We consider Hölder continuous circulant 2 × 2 matrix functions G2 defined on the AhlforsDavid regular boundary Γ of a domain Ω in R2n. The main goal is to study under which conditions such a function G2 can be decomposed as G 1 2 G 1 2 − G2 , where the components G2 ± are extendable to two-sided H-monogenic functions in the interior and the exterior of Ω, respectively. H-monogenicity is a conce...

Today's 'theory of mind' (ToM) concept is rooted in the distinction of nineteenth-century philosopher William Clifford between 'objects' that can be directly perceived and 'ejects', such as the mind of another person, which are inferred from one's subjective knowledge of one's own mind. George Romanes, a founder with Charles Darwin of the discipline of comparative psychology, considered the min...

Clifford analysis offers a higher dimensional function theory studying the null solutions of the rotation invariant, vector valued, first order Dirac operator ∂. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure J on Euclidean space and a corresponding second Dirac operator ∂J , leading to the system of equations...

It is rigorously shown how the Extended Relativity Theory in Clifford spaces (C-spaces) can explain the variable radial dependence ap(r) of the Pioneer anomaly; its sign (pointing towards the sun); why planets don′t experience the anomalous acceleration and why the present day value of the Hubble scale RH appears. It is the curvature-spin coupling of the planetary motions that hold the key. The...

In this paper we examine a new class of five dimensional (5D) exact solutions in extra dimension gravity possessing Lie algebroid symmetry. The constructions provide a motivation for the theory of Clifford nonholonomic algebroids elaborated in Ref. [1]. Such Einstein–Dirac spacetimes are parametrized by generic off–diagonal metrics and nonholonomic frames (vielbeins) with associated nonlinear c...

This essay will present a brief outline of the theory of Clifford algebras, together with a small amount of material about quadratic forms. I follow loosely the well known book Geometric algebra by Emil Artin, but with elegant modifications that I saw originally in some lecture notes by Raoul Bott, (dating from 1962), subsequently included in [Atiyah-Bott-Shapiro:1964]. In the first section, I’...

In previous work the framework for a hypercomplex function theory in superspace was established and amply investigated. In this paper a Cauchy integral formula is obtained in this new framework by exploiting techniques from orthogonal Clifford analysis. After introducing Clifford algebra valued surfaceand volume-elements first a purely fermionic Cauchy formula is proven. Combining this formula ...

The idea that spacetime has to be replaced by Clifford space (C-space) is explored. Quantum field theory (QFT) and string theory are generalized to C-space. It is shown how one can solve the cosmological constant problem and formulate string theory without central terms in the Virasoro algebra by exploiting the peculiar pseudo-Euclidean signature of C-space and the Jackiw definition of the vacu...

Clifford analysis is a higher dimensional function theory and refinement of harmonic analysis. We will extend to the infinite-dimensional setting. To do that, we have introduce algebras Dirac operator. The paper's main result Wiener chaos decomposition in apply white noise Brownian motion.

We define and study the ternary analogues of Clifford algebras. It is proved that the ternary Clifford algebra with N generators is isomorphic to the subalgebra of the elements of grade zero of the ternary Clifford algebra with N+1 generators. In the case N = 3 the ternary commutator of cubic matrices induced by the ternary commutator of the elements of grade zero is derived. We apply the terna...