acta de resumenes lxxxiv encuentro anual sociedad de...
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ACTA DE RESUMENES
LXXXIV Encuentro AnualSociedad de Matemática de Chile
2015
Conferenciasy
Sesiones Invitadas
Sociedad de Matemática de Chile
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Comité Científico:
Carlos Conca (Universidad de Chile)
Víctor González (Universidad Técnica Federico Santa María)
Milton Jara Valenzuela (IMPA BRASIL)
Sebastián Lorca (Universidad de Tarapacá)
Andrés Navas (Universidad de Santiago de Chile)
María Ofelia Ronco (Universidad de Talca)
Rubí E. Rodríguez (Universidad de la Frontera)
Jairo da Silva Bochi (Pontificia Universidad Católica de Chile )
Comité Organizador:
Hernán Burgos (UFRO)
Angel Carocca (UFRO)
Ana Cecilia de la Maza (UFRO)
Elena Olivos (UFRO)
Sociedad de Matemática de Chile, www.somachi.clCanadá 253 Departamento F. Providencia, Santiago. Chile.
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Introducción
El Encuentro de la SOMACHI es un evento de carácter nacional que se realiza anualmentey que tiene como uno de sus objetivos reunir por un par de días a los académicos, profesorese investigadores de la disciplina para que intercambien sus conocimientos y experiencias.Además en este encuentro la comunidad matemática chilena elige directiva y planifica ini-ciativas futuras.
Este año el Comité Científico ha estimado conveniente mostrar a la comunidad algunasde las contribuciones que hizo el matemático recientemente fallecido John Nash (1928-2015),Premio de Teoría John von Neumann 1978, Premio Nobel de Economía 1994, Premio LeroyP. Steele de la Sociedad Americana de Matemáticas 1999, Medalla de la Doble Hélice delLaboratorio Cold Spring Harbor 2010, Premio Abel 2015. Para ello se han programado unaserie de Conferencias Plenarias dictadas por expertos en sus trabajos, que esperamos seande provecho para el desarrollo de nuestra matemática.
La SOMACHI también se ha preocupado fuertemente de fomentar la participación deestudiantes de pre y post grado en las distintas sesiones de trabajo; esperamos que sea unaexperiencia provechosa para los alumnos que han comenzado su formación matemática.
El Comité Organizador agradece el valioso aporte de los organizadores y de los coordi-nadores de sesión, sin cuya colaboración este encuentro no habría sido posible. En estasactas se plasma un resumen de los trabajos presentados.
Comité OrganizadorLXXXIV Encuentro
Sociedad de Matemática de Chile
PucónRegión de la Araucanía26-28 Noviembre 2015
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Sesiones Invitadas:
Álgebra, Nicolás Libedinsky (UChile)
Análisis Funcional y Aplicaciones, M. A. Astaburuaga y Víctor H. Cortés(PUC)
Análisis No Arquimediano, Elena Olivos (UFRO)
Análisis Numérico, Mauricio Sepúlveda (UCONCE)
Ecuaciones Diferenciales Parciales, Claudio Muñoz (UChile)
Geometría, Maximiliano Leyton (UTALCA)
Matemática Discreta, José Soto (UChile)
Modelos Matemáticos de Sistemas Biológicos, Fernando Córdova (UCM)
Optimización, Luis Briceño (UTFSM)
Problemas Inversos y Control de EDP, Rodrigo Lecaros (UChile)
Sistemas Dinámicos, Irene Inoquio (UACH)
Teoría de Números, Amalia Pizarro (UVALPO)
Estudio de Clases. Método Japonés, Carlos Cabezas (UCM)
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ContentsIntroducción 3
Conferencias Plenarias . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Los trabajos de John Nash en Teoría de Juegos 17Mario Bravo
Nash´s imbedding problem for Riemannian manifolds 18Mauricio Godoy
Los trabajos geométricos de John Nash 19Mark Spivakovsky
Diffusive models and their intrinsic regularity theories 20Eduardo Teixeira
Análisis Funcional y Aplicaciones . . . . . . . . . . . . . . . . . . . . 22
Almost exponential decay for a system of Schrödinger equations 23M.A. Astaburuaga C. Fernández
Instability of eigenvalues for unitary perturbations 24Víctor H. Cortés
Breather solutions of a 1D non-linear Shrödinger equation 25Matías Courdurier
The topology of Chiral vector bundles: topological insulators of type AIII 26Giuseppe De Nittis Kiyonori Gomi
Un modelo simple de una situación biológica 27Manuel Elgueta
Exact β function and DGLAP-BFKL duality in a supersymmetric gaugetheory 28
Igor Kondrashuk
Teorema de Baillon para una ecuación integro differenciales 30Juan C. Pozo Octavio Vera
On the fractional Schrödinger equation on a Hilbert space 31Humberto Prado
A characterization of universally starlike functions 32Andrew Bakan Stephan Ruscheweyh Luis Salinas
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Aproximación de soluciones acotadas de ecuaciones diferenciales con argu-mento constante a trozos del tipo generalizado con impulsos 34
Manuel Pinto Ricardo Torres
Álgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Algunas propiedades del álgebra de símbolos pseudodiferenciales formales 38Jarnishs Beltran Enrique G. Reyes
Módulos de tipo FP-infinito y la Categoría Estable de Módulos de un Anillo 39Daniel Bravo
Macdonald polynomials in superspace and the 6 vertex model 40Luc Lapointe
Teoría de representaciones graduada del álgebra de blob y el cálculo deSoergel en dos colores 41
David Plaza
Groupoidal Gelfand Models in Group Theory 42Jorge Soto Andrade
Algebraic structures on stellohedra and pterahedra 43L. Berry S. Forcey M. Ronco P. Showers
Teoría de representaciones del álgebra de Yokonuma-Hecke y del álgebra debraids and ties 44
Steen Ryom-Hansen
Análisis No Arquimediano . . . . . . . . . . . . . . . . . . . . . . . . . 46
C-álgebra de operadores lineales continuos definidos en c0(I) 47J. Aguayo M. Nova J. Ojeda
Espacios y cuerpos residuales en espacios de tipo Hilbert sobre cuerpos convaluaciones no-arquimedianas 48
Herminia Ochsenius Elena Olivos
Medida Espectral sobre Algebras de Operadores definidos en c0(N) 49J. Aguayo M. Nova J. Ojeda
Espacios de Banach sobre cuerpos con valuación discreta 50Elena Olivos Herminia Ochsenius
An o-minimal approach to definability of functions in extensions of C 51Javier Utreras
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Análisis Numérico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Mathematical and numerical analysis for a nonlocal reaction diffusion system 53V. Anaya M. Bendahmane M. Langlais M. Sepúlveda
A kind of Mixed Finite Element Variational Formulation for a ParabolicProblem 54
Mauricio Barrientos Karina Vilches
Discontinuos Galerkin scheme for Helmholtz equation 55Tomás Barrios Rommel Bustinza
Hydraulic parameter estimation under non-saturated flow conditions in cop-per heap leaching 56
Emilio Cariaga Rubén Martínez Mauricio Sepúlveda
Torque free rigid body motion: an elliptic function primer 57Roberto León Luis Salinas Claudio Torres
Analysis of an augmented pseudostress-based mixed formulation for a non-linear Brinkman model of porous media flow 58
Gabriel N. Gatica Luis F. Gatica Filánder A. Sequeira
A mixed FEM for a vorticity based formulation of the Brinkman problem 59David Mora
Finite VolumeMethod Improved Sequential Solution Applied To Two-DimensionalConvective-Diffusive Heat Transfer 60
Nelson O. Moraga Juan Jaime
On the analyses of three DG schemes for Stokes problem based on velocity-pseudostress formulation 62
Tomás P. Barrios Rommel Bustinza Felipe Sánchez
Convergence of a finite volume scheme for a sorption-coagulation equation 63Erwan Hingant Mauricio Sepúlveda
Numerical Analysis Of Non-Newtonian Flows In Liquid Foods, SolidificationCasting And Polymer Injection Molding 64
Nelson O. Moraga Edgardo J. Tabilo
Numerical Challenges for a Numerical Simulation of a Coupled Model forGrain Growth in 2D 66
Claudio E. Torres Alejandro Sazo
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On Nonlinearly and Linearly Implicit IMEX Runge-Kutta methods for aclass of degenerate convection-diffusion problems 68
Luis-Miguel Villada Osorio
Ecuaciones en Derivadas Parciales . . . . . . . . . . . . . . . . . . . 70
Stability of mKdV breathers in the energy space and numerical results 71Miguel Angel Alejo
Sobolev and Hardy-Sobolev type inequalities 72Hernán Castro
On the control of the stabilized Kuramoto-Sivashinsky system by a singleforce 73
Eduardo Cerpa
A Dirichlet problem involving the divergence operator 74Gyula Csató
Solvability of fractional problems with supercritical drifts 75Gonzalo Dávila
Formación de singularidades para el flujo de mapas armónicos 76Juan Dávila
Solutions to a supercritical elliptic problem 77Jorge Faya
Entire sign changing solutions with finite energy to the fractional Yamabeequation 78
Danilo Garrido Monica Musso
Multiplicity of solutions for some semilinear problems involving nonlinearitieswith zeros 80
Leonelo Iturriaga Jorge García-Melián
Kink dynamics in the φ4 model: asymptotic stability for odd perturbationsin the energy space 82
Michal Kowalczyk
Singularly perturbed PDEs and patterns with periodic profiles 83Fethi Mahmoudi
Some results for a problem from Combustion 84Alejandro Omón Arancibia
Desigualdades de restricción de Fourier: existencia y no existencia de máxi-mos y cálculo de la mejor constante 86
René Quilodrán
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Stationary harmonic functions whose Laplacian is a Radon measure 87Rémy Rodiac
Fractional mean curvature flow 88Mariel Sáez Enrico Valdinoci
Moderate solutions of semilinear elliptic equations with Hardy potential 89PHUOC-TAI NGUYEN
On the Cp′-regularity conjecture 90Eduardo Teixeira
Geometría . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Linear systems on IHS manifolds 92Michela Artebani
Group actions on Riemann surfaces up to topological equivalence. 93Antonio Behn
Acción de grupos en superficies y variedades abelianas 94Angel Carocca
Sobre la existencia de acciones de grupos elementales en Superficies de Rie-mann 95
Mariela Carvacho
Dual families of Calabi-Yau varieties 96Paola Comparin
On singular varieties with smooth subvarieties 98M. R. Gonzalez-Dorrego
The 4-prims family. 99Víctor González Aguilera Gustavo Labbé Morales
Automorphisms of non-cyclic p-gonal surfaces 100R. A. Hidalgo A. F. Costa
Lines on cubic hypersurfaces over finite fields 101Antonio Laface
Authomorphims of graphs and Riemann surfaces 103Alexander D. Mednykh
On Jacobian of circular graphs 104Ilya A. Mednykh
Curvas de Tipo Fermat y sus Jacobianas 105Jaime Pinto
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Automorphisms Of Pseudo-Real Riemann Surfaces 106S. Quispe R. A. Hidalgo
Superficies Algebraicas: Uniformización y Aritmeticidad 107Sebastián Reyes-Carocca
Fixed points and rational representations of actions in abelian varieties 108Rubí E. Rodríguez
Familias de Jacobianas completamente descomponibles y subvariedades es-peciales de Ag. 109
Anita M. Rojas
Small degree covers and reducible hyperplane sections 111Andrea L. Tironi
Modelos Matemáticos de
Sistemas Biológicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Medidas de conservación ex situ de tipo impulsivo: Un enfoque metapobla-cional a través del modelo clásico de Levins 113
Sandra Araya Crisóstomo Héctor Rojas-Castro
Consecuencias sobre la abundancia poblacional del Efecto Allee en hábitatsbajo fragmentación 115
Rodrigo Del Valle Fernando Córdova-Lepe
Control epidemiológico optimal por hospitalización impulsiva 116M. Eugenia Solís Fernando Córdova-Lepe
Dinámica de la distribución genotípica bajo mortalidad diferenciada porrasgos fenotípicos 117
Héctor Rojas-Castro Fernando Córdova-Lepe
Observaciones a la aproximación de L.A. Segel para las ecuaciones del SistemaLigando-Receptor 119
Fernando Córdova-Lepe
Neurodidactics: Analysis of Cellular Neural Network Models 120Kuo-Shou Chiu Fernando Córdova-Lepe
A vaccine-age structured model to study the effect of a pre-erythrocyticvaccine on malaria prevalence 123
Katia Vogt Geisse Calistus Ngonghala Zhilan Feng
Modelación del cambio en la interacción de poblaciones biológicas. Estudiode un caso 125
Marcelo E. Alberto et al.
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Un Modelo Estocástico de Biorrectar de Autociclado 126Ana Venegas Ricardo Castro Fernando Córdova
Dinámica de un modelo tritrófico con una respuesta funcional monotónicano-diferenciable 127
Viviana Rivera Pablo Aguirre
Dinámica en el modelo de depredación de Holling-Tanner considerando in-terferencia entre los depredadores 128
Adrián Cecconato Eduardo González-Olivares
Modelling and stability analysis of a microalgal pond with nitrification 130F. Mairet H. Ramírez C A. Rojas-Palma
Mathematical approach regarding the environmental effects upon trait di-versity in a cell population. 132
Karina Vilches Ponce
On a nonlinear problem from catalysis: existence, multiplicity and qualitativebehaviour 133
Alejandro Omón Arancibia Gonzalo Robledo Veloso
A stochastic disease transmission in an epidemic model considering a hyper-bolic incidence rate 135
A. Christen M. A. Maulén E. González-Olivares M. Curé
Sistemas Dinámicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Aspectos recientes de la Conjetura de Palis 138Alma Armijo
Injectivity, Global and Almost Global Stability of Hurwitz Vector Fields. 139Álvaro Castañeda Víctor Guíñez
Toeplitz and strong orbit equivalence 140Maryam Hosseini
Dimensión de Hausdorff de los conjuntos de Borel-Bernstein 141Felipe Pérez
Flexibility of some groups of homeomorphisms of the line 142Cristobal Rivas
A linearization result for DEPCA systems 143Manuel Pinto Gonzalo Robledo
Shearer’s inequality and the Infimum Rule 144Pierre Paul Romagnoli
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Polinomios de Fibonacci y Componentes Errantes 145Eugenio Trucco
Matemática Discreta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Estudio de un modelo de evasión en el transporte público 147Bastián Bahamondes Pizarro
Resource Augmentation Algorithm for Single Machine Scheduling with Job-Dependent Convex Cost 148
Rodrigo A. Carrasco
Avances recientes en la resolución exacta del problema de vendedor viajero149Daniel Espinoza William Cook Marcos Goycoolea
Quasirandom hypergraphs and subsets with small Fourier coefficients 150Hiê.p Hàn
Efficient Implementation of Carathéodory’s Theorem for a Simple SchedulingPolytope 151
Ruben Hoeksma
Maximum number of colourings without monochromatic Schur triples 152Andrea Jimenez
On-line list coloring of random graphs 153Dieter Mitsche
Computing income taxes under the new Chilean tax regime: Graphs, MarkovChains and Algorithms. 154
Javiera Barrera Eduardo Moreno Sebastián Varas
Optimización . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Chance-constrained problems and rare events: an importance sampling ap-proach 156
J. Barrera, T. Homem-de-Mello, E. MorenoB. Pagnoncelli, G. Canessa
Nonsmooth Lur’e Dynamical Systems in Hilbert Spaces 157Samir Adly Abderrahim Hantoute Ba Khiet Le
Stochastic Topology Design Optimization for Continuous Elastic Materials 158Miguel Carrasco Benjamín Ivorra Angel Manuel Ramos
Existence and approximation of generalized Lagrange multipliers for varia-tional problems under uniform constraints on the gradient. 159
Felipe Alvarez Salvador Flores
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Condiciones de Optimalidad en Problemas de Control Óptimo Discreto 161C. Isoton M.A. Rojas-Medar V. Vivanco L. dos Santos
Proximal Distances over Symmetric Cones 163Julio López Erik Papa
Stability in Generalized Nash Equilibrium Problems with nonsmooth payofffunctions, application to Electricity market 164
Matthieu Maréchal Rafael Correa
Boosting Topic Models for Text Analysis 166Marcelo Mendoza
A Decomposition Method for Two-Stage Stochastic Programs with Risk-Averse Utilities 168
Tito Homem-de-Mello Sebastian Arpon Bernardo Pagnoncelli
A primal-dual mix algorithm for convex non-differentiable structured opti-mization in Hilbert spaces 170
Cesare Molinari Juan Peypouquet
Resultados sobre convexidad de la imagen de funciones cuadráticas 171Felipe Opazo Lagos
Generación de benchmark de fondos para el sistema de pensiones en Chile,un enfoque basado en optimización estocástica 172
Daniel Espinoza G. Giorgiogiulio Parra De B.
Assessing Fishery Management and Recovery Strategies through ViabilityTheory 175
Héctor Ramírez
Optimal feedback synthesis and minimal time function for the bioremediationof water resources with two patches 176
H. Ramírez C. A. Rapaport V. Riquelme
Dualidad en optimización vectorial 178M. A. Rojas-Medar L. Batista dos Santos Camila Isoton
Comparision of MINC and MRMT configurations:Effects of spatial structure and biomass diffusion 180
A. Rapaport H. Ramírez A. Rojas-Palma J. de Dreuzzy
Optimization of the concentration changes in a chemostat with one species 182Térence Bayen Jérôme Harmand Matthieu Sebbah
Problemas Inversos y Control de EDP . . . . . . . . . . . . . . . . 184
On the cost of null controllability of some linear partial differential equations185Nicolás Carreño
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On the control of the improved Boussinesq equation 186Eduardo Cerpa
An Inverse Problem for the Helmhotz Equation in a Layered Media. 187Matías Courdurier
Detection of Several Obstacles in a Stokes Flow: A mixed approach 188Matías Godoy Campbell
Controllability of coupled systems with Schrödinger equations. 189Alberto Mercado Saucedo
An ADER type scheme for evolving differential operators 190G. Montecinos J. C. López R. LecarosJ. Ortega E. F. Toro
Stability numbers to Timoshenko’s system with shear boundary dissipation192Margareth Alves Jaime E. Muñoz Rivera Mauricio Sepúlveda
Teoría de Números . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Ramas y extensiones de cuerpos 194L. Arenas-Carmona
Un retículo Hermiteano central 195Ana Cecilia de la Maza Remo Moresi
Distribución asintótica de puntos de Hecke sobre Cp 196Sebastián Herrero Miranda
Optimal bounds for Büchi’s problem in modular arithmetic 197Pablo Sáez Xavier Vidaux Maxim Vsemirnov
Una conexión entre la propiedad de Northcott y la indecidibilidad en anillosde enteros totalmente reales 198
Xavier Vidaux
Estudio de Clases. Método Japonés . . . . . . . . . . . . . . . . . 199
Promoviendo el desarrollo de habilidades del pensamiento matemático enestudiantes del sistema escolar: Una experiencia en el complejo educacionalla Granja de Cajón 200
Pamela Alarcón Valeria Carrasco Ciro GonzálezTeresa Sanhueza
Análisis de idoneidad didáctica del método japonésdesde un enfoque onto-semiótico de la instrucción matemática 202
Carlos Cabezas Pedro Arteaga
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Problemas incorrectos como medio para desarrollar aprendizaje profundo 203Hugo Caerols Katia Vogt Geisse
El aprendizaje del cálculo diferencial bajo un diseño curricular Modular 204Elías Irazoqui Becerra
Estudio de clases: hacia una alianza de la universidad con las escuelas 206Soledad Estrella Sergio Morales Raimundo Olfos
Avances y retrocesos en el Estudio de Clases (en el norte de Chile) 207Eliseo Martínez Herrera.
Comunidades de Aprendizaje GEC 208Soledad Estrella Sergio Morales Raimundo OlfosMaria Tapia
La probabilidad en el aula de educación básica.Un estudio de caso sobre los primeros elementos linguísticos 210
Claudia Vásquez Ortiz
Estudio de clases en didáctica de la matemática: proceso reflexivo de losestudiantes de pedagogía en Educación Básica en la U. Santo Tomás 212
Pierina Zanocco Soto
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Conferencias Plenarias
CONFERENCIAS
PLENARIAS
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Los trabajos de John Nash en Teoría de JuegosMario Bravo
Abstract
A comienzos de los años 50, los trabajos de John Nash constituyeron avances funda-mentales tanto en la teoría de juegos cooperativos como no-cooperativos. Las nocionesde equilibrio definidas por Nash y la variedad de interacciones a las que se aplicancambiaron definitivamente la ciencia económica moderna. En esta charla daremos unaintroducción a los aspectos principales del trabajo de Nash en este ámbito. Además,discutiremos cómo estas ideas han impactado en lo más diversos campos de investi-gación.
Universidad de Santiago de Chile. e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Nash´s imbedding problem for Riemannianmanifolds
Mauricio Godoy
Abstract
Imbedding problems are one of the most natural questions in the evolution of mathemat-ics. When new concepts are defined, they are often concrete examples of a phenomenon;afterward they are given abstract formulations that, in principle, generalize the par-ticular situations studied before; and not too long after that, the question "does theabstract definition include new examples ? " arises.Nash´s imbedding is a fundamental result in Riemannian geometry that exemplifies thedifficulties that may appear when answering the last question. Employing approxima-tion techniques coming from PDEs and geometric analysis, J. F. Nash proved in 1956that any Ck Riemannian manifold M (3 ≤ k ≤ ∞) can be Ck isometrically imbeddedin RN , where N depends quadratically on the dimension of M , if M is compact, andcubically if M is not compact.In this talk I will give an overview of the problem and its historical background, asketch of Nash´s proof and some later results improving dimensions (e.g., Gromov´sreduction to N = 5 for surfaces instead of Nash´s N = 17).
e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Los trabajos geométricos de John NashMark Spivakovsky
Abstract
En esta conferencia trataremos de resumir las principales contribuciones de John Nash ala geometría, realizadas durante casi dos décadas entre 1950 y 1968. Nos concentraremossobre los siguientes resultados y construcciones:
• El teorema del encaje de una variedad diferencial como una componente conexade una variedad algebraica real.
• Dos versiones del célebre teorema de Nash de encaje. La primera versión, el teo-rema de Nash–Kuiper, dice que todo encaje débilmente contráctil de una variedadC1 de dimensión m en el espacio Euclideano de dimensión n > m puede seraproximado arbitráariamente bien par un encaje C1 isométrico. En el teoremaoriginal de Nash n era mayor o igual a m+ 2, pero Kuiper mejoró la cota paraobtener n > m. La segunda versión, mucho más difícil de demostrar, tambiénpublicada en los Annals of Mathematics, dice que para k entre 3 e infinito todavariedad M de clase Ck y dimensión m admite un encaje Ck isométrico enel espacio Euclideano de dimensión n, donde n ≤ m(3m+11)
2 si M es compactay n ≤ m(m+1)(3m+11)
2 si M no es compacta.
La mayor parte de la conferencia será dedicada a las contribuciones de Nash enlos años sesenta del siglo XX:
• Explosión de Nash como un método para estudiar geometría y topología de sin-gularidades y un método conjetural para construir una desingularización canónicade variedades algebraicas (y analíticas) en característica cero.
• El problema de Nash sobre el espacio de arcos de una variedad algebraica singular.
e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Diffusive models and their intrinsic regularitytheories
Eduardo Teixeira
Abstract
Diffusive processes appear naturally in the mathematical formulation of a number ofmodels in pure and applied sciences, ranging from problems in physics, biology, eco-nomics, probability, differential geometry, etc. Among the most fundamental questionspertaining to the mathematical treatment of such models, understanding their intrinsicregularity theories has a central importance. In this talk I will present an overviewof the PDE theory for diffusive models and will discuss the regularity properties ofsolutions of such equations.
e-mail: [email protected]
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SESIONES
INVITADAS
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Análisis Funcional y Aplicaciones
Encargado de Sesión: M. A. AstaburuagaVíctor H. Cortés
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Almost exponential decay for a system ofSchrödinger equations
M.A. Astaburuaga C. Fernández
Abstract
Let h =(h1h2
)be a resonant solution of a linearly coupled system of perturbed Schrödinger
equations on the half line [0,∞] with Dirichlet boundary conditions at the origin. Thevector h is a generalized eigenvector of the Hamiltonian of the system H. This meansthat Hh = k2h, k a complex number with Imk < 0, and h satisfies an outgoing con-dition at ∞. Since h is not square integrable we truncate h to an interval containingthe support of the perturbation and show that if the resonance k is close to the realaxis the survival probability of the truncated solution has an approximate exponentialbehaviour in time.
References[1] Astaburuaga, M.A., Covian, P.; Fernández, C. Behaviour of the survival probability in some
one dimensional problems. J. Math Phys., 43 (2002) 4571-4581.
[2] Cattaneo, L., Graf, G. M., and Hunziker, W., A general resonance theory based on on mourre’sinequality, Ann. Henri Poincaré, 7 (2006), 583-601.
[3] Cycon, H. L. , Froese, R. G., Kirsch, W. , and Simon, B., Schrodinger Operators, Springer,1987.
[4] King, C., Exponential decay near resonances, Letters in math.Phys. 23, (1991), 215-222.
[5] Lavine, R., Spectral density and sojourn times, Atomic Scattering Theory (J. Nutall, ed.), U.of Western Ontario, London, Ontario, 1978.
[6] Lavine R., Exponential Decay, Diff.Eq. and Math. Phys., Proceedings of the Int, ConferenceU. of Alabam at Birmingham, 132-142, 1995.
Facultad de Matemáticas. Pontificia Universidad Católica de ChileFondecyt No. 1141120, ACT-1112 , email: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Instability of eigenvalues for unitaryperturbationsVíctor H. Cortés
Abstract
In this paper we deal with the problem of instability of eigenvalues of a family of unitaryoperators acting on a separable Hilbert space H by describing the corresponding FermiGolden Rule for a one parameter family Uβ : |β| < β0. We apply it to prove that closeto an eigenvalue of U0, embedded in the continuous spectra , there are not eigenvaluesof Uβ for β small.
Following [1], [2] we show that there is a close relation between this behavior andthe existence of a local commutator for the family Uβ .
This is a joint work with Astaburruaga and Bourget, Pontificia Universidad Católicade Chile.
References[1] M.A. Astaburuaga, O. Bourget, V.H. Cortés, C. Fernández, Floquet Operators without singular
continuous spectrum, J. Funct. Anal. 238 (2006) no. 2, 489-517.
[2] M.A. Astaburuaga, O. Bourget, V. Cortés, Commutations relations for unitary operators I , J.Funct. Anal. , 268(8) (2015), pp. 2188-2230.
[3] M.A. Astaburuaga, O. Bourget, V. Cortés, Commutations relations for unitary operators II, J.Approx Theory, (2015) , pp. 63-94.
[4] V. Georgescu, C. Gérard, J.S. Moller, Commutators, C0-semigroups and resolvent estimates,J. Funct. Anal. 216 (2004) no. 2, 303-361.
[5] A. Jensen, E. Mourre, P. Perry, Multiple commutator estimates and resolvent smoothness inquantum scattering theory, Ann. Inst. Henri Poincaré 41 (1984) no. 2, 207-225.
[6] T. Kato, Perturbation Theory for Linear Operators, Springer, 1980.
[7] M. Levitin, L. Parnovski, Commutators, spectral trace identities, and universal estimates foreigenvalues, J. Funct. Anal. 192 (2002), no. 2, 425-445.
Facultad de Matemáticas. Pontificia Universidad Católica de ChileFondecyt 1120786, ACT-1112, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Breather solutions of a 1D non-linearShrödinger equation
Matías Courdurier
Abstract
Non-linear phenomena in general and non-linear Schrödinger equation in particular,appear in various fields of theoretical and applied physics. In this talk, we look at thenon-linear Schrödinger equation
i∂tu = −1
2∂2xu+ V (x, u)
with a specific space dependent non-linear term V (x, u) and present an peculiar familyof breather solutions.
