Topics in mathematics

Rings, examples and basic properties, invertible elements and zero divisors, Wedderburn theorem, characteristics, homomorphisms, ideals (principal, prime and maximal ideals), quotient ring, isomorphism theorems.

Fields, constructible numbers, field extensions, splitting fields, finite fields.

Fourier series, expansion as sum of sines and cosines, complex expansion, convergence, phase portrait, applications (heat/wave equation, signal analysis,…).

Fourier series, existence, basic properties (linearity, translation, shift, differentiation,…), inverse transform, convolution. Discrete Fourier transform, fast Fourier transform, applications.

Other mathematical topics of lecturer’s choosing.

E. Kreyszig: Advanced Engineering Mathematics, 9th ed., Weiley (2006)
S. Coutinho: The Mathematics of Ciphers: Number Theory and RSA Cryptography, A.K. Peters ltd.,1999
T. W. Judson: Abstract Algebra Theory and Applications (2013), http://abstract.ups.edu/

Students learn about concepts and techniques in Algebra (rings and fields) and Mathematical Analysis (Fourier series and transform) that have a wide specter of applications in Computer Science (coding, cryptohgraphy, signal analysis, data compression, image processing,…). Students familiarise with selected mathematical problems and gain basic problem-solving skills.

Knowledge and understanding of basic definitions, theorems and of selected applications.
Application: course content is essential for the understanding of the applications of mathematical methods in Computer Science.
Reflection: Understanding the theory and its relation to the applications.
Transferable skills: familarity with the the literature and other sources, ability to identify and solve the problems, critical analysis.

Lecture and exercises.

Written exam with oral defense

Petar Pavešić:
– PAVEŠIĆ, Petar CONNER, Gregory R., HERFORT, Wolfgang, KENT, Curtis, PAVEŠIĆ, Petar. Recognizing the second derived subgroup of free groups. Journal of algebra, ISSN 0021-8693, Dec. 2018, vol. 516, str. 396-400.

– PAVEŠIĆ, Petar A topologist's view of kinematic maps and manipulation complexity. V: GRANT, Mark (ur.). Topological complexity and related topics : Mini-Workshop Topological Complexity and Related Topics, February 28 - March 5, 2016, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany, (Contemporary mathematics, ISSN 0271-4132, 702). Providence: American Mathematical Society.

– PAVEŠIĆ, Petar. Complexity of the forward kinematic map. Mechanism and Machine Theory, ISSN 0094-114X. [Print ed.], 2017, vol. 117, str. 230-243.

– PAVEŠIĆ, Petar. Splošna topologija, (Izbrana poglavja iz matematike in računalništva, 43). Ljubljana: DMFA - založništvo, 2008. VI, 89 str., ilustr. ISBN 978-961-212-205-8 [COBISS-SI-ID 240425984]

Primož Potočnik:
– POTOČNIK, Primož. Edge-colourings of cubic graphs admitting a solvable vertex-transitive group of automorphisms. Journal of combinatorial theory. Series B, ISSN 0095-8956, 2004, vol. 91, no. 2, str. 289-300 [COBISS-SI-ID 13087321]
– POTOČNIK, Primož, SPIGA, Pablo, VERRET, Gabriel. Cubic vertex-transitive graphs on up to 1280 vertices. Journal of symbolic computation, ISSN 0747-7171, 2013, vol. 50, str. 465-477 [COBISS-SI-ID 16520537]
– POTOČNIK, Primož. Tetravalent arc-transitive locally-Klein graphs with long consistent cycles. European journal of combinatorics, ISSN 0195-6698, 2014, vol. 36, str. 270-281 [COBISS-SI-ID 16862041]