This dissertation is devoted to the study of the properties of monogenic (continuously differentiable and differentiable in the corresponding sense) functions with values in hypercomplex systems, which are commutative and non-commutative (in particular, the Clifford algebra) finite-dimensional algebras over field K (of real R or complex C numbers). The application of these properties to find solutions of partial differential equations (PDEs) and linear systems of PDEs, by using respective algebraic and analytical methods is considered. In addition, the thesis also studies polynomial equations in the Segre algebra B_8 (R). We studied the basic properties of a probability measure P_(W_4 ) with values in algebra bihyberbolic numbers W_4 and a measure ω with values in algebra quaternions H.
In Chapter 1 we review results of previous researches related to the main topic of the thesis, providing theoretical information and algebraic constructions that are used.
In Chapter 2: a method for finding solutions of polynomial equations with coefficients taking values in the Segre algebra B_8 (R), which is a real eight-dimensional representation of the Segre algebra of complex quaternion B_4 (C), is developed; by using of the algebraic and analytical method for case of commutative algebras, a formula of the generalized density function f(t,x) of the distribution of random one-dimensional motion x(t) satisfying the sixth-order PDE (generalized telegraphic equation) is found, partial solutions of the fourth-order PDE (the so-called generalized biwave equation) were studied, the application of this method to of find of partial solutions of linear systems of PDEs was shown; by using of the algebraic and analytical method in case of non-commutative algebras, the expansion of the monogenic function f(∙) (continuously differentiable and left-differentiable in the sense of eigenvectors of the generalized Cauchy-Riemann operator D, i. e. Df(∙)=0) with values in the Clifford algebra 〖Cl〗_(p,q)^R (p+q=d+1) generated by (d+1)-dimensional linear space E^(d+1),d=0,1,…, over the field R, into a series of Fueter-type polynomials, was found; examples of applications of an expansion 〖Cl〗_(p,q)^R-valued function in the series to finding partial solutions of second-order PDEs, are shown.
In Chapter 3: the analogue of the classical real-valued probability measure P in the case where this measure takes values in the algebra of bihyperbolic numbers W_4, was studied; the basic properties of the bihyperbolic-valued probability measure P_(W_4 ) and the bihyperbolic-valued random variable X_(W_4 ), are studied; a classical real-valued measure μ, is generalized out to the case of the so-called the quaternion-valued measure ω where a measure takes values in the quaternion algebra H; the basic properties of the quaternion-valued measure ω, are studied.
