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Mathematical Analysis and Numerical Computation of Volume-Constrained Evolutionary Problems Involving Free Boundaries
Švadlenka, Karel ; Feistauer, Miloslav (advisor) ; Křížek, Michal (referee) ; Knobloch, Petr (referee)
The object of study of the present thesis are evolutionary problems satisfying volume preservation condition, i.e., problems whose solution have a constant value of the integral of their graph. In particular, the following types of problems with volume constraint are dealt with: parabolic problem (heat-type), hyperbolic problem (wave-type), parabolic free-boundary problem (heat-type with obstacle) and hyperbolic free-boundary problem (degenerate wave-type with obstacle). The key points are design of equations, proof of existence of weak solutions to them and development of numerical methods and algorithms for such problems. The main tool in both the theoretical analysis and the numerical computation is the discrete Morse flow, a variational method consisting in discretizing time and stating a minimization problem on each time-level. The volume constraint appears in the equation as a nonlocal nonlinear Lagrange multiplier but it can be handled elegantly in discrete Morse flow method by restraining the set of admissible functions for minimization. The theory is illustrated with results of numerical experiments.
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Cosmology on Small Scales 2016: Local Hubble Expansion and Selected Controversies in Cosmology
Křížek, Michal ; Dumin, Y. V.
The problem of cosmological expansion on small scales (e.g., inside planetary systems) has a long history, dating back to the papers (McVittie, 1932, 1933), and quite many researchers dealt with this topic in the subsequent few decades. Although most of them concluded that the Hubble expansion should be strongly suppressed at small distances, there is no commonly-established criterion for such suppression at present. Moreover, some of the widely-used theoretical arguments against the local Hubble expansion contradict each other.
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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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