Some non- linear problems in theoretical physics

An immense variety of problems in theoretical physics are of the non-linear type. Non~linear partial differential equations (NPDE) have almost become the rule rather than an exception in diverse branches of physics such as fluid mechanics, field theory, particle physics, statistical physics and optics, and the construction of exact solutions of these equations constitutes one of the most vigorous activities in theoretical physics today. The thesis entitled ‘Some Non-linear Problems in Theoretical Physics’ addresses various aspects of this problem at the classical level. For obtaining exact solutions we have used mathematical tools like the bilinear operator method, base equation technique and similarity method with emphasis on its group theoretical aspects. The thesis deals with certain methods of finding exact solutions of a number of non-linear partial differential equations of importance to theoretical physics. Some of these new solutions are of relevance from the applications point of view in diverse branches such as elementary particle physics, field theory, solid state physics and non-linear optics and give some insight into the stable or unstable behavior of dynamical Systems The thesis consists of six chapters.

Fragile-glass behavior of a short-range p-spin model

We propose a short-range generalization of the p-spin interaction spin-glass model. The model is well suited to test the idea that an entropy collapse is at the bottom line of the dynamical singularity encountered in structural glasses. The model is studied in three dimensions through Monte Carlo simulations, which put in evidence fragile glass behavior with stretched exponential relaxation and super-Arrhenius behavior of the relaxation time. Our data are in favor of a Vogel-Fulcher behavior of the relaxation time, related to an entropy collapse at the Kauzmann temperature. We, however, encounter difficulties analogous to those found in experimental systems when extrapolating thermodynamical data at low temperatures. We study the spin-glass susceptibility, investigating the behavior of the correlation length in the system. We find that the increase of the relaxation time is accompanied by a very slow growth of the correlation length. We discuss the scaling properties of off-equilibrium dynamics in the glassy regime, finding qualitative agreement with the mean-field theory.

Studies on gaussian effective potentials in some models

In classical field theory, the ordinary potential V is an energy density for that state in which the field assumes the value ¢. In quantum field theory, the effective potential is the expectation value of the energy density for which the expectation value of the field is ¢o. As a result, if V has several local minima, it is only the absolute minimum that corresponds to the true ground state of the theory. Perturbation theory remains to this day the main analytical tool in the study of Quantum Field Theory. However, since perturbation theory is unable to uncover the whole rich structure of Quantum Field Theory, it is desirable to have some method which, on one hand, must go beyond both perturbation theory and classical approximation in the points where these fail, and at that time, be sufficiently simple that analytical calculations could be performed in its framework During the last decade a nonperturbative variational method called Gaussian effective potential, has been discussed widely together with several applications. This concept was described as a means of formalizing our intuitive understanding of zero-point fluctuation effects in quantum mechanics in a way that carries over directly to field theory.

Squared interaction matrix Sherrington-Kirkpatrick model for a spin glass

The mean-field theory of a spin glass with a specific form of nearest- and next-nearest-neighbor interactions is investigated. Depending on the sign of the interaction matrix chosen we find either the continuous replica symmetry breaking seen in the Sherrington-Kirkpartick model or a one-step solution similar to that found in structural glasses. Our results are confirmed by numerical simulations and the link between the type of spin-glass behavior and the density of eigenvalues of the interaction matrix is discussed.

Superheavy nuclei in relativistic effective Lagrangian model

Isotopic and isotonic chains of superheavy nuclei are analyzed to search for spherical double shell closures beyond Z=82 and N=126 within the new effective field theory model of Furnstahl, Serot, and Tang for the relativistic nuclear many-body problem. We take into account several indicators to identify the occurrence of possible shell closures, such as two-nucleon separation energies, two-nucleon shell gaps, average pairing gaps, and the shell correction energy. The effective Lagrangian model predicts N=172 and Z=120 and N=258 and Z=120 as spherical doubly magic superheavy nuclei, whereas N=184 and Z=114 show some magic character depending on the parameter set. The magicity of a particular neutron (proton) number in the analyzed mass region is found to depend on the number of protons (neutrons) present in the nucleus.

Factorial Hidden Markov Models

We present a framework for learning in hidden Markov models with distributed state representations. Within this framework, we derive a learning algorithm based on the Expectation--Maximization (EM) procedure for maximum likelihood estimation. Analogous to the standard Baum-Welch update rules, the M-step of our algorithm is exact and can be solved analytically. However, due to the combinatorial nature of the hidden state representation, the exact E-step is intractable. A simple and tractable mean field approximation is derived. Empirical results on a set of problems suggest that both the mean field approximation and Gibbs sampling are viable alternatives to the computationally expensive exact algorithm.

