Persistence problem of compact invariant manifold under random perturbation is considered in this dissertation. Under uniformly small random perturbation and the condition of normal hyperbolicity, the original invariant manifold persists and becomes a random invariant manifold. The random counterpart has random local stable and unstable manifolds. They could be invariantly foliated thanks to the normal hyperbolicity. Those underlie an extension of the geometric singular perturbation theory to the random case which means the slow manifold persists and becomes a random manifold so that the local global structure near the slow manifold persists under singular perturbation. A normal form for a random differential equation is obtained and this helps to prove a random version of the exchange lemma.

The n-point Steiner problem in the Euclidean plane is to find a least length path network connecting n points. In this thesis we will demonstrate how to find a least length path network T connecting n points on a closed 2-dimensional Riemannian surface of constant curvature by determining a region in the covering space that is guaranteed to contain T. We will then provide an algorithm for solving the n-point Steiner problem on such a surface.

The hyperbolic volume of a knot complement is a topological knot invariant. Futer, Kalfagianni, and Purcell have estimated the volumes of Montesinos link complements for Montesinos links with at least three positive tangles. Here we extend their results to all hyperbolic Montesinos links.

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In this thesis, we apply the methods of numerical differential geometry to several different problems in heterotic Calabi-Yau compactification. We review algorithms for computing both the Ricci-flat metric on Calabi-Yau manifolds and Hermitian Yang-Mills connections on poly-stable holomorphic vector bundles over those spaces. We apply the numerical techniques for obtaining Ricci-flat metrics to study hierarchies of curvature scales over Calabi-Yau manifolds as a function of their complex structure moduli. The work we present successfully finds known large curvature regions on these manifolds, and provides useful information about curvature variation at general points in moduli space. This research is important in determining the validity of the low energy effective theories used in the description of Calabi-Yau compactifications. The numerical techniques for obtaining Hermitian Yang-Mills connections are applied in two different fashions in this thesis. First, we demonstrate that they can be successfully used to numerically determine the stability of vector bundles with qualitatively different features to those that have appeared in the literature to date. Second, we use these methods to further develop some calculations of holomorphic Chern-Simons invariant contributions to the heterotic superpotential that have recently appeared in the literature. A complete understanding of these quantities requires explicit knowledge of the Hermitian Yang-Mills connections involved. This feature makes such investigations prohibitively hard to pursue analytically, and a natural target for numerical techniques. / Doctor of Philosophy / String theory is one of the most promising attempts to unify gravity with the other three fundamental interactions (electromagnetic, weak and strong) of nature. It is believed to give a self-consistent theory of quantum gravity, which, at low energy, could contain all of the physics that we known, from the Standard Model of particle physics to cosmology. String theories are often defined in nine spatial dimensions. To obtain a theory with three spatial dimensions one needs to hide, or ``compactify," six of the dimensions on a compact space which is small enough to have remained unobserved by our experiments. Unfortunately, the geometries of these spaces, called Calabi-Yau manifolds, and additional structures associated to them, called holomorphic vector bundles, turns out to be extremely complex. The equations determining the exact solutions of string theory for these quantities are highly non-linear partial differential equations (PDE's) which are simply impossible to solve analytically with currently known techniques. Nevertheless, knowledge of these solutions is critical in understanding much of the detailed physics that these theories imply. For example, to compute how the particles seen in three dimensions would interact with each other in a string theoretic model, the explicit form of these solutions would be required. Fortunately, numerical methods do exist for finding approximate solutions to the PDE's of interest. In this thesis we implement these algorithmic techniques and use them to study a variety of physical questions associated to the attempt to link string theory to the physics observed in our experiments.

Le but de cette thèse est d'étudier les structures symplectiques dans la catégorie des variétés linéaires par morceaux (PL). La question centrale est de déterminer si toute variété symplectique lisse $(M,omega)$ peut être triangulée de manière symplectique, au sens où il existe une variété linéaire par morceaux $K$ et une triangulation $h :K -> M$ telle que $h^*omega$ est une forme symplectique constante par morceaux. Nous étudions d'abord un problème plus simple, qui consiste à trianguler les formes volumes lisses. Étant donnée une variété lisse $M$ munie d'une forme volume $Omega$, nous montrons qu'il existe une triangulation lisse $h :K -> M$ telle que $h^*Omega$ est une forme volume constante par morceaux. En particulier, les variétés symplectiques lisses de dimension 2 admettent donc des triangulations symplectiques. Étant donnée une variété symplectique fermée $(M,omega)$, nous montrons ensuite que pour certaines triangulations lisses $h :K -> M$, on peut, par une modification arbitrairement petite du complexe $K$, supposer que la forme $h^*omega$ est de rang maximal le long de tous les simplexes de $K$. Ce résultat permet d'approximer arbitrairement bien toute variété symplectique fermée par une variété symplectique PL. Nous nous intéressons finalement au cas d'une sous-variété symplectique $M$ d'un espace ambiant qui admet lui-même une triangulation symplectique. Nous montrons qu'il est possible de construire un cobordisme entre la sous-variété $M$ considérée et une approximation lisse par morceaux de celle-ci, triangulée par un complexe symplectique. / In this thesis, we study symplectic structures in a piecewise linear (PL) setting. The central question is to determine whether a smooth symplectic manifold can be triangulated symplectically, in the sense that there exists a triangulation $h :K -> M$ such that $h^*omega$ is a piecewise constant symplectic form on $K$. We first focus on a simpler related problem, and show that any smooth volume form $Omega$ on $M$ can be triangulated. This means that there always exists a triangulation $h :K -> M$ such that $h^*Omega$ is a piecewise constant volume form. In particular, symplectic surfaces admit symplectic triangulations. Given a closed symplectic manifold $(M,omega)$, we then prove that there exists triangulations $h :K -> M$ for which the piecewise smooth form $h^*omega$ has maximal rank along all the simplices of $K$. This result allows to approximate arbitrarily closely any closed symplectic manifold by a PL one. Finally, we investigate the case of a symplectic submanifold $M$ of an ambient space which is itself symplectically triangulated, and give the construction of a cobordism between $M$ and a piecewise smooth approximation of $M$, triangulated by a symplectic complex. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

The master’s thesis is focused on creating a thermodynamic model of the diesel engine for heavy commercial vehicle. After the model was created, new intake and exhaust pipes are designed based on simulations. Those proposed pipes variants are compared with the series design. After evaluating individual variants, the final version of the new pipes is selected and 3D model for this version is created.