Une partition d'un entier positif est une façon d'écrire ce nombre comme une somme d'entiers strictement positifs où l'ordre des termes ne compte pas. Plusieurs généralisations des partitions ont été étudiées, parmi lesquelles les surpartitions, qui sont des partitions où l'on peut surligner la dernière occurrence d'un nombre, les paires de surpartitions ou encore les partitions n-colorées, qui sont liées à un modèle de physique statistique. Dans cette thèse, on généralise aux paires de surpartitions les identités d'Andrews-Gordon, qui sont une extension d'un résultat classique de la théorie des partitions : les identités de Rogers-Ramanujan. Pour cela, on définit deux classes de séries hypergéométriques basiques et on montre que ce sont les séries génératrices des paires de surpartitions vérifiant différents types de conditions (multiplicités, rangs successifs, dissection de Durfee) et de certains chemins du plan. On montre également que pour certaines valeurs des paramètres, ces séries peuvent s'écrire comme des produits infinis, ce qui conduit à plusieurs identités de type Rogers-Ramanujan. La démonstration utilise diverses méthodes combinatoires et analytiques. On définit enfin une généralisation des partitions n-colorées, les surpartitions n-colorées, et on les utilise pour interpréter combinatoirement certaines séries multiples et démontrer d'autres identités de type Rogers-Ramanujan.

We show that results on the characterization of admissible quantizations, which have been derived in Potzelberger [3], have to be modified in case the probability distribution has linear components. Furthermore, we provide an example, where the limit of optimal quantizations is not admissible. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Let P be a Borel probability measure on Rd. We characterize the maximal elements p E M(P,m) with respect to the Bishop-De Leeuw order, where p E M(P, m) if and only if p P and [supp(p)] m. The results obtained have important consequences for statistical inference, such as tests of homogeneity or multivariate cluster analysis and for the theory of comparison of experiments. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

No description available.

Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The primary purpose of this paper is to use generating functions to prove other vector partition identities that parallel results of integer partitions.

No description available.

