Resumen: En este artículo, construimos una sucesión similar a la de Newton para acercarnos al cero de una función diferenciable en el sentido Fréchet no lineal definida en un espacio de Hilbert. Esta nueva sucesión utiliza el concepto del adjunto del operador, que hace que el proceso iterativo sea más manejable en la práctica en comparación al desarrollado por Kantorovich que requiere el cálculo del operador inverso en cada iteración. Dado que el cálculo del operador adjunto es más fácil en comparación con el cálculo del operador inverso que en la práctica equivale a resolver un sistema de ecuaciones, nuestra nuevo método hace que el cálculo del término de nuestra nueva sucesión sea más fácil y conveniente para la aproximación numérica. Proporcionamos un teorema de convergencia a priori de esta sucesión, donde usamos unas hipótesis equivalentes a las construidas por Kantorovich, y mostramos que nuestra nueva sucesión iterativa converge hacia la solución.

Abstract: In this paper, we build a Newton-like sequence to approach the zero of a nonlinear Fréchet differentiable function defined in Hilbert space. This new iterative sequence uses the concept of the adjoint operator, which makes it more manageable in practice compared to the one developed by Kantorovich which requires the calculation of the inverse operator in each iteration. Because the calculation of the adjoint operator is easier compared to the calculation of the inverse operator which requires in practice solving a system of linear equations, our new method makes the calculation of the term of our new sequence easier and more convenient for numerical approximations. We provide an a priori convergence theorem of this sequence, where we use hypotheses equivalent to those constructed by Kantorovich, and we show that our new iterative sequence converges towards the solution.

Abstract This paper addresses a class of sequential optimization problems known as Markov decision processes. These kinds of processes are considered on Euclidean state and action spaces with the total expected discounted cost as the objective function. The main goal of the paper is to provide conditions to guarantee an adequate Moreau-Yosida regularization for Markov decision processes (named the original process). In this way, a new Markov decision process that conforms to the Markov control model of the original process except for the cost function induced via the Moreau-Yosida regularization is established. Compared to the original process, this new discounted Markov decision process has richer properties, such as the differentiability of its optimal value function, strictly convexity of the value function, uniqueness of optimal policy, and the optimal value function and the optimal policy of both processes, are the same. To complement the theory presented, an example is provided.

ABSTRACT In this article, we consider a class of nonlinear parabolic equations. We use an integral representation combined with a sort of fixed point theorem to prove the existence of classical solutions for the initial value problem (1.1), (1.2). We also obtain a result on continuous dependence on the initial data. We propose a new approach for investigation for existence of classical solutions of some classes nonlinear parabolic equations.

RESUMEN En este artículo, consideramos una clase de ecuaciones parabólicas nolineales. Usamos una representación integral combinada con una especie de teorema de punto fijo para probar la existencia de soluciones clásicas para el problema de valor inicial (1.1), (1.2). También obtenemos un resultado sobre la dependencia continua de la data inicial. Proponemos una estrategia nueva para la investigación de la existencia de soluciones clásicas de algunas clases de ecuaciones parabólicas nolineales.

Abstract Piecewise-linear models constitute an attractive alternative to construct a function whose graph fits a finite set of discrete points. These models are preferably selected over other approximation strategies like polynomials or splines. Although there are several piecewise-linear models reported in literature, the so-called High Level Canonical has the remarkable advantage of emerging from a well-structured algorithmic methodology to efficiently determine the parameters of any given piecewise-linear function. However, as it happens in all other piecewise-linear models, it also has the problem of lack of differentiability at the breakpoints. In order to solve this problem, an approach based on an exponential approximation of the basis-function is proposed as a strategy to transform the High Level Canonical piecewise-linear model into a smooth-piecewise one. This mathematical transformation ensures the existence and continuity of the nth-order derivatives of the resulting smooth model. Besides of this, it is observed that by applying the piecewise-linear to smooth transformation, the number of terms of the resulting smooth representation can significantly be reduced due to a great number of them can be approximated by a line equation. In order to verify the effectiveness of this proposal, numerical simulations performed on one-dimensional and two-dimensional functions are reported.

Se estudia la existencia y unicidad de soluciones para problemas de valor en la frontera asociadas a ecuaciones diferenciales de funciones con valores en intervalos, usando la derivada de Hukuhara y algunos teoremas de punto fijo de funciones débilmente contractivas definidas en conjuntos parcialmente ordenados.

