Mathematics IV

Enrollment into the second academic year.

The prerequisite for the theoretical exam is a positive result of the written exam.

Homework may also be one of obligations.

Simple boundary problems for the DE y''+ky=0.
Partial differential equations (PDE), the Fourier method (separation of variables): heat flow (diffusion) PDE , Laplace's equation, the wave equation.

Fourier transforms: L_1 space, inverse transforms, convolution. Examples from mathematical physics.

Holomorphic functions, the Cauchy-Riemann equations, contour integrals, Cauchy's formula for the disk, power series expansions, analytic functions. Logarithm, winding number, general Cauchy's formula. Zeros of analytic functions. Liouville's theorem. Laurent series, isolated singularities, the Residue theorem.

Moebius transformations. Harmonic functions:
Poisson's formula for the disk. The Gamma function in C.

Solving linear second order DE using power series, the Method of Frobenius. Bessel's DE and Bessel functions.

Classifying linear second order PDEs. Wave equation – the d'Alembert solution.

Linear second order differential operators (DO): the Sturm–Liouville problem. Bessel's differential operator. Legendre polynomials, the Associated Legendre functions, spherical harmonics.
Green's Second and Third identity.

E. Zakrajšek, Analiza III, Matematični rokopisi 21, DMFA-založništvo, Ljubljana 2002.
Suhadolc, Metrični prostor, Hilbertov prostor, Fourierova analiza, Laplaceova transformacija, Matematični rokopisi 23,  DMFA, Ljubljana, 1998.

Večina snovi je v (most of the course material is in):
W. Kaplan, Advanced Calculus, Addison-Wesley, Boston 2003.

Pri sestavljanju predavanj so bile uporabljene naslednje knjige (These books were used in compiling the course):

M. Braun, Differential Equations and Their Applications, 4th ed. Applied mathematical sciences 15, Springer-Verlag, New York 1993.
V. A. Zorich, Mathematical Analysis II, Universitext, Springer Verlag, Berlin Heidelberg 2004.
K. Jaenich, Analysis fuer Physiker und Ingenieure, Funktionentheorie, Differentialgleichungen, Spezielle funktionen, 3. Aufl., Springer Lehrbuch, Springer-Verlag, Berlin Heidelberg 1995.
S. Hassani, Mathematical Physics, A Modern Introduction to its Foundations, Springer-Verlag, New York 1999. (V poštev pride le majhen del te obsežne knjige – we need just a fraction of this book.)
Tai L. Chow: Mathematical Methods for Physicists: A Concise Introduction , Cambridge University Press, Cambridge 2000.
D. Bak, D. J. Newman, Complex analysis, Undergraduate texts in Mathematics, 3rd ed., Springer-Verlag, New York 2010.
Priročnik (Handbook):
E. Kreyszig, Advanced Engineering Mathematics, 10th ed., Wiley, New York 2011.
Naloge (problems and solved problems):

B. Hvala, Zbirka izpitnih nalog iz analize z namigi, nasveti in rezultati, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, 2000.
M. Dobovišek, Rešene naloge iz Analize II, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, 2001.
J. Cimprič, Rešene naloge iz Analize III, Izbrana poglavja iz matematike in računalništva, DMFA-založništvo, Ljubljana, 2001.
M. Spiegel: Schaum's Outline of Advanced Mathematics for Engineers and Scientists
(Schaum's Outline Series), McGraw-Hill, New York 2009.

Students learn advanced topics in Analysis: partial differential equations (PDE), Fourier transforms, analytic functions, Bessel functions, orthogonal polynomials.

Knowledge and understanding:
We expect that students know important definitions and theorems, understand (and ideally be able to replicate) at least the easier proofs, and be able to apply this knowledge , e.g. in mathematical physics

Application:
The material of this course is indispensable for Mechanics, Mathematical physics and other courses in the program.

Reflection:
Students master some advanced topics in Mathematical Analysis and are able to apply them in physics.

Transferable skills:
Students learn to understand the usefulness of the abstract approach, are able to connect the acquired knowledge with what they already mastered. They also learn to use other written sources and the internet. They are able to identify and solve problems, hand in homework on time, and memorize the important topics.

Lectures, tutorials, homework (optional).

Written exam or 2 midterm exams instead of the written exam
Oral exam or theoretical test, homework
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

prof. dr. P. Legiša:

• P. Legiša, Adjacency preserving mappings on real symmetric matrices. Math.
  commun., Croat. Math. Soc., Divis. Osijek, 2011, vol. 16, no. 2, 419-432.

• P. Legiša, Automorphisms of Mn, partially ordered by the star order. Linear
  multilinear algebra, 2006, vol. 54, no. 3, 157-188.

• P. Legiša, Automorphisms of Mn, partially ordered by rank subtractivity
  ordering. Linear algebra appl. 2004, vol. 389, 147-158.
  prof. dr. B. Magajna:

• B. Magajna, Sums of products of positive operators and spectra of Lüders
  operators, Proc. Amer. Math. Soc. 141 (2013) 1349-1360.

• B. Magajna, Fixed points of normal completely positive maps on B(H), J. Math.
  Anal. Appl. 389 (2012) 1291-1302.
  81

• B. Magajna, The Haagerup norm on the tensor product of operator modules,
  J. Funct. Anal. 129 (1995) 325-348.
  prof. dr. J. Mrčun:

• I. Moerdijk, J. Mrčun: On the developability of Lie subalgebroids. Adv. Math.
  210 (2007), 1-21.

• J. Mrčun: On isomorphisms of algebras of smooth functions. Proc. Amer. Math.
  Soc. 133 (2005), 3109-3113.

• I. Moerdijk, J. Mrčun: On integrability of infinitesimal actions. Amer. J. Math.
  124 (2002), 567-593.