The work in this thesis is concerned with the study of dynamics, scattering and stability of solitons in planar models, i.e. where spacetime is (2+l)-dimensionaI. We consider both integrable models, where exact solutions can be written in closed form, and nonintegrable models where approximations and numerical methods must be employed. For theories that possess a topological lower bound on the energy, there is a useful approximation in which the kinetic energy is assumed to remain small. All these approaches are used at various stages of the thesis. Chapters 1 and 2 review the planar models which are the subjects of this thesis. Chapters 3 and 4 are concerned with integrable chiral equations. First we exhibit an infinite sequence of well-defined conserved quantities and then we construct exact soliton and soliton-antisoliton solutions using analytical methods. We find that there exist solitons that scatter in a different way to those previously found in integrable models. Furthermore, this soliton scattering resembles very closely that found in nonintegrable models, thereby providing a link between the two classes. Chapter 5 develops a numerical simulation based on topological arguments, which is used in a study of soliton stability in the (unmodified) 0(3) model. This confirms that the sohtons are unstable, in the sense that their size is subject to large changes. The same results are obtained by using the slow-motion approximation.

Some heuristic arguments are given in support of the following conjecture: If the Riemann Hypothesis (RH) does not hold then the number of zeroes of the Riemann zeta function with real part σ >  ½ is infinite.

En esta tesis se estudian álgebras autoinyectivas. Una subfamilia muy importante de las mismas la constituyen las llamadas álgebras Frobenius, que fueron introducidas por F. G. Frobenius en 1903. En el presente trabajo se establece un paralelismo entre este desarrollo clásico y el actual. Este enfoque nos permitió dar demostraciones alternativas de caracterizaciones conocidas para álgebras Frobenius y simétricas. Se presentan aquí las nociones de automorfismo de Nakayama, funtor de Nakayama y permutación de Nakayama, estableciendo la relación existente entre ellas. El carcaj de la extensión trivial T(A) de un álgebra de dimensión finita A = kQ=I fue descripto por Fernández y Platzeck en [FP]. En este trabajo describimos las relaciones de T(A) cuando A es un álgebra monomial, con lo que se obtiene una presentación de T(A) como cociente del álgebra de caminos de un carcaj por un ideal admisible de relaciones. Cuando A es además un álgebra gentil resolvemos el problema recíproco: dada un álgebra B = kQB=IB, determinar si B es la extensión trivial de un álgebra gentil. Caracterizamos tales álgebras B en base a propiedades de sus ciclos y mostramos cómo encontrar todas las álgebras gentiles A tales que T(A) _= B. Demostramos que las extensiones triviales de álgebras gentiles coinciden con las álgebras de grafo de Brauer con multiplicidad 1 en todos sus vértices, resultado obtenido por S. Schroll en [S] con otros métodos. / In this thesis we study selfinjective algebras. An important subfamily of this class of algebras consists of the so called Frobenius algebras, which were introduced by Frobenius in 1903. We establish here a parallel between the classical development and the present one. With this approach we were able to give alternative proofs of known characterizations of Frobenius and symmetric algebras. We recall the notions of Nakayama automorphism, Nakayama functor and Nakayama permutation and study the relationship between them. The quiver of the trivial extension T(A) of a finite dimensional algebra A = kQ=I was described by Fernández and Platzeck in [FP]. In this work we describe the ideal of relations for T(A) in case A is a monomial algebra. Thus we obtain a presentation for T(A) as a quotient of the path algebra of a quiver by an admissible ideal of relations. When A is, moreover, a gentle algebra, we solve the converse problem: given an algebra B = kQB=IB, determine whether B is the trivial extension of a gentle algebra. We characterize such algebras B through properties of the cycles of their quiver, and show how to obtain all gentle algebras A such that T(A) -= B. We prove that trivial extensions of gentle algebras coincide with Brauer graph algebras with multiplicity one in all vertices in the associated Brauer graph, result proven by S. Schroll in [S].

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Conner and Spencer used ultrafilters to construct homomorphisms between fundamental groups that could not be induced by continuous functions between the underlying spaces. We use methods from Shelah and Pawlikowski to prove that Conner and Spencer could not have constructed these homomorphisms with a weak version of the Axiom of Choice. This led us to define and examine a class of pathological objects that cannot be constructed without a strong version of the Axiom of Choice, which we call the class of inscrutable objects. Objects that do not need a strong version of the Axiom of Choice are scrutable. We show that the scrutable homomorphisms from the fundamental group of a Peano continuum are exactly the homomorphisms induced by a continuous function.We suspect that any proposed theorem whose proof does not use a strong Axiom of Choice cannot have an inscrutable counterexample.

This thesis studies some classes of non-periodic branch groups. In particular their growth, relations between elements and their Hausdorff dimensions.

Nuclei/Cell detection is usually a prerequisite procedure in many computer-aided biomedical image analysis tasks. In this thesis we propose two automatic nuclei/cell detection frameworks. One is for nuclei detection in skeletal muscle fiber images and the other is for brain tumor histopathological images. For skeletal muscle fiber images, the major challenges include: i) shape and size variations of the nuclei, ii) overlapping nuclear clumps, and iii) a series of z-stack images with out-of-focus regions. We propose a novel automatic detection algorithm consisting of the following components: 1) The original z-stack images are first converted into one all-in-focus image. 2) A sufficient number of hypothetical ellipses are then generated for each nuclei contour. 3) Next, a set of representative training samples and discriminative features are selected by a two-stage sparse model. 4) A classifier is trained using the refined training data. 5) Final nuclei detection is obtained by mean-shift clustering based on inner distance. The proposed method was tested on a set of images containing over 1500 nuclei. The results outperform the current state-of-the-art approaches. For brain tumor histopathological images, the major challenges are to handle significant variations in cell appearance and to split touching cells. The proposed novel automatic cell detection consists of: 1) Sparse reconstruction for splitting touching cells. 2) Adaptive dictionary learning for handling cell appearance variations. The proposed method was extensively tested on a data set with over 2000 cells. The result outperforms other state-of-the-art algorithms with F1 score = 0.96.

Diploma thesis is about stationary solutions to reaction-diffusion system of the activator-inhibitor type on a one-dimensional domain. Three homogeneous boudary value problems are studied---with pure Neumann boudary conditions, with mixed Neumann-Dirichlet boudary conditions and with Neumann conditions on the boundary where simultaneously an additional homogeneous condition is prescribed in a given point in the interior of the domain. For all three boudary value problems the existence of so-called critical points (diffusion parameters, for which a non-trivial solution exists) is proved.