This is a joint work with Olivier Bourget and Claudio Fernandez, Pontificia Univer-sidad Católica de Chile.
Facultad de Matemáticas. Pontificia Universidad Católica de Chilee-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
The topology of Chiral vector bundles:topological insulators of type AIII
Giuseppe De Nittis Kiyonori Gomi
Abstract
The classification of topological states of matter is an important hot topic in math-ematical physics. In this talk I will describe a new approach to the classification oftopological quantum systems in class AIII which is based on the study of a new cat-egory of vector bundles. The objects of this category, the chiral vector bundles, arepairs constituted by a complex vector bundle along with one of its automorphisms.We provide a classification for the homotopy equivalence classes of these objects whichis based on the construction of a suitable classifying space. The computation of thecohomology of the latter allows us to introduce a proper set of characteristic cohomol-ogy classes: Some of those just reproduce the ordinary Chern classes but there arealso new odd-dimensional classes which take care of the extra topological informationintroduced by the chiral structure. Chiral vector bundles provide the proper geomet-ric model for topological quantum systems in class AIII, namely for systems endowedwith a (pseudo-)symmetry of chiral type. The classification of the chiral vector bundlesover sphere and tori (explicitly computable up to dimension 4) recover the commonlyaccepted classification for topological insulators of class AIII which is usually based onthe K-group K1. However, this new classification turns out to be even richer since ittakes care also for the possibility of non trivial Chern classes.
References[1] De Nittis, G.; Gomi, K.: Chiral vector bundles: A geometric model for class AIII topological
quantum systems. arXiv:1504.04863, 2015
Pontificia Universidad Católica, Santiago, Chile, e-mail: [email protected] University, Nagano, Japan, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Un modelo simple de una situación biológicaManuel Elgueta
Abstract
Se estudiará una ecuación proveniente de la modelación de una situación en biología,con énfasis en la existencia, unicidad de soluciones y la estabilidad de sus solucionesestacionarias.
Facultad de Matemáticas. Pontificia Universidad Católica de ChileFondecyt No. 1150028 , e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Exact β function and DGLAP-BFKL duality ina supersymmetric gauge theory
Igor Kondrashuk
Abstract
We consider a couple of integro-differential equations which can solved analytically. Wepropose a new method based on a complex analysis to find the analytic solution. Weapply this result to a model for QCD dynamics in which DGLAP and BFKL equationsare a couple of the integro-differential equations of this type. The solution has beenfound due to possibility to solve analytically the corresponding duality equations. Thisduality relates DGLAP and BFKL kernels in a model of singlet parton evolution with adominant eigenvalue. The case of N = 1 supersymmetric gauge theory is considered indetail because the running of a gauge coupling is known exactly. The complete structureof the solution is determined by the properties of the Lambert function. This result isobtained in collaboration with Gustavo Alvarez (Departamento de Fisica, Udec), BerndKniehl (DESY, Hamburg) and Gorazd Cvetic (UTFSM, Departamento de Fisica).
References[1] V. N. Gribov and L. N. Lipatov, “Deep inelastic e p scattering in perturbation theory,”
Sov. J. Nucl. Phys. 15 (1972) 438 [Yad. Fiz. 15 (1972) 781].
[2] V. N. Gribov and L. N. Lipatov, “e+ e- pair annihilation and deep inelastic e p scatteringin perturbation theory,” Sov. J. Nucl. Phys. 15 (1972) 675 [Yad. Fiz. 15 (1972) 1218].
[3] L. N. Lipatov, “The parton model and perturbation theory,” Sov. J. Nucl. Phys. 20(1975) 94 [Yad. Fiz. 20 (1974) 181].
[4] L. N. Lipatov, “Reggeization of the Vector Meson and the Vacuum Singularity in Non-abelian Gauge Theories,” Sov. J. Nucl. Phys. 23 (1976) 338 [Yad. Fiz. 23 (1976) 642].
[5] V. S. Fadin, E. A. Kuraev and L. N. Lipatov, “On the Pomeranchuk Singularity inAsymptotically Free Theories,” Phys. Lett. B 60 (1975) 50.
[6] E. A. Kuraev, L. N. Lipatov and V. S. Fadin, “Multi - Reggeon Processes in the Yang-Mills Theory,” Sov. Phys. JETP 44 (1976) 443 [Zh. Eksp. Teor. Fiz. 71 (1976) 840].
The work of I.K. was supported in part by Fondecyt (Chile) Grants Nos. 1040368, 1050512, and 1121030,by DIUBB (Chile) Grant Nos. 153209 GI/C and 125009. e-mail: [email protected]
28
[7] E. A. Kuraev, L. N. Lipatov and V. S. Fadin, “The Pomeranchuk Singularity in Non-abelian Gauge Theories,” Sov. Phys. JETP 45 (1977) 199 [Zh. Eksp. Teor. Fiz. 72(1977) 377].
[8] I. I. Balitsky and L. N. Lipatov, “The Pomeranchuk Singularity in Quantum Chromo-dynamics,” Sov. J. Nucl. Phys. 28 (1978) 822 [Yad. Fiz. 28 (1978) 1597].
[9] Y. L. Dokshitzer, “Calculation of the Structure Functions for Deep Inelastic Scatteringand e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics.,” Sov.Phys. JETP 46 (1977) 641 [Zh. Eksp. Teor. Fiz. 73 (1977) 1216].
[10] G. Altarelli and G. Parisi, “Asymptotic Freedom in Parton Language,” Nucl. Phys. B126 (1977) 298.
[11] G. Altarelli, R. D. Ball and S. Forte, “Singlet parton evolution at small x: A Theoreticalupdate,” hep-ph/0001157.
[12] L. Euler, "De serie Lambertina Plurimisque eius insignibus proprietatibus," Acta Acad.Scient. Petropol. 2 (1783) 29-51. Reprinted in: L. Euler, Opera Omnia, Series Prima,Vol. 6, “Commentationes Algebraicae”, Teubner, Leipzig, Germany, 1921, pp. 350-369.
[13] J.H. Lambert, "Observationes variae in Mathes in Puram." Acta Helvitica, physico-mathematico-anatomico-botanico-medica 3 (1758) 128-168.
[14] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, "On theLambert W function," Adv. Comput. Math. 5 (1996) 329-359.
[15] D. R. T. Jones, “More On The Axial Anomaly In Supersymmetric Yang-mills Theory,”Phys. Lett. B 123 (1983) 45.
[16] V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, “Exact Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories from Instanton Calculus,” Nucl.Phys. B 229 (1983) 381.
[17] G. Cvetic and I. Kondrashuk, “Explicit solutions for effective four- and five-loop QCDrunning coupling,” JHEP 1112 (2011) 019 [arXiv:1110.2545 [hep-ph]].
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Teorema de Baillon para una ecuación integrodifferenciales
Juan C. Pozo Octavio Vera
Abstract
Sea X un espacio de Banach, A : D(A) ⊆ X → X un operador lineal cerrado generadorde un C0-semigrupo y g : R → R una función de variación acotada en [0, c] para todoc > 0. En este trabajo mostramos que siempre existe una función continua f tal que elproblema
u′(t) = Au(t) +
∫ t
0g(t− s)Au(s)ds+ f(t), t ∈ [0, τ ],
u(0) = x ∈ X,
(1)
no admite solución a menos que A sea un operador acotado o X contenga un subespaciocerrado isomorfo a c0.
Considerando g ≡ 0, este problema ha sido abordado en varios trabajos [1, 2, 3] yes conocido como el Teorema de Baillon sobre regularidad maximal.
References[1] J. B. Baillon, Caractére borné de certains générateurs de semi-groupes linéaires dans les espaces
de Banach, C. R. Acad. Sci. Paris 290 (1980), 757–760.
[2] B. Eberhardt, G. Greiner, Baillon’s theorem on maximal regularity. Positive operators andsemigroups on Banach lattices (CuraÃğao, 1990). Acta Appl. Math. 27 (1992), no. 1-2, 47–54.
[3] C. C. Travis, Differentiability of weak solutions to an abstract inhomogeneous differential equa-tion, Proc. Amer. Math. Soc. 82 (3) (1981), 425–430.
Pozo, Proyecto FONDECYT 3140103, e-mail: [email protected], Proyecto FONDECYT 1121120, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On the fractional Schrödinger equation on aHilbert spaceHumberto Prado
Abstract
Let A be a given self-adjoint operator on a Hilbert space H. Then by an applicationof the spectral theorem we prove existence and uniqueness of strong solutions for the
linear fractional Schrödinger equation∂αu
∂tα(t) = (−i)αAu(t) with the initial condition
u(0) = u0, in which 0 < α < 1. We show existence of an operator solution familyUα(t)t≥0. Furthermore, the solution is defined as uα(t) = Eα((−it)αA)u0 in whichuα(t) is obtained by means of the functional calculus defined by the the Mittag-Lefflerfunction.
Additionally we show that the operators Uα(t)t≥0 have a continuous dependenceon the parameter α. Moreover at the limit when α→ 1 we obtain that U1(t) is equalsto the unitary group e−itA whose infinitesimal generator is the self adjoint operator A.This work is in collaboration with P. Górka from Warsow University of Technology, andJ. Trujillo from University of La Laguna, Tenerife.
Departamento de Matemáticas y Ciencias de la Computación. Universidad de Santiago de ChileFONDECYT grant No. 1130554, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A characterization of universally starlikefunctions
Andrew Bakan Stephan Ruscheweyh Luis Salinas
Abstract
This communication deals with extensions of the classical Polya-Schoenberg conjectureabout the convolution invariance (Hadamard product invariance) of convex univalentfunctions in the unit disk. There is a continuous passage of function classes convex (andsimilar classes starlike, pre-starlike, etc.) in the unit disk to those in the slit domainΛ := C \ [1,∞), and it has been shown in the papers [1, 3, 4, 5] that the theory ofconvex –resp. starlike, pre-starlike, etc.– functions analytic in Λ (understood in theright sense) is formally very similar to the one in D.
A function F with F (0) = 0, F ′(0) = 1 and analytic in Λ is called universallystarlike if it maps every circular subdomain of Λ containing the origin univalently ontoa starlike domain with respect to the origin. Here, the word universal comes fromthe fact that these functions represent the universal multipliers (with respect to theHadamard product) of the classes of starlike analytic functions in arbitrary circularsubdomains of Λ. A universally starlike function f is a Pick function which can berepresented in either of the next two forms:
f(z) = z exp
∫[0,1]
log1
1− tzdµ(t)
=
∫[0,1]
z
1− tzdρ(t) , z ∈ Λ ,
where µ and ρ are probability measures on [0, 1]. Note that there is a one-to-one relationbetween f and µ (a.e.) but this is not the case between f and ρ. An interesting questionis to identify those functions ρ which describe a universally starlike f (and those whichdon’t). The purpose of this note is to communicate a recent solution of this problemobtained by Bakan, Ruscheweyh and Salinas [2].
References[1] A. Bakan, St. Ruscheweyh, L. Salinas, Universal convexity and universal starlikeness of poly-
logarithms. Proc. Amer. Math. Soc., Vol. 143 Nr. 2 (2015), 717–729.
Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 01601, Ukraine, e-mail:[email protected]
Partially supported by FONDECYT 1150810. Mathematisches Institut, Universitat Würzburg, 97074Würzburg, Germany, e-mail: [email protected]
Partially supported by FONDECYT 1150810. CCTVal and Departamento de Informática, UTFSM,2390123 Valparaíso, Chile, e-mail: [email protected]
32
[2] A. Bakan, St. Ruscheweyh, L. Salinas, Universally starlike and Pick functions. Pp. 1-37.Preprint, UTFSM, Valparaíso, 2015.
[3] S. Ruscheweyh, Some properties of prestarlike and universally prestarlike functions. Journal ofAnalysis 15 (2007), 247–254
[4] S. Ruscheweyh, L. Salinas and T. Sugawa, Completely monotone sequences and universallyprestarlike functions. Israel J. Math. Vol. 171 Nr. 1 (2009), 285–304.
[5] St. Ruscheweyh, L. Salinas Universally Prestarlike Functions as Convolution Multipliers. Math-ematische Zeitschrift 263 (3) (2009), 607–617.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Aproximación de soluciones acotadas deecuaciones diferenciales con argumento
constante a trozos del tipo generalizado conimpulsos
Manuel Pinto Ricardo Torres
Abstract
En [1], M.U.Akhmet consideró la ecuación
x′ (t) = f (t, x (t) , x (γ (t)))
donde γ (t) es un argumento constante a trozos del tipo generalizado. Es decir, dadas(tk)k∈Z y (ζk)k∈Z tales que tk < tk+1 , ∀k ∈ Z con limk→±∞ tk = ±∞ y tk ≤ ζk ≤ tk+1,se tiene que ∀t,∃k (t) ∈ Z tal que si t ∈ Ik = [tk, tk+1) , entonces
γ(t) = ζk.
Estas ecuaciones son llamadas Ecuaciones Diferenciales con Argumento Constante aTrozos del Tipo Generalizado (DEPCAG), las cuales poseen soluciones continuas, apesar de que γ(t) no lo sea. En los extremos de los intervalos de constancia, estasecuaciones generan una ley recursiva, la cual da origen a una ecuación discreta. Espor esto que estas ecuaciones corresponden al tipo híbridas, ya que combinan tantopropiedades de ecuaciones discretas como de continuas.
En una DEPCAG, al no considerar la continuidad en los extremos de los intervalosIk = [tk, tk+1); es decir, al considerar una condición de salto en dichos puntos, seda origen a las Ecuaciones Diferenciales con Argumento Constante a Trozos del tipoGeneralizado con Impulsos. (IDEPCAG)
x′(t) = f (t, x (t) , x (γ (t))) , t 6= tk
∆x|t=tk = Qk(x(t−k)), t = tk. (1)
En cada intervalo Ik se satisface la ecuación diferencial ordinaria
x′ (t) = f(t, x (t) , x (ζk)).
Para t = tk, la solución satisface la ley discreta de salto
∆x|t=tk = x(tk)− x(t−k ) = Qk(x(t−k )
),
M. Pinto agradece el apoyo del proyecto FONDECYT 1120709, e-mail: [email protected]. Torres agradece el apoyo del proyecto FONDECYT 1120709, e-mail: [email protected]
34
donde asumiremos que el límite lateral izquierdo
x(t−k ) = limt→tkt<tk
x(t)
existe ∀tk con k ∈ N y x(t+k)
= x (tk) está definido por
x (tk) = x(t−k ) +Qk(x(t−k )
),
(Ver ([1, 12, 14])En esta ocasión, estableceremos condiciones para producir aproximaciones de solu-ciones de sistemas impulsivos del tipo CNN (Cellular Neural Networks), por medio desoluciones IDEPCAG utilizando un argumento constante a trozos que aproxime a laidentidad. Es decir, se aproximará el sistema semilineal impulsivo
y′i(t) = −ai(t)yi(t) +Hi(t, y(t)), t 6= tk
∆yi = −qi,kyi(t−k ) + Ii,k(yi(t−k )), t = tk (2)
y (t0) = y0, t = t0
donde i = 1, 2, . . . ,m, y k ∈ N, mediante el sistema IDEPCAG
z′i(t) = −ai(t)zi(t) +Hi(t, z(γ(t))), t 6= γ(tk)
∆zi|t=γ(tk) = −qi,kzi(γ(tk)−) + Ii,k(zi(γ(tk)
−)), t = γ(tk) (3)z (ζ0) = z0, ζ0 = γ(t0)
con i = 1, 2, . . . ,m, k ∈ N, donde
Hi(t, y(t)) =m∑j=1
bij(t)fj(yj(t)) + ci(t),
tal que qi,k 6= 1, ai(t), qi,k > 0∀t ∈ R+, ∀i ∈ [1,m] y
γ(t) =
[t
δ
]δ, k ∈ N, δ > 0.
(Ver [3, 5, 9, 10, 11]). Se demostrará que al considerar tal función γ (t) (la cual convergeuniformemente a la identidad cuando δ → 0), además de condiciones del tipo Lipschitzsobre Hi(t, y(t)) y de estabilidad del sistema impulsivo lineal asociado a (2),∫ t
sai(u)du+
∑s≤tk<t
ln(1 + qi,k) ≥ σ(t− s), σ > 0, ∀i ∈ [1,m],
entoncessup
t∈[t0,∞)|y(t)− z(t)| → 0, δ → 0
ysup
t∈[t0,∞)|y(tr)− z(ζr)| → 0, δ → 0.
Este método fue propuesto por I.Györi en [7], para la ecuación escalar
x′(t) = f(t, x(t), x(t− τ))
conf (t, x (t− τ)) = βx (t− τ)
bajo ciertas condiciones de integrabilidad y sobre intervalos compactos de [0,∞) (Ver[4, 8]). Los resultados obtenidos son completamente nuevos y extienden, en el casoacotado, a los realizados en [6] para ecuaciones del tipo DEPCAG (Ver [13]).
35
References[1] M.U. Akhmet, Nonlinear Hybrid Continuous/Discrete-Time Models. Atlantis Press,
Amsterdam-Paris (2011).
[2] M.U. Akhmet, Principles of Discontinuous Dynamical Systems. Springer, New York, Dordrecht,Heidelberg, London. (2010).
[3] M.U. Akhmet, E.Yilmaz. Impulsive Hopfield-type neural network system with piecewise constantargument. Nonlinear Analysis: Real World Applications. 11:2584-2593. (2010).
[4] K.L.Cooke, I.Györi. Numerical approximation of the solutions of delay differential equationson an infinite interval using piecewise constant arguments. Computer and Mathematics withapplications, 28(1-3):81-92. (1994).
[5] A.Coronel, M.Pinto, R.Torres. Exponential periodic attractor of impulsive Hopfield-type neuralnetwork system with piecewise constant argument. En preparación.
[6] L.González. Aproximación de soluciones casi periódicas de ecuaciones diferenciales medianteargumento constante a trozos. Tesis de Magister. Facultad de Ciencias. Universidad de Chile.(2013).
[7] I.Györi. On approximation of the solutions of delay differential equations by using piecewiseconstant arguments. Internat. J. Math. Sci. 14(1). pp. 111-126. (1991).
[8] I.Györi. On numerical approximation using differential equations with piecewise-constant argu-ments. Periodica Mathematica Hungarica, (56)-1, pp 55-69 (2008).
[9] J.J.Hopfield. Neural networks and physical systems with emergent collective computational abil-ities. Proc.Nat.Acad.Sci. U.S.A. 79 (1982).
[10] M.Pinto, G. Robledo. Existence and stability of almost periodic solutions in impulsive neuralnetwork models. Applied Mathematics and Computation, 217(8):4167-4177, (2010).
[11] M.Pinto, R.Torres. Approximation of bounded solutions of an impulsive differential system usingpiecewise constant arguments. En preparación.
[12] A.M. Samoilenko, N.A. Perestyuk. Impulsive Differential Equations. World Scientific, Singapore(1995).
[13] R. Torres. Ecuaciones diferenciales con argumento constante a trozos del tipo generalizado conimpulsos. Tesis de Magister. Facultad de Ciencias. Universidad de Chile. (2015).
[14] J. Wiener. Generalized Solutions of Functional Differential Equations. World Scientific, Singa-pore (1993).
36
Álgebra
Encargado de Sesión: Nicolás Libedinsky
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Algunas propiedades del álgebra de símbolospseudodiferenciales formalesJarnishs Beltran Enrique G. Reyes
Abstract
Se presentará el álgebra de Lie ΨDO de símbolos pseudodiferenciales formales en una yvarias variables. Se mostrará la construcción de una derivación exterior que induce unaextension central del álgebra de Lie ΨDO. También se presentará el álgebra de símbo-los pseudodiferenciales formales torcidos y se estudiarán sus extensiones centrales. Seconstruyen jerarquías de extensiones centrales de álgebras de Lie de dimensión infinita,generalizando de esta manera trabajos previos de Khesin [2]. Se construirán triplesde Manin para estas álgebras, y se observará (motivado por de Bajo et al [3]) quela construcción de estas álgebras de símbolos pseudodiferenciales fomales proporcionaejemplos de álgebras de Lie simplécticas cuadráticas en dimension infinita.
References[1] Jarnishs Beltran and Enrique G. Reyes, âĂIJFormal Pseudodifferential Operators
in One and Several Variables, Central Extensions, and Integrable Systems,âĂİ Ad-vances in Mathematical Physics, vol. 2015, Article ID 210346, 16 pages, 2015.doi:10.1155/2015/210346
[2] B. A. Khesin, âĂIJA hierarchy of centrally extended algebras and the logarithm of thederivative operator,âĂİ International Mathematics Research Notices, no. 1, pp. 1âĂŞ5,1992
[3] Bajo, I., Benayadi, S. and Medina, A. âĂIJ Symplectic structures on quadratic LiealgebrasâĂİ, Journal of Algebra, 316(1), 174-188, 2007
Centro de Investigacion en Complejidad Social, Universidad del Desarrollo e-mail:[email protected]
Universidad de Santiago de Chile , e-mail: :[email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Módulos de tipo FP-infinito y la CategoríaEstable de Módulos de un Anillo
Daniel Bravo
Abstract
Decimos que un R-módulo (izquierdo) M , sobre un anillo unitario R, es de Tipo FP∞,si M tiene una resolución por módulos libres finitamente generados; denotamos porFP∞ a la clase de estos módulos.
En esta charla, hablaremos de las propiedades de FP∞, en particular, como estaclase de módulos se comporta en el contexto de anillos Noetherianos [Lam01], coherentes[Gla89] y finalmente para cualquier anillo en general [Bie81]. Mostraremos como estaclase de módulos nos permite definir una generalización de la categoría estable demódulos [BGH14].
A lo largo de la charla algunos resultados sobre a la clase de módulos módulosfinitamente n-presentados, denotados por FPn, serán presentados [BP15].
References[Bie81] Robert Bieri, Homological dimension of discrete groups, second ed., Queen Mary
College Mathematical Notes, Queen Mary College Department of Pure Mathematics,London, 1981.
[BGH14] D. Bravo, J. Gillespie, and M. Hovey. The stable module category of a generalring. preprint, arXiv:1405.5768v1, 2014.
[BP15] D. Bravo and M. Perez. Finiteness conditions and cotorsion pairs. preprint,arXiv:1510.08966, 2015.
[Gla89] S. Glaz. Commutative Coherent Rings. Lecture Notes in Mathematics. SpringerBerlin Heidelberg, 1989.
[Lam01] T. Y. Lam. A First Course in Noncommutative Rings. Graduate Texts in Mathe-matics. Springer New York, 2001.
Universidad Austral de Chile e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Macdonald polynomials in superspace and the 6vertex model
Luc Lapointe
Abstract
The Macdonald polynomials in superspace are symmetric polynomials involving com-muting and anticommuting variables that generalize the Macdonald polynomials. Wewill describe how the combinatorics of the Macdonald polynomials extends to super-space. We will focus in particular on how the partition function of the 6 vertex modelarises in the Pieri rules for the Macdonald polynomials in superspace.
Proyecto Fondecyt #1130696, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Teoría de representaciones graduada del álgebrade blob y el cálculo de Soergel en dos colores
David Plaza
Abstract
En esta charla estudiaremos la teoría de representaciones graduada del álgebra de blobbn(q,m). En primer lugar, explicaremos como podemos equipar a bn(q,m) con la es-tructura de álgebra celular graduada [PR14]. Usando esta construcción podemos definiry calcular explicitamente los números de descomposición graduados de bn(q,m) [P13].Estos resultados corresponden a un trabajo en conjunto con Steen Ryom-Hansen.
Estos números coinciden con los polinomios de Kazhdan-Lusztig asociados al grupodiedral infinito. Si el tiempo lo permite, formularemos una conjetura que relaciona elálgebra de blob con la categoría de bimódulos de Soergel en dos colores, que explicaríaa nivel categórico la coincidencia combinatoria entre los números de descomposición debn(q,m) y los polinomios de Kazhdan-Lusztig.
References[PR14] Plaza, D., & Ryom-Hansen, S. (2014). Graded cellular bases for TemperleyâĂŞLieb algebras
of type A and B. Journal of Algebraic Combinatorics, 40(1), 137-177.
[P13] Plaza, D. (2013). Graded decomposition numbers for the blob algebra. Journal of Algebra,394, 182-206.
Parcialmente financiado por FONDECYT-Postdoctorado 3140612,e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Groupoidal Gelfand Models in Group TheoryJorge Soto Andrade
Abstract
We have conjectured [2] that for any finite group of Lie type G some canonical G-set may be found from which a Gelfand Model [4, 5, 2, 1] for G may be obtained bygeometric induction from a linear character of the associated motion groupoid. Thisconjecture has been proved for dihedral groups, the symmetric groups [3]and the pro-jective general linear group of rank 2. The case of the general linear group is work inprogress. This construction via groupoids works even for groups that have no involutionmodel (like GL(2, q)) in the sense of Bump and Ginzburg [5]. We have also conjecturedthat a Gelfand Model of a (finite) group G always lies in the Green ring Green(G) ofG (generated by all permutation representations of G) and that it may be realised as atop cohomology space of some G-set for a suitable equivariant cohomology theory. Wepresent some recent counter examples [2] and discuss the domain of validity and vari-ants of these conjectures. We conjecture moreover that not only Gelfand Models butall symmetric functions of the irreducible representations of G lie in Green(G). Thenthe irreducible representations of G would appear as the roots in the representationring R(G) of G of a polynomial equation with coefficients in Green(G), whose degreeis the number of conjugacy classes of G. This is joint work with Anne-Marie Aubertand Antonio Behn [2]
References[1] J.-L. Aguado and J.O. Araujo, A Gel’fand model for the symmetric group, Communications in
Algebra 29 (4), pp. 1841–1851 (2001).
[2] A.-M. Aubert, A. Behn, J. Soto-Andrade, Groupoids, Geometrical Induction and GelfandModels, preprint (2015).
[3] V. Kodiyalam and D.-N. Verma, A natural representation model for symmetric groups,arXiv:math.RT/0402216.
[4] J. Soto-Andrade, Geometrical Gel’fand Models, Tensor Quotients and Weil Representations,Proc. Symp. Pure Math., 47 (1987), Amer. Math. Soc., 305-316.
[5] D. Bump, D. Ginzburg, Generalized Frobenius Schur numbers, J. of Algebra 2¯78 (2004) 294Ð313
Supported by Fondecyt Project 1140510, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Algebraic structures on stellohedra andpterahedra
L. Berry S. Forcey M. Ronco P. Showers
Abstract
Stellohedra and pterahedra are families of polytopes, which may be obtained by thecontraction of certain faces of permutohedra (see [3]). Both families are examples of M.Carr and S. Devadoss graph associahedra (see [1]), they correspond to the polytopesK(Gn) associated to graphs which are suspensions of other finite graphs. The stellohe-dron of dimension n is the polytope K(Stn) associated to the suspension of the trivialgraph with n nodes, while the pterahedra of dimension n is the polytope associated tothe suspension of the line graph with n nodes.
We study the simplicial complex of faces of the stellohedron and the permutohedronin terms of differents types of planar trees , and describe an extension of the Tamariorder in this case.
Moreover, we defined associative products of degree −1 on the vector spaces spannedby the nodes of both families of polytopes. These notion uses the Hopf algebra structureon the space spanned by the set of all permutations, introduces by C. Malvenuto and C.Reutenauer in [2]. Our construction suggest a simple way to defined algebraic structureson the spaces spanned by the faces of Carr and Devadoss’s polytopes of the suspensionsof certain families of graphs.
References[1] M. Carr, S. Devadoss, Coxeter complexes and graph associahedra, Topology Appl. 153(12)
(2006) 2155-2168.
[2] C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomondescent algebra, J. of Algebra 177 (3) (1995) 967-982.
[3] A. Tonks, Relating the associahedron and the permutohedron, in Operads: Proceedings of Re-naissance Conferences (Hartford, CT/Luminy, 1995), vol. 202 of Contemp. Math., Amer. Math.Soc., Providence, RI (1997) 33-36.