Exercise sheet 3 pdf

Exercises and solutions in PDF

Exercise sheet 3

Exercises and solutions in LaTex

La construcción de sentido del inicio de un cambio estratégico en las organizaciones: el caso de la Universidad del Rosario

A través de un caso de estudio se explora cómo la construcción de sentido de un grupo de directivos, bajo una misma inspiración, generó el inicio de un cambio estratégico en una prestigiosa y reconocida universidad colombiana, la Universidad del Rosario. Una institución que en un momento determinado notó que estaba siendo percibida dentro del sector de la educación superior como pequeña, estática en el avance de algunas disciplinas del conocimiento y conservadora; en otras palabras, que estaba perdiendo el reconocimiento que usualmente la había acompañado. A través del estudio de este caso se utilizó la técnica de análisis de discurso para comprender la construcción de sentido del inicio de un cambio estratégico en las organizaciones. Esta técnica permitió analizar la información cualitativa derivada de las entrevistas que se realizaron en profundidad a la cúpula de directivos de la institución y a algunos destacados representantes del sector de la Educación Superior en Colombia. Los resultados sugieren que se hicieron presentes, efectivamente, algunas condiciones específicas que marcaron el inicio de un cambio estratégico en la institución y un viraje en su identidad e imagen. Hechos que se sustentaron en los miembros de un equipo que procuró interpretar y comprender los cambios existentes en el entorno global y local, y asimilar, igualmente, algunos destacados retos que se planteaban por aquella época, al interior de la propia Universidad

Caracterización de la cultura organizacional y lineamientos de intervención para la implementación de procesos de cambio en las organizaciones. Caso empresa sector financiero

El presente Trabajo de Grado busca caracterizar la cultura organizacional de una empresa del sector Financiero en Colombia y realizar orientaciones de acciones para el cambio organizacional de acuerdo con la estrategia de perdurabilidad establecida por la Alta Dirección de dicha empresa. Para este fin, se realiza una cuidadosa revisión y actualización del estado del arte de los conceptos clave ¨Cultura Organizacional¨ y ¨Cambio Organizacional¨. Es de resaltar que para el primero de ellos, se toma como punto de partida el estado del arte sobre Cultura Organizacional realizado por el profesor Carlos Eduardo Méndez Álvarez y cuyo marco temporal abarca desde los orígenes del concepto en el siglo XIX hasta el año 2006. Asimismo, luego de una cuidadosa revisión de los Modelos de Cambio Organizacional existentes y de la realidad de la empresa objeto de estudio, se adopta el Modelo ADKAR que consta de cinco fases: Conciencia del Cambio, Deseo, Conocimiento, Capacidad – Habilidad y Refuerzo. Asimismo, a partir de la construcción de un fundamento teórico sólido y a través de la aplicación de la metodología para describir la Cultura Organizacional en Colombia MEDECO se busca una aproximación a la Cultura Organizacional de la empresa objeto de estudio con el fin de describir e identificar los rasgos predominantes de su cultura organizacional y entregar una propuesta final con los rasgos necesarios que alientan la consecución exitosa de los procesos de cambio.

Automatic generation of active coordinates for quantum dynamics calculations: application to the dynamics of benzene photochemistry

A new practical method to generate a subspace of active coordinates for quantum dynamics calculations is presented. These reduced coordinates are obtained as the normal modes of an analytical quadratic representation of the energy difference between excited and ground states within the complete active space self-consistent field method. At the Franck-Condon point, the largest negative eigenvalues of this Hessian correspond to the photoactive modes: those that reduce the energy difference and lead to the conical intersection; eigenvalues close to 0 correspond to bath modes, while modes with large positive eigenvalues are photoinactive vibrations, which increase the energy difference. The efficacy of quantum dynamics run in the subspace of the photoactive modes is illustrated with the photochemistry of benzene, where theoretical simulations are designed to assist optimal control experiments

Polydispersity-induced macrophase separation in diblock copolymer melts

The effect of A-block polydispersity on the phase behavior of AB diblock copolymer melts is examined using a complete self-consistent field theory treatment that allows for fractionation of the parent molecular-weight distribution. In addition to observing the established shift in phase boundaries, we find the emergence of significant two-phase coexistence regions causing, for instance, the disappearance of the complex phase window. Furthermore, we find evidence that polydispersity relieves packing frustration, which will reduce the tendency for long-range order.

Architectural effect on the surface tension of an ABA triblock copolymer melt

The excess surface energy of lamellae formed by an ABA triblock copolymer melt oriented parallel to a neutral surface is evaluated using self-consistent field theory (SCFT). Consistent with experiments and previous SCFT calculations, we find a preference for the A-rich domains at the surface, which can only be attributed to the architectural asymmetry between the A and B blocks. The behavior was previously attributed to a loss of bridging configurations that occurs when the B-domain resides at the surface. Here we demonstrate that it is actually the presence of chain ends that reduces the excess surface energy of an A-rich domain relative that of a B-rich domain.

Inverse problems in dynamic cognitive modeling

Inverse problems for dynamical system models of cognitive processes comprise the determination of synaptic weight matrices or kernel functions for neural networks or neural/dynamic field models, respectively. We introduce dynamic cognitive modeling as a three tier top-down approach where cognitive processes are first described as algorithms that operate on complex symbolic data structures. Second, symbolic expressions and operations are represented by states and transformations in abstract vector spaces. Third, prescribed trajectories through representation space are implemented in neurodynamical systems. We discuss the Amari equation for a neural/dynamic field theory as a special case and show that the kernel construction problem is particularly ill-posed. We suggest a Tikhonov-Hebbian learning method as regularization technique and demonstrate its validity and robustness for basic examples of cognitive computations.