In classical invariant theory, the Gröbner base of the ideal of syzygies and the normal forms of polynomials of invariants are two core contents. To improve the performance of invariant theory in symbolic computing of classical geometry, advanced invariants are introduced via Clifford product [5]. This paper addresses and solves the two key problems in advanced invariant theory: the Gröbner bas...

This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar to the usual theory of vector functions of a real variable, has some subtle issues which make its presentation worhtwhile.We refer in particular to the deri...

Knill introduced a generalization of stabilizer codes, in this note called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the abstract error group is given by an extraspecial p-group. Suppose that the abstract error group has an abelian index group, then we show that a Clifford...

Clifford codes are a class of quantum error control codes that form a natural generalization of stabilizer codes. These codes were introduced in 1996 by Knill, but only a single Clifford code was known, which is not already a stabilizer code. We derive a necessary and sufficient condition that allows to decide when a Clifford code is a stabilizer code, and compile a table of all true Clifford c...

Knill introduced a generalization of stabilizer codes, in this note called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the abstract error group has an abelian index group. In particular, if the errors are modelled by tensor products of Pauli matrices, then the associated Cli...

It is proposed how the Extended Relativity Theory in C-spaces (Clifford spaces) allows a unified formulation of point particles, strings, membranes and p-branes, moving in ordinary target spacetime backgrounds, within the description of a single polyparticle moving in C-spaces. The degrees of freedom of the latter are provided by Clifford polyvector-valued coordinates (antisymmetric tensorial c...

Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra G ≡ Cl3,0 as a computational model in the Maxima c...

The purpose of this note is to present a proof of the Bott periodicity theorem that is based on the periodicity of Clifford algebras. Such a proof was first predicted in [2], and then constructed in [8] and in [3]. Here, we give another proof along the same lines as [8], but based on a different model of K-theory. In order to simplify the notation, we only present the periodicity for KO theory....

We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable map f : X → Y (not necessarily K-oriented). The push-forward map generalizes the push-forward map in ordinary K-theory for any K-oriented map and the Atiyah-Singer index theorem of Dirac operators on Clifford modules. For D-branes satisfyin...