Ce travail doctoral s'intéresse aux positionnements identitaires des enseignants et des élèves induits par la relation que la scène sociale en Éducation Physique et Sportive instaure entre eux. Ces positionnements sont abordés par le biais des théories liées à la catégorisation sociale et plus précisément la théorie de l'identité sociale, la théorie de l'auto-catégorisation et les partitions sociales. Si les partitions sociales sont un prolongement des deux premières théories, elles possèdent également leurs propres fondements théoriques, issus de la psychologie sociale du langage, qui offrent de nouvelles perspectives liées à l'approche de la catégorisation sociale. Un premier travail de réflexion a ainsi été mené pour proposer une relecture de certains principes de la catégorisation sociale. À la suite de cette construction théorique, trois études principales ont été réalisées afin de déterminer le poids des pré-jugements portés par la scène sociale de la classe sur les processus de catégorisation sociale lors de la mobilisation des identités d'enseignants et d'élèves en EPS. Dans un contexte où l'enseignant est institutionnellement le dominant hiérarchique, les élèves perçoivent-ils des espaces identitaires leur permettant d'occuper une position sociale satisfaisante au sein de cette relation imposée? La première étude, s'articulant autour de trois sous-études, cherche à déterminer le poids de la scène sociale sur les effets de catégorisation sociale. Dans une scène sociale à l'avantage des hommes (football), les femmes, contrairement à ce qui était attendu, cherchent à établir une position sociale favorable pour leur groupe par le jeu de la catégorisation sociale. La deuxième étude s'intéresse aux différentes insertions identitaires sur lesquelles les enseignants et les élèves en EPS peuvent s'appuyer lors de leurs interactions. Les résultats montrent, à partir de l'analyse des verbatim d'une séance d'EPS, que si le contexte prédétermine leurs relations sur la base de leurs rôles sociaux, d'autres espaces identitaires sont co-construits et investis sans pour autant remettre en cause les règles de la scène sociale. Enfin, la troisième étude propose une cartographie des positionnements identitaires qu'implique la relation intergroupe enseignants-élèves en EPS. À partir de l'utilisation de l'outil RepMut, il s'avère qu'au niveau catégoriel, seuls les enseignants sont en mesure de se positionner de façon satisfaisante. En effet, le rôle d'élève qu'instaure la relation à l'enseignant ne permet aux élèves de bénéficier d'une position catégorielle que s'ils usent d'une dimension de comparaison différente de celle établie par la scène sociale. En conclusion, ce travail montre qu'en EPS, et a priori de façon plus générale, la relation enseignants-élèves ne permet qu'aux premiers d'occuper une place sociale conforme à celle attendue. En effet, face à l'enseignant, il semble bien difficile pour les élèves d'investir le rôle que l'école attend d'eux. / This doctoral work is interested in the identical positionings of the teachers and the pupils led by the relation that the social scene in Physical Education establishes between them. These positionings are approached by means of the theories bound to the social categorization and more precisely the theory of the social identity, the theory of the auto-categorization and the conception of the social partitions. If the social partitions establish a continuation of the first two theories, they also rest on their own theoretical foundation, based on social psychology of the language. The goal of the first work was to propose a review of certain principles of the social categorization. This proofreading leaned in particular on the idea that the positionings of every group co-build themselves from the relation intergroup, and that the positions adopted by each are not necessarily symmetric. Following this theoretical construction, three main studies were led to determine the weight of pre-judgments carried by the social scene of the classroom on the processes of social categorization during the mobilization of the identities of teachers and pupils in Physical Education. More precisely, in a context where the teacher is institutionally the hierarchical dominant of the pupils, the pupils do perceive identical spaces allowing them to occupy a satisfactory social position within this relation. The first study, articulating around three sub-studies, tries to determine the weight of the social scene on the effects of social categorization. Contrary to what was expected in this social scene to the advantage of the men (soccer), the women manage to establish a favorable social position for their group by the game of the social categorization. The second study focuses on the various identical insertions on which teachers and pupils in Physical Education can lean during their interactions. The results show, from the analysis of the verbatim of a session of Physical Education that, if the context predetermines their relations on the basis of their social roles, other identical spaces are co-built and invested without questioning the rules of the social scene. Finally, the third study proposes a mapping of the identical positionings which implies the relation intergroup teachers-pupils in physical education. The tool RepMut allows to highlight that at the category-specific level, only teachers are capable of positioning in a satisfactory way. Indeed, the relation to the teacher does not allow pupils to benefit from a satisfactory category-specific position, if it is not by using dimensions of comparison different from that established by the social scene. In conclusion, this work shows that in Physical Education, and prejudice in a more general way, the relation teachers-pupils allows only the first ones to occupy a social place in compliance with that waited. Indeed, in front of the teacher, it seems difficult for pupils to invest the role for which the school expects from them.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / In this thesis, various partition functions with respect to Rogers-Ramanujan identities and George Andrews' partitions with initial repetitions are studied. Agarwal and Goyal gave a three-way partition theoretic interpretation of the Rogers- Ramanujan identities. We generalise their result and establish certain connections with some work of Connor. Further combinatorial consequences and related partition identities are presented. Furthermore, we re ne one of the theorems of George Andrews on partitions with initial repetitions. In the same pursuit, we construct a non-diagram version of the Keith's bijection that not only proves the theorem, but also provides a clear proof of the re nement. Various directions in the spirit of partitions with initial repetitions are discussed and results enumerated. In one case, an identity of the Euler-Pentagonal type is presented and its analytic proof given. / M T 2016

Neste trabalho, com base em representações por matrizes de duas linhas para alguns tipos de partição (algumas já conhecidas e outras novas), identificamos propriedades sugeridas por classificá-las de acordo com a soma dos elementos de sua segunda linha. Esta soma sempre fornece alguma propriedade da partição relacionada. Se considerarmos versões sem sinal de algumas funções Mock Theta, seu termo geral pode ser interpretado como função geradora para algum tipo de partição com restrições. Para retornar aos coeficientes originais, é possível definir um peso para cada matriz e depois somá-las para contá-los. Uma representação análoga para essas partições nos permite observar propriedades sobre elas, novamente por meio de uma classificação referente à soma dos seu elementos da segunda linha. Esta seriação é feita por meio de tabelas criadas pelo software matemático Maple, as quais nos sugerem padrões e identidades relacionadas com outros tipos de partições conhecidas e, muitas vezes, encontrando uma fórmula fechada para contá-las. Tendo as conjecturas obtidas, elas são provadas por meio de bijeções entre conjuntos ou por contagem. / In this work, based on representations by matrices of two lines for some kind of partition (some already known and other new ones), we identify properties suggested by classifying them according to the sum of its second line. This sum always provides some properties of the related partition. If we consider unsigned versions of some Mock Theta Functions, its general term can be interpreted as generating function for some kind of partition with restrictions. To come back to the original coefficients, you can set a weight for each array and so add them to evaluate the coefficients. An analogous representation for partitions allows us to observe properties, again by classificating them according to the sum of its elements on the second row. This classification is made by means of tables created by mathematical software Maple, which suggest patterns, identities related to other known types of partitions and often, finding a closed formula to count them. Having established conjectured identities, all are proved by bijections between sets or counting methods.

