# Institute of Information Engineering, Automation, and Mathematics

Project number:
SK-CZ-0098-07
Title of the project:
Sk-Cz cooperation: Number theory and its applications
Project type:
Cooperations
Project duration (start):
00.00.2008
Project duration (end):
00.00.2009

Title: Number theory and its applications
Number: APVV SK-CZ-0098-07
Period: 2008-2009
Principal investigator: Doc. RNDr. Oto Strauch, DrSc
Scientific co-workers: Doc. RNDr. Vladimír Baláž, PhD
Partners: SAV Bratislava, Ostravska Univerzita, Akadémia vied ČR

Anotation:

Development of Number Theory and its applications in the following particular fields: Dynamic systems and applications in uniform distribution, densities for natural numbers, uniform distribution of sequences, quantitative measure for distribution of nets, random numbers generation, quasi Monte Carlo methods and distribution functions.

Keywords:

sequences, dynamic systems, densities, quasi–Monte Carlo methods, uniform distribution, distribution function, random numbers generation

Goals of project:
The aim of the project in the narrow international cooperation is to find new results and their applications in the following fields of Number theory:

1. Theory of generation of pseudorandom numbers, 2. Theory of densities for natural numbers. Both of these theories are connected by means of Theory of uniformly distributed sequences which definition arises from the notion of density.

One of many applications is numerical integration by means of quasi–Monte Carlo method. Further, we precisely specify our aims. A random numbers is a physical process of measurement in which any further measurement is statistically independent. Generator of pseudorandom numbers is exactly defined algorithm by means of which we compute the sequence so called pseudorandom number sequence, thus it is deterministic and moreover it satisfies some properties which are not fixed determined. There is no general definition of pseudorandomnes. The pseudorandom number sequence can be infinite (i. e. theoretically, the computation is unbounded) or finite (i. e. after finite number of steps it starts to repeat). For an infinite sequence in the interval we require the sequence together with the sequence of couples, triples, etc. to be uniformly distributed, i.e. to be so-called completely uniformly distributed. (D.E. Knuth condition for pseudorandomnes). Further, there are requirements for its discrepancy, we require the sequence to be so-called low discrepancy sequence.

For discrete sequence Ch. Mauduit a A. Sárkózy (1997) introduce new measures of pseudorandomnes, so-called well-distribution measure a correlation measure. The most important and known pseudorandom sequences are: finite pseudorandom sequences – linear congruential sequences, (n,m,s)–nets, LFSR sequences, quadratic congruential sequences, power congruential sequences (especially Blum–Blum–Shub generator), inverse congruential sequences, binary sequences e.g. Champernowne sequence, Thue–Morse sequence, Rudin–Shapiro sequence etc). Combining these sequences, our first problem is to find new pseudorandom generators. The second problem is to find new tests for pseudorandomnes and compare their effect with classical statistical tests. Specially, we will investigate the combination of van der Corput sequence and a quadratic generator. Further by means of theory of dynamic systems we will investigate sequences generated by the function sun on digits, which contains weighted coefficients. Using theory of dynamic systems for this sequence we will construct dynamical system, which will be ergodic, biparametric hence we obtain uniform distribution of such sequence. We will also study relations between maximum on digits function and counting function (both are increasing) it goes out from empirical observation, that if one of them is increasing more quickly the other one more slowly and vice versa. Further task is concerning of block sequences, which provide one of the main tools for the construction the sequences with prescribed distribution properties. There were find some criteria on uniform distribution and also basic properties of the set of all distribution functions of a special block sequence xn = (x1/xn,x2/xn,....xn/xn). The problem is to find further properties of the set of all distribution functions and further criteria of everywhere density of xm/xn.

2. The density on the set of natural numbers is every nonnegative measure. The basic density is so called the asymptotic density further we have Buck density, logarithmic density weight density, density with respect to summation methods, specially, densities with respect to matrix summation methods. There is also known Abel density, zeta density, Schnirelman density. Using these densities, it is possible to define different types of uniform distributions. By means of density we can define also convergence. This is a special case of the so-called I–convergence with respect to ideal I of null sets. In this way we can define the statistical convergence, the uniform convergence, Iu–convergence with respect to a density u. Further types of convergences are almost convergence, strong p–Ceasaro convergence, φ–convergence and so on. Between sets of all convergent sequences are studied some inclusion and Baire category. Our aim is to find new relations among uniformly distributed sequences with respect to different densities and also to find new relations among different types of sets of convergent sequences and find new properties of arithmetic and multiplicative functions for instance Ω, ω, ordp(n) and others.