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## Basis-free solution to Sylvester equation in Clifford algebra of arbitrary dimension

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.

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras G_{p,q} of vector space of dimension n = p + q. For the first time, we present explicit formulas for all characteristic polynomial coefficients in the case n = 5. The formulas involve only the operations of geometric product, summation, and operations of conjuga- tion. For the first time, we present an analytic proof of the corresponding formulas in the case n = 4. We present some new properties of the operations of conjugation and grade projection and use them to obtain the main results of this paper. The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented explicit formulas for characteristic polynomial coefficients can also be used in symbolic computation.

9TH INTERNATIONAL CONFERENCE ON MATHEMATICAL MODELING: Dedicated to the 75th Anniversary of Professor V.N. Vragov

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.

We calculate characteristic polynomials of operators explicitly represented as polynomials of rank $1$ operators. Applications of the results obtained include a generalization of the Forman--Kenyon's formula for a determinant of the graph Laplacian and also provide its level $2$ analog involving summation over triangulated nodal surfaces with boundary.

Let (M,I,J,K) be a hyperkahler manifold, and Z⊂(M,I) a complex subvariety in (M,I). We say that Z is trianalytic if it is complex analytic with respect to J and K, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures (M,I,J′,K′) containing I. For a generic complex structure I on M, all complex subvarieties of (M,I) are absolutely trianalytic. It is known that a normalization Z′ of a trianalytic subvariety is smooth; we prove that b2(Z′) is no smaller than b2(M) when M has maximal holonomy (that is, M is IHS). To study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of the hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold X is a k-dimensional space R of closed 2-forms on X which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in R is a smooth, non-degenerate quadric hypersurface in R. We consider absolutely trianalytic tori in a hyperkahler manifold M of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where k=b2(M). We show that the tangent bundle TX of a k-symplectic manifold is a Clifford module over a Clifford algebra Cl(k−1). Then an absolutely trianalytic torus in a hyperkahler manifold M with b2(M)≥2r+1 is at least 2r−1-dimensional.

We discuss some well-known facts about Clifford algebras: matrix representations, Cartan’s periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in <span data-mathml="nn dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.

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.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.