Proyectos

Esta propuesta está dedicada al estudio de problemas locales y no locales, elípticos y parabólicos, en Ecuaciones en Derivadas Parciales (EDP). Se espera obtener resultados de existencia para los problemas planteados, son 3 los problemas que se quieren estudiar: El primer modelo involucra al Laplaciano Fraccionario con singularidades no lineales (Ecuación Fraccionaria de Burger ver (3)). En este problema queremos probar la existencia y la no unicidad de soluciones débiles de (3), teniendo en cuenta que las soluciones de entropía son soluciones débiles, el primer paso será probar existencia de soluciones de entropía, para esto usaremos el método de sub y supersoluciones, donde probaremos los resultados de los principios de Comparación y L1-Contracción. Una vez teniendo la existencia de solución de entropía, pasaremos a construir una solución débil que no sea solución de entropía, donde esta solución débil se obtendrá como límite de soluciones a problemas regularizados estacionarios, en donde usaremos métodos variacionales para resolver el problema regularizado. El segundo problema, es una extensión al caso no local del problema estudiado en [L. Jeanjean and V. Radulescu, Nonhomogeneous quasilinear elliptic problems: linear and sublinear cases, Journal d'Analyse Mathématique (2021)]. Para probar existencia de soluciones aplicaremos estudios de mínimos locales y el teorema del paso de montaña para el funcional de energía asociado. Para finalizar nuestra propuesta, queremos encontrar soluciones periódicas para sistemas de EDP de cuarto orden (tipo Fisher-Kolmogorov generalizado, ver (9). Siguiendo las técnicas en [P. Smyrnelis, Connecting orbits in Hilbert Spaces and application to PDEs, Comm. Pure Appl Anal 19 (5): 2897--2818 (2020)], usaremos métodos variacionales para probar la existencia de órbitas conectadas en espacios de Hilbert. Construiremos órbitas periódicas usando la construcción desarrollada por Alessio, Montecchiari y Zúñiga en [F. Alessio and P. Montecchiari and A. Zuniga, Prescribed energy connecting orbits for gradient systems, Discr. Cont. Dyn. Systems 39 (8): 4895--4928 (2019].

Esta propuesta está dedicada al estudio de problemas locales y no locales, elípticos y parabólicos, en Ecuaciones en Derivadas Parciales (EDP). Se espera obtener resultados de existencia para los problemas planteados, son 3 los problemas que se quieren estudiar: El primer modelo involucra al Laplaciano Fraccionario con singularidades no lineales (Ecuación Fraccionaria de Burger ver (3)). En este problema queremos probar la existencia y la no unicidad de soluciones débiles de (3), teniendo en cuenta que las soluciones de entropía son soluciones débiles, el primer paso será probar existencia de soluciones de entropía, para esto usaremos el método de sub y supersoluciones, donde probaremos los resultados de los principios de Comparación y L1-Contracción. Una vez teniendo la existencia de solución de entropía, pasaremos a construir una solución débil que no sea solución de entropía, donde esta solución débil se obtendrá como límite de soluciones a problemas regularizados estacionarios, en donde usaremos métodos variacionales para resolver el problema regularizado. El segundo problema, es una extensión al caso no local del problema estudiado en [L. Jeanjean and V. Radulescu, Nonhomogeneous quasilinear elliptic problems: linear and sublinear cases, Journal d'Analyse Mathématique (2021)]. Para probar existencia de soluciones aplicaremos estudios de mínimos locales y el teorema del paso de montaña para el funcional de energía asociado. Para finalizar nuestra propuesta, queremos encontrar soluciones periódicas para sistemas de EDP de cuarto orden (tipo Fisher-Kolmogorov generalizado, ver (9). Siguiendo las técnicas en [P. Smyrnelis, Connecting orbits in Hilbert Spaces and application to PDEs, Comm. Pure Appl Anal 19 (5): 2897--2818 (2020)], usaremos métodos variacionales para probar la existencia de órbitas conectadas en espacios de Hilbert. Construiremos órbitas periódicas usando la construcción desarrollada por Alessio, Montecchiari y Zúñiga en [F. Alessio and P. Montecchiari and A. Zuniga, Prescribed energy connecting orbits for gradient systems, Discr. Cont. Dyn. Systems 39 (8): 4895--4928 (2019].

