ALGUNOS ASPECTOS DE LA PREHISTORIA DE LA MATEMÁTICA

Entre las primeras actividades del hombre prehistórico se pueden nombrar la conservación del fuego, la creación de trampas para cazar animales, la construcción de casas y tumbas. También el cálculo de distancias con su cuerpo y sus pasos, el grabado de escenas en sus cavernas, la observación del movimiento de los astros y las direcciones espaciales. En esas actividades están prefigurados los conceptos básicos de la matemática: número, medida, orden. Si bien la Historia comienza con la escritura, existen numerosas conjeturas acerca del quehacer humano en la Prehistoria. Se exponen algunas de ellas relacionadas con conceptos matemáticos, basadas en los registros disponibles más antiguos

AN ANALYTICAL NON-STATIONARY DISPERSION-DEPOSITION MODEL

In this work an atmospheric dispersion-deposition model to describe the radionuclides concentration after a hypothetical nuclear accident is presented. The developed model has two parts. The first one describes the concentration due to the material released after an arbitrary period from a given source. The second one describes the concentration change during the current period due to the dispersion of the material that is in the atmosphere. Here the solution for the first part is presented. A non-stationary analytical solution is obtained from the atmospheric diffusion equation. The model solves the transport (convection-diffusion) equation in which the contaminant settling is explicitly incorporated. Regarding the boundary conditions, it is assumed a null diffusion through the mixed layer top and an albedo at the ground level. From the general solution obtained for the contaminant concentration the Gaussian plume formula is derived as a particular case when it is assumed a stationary point source, vd =0 (total reflection) and vs,=0 (without settling). Assuming vd ≠ 0 and vs,= 0, is compared with the Source Depletion Model (SoDM) and with the Surface Depletion Model (SuDM). The agreement is excellent when comparing with the SuDM, which is the exact solution for the diffusion equation when the settling is modeled only in the boundary condition. Two additional cases are presented, the former is the steady state solution: vd ≠ 0 and vs, ≠ 0, and the last one is a non-stationary simulation with a transport time of 3600 s with vd ≠ 0 and vs, ≠ 0. For the last case, two Erf functions appear in the solution as result of time integration that model the plume front traveling in the atmosphere. This new contaminant transport model describes the concentration evolution in a more realistic way, representing the plume falling. This is an improvement respect to the known dispersion-deposition models (SoDM, SuDM)

Approximations by finite elements, shape sensitivity analysis and functional analysis

Este proyecto continúa el proyecto 06/G512 con título Resolución numérica de ecuaciones diferenciales, problemas de optimización y temas de análisis relacionados desarrollado en el período 2016-2018. Motivado por problemas en los que estructuras elásticas se encuentran sumergidas en un fluido provenientes de la ingeniería nuclear (como por ejemplo, elementos combustibles sumergidos el fluido refrigerante) se estudia el análisis espectral de su formulación como operadores compactos no-autoadjuntos que surgen en problemas de vibraciones de sistemas fluido-estructura. Se estudiarán problemas de optimización relacionados con la forma y topología de materiales elásticos anisotrópicos, se implementará en un código la derivada topológica y se analizarán distintas funciones objetivos a minimizar o maximizar según corresponda. Se realizarán aplicaciones en problemas médicos provenientes de lesiones óseas. Por otro lado, se han estudiado dentro del proyecto BNCT (que estudia la Terapia de Captura Neutrónica en Boro), en forma experimental, sobre la temperatura de la piel de la bolsa del cachete de hamsters, inoculados con tumores, en el tratamiento de cáncer bucal en hámster. La temperatura superficial esta controlada por la perfusión de la sangre, la actividad metabólica y la conducción de calor entre la piel y el medio ambiente. Estas propiedades fisiológicas pueden cambiar dentro del tejido tumoral, lo cual modifica la temperatura superficial. Utilizando esta temperatura superficial, se pueden plantear problemas inversos para determinar ciertos parámetros relevantes del problema, como por ejemplo la conductividad térmica.This project is a continuation of project 06 / G512 with the title Numerical resolution of differential equations, optimization problems and related analysis topics developed in the period 2016-2018. Motivated by problems in which elastic structures are immersed in a fluid (for example, fuel elements immersed in the cooling fluid), the modes of vibration and resonance frequencies are studied, establishing their formulation as compact non-self-adjoint operators. Optimization problems related to the shape and topology of anisotropic elastic materials will be studied, the topological derivative will be implemented in a code and different objective functions will be analyzed to minimize or maximize according to the cases considered. Applications will be made in medical problems from bone injuries. On the other hand, they have been studied in the BNCT project (which studies Boron Neutron Capture Therapy), experimentally, on the skin temperature of the bag of hamsters' cheeks, inoculated with tumors, in the treatment of cancer oral in hamster. The surface temperature is controlled by blood perfusion, metabolic activity and heat conduction between the skin and the environment. These physiological properties can change within the tumor tissue, which modifies the surface temperature. Using this surface temperature, inverse problems can arise to determine certain relevant parameters of the problem, such as thermal conductivity

