Spectral clustering is often used to partition a data set into a specified number of clusters. Both the unweighted and the vertex-weighted approaches use eigenvectors of the Laplacian matrix of a graph. Our focus is on using vertex-weighted methods to refine clustering of observations. An eigenvector corresponding with the second smallest eigenvalue of the Laplacian matrix of a graph is called a Fiedler vector. Coefficients of a Fiedler vector are used to partition vertices of a given graph into two clusters. A vertex of a graph is classified as unassociated if the Fiedler coefficient of the vertex is close to zero compared to the largest Fiedler coefficient of the graph. We propose a vertex-weighted spectral clustering algorithm which incorporates a vector of weights for each vertex of a given graph to form a vertex-weighted graph. The proposed algorithm predicts association of equidistant or nearly equidistant data points from both clusters while the unweighted clustering does not provide association. Finally, we implemented both the unweighted and the vertex-weighted spectral clustering algorithms on several data sets to show that the proposed algorithm works in general.

Cette thèse porte sur certaines extensions des automates pondérés, et étudie les séries qu’ils réalisent en fonction de la nature des poids.Ces extensions se distinguent par les mouvements supplémentaires autorisés à la tête de lecture de l’automate : retour au début du mot pour les automates circulaires, changement de sens de lecture pour les automates boustrophédons.Dans le cas général, les automates pondérés circulaires sont plus puissants que les automates unidirectionnels classiques, et moins puissants que les boustrophédons.On introduit de plus les expressions de Hadamard, qui sont une extension des expressions rationnelles et qui permettent de dénoter le comportement des automates circulaires. Les aspects algorithmiques de cette conversion sont étudiés dans le cas où les poids appartiennent à un semi-anneau rationnellement additif.On montre que lorsque les poids sont des nombres rationnels, réels ou complexes, les automates circulaires sont aussi expressifs que les boustrophédons.Enfin, si les poids forment un bi-monoïde localement fini, les automates boustrophédons ne sont pas plus expressifs que les automates pondérés classsiques. / This thesis deals with some extensions of weighted automata,and studies the series they can realisedepending on the nature of their weigths.These extensions are characterised by howthe input head of the automaton is allowed to move:rotating automata can go back at the beginning of the word,and two-way automata can change the reading direction.In the general setting, weigthed rotating automata are morepowerful than classical one-way automata, and less powerfulthan two-way ones.Moreover, we introduce Hadamard expressions,which are an extension of rational expressions and can denotethe behaviour of rotating automata.The algorithms for this conversion are studied when the weights belong toa rationally additive semiring.Then, rotating automata are shown as expressive as two-way automatain the case of rational, real or complex numbers.It is also proved that two-way and one-way automataare equivalent when weighted on a locally finite bimonoid.

Title: Development of Czech Floorball between years 2007 and 2018 Objectives: The main goal of this thesis is to create a development plan of Czech floorball for the years 2020 to 2030. The partial goal is to analyze the re- course of this plan; the supporting goal is to evaluate the development plans from year 2008 to 2018. Methods: This thesis performs analyses of data from internal and external envi- ronment, material, financial, personal and intangible resources, using PESTLE and situation analysis. The analyses are applied on data col- lected from primary and secondary resources. Evaluation of the development plan for the reviewed period is per- formed using the method of weighted averages of fulfilled partial goals. Results: The offered development plan for the years 2020 to 2030 includes 134 individual goals. 49 partial goals were not modified; parameters for evaluation were suggested in 34 goals; 22 goals were modified or clarified; 29 goals were added. Overall, 63 % of the development plan was modified. According to overall weighted average, at 76 %, Development and youth is the most successfully fulfilled area. Next, at 69 %, is the area of Communication, media, and marketing, followed by Finance and economics at 68 %; Success and attractiveness is at 62 %, Education and metho-dology is...

Title: Actuarial and Exposure-based Models for Hail Peril Author: Bc. Matúš Drobuliak Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michal Pešta, Ph.D., Department of Probability and Mathe- matical Statistics Abstract: This thesis covers an introduction to catastrophe modelling and focuses on statistical methods for extreme events. This includes methods of estimating parameters of claim distribution with a focus on probability weighted moments estimation technique. Furthermore, times series modelling, skew t-distribution, and two model clustering techniques are examined as well. This is later utilised in the practical application part of this thesis, which uses real data provided by an insurance company operating in the Czech Republic. Probability distribution fitting of large claims caused by hailstorms and Monte Carlo simulation of future losses are demonstrated later. Keywords: Catastrophe modelling, Hail peril, Probability weighted moments, Extreme events, ARMA-GARCH, Monte Carlo simulation iii

Nesse trabalho, estudamos a equação de Schrödinger não-linear com uma função potencial delta atrativa. As soluções para essa equação tem uma componente localizada e uma dispersiva. Além de estudar o comportamento das soluções dessa equação em espaços de Sobolev clássicos, mostramos algumas propriedades do grupo unitário em espaços Lp, L2 com peso, Sobolev com peso e assim obtemos alguns resultados de boa colocação local e global das soluções. O ponto central desta tese é mostrarmos a existência de uma variedade invariante centro que irá consistir de órbitas periódicas no tempo. / In this work, we study the nonlinear Schrodinger equation with an attractive delta function potential.The solutions to this equation have a localized and a dispersive component. In addition to studying the behavior of solutions of this equation in classical Sobolev space, we show some properties for the unitary group in Lp, weighted L2 and Sobolev spaces and so we get some results of local and global well-posedness of solutions. The central theme this thesis is to show the existence of a center invariant manifold, which will consist of time-periodic orbits.

We consider a longitudinal system in which transitions between the states are governed by a discrete time finite state space stochastic process X. Our aim, using aggregated sample survey data of the form typically collected by official statistical agencies, is to undertake model based inference for the underlying process X. We will develop inferential techniques for continuing sample surveys of two distinct types. First, longitudinal surveys in which the same individuals are sampled in each cycle of the survey. Second, cross-sectional surveys which sample the same population in successive cycles but with no attempt to track particular individuals from one cycle to the next. Some of the basic results have appeared in Davis et al (2001) and Davis et al (2002).¶ Longitudinal surveys provide data in the form of transition frequencies between the states of X. In Chapter Two we develop a method for modelling and estimating the one-step transition probabilities in the case where X is a non-homogeneous Markov chain and transition frequencies are observed at unit time intervals. However, due to their expense, longitudinal surveys are typically conducted at widely, and sometimes irregularly, spaced time points. That is, the observable frequencies pertain to multi-step transitions. Continuing to assume the Markov property for X, in Chapter Three, we show that these multi-step transition frequencies can be stochastically interpolated to provide accurate estimates of the one-step transition probabilities of the underlying process. These estimates for a unit time increment can be used to calculate estimates of expected future occupation time, conditional on an individual’s state at initial point of observation, in the different states of X.¶ For reasons of cost, most statistical collections run by official agencies are cross-sectional sample surveys. The data observed from an on-going survey of this type are marginal frequencies in the states of X at a sequence of time points. In Chapter Four we develop a model based technique for estimating the marginal probabilities of X using data of this form. Note that, in contrast to the longitudinal case, the Markov assumption does not simplify inference based on marginal frequencies. The marginal probability estimates enable estimation of future occupation times (in each of the states of X) for an individual of unspecified initial state. However, in the applications of the technique that we discuss (see Sections 4.4 and 4.5) the estimated occupation times will be conditional on both gender and initial age of individuals.¶ The longitudinal data envisaged in Chapter Two is that obtained from the surveillance of the same sample in each cycle of an on-going survey. In practice, to preserve data quality it is necessary to control respondent burden using sample rotation. This is usually achieved using a mechanism known as rotation group sampling. In Chapter Five we consider the particular form of rotation group sampling used by the Australian Bureau of Statistics in their Monthly Labour Force Survey (from which official estimates of labour force participation rates are produced). We show that our approach to estimating the one-step transition probabilities of X from transition frequencies observed at incremental time intervals, developed in Chapter Two, can be modified to deal with data collected under this sample rotation scheme. Furthermore, we show that valid inference is possible even when the Markov property does not hold for the underlying process.

Smoothed Particle Hydrodynamics (SPH) is analysed as the weighted residual method. In particular the analysis focuses on the collocation aspect of the method. Using Monte Carlo experiments we demonstrate that SPH is highly sensitive to node disorder, especially in its symmetrised energy and momentum conserving form. This aspect of the method is related to low [Beta] MHD instabilities observed by other authors. A remedy in the form of the Weighted Differences Method is suggested, which addresses this problem to some extent, but at a cost of losing automatic conservation of energy and momentum. ¶ The Weighted Differences Method is used to simulate propagation of Alfven and magnetosonic wave fronts in [Beta] = 0 plasma, and the results are compared with data obtained with the NCSA Zeus3D code with the Method of Characteristics (MOC) module. ¶ SPH is then applied to two interesting astrophysical situations: accretion on to a white dwarf in a compact binary system, which results in a formation of an accretion disk, and gravitational collapse of a magnetised vortex. Both models are 3 dimensional. ¶ The accretion disk which forms in the binary star model is characterised by turbulent flow: the Karman vortex street is observed behind the stream-disk interaction region. The shock that forms at the point of stream-disk interaction is controlled by the means of particle merges, whereas Monaghan-Lattanzio artificial viscosity is used to simulate Smagorinsky closure. ¶ The evolution of the collapsing magnetised vortex ends up in the formation of an expanding ring in the symmetry plane of the system. We observe the presence of spiralling inward motion towards the centre of attraction. That final state compares favourably with the observed qualitative and quantitative characteristics of the circumnuclear disk in the Galactic Centre. That simulation has also been verified with the NCSA Zeus3D run. ¶ In conclusions we contrast the result of our Monte Carlo experiments with the results delivered by our production runs. We also compare SPH and Weighted Differences against the new generation of conservative finite differences methods, such as the Godunov method and the Piecewise Parabolic Method. We conclude that although SPH cannot match the accuracy and performance of those methods, it appears to have some advantage in simulation of rotating flows, which are of special interest to astrophysics.

