Related papers: "Pentagonal Algebra" and Four-Simplex Equation
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of…
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
The computation of matrix functions is a well-studied problem. Of special importance are the exponential and the logarithm of a matrix, where the latter also raises existence and uniqueness questions. This is particularly relevant in the…
Simplicial approximation and the ideas associated with the Regge calculus.provide a concrete way of implementing a sum over histories formulation ofquantum gravity. A four-dimensional simplicial geometry is made up of flat four-simplices…
A very elementary introduction to quantum algebras is presented and a few examples of their physical applications are mentioned.
This survey aims to collect the main results of the theory of the set-theoretical solutions to the pentagon equation obtained up to now in the literature. In particular, we present some classes of solutions and raise some questions.
This work provides explicit characterizations and formulae for the minimal polynomials of a wide variety of structured $4\times 4$ matrices. These include symmetric, Hamiltonian and orthogonal matrices. Applications such as the complete…
We introduce the notion of quadri-algebras. These are associative algebras for which the multiplication can be decomposed as the sum of four operations in a certain coherent manner. We present several examples of quadri-algebras: the…
This study examines Quaternion Dirac solutions for an infinite square well. The quaternion result does not recover the complex result within a particular limit. This raises the possibility that quaternionic quantum mechanics may not be…
This work provides a quaternioinc reprsentation for real symplectic matrices in dimension four, analogous to the pair of unit quaternions representation for special orthogonal matrices. In the process of finding formulae for this…
This paper fails to derive quantum mechanics from a few simple postulates. But it gets very close --- and it does so without much exertion. More exactly, I obtain a representation of finite-dimensional probabilistic systems in terms of…
This article investigates the two-parameter quantum matrix algebra at roots of unity. In the roots of unity setting, this algebra becomes a Polynomial Identity (PI) algebra and it is known that simple modules over such algebra are…
We construct a solution to pentagon equation with anticommuting variables living on two-dimensional faces of tetrahedra. In this solution, matrix coordinates are ascribed to tetrahedron vertices. As matrix multiplication is noncommutative,…
We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of…
In this paper we present efficient computational and symbolic algorithms for solving a nearly pentadiagonal linear systems. The implementation of the algorithms using Computer Algebra Systems (CAS)such as MAPLE, MACSYMA, MATHEMATICA, and…
We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative…
A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum…