Title: Some topics of topological measure theory with application in stochastic analysis Author: Pavel Kříž Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Josef Štěpán, DrSc., Department of Probability and Mathematical Statistics Abstract: This work studies identifications of values of probability limits based on trajectories of convergent (random) sequences. The key concept is the so called Probability Limit Identification Function (PLIF). The main concern is focused on the existence of PLIFs, mainly those, which are measurable and adapted. We also study in more detail special cases, when the convergence in probability and the convergence almost surely coincide. Furthermore, possible applications of the PLIF concept in stochastic analysis (path-wise representations of stochastic integrals and weak solutions of the stochastic differential equations), as well as in estimation theory (the existence of strongly consistent estimators) are outlined. The achieved results are based on analyses of the topologies on spaces of measures, spaces of random variables and spaces of real-valued functions. Keywords: Probability Limit, Identification, Almost-sure Convergence 1

This study reexamines Lakoff’s (1973) claim that women use more hedges than men is true. Because of the vast number of hedges, this study focuses on two hedges: I think and I’m sure. It also investigates how the included hedges are used by men and women to express belief and opinion. The study has been carried out with the help of a corpus called British National Corpus 2014 (BNC2014). From this database, authentic conversations that include these hedges in clause-final position have been extracted. By using the extracted and processed data, a conclusion can be drawn regarding similarities and differences in how often men and women use these hedges and in what context they are used. The results show that Lakoff’s (1973) claim has a certain truth to it, since 63.0% of the valid I think tokens and 67.6% of the valid I’m sure tokens were produced by women. As for the expression of belief or opinion, the results points towards I think and I’m sure upholding traditional gender traits.

This master's thesis delivers a comprehensive view at leasing business processes running in small and medium-sized leasing companies including the feasibility of their support through an imple-mentation of leasing application software. The first objective of this thesis is the design of an application software system, which will cover general leasing process in small and medium-sized leasing companies for the purpose of development packaged application software system suitable for implementation into small and medium-sized leasing companies. The second objective is a design of a specific implementation of a software system defined in previous objective of thesis. The design of an implementation is based on Microsoft Dynamics Sure Step project management methodology, which was adapted for the purpose of implementation of designed leasing application software into medium-sized leasing company. The first part of the thesis introduces basic concepts occurring throughout whole thesis for a clear understanding of the purpose of the practical work and its processing. The second part describes the Microsoft Dynamics platform as a variety of business software tools suitable for departmental development of leasing solution. This platform includes project management methodology suitable for the business software tools. The third part contains a design of an application software covering general leasing process. The fourth part describes the design of a specific implementation of a leasing application software system defined in previous part of thesis into medium-sized leasing company.

Structural Health Monitoring (SHM) systems were developed to evaluate the integrity of a system during operation, and to quickly identify the maintenance problems. They will be used in future aerospace vehicles to improve safety, reduce cost and minimize the maintenance time of a system. Many SHM systems were already developed to evaluate the integrity of plates and used in marine structures. Their implementation in manufacturing processes is still expected. The application of SHM methods for complex geometries and welds are two important challenges in this area of research. This research work started by studying the characteristics of piezoelectric actuators, and a small energy harvester was designed. The output voltages at different frequencies of vibration were acquired to determine the nonlinear characteristics of the piezoelectric stripe actuators. The frequency response was evaluated experimentally. AA battery size energy harvesting devices were developed by using these actuators. When the round and square cross section devices were excited at 50 Hz frequency, they generated 16 V and 25 V respectively. The Surface Response to Excitation (SuRE) and Lamb wave methods were used to estimate the condition of parts with complex geometries. Cutting tools and welded plates were considered. Both approaches used piezoelectric elements that were attached to the surfaces of considered parts. The variation of the magnitude of the frequency response was evaluated when the SuRE method was used. The sum of the square of the differences was calculated. The envelope of the received signal was used for the analysis of wave propagation. Bi-orthogonal wavelet (Binlet) analysis was also used for the evaluation of the data obtained during Lamb wave technique. Both the Lamb wave and SuRE approaches along with the three methods for data analysis worked effectively to detect increasing tool wear. Similarly, they detected defects on the plate, on the weld, and on a separate plate without any sensor as long as it was welded to the test plate.