Our brain, perhaps the most sophisticated and mysterious part of the human body, to some extent, determines who we are. However, it’s a vulnerable organ. When subjected to an impact, such as a traffic accident or sport, it may lead to traumatic brain injury (TBI) which can have devastating effects for those who suffer the injury. Despite lots of efforts have been put into primary injury prevention, the number of TBIs is still on an unacceptable high level in a global perspective. Brain edema is a major neurological complication of moderate and severe TBI, which consists of an abnormal accumulation of fluid within the brain parenchyma. Clinically, local and minor edema may be treated conservatively only by observation, where the treatment of choice usually follows evidence-based practice. In the first study, the gravitational force is suggested to have a significant impact on the pressure of the edema zone in the brain tissue. Thus, the objective of the study was to investigate the significance of head position on edema at the posterior part of the brain using a Finite Element (FE) model. The model revealed that water content (WC) increment at the edema zone remained nearly identical for both supine and prone positions. However, the interstitial fluid pressure (IFP) inside the edema zone decreased around 15% by having the head in a prone position compared with a supine position. The decrease of IFP inside the edema zone by changing patient position from supine to prone has the potential to alleviate the damage to axonal fibers of the central nervous system. These observations suggest that considering the patient’s head position during intensive care and at rehabilitation should be of importance to the treatment of edematous regions in TBI patients. In TBI patients with diffuse brain edema, for most severe cases with refractory intracranial hypertension, decompressive craniotomy (DC) is performed as an ultimate therapy. However, a complete consensus on its effectiveness has not been achieved due to the high levels of severe disability and persistent vegetative state found in the patients treated with DC. DC allows expansion of the swollen brain outside the skull, thereby having the potential in reducing the Intracranial Pressure (ICP). However, the treatment causes stretching of the axons and may contribute to the unfavorable outcome of the patients. The second study aimed at quantifying the stretching and WC in the brain tissue due to the neurosurgical intervention to provide more insight into the effects upon such a treatment. A nonlinear registration method was used to quantify the strain. Our analysis showed a substantial increase of the strain level in the brain tissue close to the treated side of DC compared to before the treatment. Also, the WC was related to specific gravity (SG), which in turn was related to the Hounsfield unit (HU) value in the Computerized Tomography (CT) images by a photoelectric correction according to the chemical composition of the brain tissue. The overall WC of brain tissue presented a significant increase after the treatment compared to the condition seen before the treatment. It is suggested that a quantitative model, which characterizes the stretching and WC of the brain tissue both before as well as after DC, may clarify some of the potential problems with such a treatment. Diffusion Weighted (DW) Imaging technology provides a noninvasive way to extract axonal fiber tracts in the brain. The aim of the third study, as an extension to the second study was to assess and quantify the axonal deformation (i.e. stretching and shearing)at both the pre- and post-craniotomy periods in order to provide more insight into the mechanical effects on the axonal fibers due to DC. Subarachnoid injection of artificial cerebrospinal fluid (CSF) into the CSF system is widely used in neurological practice to gain information on CSF dynamics. Mathematical models are important for a better understanding of the underlying mechanisms. Despite the critical importance of the parameters for accurate modeling, there is a substantial variation in the poroelastic constants used in the literature due to the difficulties in determining material properties of brain tissue. In the fourth study, we developed a Finite Element (FE) model including the whole brain-CSF-skull system to study the CSF dynamics during constant-rate infusion. We investigated the capacity of the current model to predict the steady state of the mean ICP. For transient analysis, rather than accurately fit the infusion curve to the experimental data, we placed more emphasis on studying the influences of each of the poroelastic parameters due to the aforementioned inconsistency in the poroelastic constants for brain tissue. It was found that the value of the specific storage term S_epsilon is the dominant factor that influences the infusion curve, and the drained Young’s modulus E was identified as the dominant parameter second to S_epsilon. Based on the simulated infusion curves from the FE model, Artificial Neural Network (ANN) was used to find an optimized parameter set that best fit the experimental curve. The infusion curves from both the FE simulations and using ANN confirmed the limitation of linear poroelasticity in modeling the transient constant-rate infusion. To summarize, the work done in this thesis is to introduce FE Modeling and imaging technologiesincluding CT, DW imaging, and image registration method as a complementarytechnique for clinical diagnosis and treatment of TBI patients. Hopefully, the result mayto some extent improve the understanding of these clinical problems and improve theirmedical treatments. / QC 20120201

In this study I analyse the performance of a democratic decision-making rule: the weighted majority rule. It assigns to each voter a number of votes that is proportional to her stakes in the decision. It has been shown that, for collective decisions with two options, the weighted majority rule in combination with self-interested voters maximises the common good when the latter is understood in terms of either the sum-total or prioritarian sum of the voters’ well-being. The main result of my study is that this argument for the weighted majority rule — that it maximises the common good — can be improved along the following three main lines. (1) The argument can be adapted to other criteria of the common good, such as sufficientarian, maximin, leximin or non-welfarist criteria. I propose a generic argument for the collective optimality of the weighted majority rule that works for all of these criteria. (2) The assumption of self-interested voters can be relaxed. First, common-interest voters can be accommodated. Second, even if voters are less than fully competent in judging their self-interest or the common interest, the weighted majority rule is weakly collectively optimal, that is, it almost certainly maximises the common good given a large numbers of voters. Third, even for smaller groups of voters, the weighted majority rule still has some attractive features. (3) The scope of the argument can be extended to decisions with more than two options. I state the conditions under which the weighted majority rule maximises the common good even in multi-option contexts. I also analyse the possibility and the detrimental effects of strategic voting. Furthermore, I argue that self-interested voters have reason to accept the weighted majority rule.

