Title: Mathematical analysis of equations describing the flow of compressible heat conducting fluids Author: Šimon Axmann Department: Mathematical Institute of Charles University Supervisor: doc. Mgr. Milan Pokorný, Ph.D., Mathematical Institute of Charles University Abstract: The present thesis is devoted to the mathematical analysis of equa- tions describing the flow of viscous compressible newtonian fluid in various time regimes. In particular, we present existence results for three problems arising as special cases of a general model derived in the introductory part. The first chap- ter deals with time-periodic solutions to the full Navier-Stokes-Fourier system for heat-conducting fluid. The second chapter contains the proof of existence of steady solutions to a system arising from phase field model for two-phase com- pressible fluid. Finally, in the last section we study steady strong solutions to the Navier-Stokes equations under the additional assumption that the fluid is suffi- ciently dense. For each problem a different concept of the solution is considered, on the other hand in all cases an essential role is played by the crucial quantity effective viscous flux. Keywords: compressible Navier-Stokes system; weak solution; entropy variational solution; large data

L'inférence bayésienne répond aux questions clés de la perception, comme par exemple : "Que faut-il que je crois étant donné ce que j'ai perçu ?". Elle est donc par conséquent une riche source de modèles pour les sciences cognitives et les neurosciences (Knill et Richards, 1996). Cette thèse de doctorat explore deux modèles bayésiens. Dans le premier, nous explorons un problème de codage efficace, et répondons à la question de comment représenter au mieux une information probabiliste dans des neurones pas parfaitement fiables. Nous innovons par rapport à l'état de l'art en modélisant une information d'entrée finie dans notre modèle. Nous explorons ensuite un nouveau modèle d'observateur optimal pour la localisation d'une source sonore grâce à l’écart temporel interaural, alors que les modèles actuels sont purement phénoménologiques. Enfin, nous explorons les propriétés de l'algorithme d'inférence approximée "Expectation Propagation", qui est très prometteur à la fois pour des applications en apprentissage automatique et pour la modélisation de populations neuronales, mais qui est aussi actuellement très mal compris. / Bayesian inference answers key questions of perception such as: "What should I believe given what I have perceived ?". As such, it is a rich source of models for cognitive science and neuroscience (Knill and Richards, 1996). This PhD manuscript explores two such models. We first investigate an efficient coding problem, asking the question of how to best represent probabilistic information in unrealiable neurons. We innovate compared to older such models by introducing limited input information in our own. We then explore a brand new ideal observer model of localization of sounds using the Interaural Time Difference cue, when current models are purely descriptive models of the electrophysiology. Finally, we explore the properties of the Expectation Propagation approximate-inference algorithm, which offers great potential for both practical machine-learning applications and neuronal population models, but is currently very poorly understood.

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Abstract Cluster analysis has become one of the main tools used in extracting knowledge from data, which is known as data mining. In this area of data analysis, data of large dimensions are often processed, both in the number of objects and in the number of variables, which characterize the objects. Many methods for data clustering have been developed. One of the most widely used is a k-means method, which is suitable for clustering data sets containing large number of objects. It is based on finding the best clustering in relation to the initial distribution of objects into clusters and subsequent step-by-step redistribution of objects belonging to the clusters by the optimization function. The aim of this Ph.D. thesis was a comparison of selected variants of existing k-means methods, detailed characterization of their positive and negative characte- ristics, new alternatives of this method and experimental comparisons with existing approaches. These objectives were met. I focused on modifications of the k-means method for clustering of large number of objects in my work, specifically on the algorithms BIRCH k-means, filtering, k-means++ and two-phases. I watched the time complexity of algorithms, the effect of initialization distribution and outliers, the validity of the resulting clusters. Two real data files and some generated data sets were used. The common and different features of method, which are under investigation, are summarized at the end of the work. The main aim and benefit of the work is to devise my modifications, solving the bottlenecks of the basic procedure and of the existing variants, their programming and verification. Some modifications brought accelerate the processing. The application of the main ideas of algorithm k-means++ brought to other variants of k-means method better results of clustering. The most significant of the proposed changes is a modification of the filtering algorithm, which brings an entirely new feature of the algorithm, which is the detection of outliers. The accompanying CD is enclosed. It includes the source code of programs written in MATLAB development environment. Programs were created specifically for the purpose of this work and are intended for experimental use. The CD also contains the data files used for various experiments.