Structural health monitoring (SHM) systems generally install low cost excitation component and/or sensors to the machines or buildings permanently to monitor the health of it. Non Destructive Evaluation (NDE) systems use high cost sensors to perform the inspection of structures. It would be advantageous to inspect the some aerospace structures and parts in vacuum environment by using the remotely monitoring systems such as laser vibrometer. In this study, a scanning laser vibrometer is used to detect the location of the problem at the structure. For data collection and analysis recently, developed surface response to excitation (SuRE) method was used. First the software of the scanning laser vibrometer was used to evaluate the capabilities of the system. The vibration of simple structures such as the cone of a speaker and a beam was tested when they were excited below 200 Hz and the mode shapes were studied. Later, the scanning laser vibrometer was used for implementation of the SuRE method. The surface vibration of beams and plates were monitored while their surfaces were excited at high frequencies with a piezoelectric exciter. External force was applied to one point of the surface of the considered structures. The scanning laser vibrometer evaluated the vibration of different points on the structure according to given program and obtained the frequency response before and after the external load was applied. The sum of the squares of the differences of the frequency responses were obtained for each point on the surface and presented with a contour plot. Use of the scanning laser vibrometer was convenient at the test conditions. The calculated mode shapes were very similar to the expected ones when the simple structures were tested. The locations of the external forces were identified correctly when the beams and plates were tested. The study indicated that the scanning laser vibrometer and SuRE method may be used for identification of defects and/or loose fasteners.

Volume 14 of publication series EINGEBETTETE, SELBSTORGANISIERENDE SYSTEME is devoted to the structure oriented evaluation of e-learning. For future knowledge society, beside creation of intelligent technologies, adapted methods of knowledge transfer are required. In this context e-learning becomes a key technology for development of any education system. E-learning is a complex process into which many different groups with specific tasks and roles are included. The dynamics of an e-learning process requires adjusted quality management. For that corresponding evaluation methods are needed. In the present work, Dr.Tudevdagva develops a new evaluation approach for e-learning. The advantage of her method is that in contrast to linear evaluation methods no weight factors are needed and the logical goal structure of an elearning process can be involved into evaluation. Based on general measure theory structure oriented score calculation rules are derived. The so obtained score function satisfies the same calculation rules as they are known from normalised measures. In statistical generalisation, these rules allow the structure oriented calculation of empirical evaluation scores based on checklist data. By these scores the quality can be described by which an e-learning has reached its total goal. Moreover, a consistent evaluation of embedded partial processes of an e-learning becomes possibly. The presented score calculation rules are part of a eight step evaluation model which is illustrated by pilot samples. U. Tudevdagva’s structure oriented evaluation model (SURE model) is by its embedding into the general measure theory quite universal applicable. In similar manner, an evaluation of efficiency of administration or organisation processes becomes possible.

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Volume 14 of publication series EINGEBETTETE, SELBSTORGANISIERENDE SYSTEME is devoted to the structure oriented evaluation of e-learning. For future knowledge society, beside creation of intelligent technologies, adapted methods of knowledge transfer are required. In this context e-learning becomes a key technology for development of any education system. E-learning is a complex process into which many different groups with specific tasks and roles are included. The dynamics of an e-learning process requires adjusted quality management. For that corresponding evaluation methods are needed. In the present work, Dr.Tudevdagva develops a new evaluation approach for e-learning. The advantage of her method is that in contrast to linear evaluation methods no weight factors are needed and the logical goal structure of an elearning process can be involved into evaluation. Based on general measure theory structure oriented score calculation rules are derived. The so obtained score function satisfies the same calculation rules as they are known from normalised measures. In statistical generalisation, these rules allow the structure oriented calculation of empirical evaluation scores based on checklist data. By these scores the quality can be described by which an e-learning has reached its total goal. Moreover, a consistent evaluation of embedded partial processes of an e-learning becomes possibly. The presented score calculation rules are part of a eight step evaluation model which is illustrated by pilot samples. U. Tudevdagva’s structure oriented evaluation model (SURE model) is by its embedding into the general measure theory quite universal applicable. In similar manner, an evaluation of efficiency of administration or organisation processes becomes possible.

