Studieninhalte

Students will acquire familiarity with basic mathematical structures, and elementary concepts of mathematical computation.

• Basic mathematical structures:

o Sets, functions, relations

o Real numbers, natural numbers, integers

o Absolute value, minimum and maximum

o Groups and fields

• Arithmetic on a computer:

o Floating Point Representation, Cancellation of Leading Digits and b-adic Representation

o Practical Floating Point systems

o Use of the Python programming language and the NumPy library

o Absolute and relative error, condition number, Landau O-notation

• Fundamental algorithms:

o Polynomial interpolation

o Divided Differences

o Linear Systems and Gaussian Elimination

o LU-Decomposition and Matrix norms

o Symmetry and Cholesky

o Linear Independence and Orthogonality

o Cubic Splines

o Approximation and Normal equations

Students acquire basic competences in discrete structures, probability theory and basic statistics. The topics are

• Logic and sets

• Sorting algorithms and complexity

• Graphs and trees

• Graph colouring and algorithms on graphs

• Probability spaces and probability

• Conditional probability and independency

• Random variables and discrete distributions

• Expectation and variance

• Elementary statistics

The student learns the basics of one-dimensional real analysis.

Content:

• construction of the real numbers,

• sequences (convergence and convergence criteria),

• series (convergence and convergence criteria),

• functions of one real variable,

• continuity, uniform, and Lipschitz continuity,

• complex numbers,

• function types: exponential, logarithm, sine, cosine, …

• differentiability and differentiation, differentiation rules,

• introductory examples for an ordinary differential equation (exp(x)),

• Taylor’s theorem,

• Riemann-Integral, fundamental theorem of calculus,

• function sequences, uniform convergence.

Students acquire basic competences in linear algebra:

• Fields and complex numbers

• Matrices and vectors

• Matrix products

• Subspaces

• Linear independence and bases

• Dimension

• Matrices and linear maps

• Change of basis

• Determinants (characterization, existence, properties, permutations)

• Endomorphisms (trace, eigenvalues and eigenvectors)

• Affine spaces and quotient vector spaces

The student learns how to differentiate and integrate functions in several variables.

Content:

· Normed, metric, and topological spaces,

· convergence in metric spaces,

· closedness, compactness, connectedness,

· differentiability and differentiation of functions in several variables,

· classical differential operators (nabla, rot, curl),

· inverse and implicit function theorem,

· integration on curves, surfaces, and other geometric bodies,

· integral theorems

Students acquire basic competences in linear algebra:

• Symmetry and groups

• Bilinear forms

• Euclidean spaces

• orthonormal bases

• Self adjoint endomorphism

• Quadratic forms

• Unitary spaces

• Jordan normal form

• Duality

The student learns advanced topics of analysis such as measure and integration theory and complex analysis.

Content:

• Rings, sigma-algebras, contents, and measures,

• measurable functions and Lebesgue integration, L^p-spaces,

• convergence theorems (Beppo-Levi theorem, Fatou’s lemma, Lebesgue’s dominated convergence theorem),

• analysis in the complex plane (holomorphic functions, power series expansions, integration along curves),

• Cauchy’s integral theorem and formula,

• maximum principle,

• isolated singularities and meromorphic functions,

• Laurent series expansions,

• residue calculus

This is a first course on probability theory, starting from the basics and going on to the most important and useful topics

in modern probability. Topics to be covered:

• Probability spaces, random variables, expectation

• The classical laws of probability

• Independence and the Borel-Cantelli lemmas

• Sums of independent random variables

• Convergence of random variables

• The law of large numbers

• Convergence in law

• Characteristics functions

• The central limit theorem

• Markov chains

• Random walks

• Recurrence and transience

• Stationary and reversible measures

• Convergence to the stationary measure

• Markov chain Monte Carlo

• Introduction to statistics: estimators, confidence intervals, hypothesis testing

This "Numerics" course focuses on computer-based methods for solving mathematical problems that arise in various scientific and engineering applications. The primary objective is to develop a fundamental understanding of the construction of numerical methods. A significant portion of the course is dedicated to analyzing these methods to determine their usability, limitations, and applicability, with particular emphasis on three key aspects: (1) assessing the accuracy provided by different methods, (2) evaluating their efficiency, and (3) addressing issues of stability. The course covers a range of standard methods commonly used in numerical computation, including:

A) direct solution methods for linear systems

B) Iterative solution methods for linear systems

C) Root finding techniques for nonlinear equations

D) Interpolation and approximation methods for functions

E) Numerical integration and quadrature.

F) Numerical solution for ordinary differential equations.

An essential component of numerical analysis involves the practical implementation of the methods discussed in the course. This hands-on approach allows students to gain firsthand experience with the challenges related to accuracy, computational effort, and stability. The coursework includes computational experiments, which students will conduct using the Octave/MATLAB or Python programming language or another language of their choice.

