We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant.

This text focuses on the relation of information and image, hence Information Theory and Image Analysis, as well as visualization of information and methods of visual analytics focusing on analysis of art works. It also concentrates on evolution of digital image and related new type of perception and artificial aesthetics. We narrow the broader topic of the image and image information to the abstract art, namely the work of Czech painter Frantisek Kupka, which is used as input in the experiment presenting original method of image analysis using the function of information entropy (Rényi entropy). This approach was used for the first time for analysis of art works with the aim to obtain the comparaison of natural and artificial classification of image information. We chose the work of abstract art not only with regard to given history of grammatics of abstract forms and its relation to the digital image, but also as an emblematic example of effective gaining of information from complex environment. Work thus summarizes historical context of evolution of digital image and theoretical reflection of contemporary image analytics and others techniques relevant to the image information and emphasizes relation of abstract art to the natural and simulated complexity.

The master's thesis deals with sense disambiguation of Czech words. Reader is informed about task's history and used algorithms are introduced. There are naive Bayes classifier, AdaBoost classifier, maximum entrophy method and decision trees described in this thesis. Used methods are clearly demonstrated. In the next parts of this thesis are used data also described.  Last part of the thesis describe reached results. There are some ideas to improve the system at the end of the thesis.

Feature selection or variable selection is an important step in different machine learning tasks. In a traditional approach, users specify the amount of features, which shall be selected. Afterwards, algorithm select features by using scores like the Joint Mutual Information (JMI). If users do not know the exact amount of features to select, they need to evaluate the full learning chain for different feature counts in order to determine, which amount leads to the lowest training error. To overcome this drawback, we extend the JMI score and mitigate the flaw by introducing a stopping criterion to the selection algorithm that can be specified depending on the learning task. With this, we enable developers to carry out the feature selection task before the actual learning is done. We call our new score Historical Joint Mutual Information (HJMI). Additionally, we compare our new algorithm, using the novel HJMI score, against traditional algorithms, which use the JMI score. With this, we demonstrate that the HJMI-based algorithm is able to automatically select a reasonable amount of features: Our approach delivers results as good as traditional approaches and sometimes even outperforms them, as it is not limited to a certain step size for feature evaluation.

The discovery of the entropy production paradox (Hoffmann et al., 1998) raised basic questions about the nature of irreversibility in the regime between diffusion and waves. First studied in the form of spatial movements of moments of H functions, pseudo propagation is the pre-limit propagation-like movements of skewed probability density function (PDFs) in the domain between the wave and diffusion equations that goes over to classical partial differential equation propagation of characteristics in the wave limit. Many of the strange properties that occur in this extraordinary regime were thought to be connected in some manner to this form of proto-movement. This paper eliminates pseudo propagation by employing a similar evolution equation that imposes spatial unimodal symmetry on evolving PDFs. Contrary to initial expectations, familiar peculiarities emerge despite the imposed symmetry, but they have a distinct character.

Many complexity measures are defined as the size of a minimal representation in a specific model class. One such complexity measure, which is important because it is widely applied, is statistical complexity. It is defined for discrete-time, stationary stochastic processes within a theory called computational mechanics. Here, a mathematically rigorous, more general version of this theory is presented, and abstract properties of statistical complexity as a function on the space of processes are investigated. In particular, weak-* lower semi-continuity and concavity are shown, and it is argued that these properties should be shared by all sensible complexity measures. Furthermore, a formula for the ergodic decomposition is obtained. The same results are also proven for two other complexity measures that are defined by different model classes, namely process dimension and generative complexity. These two quantities, and also the information theoretic complexity measure called excess entropy, are related to statistical complexity, and this relation is discussed here. It is also shown that computational mechanics can be reformulated in terms of Frank Knight''s prediction process, which is of both conceptual and technical interest. In particular, it allows for a unified treatment of different processes and facilitates topological considerations. Continuity of the Markov transition kernel of a discrete version of the prediction process is obtained as a new result.