NSA Grant H98230-14-0121, e-mail:[email protected] Regular 1130939, e-mail: [email protected] Grant H98230-14-0121, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Teoría de representaciones del álgebra deYokonuma-Hecke y del álgebra de braids and
tiesSteen Ryom-Hansen
Abstract
En esta charla presentaremos resultados obtenidos en colaboración con Jorge Espinoza.El álgebra de Yokonuma-Hecke YHn se ha estudiado intensamente en los últimos años,sobre todo del punto de vista de teoria de nudos. Estudiaremos la teoría de repre-sentaciones de YHn y del álgebra de ’braids and ties’ En introducida por Aicardi yJuyumaya. El álgebra En tiene una base parametrizada por particiones conjuntistas,lo que indica una relación con el álgebra de particiones Pn, definida por V. Jones yP. Martin. Sin embargo, en general la relación todavía no está clara. Construimosmódulos espacios tensoriales para ambas álgebras y demostramos que son fieles. Estonos permite construir un isomorphismo concreto entre YHn y el álgebra modificada deAriki-Koike, introducida por Shoji. Tambien obtenemos de esta manera la presentacionusada por Lusztig. Al final de la charla demostraremos que En es una álgebra celular.
References[1] F. Aicardi, J. Juyumaya, Markov trace on the algebra of braids and ties, arXiv:1408.5672, a
aparecer, Moscow Mathematical Journal.
[2] E. O. Banjo, The Generic Representation Theory of the Juyumaya Algebra of Braids and Ties,Algebras and Representation Theory 16(5) (2013), 1385-1395.
[3] M. Chlouveraki, L. Poulain d’Andecy, Representation theory of the Yokonuma-Hecke algebra,Advances in Mathematics 259 (2014), 134-172.
[4] M. Chlouveraki, J. Juyumaya, K. Karvounis, S. Lambropoulou, Identifying the invariants forclassical knots and links from the Yokonuma-Hecke algebras, arXiv:1505.06666.
[5] M. Chlouveraki, S. Lambropoulou, The Yokonuma-Hecke algebras and the HOMFLYPT poly-nomial, J. Knot Theory and its Ramifications 22 (14) (2013) 1350080
[6] J. Espinoza, S. Ryom-Hansen, Cell structures for the Yokonuma-Hecke algebra and the algebraof braids and ties, arXiv:1506.00715, 1-35.
[7] J. Juyumaya, Sur les nouveaux générateurs de l’algèbre de Hecke H(G,U,1). (French) On newgenerators of the Hecke algebra H(G,U,1), J. Algebra 204 (1998)(1), 49-68.
FONDECYT 1121129, e-mail: [email protected]
44
[8] J. Juyumaya, S. Lambropoulou, p-Adic framed braids II, Advances in Mathematics 234 (2013),149-191.
[9] G. Lusztig, Character sheaves on disconnected groups, VI, Represent. Theory (electronic) 8(2004), 377-413.
[10] S. Ryom-Hansen, On the Representation Theory of an Algebra of Braids and Ties, J. AlgebraComb., 33 (2011), 57-79.
[11] T. Shoji, A Frobenius formula for the characters of Ariki-Koike algebras, J. Algebra, 221 (1999),293-314.
45
Análisis No Arquimediano
Encargado de Sesión: Elena Olivos
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
C-álgebra de operadores lineales continuosdefinidos en c0(I)
J. Aguayo M. Nova J. Ojeda
Abstract
Sea I un conjunto arbitrario de índices. Se estudia una clase de operadores linealescontinuos definidos sobre
c0 (I) :=
(xi)i∈I : xi ∈ K, lim
i∈Ixi = 0
,
la cual resulta ser una C-álgebra de Banach conmutativa con unidad. Se muestra quebajo ciertas condiciones, esta álgebra es isométricamente isomorfa a un espacio de fun-ciones continuas definidas sobre un conjunto compacto particular.
References[1] J. Aguayo, M. Nova and J. Ojeda, Spectral measures on C-algebras of operators in c0 (N).
Artículo sometido.
[2] A. van Rooij, Non-archimedean Functional Analysis, Marcer-Dekker, New York, 1978.
Departamento de Matemática, Universidad de Concepción. e-mail: [email protected] de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción.
e-mail: [email protected] de Matemática, Universidad de Concepción. e-mail: [email protected] trabajo es parcialmente financiado por Proyecto VRID N 214.014.038-1.0IN
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Espacios y cuerpos residuales en espacios de tipoHilbert sobre cuerpos con valuaciones
no-arquimedianasHerminia Ochsenius Elena Olivos
Abstract
Sea K un cuerpo completo en una valuación no arquimediana de rango infinito. Unespacio tipo Hilbert según la norma (NHS) es un K-espacio de Banach de tipo contableen el cual todo subespacio cerrado admite un complemento ortogonal en el sentido dela norma. Cuando esta norma proviene de una forma bilineal simétrica, se habla deespacios tipo Hilbert según la forma (FHS), que son espacios ortomodulares. La teoríade operadores en estos espacios usa como una herramienta importante la reducción aespacios residuales sobre cuerpos residuales. Se expondrá su construcción y ejemplosde teoremas sobre operadores que surgen de su aplicación, correspondientes a trabajosconjuntos con H. Keller, E. Olivos y W. Schikhof.
Depto. de Matemática y Est. Universidad de La Frontera. e-mail: [email protected]. de Matemática y Est. Universidad de La Frontera. e-mail: [email protected] a Proyecto DIUFRO DI15-0043
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Medida Espectral sobre Algebras de Operadoresdefinidos en c0(N)
J. Aguayo M. Nova J. Ojeda
Abstract
En un contexto no-arquimediano, se estudia el espacio Gelfand de ciertas álgebras deoperadores lineales definidos sobre el espacio c0(N), las cuales resultan ser C-álgebrasde Banach conmutativas con unidad. Se muestra que bajo ciertas condiciones, estasálgebras son isometricamente isomorfas a un espacio de funciones continuas definidassobre conjunto compacto particular. La isometría entre álgebras antes mencionadapreserva elementos idempotentes y permite definir la medida asociada, la cual es cono-cida como medida espectral. Se muestra, además, que cada elemento del álgebra deBanach descrita en este estudio, puede ser representada como la integral de algunafunción continua definida con esta medida.
References[1] J. Aguayo y M. Nova, Non-archimedean Hilbert like Spaces, Bull. Belg. Math. Soo., Vol. 14,
pp. 787-797, 2007.
[2] J. Aguayo, M. Nova y K. Schamseddine, Characterization of Compact and Self-adjoint oper-ators on Free Banach Spaces of countable type over the complex Levi-Civita field, Journal ofMathematical Physics, Vol. 54(2), 2013.
[3] V. Berkovich, Spectral Theory and analytic geometry over non-archimedean fields, Mathemati-cal Surveys and Monograph, Number 33, AMS, 1990.
[4] B. Diarra, Bounded linear operators on ultrametric Hilbert spaces, Afr. Diaspora J. Math., Vol.8(2), pp. 173-181, 2009.
[5] L. Narici y E. Beckenstein, A non-Archimedean Inner Product, Contemporary Mathematics,Vol. 384, pp. 187-202, 2005.
[6] A. Van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, New York, 1978.
[7] M. Vishik, Non-Archimedean spectral Theory, J. Soviet Math., Vol. 30, pp. 2513-54, 1985.
Departamento de Matemática, Universidad de Concepción. e-mail: [email protected] de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción.
e-mail: [email protected] de Matemática, Universidad de Concepción. e-mail: [email protected] trabajo es parcialmente financiado por Proyecto VRID N 214.014.038-1.0IN.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Espacios de Banach sobre cuerpos con valuacióndiscreta
Elena Olivos Herminia Ochsenius
Abstract
Si el grupo de valores G de un cuerpo con valuación no arquimediana K es cíclico, elG-módulo X de las normas de cualquier espacio E sobre este cuerpo tiene base convexa.Ello implica que existe un conjunto ordenado B tal que X es isomorfo a B ×G, con laacción g(b, g′) 7→ (b, gg′) y orden antilexicográfico. Como consecuencia, el espacio Ees un NHS si y solo si B es bien ordenado. En este trabajo estudiamos las propiedadesde espacios uno-ortogonal, rigidos y que contienen c0 cuando el cuerpo tiene valuacióndiscreta.
References[1] H. Ochsenius, H. and W.H. Schikhof, Banach spaces over fields with an infinite rank valuation.
In p-Adic Functional Analysis, Lecture Notes in pure and applied mathematics 207, edited byJ. Kakol, N. De Grande-De Kimpe and C. Perez-Garcia. Marcel Dekker (1999), 233-293.
[2] H. Ochsenius and W.H. Schikhof, Norm Hilbert spaces over Krull valued fields. Indag. Mathem.N.S. 17 , (1), (2006), 65-84.
[3] A.C.M. van Rooij, Non-archimedean Functional Analysis. Marcel Dekker, New York(1978).
Depto. de Matemática y Est. Universidad de La Frontera. e-mail: [email protected]. de Matemática y Est. Universidad de La Frontera. e-mail: [email protected] Proyecto DIUFRO DI15-0043
50
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
An o-minimal approach to definability offunctions in extensions of C
Javier Utreras
Abstract
In 1984, A. Pillay and C. Steinhorn introduced the concept of an o-minimal structure toobtain analytical properties of R, and some other related real closed fields, in a limitedtame topological setting. In 2001, Y. Peterzil and S. Starchenko obtained analogues ofmany results of complex analysis for the algebraic closures of these real closed fields,and in 2008 A. J. Wilkie proved that in the standard case the holomorphic functionsdefinable in this setting are, almost everywhere, obtained from a given set of functionsvia standard complex analytic operations (Schwarz reflection, differentiation, implicitdefinition and composition).
We will present a generalization of Wilkie’s result to non-Archimedean extensionsof C, using the fact that under this tame topological setting many standard resultsfrom analysis may still work, after giving some consideration to the parameters of thefunctions.
References[1] Y. Peterzil and S. Starchenko, Expansions of algebraically closed fields in o-minimal structures.
Sel. math., New ser. 7 (2001) 409–445.
[2] A. Pillay and C. Steinhorn, Definable sets in ordered structures. Bull. Amer. Math. Soc. (N.S.)11 (1984), no. 1, 159–162.
[3] J. Utreras, Model theory of holomorphic functions in an o-minimal setting. PhD Thesis, Schoolof Mathematics, The University of Manchester, 2014.
[4] A. J. Wilkie, Some local definability theory for holomorphic functions. Model Theory withApplications to Algebra and Analysis, Vol 1 (2008) LMS Lecture Note Series 349, CUP, 197-213.
e-mail: [email protected]
51
Análisis Numérico
Encargado de Sesión: Mauricio Sepúlveda
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Mathematical and numerical analysis for anonlocal reaction diffusion system
V. Anaya M. Bendahmane M. Langlais M. Sepúlveda
Abstract
This work is concerned with a model of the indirect transmission of an epidemic dis-ease between two spatially distributed host populations having non-coincident spatialdomains with nonlocal and cross-diffusion, the epidemic disease transmission occurringthrough a contaminated environment. The mobility of each class is assumed to be in-fluenced by the gradient of the other classes. We address the questions of existence ofweak solutions by using a regularization method. Moreover, we propose a finite volumescheme and proved the well-posedness, nonnegativity and convergence of the discretesolution. The convergence proof is based on deriving a series of a priori estimates andby using a general Lp compactness criterion. Finally, the numerical scheme is illustratedby some examples.
References[1] B. Andreianov, M. Bendahmane and R. Ruiz-Baier, Analysis of a finite volume method
for a cross-diffusion model in population dynamics, Mathematical Models and Methods inApplied Sciences, 21, (2011), 307–344.
[2] M. Bendahmane and M. Langlais, A reaction-diffusion system with cross-diffusion mod-elling the spread of an epidemic disease Journal of Evolution Equations, 10(4), (2010), 883–904
[3] R. Eymard, Th. Gallouët, and R. Herbin, Finite volume methods.In: Handbook ofNumerical Analysis, vol. VII, North-Holland, Amsterdam, 2000
[4] J. Simon, Compact sets in the space Lp(0, T;B). Ann. Mat. Pura Appl. (4)146, (1987), 65–96
GIMNAP and Departamento de Matemática, Universidad del Bío-Bío, Concepción, Chile. e-mail:[email protected]
Institut the Mathématiques de Bordeaux, Université de Bordeaux, France, e-mail:[email protected]
Institut the Mathématiques de Bordeaux, Université de Bordeaux, France e-mail:[email protected]
CI2MA and DIM, Universidad de Concepción, Concepción, Chile, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A kind of Mixed Finite Element VariationalFormulation for a Parabolic Problem
Mauricio Barrientos Karina Vilches
Abstract
In this work we study a parabolic problem arised from Keller-Segel model for chemo-taxis. A new formulation of the system of partial differential equations is obtainedby the introduction of a new variable, which have a mixed variational formulationstructure. At this point, the applicability of adaptive moving meshes theory is carriedout with the purpose to obtain a cheap and better description of the behavior of theparticles close to the blow up.
References[1] M.J. Baines, M.E. Hubbard and P.K. Jimack, Velocity-Based Moving Mesh Methods for
Nonlinear Partial Differential Equations, Commun. Comput. Phys., vol. 10 (3) (2011) 509–576.
[2] E.F. Keller and L.A. Segel, Traveling bands of chemotactic bacteria, J. Thoer. Biol., vol.30 (1971) 235–248.
[3] R. Marlow, M.E. Hubbard and P.K. Jimack, Moving mesh methods for solving parabolicpartial differential equations. Comput. & Fluids, vol. 46 (2011), 353–361.
[4] A. Morrocco, Numerical Simulation of Chemotactic Bacteria Aggregation via Mixed FiniteElements, ESAIM: M2AN., vol. 37 (4) (2003) 617–630.
PUCV, Pontificia Universidad Católica de Valparaíso, Chile. e-mail: [email protected] supported by Proyecto DI Regular 037.438/2015.
Departamento de Ingeniería Matemática, Universidad de Chile, Chile. e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Discontinuos Galerkin scheme for Helmholtzequation
Tomás Barrios Rommel Bustinza
Abstract
In this talk, in order to describe the different phenomena, we first apply the localdiscontinuous Galerkin (LDG for short) method to solve a Helmholtz problem in abounded domain. We establish existence, uniqueness as well as optimal rate of conver-gence, for meshes such that its meshsize is small enough. Additionaly, with the aimto obtain more flexibility in the elections of the discrete subspaces pairs, we analysedan stabilized mixed discontinuous formulation for this problem. The procedure of thestabilization is through of the addition of an appropriate Galerkin least squares term tothe mixed formulation. Following the same ideas than in the first part, we also provethe well posedness of this scheme and the optimal convergence are guaranteed for hsmall enough, too. Finally, several numerical experiments confirming the theoreticalproperties for both approaches are reported.
References[1] T.P. Barrios and R. Bustinza: An augmented discontinuous Galerkin method for elliptic
problems. Comptes Rendus de l’Academie des Sciences, Series I, vol. 344, pp. 53-58, (2007).
[2] T.P. Barrios and R. Bustinza: A priori and a posteriori error analyses of an augmenteddiscontinuous Galerkin formulation. IMA Journal of Numerical Analysis, vol 30, 4, pp. 987-1008, (2010).
[3] T.P. Barrios and R. Bustinza: An a posteriori error analysis of an augmented discon-tinuous Galerkin formulation for Darcy flow. Numerische Mathematik, vol 120, pp. 231-269,(2012).
[4] T.P. Barrios, R. Bustinza and V. Domínguez: On the discontinuous Galerkin methodfor solving boundary value problems for the Helmholtz equation: A priori and a posteriorierror analyses. Preprint 2013-13, Departamento de Ingeniería Matemática, Universidad deConcepción, (2013).
[5] A.H. Schatz: An observation concerning Ritz-Galerkin methods with indefinite bilinear forms.Mathematics of Computation, vol. 28, 128, pp. 959-962, (1974).
DMFA, Universidad Católica de la Santísima Concepción, Chile. e-mail: [email protected] and DIM, Universidad de Concepción, Concepción, Chile. e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Hydraulic parameter estimation undernon-saturated flow conditions in copper heap
leachingEmilio Cariaga Rubén Martínez Mauricio Sepúlveda
Abstract
The mathematical modeling of the unsaturated flow problem requires the simultaneousresolution of two problems: the Richards equation and the estimation of the hydraulicparameters involved in hydraulic conductivity and in the retention curve. Varioustechniques have been applied to both problems in a wide range of situations. In thisarticle, a novel implementation of the processing techniques involved in copper heapleaching is presented. Specifically, the impact of the used numerical method and theselection of the parametric family are evaluated. From a methodological point of view,a global algorithm is proposed that integrates the solutions of both problems. Finally,our computational experiments are compared with previous experimental results fromthe Chilean copper mining industry and related works.
References[1] E.Cariaga, R.Martínez, M.Sepúlveda, Hydraulic parameter estimation under non-
saturated flow conditions in copper heap leaching, Mathematics and Computers in Simulation,vol. 109, pp. 20-31, 2015.
UCT, Universidad Católica de Temuco, Chile, e-mail: [email protected], Universidad Austral de Chile, Chile, e-mail: [email protected] and DIM, Universidad de Concepción, Concepción, Chile, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Torque free rigid body motion: an ellipticfunction primer
Roberto León Luis Salinas Claudio Torres
AbstractAn analysis of the motion for a rigid box is presented when a force or torque is notapplied. The fundamental equations describing the motion of the rigid box are the wellknown equations from classical mechanics [1],∑
F = ma ,∑
MG =·HG
where the sum of all external forces acting on the body is equal to mass by the instan-taneous acceleration of the center of mass G, and the sum of torques applied to thebody is the derivative of the angular momentum of the body about its mass center G.In our case, it is considered a torque free motion, which means that the direction of the
angular momentum remains fixed with respect to a fixed coordinate system·HG = 0,
obtaining
I11·ω1 = (I22 − I33)ω2 ω3 ,
I22·ω2 = (I33 − I11)ω3 ω1 ,
I33·ω3 = (I11 − I22)ω1 ω2 ,
where I11, I22 and I33 are the principal moments of inertia, and ω1, ω2 and ω3 are theinstantaneous angular velocity of the rigid body.Solving these set of differential equations, it is obtained as solutions, the set of Jacobielliptic functions sn, cn and dn [2] for each rotational velocity of the rigid body, and itwill depend on the initial conditions which Jacobi elliptic function correspond to eachrotational velocity. An algorithm to obtain these functions is presented with the aimto get an analytical solution. A comparison between analytical and numerical solutionis presented as well.
References[1] Herbert Goldstein, Classical Mechanics, Addison-Wesley, 1980.
[2] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge University Press,1963.
CCTVal, Universidad Técnica Federico Santa María and Facultad de Ingeniería, Universidad AndresBello, e-mail: [email protected] , [email protected] , [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Analysis of an augmented pseudostress-basedmixed formulation for a nonlinear Brinkman
model of porous media flowGabriel N. Gatica Luis F. Gatica Filánder A. Sequeira
AbstractIn this work we introduce and analyze an augmented mixed finite element method forthe two-dimensional nonlinear Brinkman model of porous media flow with mixed bound-ary conditions. More precisely, we extend a previous approach for the respective linearmodel to the present nonlinear case, and employ a dual-mixed formulation in whichthe main unknowns are given by the gradient of the velocity and the pseudostress. Inthis way, and similarly as before, the original velocity and pressure unknowns are easilyrecovered through a simple postprocessing. In addition, since the Neumann boundarycondition becomes essential, we impose it in a weak sense, which yields the introductionof the trace of the fluid velocity over the Neumann boundary as the associated Lagrangemultiplier. We apply known results from nonlinear functional analysis to prove thatthe corresponding continuous and discrete schemes are well-posed. In particular, a fea-sible choice of finite element subspaces is given by Raviart-Thomas elements of orderk ≥ 0 for the pseudostress, piecewise polynomials of degree ≤ k for the gradient of thevelocity, and continuous piecewise polynomials of degree ≤ k+ 1 for the Lagrange mul-tiplier. We also derive a reliable and efficient residual-based a posteriori error estimatorfor this problem. Finally, several numerical results illustrating the performance and therobustness of the method, confirming the theoretical properties of the estimator, andshowing the behaviour of the associated adaptive algorithm, are provided.
References[1] G.N. Gatica, L.F. Gatica, A. Márquez, Analysis of a pseudostress-based mixed finite
element method for the Brinkman model of porous media flow, Numerische Mathematik, vol.126, 4, pp. 635-677, 2014.
[2] G.N. Gatica, A. Márquez and M.A. Sánchez, A priori and a posteriori error analysesof a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows, Comput.Methods Appl. Mech. Engrg., Vol 200, No 17-20, pp. 1619-1636, 2011.
CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Chile, e-mail:[email protected]
Departamento de Matemática y Física Aplicadas, Facultad de Ingeniería, Universidad Católica de laSantísima Concepción, Concepción, Chile, e-mail: [email protected]
Escuela de Matemática, Universidad Nacional de Costa Rica, Heredia, Costa Rica, e-mail:[email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A mixed FEM for a vorticity based formulationof the Brinkman problem
David Mora
Abstract
In this talk, we develop a mixed finite element method for the Brinkman equationsformulated in terms of velocity, vorticity and pressure. By employing an extensionof the Babuska-Brezzi theory, it is proved that the resulting continuous and discretevariational formulations are well-posed. In particular, we show that Raviart-Thomaselements of order k ≥ 0 for the approximation of the velocity field, piecewise continuouspolynomials of degree k+ 1 for the vorticity, and piecewise polynomials of degree k forthe pressure, yield unique solvability of the discrete problem. We establish a priorierror estimates in the natural norms. Finally, we report several numerical experimentsillustrating the behavior of the proposed scheme and confirming our theoretical results.
We will report on results obtained in collaboration with V. Anaya, R. Oyarzúa andR. Ruiz-Baier.
References[1] M. Amara, D. Capatina-Papaghiuc and D. Trujillo, Stabilized finite element method
for Navier-Stokes equations with physical boundary conditions, Math. Comp., vol. 76, 259, pp.1195–1217, 2007.
[2] V. Anaya, D. Mora and R. Ruiz-Baier, An augmented mixed finite element method for thevorticity-velocity-pressure formulation of the Stokes equations, Comput. Methods Appl. Mech.Engrg., vol. 267, pp. 261–274, 2013.
Departamento de Matemática, Universidad del Bío-Bío, Concepción, Chile, and CI2MA, Universidad deConcepción, Concepción, Chile e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Finite Volume Method Improved SequentialSolution Applied To Two-DimensionalConvective-Diffusive Heat Transfer
Nelson O. Moraga Juan Jaime
AbstractThe numerical solution of fluid mechanics with convective heat transfer problems inmost of the industrial applications is obtained by the Finite Volume Method, FVM [1].One node per volume and suitable interpolation functions for convective and diffusionterms of the governing partial differential equations are used in the FVM. The solutionof the discretized mathematical model is obtained by a sequential-iterative-implicit-procedure to solve in a segregated way the continuity, linear momentum and energynon-linear coupled PDEâĂŹs [2]. The purpose of this paper is to describe improve-ments in the efficiency of the numerical solution, estimated in terms of reduction inthe number of iterations and in savings of the CPU time needed to solve each problem,by the use of a novel sequential procedure, PSIMPLER, developed by our group [3].The numerical analysis is applied to convective-diffusive problems, in two-dimensions,with and without liquid to solid phase change in either Cartesian or Polar coordinates.The problems solved include conjugate mixed convective heat transfer and natural heatconvective cooling, with and without solidification, inside inner rectangular cavities andin the annular space between horizontal concentric cylindrical containers. Results ofthe evolution of velocity, temperature and liquid-solid interface position obtained bythe classical sequential algorithm Semi-Implicit Method for Pressure Linked Equations,SIMPLE [4] and by the improved PSIMPLER algorithm, are presented and discussedfor each problem. The omission of the corrections of the velocity components of SIM-PLE, avoided in PSIMPLER and the use to a second prediction-correction step in theproposed iterative algorithm allows the use of higher values for the under-relaxation co-efficients for the dependent variables: velocity, pressure and temperature. As a result,the robustness of sequential algorithm PSIMPLER and the computational efficiency,calculated in terms of the CPU time to achieve the desired convergence, are analyzedand discussed for each one of the problems investigated.
References[1] F. Moukalled, L. Mangani, M. Darwish, The Finite Volume Method in Computational Fluid
Mechanics: An Advanced Introduction with OpenFOAM and Mathlab (Fluid Mechanics and ItsApplications), Springer, 2015.
Departamento de Ingeniería Mecánica, Universidad de La Serena, Chile, e-mail: [email protected] en Ingeniería en Alimentos y Bioprocesos, Universidad de La Serena, ChileEl trabajo es financiado por CONICYT-Chile en Proyecto FONDECYT 1140074
60
[2] H. K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics: TheFinite Volume Method, Longman Scientific Technical, 2007.
[3] N. Moraga, S. RamÃŋrez, M. Godoy, P. Ticchione, Study of convective non-Newtonian alloysolidification in molds by the PIMPLER/Finite Volume Method, Numerical Heat Transfer,Part A, 12: 936-953, 2010.
[4] S. V. Patankar, D. B. Spalding, A calculation procedure for heat, mass and momentum transferin three-dimensional parabolic flows, International Journal Heat Mass Transfer, 15: 1787-1806,1972
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On the analyses of three DG schemes for Stokesproblem based on velocity-pseudostress
formulationTomás P. Barrios Rommel Bustinza Felipe Sánchez
Abstract
In this talk we first discuss the well posedness of a modified LDG scheme of Stokesproblem, considering a velocity-pseudostress formulation. The difficulty here relies onthe fact that the application of classical Babuška–Brezzi theory is not easy, so we pro-ceed in a non standard way. For uniqueness we apply a discrete version of Fredholm’salternative theorem, while the a priori error analysis is done introducing suitable projec-tions of exact solution. As a result, we prove that the method is convergent, and undersuitable regularity assumption on the exact solution, the optimal rate of convergenceis guaranteed. Next, we propose a second discrete formulation, by adding a div-divstabilization term, which helps to guarantee its well posedness as well as the a priorierror estimates. These are done by application of standard theory. Finally, with theaim of having more freedom to choose the approximation spaces, we propose a third DGscheme, adding another least-square type term to the previous one. As consequence,the discrete space where the velocity unknown is looking for, will change. This make usto redefine one of the parameters that define the so called numerical fluxes, in contrastwith the previous ones, and we recover the classical LDG approach for Stokes problem,considering velocity and pseudostress as unknowns.
References[1] T.P. Barrios and R. Bustinza: A priori and a posteriori error analyses of an augmented
discontinuous Galerkin formulation. IMA Journal of Numerical Analysis, vol 30, 4, pp. 987-1008, (2010).
[2] B. Cockburn, G. Kanschat, D. Schötzau and C. Schwab: Local discontinuous Galerkinmethod for the Stokes system. SIAM Journal on Numerical Analysis, vol. 40, pp. 319-343,(2002).
Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción,Chile, e-mail: [email protected]
CI2MA and DIM, Universidad de Concepción, Concepción, Chile. e-mail: [email protected], Universidad de Concepción, Concepción, Chile, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Convergence of a finite volume scheme for asorption-coagulation equationErwan Hingant Mauricio Sepúlveda
Abstract
This work is devoted to the derivation and the mathematical study of a new modelfor water-soluble polymers and metal ions interactions, which are used in chemistry fortheir wide range of applications. First, we motivate and derive a model that describesthe evolution of the configurational distribution of polymers. One of the novelty re-sides in the configuration variables which consider both, the size of the polymers andthe quantity of metal ions they captured through sorption. The model consists in anon-linear transport equation with a quadratic source term, the coagulation. Then, weprove the existence of solutions for all time to the problem thanks to classical fixedpoint theory. Next, we reformulate the coagulation operator under a conservative formwhich allows to write a finite volume scheme. The sequence of approximated solutionsis proved to be convergent (toward a solution to the problem) thanks to a L1−weak sta-bility principle. Finally, we illustrate the behaviour of the solutions using this numericalscheme and we intend to discuss on the long-time behaviour.
