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Rabin-Miller test and the choice of a basis
Franců, Martin ; Simon, Petr (advisor) ; Čunát, Vladimír (referee)
This thesis is dedicated to various choices of basis in Rabin-Miller test. Short overview of similar methods is shown and some properties of structure of the set of strong liars are proved in theoretical part. Selected innovative choices of basis are tested on the set of odd composite numbers in range of 100 and 200 000 000 and the results are compared to results of tests with usual choices of bases. Hypothesis about possible improvement of test through using basis of special form with regard to tested number is proposed. Program used for compu- tations of these results is included. The program allows user to compare results of tests with various ways of choosing basis. The second part of the thesis contains documentation of the program.
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Approximation of a non-increasing rearrangement of a function
Franců, Martin ; Pick, Luboš (advisor) ; Felcman, Jiří (referee)
The non-increasing rearrangement of a measurable real function defined on an appropriate measure space is of the enormous significance in disciplines such as theory of function spaces or interpolation theory and their applications in PDEs. Unfortunately, while it has good and widely applicable mapping properties, it is virtually impossible to calculate the non-increasing rearrangement of a concrete given function precisely. Numerical algorithms for approximation are desirable for this reason. Such method of approximation, based on interpolation by a linear spline, is presented in this thesis. In the first half of this thesis, the developed method is described, while the error estimates of the method are subject to the second part.
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Isoperimetric problem, Sobolev spaces and the Heisenberg group
Franců, Martin ; Pick, Luboš (advisor) ; Cianchi, Andrea (referee) ; Nekvinda, Aleš (referee)
In this thesis we study embeddings of spaces of functions defined on Carnot- Carathéodory spaces. Main results of this work consist of conditions for Sobolev- type embeddings of higher order between rearrangement-invariant spaces. In a special case when the underlying measure space is the so-called X-PS domain in the Heisenberg group we obtain full characterization of a Sobolev embedding. The next set of main results concerns compactness of the above-mentioned em- beddings. In these cases we obtain sufficient conditions. We apply the general results to important particular examples of function spaces. In the final part of the thesis we present a new algorithm for approximation of the least concave majorant of a function defined on an interval complemented with the estimate of the error of such approximation. 1
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Isoperimetric problem, Sobolev spaces and the Heisenberg group
Franců, Martin ; Pick, Luboš (advisor) ; Cianchi, Andrea (referee) ; Nekvinda, Aleš (referee)
In this thesis we study embeddings of spaces of functions defined on Carnot- Carathéodory spaces. Main results of this work consist of conditions for Sobolev- type embeddings of higher order between rearrangement-invariant spaces. In a special case when the underlying measure space is the so-called X-PS domain in the Heisenberg group we obtain full characterization of a Sobolev embedding. The next set of main results concerns compactness of the above-mentioned em- beddings. In these cases we obtain sufficient conditions. We apply the general results to important particular examples of function spaces. In the final part of the thesis we present a new algorithm for approximation of the least concave majorant of a function defined on an interval complemented with the estimate of the error of such approximation. 1
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Rabin-Miller test and the choice of a basis
Franců, Martin ; Simon, Petr (advisor) ; Čunát, Vladimír (referee)
This thesis is dedicated to various choices of basis in Rabin-Miller test. Short overview of similar methods is shown and some properties of structure of the set of strong liars are proved in theoretical part. Selected innovative choices of basis are tested on the set of odd composite numbers in range of 100 and 200 000 000 and the results are compared to results of tests with usual choices of bases. Hypothesis about possible improvement of test through using basis of special form with regard to tested number is proposed. Program used for compu- tations of these results is included. The program allows user to compare results of tests with various ways of choosing basis. The second part of the thesis contains documentation of the program.
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|
|
Approximation of a non-increasing rearrangement of a function
Franců, Martin ; Pick, Luboš (advisor) ; Felcman, Jiří (referee)
The non-increasing rearrangement of a measurable real function defined on an appropriate measure space is of the enormous significance in disciplines such as theory of function spaces or interpolation theory and their applications in PDEs. Unfortunately, while it has good and widely applicable mapping properties, it is virtually impossible to calculate the non-increasing rearrangement of a concrete given function precisely. Numerical algorithms for approximation are desirable for this reason. Such method of approximation, based on interpolation by a linear spline, is presented in this thesis. In the first half of this thesis, the developed method is described, while the error estimates of the method are subject to the second part.
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