It is significantly important to search for exact soliton solutions to nonlinear partial differential equations (PDEs) of mathematical physics. Transforming nonlinear PDEs into bilinear forms using the Hirota differential operators enables us to apply the Wronskian and Pfaffian techniques to search for exact solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili (KP) equation with not only constant coefficients but also variable coefficients under a certain constraint (ut + α 1(t)uxxy + 3α 2(t)uxuy)x +α 3 (t)uty -α 4(t)uzz + α 5(t)(ux + α 3(t)uy) = 0. However, bilinear equations are the nearest neighbors to linear equations, and expected to have some properties similar to those of linear equations. We have explored a key feature of the linear superposition principle, which linear differential equations have, for Hirota bilinear equations, while intending to construct a particular sub-class of N-soliton solutions formed by linear combinations of exponential traveling waves. Applications are given for the (3+1) dimensional KP, Jimbo-Miwa (JM) and BKP equations, thereby presenting their particular N-wave solutions. An opposite question is also raised and discussed about generating Hirota bilinear equations possessing the indicated N-wave solutions, and two illustrative examples are presented. Using the Pfaffianization procedure, we have extended the generalized KP equation to a generalized KP system of nonlinear PDEs. Wronskian-type Pfaffian and Gramm-type Pfaffian solutions of the resulting Pfaffianized system have been presented. Our results and computations basically depend on Pfaffian identities given by Hirota and Ohta. The Pl̈ucker relation and the Jaccobi identity for determinants have also been employed. A (3+1)-dimensional JM equation has been considered as another important example in soliton theory, uyt - uxxxy - 3(uxuy)x + 3uxz = 0. Three kinds of exact soliton solutions have been given: Wronskian, Grammian and Pfaffian solutions. The Pfaffianization procedure has been used to extend this equation as well. Within Wronskian and Pfaffian formulations, soliton solutions and rational solutions are usually expressed as some kind of logarithmic derivatives of Wronskian and Pfaffian type determinants and the determinants involved are made of functions satisfying linear systems of differential equations. This connection between nonlinear problems and linear ones utilizes linear theories in solving soliton equations. B̈acklund transformations are another powerful approach to exact solutions of nonlinear equations. We have computed different classes of solutions for a (3+1)-dimensional generalized KP equation based on a bilinear B̈acklund transformation consisting of six bilinear equations and containing nine free parameters. A variable coefficient Boussinesq (vcB) model in the long gravity water waves is one of the examples that we are investigating, ut + α 1 (t)uxy + α 2(t)(uw)x + α 3(t)vx = 0; vt + β1(t)(wvx + 2vuy + uvy) + β2(t)(uxwy - (uy)2) + β3(t)vxy + β4(t)uxyy = 0, where wx = uy. Double Wronskian type solutions have been constructed for this (2+1)-dimensional vcB model.

We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of smaller ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the defining equations of Rees rings for a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving that, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of a graded component of the Rees ring in the generic case is an approximate resolution of the same component of the Rees ring in question. We end the paper by giving some examples of explicit generation and concentration degree bounds.

It is needless to point out the necessity and the importance of the study of Pfaffian differential expressions and equations. While it is interesting to consider from the pure mathematical point of view, their applications in many branches of applied mathematics are well known. To mention a few, one may observe that they arise in connection with line integrals (example, determination of work). They provide a more rational formulation of the foundations of thermodynamics as 'developed by the Greek mathematician Caratheodory. They also arise in the problem of determining the orthogonal trajectories. In many branches of engineering and other physical sciences they appear with problems concerning partial differential equations.

The aim of this dissertation lies in establishing an o-minimal de Rham cohomology theory for smooth abstract-definable manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function, by following the classical de Rham cohomology. We can specify the o-minimal cohomology groups and attain some properties as the existence of Mayer-Vietoris sequence and the invariance under smooth abstract-definable diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must, working in a tame context that defines sufficiently many primitives, assume the validity of a statement related to Bröcker\'s problem. / O objetivo desta tese reside em estabelecer uma cohomologia de De Rham o-minimal para variedades definíveis abstratas lisas em uma expansão o-minimal do corpo ordenado dos reais, a qual admite decomposição celular lisa e define a função exponencial, seguindo a cohomologia de De Rham clássica. Além de especificarmos os grupos da cohomologia de Rham o-minimal, obtemos algumas propriedades, como a existência da sequência de Mayer-Vietoris e a invariância sob difeomorfismos definíveis abstratos lisos. Todavia, a fim de lograrmos a invariância de nossa cohomologia o-minimal sob homotopia definível abstrata devemos, além de trabalhar num contexto moderado no qual muitas primitivas são definidas, assumir a validade de uma asserção relacionada ao problema de Bröcker.

