|
|
Riemann zeta function
Čoupek, Petr ; Rokyta, Mirko (advisor) ; Zahradník, Miloš (referee)
Riemann zeta function represents an important tool in analytical number theory with various applications in quantum mechanics, probability theory and statistics. First introduced by Bernhard Riemann in 1859, zeta function is a central object of many outstanding problems. From previous results follows the importance of zeta function for further development in the field of number theory. This thesis provides basic properties of the Riemann zeta function. In particular, we prove theorems concerning the distribution of its roots outside and inside the critical strip which leads to the formulation of the Riemann hypothesis and theorems concerning the irrationality of selected values of the Riemann zeta function including the proof of the irrationality of ζ(3). 1
|
|
|
Gradientní modely
Bernát, Marek ; Kotecký, Roman (advisor) ; Zahradník, Miloš (referee)
We have investigated gradient models, one of them was a model with double-well potential and the other one a so called extended model. In dimension two we have calculated exact free energies of the disseminated edge configurations for the extended model and for arbitrary dimension we have derived bounds on these free energies. Combining these bounds with an argument on exstince of bad contours together with the estimate of the number of these contours and using the method of reflection positivity we have been able to show that at low temperatures there is a phase transition in the extended model. We have further shown that the phase transition exists also in the double-well model as long as a conjecture on estimates of mean energy holds. Besides these results the thesis also contains basic tools of statistical physics and facts from related fields, as well as basic results on gradient models, so that our work can serve as an introduction into these areas.
|
|
|
Alternative mathematical notation and its applications in calculus
Marian, Jakub ; Pick, Luboš (advisor) ; Zahradník, Miloš (referee)
We explore the possibility of formalizing classical notions in calculus without using the notion of variable. We provide a new mathematical 'language' capable of performing all classical computations (namely computing limits, finite differences, one-dimensional derivatives, and indefinite and definite integrals) without any need to introduce a variable. Equations written using our notation contain only func- tion symbols (and as such are completely rigorous and don't leave any room for vague interpretations). They also tend to be much shorter and more mathemati- cally transparent than their traditional counterparts (for example, there is no need for introduction of new symbols in integration, and definite integration is formalized in such a way that all rules (including 'substitution' rules) for indefinite integration translate directly to definite integration). We also fully formalize the Landau little-o notation in a way that makes computation of limits using it fully rigorous. 1
|
|
| |
|
| |
|
| |
|
| |
|
|
Lebesgue density theorem for Haar measure
Sterzik, Marek ; Zahradník, Miloš (referee) ; Simon, Petr (advisor)
In this work, we study Lebesgue theorem analogy in the space 2k with Haar measure and a related theorem about -k-linkedness of the measure algebra of this space. The whole text is divided in three chapters. In the first chapter we explain some important definitions and basic properties of the measure space. The Lebesgue theorem is studied in the second chapter. After the essential definition of the point of density, the major part of the chapter is dedicated to the proof of the theorem. The theorem states, that the symmetric difference between any measurable set and the set of its points of density has measure zero. In the third chapter we study the -k-linkedness theorem; a theorem which states that the measure algebra of the space 2 is -k-linked, if 2 .
|
|
|
Exponential function and Mayer expansion
Nagy, Oliver ; Zahradník, Miloš (advisor) ; Loebl, Martin (referee)
Title: Exponential function and Mayer expansion Author: Oliver Nagy Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Miloš Zahradník, CSc., Department of Mathematical Analysis Abstract: The unifying topic of this thesis is cluster expansion in statistical phy- sics. It is divided into three chapters. In the first one we present the necessary mathematical apparatus - selected topics from combinatorics, graph theory and theory of generating functions. The second one is an introduction to cluster expan- sion and abstract polymer model. Finally, in the third chapter we show a new resummation method for partition function of hard-core repulsive abstract poly- mer model. In this resummation we make use of cancellations of terms in partition function to rewrite the sum of clusters to a sum of quilted clusters, or alternati- vely as a sum of "bunches". The methods we use in this final chapter are original and may lead to some new results. Keywords: binomial and multinomial formula; power series; inclusion-exclusion principle; cluster expansion. iii
|
guest ::
