Ein Jahr an der University of Hertfordshire

Für Studenten, die das 6. Semester erfolgreich beendet haben, besteht die Möglichkeit, zwei Semester an der
University of Hertfordshire in Hatfield (ca. 1/2 Std. Bahnfahrt nördlich von London) zu studieren und dort die Diplomarbeit anzufertigen.
Nach der Rückkehr an die TFH kann dann die Diplomprüfung abgelegt werden, da die dort erbrachten Studienleistungen hier anerkannt werden.

Liste äquivalenter Lehrveranstaltungen der Studiengänge Mathematik für Studentenaustausch

Gegenübergestellt werden

- die Hauptfächer im 7. und 8. Semester
- die Fächer des 7. Semesters im Schwerpunkt Mathematik und Technik und
- die Fächer des 7. Semesters im Schwerpunkt Wirtschaftsmathematik und Statstik

an der TFH

den jeweiligen Level 3 - Jahreskursen der UH.

Legende

( 4 + 2 ) bei TFH bedeutet:    4 Std. Vorlesung
                                           + 2 Std. Übung;

( 2P ) bei UH bedeutet:   2 Zeit-Module Pflicht und
( 2 WP ) entsprechend     2 Zeit-Module Wahlpflicht;

4 + 2 für ein Semester an der TFH entspricht ungefähr 1  1/4 Zeit-Modulen für ein Jahr an der UH.

Hauptfächer
 

TFH Berlin	UH Hatfield
7. Semester	Level 3 - Jahreskurse
Numerische Mathematik 3 ( 2+ 2 )	Numerical Algorithms ( 2P )
Diplomanden-Seminar ( 2 )	Project ( 4 P )
8. Semester
Diplomarbeit

Schwerpunkt Mathematik und Technik
 

TFH Berlin	UH Hatfield
7. Semester	Level 3 - Jahreskurse
Methoden der Finiten Elemente 2 ( 2 + 2 )	PDE Models ( 2P )
Variationsmethoden ( 2 + 2 )
Mathematische Physik
Wahlpflichtfach ( 2 + 2 )	Auswahl ( 2WP ) aus folgenden Fächern: - Statistics - Digital Control - Non-linear Systems - Differential Geometry & Matrices - Linear Optimization - Modelling & Simulation

Schwerpunkt Wirtschaftsmathematik und Statistik
 

TFH Berlin	UH Hatfield
7. Semester	Level 3 - Jahreskurse
Statistik 3 ( 4 + 2 )	Statistics ( 2P )
Statistisches WP ( 2 + 2 )
Wirtschaftsmathematisches WP ( 2 + 2 )	Auswahl ( 2WP ) aus folgenden Fächern: - PDE Models - Digital Control - Non-linear Systems - Differential Geometry & Matrices - Linear Optimization - Modelling & Simulation

- Course aims and objectives

At the end of the course the student will:

a) be able to design, analyse and implement algorithms for a wide range of mathematical problems.
b) be able to evaluate algorithms with regard to their ability and complexity.
c) be able to select appropriate techniques, algorithms and data structures for efficient and
    reliable solution of particular numerical problems.

1) Matrix factorisation:
The QU decomposition using Gram-Schmidt, Givens and Householder techniques;
operation counts and error analysis. Solution of simultaneous linear equations, including overdetermined
systems. Singular value decomposition : reduction of bi-diagonal forms, calculation of singular value and vectors;
application to least squares approximation.
2) Solution of linear simultaneous equations:
conjugate direction and implicit Cholesky iterative schemes;
accuracy in computed solutions using iterative refinement or interval arithmetic;
revision of Kulisch operators especially accurate scalar products; contraction and fixed point theorems.
3) Approximation of matrix eigenvalues:
Gershgorin discs, Wayland-Hopfman theorem, the trace and Frobenius norm.
4) Matrix eigenvalue calculations:
Jacobi methods, reduction to tridiagonal form, Sturm sequences and QR method for symmetric matrices;
the nonsymmetric matrix - complex eigenvalues, repeated eigenvalues, incomplete eigenspaces,
the effect on the power method; QR method for Hessenberg matrices.
5) Nonlinear optimization:
the unconstrained problem using Newton, quasi-Newton, conjugate gradient and truncated Newton methods;
related methods for solving simultaneous nonlinear equations and least squares problems;
convergence and termination properties. The equality constrained problem using penalty function and
sequential quadratic programming techniques; extensions for inequality constraints.
6) Numerical solution of ordinary differential equations:
Multistep methods, e.g. Milne, Adams-Moulton etc.; preditor-corrector strategies; error estimates;
stability; methods for stiff problems.

- Course aims and objectives

At the end of the course the student will:

(i)    recognise the partial diffrential equations associated with diffusion and wave motion
(ii)   be able to classify partial differential equations in 2, 3 and 4 independent variables
(iii)  recognise properly-posed problems for boundary and initial value problems
(iv)  be able to use the method of characteristics for the solution of hyberpolic equations
(v)   be able to use finite difference and finite element methods for the numerical solution of
        initial and boundary value problems of parabolic and elliptic types

Modelling

Development of PDE models, with particular reference to diffusion and wave motion, in physics, engineering
and biology
e.g.  First order hyperbolic systems - traffic flow, conservation laws.
       Second order parabolic systems - mass diffusion, heat flow, random walks, lumped-parameter models
       Second order hyperbolic systems - transverse waves (strings and membranes), longitudinal waves
       (elastic rods and sound waves).
       Elliptic systems - degenerate parabolic and potential problems.

Theoretical Background

Classification of quasi-linear second order PDE's in 2, 3 and 4 independent variables and systems of first order equations.
Characteristics and canonical forms. Boundary and initial-value problems. Properly-posed problems. Adjoint operators.
Existence theorems (without proofs).
Associated eigenvalue problems.
Analytic solutions will be developed only to the extent that this aids insight into the underlying physical processes.

Computation

Finite difference methods for elliptic and parabolic equations.Stability, convergence and consistency.
The method of characteristics for hyperbolic equations.

The finite element method and the boundary element method developed as members of the class of weighted residual methods
for the approximate solution of linear and non-linear boundary and initial value problems.

- Cource aims and objectives

At the end of the course the student will:

- be able to perform initial data analysis in a variety of situations,
- understand the basic principles of statistical inference together with some basic methods
- appreciate the usefulness of computer packages in statistical analyses
- have had experience of undertaking some simple statistical investigations.

- Course content

Initial data analysis	- types of data, scales summarisations and   displays, ideas of robustness
Probability	- concepts, formal framework   definitions and basic rules   conditional probability and independence   Bayes Theorem   Expectation
Probability models	- random variables, pdf, cdf, moments   mean and variance of sums and products   specific models to include binomial, Poissons,   normal and exponential   Hypergeometric with   relevance to finite populations
Sampling distributions	- concept, empirical examination of examples,   eg sample mean, median, proportion   statement of central unit theorem
Bivariate distributions	- approached empirically through discrete models,   conditional and marginal distributions
Computer software	- use of basic statistical package, eg MINITAB
Inference	- basic principles of point and interval estimation   basic principles of hypothesis testing-types of   hypothesis and error   comon sampling distributions - t, X², F   application to following:   mean of normal, variance known and unknown   variance of normal   difference of means of two normals, paired   and unpaired   ratio of variances of two normals   proportion based on large samples   contingency tables   goodness of fit
Bivariate distributions	- method of least squares applied to linear   regression with single regressor correlation
Non-parametric methods	- meaning and need for methods   simple examples eg sign and run tests