FB04 Mathematik

Lehrinhalte
Normed vector spaces, completion; Theorem of Hahn-Banach, Theorem of Banach-Steinhaus, Open Mapping Theorem, Closed Graph Theorem; Hilbert spaces; reflexive spaces, weak convergence; Sobolev spaces, weak solution of the Dirichlet problem; spectral properties of linear operators; compact operators on Banach spaces, spectral theorem for compact operators.

Literatur
Alt: Lineare Funktionalanalysis;
Conway: A Course in Functional Analysis;
Reed, Simon: Functional Analysis: Methods of Modern Mathematical Physics I;
Rudin: Functional Analysis;
Werner: Funktionalanalysis;
Ciarlet: Functional Analysis;

Voraussetzungen
recommended: Analysis, Integration Theory, Complex Analysis, Linear Algebra or comparable prerequisites acquired in mathematics courses in engineering programmes

Online-Angebote
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Lehrinhalte
The aim of the lecture is to develop of a mathematical model, which includes both classical probability theory and quantum mechanics.

Central topics:
Random Variables and Observables
Classical and non-commutative probability spaces
Finite-dimensional operator algebras
Composed systems
Transition probabilities and completely positive operators
Examples of quantum Markov processes
Markov processes as couplings to white noise
A look back at classical Markov processes and coding theory
Some ergodic theory of stationary processes
Essentially commutative Markov processes, Levy-Chintchin-Formula and a theorem of Hunt.

Literatur
For Operator Algebas: Some chapters from, e.g.,
G. J. Murphy: C*-Algebras and Operator Theory
M. Takesaki: Theory of Operator Algebras I

For Probability: Parts of
A.N. Shiryaev:  Probability, 
oder some other book on probability and stochastic processes.

Further literature will be announced in the lecture.


 

Voraussetzungen
Functional Analysis and introductions to algebra and probability. First encounters with quantum mechanics are helpful but not necessary.

Official Course Description
See above

Zusätzliche Informationen
Further current information can be found at moodle

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