This master’s thesis describes equations of motion for dynamic model of nonholonomic constrained system, namely the trident robotic snakes. The model is studied in the form of Lagrange's equations and D’Alembert’s principle is applied. Actually this thesis is a continuation of the study going at VUT about the simulations of non-holonomic mechanisms, specifically robotic snakes. The kinematics model was well-examined in the work of of Byrtus, Roman and Vechetová, Jana. So here we provide equations of motion and address the motion planning problem regarding dynamics of the trident snake equipped with active joints through basic examples and propose a feedback linearization algorithm.

Dans cette thèse, je présente une étude que j'ai réalisée à l'université Paris-sud sous la direction de Thierry Jolicœur sur les phases des Hall Quantiques Fractionnaire (HQF) dans la géométrie du cylindre.Après une rapide introduction dans le premier chapitre, je présente dans le second quelques concepts de base de l'effet HQF et j'introduit certains aspects de la géométrie cylindrique.Le chapitre 3 est consacré à l'étude de la limite du cylindre fin, c'est à dire lorsque la circonférence du cylindre est de l'ordre de quelques longueurs magnétiques. Dans cette limite, on sait que la fonction d'onde de Laughlin au remplissage 1/q se réduit à un cristal unidimensionnel, où une orbitale sur q est occupée. Dans le but d'étudier un limite intermédiaire, nous conservons les quatre premiers termes du développement de l’Hamiltonien lorsque la circonférence est petite devant la longueur magnétique. On trouve alors une expression exacte de l'état fondamental au moyen d'opérateurs de "squeezing" ou de produits de matrices. Nous trouvons également une écriture similaire pour les quasi- trous, les quasi-électron et la branche magnétoroton.Dans les chapitres 4 et 5, je me concentre sur l'étude des excitations de bord chirales des phases de HQF. Je présente une étude microscopique de ces états de bord dans la géométrie du cylindre, lorsque les quasi-particules peuvent passer d'un bord à l'autre par effet tunnel. J'étudie d'abord dans le chapitre 4 la phase de HQF principale dont l'état fondamental est bien décrit par la fonction d'onde de Laughlin. Pour un échelle d'énergie plus faible que le gap du volume, le théorie effective est donnée par un fluide d'électrons unidimensionnel bien particulier : un liquide de Luttinger chiral. À l'aide de diagonalisations numériques exactes, nous étudions le spectre des états de bord formé de le combinaison des deux bord contre-propageant sur chacun des cotés du cylindre. Nous montrons que les deux bords se combinent pour former un liquide de Luttinger non-chiral, où le terme de courant reflète le transfert de quasi-particules entre les bords. Cela nous permet d'estimer numériquement les paramètre de Luttinger pour un faible nombre de particules, et nous trouvons une valeur cohérente avec la théorie de X. G. Wen.J'analyse ensuite dans le chapitre 5 les modes de bord des phases de HQF au remplissage 5/2. À partir une construction basée sur la Théorie des Champs Conformes (TCC), Moore et Read (Nucl. Phys. B, 1991) ont proposé que la physique essentielle de cette phase soit décrite par un état apparié de fermion composites. Une propriété importante de cet état est que ses excitations émergentes permutent sous une statistique non-abéliène. Lorsqu'elles sont localisées sur les bords, ces excitations sont décrites par un boson chiral et un fermion de Majorana. Dans la géométrie du cylindre, nous montrons que le spectre des excitations de bord est fomé des tours conformes du modèle IsingxU(1). De plus, par une méthode Monte-Carlo, nous estimons les différentes dimensions d'échelle sur des grands systèmes (environ 50 électrons), et nous trouvons des valeurs en accord avec les prédictions de la TCC.Dans le dernier chapitre de ce manuscrit, je présente un travail que j'ai réalisé à UBC (Vancouver) en collaboration avec Marcel Franz sur les phase de Hall quantiques de spin induites dans le graphène par des adatomes. Dans ce système, les adatomes induisent un couplage spin-orbite sur les électrons des la feuille de graphène et introduisent du désordre qui est susceptible de détruire le gap spectral. Nous montrons dans ce chapitre que le gap spectral est préservé lorsque des valeurs réalistes de paramètres sont usités. De plus, au moyen de calculs analytiques à base énergie et de diagonalisations numériques exactes, nous identifions un signal caractéristique dans la densité d'états locale mettant en évidence la présence d'un gap topologique. Ce signal pourrait être observé au moyen d'un microscope à effet tunnel. / I present in this thesis a study that I did in the university Paris-sud under the supervision of Thierry Jolicœur onto Fractional Quantum Hall (FQH) phases in the cylinder geometry. After a short introduction in the first chapter, I present some basic concept relative to the FQH effect in the second one and introduce some essential features relative to the cylinder geometry, useful for the chapters 3, 4, and 5. The chapter 3 is dedicated to the study of the thin cylinder limit, i.e. when the circumference of the cylinder is of the order of a few magnetic length. In this limit, it is known that the Laughlin wave function at the filling factor 1/q is reduced to a one dimensional crystal in the lowest Landau level orbitals where one every q orbitals is occupied. We Taylor expand the Hamiltonian when the circumference is small compare to the magnetic length in order to study an intermediate limit. When only the first four terms of the development are kept, it is possible to find exact representations of the ground state with "squeezing" operators or matrix products. We also find similar representations for quasiholes, quasielectrons and the magnetorton branch. These results have been published in the article Phys. Rev. B 85, 155116 (2012). In the chapter 4 and 5 I focus onto the gapless chiral edge excitations of FQH phases. I present a microscopic study of those edges states in the cylindrical geometry where quasiparticles are able to tunnel between edges. I first study the principal FQH phase at the filling fraction 1/3 whose ground state is well described by the Laughlin wave function in the chapter 4. For an energy scale lower than the bulk gap, the effective theory is given by a very peculiar one dimensional electron fluid localized at the edge: a chiral Luttinger liquid. Using numerical exact diagonalizations, we study the spectrum of edge modes formed by the two counter-propagating edges on each side of the cylinder. We show that the two edges combine to form a non-chiral Luttinger liquid, where the current term reflects the transfer of quasiparticles between edges. This allows us to estimate numerically the Luttinger parameter for a small number of particles and find it coherent with the one predicted by X. G. Wen theory. We published this work in Phys. Rev. B 86, 115214 (2012). I then analyze edge modes of the FQH phase at filling fraction 5/2 in the chapter 5. From a Conformal Field Theory (CFT) based construction, Moore and Read (Nucl. Phys. B, 1991) proposed that the essential physics of this phase is described by a paired state of composite fermions. A striking property of this state is that emergent excitations braid with non-Abelian statistics. When localized along the edge, those excitations are described through a chiral boson and a Majorana fermion. In the cylinder geometry, we show that the spectrum of edge excitations is composed of all conformal towers of the IsingxU(1) model. In addition, with a Monte Carlo method, we estimate the various scaling dimensions for large systems (about 50 electrons), and find them consistent with the CFT predictions.In the last chapter of my manuscript, I present a work that I did in UBC (Vancouver) in collaboration with Marcel Franz onto quantum spin Hall phases in graphene induced by adatoms. In this system, adatoms induce a spin orbit coupling for electrons in the graphene sheet and create some disorder which might be responsible for destruction the spectral gap. We show in this chapter and in the article [Phys. Rev. B 89, 201410(R) (2014)] that the spectral gap remains open for a realistic range of parameters. In addition, with analytical computations in the low energy approximation and numerical exact diagonalizations, we find characteristic signal in the local density of states highlighting the presence of topological gap. This signal might be observed in scanning tunneling spectroscopy experiments.