Calibration methods for quantitative on-site sampling using solid phase microextraction (SPME) were developed based on diffusion mass transfer theory. This was investigated using adsorptive polydimethylsiloxane/divinylbenzene (PDMS/DVB) and Carboxen/polydimethylsiloxane (CAR/PDMS) SPME fiber coatings with volatile aromatic hydrocarbons (BTEX: benzene, toluene, ethylbenzene, and o-xylene) as test analytes. Parameters that affected the extraction process (sampling time, analyte concentration, water velocity, and temperature) were investigated. Very short sampling times (10-300 s) and sorbents with a strong affinity and large capacity were used to ensure a 'zero sink' effect calibrate process. It was found that mass uptake of analyte changed linearly with concentration. Increase of water velocity increased mass uptake, though the increase is not linear. Temperature did not affect mass uptake significantly under typical field sampling conditions. To further describe rapid SPME analysis of aqueous samples, a new model translated from heat transfer to a circular cylinder in cross flow was used. An empirical correlation to this model was used to predict the mass transfer coefficient. Findings indicated that the predicted mass uptake compared well with experimental mass uptake. The new model also predicted rapid air sampling accurately. To further integrate the sampling and analysis processes, especially for on-site or <i>in-vivo</i> investigations where the composition of the sample matrix is very complicated and/or agitation of the sample matrix is variable or unknown, a new approach for calibration was developed. This involved the loading internal standards onto the extraction fiber prior to the extraction step. During sampling, the standard partially desorbs into the sample matrix and the rate at which this process occurs, was for calibration. The kinetics of the absorption/desorption was investigated, and the isotropy of the two processes was demonstrated, thus validating this approach for calibration. A modified SPME device was used as a passive sampler to determine the time-weighted average (TWA) concentration of volatile organic compounds (VOCs) in air. The sampler collects the VOCs by the mechanism of molecular diffusion and sorption on to a coated fiber as collection medium. This process was shown to be described by Fick's first law of diffusion, whereby the amount of analyte accumulated over time enable measurement of the TWA concentration to which the sampler was exposed. TWA passive sampling with a SPME device was shown to be almost independent of face velocity, and to be more tolerant of high and low analyte concentrations and long and short sampling times, because of the ease with which the diffusional path length could be changed. Environmental conditions (temperature, pressure, relative humidity, and ozone) had little or no effect on sampling rate. When the SPME device was tested in the field and the results compared with those from National Institute of Occupational Health and Safety (NIOSH) method 1501 good agreement was obtained. To facilitate the use of SPME for field sampling, a new field sampler was designed and tested. The sampler was versatile and user-friendly. The SPME fiber can be positioned precisely inside the needle for TWA sampling, or exposed completely outside the needle for rapid sampling. The needle is protected within a shield at all times hereby eliminating the risk of operator injury and fiber damage. A replaceable Teflon cap is used to seal the needle to preserve sample integrity. Factors that affect the preservation of sample integrity (sorbent efficiency, temperature, and sealing materials) were studied. The use of a highly efficient sorbent is recommended as the first choice for the preservation of sample integrity. Teflon was a good material for sealing the fiber needle, had little memory effect, and could be used repeatedly. To address adsorption of high boiling point compounds on fiber needles, several kinds of deactivated needles were evaluated. RSC-2 blue fiber needles were the more effective. A preliminary field sampling investigation demonstrated the validity of the new SPME device for field applications.