Number of found documents: 386
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Badatelské úlohy ve vyučování geometrii
Roubíček, Filip
2015 - Czech
Podnětem badatelských aktivit ve vyučování geometrii je geometrická úloha, která popisuje neznánou, neurčitou nebo nejednoznačnou situaci a požaduje její komplexní prozkoumání. V příspěvku je popsáno šest badatelských úloh z různých oblastí elementární geometrie. The stimulus of inquire activities in the teaching of geometry is a geometrical problem which describes an unknown, undefined or ambiguous situation and requires its complexe exploration. Six inquiry-stimulating problems of various areas of elementary geometry are described in the paper. Keywords: inquiry-based teaching; inquiry activities; geometrical problem Available at various institutes of the ASCR
Badatelské úlohy ve vyučování geometrii

Podnětem badatelských aktivit ve vyučování geometrii je geometrická úloha, která popisuje neznánou, neurčitou nebo nejednoznačnou situaci a požaduje její komplexní prozkoumání. V příspěvku je popsáno ...

Roubíček, Filip
Matematický ústav, 2015

Separable reductions and rich families in theory of Fréchet subdifferentials
Fabian, Marián
2015 - English
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. Keywords: Fréchet subdifferentials Available on request at various institutes of the ASCR
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; ...

Fabian, Marián
Matematický ústav, 2015

Why quintic polynomial equations are not solvable in radicals
Křížek, Michal; Somer, L.
2015 - English
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. Keywords: Galois theory; finite group; permutation Available in digital repository of the ASCR
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 ...

Křížek, Michal; Somer, L.
Matematický ústav, 2015

Different approaches to interface weights in the BDDC method in 3D
Čertíková, M.; Šístek, Jakub; Burda, P.
2015 - English
In this paper, we discuss the choice of weights in averaging of local (subdomain) solutions on the interface for the BDDC method (Balancing Domain Decomposition by Constraints). We try to find relations among different choices of the interface weights and compare them numerically on model problems of the Poisson equation and linear elasticity in 3D. Problems with jumps in coefficients of material properties are considered and both regular and irregular interfaces between subdomains are tested. Keywords: BDDC; interface weights; coefficient jumps Available in digital repository of the ASCR
Different approaches to interface weights in the BDDC method in 3D

In this paper, we discuss the choice of weights in averaging of local (subdomain) solutions on the interface for the BDDC method (Balancing Domain Decomposition by Constraints). We try to find ...

Čertíková, M.; Šístek, Jakub; Burda, P.
Matematický ústav, 2015

Smooth approximation spaces based on a periodic system
Segeth, Karel
2015 - English
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. Keywords: smooth interpolation; data interpolation; cubic spline interpolation Available in digital repository of the ASCR
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 ...

Segeth, Karel
Matematický ústav, 2015

On the number of stationary patterns in reaction-diffusion systems
Rybář, Vojtěch; Vejchodský, Tomáš
2015 - English
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. Keywords: diffusion driven instability; Turing patterns; classification of non-unique solutions Available in digital repository of the ASCR
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 ...

Rybář, Vojtěch; Vejchodský, Tomáš
Matematický ústav, 2015

On the quality of local flux reconstructions for guaranteed error bounds
Vejchodský, Tomáš
2015 - English
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. Keywords: a posteriori error estimates; complementarity; index of effectivity; elliptic problem Fulltext is available at external website.
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, ...

Vejchodský, Tomáš
Matematický ústav, 2015

Irregularity of turing patterns in the Thomas model with a unilateral term
Rybář, Vojtěch; Vejchodský, Tomáš
2015 - English
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. Keywords: reactin-diffusion system; Thomas model; Turing instability Available in digital repository of the ASCR
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 ...

Rybář, Vojtěch; Vejchodský, Tomáš
Matematický ústav, 2015

A note on tension spline
Segeth, Karel
2015 - English
Spline theory is mainly grounded on two approaches: the algebraic one (where splines are understood as piecewise smooth functions) and the variational one (where splines are obtained via minimization of quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the well known tension spline (called also spline in tension or spline with tension). We present the results of a 1D numerical example that show the advantages and drawbacks of the tension spline. Keywords: smooth interpolation; tension spline; Fourier transform Available in digital repository of the ASCR
A note on tension spline

Spline theory is mainly grounded on two approaches: the algebraic one (where splines are understood as piecewise smooth functions) and the variational one (where splines are obtained via minimization ...

Segeth, Karel
Matematický ústav, 2015

A parallel finite element solver for unsteady incompressible Navier-Stokes equations
Šístek, Jakub
2015 - English
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. Keywords: pressure-correction; parallel algorithms; flow around a sphere Fulltext is available at external website.
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 ...

Šístek, Jakub
Matematický ústav, 2015

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