<div>This dissertation is based on joint work with Monte Cooper and is broken into two main parts, both of which study the defining ideals of the Rees rings of determinantal and Pfaffian ideals of generic height. In both parts, we attempt to place degree bounds on the defining equations.</div><div> </div><div> The first part of the dissertation consists of Chapters 3 to 5. Let $R = K[x_{1},\ldots,x_{d}]$ be a standard graded polynomial ring over a field $K$, and let $I$ be a homogeneous $R$-ideal generated by $s$ elements. Then there exists a polynomial ring $\mathcal{S} = R[T_{1},\ldots,T_{s}]$, which is also equal to $K[x_{1},\ldots,x_{d},T_{1},\ldots,T_{s}]$, of which the defining ideal of $\mathcal{R}(I)$ is an ideal. The polynomial ring $\mathcal{S}$ comes equipped with a natural bigrading given by $\deg x_{i} = (1,0)$ and $\deg T_{j} = (0,1)$. Here, we attempt to use specialization techniques to place bounds on the $x$-degrees (first component of the bidegrees) of the defining equations, i.e., the minimal generators of the defining ideal of $\mathcal{R}(I)$. We obtain degree bounds by using known results in the generic case and specializing. The key tool are the methods developed by Kustin, Polini, and Ulrich to obtain degree bounds from approximate resolutions. We recover known degree bounds for ideals of maximal minors and submaximal Pfaffians of an alternating matrix. Additionally, we obtain $x$-degree bounds for sufficiently large $T$-degrees in other cases of determinantal ideals of a matrix and Pfaffian ideals of an alternating matrix. We are unable to obtain degree bounds for determinantal ideals of symmetric matrices due to a lack of results in the generic case; however, we develop the tools necessary to obtain degree bounds once similar results are proven for generic symmetric matrices.</div><div> </div><div> The second part of this dissertation is Chapter 6, where we attempt to find a bound on the $T$-degrees of the defining equations of $\mathcal{R}(I)$ when $I$ is a nonlinearly presented homogeneous perfect Gorenstein ideal of grade three having second analytic deviation one that is of linear type on the punctured spectrum. We restrict to the case where $\mathcal{R}(I)$ is not Cohen-Macaulay. This is a natural next step following the work of Morey, Johnson, and Kustin-Polini-Ulrich. Based on extensive computation in Macaulay2, we give a conjecture for the relation type of $I$ and provide some evidence for the conjecture. In an attempt to prove the conjecture, we obtain results about the defining ideals of general fibers of rational maps, which may be of independent interest. We end with some examples where the bidegrees of the defining equations exhibit unusual behavior.</div>

