We briefly summarize existing result in theory of minimizers of elliptic variational functionals. We introduce proof of existence and regularity such functional under assumpti- ons of quaziconvexity and izotrophic growth estimates, and discuss possible generalization to anizotropic case. Our proof is a compilation from more sources, modified in order of simplicity, readability and detailed analysis of all steps.

In my thesis I concern with the use of motoric activities in math classes. The chapters that offer a range of motoric activities were processed on the base of a questionnaire and an experiment. Some of the activities are currently being implemented in math classes by teachers of first, second and third grades at primary schools, others are part of textbooks designed for this age group. In addition to that, I offer five more possible activities, which have been recorded and interpreted within six experiments. The goal of the thesis is to demonstrate that physical movement cannot be separated from the life of six to nine years old children, and offer some motoric activities, which can potentially be contributive for development of mathematical skills and abilities.

In the present work we study the existence a properties of solution of the system of partial differential equations describing steady flow of Newtonian fluid. We consider that this system has anisotropic dissipative potential. We prove existence of weak solution to this system and its partial C1,α -regularity in 3D and full C1,α -regularity in 2D. 1

Apresentamos uma construção para a teoria das hiperfunções no espaço euclidiano seguindo a abordagem de André Martineau baseada em funcionais analíticos e aplicando um teorema de dualidade de Jean-Pierre Serre. Estudamos também o teorema de divisão de hiperfunções por funções reais-analíticas, provado em Kantor e Schapira (1971). No último capítulo, desenvolvemos alguns aspectos da teoria das hiperfunções no toro. / We present the hyperfunction theory on the Euclidean space following André Martineau\'s approach based on analytic functionals and a duality theorem due to Jean- Pierre Serre. We also study a division theorem proved in Kantor and Schapira (1971). In the last chapter, we develop some aspects of hyperfunction theory on the torus.

In this thesis we show that the finite element error for the high contrast elliptic interface problem is independent of the contrast in the material coefficient under certain assumptions. The error estimate is proved using a particularly technical proof with construction of a specific function from the finite dimensional space of piecewise linear functions. We review the multiscale finite element method of Chu, Graham and Hou to give clearer insight. We present some generalisations to extend their work on a priori contrast independent local boundary conditions, which are then used to find multiscale basis functions by solving a set of local problems. We make use of their regularity result to prove a new relative error estimate for both the standard finte element method and the multiscale finite element method that is completely coefficient independent. The analytical results we explore in this thesis require a complicated construction. To avoid this we present an adaptive multiscale finite element method as an enhancement to the adaptive local-global method of Durlofsky, Efendiev and Ginting. We show numerically that this adaptive method converges optimally as if the coefficient were smooth even in the presence of singularities as well as in the case of a realisation of a random field. The novel application of this thesis is where the adaptive multiscale finite element method has been applied to the linear elasticity problem arising from the structural optimisation process in mechanical engineering. We show that a much smoother sensitivity profile is achieved along the edges of a structure with the adaptive method and no additional heuristic smoothing techniques are needed. We finally show that the new adaptive method can be efficiently implemented in parallel and the processing time scales well as the number of processors increases. The biggest advantage of the multiscale method is that the basis functions can be repeatedly used for additional problems with the same high contrast material coefficient.

The fact that clustering is perhaps the most used technique for exploratory data analysis is only a semaphore that underlines its fundamental importance. The general problem statement that broadly describes clustering as the identification and classification of patterns into coherent groups also implicitly indicates it's utility in other tasks such as supervised learning. In the past decade and a half there have been two developments that have altered the landscape of research in clustering: One is improved results by the increased use of graph theoretic techniques such as spectral clustering and the other is the study of clustering with respect to its relevance in semi-supervised learning i.e. using unlabeled data for improving prediction accuracies. In this work an attempt is made to make contributions to both these aspects. Thus our contributions are two-fold: First, we identify some general issues with the spectral clustering framework and while working towards a solution, we introduce a new algorithm which we call "Regularity Clustering" which makes an attempt to harness the power of the Szemeredi Regularity Lemma, a remarkable result from extremal graph theory for the task of clustering. Secondly, we investigate some practical and useful strategies for using clustering unlabeled data in boosting prediction accuracy. For all of these contributions we evaluate our methods against existing ones and also apply these ideas in a number of settings.