References[1] I.S. Ciuperca, E, Hingant, L.I. Palade, L. Pujo-Menjouet, Fragmentation and
monomer lengthening of rod-like polymers, a relevant model for prion proliferation, DiscreteContin. Dyn. S. - B, vol. 17, 3, pp. 775-799, 2012.
[2] H. Hingant, M. Sepúlveda Derivation and mathematical study of a sorption-coagulationequation, Nonlinearity, vol. 28, 10, pp. 3623âĂŞ3661, 2015
[3] B. Rivas, E.D. Pereira, M. Palencia, J., Sánchez, Water-soluble functional polymersin conjunction with membranes to remove pollutant ions from aqueous solutions, Progress inPolymer Science vol. 36, 2, pp. 294-322, 2011.
UAMat, Universidad Federal de Campina Grande, Brazil, e-mail: [email protected] and DIM, Universidad de Concepción, Concepción, Chile, e-mail: [email protected]
63
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Numerical Analysis Of Non-Newtonian Flows InLiquid Foods, Solidification Casting And
Polymer Injection MoldingNelson O. Moraga Edgardo J. Tabilo
Abstract
The objective of this work is to describe the numerical analysis and solution of fluidmechanics and convective heat transfer in three industrial processes. The cases investi-gated and the numerical methods are: the Finite Element Method in the fabrication ofyoghurt in the food industry [1], the Finite Volume Method in a solidification process [2,3] and the hybrid Finite Difference-Finite Element Method in the polymer injection in-dustry [4,5]. Most of the fluids in liquid foods and in molten metals, alloys and polymersare non-Newtonians. In these applications the relation between shear stresses and de-formation rate introduces additional non-linear terms in the momentum equations thatmust be added to the non-linearity caused by the convective terms. The applicationsof non-Newtonian flows studied in this presentation are: the rotational movement ofthixotropic yoghurt in a cylindrical container, the melting of pseudo-plastic Al-1.7%Sialloy inside a square mold and injection and three-dimensional solidification of Cross-WLF polyester in a mold cooled by water. In each case the procedure starts from theconstruction of the mathematical models in terms of a system of PDEÂťs, followed bythe numerical analysis of the discrete solution obtained by the Finite Element Method,by the Finite Volume Method or by the hybrid Finite Difference-Finite Element method.Numerical convergence and solution stability obtained by successive under-relaxation ofthe dependent variables and of the effective viscosity are analyzed. Finally, a discussionof the results obtained for the variation in time and in space of the dependent variables:velocity and temperature is presented for each one of the three problems.
References[1] G. Mullineux, M. J. H. Simmons, Influence of rheological model on the processing of yoghurt,
Journal of Food Engineering, 84: 250-257, 2008.
[2] N. Moraga, R. Lemus-Mondaca, 2011, Numerical conjugate air mixed convection / non-Newtonian liquid solidification for various cavity configurations and rheological models, In-ternational Journal of Heat and Mass Transfer, 54: 5116-5125, 2011.
Departamento de Ingeniería Mecánica, Universidad de La Serena, Chile, e-mail: [email protected] en Ingeniería en Alimentos y Bioprocesos, Universidad de La Serena, ChileEl trabajo es financiado por CONICYT-Chile en Proyecto FONDECYT 1140074
64
[3] N. O. Moraga, E. F. Castillo, C. P. Garrido, Non-Newtonian annular alloy solidification inmold, Heat and Mass Transfer, 48: 1415-1424, 2012.
[4] S. C. SomÃľ, D. Delaunay, J. Jaraj, J. L. Bailleul, N. Boyard, S. Quillet, Modeling of thethermal contact resistance time evolution at polymer-mold interface during injection molding:Effect of polymersÂť solidification, Applied Thermal Energy, 84: 150-157, 2015.
[5] C. Salinas, D. Vasco, N. Moraga, 2013, Two dimensional non-Newtonian injection mouldingwith a new Control Volume FEM / Volume of Fluid, International Journal for NumericalMethods in Fluids, 71: 1509-1523, 2013.
65
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Numerical Challenges for a NumericalSimulation of a Coupled Model for Grain
Growth in 2DClaudio E. Torres Alejandro Sazo
Abstract
Mathematical analysis of numerical algorithms that model the evolution of grain struc-ture of polycrystalline materials in 2D has gained interest over the past years [1,2]. Thisinterest has been driven by disagreement between experimental data and simulated data.To create simulated data, a number of assumptions and simplifications are made. Fromthe simplifications made, two family of approaches appear: curvature driven model andvertex model. Both of them have been greatly studied but unfortunately neither ofthem has been able to fully predict correctly experimental data. In this direction, wepropose a new coupled model were we connect both approaches.
A grain structure can be defined as a collection of non-overlapping grains thatcompletely cover a bounded periodic/non-periodic 2D domain. Grains are of polygonalshapes with curved sides, such that two grains meet on a grain boundary and grainboundaries meet at grain boundary junction (also known as triple junctions).
The mathematical models are based on the total grain boundary energy:
E(t) =∑k
∫ 1
0γ(∆αk) ‖lk(s, t)‖ds,
where γ is the grain boundary energy, ∆αk is the misorientation of grains meeting at
grain boundary k, lk =d~ξk(s, t)
ds, ~ξk(s, t) is a parametric curve of the grain boundary
k. The evolution of the grain structure is modeled such that dissipation of the energy
is enforced, i.e.d
dtE(t) < 0. Thus,
d
dtE(t) = −
K∑k=1
∫ 1
0
∂
∂s(T(k)) · v(k) ds+
M∑m=1
vm ·3∑l=1
T(m,l)
which gives us the following integral-differential system of equations:
Departamento de Informática and CCTVal, Universidad Técnica Federico Santa María, Valparaíso,Chile. e-mail: [email protected]
Departamento de Informática, Universidad Técnica Federico Santa María , Valparaíso, Chile. e-mail:[email protected]
66
x(k)i (t) = µ
∫ 1
0
∂
∂s
(T(k)(s, t)
)φi(s) ds , i 6= 1, N
xm(t) = λ3∑l=1
(−T(m,l) +
∫ 1
0T(m,l)(s, t)φ1,m(s) ds
)In this talk we will briefly discuss several numerical challenges we have encounter
for the numerical simulation [3], such as: derivative of unitary vectors, convergence,among others.
keywords: mathematical modelling, numerical analysis, spectral method, grain-growth.
References[1] C. E. Torres and M. Emelianenko and D. Golovaty and D. Kinderlehrer and S. Ta’asan, Nu-
merical Analysis of the Vertex Models for Simulating Grain Boundary Networks, SIAM Journalon Applied Mathematics, Vol. 75 Nr. 2 (2015), 762-786.
[2] Kinderlehrer, David and Livshits, Irene and Ta’asan, Shlomo, A Variational Approach to Mod-eling and Simulation of Grain Growth, SIAM Journal on Scientific Computing, Vol. 28 Nr. 5(2006), 1694-1715.
[3] Trefethen, Lloyd N., Spectral methods in MatLab, Society for Industrial and Applied Mathe-matics, 2000.
67
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On Nonlinearly and Linearly Implicit IMEXRunge-Kutta methods for a class of degenerate
convection-diffusion problemsLuis-Miguel Villada Osorio
AbstractMulti-species kinematic flow models with strongly degenerate diffusive corrections giverise to systems of nonlinear convection-diffusion equations of arbitrary size. Aplicationsto these systems include models of polydisperse sedimentation and multi-class trafficflow. Implicit-explicit (IMEX) Runge-Kutta (RK) methods [1] are suitable for the solu-tion of these convection-diffusion problems since the stability restrictions, coming fromthe explicitly treated convective part, are much less severe than those that would bededuced from an explicit treatment of the diffusive term. These schemes usually com-bine an explicit Runge-Kutta scheme for the time integration of the convective partwith a diagonally implicit one for the diffusive part. In [4], a nonlinear implicit IMEX-RK scheme of this type is proposed, where the nonlinear and non-smooth systems ofalgebraic equations arising in the implicit treatment of the degenerate diffusive partare solved by smoothing of the diffusion coefficients combined with a Newton-Raphsonmethod with line search. A particularly efficient variant of these schemes, so-calledlinearly implicit IMEX-RK schemes [3], arise from discretizing the diffusion terms in away that more carefully distinguishes between stiff and nonstiff dependence, such thatin each time step only a linear system needs to be solved. In this talk a serie of exam-ples of polydisperse sedimentation [2] and multi-class traffic flow [5] it is demonstratedthat these linearly implicit IMEX-RK schemes approximate the same solutions as thenonlinearly implicit versions, and in many cases these schemes are more efficient.
This contribution is based on a serie of joint works [3, 4, 5] with R. Bürger (Univer-sidad de Concepción), P. Mulet (Universitat de València, Spain) and S. Boscarino, G.Russo (University of Catania, Italy).
References[1] U. Ascher, S. Ruuth, and R. Spiteri, Implicit-explicit Runge-Kutta methods for time-
dependent partial differential equations. Applied Numerical Mathematics, vol. 25, pp. 151–167,(1997).
[2] S. Berres, R. Bürger, K.H. Karlsen, and E.M. Tory. Strongly degenerate parabolic-hyperbolic sys-tems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math., 64 (2003),41–80.
CI2MA & GIMNAP-Departamento de Matemática, Universidad del Bío-Bío, Chile e-mail:[email protected]
68
[3] S. Boscarino, R. Bürger, P. Mulet, G. Russo and L.M. Villada. Linearly implicit IMEX Runge-Kutta methods for a class of degenerate convection-diffusion problems. SIAM J. Sci. Comput.,37 (2015), pp. B305–B331.
[4] R. Bürger, P. Mulet, and L.M. Villada. Regularized nonlinear solvers for IMEX methods appliedto diffusively corrected multi-species kinematic flow models. SIAM J. Sci. Comput., 35 (2013),B751–B777.
[5] R. Bürger, P. Mulet, and L.M. Villada. A diffusively corrected multiclass Lighthill-Whitham-Richards traffic model with anticipation lengths and reaction times. Adv. Appl. Math. Mech.,5 (2013), pp. 728-758.
69
Ecuaciones en Derivadas Parciales
Encargado de Sesión : Claudio Muñoz
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Stability of mKdV breathers in the energy spaceand numerical results
Miguel Angel Alejo
Abstract
In this talk I will show some recent results about the H1 stability of breather solutionsof mKdV. I will also present Bäcklund transformations for the mKdV and I will showsome numerical results about the study of the spectra of linearized operators.
e-mail: [email protected]
71
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Sobolev and Hardy-Sobolev type inequalitiesHernán Castro
Abstract
In this talk we ask ourself about the validity of the following family of inequalities ofthe type (∫
Ω|u(x)|qv(x)dx
)1/q
≤ C(∫
Ω|∇u(x)|pw(x)dx
)1/p
,
where v and w are weights functions and Ω is some subset of RN , usually Ω = RN ,Ω = RN+ (the half-space), or Ω = (RN )+ (the positive cone).
There are some famous weights that have been known and vastly used for a longtime, for instance v(x) = w(x) = 1 (Sobolev inequality), v(x) = |x|−q, w(x) = 1 (Hardyinequality), and v(x) = |x|b, w(x) = |x|a (Hardy-Sobolev, Caffarelli-Kohn-Nirenberg),and in all these cases we have a range of values of p and unique critical exponent q = p∗
for which the inequality holds.In a recent work, Cabré and Ros-Oton (2013) considered the case identical monomial
weights v(x) = w(x) = xa11 · . . . ·xaNN establishing the validity of such inequality in when
ai ≥ 0 for all p ≥ 1. They show that if |A| = a1 + a2 + . . .+ aN , then p∗ = N+|A|N+|A|−p .
Our goal is to extend the result of Cabré and Ros-Oton to obtain a Hardy-Sobolevtype inequality for monomial weights, that is to consider weights of the form
v(x) = xb11 · . . . · xbNN and w(x) = xa11 · . . . · x
aNN .
IMAFI, Universidad de Talca, e-mail: [email protected]
72
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On the control of the stabilizedKuramoto-Sivashinsky system by a single force
Eduardo Cerpa
Abstract
This talk presents a control problem for a one-dimensional nonlinear parabolic system,which consists of a Kuramoto-Sivashinsky (KS) equation coupled to a heat equation.We consider a distributed control force supported in an interior open subset of thedomain and acting only on either the KS equation [1] or the heat equation [2]. Thelocal null-controllability of the system is proven. The proof is based on a Carlemanestimate for the linearized system around the origin. A local inversion theorem isapplied to get the result for the nonlinear system.
References[1] E. Cerpa, A. Mercado, A. Pazoto, Null controllability of the stabilized Kuramoto-
Sivashinsky system with one distributed control, SIAM J. Control Optim., Vol. 53, No. 3, 2015,pp. 1543-1568.
[2] N. Carreño, E. Cerpa, Local controllability of the stabilized Kuramoto-Sivashinsky systemby a single control acting on the heat equation, under review.
Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile, e-mail:[email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A Dirichlet problem involving the divergenceoperator
Gyula Csató
Abstract
Consider the following classical and broadly treated problem: Given a function f onΩ ⊂ Rn, find a vector field u such that
div u = f in Ω
u = 0 on ∂Ω.
It is obvious that a necessary condition is∫
Ω f = 0. This is also a sufficient condition.Let us generalize the diffirential operator and introduce the boundary value problem
div u+ 〈a, u〉 = f in Ω
u = 0 on ∂Ω,
where a is a given vector field and 〈 , 〉 is the scalar product. What is now the necessaryand sufficient condition for solvability? What is the expected regularity result? Theanswer is easy, if a is of the special form a = gradA. We present some results andconjectures about the general case. This is joint work with B. Dacorogna appearing inthe following reference:
References[1] Csató G. and Dacorogna B., A Dirichlet problem involving the divergence operator, Ann. Inst.
H. Poincaré Anal. Non LinÃľaire, doi:10.1016/j.anihpc.2015.01.006, to appear.
Gyula Csató . e-mail: [email protected]
74
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Solvability of fractional problems withsupercritical drifts
Gonzalo Dávila
Abstract
We study the existence of viscosity solutions u : RN → R of quasilinear integro-differential equations given by the following model
(−∆)su+ g(|Du|) = f in Ω,
subject to the exterior Dirichlet condition
u = ϕ in Ωc,
with f ∈ C(Ω), ϕ ∈ C(Ωc) and bounded. Here g ∈ C1(R+) is increasing with g(0) = 0,and for t large we have g(t) ∼ tp with p ≥ 2s, that is we are in a critical or supercriticalregime.
We prove that under suitable conditions there are viscosity solutions attaining theboundary data in the classical sense. As a by product of our proof we also get boundaryregularity for the viscosity solution.
This is a joint work with Alexander Quaas (UTFSM) and Erwin Topp (UTFSM).
Universidad Santa María
75
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Formación de singularidades para el flujo demapas armónicos
Juan Dávila
Abstract
Estudiamos formación de singularidades en tiempo finito para el flujo de mapasarmónicos de un dominio en el plano a valores en la esfera 2-dimensional. Encontramosel perfil y la tasa de reviente para ciertas condiciones iniciales, y la estabilidad deeste fenómeno. Anteriormente van den Berg, Hulshof y King (2003) habían encontradoformalmente la tasa de reviente, y Raphael y Schweyer (2013) probaron estos resultadosen el caso radial en todo el plano, y para mapas con corotación 1.
Este trabajo es en colaboración con Manuel del Pino (U. de Chile) y Juncheng Wei(UBC).
Financiamiento de Fondecyt 1130360, Fondo Basal CMMMillenium Nucleus CAPDE NC130017, e-mail: [email protected]
76
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Solutions to a supercritical elliptic problemJorge Faya
Abstract
We consider the supercritical problem
−∆v = |v|p−2v in Θ, v = 0 on ∂Θ, (1)
in a bounded smooth domain Θ ⊂ RN , N ≥ 3, for p > 2NN−2 . In some cases one can
reduce this problem to a critical problem of the form
−∆u = Q(x)|u|4
n−2u in Ω, u = 0 on ∂Ω, (2)
in a domain Ω ⊂ RN of lower dimension by means of a Hopf map and some symmetryconsiderations. We shall present some existence and multiplicity results for problem (2)which will give rise to solutions of problem (1) in some particular cases. This is jointwork with Mónica Clapp and Angela Pistoia.
CMM, Universidad de Chile
77
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Entire sign changing solutions with finite energyto the fractional Yamabe equation
Danilo Garrido Monica Musso
Abstract
We are interested in the existence of finite-energy sign-changing solutions to the frac-tional Yamabe type equation in Rn,
(−∆)su = γ |u|p−1u in Rn (1)
where n ≥ 3, p is the fractional critical Sobolev exponent p = n+2sn−2s and γ > 0 is a
constant chosen for normalization purposes. For any s ∈ (0, 1), (−∆)s is the nonlocaloperator defined as
(−∆)s(x) = c(n, s)P.V.
∫Rn
u(x)− u(y)
|x− y|n+2sdy = c(n, s) lim
ε→0+
∫Rn\B(x,ε)
u(x)− u(y)
|x− y|n+2sdy,
where P.V. stands for the principal value and c(n, s) = π−(2s+n2
) Γ(n2
+s)
Γ(−s) .
Finite energy sign-changing solutions to (1), are poorly understood. The purpose ofthis paper is to give a first example of finite-energy sign-changing solutions to (1), in alldimensions n ≥ 3, and for s ∈ (1
2 , 1): we build a solution to Equation (1) which lookslike the soliton U surrounded by k negative copies U properly scaled and distributedalong the vertices of a regular polygon with radius 1. Our main result is the following
TheoremLet n ≥ 3 and s ∈ (1
2 , 1). Write Rn = C × Rn−2 and let ξkj = (e2jπik , 0), j = 1, . . . , k.
Then for any sufficiently large k there is a finite energy solution to Problem (1) of theform
uk(x) = U(x)−k∑j=1
µ−n−2s
2k U
(µ−1k (x− ξj)
)+ o(1),
where
µk =[2n−2s
2∑∞
j=1 j2s−n]−1
k2(1 + o(1))
The first author is partially supported by Mecesup Grant 0711 and VRI Scholarship, e-mail:[email protected]
Partially supported by Fondecyt Grant 1120151 and Millennium Nucleus Center for Analysis of PDE,NC130017, e-mail: [email protected]
78
References[1] L. Caffarelli L, L. Silvestre, An extension problem related to the fractional Laplacian. Commun.
Part. Diff. Eqns 32 (2007) 1245–1260.
[2] E. Carlen, M. Loss. Extremals of functionals with competing symmetries. J. Funct. Anal., 88(2)(1990) 437ÃćâĆňâĂIJ-456.
[3] A. Chang, M. González. Fractional Laplacian in conformal geometry. Adv. Math., 226(2) (2011)1410ÃćâĆňâĂIJ-1432.
[4] W. Chen, C. Li, B. Ou. Classification of solutions for an integral equation. Comm. Pure Appl.Math., 59(3) (2006) 330ÃćâĆňâĂIJ-343.
[5] J. Dávila, M. del Pino, Y. Sire. Nondegeneracy of the bubble in the critical case for nonlocalequations. Proc. Amer. Math. Soc., 141 (2013), no. 11, 3865ÃćâĆňâĂIJ-3870.
[6] E. Di Nezza, G. Palatucci, E. Valdinoci. HitchhikerÃćâĆňâĎćs guide to the fractional Sobolevspaces.Bull. Sci. Math., 136 (5), (2012) 521ÃćâĆňâĂIJ-573.
[7] M. del Pino, M. Musso, F. Pacard, A. Pistoia. Large Energy Entire Solutions for the YamabeEquation. Journal of Differential Equations 251, (2011), no. 9, 2568–2597.
[8] M. del Pino, M. Musso, F. Pacard. A. Pistoia. Torus action on Sn and sign changing solutionsfor conformally invariant equations. Annali della Scuola Normale Superiore di Pisa, Cl. Sci.(5) 12 (2013), no. 1, 209–237.
[9] F. Fang. Infinitely many non-radial sign-changing solutions for a Fractional Laplacian equationwith critical nonlinearity. ArXiv:1408.3187, 2014.
[10] R. Frank, E. Lieb. Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality.Calc.Var. Partial Differential Equations, 39(1-2) (2010), 85ÃćâĆňâĂIJ-99.
[11] R. Frank, E. Lieb. A new, rearrangement-free proof of the sharp Hardy-Littlewood- Sobolevinequality. In Spectral theory, function spaces and inequalities, volume 219 of Oper. TheoryAdv. Appl., pages 55ÃćâĆňâĂIJ67. BirkhÃĆÂĺauser/Springer Basel AG, Basel, 2012.
[12] M. González, R. Mazzeo, Y. Sire. Singular solutions of fractional order conformal Laplacians.J. Geom. Anal., 22(3) (2012) 845ÃćâĆňâĂIJ-863.
[13] M. González, J. Qing. Fractional conformal Laplacians and fractional Yamabe problems. Anal-ysis and PDE, to appear.
[14] Y. Li. Remark on some conformally invariant integral equations: the method of moving spheres.J. Eur. Math. Soc. (JEMS), 6(2) (2004) 153ÃćâĆňâĂIJ180.
[15] Y. Li, M. Zhu. Uniqueness theorems through the method of moving spheres. Duke Math. J.,80(2) (1995) 383ÃćâĆňâĂIJ417.
[16] E. Lieb. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. ofMath. (2), 118 (2), (1983) 349ÃćâĆňâĂIJ- 374.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Multiplicity of solutions for some semilinearproblems involving nonlinearities with zeros
Leonelo Iturriaga Jorge García-Melián
Abstract
In this talk we consider the semilinear elliptic problem−∆u = λf(u) in Ωu = 0 on ∂Ω
where f is a nonnegative, locally Lipschitz continuous function with r positive zeros, Ωis a smooth bounded domain and λ > 0 is a parameter. We show that for large enoughλ there exist 2r positive solutions, irrespective of the behavior of f at zero or infinity,provided only that f verifies a suitable non integrability condition near each of its zeros,thereby generalizing previous known results. The construction of the solutions rely onthe sub and supersolutions method and topological degree arguments, together withthe use of a new Liouville theorem which is an extension of recent results to this typeof nonlinearities.
References[1] S. Alarcón, J. García-Melián, A. Quaas, Optimal Liouville theorems for supersolutions
of elliptic equations involving the Laplacian, submitted for publication.
[2] S. Alarcón, L. Iturriaga, A. Quaas, Existence and multiplicity results for Pucci’s oper-ators involving nonlinearities with zeros. Calc. Var. Partial Differential Equations 45 (2012),no. 3-4, 443–454.
[3] S. N. Armstrong, B. Sirakov, Nonexistence of positive supersolutions of elliptic equationsvia the maximum principle, Comm. Part. Diff. Eqns. 36 (2011), 2011–2047.
[4] B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear ellipticequations, Comm. Pure Appl. Math. 34 (1981), 525–598.
Departamento de Matemática, Universidad Técnica Federico Santa María. e-mail:[email protected]
Departamento de Análisis Matemático, Universidad de La Laguna.C/. Astrofísico Francisco Sánchez s/n, 38271 – La Laguna, SPAINandInstituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica,Molecular y Fotónica, Facultad de Física, Universidad de La Laguna, e-mail: [email protected]
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[5] B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,Comm. Partial Differential Equations 6 (1981), 883–901.
[6] L. Iturriaga, S. Lorca, E. Massa, Positive solutions for the p-Laplacian involving criticaland supercritical nonlinearities with zeros. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010),763–771.
[7] L. Iturriaga, E. Massa, J. Sánchez, P. Ubilla, Positive solutions of the p-Laplacianinvolving a superlinear nonlinearity with zeros, J. Diff. Eqns. 248 (2010), 309–327.
[8] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev.24 (4) (1982), 441–467.
[9] A. Quaas, B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci’sOperator, Comm. Partial Differential Equations 31 (2006), 987–1003.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Kink dynamics in the φ4 model: asymptoticstability for odd perturbations in the energy
spaceMichal Kowalczyk
Abstract
We consider a classical equation
φtt − φxx = φ− φ3, (t, x) ∈ R× R
known as the φ4 model in one space dimension. The kink, defined byH(x) = tanh(x/√
2),is an explicit stationary solution of this model. From a result of Henry, Perez andWreszinski it is known that the kink is orbitally stable with respect to small perturba-tions of the initial data in the energy space. In this paper we show asymptotic stabilityof the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptoticstability of solitons for the generalized Korteweg-de Vries equations. However, this ap-proach has to be adapted to additional difficulties, pointed out by Soffer and Weinsteinin the case of general nonlinear Klein-Gordon equations with potential: the interactionsof the so-called internal oscillation mode with the radiation, and the different rates ofdecay of these two components of the solution in large time.
e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Singularly perturbed PDEs and patterns withperiodic profiles
Fethi Mahmoudi
Abstract
We consider a class of singularly perturbed equations in planar domains: as the singularperturbation parameter tends to zero, we exhibit a family of solutions concentratingat the boundary with asymptotically periodic profile. As solutions with uniform pro-file at the boundary were known to exist, the result here reflects the phenomenon ofTuring’s instability, which triggers formation of inhomogeneous structures from morehomogeneous ones.
CMM and Universidad de Chile [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Some results for a problem from CombustionAlejandro Omón Arancibia
Abstract
This work presents results concerning the Perturbed Gelfand Problem, given by
∂v
∂t= ∆v + λ ev/(1+ε v) in ]0, T ) × Ω, (1)
v(0, x) = v0(x) in Ω,
v = 0 on ]0, τ) × ∂ Ω,
and an Arrhenius problem of the type
∂v
∂t= ∆v + λ (1 + εv)q ev/(1+ε v) in ]0, T ) × Ω, (2)
v(0, x) = v0(x) in Ω,
v = 0 on ]0, τ) × ∂ Ω
with q > 0 and q < 0. The sign of q, including zero, depends on the character of thechemical reaction under modelling, as both (1) and (2) are motivated by CombustionTheory, see [5] or [11]. In particular, when ε 0, it is recovered the classical GelfandProblem.
Within the results to present there are multiplicity ones in the case of thesteady problems, and blow-up results for the evolution problem. Numerical resultswill be also presented as part of the study, in particular bifurcation diagrams and alsocomputation of the Morse Index.
References[1] E. Ash, B. Eaton, K. Gustafson: Counting the number of solutions in combustion and reactive
flow problems; Z. angew. Math. Phys., vol. 41 (1981), pp. 558-578.
[2] J. Bebernes, D. Eberly: Mathematical Problems from Combustion Theory ; Applied Mathemat-ical Sciences vol. 83, Springer-Verlag (1989).
[3] R. Bellman, J. Bentsman, S. Meerkov: Vibrational control of systems with Arrhenius dymanics;J. Math. Ann. Appl. vol 91 (1983), pp. 152-191.
[4] W. Börsch-Supan: On the stability of bifurcation branches in thermal ignition; Z. angew. Math.Phys., vol. 35 (1984), pp. 332-344.
Dirección de Investigación-UFRO, e-mail: [email protected]
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[5] J. Buckmaster, G.S.S. Ludford: Theory of Laminar Flames; Cambridge Monographs on Me-chanics and Mathematics, Cambridge University Press (1982).
[6] Y. Du, Y. Lou: Proof os a conjecture for the perturbed Gelfand Equation from Combustiontheory ; J. Diff. Equations, vol. 173 (2001), pp. 213-230.
[7] Y. Du: Exact multiplicity and S-shape bifurcation curve for some semilinear elliptic problemsfrom combustion theory ; SIAM J. Math. Anal., vol. 32-4 (2000), pp. 707-733.
[8] G. Gavallas: Nonlinear Differential Equations of Chemically Reacting Systems; Springer Tracksin Natural Philosophy vol. 17 (1969).
[9] V. Giovangigli: Modélisation numérique de la chimie complexe; Images del Mathématiques,Modélisation de la Combustion, edited by H. Berestycki, C-M- Brauner, P. Clabin, C. Schidt-Lainé, CNRS-France (1996).
[10] A. Kapila: Asymptotic Treatment of Chemically Reacting Systems; Applied Mathemtical SeriesPITMAN (1983).
[11] A. Liñán, F. Williams: Fundamental Aspects of Combustion; The Oxford Engineering ScienceSeries, vol. 33 (1993), Oxford University Press.
[12] P.-L. Lions: Asymptotic behavior of some nonlinear heat equations; Physica 5D (1982), pp.293-306.
[13] D.H. Sattinger: Topics in Stability and Bifurcaton Theory ; Lectures Notes in Mathematics,vol. 309 (1973), Springer-Verlag.
[14] K. Taira, K. Umezu: Semilinear elliptic boundary-value problems in chemical reactor theory ;J. Diff. Eqns., vol. 142 (1998), pp. 434-454.
[15] K. Taira: Semilinear elliptic boundary-value problems in combustion theory ; Proc. Royal Soc.Edimburgh, vol. 132A (2002), pp. 1453-1476.
[16] H. Wiebers: S-shape bifurcation of nonlinear elliptic boundary value problems; Math. Ann. 270(1985), pp. 555-570.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Desigualdades de restricción de Fourier:existencia y no existencia de máximos y cálculo
de la mejor constanteRené Quilodrán
Abstract
La transformada de Fourier de una función integrable en Rd es una función continua quetiende a cero en infinito y por lo tanto tiene sentido restringirla a una variedad S, comola esfera, el paraboloide, etc., mientras que para una función de L2 la transformada deFourier puede ser cualquier función de L2 y por lo tanto no tiene sentido, en general,restringirla a un conjunto de medida cero. De aquí nace la pregunta de si es posible,en algún sentido, restringir la transformada de Fourier de una función de Lp, parap ∈ [1, 2], más precisamente, si existe una desigualdad
‖f∣∣S‖Lq ≤ C‖f‖Lp .
Hablaremos de esta desigualdad, su relación con desigualdades de Strichartz para EDPsclásicas, de la mejor constante C y de la existencia y no existencia de funciones quemaximizan ‖f
∣∣S‖Lq/‖f‖Lp .
Universidad de los Lagos, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Stationary harmonic functions whose Laplacianis a Radon measure
Rémy Rodiac
Abstract
Let Ω ⊂ R2 be a bounded set. In this talk I will study local regularity properties ofRadon measures µ and functions h ∈ H1(Ω) such that
∆h = µ (1)
and the quantityωh := (∂xh)2 − (∂yh)2 − 2i(∂xh)(∂yh) (2)
is holomorphic in Ω. This problem is related to limiting vorticities measures of Ginzburg-Landau system as shown by Sandier and Serfaty. It is also linked to vorticity measureof the time independent Euler system in fluid mechanics and to limiting vorticity mea-sures of system of point vortices. I will prove that, locally, near almost every point ofthe domain, h can be written as h = |H| for some smooth harmonic function H. Inparticular we deduce that the measure ∆h is concentrated on lines, which are sets ofzeros of harmonic functions.
e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Fractional mean curvature flowMariel Sáez Enrico Valdinoci
Abstract
In this talk I will discuss a fractional analog to the classical mean curvature flow.Namely, we consider the evolution of surfaces with normal speed equal to the fractionalmean curvature and analyze their behavior under suitable assumptions. I will discussin more depth the evolution of graphical hyper-surfaces, which is an important modelin the local case.
This is joint work with Enrico Valdinoci
Pontificia Universidad Católica de Chile, partially funded by Fondecyt 1150014 e-mail:[email protected]
Weierstrass Institut für Angewandte Analysis und Stochastike, e-mail:[email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Moderate solutions of semilinear ellipticequations with Hardy potential
PHUOC-TAI NGUYEN
Abstract
Let Ω be a bounded smooth domain in RN . We study positive solutions of equation(E) −Lµu+ uq = 0 in Ω where Lµ = ∆ + µ
δ2, µ > 0, q > 1 and δ(x) = dist (x, ∂Ω). A
positive solution of (E) is moderate if it is dominated by an Lµ-harmonic function. Ifµ < CH(Ω) (the Hardy constant for Ω) every positive Lµ- harmonic functions can berepresented in terms of a finite measure on ∂Ω via the Martin representation theorem.However the classical measure boundary trace of any such solution is zero. We introducea notion of normalized boundary trace by which we obtain a complete classification ofthe positive moderate solutions of (E) in the subcritical case, 1 < q < qµ,c. (The criticalvalue depends only on N and µ.) For q ≥ qµ,c we show that there exists no moderatesolution with an isolated singularity on the boundary.
Pontificia Universidad Catolica de Chile, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On the Cp′-regularity conjectureEduardo Teixeira
Abstract
C1,α regularity estimate for solutions of the p-Poisson equation with bounded source,∆pu = f(x) ∈ L∞, has been well established since early 1980’s. Determining theoptimal (universal) Hölder exponent α has been a tantalizing question since then, withmajor implications to the PDE theory and its applications. The explicit example,∆p(|x|p
′) = cte sets the ground for what has been termed the Cp′-regularity conjecture:
The optimal Hölder continuity exponent for the gradient of a function whose p-laplacianis bounded is 1
p−1 , at least when p > 2. In this talk I will discuss a recent proof ofthe Cp′-regularity conjecture in dimension two. This is a joint work with Araujo andUrbano.
Universidade Federal do Ceara, Brasil, e-mail: [email protected]
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Geometría
Encargado de Sesión : Maximiliano Leyton
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Linear systems on IHS manifoldsMichela Artebani
Abstract
A classical result in the theory of surfaces is that any complex K3 surface S which hasan ample and base point free class h ∈ H2(S,Z) with h2 = 2 is a double cover of theprojective plane branched along a smooth sextic curve [5]. By a result of Beauvilleand Fujiki [1, 3] the cohomology group H2(X,Z) of a holomorphic symplectic manifold(or IHS) has a lattice structure given by a non-degenerate quadratic form qX whichgeneralizes the intersection pairing for K3 surfaces. It is thus natural to ask for ageometric characterization of IHS manifolds carrying a class h ∈ H2(X,Z) with qX(h) =2. In [4] O’Grady conjectured that general IHS should behave as K3 surfaces, moreprecisely that if X is a generic deformation of the Hilbert scheme S[r] for some K3surface S, equipped with an ample class h with qX(h) = 2, then h is base point freeand the associated morphism φh : X → PN is a double cover.
In a joint work in progress with Samuel Boissière and Alessandra Sarti we areexploring this problem in case X = S[2], where S is a projective K3 surface of Picardnumber one. A nice example by Beauville shows that in this case O’Grady conjectureis false [2]. In this talk I will give a short introduction to IHS manifolds and to theseproblems and conjectures on their linear systems.
References[1] Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential
Geom. 18, no. 4, 755–782.
[2] Beauville, A., Some remarks on Kähler manifolds with c1 = 0, Classification of algebraic andanalytic manifolds (Katata, 1982), 1983, 1–26.
[3] Fujiki, A., On the de Rham cohomology group of a compact Kähler symplectic manifold, Alge-braic geometry, Sendai, 1985 (Advanced Studies in Pure Mathematics 10), ed. T. Oda, NorthHolland, 1987, 105–165.
[4] O’Grady K.G., Involutions and linear systems on holomorphic symplectic manifolds, Geom.Funct. Anal. 15 (2005), no. 6, 1223Ð1274.
[5] Saint-Donat, B., Projective models of K3 surfaces, Amer. J. Math. 96 (1974) 602–639.
Proyecto Fondecyt Regular N. 1130572, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Group actions on Riemann surfaces up totopological equivalence.
Antonio Behn
Abstract
Let G be a finite group acting via holomorphic maps on a compact Riemann surfaceX of genus g ≥ 2 with signature (0;m1, . . . ,mr). By Riemann’s Existence Theorem,this action is described by a generating vector ν = (g1, . . . , gr) of elements in G havingproduct 1 and gi of order mi. Conversely, orbits by the action of Aut(G) × Br on theset of generating vectors corresponding to a fixed signature give rise to topologicallynon-equivalent actions, called Hurwitz equivalence classes. To produce families of cov-erings, and to study their images in different moduli spaces, is of interest for manyproblems in algebraic geometry. Therefore to compute Hurwitz equivalence classes isan experimental tool that is required in different settings.
Working with Anita Rojas and Camila Muñoz we developed scripts in SAGE thatcompute the orbits of generating vectors for a given group and signature. These scriptswere part of Camila’s Master Thesis. We show several applications, including a list ofall the group actions on genus 5, up to topological equivalence.
These and other related SAGE programs can be found at http://geometry.uchile.cl
References[1] A. Behn, R. E. Rodriguez and A. M. Rojas, Adapted hyperbolic and symplectic representations
for group actions on Riemann surfaces. Journal of Pure and Applied Algebra 217 (2013), 409-426. https://sites.google.com/a/u.uchile.cl/polygons/
[2] A. Behn, C. Muñoz and A. M. Rojas, Classification of topologically non-equivalent actionsusing generating vectors, a SAGE Package.
Partially supported by Fondecyt Grant 1140507, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Acción de grupos en superficies y variedadesabelianas
Angel Carocca
Abstract
En esta charla abordaremos diferentes aspectos de acciones de grupos finitos en super-ficies de Riemann y variedades abelianas, con interés en las representaciones linealesinducidas por tales acciones. Presentaremos algunos resultados sobre la acción linealinducida en el espacio de Riemann-Roch.
Universidad de La Frontera, Temuco, Chile. Parcialmente financiado por ACT1415 y FONDECYTe-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Sobre la existencia de acciones de gruposelementales en Superficies de Riemann
Mariela Carvacho
Abstract
Estudiamos acciones de Zkp, el p−grupo elemental abelidano de rango k, en superficiesde Riemann compactas de género g > 1. Por simplicidad tomamos p = 2 y asumimosque la acción tiene R ≥ 3 puntos de ramificación sobre la esfera. En esta charlamostraremos condiciones numeéricas necesarias y suficientes para construir vectoresgeneradores asegurando la existencia de tales acciones.Este es un trabajo en conjunto con Anthony Weaver.
e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Dual families of Calabi-Yau varietiesPaola Comparin
Abstract
In a joint work with Michela Artebani (Universidad de Concepción) and Robin Guilbot(University of Warsaw) we present a duality between families of Calabi-Yau hyper-surfaces in Q-Fano toric varieties with canonical singularities. This is based on thedefinition of good pairs of polytopes. We also show how this duality generalizes bothBatyrev and Berglund-Hübsch-Krawitz (BHK) constructions of mirror Calabi-Yau va-rieties.
Among the constructions of mirror pairs of Calabi-Yau varieties, we consider the dualitypresented by Batyrev in [2] and the construction by Berglund, Hübsch and Krawitz [3] ,[5](BHK for short).
Batyrev in [2] provides pair of families of Calabi-Yau hypersurfaces in Fano toric varieties.Let ∆ be a reflexive polytope and let X∆ be the associated variety, which is Fano. Thefamily F(∆) of anticanonical hypersurfaces of X∆ is a family of Calabi-Yau varieties. If∆ is reflexive, the same holds for the polar polytope ∆∗, so that the construction can berepeated to obtain a dual family F(∆∗) of anticanonical Calabi-Yau hypersurfaces in X∆∗ .Moreover, Batyrev proves that the general members X, X∗ of the dual families F(∆) andF(∆∗) satisfy the topological mirror test:
hp,qst (X) = hn−p,qst (X∗)
where n is the dimension of X and X∗ and hp,qst denotes the string-theoretic Hodge numbers.Another well-known construction of mirror pairs of Calabi-Yau hypersurfaces is due to
Berglund and Hübsch [3] and was later refined by Krawitz [5]. In this case, a Calabi-Yauvariety is defined by a transverse polynomial W in P(w) = P(w0, . . . , wn) and a symplecticgroup G. The number of monomials inW is chosen equal to the number of variables (Delsartetype) so that W can be taken of the form
W =n∑i=0
n∏i=0
xaijj , aij ∈ Z≥0
and this defines a square matrix AW = (aij) of exponents which is assumed invertible overQ. Then one takes the quotient of W = 0/G. Under the assumption that the total degreed of W is d =
∑ni=0wi, this construction provides a Calabi-Yau hypersurface in P(w)/G.
Taking the transposed matrix ATW and with a suitable definition of transposed group GT
Universidad de Concepción, Proyecto Postdoctorado Fondecyt 3150015, e-mail: [email protected]
96
(see [5]), one can repeat the construction obtaining a dual hypersurface. Chiodo and Ruan[4] prove that such a construction gives pairs of Calabi-Yau hypersurfaces which satisfy thetopological mirror test on Hodge numbers.
In [1] we present a duality of Calabi-Yau hypersurfaces in Q-Fano toric varieties withcanonical singularities (w.c.s) which generalizes the two constructions shown above. Thekey definition is the following:Definition. Let ∆1,∆2 be two polytopes. The pair (∆1,∆2) is a good pair if ∆1,⊂ ∆2 and∆1 and ∆∗2 are canonical polytopes, i.e the origin is their only interior lattice point.
Observe that, if (∆1,∆2) is a good pair, the same hold for the polar pair (∆∗2,∆∗1).
Definition 1, together with the following Theorem, allows to define families of Calabi-Yauvarieties in Q-Fano toric varieties w.c.s.Theorem 1. If X is a Q-Fano toric varieties w.c.s. and ∆ is a canonical polytope containedin the anticanonical polytope of X, let F(∆) be the family of anticanonical hypersurfaceswhose Newton polytope is ∆. Then the general element in F(∆) is a Calabi-Yau variety.
If (∆1,∆2) is a good pair and F(∆1) is the family of anticanonical hypersurfaces in X∆2
having ∆1 as Newton polytope, then by the theorem the general member of F(∆1) is Calabi-Yau. Similarly, the general element in F(∆∗2) is a Calabi-Yau hypersurface inX∆∗1
. Of course,observe that if ∆1 = ∆2, the duality of good pairs coincides with Batyrev construction.
We can now prove that the duality of good pairs is a generalization of BHK construction.In fact, in [1] we define a generalized version of the BHK construction, meaning that weallows polynomials that are not of Delsarte type in toric varieties that are Q-Fano w.c.s.Associated to a matrix and a group (AW , G), we construct a pair of polytopes (∆1,∆2). Thesame can be done for the trasponsed pair (ATW , G
T ), obtaining the pair (∆T1 ,∆
T2 ). Theorem
2 shows that the duality of good pairs is a generalization of the generalized version of theBHK construction.Theorem 2. In the previous setting, the pair (∆1,∆2) is a good pair and (∆T
1 ,∆T2 ) is the
polar pair (∆∗2,∆∗1).
References[1] M. Artebani, P. Comparin, R. Guilbot, Families of Calabi-Yau hypersurfaces
in Q-Fano toric varieties, arXiv:1501.05681.
[2] V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric va-rieties, J. Algebraic Geom. 3 (1994), no. 3, 493–535.
[3] P. Berglund, T. Hübsch, A generalized construction of mirror manifolds, Nuclear Phys. B 393(1993), no. 1-2, 377–391.
[4] A. Chiodo, Y. Ruan, LG/CY correspondence: the state space isomorphism, Adv. Math 227(2011), no. 6, 2157–2188.
[5] M. Krawitz, FJRW rings and Landau-Ginzburg mirror symmetry, ProQuest LLC, Ann Arbor,MI, 2010. Thesis (Ph.D.) - University of Michigan.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On singular varieties with smooth subvarietiesM. R. Gonzalez-Dorrego
Abstract
Let k an algebraically closed field, char k = 0. Let Z be a reduced irreducible nonsin-gular subvariety of a normal n-fold X with certain type of singularities, such that Zintersects Sing(X). We study the singularities of X through which Z passes.
Universidad Autónoma de Madrid, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
The 4-prims family.Víctor González Aguilera Gustavo Labbé Morales
Abstract
LetMg be the moduli space of smooth curves of genus g and Mg be the set of stablecurves. The set Mg can be endowed with a structure of projective complex variety andcontainsMg as a dense open variety.
The 4-prism is a stable graph thus it determine a terminal stable curve (or a nodedRiemann surface) in M5, therefore it is a limit of a 1-dimensional family of smoothalgebraic curves (or Riemann surfaces) in M5. In this short note we give a Fuchsiandescription and an algebraic description of this family. We also describe the otherspoints of this family that belong to M5 −M5.
References[1] Bers, L.On spaces of Riemann surfaces with nodes. Bulletin of the AMS. 80, Number 6,
1219-1222, 1974.
[2] Costa, A. and González Aguilera, V. Limits of equisymmetric 1-complex dimensionalfamilies of Riemann surfaces. Preprint 2015.
[3] Deligne, P. and Mumford, D. The irreducibility of the space of curves. PublicationsMathématiques de L’I.H.E.S. 36 , 75-109, 1965.
e-mail: [email protected]: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Automorphisms of non-cyclic p-gonal surfacesR. A. Hidalgo A. F. Costa
Abstract
In this talk I will be concerned with holomorphic automorphisms of a non-cyclic p-gonalRiemann surface S of genus g > (p − 1)2, where p ≥ 3 is a prime integer. We will seethat the order of any automorphism is bounded above by 2(g + p − 1) and that thismaximal order is attained for infinitely many genera. This generalizes the similar resultfor the particular case p = 3 recently obtained by Costa-Izquierdo in [2]. Moreover, wealso observe that the full group of holomorphic automorphisms of S is either the trivialgroup or is a finite cyclic group or a dihedral group or one of the Platonic groups A4,A5 and Σ4. Examples in each case exists for infinitely many values of g. In the casethat S admits a holomorphic automorphism of order 2(g + p − 1), then its full groupof automorphisms is the cyclic group generated by it and every p-gonal map of S isnecessarily simple.
As a first consequence, we note that each pair (S, π), where S is a Riemann surfaceof genus g ≥ 2 and π is a non-cyclic p-gonal map, can be defined over its field of moduli.Also, if g > (p− 1)2 and the group of automorphisms of the non-cyclic p-gonal surfaceS is different from a non-trivial cyclic group, then S can be also be defined over its fieldof moduli.
A second consequence is that every dessin d’enfant of prime degree is definable overits field of moduli.
References[1] A. F. Costa and R. A. Hidalgo. Automorphisms of non-cyclic p-gonal surfaces. In preparation.
[2] A. F. Costa and M. Izquierdo. Maximal order automorphisms of trigonal Riemann surfaces. J.Algebra 323 (2010), 27–31.
Partially supported by FONDECYT 1150003, e-mail: [email protected]: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Lines on cubic hypersurfaces over finite fieldsAntonio Laface
Abstract
Let Fq be the finite field with q elements and let X be a smooth n-dimensional cubichypersurface of Pn+1
Fq . We address the problem of deciding if X contains a line definedover the base field Fq. If n = 2 and there exists a non-cube a ∈ Fq, then the surface ofequation x3
1 + x32 + x3
3 + ax34 = 0 does not contain lines. If n ≥ 3 we show that smooth
cubic hypersurfaces contain a line in each of the following cases:
• n = 3 and q ≥ 11;
• n = 4, and q = 2 or q ≥ 5;
• n ≥ 5.
The proof for n = 3 or 4 is more interesting and goes as follows. Let F (X) be the Fanoscheme of lines of X. Using a recent formula of Galkin–Shinder [GS] which relates thenumber of Fq-points on F (X) with the number of Fq- and Fq2-points on X we findthe zeta function of F (X). Then we make use of Weil conjectures [D1, D2] to showthat X always contains Fq-lines when q ≥ 11 and n = 3 or q ≥ 5 and n = 4. UsingMagma [BC], we produce examples of smooth cubic threefolds containing no lines forq ∈ 2, 3, 4, 5 leaving only the cases where q ∈ 7, 8, 9 open, at least when X issmooth.
If X is a threefold which admits mild singularities, i.e. one singular point of typeA1 or of type A2, the geometry of its Fano scheme F (X) is closely related to that ofa smooth genus-4 curve ([CG, KvG]; see also [GS, Example 5.8]). Using the results of[HLT] on pointless curves of genus 4, we prove that X always contains Fq-lines whenq ≥ 4 and produce examples for q ∈ 2, 3 where X contains no Fq-lines.
This is joint work [DLR] with O. Debarre and X. Rolleau.
References[BC] Bosma, W., Cannon, J., Playoust, C., The Magma algebra system. I. The user language, J.
Symbolic Comput. 24 (1997), 235–265,
[CG] Clemens, C.H., Griffiths, P.A., The intermediate Jacobian of the cubic threefold, Ann. ofMath. 95 (1972), 281–356.
[DLR] Debarre, O., Laface, A., Roulleau, X., Lines on cubic hypersurfaces over finite fields eprintarXiv:1510.05803
Proyecto Fondecyt regular n. 1150732, e-mail: [email protected]
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[D1] Deligne, P., La conjecture de Weil I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–308.
[D2] Deligne, P., Théorie de Hodge II, Publ. Math. Inst. Hautes Études Sci. 40 (1972), 5–57.
[GS] Galkin, S., Shinder, E., The Fano variety of lines and rationality problem for a cubic hyper-surface, eprint arXiv:1405.5154v2
[HLT] Howe, E.W., Lauter, K.E., Top, J., Pointless curves of genus three and four, in Arithmetic,geometry and coding theory (AGCT 2003), 125–141, Sémin. Congr. 11, Soc. Math. France,Paris, 2005.
[KvG] Kouvidakis, A., van der Geer, G., A note on Fano surfaces of nodal cubic threefolds, inAlgebraic and arithmetic structures of moduli spaces, Sapporo 2007, 27–45, Adv. Stud. PureMath. 58, Math. Soc. Japan, Tokyo, 2010.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Authomorphims of graphs and Riemann surfacesAlexander D. Mednykh
Abstract
We give a short survey of old and new results about automorphism groups and branchedcoverings of graphs. The latter notion was introduced independently by T. D. Parsons,T. Pisanski, P. Jackson (1980), H. Urakawa (2000), B. Baker, S. Norine (2009) andothers. The branched covering of graphs are also known as harmonic maps or verticallyholomorphic maps of graphs. The main idea of the present talk is to create a paral-lel between classical results on branched covering of Riemann surfaces and those forgraphs. We introduce the notion of harmonic action on a graph and discuss the Hur-witz type theorems for the groups acting harmonically. These results can be regardedas discrete analogues of the well known theorems by Hurwitz and Accola-Maclachlan.They, respectively, give sharp upper and lower bounds for the order of an automorphismgroup acting on a Riemann surface.
We present discrete versions of theorems by Wiman (1895), Oikawa (1956) andArakawa (2000), which sharpen the Hurwitz upper bound for various classes of groupsacting on a Riemann surface of given genus.
Then we define a hyperelliptic graph as two fold branched covering of a tree anda γ- hyperelliptic graph as two fold branched covering of a graph of genus γ. A fewdiscrete versions of the well-known results on γ-hyperelliptic Riemann surface will begiven.
Sobolev Institute of Mathematics, Novosibirsk, and Institute of Mathematics, Siberian Federal Univer-sity, Krasnoyarsk, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On Jacobian of circular graphsIlya A. Mednykh
Abstract
We consider Jacobians of graphs as discrete analogues of Jacobians of Riemann surfaces.More precisely, Jacobian of graph is an Abelian group generated by flows satisfyingthe first and the second Kirchhoff rules. One also can define a circulant graph asthe Cayley graph of a cyclic group. The family of circulant graphs is quite wide.It includes complete graphs, cyclic graphs, antiprism graphs, even prism graphs andMoebius ladder graph. We propose a new method to find the structure of Jacobiansfor a large subfamily of circulant graphs.
Sobolev Institute of Mathematics, Novosibirsk, and Institute of Mathematics, Siberian Federal Univer-sity, Krasnoyarsk, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Curvas de Tipo Fermat y sus JacobianasJaime Pinto
Abstract
En esta charla se hablará sobre cierto tipo de superficies de Riemann compactas conacciones de grupos de la forma G = Zn o Zn, donde n es una potencia de primoimpar. Mencionaremos una representación de estas superficies como curvas planasafines cíclicas n-gonales, y estudiaremos la acción de G en sus variedades Jacobianasasociadas, mostrando las representaciones irreducibes complejas de G para determinaruna descomposición isógena de dichas Jacobianas.
References[1] E. Bujalance, F.J. Cirre, M. Conder, On Extendability of Group Actions on Compact
Riemann Surfaces, Trans. Amer. Math. Soc. 355 (2003), 1537-1557.
[2] H. Lange, S. Recillas, Abelian Varieties with group action, J. Reine Angew. Math. 575(2004), 135Ű155.
[3] A. Rojas, Group Actions on Jacobian Varieties, Rev. Mat. Iberoamericana 23 (2007), 397-420.
[4] J-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, 1977.
[5] A. Wooton, Defining equations for cyclic prime covers of the Riemann Sphere, Israel Journalof Mathematics 157 (2007) ,103-122.
U. de Chile. e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Automorphisms Of Pseudo-Real RiemannSurfaces
S. Quispe R. A. Hidalgo
Abstract
Let S be a pseudo-real Riemann surface of genus g with conformal automorphism groupAut(S) and let H be a subgroup of Aut(S) so that S/H has genus zero.
In this talk, we state that if H is unique within its signature (i.e., for every subgroupK of Aut(S) isomorphic to H and with S/K of the same signature as S/H it holds thatH = K), then the number of cone points of S/H of the same order is even, and thecocient group Aut(S)/H is either trivial or cyclic. This result is applied to pseudo-realhyperbolic generalized Fermat curve of type (k, n) [2, 3, 4], also this results generalizesthe result obtains by E. Bujalance and A. Costa for Cyclic p-gonal pseudo-real Riemannsurfaces in [1].
References[1] E. Bujalance and A. F. Acosta. Automorphism Groups of cyclic p-gonal pseudo-real Riemann
surfaces. http://arxiv.org/abs/1503.04139
[2] G. González-Diez, R. A. Hidalgo and M. Leyton. Generalized Fermat curves. Journal of Algebra321, 2009, 1643–1660.
[3] R. A. Hidalgo. Non-hyperelliptic Riemann surfaces with real field of moduli but not definableover the reals. Archiv der Mathematik 93 (2009), 219–222.
[4] R. A. Hidalgo, A. Kontogeorgis, M. Leyton and P. Paramantzoglou. Automorphisms of theGeneralized Fermat Curves. http://arxiv.org/abs/1409.3063
[5] R. A. Hidalgo and S. Quispe. Automorphisms of pseudo-real Riemann surfaces. Preprint 2015.
Partially supported by Project FONDECYT 3140050, e-mail: [email protected] supported by Project FONDECYT 1150003, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Superficies Algebraicas: Uniformización yAritmeticidad
Sebastián Reyes-Carocca
Abstract
Una variedad algebraica (compleja, proyectiva y lisa) X es aritmética si existen poli-nomios homogéneos con coeficientes algebraicos definiendo una variedad isomorfa a X.El teorema de Belyi [1] caracteriza todas las curvas algebraicas (equivalentemente, su-perficies de Riemann compactas) que son aritméticas. Para el caso dos-dimensional,si X es una fibración de Kodaira entonces la aritmeticidad está completamente deter-minada por (la clase de isomorfía) de su cobertor universal [2]. Generalizando estehecho, en esta charla veremos cómo para una superficie algebraica la propiedad de seraritmética puede ser distinguida en el cobertor universal, cuando ésta corresponde alespacio total de una familia de superficies de Riemann de tipo finito. Luego aplicaremoslo anterior para proveer una caracterización de aritmeticidad para cualquier superficiealgebraica de tipo general en términos de los cobertores universales de sus abiertos deZariski.
Éste es un trabajo en conjunto con Gabino González-Diez.
References[1] Belyi, G., On Galois extensions of a maximal cyclotomic field, Math. USSR Izv. 14 (1980),
247-256.
[2] González-Diez, G., y Reyes-Carocca, S., The arithmeticity of a Kodaira fibration isdetermined by its universal cover, Comment. Math. Helv. 90 (2015), 429-434.
Partially supported by Spanish MEyC Grant MTM 2012-31973, Becas Chile and Universidad de LaFrontera, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Fixed points and rational representations ofactions in abelian varieties
Rubí E. Rodríguez
Abstract
In this talk we will present some known and new results on the relations between fixedpoints of an endomorphism of an abelian variety and its rational representation.
We will also discuss the natural generalization to the rational representation of agroup or a Hecke algebra acting on an abelian variety.
Universidad de La Frontera, Temuco, Chile. Partially supported by ACT1415 and FONDECYT e-mail:[email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Familias de Jacobianas completamentedescomponibles y subvariedades especiales de
Ag.
Anita M. Rojas
Abstract
Sea G un grupo finito actuando en género g con firma m = [0; s1, . . . , sr], y vectorgenerador θ = (g1, . . . , gr). Para un par fijo (m, θ), y moviendo los puntos rama en P1
del cubrimiento total π : X → P1 = X/G, se obtiene una familia de dimensión (r − 3)de tales cubrimientos, junto a una familia J (G,m, θ) de la misma dimensión de lascorrespondientes variedades Jacobianas JX. Para referencias vea [2] o [5]. Por otrolado, el grupo simpléctico Sp(2g,Z) actúa en el semiespacio superior de Siegel Hg, yAg = Sp(2g,Z) \ Hg es un espacio analítico complejo que parametriza las variedadesabelianas principalmente polarizadas de dimensión g módulo isomorfismo.
Denote por Z(G,m, θ) la clausura de J (G,m, θ) en Ag. La acción de G en X, yen su Jacobiana JX, induce una representación simpléctica ρ : G → Sp(2g,Z) de G.Sea HG
g el conjunto de puntos fijos de G en Hg (vea [1] para más detalles). En [2,Thms. 1.4, 3.9, Lemma 3.8] hay una caracterización simple de condiciones bajo lascuales Z(G,m, θ) es una subvariedad especial. Su criterio es como sigue, si la dimensiónde HG
g es igual a la dimensión de J (G,m, θ), la cual es r − 3, entonces Z(G,m, θ) esuna subvariedad especial de Ag que está contenida en la clausura Tg del lugar de Torelli(o Jacobiano), and que intersecta no trivialmente al lugar de Torelli T 0
g .Adicionalmente, en [3, Question 6.6] los autores preguntan por subvariedades espe-
ciales de Tg de dimensión positiva, tales que la variedad abeliana correspondiente alpunto genérico es isógeno a un producto de curvas elípticas; esto es, completamentedescomponible.
En [4] encontramos varios ejemplos de variedades Jacobianas completamente des-componibles, incluyendo varias familias. Tenemos así un amplio escenario donde buscarsubvariedades especiales de Ag: Dado un par (m, θ) para un G fijo, usando [1] se puedecalcular la dimensión de HG
g , aunque es computacionalmente caro, además se conocela dimensión de la familia J (G,m, θ). Si ambas coinciden, la clausura de dicha familiatendrá las propiedades deseadas.
En esta charla explicaremos los conceptos involucrados en estas preguntas y mostra-remos ejemplos de subvariedades especiales correspondientes a familias de variedadesJacobianas completamente descomponibles, estas fueron desarrolladas en trabajo con-junto con Jennifer Paulhus de Grinnell College y se utiliza la técnica de descomposiciónsegún el álgebra de grupo.
Fondecyt Regular 1140507, e-mail: [email protected]
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References[1] A. Behn, R. E. Rodríguez, A.M. Rojas, ‘Adapted Hyperbolic Polygons and Symplectic Rep-
resentations for group actions on Riemann surfaces’, J. Pure Appl. Alg. 217 (2013) 409–426.http://www.geometry.uchile.cl
[2] P. Frediani, A. Ghigi, M. Penegini, ‘Shimura varieties in the Torelli locus via Galois coverings’,Geometriae Dedicata (2015) 1–16.
[3] B. Moonen , F. Oort, ‘The Torelli locus and special subvarieties’, Handbook of Moduli, 2 (2011)549–594.
[4] J. Paulhus , A. M. Rojas, ‘Completely decomposable Jacobian varieties in new genera’, Preprint(2015).
[5] H. Völklein, ‘Groups as Galois groups’, Cambridge studies in advanced mathematics 53. Cam-bridge University Press, 1996.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Small degree covers and reducible hyperplanesections
Andrea L. Tironi
Abstract
Let X be a smooth complex projective variety with dimX ≥ 4 and let L be an ampleline bundle on X. Assume that there is a reducible divisor A = A1 + ... + Ar ∈ |L|,r ∈ Z≥1, such that there exist finite surjective morphisms πi : Ai → Yi of degree di ≤ 3for every i = 1, ..., r. We classify pairs (X,L) as above under the assumption that Yi is avery special complex projective manifold with Pic(Yi) ∼= Z[OYi(1)] for every i = 1, ..., r.
References[1] A.L. Tironi, Varieties of Picard rank one as components of ample divisors, Osaka J. Math. 52
(2015), no. 3, 601–616.
This work is partially supported by Proyecto VRID N. 214.013.039-1.OIN, e-mail: [email protected]
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Modelos Matemáticos de
Sistemas Biológicos
Encargado de Sesión : Fernando Córdova
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Medidas de conservación ex situ de tipoimpulsivo: Un enfoque metapoblacional a través
del modelo clásico de LevinsSandra Araya Crisóstomo Héctor Rojas-Castro
Abstract
La Biología de Metapoblaciones estudia los efectos de la dinámica local de pobla-ciones sobre la persistencia regional de una especie y, por lo tanto, ha alcanzado sumáxima aplicación en la Biología de la conservación (Hanski y Simberloff, 1977), ver[2].
Consideremos una metapoblación, que como tal, se encuentra distribuida en frag-mentos o parches, ver [3], [5]. Suponiendo que las tasas de colonización y de extinciónson parámetros constantes que no se ven afectados por las propiedades que determinadoparche pueda tener, entonces la dinámica de la metapoblación puede modelarse a travésdel modelo planteado por Richard Levins en su trabajo “Some demographic and geneticconsequences of environmental heterogeneity for biological control", ver [4]:
u′(t) = mu(t)
1− u(t)
v
− g u(t), (1)
donde u corresponde a la cantidad de parches ocupados, v al total de parches que esposible ocupar y m y g son las tasas de colonizacón y extinción respectivamente.
Para este trabajo, supondremos que el modelo (1) describe la dinámica de unametapoblación que está extinguiéndose (m < g) y que con el propósito de evitar quela población total se extinga completamente, se aplica como medida de conservación lacolonización artificial de K parches cada τ unidades de tiempo, es decir, cada ciertacantidad de tiempo (fija y determinada con anterioridad) se colonizarán K parchesvacíos con la cantidad necesaria de individuos, para que éste se considere como ocupado(Conservación Ex-Situ, ver [1]).
Matemáticamente, la incorporación de esta medida de conservación puede expresarsemediante el sistema de ecuaciones diferenciales impulsivo:
u(t) = mu
1− uv
− g u, si t 6= k τ,
u(t+) = u(t) +K, si t = k τ.(2)
A partir del sistema (2), se obtienen y presentan resultados analíticos y de simu-lación, a fin de establecer la dinámica de largo plazo.
Universidad Católica del Maule, e-mail: [email protected] Católica del Maule, e-mail: [email protected]
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References[1] Conabio. (2009). Capital natural de México, vol. II : Estado de conservación y tendencias de
cambio. Comisión Nacional para el Conocimiento y Uso de la Biodiversidad, México.
[2] Hanski, I. y Simberloff, D. (1997). The metapopulation approach, its history, conceptual domain,and application to conservation. En Metapopulation biology: ecology, genetics, and evolution.Academic Press, San Diego, USA.
[3] Hanski, I. (1999). Metapopulation ecology. Oxford University Press, New York, USA.
[4] Levins, R. (1969). Some Demographic and Genetic Consequences of Environmental Hetero-geneity for Biological Control. Bulletin of the Entomological Society of America. Vol 15, pp:237-240.
[5] McCullough, D. (1996). Metapopulations and Wildlife Conservation. Island Press, Washington.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Consecuencias sobre la abundancia poblacionaldel Efecto Allee en hábitats bajo fragmentación
Rodrigo Del Valle Fernando Córdova-Lepe
Abstract
En este trabajo deducimos un modelo de crecimiento poblacional de una especieel cual considera una fragmentación progresivo de su hábitat [3, 5, 6], proceso quefinalmente divide la población en dos parches. La novedad con respecto a Del Valle &Córdova-lepe [2] es asumir la presencia de una relación positiva entre la tasa per cápitade crecimiento y la densidad poblacional a bajas densidades, esto es, un Effecto Allee,ver [1, 4]. Este efecto, por sí solo aumenta las posibilidades de extinción de la especieen cuestión. Nuestro interés está en estudiar las consecuencias sobre la dinámica dela abundancie de la concurrencia simultánea de Efecto Allee y la fragmentación, estaúltima en el sentido de sin pérdida de hábitat.
References[1] Berec, L., D. S. Boukal, and M. Berec 2001, Linking the Allee effect, sexual reproduction and
temperature-dependence sex determination via spatial dynamics, American Naturalist Vol. 157,217-230.
[2] Del Valle, R. & Córdova-Lepe, F. A mathematical model of a single population with habitatfragmentation in progress, Proceeding of CMMSE 2014,Vol II, 429-440, 2014.
[3] Fahrig L. Efects of habitat fragmentation on biodiversity, Annual Review of Ecology, Evolutionsand Systematics, 34, 487-515, 2003.
[4] Fowler, M. S. and G. D. Ruxton, 2002, Population dynamic consequences on Allee effects,Journal of Theoretical Biology, Vol. 215, 39-46.
[5] Hurrison, S. & Bruna, E. Habitat fragmentation and large scale conservation: what do we knowfor sure?, Ecography, 22, 225-232, 1999.
[6] K.W. Herbener, S.J. Tavener & N.T. Hobbs: The distinct efects of habitat fragmentation onpopulation size, Theor. Ecol., 5, 73-82, 2012.
Universidad Católica del Maule, Proyecto Interno 434171 (2015-2016). e-mail: [email protected] Católica del Maule, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Control epidemiológico optimal porhospitalización impulsiva
M. Eugenia Solís Fernando Córdova-Lepe
Abstract
Este trabajo formula una problemática de control optimal para la erradicación deuna enfermedad contagiosa desde un sistema productivo de animales (granja). Se con-sidera que un brote de una enfermedad tipo SIS (Susceptibles -Infecciosos-Susceptibles)es controlada mediante una estrategia denominada hospitalización impulsiva, ver [3].Esto es, una secuencia de remociones temporales de una fracción (ρk en la k–ésimahospitalización) de los animales infecciosos hacia sitios de aislamiento y cuidados espe-ciales, para en un tiempo posterior retornar al grupo susceptible. El problema centrales encontrar la sucesión ρk que minimiza las pérdidas. El contexto matemático es lamaximización de un funcional de una variable ligada a una ecuación diferencial impul-siva.
References[1] Diekmann and Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Build-
ing, Analysis and Interpretation, Wiley, Chichester. (2000).
[2] Shulaw and Bowman On-Farm Biosecurity: Traffic Control and Sanitation. Extension Fact-Sheet, Veterinary Preventive Medicine 6 (2001).
[3] Córdova-Lepe, Del-Valle and Solís. Impulsive hospitalization: Epidemiological control on farms.In Proceedings of 2013 CMMSE. Cabo de Gata, AlmerÃŋa, Spain June 24-27, 2013. V2, pp.444-455. I. P. Hamilton & J. Vigo-Aguiar Eds. (2013).
Universidad Católica del Maule, e-mail: [email protected] Católica del Maule, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Dinámica de la distribución genotípica bajomortalidad diferenciada por rasgos fenotípicos
Héctor Rojas-Castro Fernando Córdova-Lepe
Abstract
Este trabajo considera una población mendeliana, cerrada, dipliode y panmíctica, lacual admite un modelo de crecimiento malthusiano. Si A y a son los dos posibles alelospara un locus determinado, compartimentamos la población en tres tipos de individuossegún su genotipo AA, Aa y aa para el locus en estudio. Denotamos por PAA(·), PAa(·)y Paa(·) las abundancias de las respectivas subpoblaciones genotípicas y por P (·) laabundancia de la población total.
Mediante las frecuencias genotípicas, es decir, x(·) = PAA(·)/P (·), y(·) = PAa(·)/P (·)y z(·) = Paa(·)/P (·), se definen las frecuencias alélicas o frecuencias génicas, fA(·) yfa(·), como la proporción que se observa de un alelo específico respecto al conjunto de losque pueden ocupar un locus determinado en la población. Entoces, fA(t) = x(t)+y(t)/2y fa(t) = z(t) + y(t)/2, t ≥ 0.
Supongamos que la población es un recurso (e.g., pesquero) de consumo humano.Asuminos que se alterna periodos de veda con instantes de captura en los cuales ciertaporción es capturada. La cantidad de individuos capturados por unidad de tiempo esdistinta en cada una de las subpoblaciones genotípicas, pues expresan una característicafenotípica que facilita o dificulta la captura. En [5] se describen ejemplos en que lacaptura de algún recurso pesquero ha implicado una variación en la diversidad genéticade la población. Por ejemplo, se comenta el caso del Coregonus lavaretus en el lagoFemund de Noruega, ver [4], que presenta tres formas distintas, y una estrategia puntualde pesca aumentó la capturabilidad de los individuos de una de estas tres formas, loque ocacionó una disminución de estos individuos y afectó la variabilidad genética dela población total.
Consideramos captura impulsiva, ver [2, 3], esto es, ocurre en una secuencia crecientede instantes tk igualmente espaciados, i.e., tk+1 − tk = τ , cierto τ > 0, para cadak ≥ 0. Además, en cada momento de captura y para cada subpoblación genotípica locosechado por unidad de biomasa es proporcional al esfuerzo de pesca ejercido (E), esdecir, la Hipótesis de Schaefer, pero con capturabilidad diferenciada qi, i ∈ 1, 2, 3.
Universidad Católica del Maule, e-mail: [email protected] Católica del Maule, e-mail: [email protected]
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En resumen, tenemos el modelo diferencial impulsivo siguiente:
P ′AA (t ) = α f2A(t)P (t)− µPAA(t),
P ′Aa (t ) = 2α fA(t)fa(t)P (t)− µPAa(t),P ′a a (t ) = α f2
a (t)P (t)− µPa a(t).
t 6= tk,
PAA(t+) = (1− q1E)PAA(t),PAa(t
+) = (1− q2E)PAa(t),Pa a(t
+) = (1− q3E)Pa a(t),
t = tk,
(1)
donde α es la tasa de natalidad y µ la tasa de mortalidad natural.
Se presentan resultados de umbral (analíticos y de simulación) para resolver ladinámica de largo plazo del sistema (1), los cuales son interpretados en el contextoespecificado de diversidad genética final.
References[1] Clark, C. (1990). Mathematical Bioeconomics: The Optimal Management of Renewable Re-
sources. John Wiley and Sons.
[2] Córdova, F., Pinto, M. (2002). Mathematical Bioeconomics. Explotation of resources andpreservation. Cubo Mat. Educ., 239, 49–54.
[3] Córdova, F., Del Valle, R., Robledo, G. (2012). A pulse fishery model with closures asfunction of the catch: Conditions for sustainability. Mathematical Biosciences, 239, 169–177.
[4] Sandlund, O., Naesje, T. (1989). Impact of a pelagic gill-net fishery on the polymorphicwhitefish (Coregonus lavaretus L.sl.) population in Lake Femund, Norway. Fisheries Research,7, 85–97.
[5] Smith, P. (1994). Genetic diversity of marine fisheries resources: possible im-pacts of fishing. FAO Fisheries Technical Paper, 344, 53p. Disponible en:http://www.fao.org/docrep/003/V4865E/V4865E00.HTM
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Observaciones a la aproximación de L.A. Segelpara las ecuaciones del Sistema
Ligando-ReceptorFernando Córdova-Lepe
Abstract
Todos los organismos a diferentes escalas de organización presentan sistemas sen-soriales bastantes comparables. Estos son sistemas que responden a señales, pero quepresentan adaptación cuando están expuestos a estos estímulos por un tiempo prolon-gado, esto es, las respuestas del sistema deben decrecer en magnitud o simplemente noexistir.
En este trabajo realizamos algunas observaciones al modelo de adaptación exactade Segel et al [1] y pretendemos una versión impulsiva del mismo. Nuestra hipótesis esque la aproximación con esta técnica matemática facilita el análisis del modelo.
En nuestro sistema, tipo Receptor-Ligando, concurren dos estados libres, un re-ceptor R y su covalente D, junto con sus correlativos estados ligados RL y DL. Lasreacciones que representan la asociación-disociación del ligando L (verticales), se con-sideran comparativamente más rápidas que las reacciones de conversión entre estadoslibres o entre estados ligados (horizontales), con ambas definidas por leyes de acción demasas simples. La naturaleza híbrida está en que: (a) la impulsividad (pulso) se incor-pora en la modelización de las reacciones verticales y (b) la escala ordinaria (continua)en las reacciones horizontales.
References[1] Segel, L.A., Goldbeter, A., Devreotes, P.N., & Knox, B.E. (1986). A mechanism for exact
sensory adaptation based on receptor modification. J. theor. Biol. 120: 151-179.
[2] Acerenza, L., Arocena, M., Graña, M. & Ortega, F. (2002).Modelos modulares de procesos celu-lares. Procesos biofísicos complejos. Simposio sobre complejidad biológica. Julio A. Hernándezand Andrés Pomo editores.DIRAC Fcultad de Ciencias Universidad de la República.
[3] Arocena, M. & Acerenza, L. (2004). Necessary conditions for a minimal model of receptor toshow adaptative response over a widw range of levels of stimulus. J. Theor. Biol. 229: 45-57.
Universidad Católica del Maule. e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Neurodidactics: Analysis of Cellular NeuralNetwork Models
Kuo-Shou Chiu Fernando Córdova-Lepe
Abstract
Cellular neural networks [9,10] have been extensively studied in past years and foundmany applications in different areas such as pattern recognition, associative memory,and combinatorial optimization. Such applications heavily depend on the dynamicalbehaviors. It is well known that studies on neural dynamical systems not only involvediscussion of stability property, but also involve other dynamics behaviors such as pe-riodic solution, bifurcation, chaos et al. Recently, the properties of periodic oscillatorysolutions are of great interest because one has been found many network such as hu-man brain are often in periodic oscillatory or even chaotic state. Many results forthe existence of their periodic solutions and the exponential convergence properties forcellular neural networks have been reported in the literatures. See, for instance, Refs.[2,7,8,11,12,15] and references cited therein.
Most neural networks can be classified into two types, continuous or discrete. How-ever, many real world systems and natural processes cannot be categorized into one ofthem. They display characteristics both continuous and discrete styles. For instance,some biological neural networks in biology, bursting rhythm models in pathology andoptimal control models in economics are characterized by abrupt changes of state. Theseare the familiar impulsive phenomena.
Other examples can also be found in information science, electronics, automaticcontrol systems, computer networking, artificial intelligence, robotics, and telecommu-nications, etc. Such a kind of phenomena, in which sudden and sharp changes oftenoccur in a continuous process, which cannot be well described by pure continuous orpure discrete models. Therefore, it is important and, in effect, necessary to study anew type of neural networks - neural networks with piecewise constant arguments -as an appropriate description of these phenomena. See, for instance, Refs. [1,6] andreferences cited therein.
Differential equations with piecewise constant arguments (briefly DEPCA) arise inan attempt to extend the theory of functional differential equations with continuousarguments to differential equations with discontinuous arguments. This task is of con-siderable applied interest, because DEPCA can be seen in many of the phenomena inthe real world. These phenomena may often be modeled by piecewise constant systemswith corresponding differential equations containing piecewise constant arguments.
Partially supported by FIE 11-14 DIUMCE, Departamento de Matemática, Facultad de Ciencias Bási-cas, Universidad Metropolitana de Ciencias de la Educación, Santiago, Chile, e-mail: [email protected],[email protected]
120
Let Z, N and R be the set of all integer, natural and real numbers, respectively. Fixa real sequence ti, i ∈ Z, such that ti < ti+1 for all i ∈ Z, ti → ±∞ as i → ±∞. Letγ : R → R be a step function given by γ(t) = ti for t ∈ Ii = [ti, ti+1) and consider theDEPCA with this general γ. In this case we speak of DEPCA of general type, in shortDEPCAG. The theory of DEPCAG has been developed by few authors [3-6,13,14].
The main purpose in this talk is to study stability of periodic solutions for neuralnetworks with a general piecewise constant argument and impulse effects (IDEPCAG):
xi(t) = −aixi(t) +n∑j=1
bijfj(xj(t)) + cijgj(xj(γ(t)))+ di
where n denotes the number of neurons in the network, xi(t) corresponds to the stateof the i–th unit at time t, fj(xj(t)) and gj(xj(γ(t))) denote, respectively, the measuresof activation to its incoming potentials of the unit j at time t. Moreover, ai > 0 andbij , cij , Ii are positive real numbers; bij denotes the synaptic connection weight of theunit j on the unit i at time t, cij denotes the synaptic connection weight of the unit jon the unit i at time γ(t), di is the input from outside the network to the unit i.
By using a fixed point theorem combined with Green’s function and some analysistechniques, some new sufficient conditions are obtained ensuring existence, uniquenessand global exponential stability of periodic solution of nonautonomous cellular neuralnetworks with piecewise constant argument of generalized type. The results given hereextend and improve the earlier publications. Moreover, connection with the neuro-didactics will be discussed.
References[1] M. U. Akhmet and E. Yilmaz, Impulsive Hopfield type neural network system with piecewise
constant argument, Nonlinear Analysis: Real World Applications, 11 (2010) 2584-2593.
[2] J. Cao, Global asymptotic stability of neural networks with transmission delays., InternationalJournal of Systems Science, 31 (2000) 1313-1316.
[3] Kuo-Shou Chiu and M. Pinto, Periodic solutions of differential equations with a general piece-wise constant argument and applications, E. J. Qualitative Theory of Diff. Equ., 46 (2010),1-19.
[4] Kuo-Shou Chiu and M. Pinto, Variation of parameters formula and Gronwall inequality fordifferential equations with general piecewise constant arguments, Acta Math. Appl. Sin. Engl.Ser., 27, No. 4 (2011), 561-568.
[5] Kuo-Shou Chiu, Stability of oscillatory solutions of differential equations with a general piece-wise constant argument, E. J. Qualitative Theory of Diff. Equ., 88 (2011), 1-15.
[6] Kuo-Shou Chiu and F. Córdova, Stability of periodic solutions for neural networks with ageneral piecewise constant argument. In preparation.
[7] S. Xu, Y. Chu and J. Lu, An analysis of global asymptotic stability of delayed cellular neuralnetworks, IEEE Transactions on Neural Networks, 13 (2002) 1239-1242.
[8] S. Xu, Y. Chu and J. Lu, Global exponential stability of delayed Hopfield neural networks, 14(2001) 977-980.
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[9] L.O. Chua and L. Yang, Cellular neural networks: theory., IEEE Trans. Circuits Systems 35(10) (1988) 1257-1272..
[10] L.O. Chua and L. Yang, Cellular neural networks: theory., IEEE Trans. Circuits Systems 35(10) (1988) 1273-1290.
[11] H. Huang, J. Cao and J. Wang, Global exponential stability and periodic solutions of recurrentneural networks with delays, Physics Letters A, 298 (2002) 393-404.
[12] S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-timecellular neural networks with delays, Applied Mathematics and Computation, 135(2003), 17-38.
[13] M. Pinto, Asymptotic equivalence of nonlinear and quasilinear differential equations with piece-wise constant arguments, Mathematical and Computer Modelling, 49 (2009), 1750-1758.
[14] M. Pinto, Cauchy and Green matrices type and stability in alternately advanced and delayeddifferential systems, J. Difference Eqs. Appl., 17 (2)(2011), 235-254.
[15] S. Xu, Y. Chu and J. Lu, New results on global exponential stability of recurrent neural networkswith time-varying delays, Physics Letters A, 352 (2006) 371-379.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A vaccine-age structured model to study theeffect of a pre-erythrocytic vaccine on malaria
prevalenceKatia Vogt Geisse Calistus Ngonghala Zhilan Feng
Abstract
A deterministic compartmental malaria model will be presented to study the effectsof a pre-erythrocytic vaccine on malaria dynamics in Africa. Malaria is a parasiticdisease transmitted to humans through bites of infected mosquitoes and represents animportant international health problem. The model divides the human population intosusceptible, infected and vaccinated individuals, separating the latter into two vacci-nated classes, the first consisting of individuals that received initial vaccination dosesand the second of those who received an additional booster dose. All epidemiologicalclasses vary with chronological time and vaccinated individuals in both compartmentsare additionally structured by vaccine-age. A vaccine-age dependent transition ratebetween vaccinated classes makes it possible to model a minimum vaccine-age requiredfor receiving the booster dose. The mosquito population is divided into two compart-ments: susceptible and infected. The model consists of a coupled system of ordinarydifferential equations and first order partial differential equations. An expression forthe control reproduction number R will be derived. The local and global stability ofthe disease free equilibrium, conditions for existence of endemic equilibria, as well asthe dynamics of the system under different vaccination policies will be discussed. Inparticular, the model exhibits backward bifurcation dynamics, indicating that R = 1is no longer the threshold value for disease eradication.
References[1] E. Bache et al., “Efficacy and safety of RTS, S/AS01 malaria vaccine with or without a booster
dose in infants and children in Africa: final results of a phase 3, individually randomised,controlled trial,” Lancet, vol. 386, no. 9988, 2015.
[2] S. Mandal and R. Sarkar and S. Sinha “Mathematical models of malaria-a review,” MalariaJournal, vol. 10, no. 202, 2011.
Universidad Adolfo Ibáñez, e-mail: [email protected] of Global Health and Social Medicine, Harvard Medical School, e-mail:
[email protected] of Mathematics, Purdue University, e-mail: [email protected]
123
[3] C. M. Kribs-Zaleta and J. X. Velasco-Hernández, “A simple vaccination model with multipleendemic states,” Mathematical biosciences, vol. 164, no. 2, pp. 183–201, 2000.
[4] P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold en-demic equilibria for compartmental models of disease transmission,” Mathematical Biosciences,vol. 180, pp. 29–48, 2002.
[5] F. Brauer and C. Castillo-Chavez, “Mathematical models in population biology and epidemi-ology,” Springer, 2011.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Modelación del cambio en la interacción depoblaciones biológicas. Estudio de un caso
Marcelo E. Alberto et al.
Abstract
Se estudia un caso de cambio en el tipo de interacción de comensalismo a amensalismo.Se utiliza una variante del modelo de cambio en la interacción presentado por Hernán-dez [1] y Hernández y Barradas [2]. x′(t) = r1x(t)
1− x(t)
K1
y′(t) = r2y(t)
1− y(t)
K2+ bx(t)−x2(t)
1+cx2(t)x(t)K2
En un cultivo de vid se considera el sistema agronómico de dos componentes: vid y
verdeo; se realizan mediciones de biomasa de ambos componentes y se estima estadís-ticamente el tipo de interacción entre ambas por Regresión de Poisson. Los datos sontambién utilizados para la estimación de los parámetros del modelo matemático en elcaso estudiado. Se presenta además el análisis de estabilidad del modelo.