This course will cover the foundations of abstract algebra, focusing on the notions of group, ring and field. For each object we will present basic concept and some applications. In the case of groups: quotients, isomorphism theorems, the symmetric group, Sylow theorems. For rings: Ideals, polynomial rings, Principal Ideal Domains and some notions of divisibility. For fields: field extensions, splitting fields.

By the end of this module, students will be able to ….

• formulate ODE models of various phenomena,

• appreciate the role and importance of ODEs in science and technology,

• recall the main categories of ODEs, and classify an unfamiliar ODEs according to these categories,

• solve ODEs analytically, in many of the cases where this is possible,

• solve ODEs numerically,

• prove basic facts about ODEs and their solutions,

• recognize when theory guarantees the existence and uniqueness of solutions to an ODE,

• prove the theorems providing these theoretical guarantees,

• explain the main ways in which existence or uniqueness can fail to hold,

• produce creative examples and counterexamples.

This is a first course on functional analysis, which is analysis in infinite dimensions. Contents:

• Review of metric and normed spaces

• Hilbert spaces: definition and basic properties, the Riesz lemma, orthonormal bases

• Banach spaces: examples and basic properties, the Hahn-Banach theorem, dual spaces, the Baire category theorem, closed operators

• Bounded operators: notions of convergence, adjoints, the spectrum, compact operators, spectral theory for compact operators

• Unbounded operators

This course provides an introduction to both the theory of partial differential equations (PDEs) and the numerical methods used for their solution in various scientific and engineering applications. The primary objective is to develop a solid foundation in the formulation and analysis of numerical techniques. A significant focus is placed on understanding the core principles that determine the usability, limitations, and applicability of these methods, with particular emphasis on three key aspects: (1) accuracy, (2) efficiency, and (3) stability.

The course covers a range of standard numerical methods, including:

• Analytical solutions

• Functional analysis and weak formulation

• Finite difference method,

• Finite element method,

• Time-dependent problems.

A practical component is integrated into the course, enabling students to implement and test the numerical methods discussed. Through hands-on computational experiments using MATLAB, students will explore challenges related to accuracy, computational cost, and stability, reinforcing their theoretical understanding with practical insights.

The goal of the seminar is to understand and present an advanced mathematical topic in a mathematical paper and

in an online presentation to an audience unfamiliar with the topic.

Examples of topics:

o Probability theory

o Mathematics of statistical physics

o Machine learning

o Number theory

o Computer formalisation of proofs

o Geometry

o Dynamical systems

o Systems and control theory

o Optimization

o Modeling, model reduction, system identification

o Applied and numerical linear algebra (structured and nonlinear eigenvalue problems, matrix equations and inequalities)

o Differential-algebraic equations

o Optimization

o Numerical Methods for partial differential equations

o Fast iterative solution methods for large scale systems

o Multilevel and domain decomposition methods

o High Performance Computing

o Scientific Machine Learning

o Computational mechanics and fluid structure interaction

o Computational medicine

o Computational geology

(AS26)

This two-semester module consists of a seminar followed by a bachelor thesis. In the guided seminar, students will read, summarize, and present a mathematically advanced paper with the aim to consolidate the essential competence of mathematicians: “Comprehend – Transmit – Write up”. This is the basis for the bachelor thesis.

• Ordinary differential equations

• First order systems, existence, uniqueness

• Applications : prey-predators, population dynamics, finance...

• Resolution of typical cases, linear systems

• Celestial mechanics

• Diffusion and heat equation, weak formulations

• Wave equation

• Elasticity

• Fluid mechanics

• Numerical approximations

Students acquire basic competences in differential geometry:

• Fundamental notions ((co-)tangent space, smooth maps, differential, vector fields, gradient, exterior derivative)

• Curves (unit speed curves, curves in the plane, local & global geometric properties of curves)

• Surfaces (embedded surfaces, tangent planes, orientation, geodesics, covariant derivative, curvature)

• Intrinsic geometry (Gauss-Codazzi equations, curvature tensor & theorema egregium, geodesic curvature, Gauss—Bonnet theorem)

• Further topics (differential forms, hyperbolic geometry, green’s theorem)

Elementary number theory: prime factorisation of integers; congruences and modular arithmetic; applications to cryptography (RSA algorithm). Quadratic residues, Legendre and Jacobi symbols. Proof of quadratic reciprocity via Gauss sums.

Algebraic number theory: Gaussian integers, applications to sums of two squares. Number fields, algebraic integers; unique factorisation of ideals. Pell’s equation and units in real quadratic fields. Finiteness of the ideal class group (statement)

By the end of this module, students will be able to ….

• Solve constrained and unconstrained optimization problems by hand and using numerical optimization algorithms.

• Be able to mathematically analyze important numerical optimization algorithms.

• Have a comprehensive theoretical understanding of the basic framework of supervised machine learning.

• Be able to describe, implement, and in some cases mathematically analyze, several machine learning algorithms for supervised learning, including linear, logistic regression and neural networks.

Students can choose from a catalogue of modules from the study courses of the other Faculties offered by FernUni Schweiz. The available modules from the Faculties of Economics, Psychology, Law and History are published every semester. The modules are given in the language of the study course.