The subject of the present thesis is the Chern-Simons-Schrödinger system, which is a gauge-covariant Schrödinger system in two spatial dimensions with a long-range electromagnetic field. The present thesis studies two aspects of the system: that of well-posedness and that of the long-time behaviour. The first main result of the thesis concerns the large-data well-posedness of the initial-value problem for the Chern-Simons-Schrödinger system. We impose the Coulomb gauge to remove the gauge-invariance, in order to obtain a well-defined initial-value problem. We prove that, in the Coulomb gauge, the Chern-Simons-Schrödinger system is locally well-posed in the Sobolev spaces $H^s$ for $s\ge 1$, and that the solution map satisfies a weak Lipschitz continuity estimate. The main technical difficulty is the presence of a derivative nonlinearity, which rules out the naive iteration scheme for proving well-posedness. The key idea is to retain the non-perturbative part of the derivative nonlinearity in the principal operator, and to exploit the dispersive properties of the resulting paradifferential-type principal operator, in particular frequency-localised Strichartz estimates, using adaptations of the $U^p$ and $V^p$ spaces introduced by Koch and Tataru in other contexts. The other main result of the thesis characterises the large-time behaviour in the case where the interaction potential is the defocusing cubic term. We prove that the solution to the Chern-Simons-Schrödinger system in the Coulomb gauge, starting from a localised finite-energy initial datum, will scatter to a free Schrödinger wave at large times. The two crucial ingredients here are the discovery of a new conserved quantity, that of a pseudo-conformal energy, and the cubic null structure discovered by Oh and Pusateri, which reveals a subtle cancellation in the long-range electromagnetic effects. By exploiting pseudo-conformal symmetry, we also prove the existence of wave operators for the Chern-Simons-Schrödinger system in the Coulomb gauge: given a localised finite-energy final state, there exists a unique solution which scatters to that prescribed state.

Title: Weak solutions for a class of nonlinear integrodifferential equations Author: Ivan Soukup Department: Department of mathematical analysis Supervisor: RNDr. Tomáš Bárta, Ph.D. Supervisor's e-mail address: tomas.barta@mff.cuni.cz Abstract: The work investigates a system of evolutionary nonlinear partial integrodifferential equations in three dimensional space. In particular it stud- ies an existence of a solution to the system introduced in [1] with Dirichlet boundary condition and initial condition u0. We adopt the scheme of the proof from [9] and try to avoid the complications rising from the integral term. The procedure consists of an approximation of the convective term and an ap- proximation of the potentials of both nonlinearities using a quadratic function, proving the existence of the approximative solution and then returning to the original problem via regularity of the approximative solution and properties of the nonlinearities. The aim is to improve the results of the paper [1]. 1

Functional magnetic resonance imaging has been recently the most common tool for examining the neural activity in human and animals. The goal of a typical data-mining challenge is the localisation of brain areas activated during a cognitive task which is usually performed using a linear model or correlation methods. For this purpose several authors have proposed the use of methods evaluating signal complexity which could possibly overcome some of the shortcomings of the standards methods due to their independence on a priori knowledge of data characteristics. This work explains possibilities of using such methods including aspects of their configuration and it proposes an evaluation of performance of the methods applied on simulated data following expected biological characteristics. The results of the evaluation of performance showed little advantage of these methods over the standard ones in cases when the standard methods were possible to apply. However, some of the methods evaluating signal complexity were found useful for determining the regularity of signals which is a feature that cannot be assessed by the standard methods. Optimal parameters of the methods evaluating signal regularity were determined on simulated data and finally the methods were applied on the data examining emotional processing of...

Title: Regularity criteria for instationary incompressible Navier-Stokes equations Author: Šimon Axmann Institute: Mathematical Institute of Charles University Supervisor: doc. Mgr. Milan Pokorný, Ph.D., Mathematical Institute of Charles University Abstract: In the present thesis we study the global conditional regularity of weak solutions to the Cauchy problem for instationary incompressible Navier-Stokes equations in three space dimensions. In the first section, we present an overview of known conditions implying the full regularity of the equations under conside- ration. For the sake of clarity, we expose only the regularity criteria on the scale of Lebesgue spaces, especially in terms of the velocity and its components, the gradient of the velocity and its components, the pressure and the vorticity. In the subsequent sections, we generalize four regularity criteria using two different techniques. We are able to replace one velocity component or its gradient, consi- dered in the known results, by a projection of the velocity into a general vector field. For the purpose of the second method, we also generalize the multiplicative Gagliardo-Nirenberg inequality.