In this thesis we investigate three projects within in the field of KPZ universality class and integrable probability. The first project studies the weak KPZ universality for half-space directed polymers in dimension 1+1. This is the half-space analogue of the full-space polymers studied in [AKQ]. The novelty is the extra random environment introduced at the boundary. The new technical challenges are the estimates for half-space heat kernels which are super-probability measures and accurate estimates on visits to origin (weighted by the boundary randomness) for a simple symmetric walk in dimension 1 and 2 respectively. The second project introduces a framework to prove tightness of a sequence of discrete Gibbsian line ensembles $\mathcal{L}^N = \{\mathcal{L}_k^N(u), k \in \mathbb{N}, u \in \frac{1}{N}\mathbb{Z}\}$, which is a countable collection of random curves. The sequence of discrete line ensembles $\mathcal{L}^N$ we consider enjoys a resampling invariance property, which we call $({\ensuremath{\mathbf{H}}}^N,\textup{H}^{\textup{RW},N})$-Gibbs property. We assume that $\mathcal{L}^N$ satisfies technical assumptions A1-A4 on $({\ensuremath{\mathbf{H}}}^N, {\textup{H}^{\textup{RW},N}})$ and the assumption that the lowest labeled curve with a parabolic shift, $\mathcal{L}_1^N(u) + \frac{u^2}{2}$, converges weakly to a stationary process in the topology of uniform convergence on compact sets. Under these assumptions, we prove our main result Theorem 3.1.13 that $\mathcal{L}^N$ is tight as a sequence of line ensembles and that the $\ensuremath{\mathbf{H}}$-Brownian Gibbs property holds for all subsequential limit line ensembles with $\ensuremath{\mathbf{H}}(x)= e^x$. As an application of Theorem 3.1.13, under the weak noise scaling, we show that the scaled log-gamma line ensemble $\overbar{\mathcal{L}}^N$ is tight, which is a sequence of discrete line ensembles associated with the log-gamma polymer model via the geometric RSK correspondence. The $\ensuremath{\mathbf{H}}$-Brownian Gibbs property (with $\ensuremath{\mathbf{H}}(x) = e^x$) of its subsequential limits also follows. The third project proves an analogue of the classical Komlós-Major-Tusnády (KMT) embedding theorem for random walk bridges and it serves as a key technical input for the second project. The random bridges we consider are constructed through random walks with i.i.d jumps that are conditioned on the locations of their endpoints. We prove that such bridges can be strongly coupled to Brownian bridges of appropriate variance when the jumps are either continuous or integer valued under some mild technical assumptions on the jump distributions. Our arguments follow a similar dyadic scheme to KMT's original proof, but they require more refined estimates and stronger assumptions necessitated by the endpoint conditioning. In particular, our result does not follow from the KMT embedding theorem, which we illustrate via a counterexample.

No description available.

No description available.

The aim of this work is to discuss the effects found in static and dynamic wormholes that occur as a solution of Einstein equations in general relativity. The ground is prepared by presentation of "faster than light" effects, historical perspective, wormhole definition and contemporary directions in wormhole research. Then the focus is narrowed to Morris-Thorne framework for static spherically symmetric wormhole. Energy conditions being a fundamental component in wormhole physics are discussed in detail, their definition, most common violations and attempts to exocit matter quantification. Two types of dynamic wormholes, evolving and rotating, together with their variations are considered. Computer algebra and Cartan calculus are used to obtain detailed solutions. / Mathematical Sciences / M.Sc. (Applied Mathematics)

The main objects of study of the thesis are two classes of special vector configurations appeared in the geometry and the theory of integrable systems. In the first part we consider a special class of vector configurations known as the V-systems, which appeared in the theory of the generalised Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known, but their classification is an open problem. We derive the relations describing the infinitesimal deformations of V-systems and use them to study the classification problem for V-systems in dimension 3. In particular, we prove that the isolated cases in Feigin-Veselov list admit only trivial deformations. We present the catalogue of all known 3D V-systems including graphical representations of the corresponding matroids and values of v-functions. In the second part we study the vector configurations, which form vertex sets for a new class of polyhedra called affine B-regular. They are defined by a 3-dimensional analogue of the Buffon procedure proposed by Veselov and Ward. The main result is the proof of existence of star-shaped affine B-regular polyhedron with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdière and Lovász.

In this dissertation we study the anisotropic Manev problem that describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic correction term. The dynamic of the system under discussion is very complicated and we use various methods to find a qualitative description of the flow. One of the strategies we use is to study the collision and near collision orbits. In order to do that we utilize McGehee type transformations that lead to an equivalent analytic system with an analytic energy relation. In these new coordinates the collisions are replaced by an analytic two-manifold: the so called collision manifold. We focus our attention on the heteroclinic orbits connecting fixed points on the collision manifold and on the homoclinic orbit to the equator of the mentioned manifold. We prove that as the anisotropy is introduced only four heteroclinic orbits persist and we show the exixtence of infinitely many transversal homoclinic orbits using a suitable generalization of the Poincaré-Melnikov method. Another strategy we apply is to study the symmetric periodic orbits of the system. To tackle this problem we follow two different approaches. First we apply the Poincaré continuation method and we find symmetric periodic orbits for small values of the anisotropy. Then we utilize a direct method of the calculus of variations, namely the lower semicontinuity method, and we prove the existence of symmetric periodic orbits for any value of the anisotropy parameter. In the last chapter we use the Killing's equation in an unusual way to prove that the anisotropic Kepler problem (that can be considered a particular case of the Manev) does not have first integrals linear in the momentum. / Graduate

Abstract not available

Dans ce travail de thèse des ingrédients essentiels pour la description théorique de la dynamique des réactions de fusion et fission nucléaires sont étudiés, tel que le potentiel d'interaction entre noyaux cible et projectile pour le processus de fusion et l'énergie de déformation dans un espace multidimensionnel pour la fission. Nous avons évalué en particulier l'importance de la différence des distributions de densités protons et neutrons. Pour le processus de fusion, le potentiel d'interaction entre les noyaux peut être déterminé à travers des densités obtenues d'une façon auto-consistante par des calculs variationnels semi-classiques à partir d'une interaction nucléon-nucléon effective de type Skyrme. Les barrières de fusion ainsi obtenues permettent d'évaluer des sections efficaces de fusion dans le cadre du formalisme de Langevin. Pour le processus de fission il est essentiel de tenir compte de la grande richesse de formes nucléaires qui apparaissent tout au long du chemin de fission de l'état fondamental jusqu'au point de scission. Nous montrons qu'une paramétrisation tenant compte de l'élongation, ainsi que de la possible constriction, asymétrie gauche-droite et non-axialité du noyau, est effectivement capable dans le cadre de notre approche macroscopique-microscopique de donner une description précise de ce phénomène. On peut ainsi enrichir l'expression de l'énergie de type goutte liquide par un terme qui décrit la variation de l'énergie de liaison nucléaire due aux différentes déformations des distributions protons et neutrons. La réduction des hauteurs des barrières de fission qui en résulte est seulement de l'ordre du MeV, mais il peut facilement en résulter un changement de la section efficace de fission d'un ordre de grandeur et ainsi jouer un rôle capital pour la stabilité des noyaux super-lourds ou des noyaux exotiques

sans

Currently, one application of the mean-field methods in nuclear physics is the investigation of exotic nuclear symmetries. This is related, in particular, to the study of nuclear rotation about an axis tilted with respect to the principal axes of the mass distribution in the Tilted-Axis Cranking (TAC) model. The present work presents one of the first TAC calculations performed within fully self-consistent methods. The Hartree-Fock method with the Skyrme effective two-body interaction has been used. A computer code has been developed that allows for the breaking of all spatial symmetries of the solution. As a first application, calculations for the magnetic bands in 142Gd and for the chiral bands in 130Cs, 132La, 134Pr, and 136Pm have been carried out. The appearance of those bands is due to a new mechanism of breaking the spherical symmetry and to the spontaneous breaking of the chiral symmetry, respectively. The self-consistent solutions for 142Gd confirm the important role of the shears mechanism in generating the total angular momentum. However, the agreement with experimental data is not satisfactory, probably due to the lack of the pairing correlations in the calculations or to the possibly overestimated deformation. The results obtained for 132La constitute the first fully self-consistent proof that the nuclear rotation can attain a chiral character. It has been shown that the chiral rotation can only exist above a certain critical angular frequency. It has also been checked that the terms of the Skyrme mean field odd under the time reversal have no qualitative influence on the results.