Chile, Peru and France, as well as many countries in South America and Europe, share a very similar systems to deal with their electricity markets. In parallel, all three countries (together with the rest of the world) are being affected by climate change in many aspects, such as scarcity of water, intense droughts, pollution and the greenhouse effect, the necessity of new energy sources, just to name a few. To face these challenges, we need new technology coming from many fields of science. One of such fields is mathematics and in particular, stochastic optimization and game theory. These theoretical fields allow us to model economic interactions, management solutions, optimal design and operations, among many other relevant aspects of Natural Resources and Energy Management. In the present project, we propose to develop new theoretical and numerical advances in four research lines, concerning Stochastic Optimization and Game Theory. Namely, we will work on: 1) Continuity-like properties in Equilibrium problems; 2) Regularity in Generalized Equilibrium problems; 3) Bilevel games with decision-dependent uncertainty; and 4) Algorithms and mechanism design in learning games. The four research lines are strongly motivated by the aforementioned applications.

Chile, Peru and France, as well as many countries in South America and Europe, share a very similar systems to deal with their electricity markets. In parallel, all three countries (together with the rest of the world) are being affected by climate change in many aspects, such as scarcity of water, intense droughts, pollution and the greenhouse effect, the necessity of new energy sources, just to name a few. To face these challenges, we need new technology coming from many fields of science. One of such fields is mathematics and in particular, stochastic optimization and game theory. These theoretical fields allow us to model economic interactions, management solutions, optimal design and operations, among many other relevant aspects of Natural Resources and Energy Management. In the present project, we propose to develop new theoretical and numerical advances in four research lines, concerning Stochastic Optimization and Game Theory. Namely, we will work on: 1) Continuity-like properties in Equilibrium problems; 2) Regularity in Generalized Equilibrium problems; 3) Bilevel games with decision-dependent uncertainty; and 4) Algorithms and mechanism design in learning games. The four research lines are strongly motivated by the aforementioned applications.

Overview: Mechanical wave propagation physics is related to traumatic brain injury mechanisms. For instance, nonlinear shear waves can form in the brain progressively with propagation, amplifying the acceleration locally. This phenomenon is compatible with axonal brain injury in which the lesions are localized far from the impact region. Within the brain, not only shear waves propagate. Especially when considering the brain is full of folds and surfaces, including the gray-white matter interface, which can be seen in Fig. 1. The abundance of interfaces or surfaces makes us hypothesize that surface waves might be crucial for describing the biomechanics of traumatic brain injury. The surface waves are named after the nature of the interface. A wave propagating in a solid-vacuum interface is known as a Rayleigh wave, a wave propagating in a solid-fluid interface is known as a Scholte wave, and a wave propagating in a solid-solid interface is known as a Stoneley wave. This last might propagate within an interface formed by two types of soft tissue. An example of this is the interfaces formed by the white and gray matter in the brain (see Fig. 1). Surface waves, although confined to a surface, can penetrate up to a wavelength. In the context of soft tissues, the typical frequencies of elastic waves that propagate are in the range of 10 to 300 Hz. At these frequencies, the wavelengths are on the order Figure 1: Section of a temporal of centimeters, creating the necessity of studying these waves at brain lobe. Image obtained from depth. Surface waves are not explored sufficiently in incompressible soft the visible human project [1]. solids yet. We recently measured Scholte waves at depth in these materials. However, we are not aware of measurements of Stoneley waves at depth in incompressible soft solids like brain matter or gelatin. The lack of this experimental evidence is due to the challenges of measuring deformation in opaque materials without disrupting the medium. Thus, the general objective of this proposal is to detect, describe and characterize the propagation of Stoneley waves in interfaces formed by two incompressible tissue-mimicking materials using Ultrafast Ultrasound elastography-related techniques. Methodology: Advanced ultrasound imaging techniques implemented on a highly customized ultrasound imaging platform designed for high frame-rate imaging will be used to characterize fundamental Stoneley wave physics propagating at the interfaces between two soft solids. We first will perform experiments in flats and simple interfaces to obtain the parameter space (shear modulus, density, and prestress) in which planar Stoneley waves exist. Then, we will explore the effects of the bonding condition between the two mediums on the dispersion relation. Third, we will investigate the interaction of the shape of the interface on the wave propagation, and lastly, we will intend to propagate Stonely waves into 3D inclusions. These observations will be achieved with a number of steps that integrate advancements in ultrasound imaging, algorithms that measure the deformation, and modeling. Custom two-dimensional and three-dimensional imaging sequences, designed for displacement tracking, will be implemented for a dedicated Linear and Matrix array transducer that has 128 or 1024 elements and can reach a spatial resolution of 200microns at a very high framerate in the order of 10000 frames per second (2D or 3D frames respectively). Expected results: The results of this proposal will elucidate the conditions that the two soft solids need to propagate Stoneley waves. These conditions refer to the combination of mechanical properties of the materials, such as shear modulus and density, and the prestress field needed. We expect to establish the effect that the bonding condition between the two soft solids has on the nature of the Stoneley wave. In particular, we will monitor how the Stoneley wave speed and dispersion change with different bonding conditions. We believe this phenomenology has implications in imaging technology, tumor diagnosis, and brain injury biomechanics.

