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Of all publications in the section: 11
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Article
Shirokov D. Advances in Applied Clifford Algebras. 2010. Vol. 20. No. 2. P. 411-425.

In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudo-unitary groups. Our main techniques are Clifford algebras. We have found 12 types of subalgebras of Lie algebras of pseudo-unitary groups.

Article
Shirokov D. Advances in Applied Clifford Algebras. 2021. Vol. 31. P. 1-19.

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts ofcharacteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case n = 4, the proofs for the case n = 5 and the case of arbitrary dimension n. The results can be used in symbolic computation.

Article
Shirokov D. Advances in Applied Clifford Algebras. 2019. Vol. 29. No. 50. P. 1-12.

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 use the method of averaging in Clifford’s geometric algebra previously proposed by the author. 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.

Article
Shirokov D. Advances in Applied Clifford Algebras. 2015. Vol. 25. No. 1. P. 227-244.

We formulate generalizations of Pauli’s theorem on the cases of real and complex Clifford algebras of even and odd dimensions. We give analogues of these theorems in matrix formalism. Using these theorems we present an algorithm for computing elements of spin groups that correspond to elements of orthogonal groups as double cover.

Article
Shirokov D. Advances in Applied Clifford Algebras. 2018. Vol. 28. No. 3. P. 1-16.

We present a new class of covariantly constant solutions of the Yang–Mills equations. These solutions correspond to the solution of the field equation for the spin connection of the general form.

Article
Shirokov D. Advances in Applied Clifford Algebras. 2012. Vol. 22. No. 2. P. 483-497.

In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous papers. On the basis of new classification of Clifford algebra elements, it is possible to reveal and prove a number of new properties of Clifford algebras. We use k-fold commutators and anticommutators. In this paper we consider Clifford and exterior degrees and elementary functions of Clifford algebra elements.

Article
Shirokov D. Advances in Applied Clifford Algebras. 2017. Vol. 27. No. 1. P. 149-163.

In this paper we consider different operators acting on Clifford algebras. We consider Reynolds operator of Salingaros’ vee group. This operator “average” an action of Salingaros’ vee group on Clifford algebra. We consider conjugate action on Clifford algebra. We present a relation between these operators and projection operators onto fixed subspaces of Clifford algebras. Using method of averaging we present solutions of system of commutator equations.

Article
Shirokov D. Advances in Applied Clifford Algebras. 2021. Vol. 31.

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras — subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras — graded central simple algebras or graded central simple algebras with involution.

Article
Shirokov D. Advances in Applied Clifford Algebras. 2012. Vol. 22. No. 1. P. 243-256.

We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements.