<p>U ovoj tezi dokazane su teoreme o nepokretnoj tački koje predstavljaju jednoznačna i vi&scaron;eznačna uop&scaron;tenja Banahovog prin&shy;cipa kontrakcije u verovatnosnim metričkim i fazi metričkim pros&shy;torima. Dokazana je teorema koja predstavlja uop&scaron;tenje teoreme o nepokretnoj tački za verovatnosnu ^-kontrakciju / :&nbsp; S &mdash;* S,gde je&nbsp; ( S ,&nbsp; J7, T ) kompletan Mengerov prostor. Uveden je pojam jake (6n)-kontrakcije i dokazana je teorema koja predstavlja uop&scaron;tenje teoreme Sehgala i Bharuche-Reid kada je preslikavanje / :&nbsp; S &mdash;&gt; S jaka (6n)-kontrakcija. Teorema Caristija, koja predstavlja jedan od najvažnijih rezultata za teoriju nepokretne tačke i nelinearnu analizu uop&scaron;tena je u kompletnom Mengerovom prostoru&nbsp; (S ,&nbsp; F , T ), gde je t-norma&nbsp; T H -tipa. Kako Mengerovi prostori pripadaju klasi kvazi-uniformnih prostora dokazana je teorema o nepokretnoj tački tri preslikavanja u jednoj specijalnoj klasi kvazi-uniformnih prostora. Dokazana je teorema o nepokretnoj tački koja predstavlja verovatnosno uop&scaron;tenje Nadlerove g-kontrakcije za tri preslikavanja kao i uop&scaron;tenje Hiksovog principa kontrakcije za tri preslikavanja. Teorija<br />kontraktora, koju je uveo M. Altman, odnosi se na re&scaron;avanje nelin&shy;earnih operatorskih jednačina u Banahovim prostorima. U tezi su dokazane teoreme koje obezbeđuju postojanje i jedinstvenost re&scaron;enja za nelinearne operatorske jednačine sa jednoznačnim i vi&scaron;eznačnim operatorom u nearhimedovskim Mengerovim verovatnosnim normi&shy; ranim prostorima.</p> / <p>In this thesis fixed point theorems which present singleval&shy;<br />ued and multivalued generalization of Banach contraction principle<br />in probabilistic metric and fuzzy metric spaces are proved. The<br />theorem which presents generalization of fixed point theorem for<br />probabilistic g-contraction / :&nbsp; S &mdash;* S is proved, where ( S ,&nbsp; J7, T ) is<br />complete Menger space. A notion of strong (&pound;&gt;n)-contraction is in&shy;<br />troduced and the theorem which presents a generalization of Sehgal<br />and Bharucha-Raid theorem when the mapping / :&nbsp; S&nbsp; &mdash;&raquo;&nbsp; S is strong<br />(6n)-contraction is proved. Caristi&rsquo;s theorem, which presents one of<br />the most imortant results for the fixed point theorem and nonlinear<br />analysis is generalized in complete Menger space&nbsp; (S, J-, T ), where<br />t-norm&nbsp; T is of&nbsp; H -type. As Menger&rsquo;s spaces belong to the class of<br />quasi-uniformizable spaces, the fixed point theorem for three map&shy;<br />pings in one special class of quasi-uniformizable spaces is proved.<br />The fixed point theorem which presents a probabilistic generaliza&shy;<br />tion of Nadler g-contraction for three mappings is proved as well<br />as the generalization of Hicks&rsquo;s contraction principle for three map&shy;<br />pings. The theory of contractor, which was introduced by M. A lt&shy;<br />man refers to solving nonlinear operator equations in Banach spaces.<br />This thesis proves the theorems which provide the existence and<br />uniqueness of the solutions for nonlinear operator equations with<br />singlevalued and multivalued operators in nonarhimedian Menger&rsquo;s<br />probabilistic normed spaces</p>

