DOLNÍK, M. Lerchova věta v teorii časových škál a její důsledky pro zlomkový kalkulus [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2017.

Posudek vedoucího

Kisela, Tomáš

The goals of the thesis were two: to formulate and prove Lerch's Theorem for the nabla Laplace transform (nLT) on time scales and to discuss its consequences for introduction of fractional calculus to the time scales theory. Both the goals were achieved, in particular the one concerning Lerch's Theorem in significant manner. The thesis suffers from many shortcomings in the field of typography, stylistics, spelling and logical organization from the viewpoint of scientific work. However, these aspects are outweighed by exceptional results (wrt usual level of master's theses) achieved mostly by independent work with very little need for supervision. Authors are often tempted by both time scales theory and fractional calculus to straightforwardly accept results known from ordinary calculus which might easily lead to false conlusions. One of such false conclusions is exposed in this thesis. The key part is Sec. 3.4.1 where several examples of time scales with non-unique inverse nLT are presented including particular calculations. This actually breaks the implicit assumption of uniqueness of inverse nLT which is utilized in research papers published by several authors even in impacted journals. Moreover, Sec. 3.4.2 and 3.4.3 outline few classes of time scales where uniqueness of the inverse nLT is proved. The results of this thesis will be, after proper adjustments, submitted for publication in some mathematical journal.

Posudek oponenta

Nechvátal, Luděk

The topic of the thesis belongs to the area of time scale calculus – a relatively modern and growing field of mathematics unifying continuous and discrete calculus. Despite many achievements in this theory during the last two decades, well-defined analogues of some crucial notions known from classical calculus are still missing. For instance, a power function on arbitrary time scales belongs among such open problems. The power function plays a crucial role in fractional calculus – another rapidly developing theory. It seems that the Laplace transform is a key tool to obtain a “right” definition of the mentioned power function. The presented work focuses on definition of the Laplace transform on a general time scale and especially on its uniqueness (the student follows and extends his work in the bachelor thesis). The text is written in (rather bad) English (systematic use of incorrect terms, misprints, incorrect grammar). Overall, a formal aspect of the work is not a strong point. I also have noticed inaccuracies in the mathematical formulae (e.g., in Definitions 3.1.3 and 3.2.1; when talked about a domain of convergence in Examples 3.2.1 and below, one should say “a necessary and sufficient condition”; sometimes the argument of a function is used and sometimes is not in the same formulae; and others). On the other hand, I have to appreciate that the thesis contains some original and nontrivial results, I believe that after some refinements these could be published in an appropriate scientific journal. I would say that the thesis concerns unnecessarily too many topics, sometimes it is difficult to stay concentrated on what is important. The goals of the thesis have been fulfilled. I recommend the thesis to be defended and classify it by degree C (good).

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