References[1] Hernández M.J., Barradas I. Variation in the outcome of population interactions: bifurcations
and catastrophes. - J. Math. Biol. 46, 571-594. 2003
[2] Hernández M. J. - Dynamics of transitions between population interactions: a nonlinear inter-action alpha-function defined. Proc. R. Soc. Lond. - Vol. 265 no. 1404. 1433-1440. 1998
[3] Gillman, M., Hails, R. An introduction to ecological modelling: putting practice into theory.Oxford: Blackwell Scientific Publications - 1997
[4] May, R. M. Models for two interacting populations. InTheoretical ecology: principles andapplications, 2nd edn (ed. R. M. May), pp. 78-104. Sunderland, MA: Sinauer - 1981.
[5] Addicott, J. F. - Stability properties of 2-species models of mutualism: simulation studies.Oecologia 49, 42-49 - 1981
[6] Wolin, C. L., Lawlor, L. R. - Models of facultative mutualism: density eÂąects. Am. Nat. 124,843-862 - 1984 .
Bageta C.R. - Cecconato A. - Nodaro V. - Bevaqua A. - Garriga M. - Tirador M. - Sartor, C. -Quiroga,R. Fac. Ciencias Agrarias. Fac. Ciencias Exactas. Universidad Nacional de Cuyo, Argentina. e-mail:[email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Un Modelo Estocástico de Biorrectar deAutociclado
Ana Venegas Ricardo Castro Fernando Córdova
Abstract
En el presente trabajo se analizará un modelo de biorreactor en donde la tasa de con-sumo es perturbada por un ruido estocástico. El modelo estudiado corresponde a unode autociclado con una acción de vaciado - llenado rápida, lo anterior cuando el sustratoalcanza un cierto nivel poblacional predeterminado. Las preguntas matemáticas se cen-trarán en demostrar existencia y unicidad del proceso solución del modelo y tambiénen el estudio probabilístico de la distribución de los tiempos de vaciado.
References[1] Brown WA, Cooper DG. Self-cycling fermentation applied to antibacterial oxaloacetic RAG-1.
Applied and Environment Microbiology 1991; 57:2901-2906.
[2] Brown WA. The self-cycling fermentor: development, applications, and future opportunities.Recent Research Developments in Biotechnology & Bioengineering 2001; 4:61-90.
[3] Van Walsum GP, Cooper DG. Self-cycling fermentation in a stirred tank reactor. Biotechnologyand Bioengineering 1993; 42:1175-1180.
[4] Samoilenko AM, Perestyuk NA. Impulsive Differential Equations. World Scientific: River EdgeNJ, 1995.
[5] Smith RJ, Wolkowicz GSK. Analysis of a model of the nutrient driven self-cycling fermentationprocess. Dynamics of Continuous, Discrete & Impulsive Systems. Series B. Applications &Algorithms 2003; 11:239-265.
[6] Cordova-Lepe F. Advances in a theory of impulsive differential equations at impulsive-dependent times. In BIOMAT 2006, International Symposium on Mathematical and Compu-tational Biology, Mondaini RP, Dilao R (eds).World Scientific: Hackensack NJ, 2007; 343-357.
[7] Øksendal, Bernt. Stochastic differential equations. Springer Berlin Heidelberg, 2003.
Universidad del Bío-Bío. e-mail: [email protected] Tecnológica Metropolitana [email protected] Católica del Maule e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Dinámica de un modelo tritrófico con unarespuesta funcional monotónica no-diferenciable
Viviana Rivera Pablo Aguirre
Abstract
En este trabajo se estudia un modelo que describe la interacción de tres especies.Es un sistema tritrófico: la especie 1 es depredador (generalista) de la especie 2 y3, mientras que la especie 2 es depredador (específico) de la especie 3, siguiendo unarespuesta funcional no diferenciable del tipo Holling II [2].El modelo está dado por un sistema de ecuaciones diferenciales ordinarias, el cualse estudia cualitativamente: se demuestra la unicidad de soluciones en el sistema nodiferenciable, la existencia de una bifurcación de Hopf [4] y una bifurcación de HopfGeneralizada [3] . Respaldados por la teoría de bifurcaciones explicaremos los cambiostopológicos del modelo, los cuales representan situaciones o fenómenos concretos en ladinámica poblacional [1]. Los resultados analíticos son complementados con un análisisnumérico de bifurcaciones, realizado con el paquete Matcont [3].
References[1] E. Sáez and E. González-Olivares, Dynamics on a predator-prey model, SIAM Journal
of Applied Mathematics 59, 1999, pp. 1867-1878.
[2] P. Turchin, Complex population dynamics. A theoretical/empirical synthesis, Mongraphs inPopulation Biology 35, Princeton University Press, 2003.
[3] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory Second Edition, Springer.
[4] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcationof Vector Fields. Second Edition, 1985.
Programa de Incentivos a la Iniciación Científica (PIIC), DGIP USM. e-mail:[email protected]
e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Dinámica en el modelo de depredación deHolling-Tanner considerando interferencia entre
los depredadoresAdrián Cecconato Eduardo González-Olivares
Abstract
En este trabajo se analiza el modelo predador-presa Holling-Tanner modificado [7, 8],al suponer que las funciones de crecimiento poblacional de ambas especies son del tipologístico [7, 9] y que la capacidad de soporte de la especie depredadora es proporcionalal tamaño poblacional de las presas Asimismo, se asume para el modelo propuestoque el depredador es especialista y que su respuestas funcional es hiperbólica, un tipoparticular de las respuestas funcionales del tipo Holling II [9]
La característica distintiva que el presente desarrollo pretende modelizar está aso-ciada a la competencia intraespecífica de los individuos de la especie depredadora almomento de atacar una presa, fenómeno que es referido como interferencia entre losdepredadores [5].
El modelo es descrito por el siguiente sistema de ecuaciones diferenciales autónomasdel tipo Kolmogorov [4]:
Xµ :
dxdt = r
(1− x
K
)x− qx
x+ayp
dydt = s
(1− y
nx
)y
donde, x = x (t) e y = y (t) representan los tamaños poblacionales de las presasy los depredadores respectivamente, en función del tiempo, para t ≥ 0 (medidos endensidad, biomasa o cantidad de individuos).
Por su parte, los parámetros son todos positivos, con µ = (r,K, q, a, s, n, p) ∈ R+6 ×
]0, 1[ y ellos tienen diferentes significados ecológicos [1]Para simplificar los cálculos se utiliza un sistema de ecuaciones diferenciales topológi-
camente equivalente al original [6], determinando una región de invarianza, probando elacotamiento de las soluciones y determinando condiciones para la existencia de puntosde equilibrio al interior del primer cuadrante, y que el sistema puede presentar una, doso tres singularidades positivas.
Los resultados obtenidos serán comparados con los resultados obtenidos en [2] y [8]
Instituto Superior del Profesorado San Pedro Nolasco, Universidad del Aconcagua, Mendoza, Argentina.e-mail: [email protected]
Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Chile. e-mail:[email protected]
128
References[1] C. Arancibia-Ibarra and E. González-Olivares, A modified Leslie-Gower predator-prey model
with hyperbolic functional response and Allee effect on prey, In R. Mondaini (Ed.) BIOMAT2010 International Symposium on Mathematical and Computational Biology, World Scienti.c Co.Pte. Ltd., Singapore, 2011 146-162.
[2] M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-preymodel with modi.ed Leslie-Gower and Holling-type II schemes, Applied Mathematics Letters,16 (2003) 1069-1075.
[3] C. Chicone, Ordinary differential equations with applications (2nd edition), Texts in AppliedMathematics 34, Springer 2006.
[4] H. I. Freedman, Deterministic mathematical models in Population Ecology, Marcel Dekker, 1980.
[5] H.I. Freedman, Stability analysis of a predator-prey system with mutual interference and densitydependent death rates, Bulletin of Mathematical Biology 41 (1979) 67-78.
[6] E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma and J. D. Flores, Dynamical complexitiesin the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, AppliedMathematical Modelling 35 (2011) 366-381.
[7] R. M. May, Stability and complexity in model ecosystems, Princeton University Press 1974.
[8] E. Sáez and E. González-Olivares, Dynamics on a predator-prey model. SIAM Journal of AppliedMathematics 59 (1999) 1867-1878.
[9] P. Turchin, Complex population dynamics. A theoretical/empirical synthesis, Mongraphs in Pop-ulation Biology 35 Princeton University Press 2003.
129
LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Modelling and stability analysis of a microalgalpond with nitrification
F. Mairet H. Ramírez C A. Rojas-Palma
Abstract
Microalgae culture fed with ammonium may face the presence of nitrifying bacteria.The aim of this manuscript is to propose and analyze a nonlinear system which rep-resents the dynamics of these two species (microalgae and nitrifying bacteria) in com-petition for nitrogen (present as ammonium and nitrate produced by nitrification) ina continuous process. The existence and local stability of system equilibria is studied.Reduction by conservation principle, perturbed systems and Lyapunov methods areused to provide sufficient conditions for the global asymptotic stability of the systemequilibria. Finally, we illustrate our analysis with a case study, showing which operatingconditions (dilution rate and pond depth) can promote the presence of nitrifiers withmicroalgae.
References[1] J. Alex, H. Pastagiya, and N. Holm. First results of the development of a combined
high rate biomass-algal model for wastewater treatment applications. In Proceedings of 2ndIWA/WEFWastewater Treatment Modelling Seminar, Mont-Sainte-Anne, Canada, 2010.
[2] R. A. Armstrong and R. McGehee. Competitive Exclusion. The American Naturalist,115(2):151–170, 1980.
[3] D. Arrowsmith and C. Place. An Introduction to Dynamical Systems. Cambridge UniversityPress, 1990.
Inria BIOCORE, BP93, 06902 Sophia-Antipolis Cedex, France email:[email protected]@inria.fr.
Centro de Modelamiento Matemático (CNRS UMI2807), FCFM, Universidad de Chile, Chile email:[email protected], [email protected]
Departamento de Ingeniería Matemática, FCFM, Universidad de Chile email:[email protected]@dim.uchile.cl
Ecole Doctoral I2S, Université de Montpellier, Place Eugéne Bataillon 34095, Montpellier, France.Partially supported by CONICYT Anillo ACT 1106-ACPA (H. Ramírez), ANR-14-CE04-0011 Phycover
(F. Mairet), CONICYT doctoral grant and CONICYT PAI/ Concurso Nacional Tesis de Doctorado en laEmpresa, convocatoria 2014, 781413008 (A. Rojas-Palma), the BASAL Project (Centro de ModelamientoMatemático, Universidad de Chile), Math-Amsud N 15 MATH-02 and project BIONATURE of CIRIC,INRIA-Chile.
130
[4] M. M. Ballyk, C. C. McCluskey, and G. S. K.Wolkowicz. Global analysis of competition for per-fectly substitutable resources with linear response. Journal of Mathematical Biology, 51(4):458–490, 2005.
[5] M. M. Ballyk and G. S. K.Wolkowicz. Exploitative Competition in the Chemostat for TwoPerfectly Substitutable Resources. Mathematical Biosciences, 118:127–180, 1993.
[6] C. Chicone. Ordinary Differential Equations with Applications. Texts in Applied Mathematics.Springer, 2006.
[7] P. De Leenheer, S. A. Levin, E. D. Sontag, and C. A. Klausmeier. Global stability in a chemostatwith multiple nutrients. Journal of Mathematical Biology, 52(4):419–438, 2006.
[8] P. Gajardo, F. Mazenc, and H. RamÃŋrez. Competitive exclusion principle in a model ofchemostat with delays. Dynamics of Continuous, Discrete and Impulsive Systems Ser. A: Math.Anal., 16(4a):253–272, 2009.
[9] S. B. Hsu. A survey of constructing Lyapunov functions for mathematical models in populationbiology. Taiwanese J. Math., 9(2):151–173, 2005.
[10] S. B. Hsu, K. S. Cheng, and S. P. Hubbell. Exploitative Competition of Microorganisms forTwo Complementary Nutients in Continuous Cultures. SIAM J. Appl. Math., 41(3):422–444,Dec. 1981.
[11] S. B. Hsu and P. Waltman. A survey of mathematical models of competition with an inhibitor.Mathematical Biosciences, 187:53–91, 2004.
[12] H. Khalil. Nonlinear Systems: Pearson New International Edition. Always learning. PearsonEducation, Limited, 2013.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Mathematical approach regarding theenvironmental effects upon trait diversity in a
cell population.Karina Vilches Ponce
Abstract
The environmental effects in phenotype and trait diversity were explained extensivelyby ecologists, biologists and zoologists obtained different versions of its real influence inthe evolutionary and natural selection. We propose to study the asymptotic behavior ofa no local transport equation that models the environmental influence on trait evolutionperhaps over time in some populations using a viscosity ε and the anzats e
uε . Concluding
in a mathematical sense: If there is a stable environment then the trait variability in allpopulations is neglected.
References[1] Barles.G, Perthame. B. Dirac concentrations in Lotka-Volterra parabolic PDEs. Indiana Univ.
Math. J. 57(7) 2008, 3275–3301
[2] Barles. G, Mirrahimi.S, Perthame.B. Concentration in Lotka-Volterra parabolic or integralequations: a general convergence result.
[3] Calsina.A, Cuadrado. S. Small mutation rate and evolutionary stable strategies in infinitedimensional adaptive dynamics. J. Math. Biol. 48, 135-159 (2004).
[4] Cuadrado. S. Adaptive dynamics in an infinite dimensional setting. PHD Thesis.
[5] Desvillettes.L, Jabin.P-E, Mischeler. S, Raoul. G. On selection dynamics for continuous struc-tured populations. Commun. Math. Sci. Vol. 6, N. 3, pp. 729-747.
[6] Diekmann. O, Jabin P.-E, Mischler. S and Perthame. B. The dynamics of adaptation: Anilluminating example and a Hamilton-Jacobi approach. Th. Pop. Biol., 67(4) (2005) 257–271.
[7] Logan.J.D, Ledder.G, Wolesensky.W. Type II functional response for continuous, physiologi-cally structured models. Journal of Theoretical Biology (2009).
[8] Lorz. A, Mirrahimi. S and Perthame. B Dirac mass dynamics in a multidimensional nonlocalparabolic equation. (2010)
e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On a nonlinear problem from catalysis:existence, multiplicity and qualitative behaviour
Alejandro Omón Arancibia Gonzalo Robledo Veloso
Abstract
This work studies a model of chemical reactor with an Arrhenius dependence on tem-perature, this is a nonlinear ordinary system of two variables: one which correspondsto the normalized concentration of the reacting species, and the temperature. Thesystem also has a non trivial dependence on the parameters involved: the Damköhlernumber, the activation energy and the heat release. Different behaviour is identified interm of the value of this parameters with respect to some nonlinear algebraic equations.
Within the topics studied there can be mentioned: number of steady solutions(multiplicity), stability of the steady solutions, (non) existence of periodic solutions ina neighborhood of the steady solution. One of the main improvements of in all pre-vious mentioned topics is the fact that the presented analysis is extended for the caseof the activation energy strictly positive, which since the formulation of the problemin reference [5] up to recent references like [6] is neglected, as in all the literature isassumed the hypothesis that the activation energy is zero. Numerical test showing thebehaviour of the system is also presented.
References[1] R. Aris: On some diagrams of chemical reaction engineering ; Chaos, vol. 9-1 (1999),pp. 1-14
[2] D. Cohen, R. Alexander: Chemical reactor theory and problems in diffusion; Physica D, vol.(1986), pp. 122-141.
[3] J. Guckenheimer: Multiple bifurcation problems for chemical reactors; Physica D, vol (1986),pp. 1-20.
[4] V. Patil, S. Subramanian, V. Balakotaiah: Singular theory approach for calculating the runawayboundaries of heterogeneous reactor models; Ind. Eng. Chem. Res., vol. 36 (1997), pp. 3230-3241.
[5] A. Poore: A model equation arising from chemical reactor theory Arch. Rat. Mech. Anal., vol.52-4 (1973), pp. 358-388.
e-mail: [email protected] Robledo Veloso, e-mail: [email protected]
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[6] M.Z. Solórzano, W.H. Ray: Multiplicity and stability of chemical reactors with evaporatingcooling Ind. Eng. Chem. Res., vol 47 (2008), pp. 9025-9039.
[7] A. Uppal, W.H. Ray: On the dynamic behavior of continuous stirred tank reactors; Chem. Eng.Sci., vol. 29 (1974), pp. 967-985.
[8] G.A. Viswanathan, D. Luss: Hot zones formation and dynamics in long adiabatic packet-bedreactors; Ind. Eng. Chem. Res., vol. 45 (2006), pp. 7057-7066.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A stochastic disease transmission in an epidemicmodel considering a hyperbolic incidence rate
A. Christen M. A. Maulén E. González-Olivares M. Curé
Abstract
In this paper a stochastic SI epidemic model is analyzed, which is based on the modelproposed by Roberts and Saha [?], considering a hyperbolic type nonlinear incidencerate. According to our knowledge this incidence rate has not been previously used forthis type of epidemic models. Although this kind of rate has receive more attention inlast year for being more realistic. Assuming the proportion of infected population varieswith time, a new model described by an ordinary differential equation is presented,which is analogous to an equation describing the double Allee effect. The limit of thesolution of this equation (deterministic model) is found when time tends to infinity.Then, the asymptotic behaviour of a stochastic fluctuation due to the environmentalvariation in the coefficient of disease transmission is studied. So, a stochastic differentialequation (SDE) is obtained. In the last results obtained the existence of an uniquesolution is proved. Moreover, the SDE is analysed through the associated Fokker-Planck equation to obtain the probability density function (its invariant probabilitydistribution) when the proportion of the infected population reaches steady state. Anexplicit expression for invariant measure is found together with some interesting featuresabout it. The long time behaviour of deterministic and stochastic models are comparedin simulations.
References[1] C. A. Braumann and C. Carlos, Allee effects models in randomly varying environments, Pro-
ceedings of the 13th International Conference on Computational and Mathematical Methodsin Science and Engineering CMMSE 2013, pp. 304-307.
[2] S. Busenberg and K. L. Cooke, Vertically Transmitted Diseases, Biomathematics 23, Springer-Verlag, Berlin, 1993.
[3] V. Capasso and D. Bakstein, An introduction to continuous-time stochastic processes (Secondedition). 2012. Springer.
[4] C. Chen, Y. Kang, Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence,Abstract and Applied Analysis 2014 (2014) Article ID 723825, 13 pages.
Instituto de Estadística, Pontificia Universidad Católica de Valparaíso, e-mail:[email protected] , [email protected]
Grupo de Ecología Matemática, Instituto de Matemáticas, e-mail: [email protected] de Valparaíso, e-mail: [email protected]
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[5] P. Das, D. Mukherjee and A. K. Sarkar, Study of an S-I epidemic model with nonlinear inci-dence rate: Discrete and stochastic version, Applied Mathematics and Computation 218 (2011)2509âĂŞ2515.
[6] Y. Ding, M. Xu and L. Hu, Asymptotic behavior and stability of a stochastic model for AIDStransmission, Applied Mathematics and Computation 204 (2008) 99-108.
[7] Y. Ding, M. Xu and L. Hu, Risk Analysis for AIDS control based on a stochastic model withtreatment rate, Human and Ecological Risk Assessment: An International Journal 15 (2009)765-777.
[8] D. Fan, K. Wang and L. Hong, The complete parameters analysis of the asymptotic behaviourof a logistic epidemic model with two stochastic perturbations, Mathematical Problems inEngineering Article ID 904383 Volume 2009 7 pages.
[9] E. González-Olivares, B. González-Yañez, J. Mena-Lorca and R. Ramos-Jiliberto, Modellingthe Allee effect: are the different mathematical forms proposed equivalents? In R. Mondaini(Ed.) Proceedings of the 2006 International Symposium on Mathematical and ComputationalBiology, E-papers Serviços Editoriais Ltda. Rio de Janeiro (2007) 53-71.
[10] E. González-Olivares, B. González-Yañez, J. Mena-Lorca, A. Rojas-Palma and J. D. Flores,Consequences of double Allee effect on the number of limit cycles in a predator-prey model,Computers and Mathematics with Applications 62 (2011) 3449-3463.
[11] A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, A stochastic differential equation SISepidemic model, SIAM J. Appl. Math. 71 3 (2011) 876-902.
[12] H. W. Hethcothe, The mathematics of infectious disease, SIAM Review 42 (2000) 599-653.
[13] Z. Hu, W. Ma, S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate andtreatment, Mathematical Biosciences 238 1 (2012) 12-20.
[14] P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Ex-periments. 1994. Springer.
[15] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic non-linear SIRS epidemic model, Nonlinear Analysis: Modelling and Control, 2011, Vol. 16, No. 1,59âĂŞ76.
[16] W-m. Liu, H. W. Hethcote and S. A. Levin, Dynamical behaviour of epidemiological modelswith nonlinear incidence rates, Journal of Mathematical Biology 25 (1987) 359-380.
[17] J. Zhou and H. W. Hethcote, Population size dependent incidence in models for diseases withoutimmunity, Journal of Mathematical Biology 32 (1994) 809-834.
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Sistemas Dinámicos
Encargado de Sesión : Irene Inoquio
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Aspectos recientes de la Conjetura de PalisAlma Armijo
Abstract
En los años 90 Palis conjeturó que se puede clasificar los difeomorfismos con latopología Cr. En difeomorfismos hiperbólicos y como son los que no lo son. Muchosmatemáticos han tratado de darle respuesta a esta conjetura, vamos a ver los avancesrealizados por varios autores y en particular el trabajo realizado en mi tesis de doctor-ado.
Universidad de Santiago, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Injectivity, Global and Almost Global Stabilityof Hurwitz Vector Fields.
Álvaro Castañeda Víctor Guíñez
Abstract
We give, in dimension three, a family of vector fields that are examples to Weak Markus–Yamabe Conjecture and simultaneously counterexamples to Markus–Yamabe Conjec-ture. Furthermore, we construct a family of almost Hurwitz vector fields such that theorigin is almost globally asymptotically stable by using the novel concept of densityfunctions, and we give a family of the Hurwitz vector fields, perturbing the previousfamily, that are example to Markus–Yamabe Conjecture.
References[1] L.A. Campbell, Unipotent Jacobian matrices and univalent maps, Contemp. Math. 264 (2000),
157âĂŞ-177.
[2] A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progress in Math-ematics, vol. 190, Birkhauser, Basel, 2000.
[3] A. Fernandes, C. Gutiérrez, R. Rabanal, On local diffeomorphisms of Rn that are injective,Qual. Theory of Dyn. Systems 4 (2004), 255-262.
[4] L. Markus, H. Yamabe, Global Stability Criteria for Differential Systems, Osaka Math. J. 12(1960), 305âĂŞ-317.
[5] G.H. Meisters, C. Olech, Solution of Global Asymptotic Stability Jacobian Conjecture for thePolynomial Case, Analyse Mathématique et Applications, Gauthier-Villars, Montrouge, (1988),373–381.
[6] C. Olech, On the Global Stability of an Autonomous System on the Plane, Contributions toDiff. Eq. 1 (1963), 389–400.
[7] R. Potrie, P. Monzón Local Implications of Almost Global Stability, Dynamical Systems 24(2009), 109–115.
[8] A. Rantzer, A dual to Lyapunov’s Stability Theorem, Syst.Cont.Lett. 42 (2001) 161–168.
FONDECYT Iniciación Project 11121122, e-mail: [email protected]: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Toeplitz and strong orbit equivalenceMaryam Hosseini
Abstract
We show that for any unital Dimension group, (G, u), of rank bigger than one withnon-cyclic rational subgroup and for any countable subgroup of the unit circle, Λ, thereexists a Toeplitz system (X,T ) with an invariant measure µ such that K0(X,T ) = Gand SPµ(T ) = Λ. This is a joint work with M. Isabel Cortéz at university of Santiagoand Thierry Giordano at university of Ottawa.
Universidad de Santiago, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Dimensión de Hausdorff de los conjuntos deBorel-Bernstein
Felipe Pérez
Abstract
El teorema de Borel-Bernstein [1],[2], afirma que si B > 1, entonces el conjunto E(B) =x :∈ [0, 1] : an(x) ≥ Bn i.o. , siendo an(x) el n-ésimo término de la expansión en frac-ciones contínuas de x, tiene medida de Lebesgue 0. En [3], Wang y Wu probaron que lafunción que codifica la dimensión de Hausdorff de E(B), D : B 7→ dimH E(B) definidaen (1,∞) es contínua, decreciente, y satisface limB→1D(B) = 1 y limB→∞D(B) = 1/2.Utilizando los métodos del Formalismo Termodinámico, es posible mejorar la regulari-dad de D(B) y concluir que es real analítica.
References[1] Bernstein, F. (1911). Über eine Anwendung der Mengenlehre auf ein aus der Theorie der
sakularen Storungen herruhrendes Problem. Mathematische Annalen, 71(3), 417-439.
[2] Émile Borel, M. (1909). Les probabilités dénombrables et leurs applications arithmétiques.Rendiconti del Circolo Matematico di Palermo (1884-1940), 27(1), 247-271.
[3] Wang, B. W., Wu, J. (2008). Hausdorff dimension of certain sets arising in continued fractionexpansions. Advances in Mathematics, 218(5), 1319-1339.
e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Flexibility of some groups of homeomorphismsof the line
Cristobal Rivas
Abstract
There are many result stating rigidity phenomena all across dynamical sistems. Inthis talk, I will explain a recent result obtained in collaboration with Juan Alonso andJoaquin Brum (U. de la Republica) stating that any action by homeomorphism on theline of the fundamental group of a closed surface is flexible. By this we mean that theaction considered can be approximated in a natural topology by another action which isnot semi-conjugated to the original one. This is somehow surprising since for the caseof actions on the circle, these groups admits action that are rigid : any other actionclose to it is semi-conjugated to the original one [1].
We will use this flexibility to deduce some nice result about the space of left-orderingsof the groups considered.
References[1] K. Mann, Space of surface group representation. Invent. Math. 201 (2015).
Partially supported by Proyecto Anillo 1103 and FONDECYT 1150691, e-mail:[email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A linearization result for DEPCA systemsManuel Pinto Gonzalo Robledo
AbstractWe generalize the topological equivalence results obtained by Palmer [2] and Shi &Xiong [4] to the systems of differential equations with piecewise constant argumentsγ(·):
x(t) = A(t)x(t) +A0(t)x(γ(t)) + f(t, x(t), x(γ(t))), (1)
andy(t) = A(t)y(t) +A0(t)y(γ(t)), (2)
that is, we obtain conditions ensuring the existence of a homeomorphism between thesolutions of the systems (1) and (2).
The function t 7→ γ(t) is defined with the help of two sequences tii∈Z and ζii∈Z,which satisfy:
(B1) ti < ti+1 and ti ≤ ζi ≤ ti+1 for any i ∈ Z,(B2) ti → ±∞ as i→ ±∞,
(B3) γ(t) = ζi for t ∈ [ti, ti+1),
(B4) there exists a constant θ > 0 such that
ti+1 − ti ≤ θ, for any i ∈ Z.
Some additional properties on (2) and its quasilinear perturbation will be consid-ered: a) we will assume that (2) admits a property of exponential dichotomy recentlyintroduced by Akhmet [1]. We will characterize it by using the Cauchy matrix for (2)introduced by Pinto in [3]. b) We will assume that f is Lipschitz and bounded.
References[1] M. Akhmet, Exponentially dichotomous linear systems of differential equations with piecewise
constant argument, Discontinuity, Nonlinearity, and Complexity, 1 (2012), 337–352.
[2] K.J. Palmer, A generalization of Hartman’s linearization Theorem, J. Math. Anal. Appl., 41(1973), 753–758.
[3] M. Pinto, Cauchy and Green matrices type and stability in alternately advanced and delayeddifferential systems. J. Difference Equ. Appl., 17 (2011), 721–735.
[4] J. Shi, K. Xiong, On Hartman’s linearization theorem and Palmer’s linearization theorem, J.Math. Anal. Appl., 92 (1995), 813–832.