A posteriori error estimates for non-conforming approximation of eigenvalue problems

We consider the approximation of eigenvalue problem for the Laplacian by the Crouzeix– Raviart non-conforming finite elements in two and three dimensions. Extending known techniques for source problems, we introduce a posteriori error estimators for eigenvectors and eigenvalues. We prove that the error estimator is equivalent to the energy norm of the eigenvector error up to higher order terms. Moreover, we prove that our estimator provides an upper bound for the error in the approximation of the first eigenvalue, also up to higher order terms. We present numerical examples of an adaptive procedure based on our error estimator in two and three dimensions. These examples show that the error in the adaptive procedure is optimal in terms of the number of degrees of freedom.Fil: Dari, Enzo Alberto. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Duran, Ricardo Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Padra, Claudio. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Estimaciones del error para el método de elementos finitos no conformes

Fil: Padra, Claudio. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Locating multiple tumors by moving shape analysis

A methodology to determine the unknown shape of an embedded tumor is proposed. A functional that represents the mismatch between a measured experimental temperature profile, which may be obtained by infrared thermography at skin surface, and the solution of an appropriate boundary problem is defined. Using the Pennes’s bioheat transfer equations, the temperature in a section of healthy tissue with a tumor region is modeled by a boundary problem. The functional is related to the shape of the tumor through the solution of the boundary problem, in such a way that finding the minimum of the functional form also means finding the unknown shape of the embedded tumor. The shape derivative of the functional is computed in each node of an approximation of the solution by the method of Finite Elements using similar methods considered by Pironneau. The algorithm presented include an adaptive strategy to improve the error of the objective function. Numerical results with multiple connected tumors are considered to illustrate the potential of the proposed methodologyFil: Padra, Claudio. Comision Nacional de Energia Atomica. Gerencia del Area de Investigaciones y Aplicaciones no Nucleares. Gerencia de Fisica (CAB); Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Salva, Natalia Nieves. Comision Nacional de Energia Atomica. Gerencia del Area de Investigaciones y Aplicaciones no Nucleares. Gerencia de Fisica (CAB); Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

An hp finite element adaptive scheme to solve the Poisson problem on curved domains

In this work, we introduce an hp finite element method for two-dimensional Poisson problems on curved domains using curved elements. We obtain a priori error estimates and define a local a posteriori error estimator of residual type. We show, under appropriate assumptions about the curved domain, the global reliability and the local efficiency of the estimator. More precisely, we prove that the estimator is equivalent to the energy norm of the error up to higher-order terms. The equivalence constant of the efficiency estimate depends on the polynomial degree. We also present an hp adaptive algorithm and several numerical tests which show the performance of the adaptive strategy.Fil: Armentano, Maria Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Padra, Claudio. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; ArgentinaFil: Scheble, Mario. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentin

An hp finite element adaptive scheme to solve the Laplace model for fluid-solid vibrations

In this paper we introduce an hp finite element method to solve a two-dimensional fluid-structure spectral problem. This problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid. We prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues. We define an a posteriori error estimator of the residual type which can be computed locally from the approximate eigenpair. We show its reliability and efficiency by proving that the estimator is equivalent to the energy norm of the error up to higher order terms, the equivalence constant of the efficiency estimate being suboptimal in that it depends on the polynomial degree. We present an hp adaptive algorithm and several numerical tests which show the performance of the scheme, including some numerical evidence of exponential convergence.Fil: Armentano, Maria Gabriela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Padra, Claudio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; ArgentinaFil: Rodríguez, R.. Universidad de Concepción; ChileFil: Scheble, Mario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentin

An hp Adaptive Strategy to Compute the Vibration Modes of a Fluid-Solid Coupled System

In this paper we propose an hp finite element method to solve a two-dimensional fluid-structure vibration problem. This problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid. We use a residual-type a posteriori error indicator to guide an hp adaptive algorithm. Since the tubes are allowed to be different, the weak formulation is a non-standard generalized eigenvalue problem. This feature is inherited by the algebraic system obtained by the discretization process. We introduce an algebraic technique to solve this particular spectral problem. We report several numerical tests which allow us to assess the performance of the schemeFil: Armentano, Maria Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Padra, Claudio. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (Centro Atómico Bariloche); Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Rodriguez, Rodolfo. Universidad de Concepción; ChileFil: Scheble, Mario. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (Centro Atómico Bariloche); Argentin