Aquesta tesi es centra en algorismes de clustering multiobjectiu, que estan basats en optimitzar varis objectius simultàniament obtenint una col•lecció de solucions potencials amb diferents compromisos entre objectius. El propòsit d'aquesta tesi consisteix en dissenyar i implementar un nou algorisme de clustering multiobjectiu basat en algorismes evolutius per afrontar tres reptes actuals relacionats amb aquest tipus de tècniques. El primer repte es centra en definir adequadament l'àrea de possibles solucions que s'explora per obtenir la millor solució i que depèn de la representació del coneixement. El segon repte consisteix en escalar el sistema dividint el conjunt de dades original en varis subconjunts per treballar amb menys dades en el procés de clustering. El tercer repte es basa en recuperar la solució més adequada tenint en compte la qualitat i la forma dels clusters a partir de la regió més interessant de la col•lecció de solucions ofertes per l’algorisme. / Esta tesis se centra en los algoritmos de clustering multiobjetivo, que están basados en optimizar varios objetivos simultáneamente obteniendo una colección de soluciones potenciales con diferentes compromisos entre objetivos. El propósito de esta tesis consiste en diseñar e implementar un nuevo algoritmo de clustering multiobjetivo basado en algoritmos evolutivos para afrontar tres retos actuales relacionados con este tipo de técnicas. El primer reto se centra en definir adecuadamente el área de posibles soluciones explorada para obtener la mejor solución y que depende de la representación del conocimiento. El segundo reto consiste en escalar el sistema dividiendo el conjunto de datos original en varios subconjuntos para trabajar con menos datos en el proceso de clustering El tercer reto se basa en recuperar la solución más adecuada según la calidad y la forma de los clusters a partir de la región más interesante de la colección de soluciones ofrecidas por el algoritmo. / This thesis is focused on multiobjective clustering algorithms, which are based on optimizing several objectives simultaneously obtaining a collection of potential solutions with different trade¬offs among objectives. The goal of the thesis is to design and implement a new multiobjective clustering technique based on evolutionary algorithms for facing up three current challenges related to these techniques. The first challenge is focused on successfully defining the area of possible solutions that is explored in order to find the best solution, and this depends on the knowledge representation. The second challenge tries to scale-up the system splitting the original data set into several data subsets in order to work with less data in the clustering process. The third challenge is addressed to the retrieval of the most suitable solution according to the quality and shape of the clusters from the most interesting region of the collection of solutions returned by the algorithm.

In the first part of this thesis we are concerned with the constitutive the- ory for incompressible fluids characterized by a continuous monotone rela- tion between the velocity gradient and the Cauchy stress. We, in particular, investigate a class of activated fluids that behave as the Euler fluid prior activation, and as the Navier-Stokes or power-law fluid once the activation takes place. We develop a large-data existence analysis for both steady and unsteady three-dimensional flows of such fluids subject either to the no-slip boundary condition or to a range of slip-type boundary conditions, including free-slip, Navier's slip, and stick-slip. In the second part we show that the W−1,q norm is localizable provided that the functional in question vanishes on locally supported functions which constitute a partition of unity. This represents a key tool for establishing local a posteriori efficiency for partial differential equations in divergence form with residuals in W−1,q . In the third part we provide a novel analysis for the pressure convection- diffusion (PCD) preconditioner. We first develop a theory for the precon- ditioner considered as an operator in infinite-dimensional spaces. We then provide a methodology for constructing discrete PCD operators for a broad class of pressure discretizations. The...

Företaget Nordicstation hanterar stora datamängder åt Swedbank där datalagringen sker i relationsdatabasen Microsoft SQL Server 2012 (SQL Server). Då det finns andra databashanterare designade för stora datavolymer är det oklart om SQL Server är den optimala lösningen för situationen. Detta examensarbete har tagit fram en jämförelse med hjälp av prestandatester, beträffande exekveringstiden av databasfrågor, mellan databaserna SQL Server, Cassandra och NuoDB vid hanteringen av stora datamängder. Cassandra är en kolumnbaserad databas designad för hantering av stora datavolymer, NuoDB är en minnesdatabas som använder internminnet som lagringsutrymme och är designad för skalbarhet. Resultaten togs fram i en virtuell servermiljö med Windows Server 2012 R2 på en testplattform skriven i Java. Jämförelsen visar att SQL Server var den databas mest lämpad för gruppering, sortering och beräkningsoperationer. Däremot var Cassandra bäst i skrivoperationer och NuoDB presterade bäst i läsoperationer. Analysen av resultatet visade att mindre access till disken ger kortare exekveringstid men den skalbara lösningen, NuoDB, lider av kraftiga prestandaförluster av att endast konfigureras med en nod. Nordicstation rekommenderas att uppgradera till Microsoft SQL Server 2014, eller senare, där möjlighet finns att spara tabeller i internminnet. / The company Nordicstation handles large amounts of data for Swedbank, where data is stored using the relational database Microsoft SQL Server 2012 (SQL Server). The existence of other databases designed for handling large amounts of data, makes it unclear if SQL Server is the best solution for this situation.  This degree project describes a comparison between databases using performance testing, with regard to the execution time of database queries.  The chosen databases were SQL Server, Cassandra and NuoDB. Cassandra is a column-oriented database designed for handling large amounts of data, NuoDB is a database that uses the main memory for data storage and is designed for scalability. The performance tests were executed in a virtual server environment with Windows Server 2012 R2 using an application written in Java. SQL Server was the database most suited for grouping, sorting and arithmetic operations. Cassandra had the shortest execution time for write operations while NuoDB performed best in read operations. This degree project concludes that minimizing disk operations leads to shorter execution times but the scalable solution, NuoDB, suffer severe performance losses when configured as a single-node. Nordicstation is recommended to upgrade to Microsoft SQL Server 2014, or later, because of the possibility to save tables in main memory.