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Course: Methods of Mathematical Physics

Department/Abbreviation: OPT/PGSMF

Year: 2020

Guarantee: 'prof. Mgr. Jaromír Fiurášek, Ph.D.'

Annotation: Advanced postgraduate course on methods of mathematical physics. Students will learn in detail selected advanced topics in mathematics which include functional analysis, calculus of variations, partial differential equations and integral equations, special functions and the representation theory of transformation groups, and stochastic processes.

Course review:
-Functional analysis: Topology; topological linear spaces; general theorems on linear operators; spectral analysis of linear operators; spectral analysis in the Hilbert space; integration and linear functionals; spaces of generalized functions (D', S') -Calculus of variations: The simplest problem of calculus of variations, classification of extrema, variation of a functional, Euler's equation, Legendre's condition; generalizations of the simplest equation, problems with variable end points, piecewise smooth solutions; constraint extremum and isoperimetric problem, variational problems in the parametric form; field of extremals, sufficient conditions of strong and weak extremum; linear variational problems, invariant variational problems, the first theorem of E. Noether -Partial differential equations and integral equations: Linear partial differential equations of the first order and their systems; linear partial differential equations of the second order, classification, fundamental solutions, classical and generalized initial-value problem, wave equation, heat equation, Helmholtz equation; boundary-value problems for the second order partial differential equations of elliptic type; mixed problems for the second order partial differential equations of hyperbolic and parabolic types; linear integral equations, classification, Fredholm theory, Hilbert-Schmidt theorem -Special functions and theory of representations of transformation groups: Basic notions of the representation theory; transformation groups and their representations; representations of compact and locally compact groups; particular transformation groups, corresponding special functions, functional relations (summation formulae, multiplication formulae, recurrent relations, differential equations, generating functions) -Stochastic processes: Kolmogorov's probability space, expectation value, convergences of random variables; point processes, distribution function, correlation functions; stochastic processes, hierarchy of probability densities, branching processes; Markovian processes, Markovian chains; master equation; one-step processes; Fokker-Planck equation, Langevin's equations; stochastic differential equations