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The course treats: Basic properties of Fourier series of functions in one variable. Exponential form and trigonometric form. Riemann-Lebesgue lemma. Elementary conditions for pointwise and uniform convergence. Gibbs phenomenon. Basic…

The course treats: Number theory: divisibility, prime numbers, the Euclidean algorithm, Chinese remainder theorem, modular arithmetic Sets, functions and relations, equivalence relations Combinatorics: the four cases of counting with…

The aim of the course is to give necessary knowledge of digital image analysis for further research within the area and to be able to use digital image analysis within other research areas such as computer graphics, image coding, video…

The course treats basic theory of analytic functions: Elementary properties of analytic functions in one variable. Complex differentiability and Cauchy-Riemann equations. Calculation rules. Elementary examples of analytic functions:…

The course treats: Existence and uniqueness of solutions to initial value problems for systems of ordinary differential equations. Approximation of solutions and continuous dependence on parameters. Linear systems with variable and…

The course treats variational problems with and without constraints, Euler equations with and without constraints, Legendre, Jacobi and Weierstrass necessary condition for local minimum, Hilbert invariant integral and Weierstrass…

The course treats: Number theory: the fundamental theorem of arithmetic, modular artithmetic. Groups: definition, basic examples of groups, subgroups, normal subgroups, factor groups, isomorphisms and homomorphisms, Lagrange's theorem,…

The course treats multiplicative number theoretic functions, Möbius inversion formula, properties of Euler's totient function, primitive roots and indices, quadratic residues, the Legendre symbol and its properties, the quadratic…

The course treats: quadratic forms and matrix factorisation, convexity, separating planes and Farkas’ Lemma, the theory of optimization with and without constraints, Lagrange functions, Karush-Kuhn-Tucker theory, duality, methods for…

The course comprises: Matrices and determinants. Linear spaces. Spectral theory.The Jordan normal form. Matrix factorizations. Matrix polynomials and matrix functions. Norms. Scalar products. Singular values. Normal matrices…

The degree project requires a literature survey and specialised studies. Furthermore, a number of compulsory activities are included, in the form of teaching sessions and seminars that treat e.g. scientific writing in English and Swedish,…

The degree project requires a literature review and special studies. In addition, it includes a number of compulsory elements, in the form of lessons and seminars, which deal with, among other things, scientific writing in English and…

A central part of the course is existence- and uniqueness theorems about measures defined on sigma-algebras, integration theory, conditional expectation and weak convergence in metric spaces.

The course treats: Convergence of sequences of analytic functions. Locally uniform convergence, normal families, Montel’s theorem. In-depth study of conformal mappings. Riemann sphere, stereographic projection, Möbius…

The course treats the geometry of curves and surfaces, especially in three dimensions. In particular, we study the concepts of curvature and torsion. The course covers: The geometry of curves in Euclidean space, their curvature and…

The course treats: Finitely generated groups: free groups, Nielsen-Schreier Theorem, group presentations, finitely presented groups. Free products: free products with amalgamation, HNN extensions. Solvable groups: polycyclic groups,…

The course treats: topological spaces and metric spaces with examples, product topologies, continuity of functions, connectedness, completeness, and compactness, including the Arzela-Ascoli Theorem, examples of applications and topological…

The course treats: Fourier series, Fourier transform and finite Fourier transform, L^2 convergence of Fourier series, pointwise convergence, Cesàro means and Fejer’s theorem, Weyl’s criterion, the Fourier inversion theorem, Parseval’s and…

This course treats the general notion of a measure defined on a sigma-algebra, construction of measures with help of outer measures, in particular the Lebesgue measure in Rd. These concepts are then used to define the integral of a…

The course treats: Linear systems of ordinary differential equations in the complex plane. Power series solutions. Singular points. Frobenius method. Some classical differential equations and special functions. Boundary value…