Ústav matematiky a deskriptivní geometrie

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    Use of cohesive approaches for modelling critical states in fibre-reinforced structural materials
    (MDPI, 2024-07-01) Kozák, Vladislav; Vala, Jiří
    During the operation of structures, stress and deformation fields occur inside the materials used, which often ends in fatal damage of the entire structure. Therefore, the modelling of this damage, including the possible formation and growth of cracks, is at the forefront of numerical and applied mathematics. The finite element method (FEM) and its modification will allow us to predict the behaviour of these structural materials. Furthermore, some practical applications based on cohesive approach are tested. The main effort is devoted to composites with fibres and searching for procedures for their accurate modelling, mainly in the area where damage can be expected to occur. The use of the cohesive approach of elements that represent the physical nature of energy release in front of the crack front has proven to be promising not only in the direct use of cohesive elements, but also in combination with modified methods of standard finite elements.
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    Rigidity of Holomorphically Projective Mappings of Kähler Spaces with Finite Complete Geodesics
    (MDPI, 2024-04-19) Vítková, Lenka; Hinterleitner, Irena; Mikeš, Josef
    In this work, we consider holomorphically projective mappings of (pseudo-) K & auml;hler spaces. We determine the conditions for finite complete geodesics that must be satisfied for the mappings to be trivial; i.e., these spaces are rigid.
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    Large time behavior of nonautonomous linear differential equations with Kirchhoff coefficients
    (Elsevier, 2024-03-31) Diblík, Josef; Pituk, Mihaly; Szederkényi, Gábor
    Nonautonomous linear ordinary differential equations with Kirchhoff coefficients are considered. Under appropriate assumptions on the topology of the directed graphs of the coefficients, it is shown that if the Perron vectors of the coefficients are slowly varying at infinity, then every solution is asymptotic to a constant multiple of the Perron vectors at infinity. Our results improve and generalize some recent convergence theorems.
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    Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
    (De Gruyter, 2024-02-05) Diblík, Josef; Růžičková, Miroslava
    A singular nonlinear differential equation z(sigma) dw/dz = aw + zwf(z , w), where sigma > 1, is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C if sigma is a natural number, in a Riemann surface of a rational function if sigma is a rational number, or in the Riemann surface of logarithmic function if sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a is an element of C { 0 } a, and that the function f f is analytic in a neighbourhood of the origin in C x C . Considering sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w = w (z ) w=w(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in C or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z -> 0 w (z) = 0 is proved and an asymptotic behaviour of w (z) s established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.
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    Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
    (De Gruyter, 2023-10-06) Diblík, Josef; Korobko, Evgeniya
    The article investigates a second-order nonlinear difference equation of Emden-Fowler type. New conditions with respect to parameters of equation are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation. Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward diffrences as well. Previously known results are discussed, and illustrative examples are considered.