Related papers: Symbolic Computations in Higher Dimensional Cliffo…
Computer algebra is widely used in various fields of mathematics, physics and other sciences. The simplification of tensor expressions is an important special case of computer algebra. In this paper, we consider the reduction of tensor…
In this paper we address the problem of constructing a class of representations of Clifford algebras that can be named "alphabetic (re)presentations". The Clifford algebras generators are expressed as m-letter words written with a…
Quantum multiparameter deformation of real Clifford algebras is proposed. The corresponding irreducible representations are found.
A particular orthogonal map on a finite dimensional real quadratic vector space (V,Q) with a non-degenerate quadratic form Q of any signature (p,q) is considered. It can be viewed as a correlation of the vector space that leads to a dual…
We derive a set of Clifford-algebraic formulas for two major nonlinear conformal transformations of the physical quantities related to Maxwell's equations. The superiority of these formulas over their vector-tensorial counterparts are…
Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and…
Three loop ladder and $V$-topology diagrams contributing to the massive operator matrix element $A_{Qg}$ are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable…
Because of the isomorphism ${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \kern -0.1em \ell}_{2,3}(\Bbb{R})$, it is possible to complexify the spacetime Clifford algebra ${C \kern -0.1em \ell}_{1,3}(\Bbb{R})$ by adding one additional timelike…
We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and…
Document preparation systems like LaTeX offer the ability to render mathematical expressions as one would write these on paper. Using LaTeX, LaTeXML, and tools generated for use in the National Institute of Standards (NIST) Digital Library…
In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…
We present a practical application of parallel symbolic computation in General Relativity: the calculation of curvature invariants for large dimension. We discuss the structure of the calculations, an implementation of the technique and…
We provide a survey of results from symbolic dynamics and algebraic topology relating to Grout, a new user-friendly program developed to calculate combinatorial properties and topological invariants of a large class of symbolic…
Given a fixed binary form $f(u,v)$ of degree $d$ over a field $k$, the associated \emph{Clifford algebra} is the $k$-algebra $C_f=k\{u,v\}/I$, where $I$ is the two-sided ideal generated by elements of the form $(\alpha u+\beta…
Based on a fact that complex Clifford algebras of even dimension are isomorphic to the matrix ones, we consider bundles in Clifford algebras whose structure group is a general linear group acting on a Clifford algebra by left…
We assume some standard choices for the branch cuts of a group of functions and consider the problem of then calculating the branch cuts of expressions involving those functions. Typical examples include the addition formulae for inverse…
The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation…
Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the…
Counting the number of perfect matchings in bipartite graphs, or equivalently computing the permanent of 0-1 matrices, is an important combinatorial problem that has been extensively studied by theoreticians and practitioners alike. The…
We have developed a symbolic algebra approach to automatically produce, verify, and optimize computer code for the Fast Multipole Method (FMM) operators. This approach allows for flexibility in choosing a basis set and kernel, and can…