Related papers: "Pentagonal Algebra" and Four-Simplex Equation
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
We associate a deformation of Heisenberg algebra to the suitably normalized Yang $R$-matrix and we investigate its properties. Moreover, we construct new examples of quantum vertex algebras which possess the same representation theory as…
One of the generalizations of the pentagon equation to higher dimensions is the so-called "six-term equation". Geometrically, it corresponds to one of the "Alexander moves", that is elementary rebuildings of simplicial complexes, namely,…
There exists an exact relationship between the quasi-exactly solvable problems of quantum mechanics and models of square and rectangular random complex matrices. This relationship enables one to reduce the problem of constructing…
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,…
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
As one type of incidence theory, the geometry of pentagram map seems quite classical at first. However, this is an excellent example of such a classical idea developed into a marvellous insight by some modern approach. We introduce an…
Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…
This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.
The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…
As an expansion of complex numbers, the quaternions show close relations to numerous physically fundamental concepts. In spite of that, the didactic potential provided by quaternion interrelationships in formulating physical laws are hardly…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
One of the principal obstacles on the way to quantum computers is the lack of distinguished basis in the space of unitary evolutions and thus the lack of the commonly accepted set of basic operations (universal gates). A natural choice,…
We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…
The author introduces the notion of a quantum form of an algebraic torus. In the case of diagonal algebraic torus we get the algebra of Laurent twisted polynomials. Quantum algebraic torus can be characterized in terms of exact sequences.…
Some comments are made on the matrices which serve as the basis of a quaternionic algebra. We show that these matrices are related with the quaternionic action of the imaginary units from the left and from the right.
Many authors studied numeric algorithms for solving the linear systems of the pentadiagonal type. The well-known Fast Pentadiagonal System Solver algorithm is an example of such algorithms. The current article are described new numeric and…
The satisfactory development of Quaternionic Analysis has indicated new solutions for physical and mathematical problems. It is worth mentioning the fact that quaternions possess four dimensions, and in this way they may be considered as…
In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same…