Les alliages à haute entropie (HEAs) sont des alliages métalliques composés de 5 éléments ou plus, présents en proportions equimolaire ou presque. Depuis leur apparition dans le domaine de la métallurgie, leurs propriétés intéressantes ont causé un intérêt croissant de la part de la communauté scientifique pour essayer de les comprendre et les prédire. Plusieurs de ces propriétés peuvent aussi être observées dans d'autres systèmes cristallins ayant moins d'éléments, comme les solutions solides concentrées (CSAs) composées de FeNiCr. Ce mémoire présente les effets du calcul des préfacteurs par l'approximation harmonique de la théorie de l'état de transition (hTST) sur la diffusion d'une lacune dans une CSA en FeNiCr modélisée par un algorithme Monte-Carlo cinétique (KMC) hors réseau. Ce travail est motivé par les nombreux débats qui entourent la diffusion lente dans les HEAs et autres CSAs hautement désordonnés. Bien que cette caractéristique ait été proposée et utilisée pour expliquer certaines des propriétés les plus intéressantes des HEAs, les mécanismes de diffusion dans ceux-ci sont encore mal compris. Dans des travaux précédents, il a été démontré que les préfacteurs hTST dans une CSA FeNiCr peuvent avoir des valeurs qui s'étalent sur plusieurs ordres de grandeur. En partant de ces résultats, l'influence de cette variation de préfacteurs sur la diffusion d'une lacune est étudiée. Grâce à une analyse comparative entre des simulations utilisant des préfacteurs hTST et constants, le rôle de l'entropie dans la diffusion est étudié. Plus de un millions d'évènements au total sont trouvés dans les 96 simulations effectuées dans chaque type de simulation, fournissant une base statistique solide. Ces simulations KMC ont été performées par l'algorithme d'activation-relaxation cinétique (kART) couplé au potentiel Bonny-2013 pour les calculs de surface d'énergie potentielle (PEL). Nous démontrons que, en plus de l'entropie configurationnelle, le désordre affecte aussi l'entropie vibrationnelle, et que ce phénomène peut être à la base de diverses propriétés de ces systèmes, y compris leur diffusion lente. Les résultats présentés ne peuvent être obtenus que grâce à une analyse cinétique du système. En effet, la dynamique obtenue ne peut pas être extraite directement de l'évaluation statique du PEL, car la corrélation de sélection des événements est construite à partir des contributions combinées du préfacteur et des barrières énergétiques. Cette combinaison affecte la loi de compensation qui est mesurée, selon si le calcul de cette loi est effectué avec les évènements qui sont disponibles ou avec les évènements sélectionnés. Une introduction, ainsi que deux chapitres sur les HEAs et sur la théorie de l'état de transition débutent ce travail, suivis de la méthodologie, présentée au chapitre 4, et de l'article au chapitre 5. / High entropy alloys (HEAs) are metallic alloys composed of 5 or more elements, present in equimolar or near equimolar proportions. Since their appearance in the field of metallurgy at the beginning of the XXIst century, their properties have caused a growing interest from the scientific community in order to understand and predict certain of these properties. Many of them can also be observed in other crystalline systems with fewer elements, such as concentrated solid solution (CSAs) composed of FeNiCr. This masters' thesis presents the effect that the computation of prefactors by the harmonic approximation of the transition state theory (hTST) has on the diffusion of a single vacancy in a FeNiCr CSA, simulated by a kinetic Monte Carlo algorithm (KMC). The debate around a sluggish defect diffusion, proposed as a core effect of HEAs and CSAs with high amounts of disorder motivates this research. Indeed, even though this characteristic is often used to explain some of the most interesting properties of HEAs, the diffusion mechanisms are still not fully comprehended. In a previous study, it was demonstrated that hTST prefactors span over several orders of magnitude. Based on these results, we study the impact of hTST on diffusion. Through a comparative analysis between simulations using hTST and constant prefactors, the role of entropy on diffusion is studied. More than one million events in total are found in the 96 simulations performed for each type of simulation, providing a solid statistical basis for this analysis. These KMC simulations were performed by the kinetic activation-relaxation technique (kART) coupled with the Bonny-2013 potential for potential energy landscape (PEL) calculations. We demonstrate that both disorder and configurational entropy strongly affect the vibrational entropy, and that this can be responsible for various properties of these systems, including their sluggish diffusion. Presented results can only be obtained by a kinetic study of the system. The kinetic patterns that are observed can not be obtained by only the static analysis of the PEL for the combination of both prefactors and energy barriers affects event selection. This selection of events can change the compensation law that is measured whether it is computed using available events or selected events. Two chapters on HEAs and transition state theory, as well as a chapter on the methodology are presented before these results.

La méthode de l’ansatz de Bethe est introduite et utilisée dans ce mémoire. Elle est employée afin de diagonaliser un opérateur dit de Heun. Cette méthode est appliquée en construisant directement, dans les cas des polynômes de Racah et de q–Racah, les opérateurs dynamiques à partir de leurs formes génériques et de leurs relations de commutation. Il devient alors possible d’obtenir les équations de Bethe, qui si elles sont respectées, conduisent à des vecteurs propres de l’opérateur de Heun. Avec cet opérateur, qui commute avec la matrice de corrélation tronquée, nous pouvons alors déterminer l’entropie d’intrication d’une chaîne fermionique basée sur les polynômes de q–Racah. / A Bethe ansatz method is introduced in this master’s thesis. This method is used to diagonalize a Heun operator. It is applied by directly building the dynamical operators from the commutation relations and their general form, in connection with the Racah and the q–Racah polynomials. We can then find the Bethe equations, and when these are satisfied, eigenvectors of the Heun operator are obtained. With this operator, which commutes with the truncated correlation matrix, it becomes possible to find the entanglement entropy of a free fermion chain based on the q–Racah polynomials.

