Поредното заседание на Общия семинар на секция "Анализ, геометрия и топология" ще се проведе на 23 септември 2014 г. от 14:00 часа в зала 478 на ИМИ.

Доклад на тема:

„On an Extension of Harmonicity and Holomorphy”

ще изнесе Prof. Julian Ławrynowicz, University of Łódź, Poland.

Поканват се всички интересуващи се.

Резюме:

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.