The aim of Computational mathematics course is to teach students how to use computational (both numerical and symbolic) methods in applications coming from various branches of mathematics.
The course covers the following subjects:
1. Polynomial algorithms: square-free factorization, polynomial factorization over finite fields, factorization of rational polynomials, monomial orders and Groebner bases;
2. Elimination theory: elimination with Groebner bases, classical elimination with resultants;
3. Inifinite summation and Gosper's algorithm;
4. Numerical integration: Monte-Carlo algorithm.
Basic linear algebra is enough. A bit of number theory can be useful but not necessary.
My aim is to present mathematical methods for quantum information processing. As in most applications it is enough to work with qubits and systems of qubits, mathematical methods originate from linear algebra, which is usually one of first curses taught. It makes quantum information accessible for very 'fresh' students. I would like to convince students that quantum information processing is useful, interesting, counter-intuitive, sometimes seemingly as mysterious as the Schroedinger cat.
Mathematical formalism of quantum mechanics.
Postulates of quantum mechanics.
Quantum information: quantum gates, no-go theorems, measurement.
Quantum entanglement: mathematical basis.
Selected applications: teleportation, dense coding.
Quantum cloning and applications.
Basic protocols for quantum cryptography: BB84, B92.
Quantum nonlocality: Bell and Leggett-Garg inequalities, contextuality.
Dynamics of quantum systems, open quantum systems.
Quantum error correction.
Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang
Lecture notes by John Preskill http://www.theory.caltech.edu/people/preskill/ph229/
The aim of the Statistics unit is to get a deep knowledge on constructing statistical models and making statistical analysis, and to improve the skills of using statistical computer packages.
The contents of this unit are the following:
1. Organising statistical analysis: collecting and data, their analysis and graphical description.
2. Linear and non-linear statistical models – estimation theory and statistical hypotheses testing.
3. Applications of linear and non-linear statistical models in econometrics and financial mathematics.
4. Parametric tests of significance involving two or more samples.
5. Conformity tests.
6. Non-parametric tests of significance involving two or more samples.
7. Applications of statistical computer software to estimation and statistical testing
The main goal of the lecture is to present basic properties of wavelet transforms and some methods of construction of wavelet bases. We will pay special attention to these wavelet transforms which have used to the analysis and the synthesis of sound signals. We also will pay special attention to structures of bases with special properties which have used to the data compression in digital transmissions.
 C.K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992.
 I. Daubechies, The wavelet transform, Time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961-1005.
 I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.
 C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628-666.
 G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston, 1994.
 D. Kozlow, Wavelets. A tutorial and a bibliography, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste, 26, supplemento (1994).
Applications in Geometry:
1. Joint characterization of Euclidean, hyperbolic and elliptic geometries.
2. Characterizations of the cross ratio.
3. A description of certain subsemigroups of some Lie groups.
Applications in Functional Analysis:
1. Analytic form of linear-multiplicative functionals in the Banach algebra of integrable functions on the real line.
2. A characterization of strictly convex spaces.
3. Some new characterizations of inner product spaces.
4. Birkhoff-James orthogonality.
5. Addition theorems in Banach algebras; operator semigroups.
1. J. Aczel & J. Dhombres, Functional equations in several variables, Cambridge University Press, Cambridge, 1989. 2. J. Aczel & S. Gołąb, Funktionalgleichungen der Theorie der Geometrischen Objekte, PWN, Warszawa, 1960. 3. J. Dhombres, Some aspects of functional equations, Chulalongkorn Univ., Bangkok, 1979. 4. D. Ilse, I. Lehman and W. Schulz, Gruppoide und Funktionalgleichungen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1984. 5. M. Kuczma, An introduction to the theory of functional equations and inequalities, Polish Scientific Publishers & Silesian University, Warszawa-Kraków-Katowice, 1985.
In this module the students, divided into teams consisting of several people, implement projects associated with the given problem.
The project consists of several phases:
1. Planning for the project. The allocation of roles and responsibilities in the team.
2. Review of available literature on the given matter.
3. Analysis of the problem, seeking methods of its solution.
4. Implementation of the solution. This phase, depending on the project, should include elements such as the analysis of empirical data, calibration, simulation and testing of the solution.
5. Preparation of the final report and presentation of results. Both the final effect and the individual phases of the project are assessed. Laboratory classes serve to current reporting and discussing work progress, and give the opportunity of obtaining assistance in the project implementation.
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