This thesis contains three main results. The first result states that, outside a slim set associated with a Gaussian process with long time memory, paths can be canonically enhanced to geometric rough paths. This allows us to apply the powerful Universal Limit Theorem in rough path theory to study the quasi-sure properties of the solutions of stochastic differential equations driven by Gaussian processes. The key idea is to use a norm, invented by B. Hambly and T.Lyons, which dominates the p-variation distance and the fact that the roughness of a Gaussian sample path is evenly distributed over time. The second result is the almost-sure uniqueness of the signatures of SLE kappa curves for kappa less than or equal to 4. We prove this by first expressing the Fourier transform of the winding angle of the SLE curve in terms of its signature. This formula also gives us a relation between the expected signature and the n-point functions studied in the SLE and Statistical Physics literature. It is important that the Chordal SLE measure in D is supported on simple curves from -1 to 1 for kappa between 0 and 4, and hence the image of the curve determines the curve up to reparametrisation. The third result is a formula for the expected signature of Gaussian processes generated by strictly regular kernels. The idea is to approximate the expected signature of this class of processes by the expected signature of their piecewise linear approximations. This reduces the problem to computing the moments of Gaussian random variables, which can be done using Wick’s formula.

Le travail présenté dans ce mémoire concerne une méthode de conception sûre de systèmes(COTS). Un COTS est un composant matériel ou logiciel générique qui est naturellement conçu pour être réutilisable et cela se traduit par une forme de flexibilité dans la mise en oeuvre de sa fonctionnalité : en clair, une même fonction peut être réalisée par un ensemble (potentiellement infini) de scénarios différents, tous réalisables par le COTS. La complexité grandissante des fonctions implémentées fait que ces situations sont très difficiles à anticiper d'une part, et encore plus difficiles à éviter par un codage correct. Réaliser manuellement une fonction composite correcte sur un système de taille industrielle, s'avère être très coûteuse. Elle nécessite une connaissance approfondie du comportement des COTS assemblés. Or cette connaissance est souvent manquante, vu qu'il s'agit de composants acquis, ou développés par un tiers, et dont la documentation porte sur la description de leur fonction et non sur sa mise en IJuvre. Par ailleurs, il arrive souvent que la correction manuelle d'une faute engendre une ou plusieurs autres fautes, provoquant un cercle vicieux difficile à maîtriser. En plus, le fait de modifier le code d'un composant diminue l'avantage lié à sa réutilisation. C'est dans ce contexte que nous proposons l'utilisation de la technique de synthèse du contrôleur discret (SCD) pour générer automatiquement du code de contrôle commande correct par construction. Cette technique produit des composants, nommés contrôleurs, qui agissent en contraignant le comportement d'un (ou d'un assemblage de) COTS afin de garantir si possible la satisfaction d'une exigence fonctionnelle. La méthode que nous proposons possède plusieurs étapes de conception. La première étape concerne la formalisation des COTS et des propriété de sûreté et de vivacité (P) en modèles automate à états et/ou en logique temporelle. L'étape suivante concerne la vérification formelle du modèle d'un(des) COTS pour l'ensemble des propriétés (P). Cette étape découvrir les états de violation des propriétés (P) appelés états d'erreur. La troisième étape concerne la correction automatique des erreurs détectées en utilisant la technique SCD. Dans cette étape génère on génère un composant correcteur qui sera assemblé au(x) COTS original(aux) pour que leur comportement général respecte les propriétés souhaitées. L'étape suivante concerne la vérification du système contrôlé pour un ensemble de propriétés de vivacité pour assurer la passivité du contrôleur et la vivacité du système. En fin, une étape de simulation est proposée pour observer le comportement du système pour quelque scénarios intéressent par rapport à son implémentation finale.