Au cours de ce travail, nous nous intéressons aux limites d'échelle de deux classes de cartes. Dans un premier temps, nous regardons les quadrangulations biparties de genre strictement positif g fixé et, dans un second temps, les quadrangulations planaires à bord dont la longueur du bord est de l'ordre de la racine carrée du nombre de faces. Nous voyons ces objets comme des espaces métriques, en munissant leurs ensembles de sommets de la distance de graphe, convenablement renormalisée. Nous montrons qu'une carte prise uniformément parmi les cartes ayant n faces dans l'une de ces deux classes tend en loi, au moins à extraction près, vers un espace métrique limite aléatoire lorsque n tend vers l'infini. Cette convergence s'entend au sens de la topologie de Gromov--Hausdorff. On dispose de plus des informations suivantes sur l'espace limite que l'on obtient. Dans le premier cas, c'est presque sûrement un espace de dimension de Hausdorff 4 homéomorphe à la surface de genre g. Dans le second cas, c'est presque sûrement un espace de dimension 4 avec une frontière de dimension 2, homéomorphe au disque unité de R^2. Nous montrons en outre que, dans le second cas, si la longueur du bord est un petit~o de la racine carrée du nombre de faces, on obtient la même limite que pour les quadrangulations sans bord, c'est-à-dire la carte brownienne, et l'extraction n'est plus requise. / In this work, we discuss the scaling limits of two particular classes of maps. In a first time, we address bipartite quadrangulations of fixed positive genus g and, in a second time, planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We view these objects as metric spaces by endowing their sets of vertices with the graph metric, suitably rescaled.We show that a map uniformly chosen among the maps having n faces in one of these two classes converges in distribution, at least along some subsequence, toward a limiting random metric space as n tends to infinity. This convergence holds in the sense of the Gromov--Hausdorff topology on compact metric spaces. We moreover have the following information on the limiting space. In the first case, it is almost surely a space of Hausdorff dimension 4 that is homeomorphic to the genus g surface. In the second case, it is almost surely a space of Hausdorff dimension 4 with a boundary of Hausdorff dimension 2 that is homeomorphic to the unit disc of R^2. We also show that in the second case, if the length of the boundary is little-o of the square root of the number of faces, the same convergence holds without extraction and the limit is the same as for quadrangulations without boundary, that is the Brownian map.

Gegenstand der vorliegenden Dissertation ist die Entwicklung von Algorithmen und Methoden mit dem Ziel, Archäologen bei der täglichen wissenschaftlichen Arbeit zu unterstützen. Im Teil I werden Ideen präsentiert, mit denen sich die extrem zeitintensive und stellenweise stupide Funddokumentation beschleunigen lässt. Es wird argumentiert, dass das dreidimensionale Erfassen der Fundobjekte mittels Laser- oder Streifenlichtscannern trotz hoher Anschaffungskosten wirtschaftlich und vor allem qualitativ attraktiv ist. Mithilfe von nicht fotorealistischen Visualisierungstechniken können dann wieder aussagekräftige, aber dennoch objektive Bilder generiert werden. Außerdem ist speziell für Gefäße eine vollautomatische und umfassende Merkmalserhebung möglich. Im II. Teil gehen wir auf das Problem der automatisierten Gefäßklassifikation ein. Nach einer theoretischen Betrachtung des Typbegriffs in der Archäologie präsentieren wir eine Methodologie, in der Verfahren sowohl aus dem Bereich des unüberwachten als auch des überwachten Lernens zum Einsatz kommen. Besonders die letzteren haben sich dabei als überaus praktikabel erwiesen, um einerseits unbekanntes Material einer bestehenden Typologie zuzuordnen, andererseits aber auch die Struktur der Typologie selbst kritisch zu hinterfragen. Sämtliche Untersuchungen haben wir beispielhaft an den bronzezeitlichen Gräberfeldern von Kötitz, Altlommatzsch (beide Lkr. Meißen), Niederkaina (Lkr. Bautzen) und Tornow (Lkr. Oberspreewald-Lausitz) durchgeführt und waren schließlich sogar in der Lage, archäologisch relevante Zusammenhänge zwischen diesen Fundkomplexen herzustellen. / The topic of the dissertation at hand is the development of algorithms and methods aiming at supporting the daily scientific work of archaeologists. Part I covers ideas for accelerating the extremely time-consuming and often tedious documentation of finds. It is argued that digitizing the objects with 3D laser or structured light scanners is economically reasonable and above all of high quality, even though those systems are still quite expensive. Using advanced non-photorealistic visualization techniques, meaningful but at the same time objective pictures can be generated from the virtual models. Moreover, specifically for vessels a fully-automatic and comprehensive feature extraction is possible. In Part II, we deal with the problem of automated vessel classification. After a theoretical consideration of the type concept in archaeology we present a methodology, which employs approaches from the fields of both unsupervised and supervised machine learning. Particularly the latter have proven to be very valuable in order to assign unknown entities to an already existing typology, but also to challenge the typology structure itself. All the analyses have been exemplified by the Bronze Age cemeteries of Kötitz, Altlommatzsch (both district of Meißen), Niederkaina (district of Bautzen), and Tornow (district Oberspreewald-Lausitz). Finally, we were even able to discover archaeologically relevant relationships between these sites.