We study the existence and uniqueness of solutions for boundary value problems associated to differential equations of interval-valued functions, by using the derivative of Hukuhara and some fixed point theorems for weakly contractive mappings defined on partially ordered sets.

O artigo trata de um problema de otimização vetorial entre espaços de Banach com restrições envolvendo cones. Usando-se uma lagrangiana que toma valores escalares e o conceito de funções subconvexas generalizadas, soluções fracamente eficientes são caracterizadas por condições do tipo ponto de sela. Os resultados, em conjunto com a noção de Hessiana generalizada (introduzida em [R. Cominetti, R. Correa, A generalized second-order derivative in nonsmooth optimization, SIAM J. Control Optim., 28 (1990), 789-809]), são aplicados para se obter condições necessárias e suficientes de segunda ordem para o caso particular em que as funcionais envolvidas são definidas em um espaço de Banach geral mas com valores em espaços de dimensão finita (sem exigir que as funções objetivo e de restrições sejam duas vezes diferenciáveis).

This article deals with a vector optimization problem with cone constraints in a Banach space setting. By making use of a real-valued Lagrangian and the concept of generalized subconvex-like functions, weakly efficient solutions are characterized through saddle point type conditions. The results, jointly with the notion of generalized Hessian (introduced in [Cominetti, R., Correa, R.: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28, 789-809 (1990)]), are applied to achieve second order necessary and sufficient optimality conditions (without requiring twice differentiability for the objective and constraining functions) for the particular case when the functionals involved are defined on a general Banach space into finite dimensional ones.

En este artículo panorámico se presentan cuatro versiones equivalentes de la propiedad de Radon-Nikodým de un espacio de Banach: el teorema de representación de Riesz, el teorema de Lewis-Stegall, un teorema sobre diferenciabilidad de funciones absolutamente continuas y una caracterización geométrica del espacio.

In this survey article, we give four equivalent classical versions of the Radon-Nikodým property for Banach spaces, namely, the Riesz representation theorem, the Lewis-Stegall theorem, a result on differentiability of absolutely continuous functions and finally, a geometric characterization of the Banach space.

A 3D geometric transformation is introduced for the nonrigid registration of medical images as an extension of a previous work carried out for two dimensions. A 3D spatial transformation is analyzed in order to guaranty the continuity, the differentiability and the one-to-one transformation by imposing constraints to the transformation parameters. It is also shown and analyzed the results when the fully automatic nonrigid registration method is applied to a CT-PET stack of the thorax with a spatial resolution of 80 x 80 x 80 and to a RM head stack with a spatial resolution of 128 x 128 x 128 pixels. The 3D geometric transformation has a spherical domain and it allows the continuity of the transformation in its boundary. This geometrical transformation can be applied to global or local ROIs (region of interest) up to a minimum diameter of three pixels. The nonrigid image registration method employs an evolutionary algorithm to obtain satisfactory global solutions while it maximizes the normalized mutual information (NMI). This approach has the disadvantage that the speed of convergence and the accuracy of the method depend on the population size of the evolutionary algorithm. Results show an improvement in the global similarity function between the target and source volumes throughout 73 transformations, from coarse to fine (3 levels of resolution), from 0.501 7 to 0.5033, using a population size of 10 individuals. 3D surface reconstructions of the thorax are also shown before and after the nonrigid registration. In addition, a simulated experiment is carried out with a RM head stack, where a unique transformation was applied. Here, it was got an improvement in the similarity criterion from 0.5046 to 0.5218.

Una tranformación geométrica tridimensional (3D) se presenta para el registro de imágenes médicas como la extensión de un trabajo previo desarrollado para dos dimensiones. La tranformación espacial 3D es analizada con la intención de garantizar la continuidad, la diferenciabilidad y la relación uniquívoca mediante la imposición de restricciones a los parámetros de la tranformación. Se demuestran y analizan los resultados cuando el registro no rígido completamente automático es aplicado a un conjunto de imágenes de tórax con una resolución espacial de 128x128x 128 pixeles adquiridas por el método de la tomografía computarizada asociada a la tomografía de emisión de positrones (PET-CT). La tranformación geométrica 3D tiene un dominio esférico que permite la continuidad de la transformación en su frontera. Esta transformación geométrica puede ser aplicada a regiones de interés locales o globales hasta un diámetro mínimo de 3 pixeles. El método de registro no rígido utiliza un algoritmo evolutivo para obtener resultados globales satisfactorios mientras que maximiza la información mutua normalizada. Esta propuesta tiene la desventaja de que la velocidad de convergencia y la precisión del método depende del tamaño de la población en el algoritmo evolutivo. Los resultados demuestran una mejora en la función de similitud global entre los volúmenes fuente y destino a través de 73 transformaciones, desde burdos hasta finos (en 3 niveles de resolución), de 0.5017 a 0.5033 usando un tamaño de población de 10 individuos. Las reconstrucciones 3D del tórax también se muestran antes y después de la transformación no rígida. Además, experimentos de simulación fueron desarrollados con imágenes tomadas de un estudio del cráneo por resonancia magnética utilizando sólo una transformación. Aquí, se obtuvo una mejora en el criterio de similitud desde 0.5046 a 0.5218.

