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## Parametrisations of Elements of Spinor and Orthogonal Groups Using Exterior Exponents

### Advances in Applied Clifford Algebras (2011-09-01) 21: 583-590 , September 01, 2011

We present new parametrizations of elements of spinor and orthogonal groups of dimension 4 using Grassmann exterior algebra. Theory of spinor groups is an important tool in theoretical and mathematical physics namely in the Dirac equation for an electron.

## On Weak Solutions to Dirac-Harmonic Equations for Differential Forms

### Advances in Applied Clifford Algebras (2017-12-01) 27: 3167-3181 , December 01, 2017

In this paper, we consider the Dirac-harmonic equations for differential forms with Dirichlet boundary data in weak sense. Using the Hodge-decomposition theory of differential forms and Browder–Minty theory of monotone operators, the existence is obtained. According to the Poincaré-type inequality of Dirac–Sobolev space, we get the higher integrability.

## Fonctions analytiques hyperboliques

### Advances in Applied Clifford Algebras (1999-06-01) 9: 91-94 , June 01, 1999

We define and study here a class of functions probably new defined on*C×IR*^{2} (where*C* is a Clifford algebra) with values in*C*×*IR* and that we call “hyperbolic analytic functions” because of their analogy with complex analytic functions.

## Comparative study of Mixed product and quaternion product

### Advances in Applied Clifford Algebras (2002-12-01) 12: 189-194 , December 01, 2002

Mixed number is the sum of a scalar and a vector. The quaternion can also be written as the sum of a scalar and a vector but the product of mixed numbers and the product of quaternions are different. Here we studied the Mixed product which is derived from the product of mixed numbers and the quaternion product which is derived from the product of quaternions. It was observed that Mixed product is more consistent with Physics than that of quaternion product.

## Sampling with Bessel Functions

### Advances in Applied Clifford Algebras (2007-08-01) 17: 519-536 , August 01, 2007

### Abstract.

The paper deals with sampling of σ-bandlimited functions in *R*^{m} with Clifford-valued, where bandlimitedness means that the spectrum is contained in the ball *B*(0, *σ*) that is centered at the origin and has radius *σ*. By comparing with the general setting, what is new in the sampling is using the explicit Bochner-type relations involving spherical harmonics and monogenics in the Clifford algebra setting. Convergence of the sampling formulas in the *L*^{2} sense and in the uniform and absolute pointwise sense are studied.

## Clifford Algebras and Matrix Factorizations

### Advances in Applied Clifford Algebras (2008-09-01) 18: 417-430 , September 01, 2008

### Abstract.

The purpose of this short note is to connect the two parts of the title, on one hand Clifford algebras and Clifford modules, on the other hand matrix factorizations of a non degenerate quadratic homogeneous polynomial. The main result states the category of Clifford modules is equivalent to a suitably defined category of Matrix factorizations of the quadratic polynomial. This shed a new light about the category of Clifford modules.

## Holditch Theorem and Steiner Formula for the Planar Hyperbolic Motions

### Advances in Applied Clifford Algebras (2010-03-01) 20: 195-200 , March 01, 2010

### Abstract.

The Steiner formula and the Holditch Theorem for one-parameter closed planar Euclidean motions [1, 5] were expressed by Müller, under the one-parameter closed planar motions in the complex sense. Also, Müller had given Holditch theorem in the complex sense, [6].

In this paper, in analogy with Complex motions as given by Müller [6], the Steiner formula and the mixture area formula are obtained under one parameter hyperbolic motions. Also Holditch theorem was expressed in the hyperbolic sense.

## A FORMULATION OF HAMILTONIAN MECHANICS USING GEOMETRIC ALGEBRA

### Advances in Applied Clifford Algebras (2000-09-01) 10: 217-223 , September 01, 2000

### Abstract.

Geometric Algebra is introduced. The basic concepts of Hamilton mechanics are derived using the Geometric Algebra. Standard Poisson bracket and standard Lagrange bracket is introduced in terms of the Geometric Algebra. The Dirac quantization map is derived in a new view, which is used to derive the plus Poisson bracket, which seems different from the previous forms by other authors.

## In Memoriam Jarolim Bureš (1942–2006)

### Advances in Applied Clifford Algebras (2009-07-01) 19: 161-162 , July 01, 2009

## Freud’s Identity of Differential Geometry, the Einstein-Hilbert Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR

### Advances in Applied Clifford Algebras (2009-02-01) 19: 113-145 , February 01, 2009

### Abstract.

We reveal in a rigorous mathematical way using the theory of differential forms, here viewed as sections of a Clifford bundle over a Lorentzian manifold, the true meaning of Freud’s identity of differential geometry discovered in 1939 (as a generalization of results already obtained by Einstein in 1916) and rediscovered in disguised forms by several people. We show moreover that contrary to some claims in the literature there is not a single (mathematical) inconsistency between Freud’s identity (which is a decomposition of the Einstein *indexed* 3-forms
$$*{\mathcal{G}^{a}}$$
in two *gauge dependent* objects) and the field equations of General Relativity. However, as we show there is an obvious inconsistency in the way that Freud’s identity is usually applied in the formulation of energy-momentum “conservation laws” in GR. In order for this paper to be useful for a large class of readers (even those ones making a first contact with the theory of differential forms) all calculations are done with all details (disclosing some of the “tricks of the trade” of the subject).