Projects

Projects: S8302 - S8305 - S8307 - S8308 -S8309 -S8310 - S8311 - S8312

Project: S8302 - Statistical Properties of Number Systems 

The general aim is to investigate the statistical properties of special functionals of general number systems and related problems in order to get a better insight to number theoretical algorithms and to specific random number generators respectively low discrepancy sequences. In particular it is planned to consider the following topics:
    1. Correlations between digits
    2. The sum-of-digits function and prime numbers
    3. Digital expansions and random sequences
 
 

Project:  S8305 - Analysis of Digital Point Sets 

In this project three groups of topics concerning distribution properties of digital (t,m,s)-nets and digital (T,s)-sequence will be studied:

1. Estimation of the L₂-(and related) discrepancy of digital point sets (especially of the Sobol-Faure-Niederreiter net).
2. Give sharp estimates for the discrepancies of  "small" digital point sets
3. Investigation of the following topics:

    a) Generalize the method of Schmid and Wolf to obtain lower bounds for the quality parameter t of digital nets
    b) Provide a theory of "shift nets"
    c) Establish a theory of a new, more sensible quality parameter for nets
    d) Determine classes of regions of bounded remainder for digital (T,s)-sequences
    e) Give a metrical  lower bound for the discrepancy of (T,s)-sequences
 

Project: S8307 - Algorithmic Diophantine Problems 

An essential part of this project is devoted to the effective solution of Dophantine equations from a theoretical as well as from an algorithmic and computational point of view. It will be dealt with specific Diophantine equations related to combinatorial questions, to classical orthogonal polynomials, Thue equations and index form equations. Within the project also a new method for the construction of elliptic curves of high rank  based on Diophantine m-tuples will be explored. The results on polynomials which are necessary for the investigation of Diophantine equations are also a useful tool for exploring the distribution properties of polynomial and linear recurring sequences of algebraic integers modula a prime ideal. These and related questions are:

1. Combinatorial and polynomial Diophantine Equations
2. Algorithmic Solution of Families of Thue Equations, Index Form Equations, and Thue Inequalities
3. Integral Points on Elliptic  Curves
4. Distribution of Arithmetic Functions and Digital Expansions
 

Project: S8308 - Quasi-Monte Carlo Methods in Finance and Insurance 

 1.   In a first part it is intended to develop new simulation techniques based on quasi-Monte Carlo methods and to apply them to problems that arise in insurance and financial mathematics. It is also intended to generalize and improve some of the underlying models in risk theory. It is aimed to improve the numerical tools for solving intergral equations and functional differential equations using quasi-Monte Carlo methods.

2.  In a second part it should be analyzed why quasi-Monte Carlo methods work in extremely high dimensions and it is intended to use the results of this analysis for effective multi-dimensional integration. In this connection especially the problem of distribution of "small" point sets in high dimensions and the concepts of weighted discrepancy and weighted function spaces have to be studied.

Project: S8309 - Combinatorial Tools for the Analysis of Number Theoretical Algorithms

Project:  S8310 - Quasi-Random Points: Theory and Software Development  

Project:  S8311 - Topological Methods in Algorithmic Number Theory and Diophantine Analysis 

Project: S8312 - Number Theoretic Methods in Cryptography and Pseudorandum Number Generation 
 
 

aktualisiert am 17.12.2002