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National Repository of Grey Literature 2 records found  Search took 0.00 seconds. 
The use of invariants of geometric transformations to solving problems
 Doubrava, Jiří ; Zamboj, Michal (advisor) ; Beran, Filip (referee)
This thesis deals with the division of bijective collinear geometric mappings. It deals with a corresponding view on its respective problems and their solutions. The main part of the thesis is a collection of solved problems in the field of plane geometry, which are divided into several groups. The thesis can be used by both mathematics teachers and students at the secondary schools. The thesis is divided into two parts; the first part is theoretical and it contains a brief introduction to the basic concepts concerning invariants of geometric transformations and a description of the structure of geometry according to Felix Klein including his Erlangen program; the second part contains the actual assignments and so- lutions of the problems. The solutions of some of the problems are supplemented by im- ages created in GeoGebra software for better illustration. The images play an important role even in the theoretical part, where they can help improve understanding of some of the more complex concepts. Some of the problems were created by the author, other prob- lems are taken from sources listed in the list of bibliography. KEYWORDS geometric transformations, groups, Erlangen program, invariant, solved problems
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Úvahy o vnitřních symetriích teorie pravděpodobnosti a možné roli Kleinovy kvartiky v základech fyziky
 Gottvald, Aleš
Probability theory features as the internal symmetries of physical laws, acting in an intrinsically 6-dimensional hyperspace. Concerning symmetries, classical thermodynamics and Klein's Erlangen program involve the same underlying idea. Probability theory is an exceptional structure, closely linked to a unique Triality symmetry and other exceptional structures in nature (symmetric group S6, Platonic solids, (2, 3, 7)-triangular group and a correspondin tessellation of a hyperbolic space, exceptional Lie groups, etc.). Exponential mapping of statistical physics is associated with Klein's quartic curve, an extremal Hurwitz surface whose 168 automorphisms may be related to Standard model of particle physics and to a highly composite number (42) of special importance for fundamental physics.
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