We study variants of P-frames and associated rings, which can be viewed as natural generalizations of the classical variants of P-spaces and associated rings. To be more precise, we de ne quasi m-rings to be those rings in which every prime d-ideal is either maximal or minimal. For a completely regular frame L, if the ring RL of real-valued continuous functions of L is a quasi m-ring, we say L is a quasi cozero complemented frame. These frames are less restricted than the cozero complemented frames. Using these frames we study some properties of what are called quasi m-spaces, and observe that the property of being a quasi m-space is inherited by cozero subspaces, dense z- embedded subspaces, and regular-closed subspaces among normal quasi m-space. M. Henriksen, J. Mart nez and R. G. Woods have de ned a Tychono space X to be a quasi P-space in case every prime z-ideal of C(X) is either minimal or maximal. We call a point I of L a quasi P-point if every prime z-ideal of RL contained in the maximal ideal associated with I is either maximal or minimal. If all points of L are quasi P-points, we say L is a quasi P-frame. This is a conservative de nition in the sense that X is a quasi P-space if and only if the frame OX is a quasi P-frame. We characterize these frames in terms of cozero elements, and, among cozero complemented frames, give a su cient condition for a frame to be a quasi P-frame. A Tychono space X is called a weak almost P-space if for every two zero-sets E and F of X with IntE IntF, there is a nowhere dense zero-set H of X such that E F [H. We present the pointfree version of weakly almost P-spaces. We de ne weakly regular rings by a condition characterizing the rings C(X) for weak almost P-spaces X. We show that a reduced f-ring is weakly regular if and only if every prime z-ideal in it which contains only zero-divisors is a d-ideal. We characterize the frames L for which the ring RL of real-valued continuous functions on L is weakly regular. We introduce the notions of boundary frames and boundary rings, and use them to give another ring-theoretic characterization of boundary spaces. We show that X is a boundary space if and only if C(X) is a boundary ring. A Tychono space whose Stone- Cech compacti cation is a nite union of closed subspaces each of which is an F-space is said to be nitely an F-space. Among normal spaces, S. Larson gave a characterization of these spaces in terms of properties of function rings C(X). By extending this notion to frames, we show that the normality restriction can actually be dropped, even in spaces, and thus we sharpen Larson's result. / Mathematics / D. Phil. (Mathematics)

<p>U ovoj tezi je definisana uop&scaron;tena konvolucija koja pripada domenu pseudo-analize i ima veliku primenu u mnogim matematičkim teorijama, npr. u proba-bilističkim metričkim prostorima, PDJ, teorijama odlučivanja, sistema, kontrole i fazi brojeva. Dokazane su bitne osobine ove operacije sa funkcijama. Dokazana je veza izmedju pseudo-konvolucija baziranih na poluprstenima različitih klasa Definisana je (5, C/)-konvolucija bazirana na uslovno distributivnom poluprstenu ([0,1], S, U)).Dat je jo&scaron; jedan vid uop&scaron;tenja konvolucije baziran na uop&scaron;tenim pseudo-operacijama.</p> / <p>In this thesis the generalized convolution have been defined. This operation with functions has applications in different mathematical theo&shy; ries, for example in Probabilistic Metric Spaces, PDE, System and Control Theory, Fuzzy numbers. Some basic properties of this operation has been proved, as well as connection between generalized convolutions based on dif&shy;ferent classes of semirings. (5, U)-convolution has been defined, as well as convolution based on generalized pseudo-operations.</p>

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)

Les travaux de ma thèse s'articulent en trois parties distinctes.Dans la première partie j'étudie les mesures de Malliavin-Shavguldize sur les difféomorphismes du cercle et de l'intervalle. Il s'agit de mesures de type « Haar » pour ces groupes de dimension infinie : elles furent introduites il a une vingtaine d'années pour permettre une étude de leur théorie des représentations. Un premier chapitre est dédié à recueillir les résultats présents dans la littérature et et les représenter dans une forme plus étendue, avec un regard particulier sur les propriétés de quasi-invariance de ces mesures. Ensuite j'étudie de problèmes de nature plus dynamique : quelle est la dynamique qu'on doit s'attendre d'un difféomorphisme choisi uniformément par rapport à une mesure de Malliavin-Shavguldize ? Je démontre en particulier qu'il y a une forte présence des difféomorphismes de type Morse-Smale.La partie suivante vient de mon premier travail publié, obtenu en collaboration avec Andrés Navas. Inspirés d'un théorème récent de Avila et Kocsard sur l'unicité des distributions invariantes par un difféomorphisme lisse minimal du cercle, nous analysons le même problème en régularité faible, avec des argument plus géométriques.La dernière partie est constituée des résultats récemment obtenus avec Mikhail Khristoforov et Victor Kleptsyn. Nous abordons les problèmes reliés à la gravité quantique de Liouville en étudiant des espaces auto-similaires qui sont la limite de graphes finis. Nous démontrons qu'il est possible de trouver des distances aléatoires non-triviales sur ces espaces qui sont compatibles avec la structure auto-similaire. / This thesis is divided into three different parts.In the first part, we study the Malliavin-Shavgulidze measure on circle and interval diffeomorphisms. They are Haar-like measures for these infinite-dimensional groups: they were introduced about twenty years ago to help to study their represantation theory. The first chapter collects the results that were obtained in the past years and in some cases we present them under a renewed point of view, with particular attention on quasi-invariance properties for this measures. Then we study some questions of dynamical nature: which is the typical dynamics that we must expect described by a diffeomorphism chosen randomly according to some Malliavin-Shavguldize measure? In particular, we prove that there is a strong presence of Morse-Smale diffeomorphisms.The third chapter comes from the published joint work with Andrés Navas. Inspired by a recent theorem by Avila and Kocsard about the uniqueness of the invariant distribution for a minimal smooth circle diffeomorphism, we analyse the same problem in low regularity, with more geometric arguments.The last part corresponds to the recent results obtained with Mikhail Khristoforov and Victor Kleptsyn. We consider problems in relation with Liouville quantum gravity, by studying self-similar metric spaces which are the limit of finite graphs. We prove that it is possible to find nontrivial random distances on these spaces which are compatible with the self-similar structure.

