1. Review of Differential Geometry 2 2. Definition of an Elliptic Operator 5 3. Properties of Elliptic Operators 7 4. Example of an Elliptic Operator 9 5. Example: The Euler Characteristic 12 6. Example: The Signature Invariant 14 7. A Theorem of Atiyah, Frank and Mayer 18 8. Clifford Algebras 20 9. A Diversion: Constructing Vector Fields on Spheres using Clifford Algebras 23 10. Topological In...

We introduce the so-called Clifford–Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related ...

We propose a new framework for constructing geometric and physical models on spacetimes provided with Lie algebroid symmetry, i.e. manifolds provided with additional anchor and generalized Lie algebra commutator structures. The approach is related to the geometry of moving nonholonomic frames with associated nonlinear connections. A strict application of such geometric methods to spinor fields ...

The present thesis introduces Clifford Algebra as a framework for neural computation. Clifford Algebra subsumes, for example, the reals, complex numbers and quaternions. Neural computation with Clifford algebras is model–based. This principle is established by constructing Clifford algebras from quadratic spaces. Then the subspace grading inherent to any Clifford algebra is introduced, which al...

Gröbner bases in polynomial rings have numerous applications in geometry, applied mathematics, and engineering. We show a few applications of Gröbner bases in robotics, formulated in the language of Clifford algebras, and in engineering to the theory of curves, including Fermat and Bézier cubics, and interpolation functions used in finite element theory.

In the present paper we give a partially negative answer to a conjecture of Ghahramani, Runde and Willis. We also discuss the derivation problem for both foundation semigroup algebras and Clifford semigroup algebras. In particular, we prove that if S is a topological Clifford semigroup for which Es is finite, then H1(M(S),M(S))={0}.

Work of Clifford, Munn and Ponizovskĭı parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees’s theorem characterizing 0-simple...

Hermitian Clifford analysis deals with the simultaneous null solutions of the orthogonal Dirac operators ∂x and its twisted counterpart ∂x|, introduced below. For a thorough treatment of this higher-dimensional function theory, we refer the reader to, for example, 1–5 . Let e1, . . . , e2n be an orthonormal basis of the Euclidean space R2n. Consider the complex Clifford algebra C2n constructed ...

Using the recently discovered Clifford statistics we propose a simple model for the grand canonical ensemble of the carriers in the theory of fractional quantum Hall effect. The model leads to a temperature limit associated with the permutational degrees of freedom of such an ensemble. We also briefly review Schur’s theory of projective representations of the permutation groups. PACS numbers: 7...

In this review we show that a Clifford algebra possesses a unique irreducible representation ; the spinor representation. We discuss what types of spinors can exist in Minkowski space-times and we explain how to construct all the supersymmetry algebras that contain a given space-time Lie algebra. After deriving the irreducible representations of the superymmetry algebras, we explain how to use ...