Yakov Krasnov

Quadratic divisors of harmonic polynomials inR n

Necessary and sufficient conditions are given for a quadratic polynomial to be a divisor of a nonzero harmonic polynomial inRn.

Complex Structures in Real Algebras. I. Two-Dimensional Commutative Case

Abstract We study complex structures in real two-dimensional commutative algebras and show their connection with homotopy properties of the multiplications. An application to the Riccati equation in rank three algebras is also discussed.

Operator Method for Solution of PDEs Based on Their Symmetries

We touch upon “operator analytic function theory” as the solution of frequent classes of the partial differential equations (PDEs).

Symmetries of cauchy—riemann—fueter equation

The objective of the present paper is to study the structure of the solution of the Cauchy-Riemann-Fueter equation and to show a technique of the symmetry operators for constructing all their solutions.

Differential Equations in Algebras

The aim of this work is to investigate how topological and dynamical properties of differential equations (in the sequel DE) are reflected in the associated algebras, as well as to show how basic algebraic concepts provide valuable insights in DE.

Idempotent Geometry in Generic Algebras

Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang SprßigTrends Math, Birkhäuser/Springer Basel AG, Basel, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.

Spectral Properties of Differential Equations in Clifford Algebras

The aim of this work is to establish a number of elementary properties about the topology and algebra of real quadratic homogeneous mappings and Riccati type ODEs occurring in Clifford Algebras. We construct an example of the quadratic vector field to show the impact of the spectral properties on qualitative theory.

Complex structures in algebra, topology and differential equations

Abstract In this paper, we present our recent results on the so-called complex structures in algebras in the context relevant to the following four problems which, at first glance, have nothing in common: (i) solubility of polynomial equations in non-associative algebras, (ii) topological/geometric properties of quadratic maps, (iii) existence of bounded solutions to quadratic differential systems, and (iv) ellipticity of the Dirac equation. The unsolved problems related to (i)–(iv) are formulated as well.

ANALYTIC FUNCTION THEORY IN ALGEBRAS

Abstract We study the widest possible generalizations of the complex analytic function theory agreeing with the Riemann and Weierstrass approaches.