Im Mittelpunkt der Arbeit stehen Alanate und Aluminiumhydride der Übergangs- und Erdalkalimetalle in Bezug auf ihre Synthese, ihre thermodynamischen Eigenschaften, ihre Zersetzungsreaktionen und ihre potentielle Eignung für reversible Wasserstoffspeicheranwendungen. Es konnten für CeAlH6, Mg(AlH4)2, Ca(AlH4)2, Sr(AlH4)2, CaAlH5 und SrAlH5 die Wärmekapazitätsfunktionen sowie die absoluten Entropien und Bildungsenthalpien bei 298,15 K bestimmt werden. Weiterhin wurden Methoden zur Näherung der genannten thermodynamischen Daten evaluiert und mit den ermittelten Werten aktualisiert und erweitert. Außerdem wurde die im Bereich der komplexen Hydride übliche Annahme, dass sich die bei der Synthese gebildeten Nebenprodukte LiCl und NaCl gegenüber den Hydriden inert verhalten, überprüft. Es konnte gezeigt werden, dass die Annahme hinsichtlich der Eigenschaften und Zersetzungsreaktionen von Mg(AlH4)2, Ca(AlH4)2 und Sr(AlH4)2 Bestand hat. In Bezug auf die Dehydrierung von CaAlH5 und SrAlH5 ist sie jedoch ungültig, wenn die Probe LiCl enthält.

L'analyse de séries temporelles biomédicales chaotiques tirées de systèmes dynamiques non-linéaires est toujours un challenge difficile à relever puisque dans certains cas bien spécifiques les techniques existantes basées sur les multi-fractales, les entropies et les graphes de récurrence échouent. Pour contourner les limitations des invariants précédents, de nouveaux descripteurs peuvent être proposés. Dans ce travail de recherche nos contributions ont porté à la fois sur l’amélioration d’indicateurs multifractaux (basés sur une fonction de structure) et entropiques (approchées) mais aussi sur des indicateurs de récurrences (non biaisés). Ces différents indicateurs ont été développés avec pour objectif majeur d’améliorer la discrimination entre des signaux de complexité différente ou d’améliorer la détection de transitions ou de changements de régime du système étudié. Ces changements agissant directement sur l’irrégularité du signal, des mouvements browniens fractionnaires et des signaux tirés du système du Lorenz ont été testés. Ces nouveaux descripteurs ont aussi été validés pour discriminer des fœtus en souffrance de fœtus sains durant le troisième trimestre de grossesse. Des mesures statistiques telles que l’erreur relative, l’écart type, la spécificité, la sensibilité ou la précision ont été utilisées pour évaluer les performances de la détection ou de la classification. Le fort potentiel de ces nouveaux invariants nous laisse penser qu’ils pourraient constituer une forte valeur ajoutée dans l’aide au diagnostic s’ils étaient implémentés dans des logiciels de post-traitement ou dans des dispositifs biomédicaux. Enfin, bien que ces différentes méthodes aient été validées exclusivement sur des signaux fœtaux, une future étude incluant des signaux tirés d’autres systèmes dynamiques nonlinéaires sera réalisée pour confirmer leurs bonnes performances. / The analysis of biomedical time series derived from nonlinear dynamic systems is challenging due to the chaotic nature of these time series. Only few classical parameters can be detected by clinicians to opt the state of patients and fetuses. Though there exist valuable complexity invariants such as multi-fractal parameters, entropies and recurrence plot, they were unsatisfactory in certain cases. To overcome this limitation, we propose in this dissertation new entropy invariants, we contributed to multi-fractal analysis and we developed signal-based (unbiased) recurrence plots based on the dynamic transitions of time series. Principally, we aim to improve the discrimination between healthy and distressed biomedical systems, particularly fetuses by processing the time series using our techniques. These techniques were either validated on Lorenz system, logistic maps or fractional Brownian motions modeling chaotic and random time series. Then the techniques were applied to real fetus heart rate signals recorded in the third trimester of pregnancy. Statistical measures comprising the relative errors, standard deviation, sensitivity, specificity, precision or accuracy were employed to evaluate the performance of detection. Elevated discernment outcomes were realized by the high-order entropy invariants. Multi-fractal analysis using a structure function enhances the detection of medical fetal states. Unbiased cross-determinism invariant amended the discrimination process. The significance of our techniques lies behind their post-processing codes which could build up cutting-edge portable machines offering advanced discrimination and detection of Intrauterine Growth Restriction prior to fetal death. This work was devoted to Fetal Heart Rates but time series generated by alternative nonlinear dynamic systems should be further considered.