Teorema del punto fijo de Banach y algunas aplicaciones

El teorema del punto fijo es aplicable a un tipo especial de sucesiones; dicho teorema es utilizado en muchas ciencias aplicadas (economía, ingeniería, informática) así como en las ciencias fundamentales (física, química, biología, etc.). El trabajo investiga la teoría del punto fijo y algunas aplicaciones del Teorema del Punto Fijo de Banach, para elaborar un documento en el que se presenten algunas aplicaciones y la teoría del punto fijo.

Aplicación de los anillos noetherianos conmutativos y las variedades algebraicas afines en la demostración del teorema de los ceros de Hilbert

Se desarrolla un estudio de todas las herramientas necesarias para llegar al teorema de los ceros de Hilbert el cual luego se demuestra en sus formas débil y fuerte. Se introducen los conceptos básicos relacionados con los anillos noetherianos y las variedades algebraicas afines que son fundamentales para el estudio del teorema de los ceros de Hilbert. Es por ello que estudiamos detenidamente el concepto de ideal primo e ideal primario, como también las distintas operaciones entre ideales, en particular la descomposición primaria de ideales. En seguida se desarrollan las demostraciones de algunos de los teoremas importantes de los anillos noetherianos, haciendo uso de la descomposición primaria de un ideal y un resultado fundamental: el teorema de la base de Hilbert. Además se desarrollan las definiciones, proposiciones, teoremas de una variedad algebraica afín y el ideal asociado a una variedad, así como también el ideal de una variedad y lo más interesante es la descomposición de ideales en variedades algebraicas afines, como la condición de cadena descendente de variedades. También se hace la aplicación de los resultados obtenidos en los capítulos anteriores, para demostrar el teorema de los ceros de Hilbert en su forma dedil así como en la forma fuerte. Finalmente adoptamos una Topología que es muy débil pero sorprendentemente útil ocupando los resultados anteriores, probando propiedades que cumple esta topología como la cerradura topológica y compacidad.

THE STOCHASTIC NAVIER STOKES EQUATIONS FOR HEAT CONDUCTING, COMPRESSIBLE FLUIDS

This dissertation is devoted to the equations of motion governing the evolution of a fluid or gas at the macroscopic scale. The classical model is a PDE description known as the Navier-Stokes equations. The behavior of solutions is notoriously complex, leading many in the scientific community to describe fluid mechanics using a statistical language. In the physics literature, this is often done in an ad-hoc manner with limited precision about the sense in which the randomness enters the evolution equation. The stochastic PDE community has begun proposing precise models, where a random perturbation appears explicitly in the evolution equation. Although this has been an active area of study in recent years, the existing literature is almost entirely devoted to incompressible fluids. The purpose of this thesis is to take a step forward in addressing this statistical perspective in the setting of compressible fluids. In particular, we study the well posedness for the corresponding system of Stochastic Navier Stokes equations, satisfied by the density, velocity, and temperature. The evolution of the momentum involves a random forcing which is Brownian in time and colored in space. We allow for multiplicative noise, meaning that spatial correlations may depend locally on the fluid variables. Our main result is a proof of global existence of weak martingale solutions to the Cauchy problem set within a bounded domain, emanating from large initial datum. The proof involves a mix of deterministic and stochastic analysis tools. Fundamentally, the approach is based on weak compactness techniques from the deterministic theory combined with martingale methods. Four layers of approximate stochastic PDE's are built and analyzed. A careful study of the probability laws of our approximating sequences is required. We prove appropriate tightness results and appeal to a recent generalization of the Skorohod theorem. This ultimately allows us to deduce analogues of the weak compactness tools of Lions and Feireisl, appropriately interpreted in the stochastic setting.

Two-dimensional Stokes flow driven by elliptical paddles

A fast and accurate numerical technique is developed for solving the biharmonic equation in a multiply connected domain, in two dimensions. We apply the technique to the computation of slow viscous flow (Stokes flow) driven by multiple stirring rods. Previously, the technique has been restricted to stirring rods of circular cross section; we show here how the prior method fails for noncircular rods and how it may be adapted to accommodate general rod cross sections, provided only that for each there exists a conformal mapping to a circle. Corresponding simulations of the flow are described, and their stirring properties and energy requirements are discussed briefly. In particular the method allows an accurate calculation of the flow when flat paddles are used to stir a fluid chaotically.

El problema dela función inversa a la luz del teorema del tubo fluorescente

Con el objeto no de introducir al estudiante universitario a la noción de función inversa sino de reorganizar ideas, darle significado a unas y resignificar otras (es decir, ayudarlo a aprehender el concepto) se elaboró un razonamiento, basado en ideas previas del alumno, que concluye en el Teorema del tubo fluorescente. Este Teorema permite, a partir del gráfico de una función biyectiva, obtener el de su inversa de un modo más sencillo y seguro que el de los textos tradicionales y, simultáneamente, aporta un claro mensaje conceptual. El cambio en la percepción del tema (en el 75 a 80% de los estudiantes) y la seducción de la inversa “instantánea” son superados por la idea (desde ahora evidente) que una función y su inversa son expresiones de una misma relación observada desde distintos puntos de vista.

Diseño y montaje de un equipo para el estudio del Teorema de la Cantidad de Movimiento

El objetivo de este proyecto es construir un banco de prácticas destinado al y análisis del impacto de un chorro contra diferentes obstáculos y estudiar dicho efecto mediante el teorema de cantidad de movimiento. Para ello se ha diseñado un prototipo basado en el banco instalado en la Escuela Técnica Superior de Ingenieros de Bilbao. Las diferentes piezas necesarias para la construcción del prototipo se realizaran mediante diferentes medios, algunas de ellas se encargaran a órganos externos a la escuela y otras se fabricaran con equipo disponible en el propio centro. Para el diseño se han tenido en cuenta parámetros y cálculos como los de la instalación hidráulica.

El Teorema de Márquez : de cómo repensamos el trabajo docente y la inclusión a partir de una experiencia de producción matemática

Fil: Conte, Rodrigo. Universidad Nacional de La Plata. Facultad de Humanidades y Ciencias de la Educación; Argentina.

Water uptake of aerosols with a focus on seeded aerosols and instrumentation techniques

This thesis focuses on the volatile and hygroscopic properties of mixed aerosol species. In particular, the influence organic species of varying solubility have upon seed aerosols. Aerosol studies were conducted at the Paul Scherrer Institut Laboratory for Atmospheric Chemistry (PSI-LAC, Villigen, Switzerland) and at the Queensland University of Technology International Laboratory for Air Quality and Health (QUT-ILAQH, Brisbane, Australia). The primary measurement tool employed in this program was the Volatilisation and Hygroscopicity Tandem Differential Mobility Analyser (VHTDMA - Johnson et al. 2004). This system was initially developed at QUT within the ILAQH and was completely re-developed as part of this project (see Section 1.4 for a description of this process). The new VHTDMA was deployed to the PSI-LAC where an analysis of the volatile and hygroscopic properties of ammonium sulphate seeds coated with organic species formed from the photo-oxidation of á-pinene was conducted. This investigation was driven by a desire to understand the influence of atmospherically prevalent organics upon water uptake by material with cloud forming capabilities. Of particular note from this campaign were observed influences of partially soluble organic coatings upon inorganic ammonium sulphate seeds above and below their deliquescence relative humidity (DRH). Above the DRH of the seed increasing the volume fraction of the organic component was shown to reduce the water uptake of the mixed particle. Below the DRH the organic was shown to activate the water uptake of the seed. This was the first time this effect had been observed for á-pinene derived SOA. In contrast with the simulated aerosols generated at the PSI-LAC a case study of the volatile and hygroscopic properties of diesel emissions was undertaken. During this stage of the project ternary nucleation was shown, for the first time, to be one of the processes involved in formation of diesel particulate matter. Furthermore, these particles were shown to be coated with a volatile hydrophobic material which prevented the water uptake of the highly hygroscopic material below. This result was a first and indicated that previous studies into the hygroscopicity of diesel emission had erroneously reported the particles to be hydrophobic. Both of these results contradict the previously upheld Zdanovksii-Stokes-Robinson (ZSR) additive rule for water uptake by mixed species. This is an important contribution as it adds to the weight of evidence that limits the validity of this rule.

Experiential knowledge representation and the design of product usability

The topic of designers’ knowledge and how they conduct design process has been widely investigated in design research. Understanding theoretical and experiential knowledge in design has involved recognition of the importance of designers’ experience of experiencing, seeing, and absorbing ideas from the world as points of reference (or precedents) that are consulted whenever a design problem arises (Lawson, 2004). Hence, various types of design knowledge have been categorized (Lawson, 2004), and the nature of design knowledge continues to be studied (Cross, 2006); nevertheless, the study of the experiential aspects embedded in design knowledge is a topic not fully addressed. In particular there has been little emphasis on the investigation of the ways in which designers’ individual experience influences different types of design tasks. This research focuses on the investigation of the ways in which designers inform a usability design process. It aims to understand how designers design product usability, what informs their process, and the role their individual experience (and episodic knowledge) plays within the design process. This paper introduces initial outcomes from an empirical study involving observation of a design task that emphasized usability issues. It discusses the experiential knowledge observed in the visual representations (sketches) produced by designers as part of the design tasks. Through the use of visuals as means to represent experiential knowledge, this paper presents initial research outcomes to demonstrate how designers’ individual experience is integrated into design tasks and communicated within the design process. Initial outcomes demonstrate the influence of designers’ experience in the design of product usability. It is expected that outcomes will help identify the causal relationships between experience, context of use, and product usability, which will contribute to enhance our understanding about the design of user-product interactions.