Objetivo general: Instalar una editorial cooperativa en la región de O’Higgins sustentada a partir de la comercialización de productos y servicios educativos desarrollados en base a evidencia científica, que permitan implementar el aprendizaje basado en juegos y que consideran la diversidad del aula. Esta propuesta incluye la instalación de un laboratorio de prototipado y producción, además del desarrollo de un paquete de productos y servicios educativos cuya comercialización permite sustentar la editorial cooperativa. 5.4.1. Objetivo Específico 1 (Editorial Cooperativa) Coordinar una red de emprendedores del área educativa para establecer una editorial cooperativa para la Región de O’Higgins para desarrollar y comercializar productos y servicios educativos lúdicos, de base científica y carácter inclusivo. Funciona bajo un modelo de cooperativa cuyos miembros son emprendedores localizados en la región de O’Higgins con interés en innovación educativa, trabajando bajo el principio de ayuda mutua. 5.4.2. Objetivo Específico 2 (Diagnóstico de Productos) Diagnosticar las necesidades de productos y servicios educativos para promover la calidad de los aprendizajes en los primeros cursos de educación básica en la región de O’Higgins. Identifica tópicos en matemáticas y lenguaje que presentan dificultades al momento de su enseñanza, describe la diversidad de estudiantes en escuelas municipales e indaga métodos de comercialización comunes. 5.4.3. Objetivo Específico 3 (Laboratorio de prototipado y Manufactura) Instalar en la Universidad de O’Higgins un laboratorio para elaborar prototipos de juegos de mesa educativos y producirlos en pequeña escala. Se adquiere maquinaria, equipamientos e insumos necesarios para que el Laboratorio de Aprendizaje Matemático (LAM) pueda prototipar juegos de mesa y, en conjunto con la Fábrica Digital O’Higgins (FabLab), producirlos con alta calidad y en pequeña escala de manera eficiente (30-100 ejemplares). 5.4.4. Objetivo Específico 4 (Desarrollo de Productos) Realizar el diseño, prototipado y manufactura de un paquete de productos educativos para fortalecer las áreas de matemática y lenguaje en los primeros años de educación básica. Diseñados en base a ciencias del aprendizaje, estos productos permiten desarrollar aprendizajes en base a juegos y actividades lúdicas considerando la diversidad de estudiantes en el aula. 5.4.5. Objetivo Específico 5 (Capacitación y Servicios) Generar un programa de capacitación para certificar a los emprendedores educativos para que provean servicios que acompañan a los productos educativos ofrecidos por la editorial. Este programa certifica que el emprendedor es experto en la aplicación de los productos educativos desarrollados. Este certificado habilita a los miembros de la editorial para ofrecer estos servicios en el mercado educativo.

Frailty is increasingly becoming an important public health challenge worldwide because it is associated with older age, and with adverse outcomes such as reduced quality of life, increased mortality rates, hospitalizations, falls, depression, and dementia. Frailty is defined as dynamic state affecting an individual who experiences losses in one or more domains of human functioning (physical, psychological, social) that are caused by the influence of a range of variables, and which increases the risk of adverse outcomes. This more integral conceptual definition promotes the collaboration of scientists, social and behavioral professionals as well as clinicians from diverse specialties. In this proposal an interdisciplinary group (Biochemistry, Geriatric, Occupational Therapist, Kinesiologist, social worker, bioengineer, statistician among others) aims to evaluate frailty in Chile with a biopsychosocial approach with the final purpose to identify and manage frailty while taking into consideration all the dimensions. Additionally, we aim to design a multidomain personalized person-base intervention for a healthy aging that can uncover a circulating microRNA biomarker panel that can allow an early-detection of frailty, leading to a new multidimensional geriatric assessment. We propose the following hypothesis: A personalized multidimensional training program reduces the frailty prevalence, increasing adherence and participation in the program among community-living older adults. This intervention will be paralleled by a distinctive miRNA profile reflecting the multiple domains of frailty, as well as improvements in diverse psychosocial traits.

