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L’objet de cette thèse est l’étude des catégories singulières des k-algèbres associatives surun anneau commutatif k. On développe la théorie de Morita pour les catégories singulières. Plus précisément, on propose une définition d’équivalence singulière à la Morita avec niveau, qui généralise la notion d’équivalence stable à la Morita introduite par Michel Broué. On montre qu’une équivalence dérivée de type standard induit une équivalence singulière à la Morita avec niveau. La deuxième partie de cette thèse est l’étude de la cohomologie de Hochschild singulière HH_sg(A,A) c’est-à-dire, l’espace des morphismes de A vers A[i] dans la catégorie singulière Dsg(A Aop) pour tous les nombres entiers i. Similaire à la cohomologie de Hochschild HH_(A,A), on montre que la cohomologie de Hochschild singulière HH_sg(A,A) est une algèbre de Gerstenhaber et donne une interprétation pour le crochet de Lie sur HH_sg(A,A) du point de vue de la théorie de PROP. On peut associer un complexe de cochaînes, qu’on appelle complexe de cochaînes de Hochschild singulières, C_sg(A,A) qui calcule la cohomologie de Hochschild singulière HH_sg(A,A). Alors on étudie une structure algébrique supérieure (e.g. l’algèbre de B1) sur C_sg(A,A) et propose une version singulière d’une conjecture de Deligne. L’objet de la troisième partie de cette thèse est de montrer que la structure d’algèbre de Gerstenhaber sur la cohomologie de Hochschild singulière est invariante par équivalences dérivées et équivalences singulières à la Morita avec niveau. L’idée de cette démonstration est analogue à l’approche développée par Keller lorsqu’il démontre que la structure d’algèbre de Gerstenhaber sur la cohomologie de Hochschild est invariante par équivalences dérivées. Similaire à la démonstration par Keller, on réalise HH_sg(A,A) avec le crochet de Lie comme une algèbre de Lie graduée du groupe algébrique gradué associé au groupe de Picard singulière sgDPic(A). / In this thesis, we are concerned with some aspects of singular categories of unitalassociative k-algebras over a commutative ring k. First, we develop a Morita theory for singular categories. Analogous to the classical Morita theory, we propose a definition of singular equivalence of Morita type with level. This follows and generalizes a definition of stable equivalence of Morita type introduced by Michel Broué. A derived equivalence of standard type induces a singular equivalence of Morita type with level. Second, we study the Hom-space from A to A[i] in the singular category Dsg(AkAop) of the enveloping algebra AkAop, where A is an associative k-projective k-algebra and i is any integer. Recall that the i-th Hochschild cohomology group HHi(A,A) can be realized as the Hom-space from A to A[i] in the bounded derived category Db(A k Aop). From this motivation, we call HomDsg(AkAop)(A,A[i]) the i-th singular Hochschild cohomology group and denote this group by HHi sg(A,A). Analogous to the Hochschild cohomology ring HH_(A,A), we prove that there is a Gerstenhaber algebra structure on the singular Hochschild ring HH_sg(A,A) and provide an interpretation of the Lie bracket from the point of view of PROP theory. We also associate a cochain complex, which we call singular Hochschild cochain complex, C_sg(A,A) to the singular Hochschild cohomology. Thenwe study the higher algebraic structures (e.g. B1-algebra) on C_sg(A,A) and propose asingular version of the Deligne conjecture. Following Keller’s approach which was developed for derived equivalences, we establish the invariance of the Gerstenhaber algebra structure which we defined on the singular Hochschild cohomology under singular equivalence of Morita type with level. In this proof, we define the singular derived Picard group sgDPic(A) of an associative algebra A and develop what we call a singular infinitesimal deformation theory. Then we realize HH_sg(A,A) as the graded Lie algebra of the ‘graded algebraic group’ associated to sgDPic(A).

International Telemetering Conference Proceedings / October 18-21, 2004 / Town & Country Resort, San Diego, California / This paper describes a new approach to object recognition by using ground-penetrating radar (GPR) imaging systems. The recognition procedure utilizes the spectral content instead of the object shape in traditional methods. To produce the identification feature of an object, the most common spectral component is obtained by singular value decomposition (SVD) of the training sets. The identification process is then integrated into the backward propagation image reconstruction algorithm, which is implemented on the FMCW GPR imaging systems.

This study examines singular readings in the Gospel of Matthew across five of the earliest extant Greek copies of Matthew: Codex Sinaiticus, Vaticanus, Ephraemi, Bezae, and Washingtonianus. In each of the selected MSS, it is determined where a spelling, word, clause, phrase, sentence, or group of sentences is different from other MSS. These “singular readings” are collected in order to shine light on what such idiosyncrasies can tell us about the MS or tendencies of the scribe who copied the MS. One of the more interesting finds is that some of our MSS add text more than they omit it, which is contrary to other studies. Apart from itacistic changes, alternate spellings are not always the most frequent type of singular reading in our MSS. The MSS have similar types of singular readings, but they often go about creating them in different ways. Conclusions are that our MSS either prefer Attic Greek to Koine (Washingtonianus) or vice versa (Sinaiticus), but two MSS (Vaticanus and Bezae) fluctuate between both grammatical standards. Our MSS typically have a high percentage of error due to parablepsis, but one MS seems to skip letters within words more often than entire words (Ephraemi). Ephraemi does not transpose words, but when the other MSS create transpositions, they all record instances where the genitive pronoun is placed prior to the word it modifies and verbs are moved forward in sentences. In addition, transpositions in Sinaiticus could have resulted from corrected leaps. Context often plays a part in the creation of singular readings, but context affects each MS differently. Nearby text seems to prompt changes in all of our MSS, but remote text such as a gospel parallel, does not often influence our scribes: Ephraemi contains the only harmonization seems to be intentional. In Sinaiticus and Washingtonianus, several readings exhibit possible interpretations of the text (but typically these do not appear to be theological changes) and they both contain readings that conflate textual variants. All of the singular readings record either a textual addition, omission, or substitution, but the MSS do not end up with the same amount of text: both Codex Vaticanus and Ephraemi add more words than they omit, whereas Codex Sinaiticus, Bezae, and Washingtonianus end up with more omissions. This final element adds a counterweight to other studies that contend MSS omit text more than they add. The examination yields few singular readings of dramatic theological import. Rather, the singular readings expose grammatical currents of the 4th-5/6th centuries, currents that are more prevalent than scribal attempts to re-present the text of Matthew.

Bekanntlich sind atmosphärische Rossbywellezüge (RWTs) Lösungen der Singular Vector Analyse eines gedämpften, barotropen Modells mit Nordwinter Grundströmen. In den SV Basen der verwendeten 40 DJF Grundströme konnten nur wenige wachsende den Rossbywellenzügen ähnliche (RWT Moden) Singulären Vektoren (SVen) gefunden werden. Die RWT Moden kommen nur in wenigen Gebieten der Erde vor. Die instabilste Mode entwickelt sich in der Region des Nordpazifiks (NPACs) innerhalb von 4 Tagen in jedem der verwendeten beobachteten DJF Grundströme. Alle anderen RWT Moden kommen nur bei Verwendung einiger der Grundströme vor. Ihre Entwicklungspfade sind eindeutig für Entwicklungszeiten bis zu 96 h und streuen für längere Zeiten. Die NPAC Mode erklärt zum Optimierungszeitpunkt 96 h bis zu 60 % der atmosphärischen kinetischen Energie (KE) auf der 300 hPa Fläche in der NPAC Region. Es konnte auch gezeigt werden, daß die Zeitreihe des beobachteten Wachstums der NPAC Mode mit dem berechneten Wachstum (den Eigenwerten) konsistent ist. Interessanterweise zeigt die NPAC-KE zum Optimierungszeitpunkt 96 h auch eine schwach signifikante Korrelation mit dem PNA-Index, die für die Optimierungszeit 144 h nicht mehr existiert. Die Ergebnisse legen die Vermutung nahe, daß die verwendeten Grundströme die Entwicklung der RWT Moden bis zu einer Entwicklungszeit von 4 Tagen dominieren und daß die finite Instabilität maßgeblich zur Entwicklung der beobachteten NPAC Rossbywellenzüge in der Atmosphäre beiträgt. Die Ergebnisse geben Hinweise darauf, daß die NPAC mode auch einen Beitrag zur Entwicklung der PNA leistet.

