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Volume I 'Algebra and Physics' is devoted to the mathematical aspects of Clifford algebras and their applications in physics. Physical applications and extensions of physical theories such as the theory of quaternionic spin, Dirac theory of electron, plane waves and wave packets in electrodynamics, and electron scattering are also presented, showing the broad applicability of Clifford geometric algebras in solving physical problems.

Treatment of the structure theory of quantum Clifford algebras, twistor phase space, introduction of a Kaluza--Klein type theory related to Finsler geometry, the connection to logic, group representations, and computational techniques--including symbolic calculations and theorem proving--round out the presentation. The unitary operator connecting the stationary Dirac and Maxwell equations is obtained. One can easily verify that these corrections are merely technical ones this is the fault of the authors, not of the journal by having a look at our other publications on this subject [1—5], where everything is o.

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  • Applications of Clifford's Geometric Algebra.
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A Fock space is constituted by states obtained by applying creation operators on a vacuum which is annihilated by all annihilation operators. In representations studied here, creation operators and annihilation operators are not Hermitian conjugates of one other, in general. Therefore, the basis vectors of the Fock space are not the Hermitian conjugates of those of the dual vector space.

In this sense, we call the representation the twisted Fock representation. In this paper, split Fibonacci and Lucas octonions are proposed and their some properties and relations are obtained. We present a method of computing elements of spin groups in the case of arbitrary dimension.

This method generalizes Hestenes method for the case of dimension 4. We present explicit formulas for elements of spin group that correspond to the elements of orthogonal groups as two-sheeted covering. These formulas allow us to compute rotors, which connect two different frames related by a rotation in geometric algebra of arbitrary dimension. In this paper, some equalities and N-fractional ordinary and partial derivatives of Riemann's zeta function are discussed. Moreover some integral representations for the functions are reported.

The factorization of the Laplacian by means of first order systems and of second order operators was considered by several authors see, e. In the paper the definition of Cauchy-Riemann system CR-system of order n is given by their symbols. This paper reveals the differences and similarities between two popular unified representations, i. Specifically, after investigating some fundamental algebraic properties of the UDQ, it is revealed that the kinematical equations represented by the UDQ and the HTM are accordant, and afterwards the direct relationship of UDQ-based error kinematical models in spatialframe and in body-frame are further discussed, with conclusion that either error kinematic model can be chosen for designing kinematical control laws.

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Finally, the comparative study on the proportional control algorithms based on the logarithmical mapping of the HTM and the UDQ shows that the UDQ-based control law is indeed higher in computational efficiency. By combining the Pauli algebra with distribution theory, a compact and conceptually simple derivation of the Stratton-Chu and Kottler-Franz equations is obtained. These are extended to freely moving integration surfaces, so that the fields due to charge distributions in arbitrary motion are represented.

A further generalization is obtained to multiple surfaces, which can be used to enclose clusters of transmitters, scatterers and receivers. Clifford algebra corresponds to Minkwoski space. We present and study a type of Riemann boundary value problems for short RBVPs for polynomially monogenic functions, i. As special cases the solutions of the corresponding boundary value problems for classical polyanalytic functions and metaanalytic functions are derived respectively. Several finite difference approximations of the Dirac operator are studied and compared.

Biquaternions and the Clifford algebra CL(2) (Video 4/14).

Main goals are finite difference Dirac operators which allow a factorization of the discrete Laplacian. We describe the fundamental solutions of the difference operators and prove convergence results in l p -spaces. Discrete versions of the Teodorescu transform are defined.

Advances in Applied Clifford Algebras

The Lipschitz semigroup is generated by all invertible and noninvertible Clifford vectors. This problem turns out to be useful in the construction of multisoliton solutions of integrable systems of nonlinear partial differential equtions. A theory of k -analytic functions on octonions is established.

The Cauchy integral formulas, Taylor series and Laurent series for the k -analytic functions are given. Moreover, we obtain the orthogonality relations for the basis of k -analytic functions. Jaime Keller passed away on January 7, We recall the pre-history of AACA.