This project aims to investigate how structural modifications of a dicationic derivative of azobenzene can affect the drug release and load capacity of its photoactive molecular aggregate. To evaluate this, three types of structural modifications are proposed. First, the introduction of functional groups on the photoactive nucleus of dicationic azobenzene is expected to shift the absorption band of the molecular photoswitch. Second, the replacement of the fluorescent organic cations over the structure of the molecular photoswitch, which confer luminescent and amphipathic properties to the system. And third, the modification of the length of the chains over the molecular photoswitch could change the aggregate size. To determine whether these potential modifications can modulate the light-induced release activity of the photoswitchable aggregate, an enzyme inhibitor will be loaded and released by illumination in the presence of the enzyme. Under this scenario, any modification of the enzymatic activity will be correlated with the drug's photorelease.

The proposal focuses on understanding the neuro-vascular aging mechanisms associated with alterations in fetal growth by intrauterine hypoxia using molecular biology and physiology as an area. The aim of the study is to demonstrate that impaired fetal growth conditions are associated with epigenetic programming of aging-related DNA methylation, chromatin remodeling, and miRNA-omic profile of junctional complex genes in the neuroendothelium, which can alter BBB integrity and permeability, increasing cerebral damage which impacts the juvenile and adulthood neurocognitive function.

Overview: Mechanical wave propagation physics is related to traumatic brain injury mechanisms. For instance, nonlinear shear waves can form in the brain progressively with propagation, amplifying the acceleration locally. This phenomenon is compatible with axonal brain injury in which the lesions are localized far from the impact region. Within the brain, not only shear waves propagate. Especially when considering the brain is full of folds and surfaces, including the gray-white matter interface, which can be seen in Fig. 1. The abundance of interfaces or surfaces makes us hypothesize that surface waves might be crucial for describing the biomechanics of traumatic brain injury. The surface waves are named after the nature of the interface. A wave propagating in a solid-vacuum interface is known as a Rayleigh wave, a wave propagating in a solid-fluid interface is known as a Scholte wave, and a wave propagating in a solid-solid interface is known as a Stoneley wave. This last might propagate within an interface formed by two types of soft tissue. An example of this is the interfaces formed by the white and gray matter in the brain (see Fig. 1). Surface waves, although confined to a surface, can penetrate up to a wavelength. In the context of soft tissues, the typical frequencies of elastic waves that propagate are in the range of 10 to 300 Hz. At these frequencies, the wavelengths are on the order Figure 1: Section of a temporal of centimeters, creating the necessity of studying these waves at brain lobe. Image obtained from depth. Surface waves are not explored sufficiently in incompressible soft the visible human project [1]. solids yet. We recently measured Scholte waves at depth in these materials. However, we are not aware of measurements of Stoneley waves at depth in incompressible soft solids like brain matter or gelatin. The lack of this experimental evidence is due to the challenges of measuring deformation in opaque materials without disrupting the medium. Thus, the general objective of this proposal is to detect, describe and characterize the propagation of Stoneley waves in interfaces formed by two incompressible tissue-mimicking materials using Ultrafast Ultrasound elastography-related techniques. Methodology: Advanced ultrasound imaging techniques implemented on a highly customized ultrasound imaging platform designed for high frame-rate imaging will be used to characterize fundamental Stoneley wave physics propagating at the interfaces between two soft solids. We first will perform experiments in flats and simple interfaces to obtain the parameter space (shear modulus, density, and prestress) in which planar Stoneley waves exist. Then, we will explore the effects of the bonding condition between the two mediums on the dispersion relation. Third, we will investigate the interaction of the shape of the interface on the wave propagation, and lastly, we will intend to propagate Stonely waves into 3D inclusions. These observations will be achieved with a number of steps that integrate advancements in ultrasound imaging, algorithms that measure the deformation, and modeling. Custom two-dimensional and three-dimensional imaging sequences, designed for displacement tracking, will be implemented for a dedicated Linear and Matrix array transducer that has 128 or 1024 elements and can reach a spatial resolution of 200microns at a very high framerate in the order of 10000 frames per second (2D or 3D frames respectively). Expected results: The results of this proposal will elucidate the conditions that the two soft solids need to propagate Stoneley waves. These conditions refer to the combination of mechanical properties of the materials, such as shear modulus and density, and the prestress field needed. We expect to establish the effect that the bonding condition between the two soft solids has on the nature of the Stoneley wave. In particular, we will monitor how the Stoneley wave speed and dispersion change with different bonding conditions. We believe this phenomenology has implications in imaging technology, tumor diagnosis, and brain injury biomechanics.