Open books for contact five-manifolds and applications of contact homology

In the first half of this thesis, we use Giroux's construction of contact open books to construct contact structures on simply connected five-manifolds. This allows us to reprove a theorem of Geiges concerning the existence of contact structures in all homotopy classes of almost contact structures on simply-connected five-manifolds. In the second part of this thesis, we give an algorithm for computing the contact homology of some Brieskorn manifolds. As an application, we construct infinitely many contact structures on the class of simply connected contact manifolds that admit nice contact forms (i.e. no Reeb orbits of degree -1, 0 or 1) and have index positivity with trivial first Chern class. In particular we give examples of simply connected five-manifolds with infinitely many contact structures

Implementation of Bourbaki's Elements of Mathematics in Coq: Part Two; Ordered Sets, Cardinals, Integers

We believe that it is possible to put the whole work of Bourbaki into a computer. One of the objectives of the Gaia project concerns homological algebra (theory as well as algorithms); in a first step we want to implement all nine chapters of the book Algebra. But this requires a theory of sets (with axiom of choice, etc.) more powerful than what is provided by Ensembles; we have chosen the work of Carlos Simpson as basis. This reports lists and comments all definitions and theorems of the Chapter ''Ordered Sets, Cardinals, Integers''. Version 3 is based on the Coq ssreflect library. Version 5 implements many properties of ordinal numbers and infinite cardinal numbers. Version 6 includes the Veblen hierarchy of ordinals, the Schütte function psi, and a bit of theory of models.Version 7 includes rational and real numbers. Versions 8 and 9 include more theorems about ordinal numbers. Version 9 includes Sperner's theorem, and corrects a mistake in the size of one. The code (including some exercises) is available on the Web, under http://www-sop.inria.fr/marelle/gaia .Nous pensons qu’il est possible de mettre dans un ordinateur l’ensemble de l’œuvre de Bourbaki. L’un des objectifs du projet Gaia concerne l’algèbre homologique (théorie et algorithmes); dans une première étape nous voulons implémenter les neuf chapitres du livre Algèbre. Au préalable, il faut implémenter la théorie des ensembles. Nous utilisons l’Assistant de Preuve Coq; les choix fondamentaux et axiomes sont ceux proposés par Carlos Simpson. Ce rapport liste et commente toutes les définitions et théorèmes du Chapitre “Ensembles ordonnés, cardinaux, nombres entiers”. La version 9 de ce document décrit la bibliothèque à la fin de l'année 2017. Une partie des exercises a été résolue. Le code est disponible sur le site Web http://www-sop.inria.fr/marelle/gai

Mathematical modeling of unsteady state photo-polymerization

Mathematical modeling of unsteady state photopolymerization in an isothermal batch reactor has been investigated. An analytical solution of the distribution of active polymers as a function of reaction time has been determined. It is verified that the quasi-stationary state approximation is valid in the photopolymerization process. The free radical polymerization of styrene is forced periodically by on-off UV light regulation to the reactor. The effect of operating condition, the ratio of light-off period to light on/off period, on the polydispersity and molecular weight distribution are theoretically and numerically investigated. Significant broadening on molecular weight distribution can be obtained during the unsteady state process. Experimental results of polydispersity are compared to a kinetic model of a nonuniformly initiated photopolymerization, which comprises a high dose rate region and a very low dose rate region. Good agreement is obtained between the model and experiments after fitting three parameters

Analysis of CDO tranche valuation and the 2008 credit crisis

Includes bibliographical references.The causes of the 2008 financial crisis were wide ranging. Some financial commentators have suggested there were significant inadequacies in the models used to price complex derivatives such as synthetic Collaterilised Debt Obligations (CDOs). We discuss the technical properties of CDOs and the modeling approaches used by CDO traders and the watchdog credit rating agencies. We look at how the pricing models fared before and during the financial crisis. Comparing our model prices to market synthetic CDO prices, we investigate how well these pricing models captured the underlying financial risks of trading in CDOs

Theory, design and applications of linear transforms for information transmission

The aim of this dissertation is to study the common features of block transforms, subband filter banks, and wavelets, and demonstrate how discrete uncertainty can be applied to evaluate these different decomposition techniques. In particular, we derive an uncertainty bound for discrete-time functions. It is shown that this bound is the same as that for continuous-time functions, if the discrete-time functions have a certain degree of regularity. This dissertation also deals with spectral modeling in filter banks. It is shown, both theoretically and experimentally, that subspectral modeling is superior to full spectrum modeling if performed before the rate change. The price paid for this performance improvement is an increase of computations. A few different signal sources were considered in this study. It is shown that the performances of AR and ARMA modeling techniques are comparable in subspectral modeling. The first is desired because of its simplicity. As an application of AR modeling, a coding algorithm of speech, namely CELP embedded in a filter bank structure was also studied. We found that there were no improvements of subband CELP technique over the full band one. The theoretical reasonings of the experimental results are also given. This dissertation also addresses the problems of what type of transform to be used and to what extent an image should be decomposed. To this aim, an objective and subjective evaluations of different transform bases were done. We propose a smart algorithm for the decomposition of a channel into its sub-channels in the discrete multitone communications. This algorithm evaluates the unevenness and energy distribution of the channel spectrum in order to get its Variable adaptive partitioning. It is shown that the proposed algorithm leads to a near optimal performance of the discrete multitone transceiver. This flexible splitting of the channel suffers less from the aliasing problem that exists in blind decompositions using fixed transforms. This dissertation extends the discrete multitone to the flexible multiband concept which brings significant performance improvements for digital communications

Producing MathML with Tralics

The software and the documentation is available at http://www-sop.inria.fr/apics/tralicsThe purpose of this paper is to show that it is possible to translate almost every mathematical formula from LaTeX syntax to XML. The document you are reading contains a great number of formulas extracted from the TeX book, the LaTeX companion, the MathML recommendation, and translated into XML by Tralics; it is available in Pdf or HTML version. The HTML version was produced using a very simple style sheet; mathematics are left unchanged, and you need a MathML-aware browser, like Firefox, or adequate plugins, and a set of fonts for the symbols or operators. The Pdf version was obtained by use of xmltex (a package by D. Carlisle that makes TeX an XML interpreter), and a great number of modifications to the file mathml2.xmt (that interprets elements in the MathML namespace). All files needed to produce this document are part of the Tralics bundle (version 2.10 or more)