Apesar dos modelos de Branas-mundo terem recebido atenção considerável nos últimos anos por fornecerem várias opções à física contemporânea, seus mecanismos não são completamente entendidos ou apropriadamente justificados. Tendo em vista tal dificuldade, neste trabalho fornecemos uma contribuição pedagógica dirigida especialmente aos alunos de pós-graduação em física, fazendo uma abordagem introdutória a um dos temas importantes em física e geometria no que diz respeito aos fundamentos da teoria de imersão de variedades. Apresentamos uma descrição do teorema de imersão de Nash de 1956 que mostra como fazer uma imersão local entre variedades Riemaniannas mantendo a regularidade e diferenciabilidade das funções de imersão, e de como isso é aplicado à física sob o ponto de vista das Branas-mundo.

Although brane-world models have gained considerable attention in the last years for providing several options for contemporary physics, their mechanisms are not completely understood or properly justified. Taking into account such difficulty, in this work we provide a pedagogical contribution, written especially for physics graduate students, and present an introductory approach to one of the important themes in physics and geometry concerning the foundations of immersion theory of manifolds. We present a description of Nash's embedding theorem of 1956 which shows how to do a local embedding between Riemannian manifolds while preserving the regularity and differentiability of the embedding functions, and how it can be applied to physics in the contex of the brane-worlds.

En este artículo se muestra a traves de ejemplos, la forma como se calculan derivadas en grupos topológicos. En cada uno de ellos se deben resolver ecuaciones diferenciales ordinarias o parciales.

This article shows how to calculate derivatives in topological groups trough examples. In each of them one should solve ordinary or partial differential equations

Neste trabalho pretende apresentar-se uma classificação dos Métodos de Penalidade existentes (salientando os Métodos de Penalidade Exacta) e descrever algumas das suas limitações e pressupostos. Esses métodos permitem resolver problemas de optimização com restrições contínuas, discretas e mistas, sem requerer continuidade, diferenciabilidade ou convexidade. A abordagem consiste em transformar o problema original, numa sequência de problemas sem restrições, derivados do inicial, possibilitando a sua resolução pelos métodos conhecidos para este tipo de problemas. Assim, os Métodos de Penalidade podem ser usados como o primeiro passo para a resolução de problemas de optimização permitindo a resolução de problemas com restrições por métodos tipicamente utilizados em problemas sem restrições. O trabalho termina com a discussão de uma nova classe de Métodos de Penalidade, para optimização não linear, que ajustam o parâmetro de penalidade dinamicamente.

In this work we present a classification of some of the existing Penalty Methods (denominated the Exact Penalty Methods) and describe some of its limitations and estimated. With these methods we can solve problems of optimization with continuous, discrete and mixing constrains, without requiring continuity, differentiability or convexity. The boarding consists of transforming the original problem, in a sequence of problems without constrains, derivate of the initial, making possible its resolution for the methods known for this type of problems. Thus, the Penalty Methods can be used as the first step for the resolution of constrained problems for methods typically used in by unconstrained problems. The work finishes discussing a new class of Penalty Methods, for nonlinear optimization, that adjust the penalty parameter dynamically.

In this note we establish a criterion to obtain classical solutions for a quasi-linear abstract retarded functional differential the infinitesimal generators of several strongly continuous semigroup of linear operators that arise in the theory of linear abstract retarded functional differential equations with infinite delay on a phase space defined axiomatically. 2000 Mathematics Subject Classification. 34K30, 34G10, 47D06

In this note we establish a result of differentiability for the mild solution of the inhomogeneous abstract Cauchy problem when the underlying space is reflexive