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Smooth approximation spaces based on a periodic system
Segeth, Karel
2015 - anglický
A way of data approximation called smooth was introduced by Talmi and Gilat in 1977. Such an approach employs a (possibly infinite) linear combination of smooth basis functions with coefficients obtained as the unique solution of a minimization problem. While the minimization guarantees the smoothness of the approximant and its derivatives, the constraints represent the interpolating or smoothing conditions at nodes. In the contribution, a special attention is paid to the periodic basis system $exp(-ii kx)$. A 1D numerical example is presented.
Klíčová slova:
smooth interpolation; data interpolation; cubic spline interpolation
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
Smooth approximation spaces based on a periodic system
A way of data approximation called smooth was introduced by Talmi and Gilat in 1977. Such an approach employs a (possibly infinite) linear combination of smooth basis functions with coefficients ...
Why quintic polynomial equations are not solvable in radicals
Křížek, Michal; Somer, L.
2015 - anglický
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.
Klíčová slova:
Galois theory; finite group; permutation
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
Why quintic polynomial equations are not solvable in radicals
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 ...
Separable reductions and rich families in theory of Fréchet subdifferentials
Fabian, Marián
2015 - anglický
We consider important properties of Fréchet subdifferentials, in particular: the non-emptiness of subdifferentials, the non-zeroness of normal cones, the fuzzy calculus, and the extremal principle; all statements being considered in Fréchet sense. Given a nonseparable Banach space X, we show how the validity of these statements is implied by the validity of them in every separable subspace of X. Such a reasoning is called “separable reduction”. We show that, behind this approach, there is a modern and powerful concept of rich subfamily of the family of all separable subspaces of X.
Klíčová slova:
Fréchet subdifferentials
Plné texty jsou dostupné na vyžádání prostřednictvím repozitáře Akademie věd.
Separable reductions and rich families in theory of Fréchet subdifferentials
We consider important properties of Fréchet subdifferentials, in particular: the non-emptiness of subdifferentials, the non-zeroness of normal cones, the fuzzy calculus, and the extremal principle; ...
Irregularity of turing patterns in the Thomas model with a unilateral term
Rybář, Vojtěch; Vejchodský, Tomáš
2015 - anglický
In this contribution we add a unilateral term to the Thomas model and investigate the resulting Turing patterns. We show that the unilateral term yields nonsymmetric and irregular patterns. This contrasts with the approximately symmetric and regular patterns of the classical Thomas model. In addition, the unilateral term yields Turing patterns even for smaller ratio of diffusion constants. These conclusions accord with the recent findings about the influence of the unilateral term in a model for mammalian coat patterns. This indicates that the observed effects of the unilateral term are general and apply to a variety of systems.
Klíčová slova:
reactin-diffusion system; Thomas model; Turing instability
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
Irregularity of turing patterns in the Thomas model with a unilateral term
In this contribution we add a unilateral term to the Thomas model and investigate the resulting Turing patterns. We show that the unilateral term yields nonsymmetric and irregular patterns. This ...
On the number of stationary patterns in reaction-diffusion systems
Rybář, Vojtěch; Vejchodský, Tomáš
2015 - anglický
We study systems of two nonlinear reaction-diffusion partial differential equations undergoing diffusion driven instability. Such systems may have spatially inhomogeneous stationary solutions called Turing patterns. These solutions are typically non-unique and it is not clear how many of them exists. Since there are no analytical results available, we look for the number of distinct stationary solutions numerically. As a typical example, we investigate the reaction-diffusion systém designed to model coat patterns in leopard and jaguar.
Klíčová slova:
diffusion driven instability; Turing patterns; classification of non-unique solutions
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
On the number of stationary patterns in reaction-diffusion systems
We study systems of two nonlinear reaction-diffusion partial differential equations undergoing diffusion driven instability. Such systems may have spatially inhomogeneous stationary solutions called ...
On the quality of local flux reconstructions for guaranteed error bounds
Vejchodský, Tomáš
2015 - anglický
In this contribution we consider elliptic problems of a reaction-diffucion type discretized by the finite element method and study the quality of guaranteed upper bounds of the error. In particular, we concentrate on complementary error bounds whose values are determined by suitable flux reconstructions. We present numerical experiments comparing the performance of the local flux reconstruction of Ainsworth and Vejchodský [2] and the reconstruction of Braess and Schröberl [5]. We evaluate the efficiency of these flux reconstructions by their comparison with the optimal flux reconstruction computed as a global minimization problem.