This thesis deals with mining of relevant information from documents and automatic splitting of multiple documents merged together. Moreover, it describes the design and implementation of software for data mining from documents and for automatic splitting of multiple documents. Methods for acquiring textual data from scanned documents, named entity recognition, document clustering, their supportive algorithms and metrics for automatic splitting of documents are described in this thesis. Furthermore, an algorithm of implemented software is explained and tools and techniques used by this software are described. Lastly, the success rate of the implemented software is evaluated. In conclusion, possible extensions and further development of this thesis are discussed at the end.

Gegenstand der vorliegenden Dissertation ist die Entwicklung von Algorithmen und Methoden mit dem Ziel, Archäologen bei der täglichen wissenschaftlichen Arbeit zu unterstützen. Im Teil I werden Ideen präsentiert, mit denen sich die extrem zeitintensive und stellenweise stupide Funddokumentation beschleunigen lässt. Es wird argumentiert, dass das dreidimensionale Erfassen der Fundobjekte mittels Laser- oder Streifenlichtscannern trotz hoher Anschaffungskosten wirtschaftlich und vor allem qualitativ attraktiv ist. Mithilfe von nicht fotorealistischen Visualisierungstechniken können dann wieder aussagekräftige, aber dennoch objektive Bilder generiert werden. Außerdem ist speziell für Gefäße eine vollautomatische und umfassende Merkmalserhebung möglich. Im II. Teil gehen wir auf das Problem der automatisierten Gefäßklassifikation ein. Nach einer theoretischen Betrachtung des Typbegriffs in der Archäologie präsentieren wir eine Methodologie, in der Verfahren sowohl aus dem Bereich des unüberwachten als auch des überwachten Lernens zum Einsatz kommen. Besonders die letzteren haben sich dabei als überaus praktikabel erwiesen, um einerseits unbekanntes Material einer bestehenden Typologie zuzuordnen, andererseits aber auch die Struktur der Typologie selbst kritisch zu hinterfragen. Sämtliche Untersuchungen haben wir beispielhaft an den bronzezeitlichen Gräberfeldern von Kötitz, Altlommatzsch (beide Lkr. Meißen), Niederkaina (Lkr. Bautzen) und Tornow (Lkr. Oberspreewald-Lausitz) durchgeführt und waren schließlich sogar in der Lage, archäologisch relevante Zusammenhänge zwischen diesen Fundkomplexen herzustellen. / The topic of the dissertation at hand is the development of algorithms and methods aiming at supporting the daily scientific work of archaeologists. Part I covers ideas for accelerating the extremely time-consuming and often tedious documentation of finds. It is argued that digitizing the objects with 3D laser or structured light scanners is economically reasonable and above all of high quality, even though those systems are still quite expensive. Using advanced non-photorealistic visualization techniques, meaningful but at the same time objective pictures can be generated from the virtual models. Moreover, specifically for vessels a fully-automatic and comprehensive feature extraction is possible. In Part II, we deal with the problem of automated vessel classification. After a theoretical consideration of the type concept in archaeology we present a methodology, which employs approaches from the fields of both unsupervised and supervised machine learning. Particularly the latter have proven to be very valuable in order to assign unknown entities to an already existing typology, but also to challenge the typology structure itself. All the analyses have been exemplified by the Bronze Age cemeteries of Kötitz, Altlommatzsch (both district of Meißen), Niederkaina (district of Bautzen), and Tornow (district Oberspreewald-Lausitz). Finally, we were even able to discover archaeologically relevant relationships between these sites.