The next meeting of the Joint Seminar of Analysis, Geometry and Topology Department will be held on September 23, 2014 at 14:00 in Room 478 of IMI.

A talk on:

„On an Extension of Harmonicity and Holomorphy”

will be delivered by Prof. Julian Ławrynowicz from University of Łódź, Poland.

Everybody is invited.

Abstract.

The concept of harmonicity and holomorphy related with the Laplace equation Δs ≡ (∂²/∂x²) + (∂²/∂y²) = 0, (x, y) є R², is extended with the use of equation

(∂/∂t)s = - Γs* + Λ(Δ +Δ_τ)s
with
Δ + Δ₁ = (∂²/∂x²) – a²(∂²/∂θ²), Δ₂ = – a²(∂²/∂θ²), Δ₃ = (∂²/∂z²) – a²(∂²/∂θ²),

Δ₄ = (∂²/∂z²) + (∂²/∂ξ²) – a²(∂²/∂θ²), Δ₅ = (∂²/∂z²) + (∂²/∂ξ²) + (∂²/∂η²) – a²(∂²/∂θ²),

where Γ and Λ are C¹-scalar functions of (x, θ) є R², …, (x, y, z, ξ, η, θ) є R⁶ for τ = 1, …,5, respectively, t є R, θ є R and x* is an arbitrary admissible function. We discuss the fundamental solutions for the equations in question (more precisely, of the corresponding linearized problem) which is a parabolic equation of the second kind. For effective solutions and τ ≡ 1, 2, 3, 4 (mod 8) it is convenient to involve the quaternionic structure, for τ ≡ 5, 6, 7, 0 (mod 8) – the paraquaternionic structure. Physically, it is natural to describe, with help of the equation involved, relaxation processes attaching (x, y, z) to the first chosen particle, (ξ, η, ζ) – to the second one, θ to temperature, entropy or order parameter, and t – to time.