Klíčová slova:
a posteriori error estimates; complementarity; index of effectivity; elliptic problem
Dokument je dostupný na externích webových stránkách.
On the quality of local flux reconstructions for guaranteed error bounds
In this contribution we consider elliptic problems of a reaction-diffucion type discretized by the finite element method and study the quality of guaranteed upper bounds of the error. In particular, ...
Talk about patterns in the mathematics classroom
Roubíček, Filip
2015 - anglický
The poster deals with the talk in the mathematics classroom which is focused on looking for relationships in a pattern. The communication of students is observed in the environment of geometrical patterns in a triangle grid and their transformation into arithmetic patterns or algebraic functions. It shows how pupils/students reason about relationships in these patterns and among these patterns, how they describe and express their generalizations in words or symbols.
Klíčová slova:
discourse; mathematics classroom; patterns
Plné texty jsou dostupné na vyžádání prostřednictvím repozitáře Akademie věd.
Talk about patterns in the mathematics classroom
The poster deals with the talk in the mathematics classroom which is focused on looking for relationships in a pattern. The communication of students is observed in the environment of geometrical ...
How primary school teachers posed problems for inquiry based mathematics education
Hošpesová, A.; Samková, L.; Tichá, Marie; Roubíček, Filip
2015 - anglický
The paper reports on a survey whose aim is to contribute to the research in the area of inquiry based mathematics education (IBME), primary school teacher training, and problem posing. The core of the study is an empirical survey with a qualitative design. We let pre-service and in-service primary school teachers pose problems which they believe enable IBME, perform the problems with pupils, and write self-reflections on this activity. We investigated the nature of the posed problems in relation to the self-reflections.
Klíčová slova:
inquiry based mathematics education; posing problems; primary school teacher training; pre-service and in-service teachers
Plné texty jsou dostupné na jednotlivých ústavech Akademie věd ČR.
How primary school teachers posed problems for inquiry based mathematics education
The paper reports on a survey whose aim is to contribute to the research in the area of inquiry based mathematics education (IBME), primary school teacher training, and problem posing. The core of the ...
A parallel finite element solver for unsteady incompressible Navier-Stokes equations
Šístek, Jakub
2015 - anglický
A parallel solver for unsteady incompressible Navier-Stokes equations is presented. It is based on the finite element method combined with the pressure-correction approach. Semi-implicit treatment of the convective term is considered, leading to five systems of linear algebraic equations to be solved in each time step. Krylov subspace iterative methods are employed for the solution of these systems with a particular emphasis on efficient parallel preconditioners. A simulation of a benchmark problem of incompressible viscous flow around a sphere at Reynolds number 300 is presented and compared with literature.
Klíčová slova:
pressure-correction; parallel algorithms; flow around a sphere
Dokument je dostupný na externích webových stránkách.
A parallel finite element solver for unsteady incompressible Navier-Stokes equations
A parallel solver for unsteady incompressible Navier-Stokes equations is presented. It is based on the finite element method combined with the pressure-correction approach. Semi-implicit treatment of ...
An application of the BDDC method to the Navier-Stokes equations in 3-D cavity
Hanek, M.; Šístek, Jakub; Burda, P.
2015 - anglický
We deal with numerical simulation of incompressible flow governed by the Navier-Stokes equations. The problem is discretised using the finite element method, and the arising system of nonlinear equations is solved by Picard iteration. We explore the applicability of the Balancing Domain Decomposition by Constraints (BDDC) method to nonsymmetric problems arising from such linearisation. One step of BDDC is applied as the preconditioner for the stabilized variant of the biconjugate gradient (BiCGstab) method. We present results for a 3-D cavity problem computed on 32 cores of a parallel supercomputer.
Klíčová slova:
incompressible fluid; Navier-Stoken equations; BDDC; Taylor-Hood finite element
Plné texty jsou dostupné v digitálním repozitáři Akademie Věd.
An application of the BDDC method to the Navier-Stokes equations in 3-D cavity
We deal with numerical simulation of incompressible flow governed by the Navier-Stokes equations. The problem is discretised using the finite element method, and the arising system of nonlinear ...
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