notion of a π-graded Hopf algebra. Then we introduce and construct a new braided monoidal category HHeYDπ over a π-graded Hopf algebra H. We introduce the notion of a π-double centr...
Let H be a crossed group-cograded Hopf quasigroup. We first introduce the notion of p-Yetter–Drinfeld quasimodule over H. If the antipode of H is bijective, we show that the category YDQ(H) of Yetter–Drinfeld quasimodules over H is a cros...
In this paper, we prove that for any pair of weak Hopf monoids H and B in a symmetric monoidal category where every idempotent morphism splits, the category of H-B-Long dimodules HBLong is monoidal. Moreover, if H is quasitriangular and B coquasitria...
Hopf algebras, as a crucial generalization of groups, have a very symmetric structure and have been playing a prominent role in mathematical physics. In this paper, let H be a dual quasi-Hopf algebra which is a more general Hopf algebra structure. A....
Let (H,μH,ΔH,αH,βH,ψH,ωH,SH) be a BiHom–Hopf algebra. First, we provide a non-trivial example of a left–left BiHom–Yetter–Drinfeld module and show that the category HHBHYD is a braided monoidal ca...
A research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings and braided group st...
Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimensi...
In this paper we present a detailed study of bonded knots and their related structures, integrating recent developments into a single framework. Bonded knots are classical knots endowed with embedded bonding arcs modeling physical or chemical bonds....
The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar constructions are available in braided monoidal categories, a...
This paper presents the total angular momentum-conserving Poincaré sphere (TAM-C PS), which offers a novel framework for efficiently characterizing a wide range of vector vortex beams. Unlike other types of Poincaré spheres, the TAM-C P...
The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has res...
Solution searching that accompanies combinatorial explosion is one of the most important issues in the age of artificial intelligence. Natural intelligence, which exploits natural processes for intelligent functions, is expected to help resolve or al...
The massive amount of available neurodata suggests the existence of a mathematical backbone underlying neuronal oscillatory activities. For example, geometric constraints are powerful enough to define cellular distribution and drive the embryonal dev...
Polymer flooding has drawn more and more attention in the world for its high incremental oil recovery factor and relative low costs compared with water flooding and other chemically enhanced oil recovery techniques. However, for many oilfields, such...
We generalize orbifold models for chords and voice leading to incorporate loudness, allowing for the modeling of resting voices, which are used frequently by composers and arrangers across genres. In our generalized setting (strictly speaking, that o...
The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot t...
Under the EU Water Framework Directive, ecological assessment and management are based on type-specific reference conditions. In the EU it may be difficult to find sites in large rivers with at least near-natural conditions, though this is not the ca...
There are frequently interactions between active folds and major rivers (mean annual water discharges > 70 m3s−1). The major river may incise across the fold, to produce a water gap across the fold, or a bevelling (or lateral planation) of t...