Dans cette thèse, nous nous sommes intéressés à l'analyse locale de systèmes différentiels linéaires singulièrement perturbés et de systèmes de Pfaff complètement intégrables et multivariés à croisements normaux. De tels systèmes ont une vaste littérature et se retrouvent dans de nombreuses applications. Cependant, leur résolution symbolique est toujours à l'étude. Nos approches reposent sur l'état de l'art de la réduction formelle des systèmes linéaires singuliers d'équations différentielles ordinaires univariées (ODS). Dans le cas des systèmes différentiels linéaires singulièrement perturbés, les complications surviennent essentiellement à cause du phénomène des points tournants. Nous généralisons les notions et les algorithmes introduits pour le traitement des ODS afin de construire des solutions formelles. Les algorithmes sous-jacents sont également autonomes (par exemple la réduction de rang, la classification de la singularité, le calcul de l'indice de restriction). Dans le cas des systèmes de Pfaff, les complications proviennent de l'interdépendance des multiples sous-systèmes et de leur nature multivariée. Néanmoins, nous montrons que les invariants formels de ces systèmes peuvent être récupérés à partir d'un ODS associé, ce qui limite donc le calcul à des corps univariés. De plus, nous donnons un algorithme de réduction de rang et nous discutons des obstacles rencontrés. Outre ces deux systèmes, nous parlons des singularités apparentes des systèmes différentiels univariés dont les coefficients sont des fonctions rationnelles et du problème des valeurs propres perturbées. Les techniques développées au sein de cette thèse facilitent les généralisations d'autres algorithmes disponibles pour les systèmes différentiels univariés aux cas des systèmes bivariés ou multivariés, et aussi aux systèmes d''equations fonctionnelles. / In this thesis, we are interested in the local analysis of singularly-perturbed linear differential systems and completely integrable Pfaffian systems in several variables. Such systems have a vast literature and arise profoundly in applications. However, their symbolic resolution is still open to investigation. Our approaches rely on the state of art of formal reduction of singular linear systems of ordinary differential equations (ODS) over univariate fields. In the case of singularly-perturbed linear differential systems, the complications arise mainly from the phenomenon of turning points. We extend notions introduced for the treatment of ODS to such systems and generalize corresponding algorithms to construct formal solutions in a neighborhood of a singularity. The underlying components of the formal reduction proposed are stand-alone algorithms as well and serve different purposes (e.g. rank reduction, classification of singularities, computing restraining index). In the case of Pfaffian systems, the complications arise from the interdependence of the multiple components which constitute the former and the multivariate nature of the field within which reduction occurs. However, we show that the formal invariants of such systems can be retrieved from an associated ODS, which limits computations to univariate fields. Furthermore, we complement our work with a rank reduction algorithm and discuss the obstacles encountered. The techniques developed herein paves the way for further generalizations of algorithms available for univariate differential systems to bivariate and multivariate ones, for different types of systems of functional equations.