A Note on Iterations-based Derivations of High-order Homogenization Correctors for Multiscale Semi-linear Elliptic Equations

This Note aims at presenting a simple and efficient procedure to derive the structure of high-order corrector estimates for the homogenization limit applied to a semi-linear elliptic equation posed in perforated domains. Our working technique relies on monotone iterations combined with formal two-scale homogenization asymptotics. It can be adapted to handle more complex scenarios including for instance nonlinearities posed at the boundary of perforations and the vectorial case, when the model equations are coupled only through the nonlinear production terms.

Unfolding Operator Method for Thin Domains with a Locally Periodic Highly Oscillatory Boundary

We analyze the behavior of solutions of the Poisson equation with homogeneous Neumann boundary conditions in a two-dimensional thin domain which presents locally periodic oscillations at the boundary. The oscillations are such that both the amplitude and period of the oscillations may vary in space. We obtain the homogenized limit problem and a corrector result by extending the unfolding operator method to the case of locally periodic media.

INVESTIGATIONS INTO MECHANISMS UNDERLYING EXTREME WAVE FORMATIONS AND COMPUTATIONALLY INTENSIVE SIMULATIONS

Various mechanisms have been proposed to explain extreme waves or rogue waves in an oceanic environment including directional focusing, dispersive focusing, wave-current interaction, and nonlinear modulational instability. The Benjamin-Feir instability (nonlinear modulational instability), however, is considered to be one of the primary mechanisms for rogue-wave occurrence. The nonlinear Schrodinger equation is a well-established approximate model based on the same assumptions as required for the derivation of the Benjamin-Feir theory. Solutions of the nonlinear Schrodinger equation, including new rogue-wave type solutions are presented in the author's dissertation work. The solutions are obtained by using a predictive eigenvalue map based predictor-corrector procedure developed by the author. Features of the predictive map are explored and the influences of certain parameter variations are investigated. The solutions are rescaled to match the length scales of waves generated in a wave tank. Based on the information provided by the map and the details of physical scaling, a framework is developed that can serve as a basis for experimental investigations into a variety of extreme waves as well localizations in wave fields. To derive further fundamental insights into the complexity of extreme wave conditions, Smoothed Particle Hydrodynamics (SPH) simulations are carried out on an advanced Graphic Processing Unit (GPU) based parallel computational platform. Free surface gravity wave simulations have successfully characterized water-wave dispersion in the SPH model while demonstrating extreme energy focusing and wave growth in both linear and nonlinear regimes. A virtual wave tank is simulated wherein wave motions can be excited from either side. Focusing of several wave trains and isolated waves has been simulated. With properly chosen parameters, dispersion effects are observed causing a chirped wave train to focus and exhibit growth. By using the insights derived from the study of the nonlinear Schrodinger equation, modulational instability or self-focusing has been induced in a numerical wave tank and studied through several numerical simulations. Due to the inherent dissipative nature of SPH models, simulating persistent progressive waves can be problematic. This issue has been addressed and an observation-based solution has been provided. The efficacy of SPH in modeling wave focusing can be critical to further our understanding and predicting extreme wave phenomena through simulations. A deeper understanding of the mechanisms underlying extreme energy localization phenomena can help facilitate energy harnessing and serve as a basis to predict and mitigate the impact of energy focusing.

POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

Tendencia editorial UR Número 3

El Boletín Tendencia Editorial es un proyecto que nació en 2010 con motivo de la Feria Internacional del Libro de Bogotá para construir y hacer visibles diferentes saberes desde la academia y la edición. Para 2014, el cambio de periodicidad coincidió con uno de los eventos más importantes para la edición universitaria, la Feria Internacional del Libro de Guadalajara donde anualmente se realiza el encuentro de Editores Universitarios Latinoamericanos, el proyecto pasó de su fase nacional a ser pensado en red. Las líneas temáticas traspasan fronteras locales y convocan, en un mismo espacio, las voces de los gestores y especialistas, cuya labor y experiencia permiten cada día mejorar la edición universitaria, lo que posibilita acabar con el mito que concibe a la Universidad como ente ajeno a la sociedad, cuando en realidad esta es la forjadora de líderes, investigadores y emprendedores.

Tendencia editorial UR Número 10

El Boletín Tendencia Editorial es un proyecto que nació en 2010 con motivo de la Feria Internacional del Libro de Bogotá para construir y hacer visibles diferentes saberes desde la academia y la edición. Para 2014, el cambio de periodicidad coincidió con uno de los eventos más importantes para la edición universitaria, la Feria Internacional del Libro de Guadalajara donde anualmente se realiza el encuentro de Editores Universitarios Latinoamericanos, el proyecto pasó de su fase nacional a ser pensado en red. Las líneas temáticas traspasan fronteras locales y convocan, en un mismo espacio, las voces de los gestores y especialistas, cuya labor y experiencia permiten cada día mejorar la edición universitaria, lo que posibilita acabar con el mito que concibe a la Universidad como ente ajeno a la sociedad, cuando en realidad esta es la forjadora de líderes, investigadores y emprendedores.

Design and synthesis of E3 ligase RNF5 inhibitors as innovative strategy to trigger mutant CFTR rescue in Cystic Fibrosis

In Cystic Fibrosis (CF) the deletion of phenylalanine 508 (F508del) in the CFTR anion channel is associated to misfolding and defective gating of the mutant protein. Among the known proteins involved in CFTR processing, one of the most promising drug target is the ubiquitin ligase RNF5, which normally promotes F508del-CFTR degradation. In this context, a small molecule RNF5 inhibitor is expected to chemically mimic a condition of RNF5 silencing, thus preventing mutant CFTR degradation and causing its stabilization and plasma membrane trafficking. Hence, by exploiting a virtual screening (VS) campaign, the hit compound inh-2 was discovered as the first-in-class inhibitor of RNF5. Evaluation of inh-2 efficacy on CFTR rescue showed that it efficiently decreases ubiquitination of mutant CFTR and increases chloride current in human primary bronchial epithelia. Based on the promising biological results obtained with inh-2, this thesis reports the structure-based design of potential RNF5 inhibitors having improved potency and efficacy. The optimization of general synthetic strategies gave access to a library of analogues of the 1,2,4-thiadiazol-5-ylidene inh-2 for SAR investigation. The new analogues were tested for their corrector activity in CFBE41o- cells by using the microfluorimetric HS-YFP assay as a primary screen. Then, the effect of putative RNF5 inhibitors on proliferation, apoptosis and the formation of autophagic vacuoles was evaluated. Some of the new analogs significantly increased the basal level of autophagy, reproducing RNF5 silencing effect in cell. Among them, one compound also displayed a greater rescue of the F508del-CFTR trafficking defect than inh-2. Our preliminary results suggest that the 1,2,4-thiadiazolylidene could be a suitable scaffold for the discovery of potential RNF5 inhibitors able to rescue mutant CFTRs. Biological tests are still ongoing to acquire in-depth knowledge about the mechanism of action and therapeutic relevance of this unprecedented pharmacological strategy.