Estruturas de dados volumétricas são de extrema utilidade em várias aplicações, e em particular na área de Visualização Científica. Essas estruturas são úteis em duas etapas do processo de visualização. A primeira é na representação de dados, isto é, de informações associadas aos valores, medidos ou simulados, os quais se deseja visualizar. A outra fase que necessita de uma estrutura de dados é a fase de exploração, ou seja, o modelo criado é usado tanto para exploração interativa quanto para a realização de simulações sobre o mesmo, por exemplo, numa cirurgia virtual. Está em desenvolvimento no ICMC, uma estrutura de dados volumétrica, chamada Singular Half-Face, que tem como característica a modelagem explícita das singularidades presentes no modelo, além de outros elementos topológicos. Este trabalho de mestrado visa testar a viabilidade da estrutura num contexto de visualização em fluxo de dados, incorporando-a à biblioteca gráfica de visualização Visualization ToolKit (VTK), que possui definição extremamente pobre da topologia dos objetos que representa. Adicionando essa nova classe e realizando sobre ela técnicas convencionais de visualização e exploração de dados, é estudada sua capacidade de apoiar todas as fases do processo de visualização. / Volumetric data structures are of extreme utility in various applications, and particularly in the area of Scientific Visualization. These structures are useful in two stages of the visualization process. The first one is in data representation, that is, in the organization of information associated with the values, measured or simulated, to be visualized. The other phase of the visualization data flow that needs a data structure is the exploration phase. It would be useful to have a model designed to be used both for interactive exploration and for simulation in a number of application, for example, in virtual surgery. A volumetric data structure, named Singular Half-Face (SHF) is being developed at the ICMC, that has as its main characteristic the explicit modeling of singularities present in the model, besides other essential topological elements. This dissertation aims at testing the viability of this structure in a context of visualization in data flow, by incorporating it in the Visualization ToolKit (VTK) class library, whose data structures are extremely poor in definition of the topology of the objects that they represent. By adding SHF to this library and carrying out conventional visualization and data exploration on it, we wish to study its support to all the phases of the visualization process.

The primary objects of study in this thesis are noncommutative surfaces; that is, noncommutative noetherian domains of GK dimension 2. Frequently these rings will also be singular, in the sense that they have infinite global dimension. Very little is known about singularities of noncommutative rings, particularly those which are not finite over their centre. In this thesis, we are able to give a precise description of the singularities of a few families of examples. In many examples, we lay the foundations of noncommutative singularity theory by giving a precise description of the singularities of the fundamental examples of noncommutative surfaces. We draw comparisons with the fundamental examples of commutative surface singularities, called Kleinian singularities, which arise from the action of a finite subgroup of SL(2; k) acting on a polynomial ring. The main tool we use to study the singularities of noncommutative surfaces is the singularity category, first introduced by Buchweitz in [Buc86]. This takes a (possibly noncommutative) ring R and produces a triangulated category Dsg(R) which provides a measure of "how singular" R is. Roughly speaking, the size of this category reflects how bad the singularity is; in particular, Dsg(R) is trivial if and only if R has finite global dimension. In [CBH98], Crawley-Boevey-Holland introduced a family of noncommutative rings which can be thought of as deformations of the coordinate ring of a Kleinian singularity. We give a precise description of the singularity categories of these deformations, and show that their singularities can be thought of as unions of (commutative) Kleinian singularities. In particular, our results show that deforming a singularity in this setting makes it no worse. Another family of noncommutative surfaces were introduced by Rogalski-Sierra-Stafford in [RSS15b]. The authors showed that these rings share a number of ring-theoretic properties with deformations of type A Kleinian singularities. We apply our techniques to show that the "least singular" example has an A1 singularity, and conjecture that other examples exhibit similar behaviour. In [CKWZ16a], Chan-Kirkman-Walton-Zhang gave a definition for a quantum version of Kleinian singularities. These require the data of a two-dimensional AS regular algebra A and a finite group G acting on A with trivial homological determinant. We extend a number of results in [CBH98] to the setting of quantum Kleinian singularities. More precisely, we show that one can construct deformations of the skew group rings A#G and the invariant rings AG, and then determine some of their ring-theoretic properties. These results allow us to give a precise description of the singularity categories of quantum Kleinian singularities, which often have very different behaviour to their non-quantum analogues.