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On the Friedmann equation for the three-dimensional hypersphere
Křížek, Michal ; Mészáros, A.
The present standard cosmological model of the evolution of our universe, is based on the Friedmann equation, which was published by Alexander Friedmann in 1922. He applied Einstein’s equations to an expanding threedimensional sphere which enabled him to avoid boundary conditions. However, his description was very brief. Therefore, the main objective of this article is to detailed a derivation of the Friedmann equation for an unknown expansion function a = a(t) representing the radius of the universe. Furthermore, we present serious arguments showing why the validity of Einstein’s equations should not be extrapolated to the entire universe.
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Anthropic principle and the local Hubble expansion
Křížek, Michal ; Somer, L.
We claim that a local expansion of the universe, whose rate is comparable with the Hubble constant H0, had an essential influence on the development of intelligent life on the Earth. We present more than 10 examples showing that some antigravitational effects of the cosmological constant are observable locally in the Solar system. It is known that the luminosity of the Sun increased approximately linearly within the last 4.5Gyr starting at 70% of its present value. We give several independent arguments showing that the average Earth-Sun distance increases about 5m/yr due to antigravitational forces and such a large recession speed cannot be explained by solar wind, tidal forces, plasma outbursts from the Sun, or by the decrease of the Solar mass due to nuclear reactions. Models based on Newtonian mechanics can explain only a few cm per year.
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Proceedings of the International Conference Applications of Mathematics 2015 : Prague, November 18-21, 2015
Brandts, J. ; Korotov, S. ; Křížek, Michal ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
Professors Ivo Babuška, Milan Práger, and Emil Vitásek are renowned experts in numerical analysis and computational methods. Their fruitful scientific careers started in Prague, at the Institute of Mathematics of the Czechoslovak Academy of Sciences (now Czech Academy of Sciences). They collaborated there on various projects including the computational analysis of the construction technology for Orlík Dam. In 1966 they published their joint book entitled Numerical Processes in Differential Equations. It is an honor for the Institute of Mathematics to host a conference on the occasion of their birthdays.
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Why quintic polynomial equations are not solvable in radicals
Křížek, Michal ; Somer, L.
We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed bz radicals, i.e., by the operations +, -, ., :, and .... Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.
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On simplicial red refinement in three and higher dimensions
Korotov, S. ; Křížek, Michal
We show that in dimensions higher than two, the popular “red refinement” tech- nique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one.
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Deset vět o pražském orloji
Křížek, Michal ; Šolcová, A. ; Somer, L.
We present ten geometrical and algebraical theorems that have a close connection with the astronomical clock (horologe) of Prague. In particular, we found that there is a remarkable relationship between the triangular numbers and the bellworks of the astronomical clock.
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