Wong Sze Hon. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 88-91). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Preliminaries --- p.1 / Chapter 1.2 --- Motivations --- p.1 / Chapter 1.3 --- Contributions --- p.3 / Chapter 1.4 --- Organization of the Thesis --- p.4 / Chapter 2 --- VLSI Physical Design Automation --- p.5 / Chapter 2.1 --- Preliminaries --- p.5 / Chapter 2.2 --- VLSI Design Cycle [1] --- p.6 / Chapter 2.2.1 --- System Specification --- p.6 / Chapter 2.2.2 --- Architectural Design --- p.6 / Chapter 2.2.3 --- Functional Design --- p.6 / Chapter 2.2.4 --- Logic Design --- p.8 / Chapter 2.2.5 --- Circuit Design --- p.8 / Chapter 2.2.6 --- Physical Design --- p.8 / Chapter 2.2.7 --- Fabrication --- p.8 / Chapter 2.2.8 --- Packaging and Testing --- p.9 / Chapter 2.3 --- Physical Design Cycle [1] --- p.9 / Chapter 2.3.1 --- Partitioning --- p.9 / Chapter 2.3.2 --- Floorplanning and Placement --- p.11 / Chapter 2.3.3 --- Routing --- p.11 / Chapter 2.3.4 --- Compaction --- p.12 / Chapter 2.3.5 --- Extraction and Verification --- p.12 / Chapter 2.4 --- Chapter Summary --- p.12 / Chapter 3 --- Recent Approaches on Circuit Partitioning --- p.14 / Chapter 3.1 --- Preliminaries --- p.14 / Chapter 3.2 --- Circuit Representation --- p.15 / Chapter 3.3 --- Delay Modelling --- p.16 / Chapter 3.4 --- Partitioning Objectives --- p.19 / Chapter 3.4.1 --- Interconnections between Partitions --- p.19 / Chapter 3.4.2 --- Delay Minimization --- p.19 / Chapter 3.4.3 --- Area and Number of Partitions --- p.20 / Chapter 3.5 --- Partitioning Algorithms --- p.20 / Chapter 3.5.1 --- Cut-size Driven Partitioning Algorithm --- p.21 / Chapter 3.5.2 --- Delay Driven Partitioning Algorithm --- p.32 / Chapter 3.5.3 --- Acyclic Circuit Partitioning Algorithm --- p.33 / Chapter 4 --- Clustering Based Acyclic Multi-way Partitioning --- p.38 / Chapter 4.1 --- Preliminaries --- p.38 / Chapter 4.2 --- Previous Works on Clustering Based Partitioning --- p.39 / Chapter 4.2.1 --- Multilevel Circuit Partitioning [2] --- p.40 / Chapter 4.2.2 --- Cluster-Oriented Iterative-Improvement Partitioner [3] --- p.42 / Chapter 4.2.3 --- Section Summary --- p.44 / Chapter 4.3 --- Problem Formulation --- p.45 / Chapter 4.4 --- Clustering Based Acyclic Multi-Way Partitioning --- p.46 / Chapter 4.5 --- Modified Fan-out Free Cone Decomposition --- p.47 / Chapter 4.6 --- Clustering Phase --- p.48 / Chapter 4.7 --- Partitioning Phase --- p.51 / Chapter 4.8 --- The Acyclic Constraint --- p.52 / Chapter 4.9 --- Experimental Results --- p.57 / Chapter 4.10 --- Chapter Summary --- p.58 / Chapter 5 --- Network Flow Based Multi-way Partitioning --- p.61 / Chapter 5.1 --- Preliminaries --- p.61 / Chapter 5.2 --- Notations and Definitions --- p.62 / Chapter 5.3 --- Net Modelling --- p.63 / Chapter 5.4 --- Previous Works on Network Flow Based Partitioning --- p.64 / Chapter 5.4.1 --- Network Flow Based Min-Cut Balanced Partitioning [4] --- p.65 / Chapter 5.4.2 --- Network Flow Based Circuit Partitioning for Time-multiplexed FPGAs [5] --- p.66 / Chapter 5.5 --- Proposed Net Modelling --- p.70 / Chapter 5.6 --- Partitioning Properties Based on the Proposed Net Modelling --- p.73 / Chapter 5.7 --- Partitioning Step --- p.75 / Chapter 5.8 --- Constrained FM Post Processing Step --- p.79 / Chapter 5.9 --- Experiment Results --- p.81 / Chapter 6 --- Conclusion --- p.86 / Bibliography --- p.88