Universidad de Chile – Departamento de Matemáticas, e-mail: [email protected] de Chile – Departamento de Matemáticas, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Shearer’s inequality and the Infimum RulePierre Paul Romagnoli
AbstractWe review subbadditivity properties of Shannon entropy, in particular, from the
Shearer’s inequality we derive the infimum ruleİ for actions of amenable groups. Webriefly discuss applicability of the infimum formula to actions of other groups. Thenwe pass to topological entropy of a cover. We prove Shearer’s inequality for disjointcovers and give counterexamples otherwise. We also prove that, for actions of amenablegroups, the supremum over all open covers of the infimum fomulaİ gives correct valueof topological entropy.
References[1] B. Bollobás and A. Thomason, Projections of bodies and hereditary properties of hypergraphs,
Bull. London Math. Soc. 27 (1995), 417–424.
[2] L. Bowen, Measure conjugacy invariants for actions of countable sofic groups,J. Amer. Math. Soc. 23 (2010), 217–245.
[3] P. Burton, Naive entropy of dynamical systems, preprint,http://arxiv.org/pdf/1503.06360v1.pdf
[4] T. Downarowicz, Entropy in dynamical systems, Cambridge University Press, New MathematicalMonographs 18, Cambridge 2011.
[5] E. Lindenstrauss, Pointwise theorems for amenable groups, Electronic Research Announcementsof AMS, 5 (1999).
[6] B. Seward, Krieger’s finite generator theorem for ergodic actions of countable groups II, preprint,http://arxiv.org/pdf/1501.03367.pdf
[7] B. Seward, private communication.
[8] B. Seward and R. Tucker-Drob, Borel structurability on the 2-shift of a countable group, preprint.http://arxiv.org/pdf/1402.4184.pdf
[9] A. Stepin and A. Tagi-Zade, Variational characterization of topological pressure of the amenablegroups of transformations (Russian), Dokl. Akad. Nauk SSSR 254 (1980), 545–549.
[10] B. Weiss, private communication.
Facultad de Ciencias Exactas, Departamento de Matemáticas, UNAB. :[email protected] author acknowledges the support of Programa Basal PFB 03, CMM, Universidad de ChileJoint work with Tomasz Downarowicz from the Institute of Mathematics, Polish Academy of Science
and Bartosz Frej from the Departament of Mathematics, Wroclaw University of Technology
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Polinomios de Fibonacci y ComponentesErrantes
Eugenio Trucco
Abstract
En el caso de una función racional
R : P1C −→ P 1
C
las componentes de Fatou periódicas bajo la acción de R fueron clasificadas por Julia,Fatou, Herman y Siegel. La pregunta de existencia o no de componentes errantes (nopre-periódicas) fue respondida por D. Sullivan en 1985 utilizando técnicas de geometríacuasi-conforme. Él demuestra que el conjunto de Fatou de una función racional no tienecomponentes errantes. En el estudio de la dinámica de funciones racionales sobre uncuerpo no arquimediano K es más conveniente estudiar la acción de la función racionalen la línea proyectiva de Berkovich asociada al cuerpo K. Esto es, estudiar
R : P1,anK −→ P 1,an
K
El resultado de Sullivan sobre la no existencia de componentes errantes no es cierto enel caso de funciones racionales sobre cuerpos no arquimedianos. Existen ejemplos defunciones racionales que presentan una componente errante en su conjunto de Fatou.Todos los ejemplos conocidos están relacionados a un fenómeno llamado ramificaciónsalvaje, una propiedad que no ocurre en los números complejos. Por lo anterior, se creeque la ramificación salvaje es necesaria para la existencia de componentes errantes. Enesta charla estudiaremos una combinatoria relacionada con los números de Fibonaccipara así comprender esta relación.
Universidad Austral de Chile, e-mail: [email protected]
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Matemática Discreta
Encargado de Sesión : José Soto
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Estudio de un modelo de evasión en eltransporte público
Bastián Bahamondes Pizarro
Abstract
El problema en estudio considera una red compuesta por un grafo dirigido G = (V,E),dos nodos s, t ∈ V , una función de costos sobre los arcos c : E 7→ Z+, y una distribu-ción de probabilidad, también sobre los arcos (pe ≥ 0 ∀ e ∈ E,
∑e∈E pe = 1). Sobre
este grafo se busca estudiar el comportamiento e interacción de dos agentes: un evasor,que desea trasladarse desde s hasta t, y un policía que se ubicará aleatoriamente sobreuno de los arcos. La dinámica es la siguiente: el evasor sabe que sobre un (y sólo un)arco de la red estará ubicado el policía, aunque su ubicación será incierta y seguirá ladistribución p. El evasor escogerá un camino entre s y t y lo recorrerá de manera quesi atraviesa el arco donde se encuentra el policía, será multado con una infracción decosto M > 0 y continuará su viaje ya sea por el camino previamente escogido (versiónno adaptativa) o por un shortest path (versión adaptativa).
Mientras que el evasor enfrenta el problema de minimizar el costo esperado delcamino que escoge, el policía busca establecer una distribución de probabilidad sobrelos arcos de manera de hacer lo más alto posible el costo del evasor, o bien maximizarla multa esperada que recauda. Se presentan resultados algorítmicos concernientes a laversión no adaptativa del problema del evasor y de complejidad referentes al problemadel policía.
Universidad de Chile, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Resource Augmentation Algorithm for SingleMachine Scheduling with Job-Dependent
Convex CostRodrigo A. Carrasco
Abstract
In this work we combine resource augmentation and alpha-point scheduling techniquesto compute approximate solutions for a general family of scheduling problems: eachjob has a convex non-decreasing cost function applied to its completion time and thegoal is to compute a schedule that minimizes the total cost subject to precedenceconstraints. We show that our algorithm is a O(1)-speed 1-approximation algorithmand our numerical experiments show that the speed-scaling ratio needed is actuallyclose to 1.
Faculty of engineering and Sciences, Universidad Adolfo Ibañez, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Avances recientes en la resolución exacta delproblema de vendedor viajero
Daniel Espinoza William Cook Marcos Goycoolea
Abstract
El problema del vendedor viajero (TSP) es uno de los problemas más clasicos en opti-mización combinatorial [?]; y uno de los problemas más usados para probar algoritmosde optimización, heurísticas y algoritmos de aproximación.
El TSP también ha sido uno de los ejemplos donde más logros ha cosechado la teoríapoliedral .
En esta charla recorreremos brevemente algunos de los hitos más importantes en laresolución práctica de instancias del TSP, con un énfasis en los avances poliedrales; asicomo también últimos desarrollos en desigualdades válidas y algoritmos de separaciónque explotan planaridad.
Universidad de Chile. e-mail: [email protected] of Waterloo, e-mail: [email protected] Adolfo Ibañez, e-mail: [email protected] han omitido las referencias por no ser compatibles con el formato.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Quasirandom hypergraphs and subsets withsmall Fourier coefficients
Hiê.p Hàn
Abstract
Quasi-randomness forms a central theme in modern discrete mathematics. Informallyspeaking quasi-random properties are deterministic properties which are characteristicfor random objects. Such exist for various discrete structures such as graphs, hyper-graphs, subsets of abelian groups etc. Being interesting for their own sake, their studyhas also revealed many connections between various areas of mathematics such as graphand hypergraph theory, number theory, geometry and also to algorithms and complex-ity. In this talk we give an introduction into the topic of quasirandomness, puttingemphasis on linear quasirandom hypergraphs and subsets of integers with small non-trvial Fourier coefficients.
Joint work with Elad Aigner-Horev.
Universidad de Chile, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Efficient Implementation of Carathéodory’sTheorem for a Simple Scheduling Polytope
Ruben Hoeksma
Abstract
In a fundamental paper in polyhedral combinatorics, Queyranne describes the completefacial structure of a classical object in combinatorial optimization, the single machinescheduling polytope. In the same paper, he answers essentially all relevant algorithmicquestions with respect to optimization and separation. In this talk, motivated byrecent applications in the design of revenue optimal incentive compatible mechanisms,we address an algorithmic question that was apparently not addressed before. Namely,we turn Caratheodory’s theorem into an algorithm, and ask to write an arbitrary pointin the scheduling polytope as a convex combination of the vertices of the polytope. Wehere give a combinatorial O(n2) time algorithm, where n is the number of jobs of thescheduling polytope. This is in fact linear in the naive encoding of the output size.We obtain this result by exploiting the fact that the scheduling polytope is a zonotope,and by the observation that its barycentric subdivision has a simple, linear description.The actual decomposition algorithm is an implementation of a method proposed byGrötschel, Lovász and Schrijver, applied to one of the subpolytopes of the barycentricsubdivision. We thereby also shed new light on an algorithm recently proposed for aspecial case, namely the permutahedron.
Universidad de Chile, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Maximum number of colourings withoutmonochromatic Schur triples
Andrea Jimenez
Abstract
We study subsets of finite abelian groups that maximize the number of r-colouringswithout monochromatic Schur triples, i.e. triples of the form (a, b, c) such that a+b = c.For r = 2, 3 and a large class of abelian groups, we show that the maximum is achievedonly by largest sum-free sets. For r > 3 this phenomenon does not persist and theproblem becomes harder. We resolve the problem for abelian groups of even order andr = 4, 5.
Joint work with Hiê.p Hàn.
Universidad de Valparaiso, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On-line list coloring of random graphsDieter Mitsche
Abstract
In this talk, the on-line list colouring of binomial random graphs G(n, p) is studied.We show that the on-line choice number of G(n, p) is asymptotically almost surelyasymptotic to the chromatic number of G(n, p), provided that the average degree d =p(n − 1) tends to infinity faster than (log log n)1/3(log n)2n2/3. For sparser graphs, weare slightly less successful; we show that if d > (log n)2+epsilon for some ε > 0, then theon-line choice number is larger than the chromatic number by at most a multiplicativefactor of C, where C ∈ [2, 4], depending on the range of d. Also, for d = O(1), theon-line choice number is by at most a multiplicative constant factor larger than thechromatic number.
Joint work with Alan Frieze, Xavier Pérez-Giménez and Pawel Prałat.
CMM, Universidad de Chile, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Computing income taxes under the new Chileantax regime: Graphs, Markov Chains and
Algorithms.Javiera Barrera Eduardo Moreno Sebastián Varas
Abstract
The tax reform act of 2014 introduced a new integrated method for income taxes,including an attributed income to shareholders of a corporation. Under this tax, Chileancorporations will attribute all their incomes to their shareholders and these will be taxedon the incomes attributed to them. This change required a new tool for computingthe taxable income of taxpayers accurately and efficiently. In this work, we show themathematical conceptualization and the solution to the problem, proving that thereis only one way to distribute incomes to taxpayers. Moreover, using the theory ofAbsorbing Markov Chains, we define a mathematical model for computing the taxableincomes of each taxpayer. In addition, we implement the mathematical model creatingan algorithm based on the properties of Absorbing Markov Chains and the Tarjan’sstrongly connected components algorithm. This allows us to compute the solutionaccurately and with efficient use of computational resources.
Faculty of engineering and Sciences, Universidad Adolfo IbañezFaculty of engineering and Sciences, Universidad Adolfo Ibañeze-mail: [email protected]
154
Optimización
Encargado de Sesión : Luis Briceño
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Chance-constrained problems and rare events:an importance sampling approachJ. Barrera, T. Homem-de-Mello, E. Moreno
B. Pagnoncelli, G. Canessa
Abstract
We study chance-constrained problems in which the constraints involve the probabilityof a rare event. We discuss the relevance of such problems and show that the existingsampling-based algorithms cannot be applied directly in this case, since they require animpractical number of samples to yield reasonable solutions. Using a Sample AverageApproximation (SAA) approach combined with importance sampling (IS) techniques,we show how variance can be reduced uniformly over a suitable approximation of thefeasibility set, and as a result the problem can be solved with much fewer samples.We provide sufficient conditions to obtain such uniform variance reduction and proveasymptotic convergence of the combined SAA-IS approach. We apply our methodologyto a telecommunications problem, find IS distributions that satisfy the conditions laidout for uniform variance reduction in that context and present numerical results toillustrate the ideas.
Universidad Adolfo Ibáñez, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Nonsmooth Lur’e Dynamical Systems in HilbertSpaces
Samir Adly Abderrahim Hantoute Ba Khiet Le
Abstract
In this paper, we study the well-posedness and stability analysis of set-valued Lur’edynamical systems in infinite-dimensional Hilbert spaces. The existence and uniquenessresults are established under the so-called passivity condition. Our approach uses aregularization procedure for the term involving the maximal monotone operator. TheLyapunov stability as well as the invariance properties are considered in detail. Onthe other hand, this work generalizes the classical composition between a maximalmonotone operator and a linear bounded mapping.
References[1] S. Adly, A. Hantoute, B. K. Le: Nonsmooth Lur’e Dynamical Systems in Hilbert Spaces, to
appear in Set-Valued and Variational Analysis.
[2] B. Brogliato, D. Goeleven: Existence, uniqueness of solutions and stability of nonsmoothmultivalued Lur’e dynamical systems, Journal of Convex Analysis, vol. 20, no. 3, pp. 881–900,2013.
[3] M. K. Camlibel, J. M. Schumacher, Linear passive systems and maximal monotone mappings,to appear in Mathematical Programming.
CMM, Universidad de Chile, e-mail: [email protected] trabajo es financiado por el Proyecto Fondecyt Postdoctorado 3150332
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Stochastic Topology Design Optimization forContinuous Elastic Materials
Miguel Carrasco Benjamín Ivorra Angel Manuel Ramos
Abstract
In this work, we develop a stochastic model for topology optimization. We find ro-bust structures that minimize the compliance for a given main load having a stochasticbehavior. We propose a model that takes into account the expected value of the com-pliance and its variance. We show that, similarly to the case of truss structures, thesevalues can be computed with an equivalent deterministic approach and the stochas-tic model can be transformed into a nonlinear programming problem, reducing thecomplexity of this kind of problems. First, we obtain an explicit expression (at the con-tinuous level) of the expected compliance and its variance, then we consider a numericaldiscretization (by using a finite element method) of this expression and finally we usean optimization algorithm. This approach allows solving design problems which includepoint, surface or volume loads with dependent or independent perturbations. We checkthe capacity of our formulation to generate structures that are robust to main loads andtheir perturbations by considering several 2D and 3D numerical examples. To this end,we analyze the behavior of our model by studying the impact on the optimized solutionsof the expected-compliance and variance weight coefficients, the laws used to describethe random loads, the variance of the perturbations and the dependence/independenceof the perturbations. Finally, the results are compared with similar ones found in theliterature for a different modeling approach.
El trabajo es financiado por el Proyecto FONDECYT 1130905Universidad de los Andes, Facultad de Ingeniería y Ciencias Aplicadas, e-mail: [email protected] Complutense de madrid, Departamento de Matemtica Aplicada, e-mail:
[email protected] Complutense de madrid, Departamento de Matemtica Aplicada, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Existence and approximation of generalizedLagrange multipliers for variational problemsunder uniform constraints on the gradient.
Felipe Alvarez Salvador Flores
Abstract
In this work we present existence results concerning generalized, measure-valued La-grange multipliers for variational problems with uniform constraints on the gradient ofthe type |∇u| ≤ 1 c.t.p. Our main technique is power-penalty, therefore our results holdunder very mild hypothesis. In particular, we do not require constraint qualificationconditions.
We study a class of constrained Dirichlet problems from the calculus of variations of thetype
infJ(v) : |T (x,∇v(x))| ≤ 1 a.e x in Ω, v = g on ∂Ω.
In particular, we prove existence and approximability of solutions and Lagrange multipliersassociated to the uniform constraint on the gradient.
We approximate the problem by a sequence of unconstrained problems penalizing theuniform constraint by a power term. Next we address the existence and approximation ofLagrange multipliers for the uniform constraint on the gradient. The underlying rationalebears some resemblances to certain methods for showing existence of Lagrange multiplierswithout recourse to separation theorems, such as the Fritz – John optimality conditions innonlinear programming.In [1] we proved the following. TheoremLet T (x, ξ) = |ξ|, and g ∈ C2(Ω) be such that ‖∇g‖∞,Ω < 1/2. Let upp≥p1 be a sequenceof solutions to the penalized problem. Let u∞ be a cluster point of upp≥p1 for the topologyof C(Ω). Under appropiate conditions on f , there exists a nonnegative Radon measuremultiplier µ such that:
1. For a nonnegative Radon measure σ, and measurable non-negative functions λ and η,
µ = λL + ησ
Moreover, λ ∈ L1(Ω).
Centro de Modelamiento Matemático, Universidad de Chile, e-mail: [email protected]
159
2. The primal-dual pair (u∞, µ) satisfies the system
−div(fξ(x, u∞,∇u∞) +∇u∞µ) + fs(x, u∞,∇u∞) = 0 in D′(Ω).
λ(x) ≥ 0 L− a.e in Ω, η(x) ≥ 0 σ − a.e in Ω.
λ(x)(|∇u∞(x)| − 1) = 0 L− a.e in Ω, η(x)(|∇u∞(x)| − 1) = 0 σ − a.e in Ω.
3. The sequence |∇up|p−1p≥p1 converges to λ in the bitting sense.
In this talk we shall discuss this result and its connection with similar ones from [2] obtainedusing infinite dimensional duality. As time permits, we shall also discuss the connectionswith overdetermined boundary value problems and its implications for the numerical analysisof such problems.
References[1] Alvarez, F., Flores, S.: Existence and approximation for variational problems under
uniform constraints on the gradient by power penalty. SIAM Journal on MathematicalAnalysis 47 (5), 3466–3487, 2015.
[2] Daniele P, Giuffrè S, Idone G, Maugeri A.: Infinite dimensional duality and applications.Math Ann 339(1):221–239, 2007.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Condiciones de Optimalidad en Problemas deControl Óptimo Discreto
C. Isoton M.A. Rojas-Medar V. Vivanco L. dos Santos
Abstract
El objetivo de este trabajo es caracterizar los puntos críticos de un problema de controlóptimo, formulado en tiempo discreto, no lineal. Para esto, extendemos la noción deKT-invexidad para problemas de programación matemática a problemas de control óp-timo discreto. Se demuestra que las condiciones necesarias de optimalidad establecidaspor el Principio del Máximo son también suficientes si y sólo si el problema es KT-invex.
Consideraremos el siguiente problema de control óptimo en tiempo discreto no lin-eal:
Minimizar Φ(x, u) =
N−1∑i=0
f (xi, ui, i)
sujeito a: xi+1 = ϕ (xi, ui, i) , i = 0, . . . , N − 1,h1(xi, i) = 0, g1(xi, i) ≤ 0, i = 0, . . . , N,h2(ui, i) = 0, g2(ui, i) ≤ 0, i = 0, . . . , N − 1,K1(x0, xN ) = 0, K2(x0, xN ) ≤ 0
(1)
donde f : Rn × Rm × [0, N − 1]→ R, ϕ : Rn × Rm × [0, N − 1]→ Rn,
h1 : Rn × [0, N ]→ Rr1 , g1 : Rn × [0, N ]→ Rs1 ,
h2 : Rm × [0, N − 1]→ Rr2 , g2 : Rn × [0, N − 1]→ Rs2 ,
K1 : Rn × Rn → Rk1 ,K2 : Rn × Rn → Rk2
son funciones continuamente diferenciables. En este caso, r1 ≤ n, r2 ≤ m e k1 ≤ 2n,
x : [0, N ]→ Rn u : [0, N − 1]→ Rm (2)x(i) 7→ xi u(i) 7→ ui (3)
xi es la variable de estado; ui es el parámetro de control; [0, N ] es el intervalo discretode la recta real, donde N ≥ 0 es el número de etapas (pasos) realizados;x := (x0, . . . , xN ) es la trayectoria; u := (u0, . . . , uN−1) es el control asociado a latrayectoria correspondiente.
Universidade Federal do Paraná, Curitiba, Brasil, e-mail: [email protected] , e-mail:[email protected]
Universidad de Tarapacá, Arica, Chile, e-mail: [email protected] Católica de la SantíÂŋsima Concepción,Concepción, Chile, e-mail: [email protected] trabajo es financiado por CAPES-PDSE, Fondecyt: 1120260, Dirección de Investigación, Universidad
Católica de la Santísima Concepción, proyecto DIN 05/2015
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Esta clase de problemas ha sido ampliamente estudiado en muchos libros y artículos,por ejemplo [2]. Un ejemplo clásico de aplicación es el llamado Problema de Estabi-lización Económica [8].
El estudio de las condiciones de optimalidad es un tema importante en AnálisisVariacional, en Optimización y también en Control Óptimo. Existen muchos artículosen la literatura que visan el estudio de las condiciones de optimalidad para estos proble-mas; véase [4]. Nuestro énfasis en este trabajo es presentar condiciones de optimalidadpara el problema de control óptimo discreto (1), para ello usaremos algunos resultadosprobados en [5].
Los resultados obtenidos se basan en la teoría desarrollada por [2] y generalizadaen [1]. Esto se hace a través de una conveniente reinterpretación del problema (PCD)como un problema de programación matemática, al cual aplicamos el formalismo deDubovitskii-Milyutin [3]. El concepto de KT-invexidad, que fue introducido por Martin[6] para problemas de programación matemática y posteriormente generalizado porOsuna-Gómez et al. [7] para problemas multiobjetivos, no sólo es interesante parala obtención de las condiciones suficientes de optimalidad; también nos proporcionauna completa caracterización de optimalidad. En este trabajo, se demuestra que unproblema de control óptimo discreto es KT-invex si y sólo si todo proceso admisibleque satisface las condiciones del Principio del Máximo es un proceso optimal. Esto es,la clase más amplia de problemas para los cuales las condiciones establecidas por elPrincipio del Máximo son, a la vez, necesarias y suficientes para la optimalidad.
References[1] Arutyunov, A. V.; Marinkovich, B. Necessary optimality conditions for discrete optimal control
problems. (Russian) Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 2005, no. 1, 43–48,57; translation in Moscow Univ. Comput. Math. Cybernet. 2005, no. 1, 38–44
[2] Boltyanskii, V. G. Optimal control of discrete systems. A Halsted Press Book. Translated fromthe Russian by Ron Hardin. John Wiley-Sons, New York-Toronto, Ont.; Israel Program forScientific Translations, Jerusalem, 1978. x+392 pp.
[3] Girsanov, I. V. Lectures on mathematical theory of extremum problems. Edited by B. T. Poljak.Translated from the Russian by D. Louvish. Lecture Notes in Economics and MathematicalSystems, Vol. 67. Springer-Verlag, Berlin-New York, 1972. iv+136 pp.
[4] Hilscher, R.; Zeidan, V. Discrete optimal control: second order optimality conditions. In honourof Professor Allan Peterson on the occasion of his 60th birthday. J. Difference Equ. Appl. 8(2002), no. 10, 875–896.
[5] Marinkovic, B.: Optimality conditions in discrete optimal control problems with state con-straints, Numerical Functional Analysis and Optimization 28.7-8 (2007): 945-955.
[6] Martin, D. H.: The essence of invexity, J. Optim. Th. Appl., vol. 47, no. 1, (1985): 65-76.
[7] Osuna-Gómez, R.; Rufián-Lizana, A.; Ruiz-Canales, P.: Invex functions and generalized con-vexity in multiobjective programming, J. Optim. Th. Appl. 98, no. 3, (1998): 651-661;
[8] Tu, Pierre NV.: Introductory Optimization Dynamics, Springer-Verlag, Berlin, New York,1991.M
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Proximal Distances over Symmetric ConesJulio López Erik Papa
Abstract
This paper is devoted to the study of proximal distances defined over symmetric cones.Specifically, our aim is to provide two ways for build them. For this, we consider twoclass of functions of real-valued satisfying some assumptions. Then, we show that itscorresponding spectrally defined function defines a proximal distance. In addition, wepresent several examples and properties of this distance. The properties are useful forthe analyze of convergence of proximal algorithms associated with a proximal distance.
References[1] A. Auslender and M. Teboulle: Interior gradient and proximal methods for convex and
conic optimization, SIAM Journal on Optimization, 16(3):697-725, 2006.
[2] S. Pan and J.S. Chen: A class of interior proximal-like algorithms for convex second-order cone programming, SIAM Journal on Optimization, 19(2):883-910, 2008.
[3] D. Sun and J. Sun: Löwner’s operator and spectral functions in euclidean jordan alge-bras, Mathematics Operations Research, 33(2):421-445, 2008.
Universidad Diego Portales, e-mail: [email protected] Federal de Rio de Janeiro, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Stability in Generalized Nash EquilibriumProblems with nonsmooth payoff functions,
application to Electricity marketMatthieu Maréchal Rafael Correa
Abstract
This talk deals with the calmness of a Generalized Nash Equilibrium Problem withnon-differentiable data. The approach consists in obtaining some error bound propertyfor the KKT system associated to the Generalized Nash Equilibrium Problem, andreturning to the primal problem thanks to the Slater constraint qualification. Weapply these results to study a smoothing method applied to the electricity market.
We have considered a perturbed Generalized Nash Equilibrium Problem (GNEP), whichconsists in finding an element x ∈ Rn = Rn1 × · · · × Rnp satisfying:
∀ν ∈ 1, · · · , p , xν ∈ arg minXν(x−ν ,u)
θν(·, x−ν , u) (1)
where 1, · · · , p denotes the set of players, xν is the strategy played by player ν, x−ν is thevector containing the strategy played by all players excepted player ν, Xν(x
−ν , u) is the setof admissible strategies for player ν depending on x−ν and on the parameter u. The functionθν(x
ν , x−ν , u) is the loss function for player ν depending on a parameter u ∈ U . We supposethat the set Xν(x
−ν , u) is described by a inequality system, that is
Xν(x−ν , u) = xν ∈ Rnν : gν(xν , x−ν , u) ≤ 0.
We denote S(u) the solution set of problem (1). This talk deals with the regularity propertiesof S, more precisely gives some sufficient conditions in order to ensure the calmness of Swhen the loss functions θν are not supposed to be differentiable.DefinitionWe say that S is calm at a point (u, x), where x ∈ S(u), if there exist constants r, L > 0such that, for all x ∈ B(x, r), for all u ∈ B(u, r),
S(u) ∩B(x, r) ⊂ S(u) +B(0, L‖u− u‖)
Instituto de ciencias basicas, facultad de ingenieria, Universidad Diego Portales, e-mail:[email protected]
Centro de Modelamiento Matematico, Universidad de Chile, e-mail: [email protected] trabajoes financiado por el Proyecto Fondecyt 3130596
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The sufficient conditions for the calmness of S are obtained from the study of the KKT systemof the Generalized Nash Equilibrium Problem and unde the Slater contraint qualification.Those results generalize in nondifferential case some results obtained [1, 2].
We use these results in order to study a smoothing method applied to electricity market.The smoothing method consists in replacing the loss functions θν by a smooth approximation.Definition Let f : Rm → R be a locally Lipschitz continuous function. We say that thefunction f : Rm × R→ Rn is a smooth approximation of f if it satisfies the following:
1. For every x ∈ Rm, f(x, 0) = f(x).
2. The function f is continuously differentiable on Rm × R \ 0.
3. For every x ∈ Rm, f is locally Lipschitz continuous around (x, 0).
4. If f is convex then f(·, u) is convex for all u ∈ R.
We consider electricity market with N producer, each producer i maximizes its revenuefunction solving the following program
P (a−i, b−i) maxAi≤ai≤Ai
Ri(ai, a−i) = aiqi(ai, a−i)− Ci(qi(ai, a−i)).
where ai is the price strategy of the producer i, a−i is the price strategy of the other pro-ducers and Ri(ai, a−i) is the revenu of the producer i.
In general the revenue function Ri is not differentiable. For numerical experiments we canreplace the revenue function Ri by a smooth approximation Ri.In this part we use the stability study about GNEP in order to derive an estimation of thedistance between the solutions of the Nash Equilibrium Problem which models the electricitymarket and the solution of the smooth approximation, and illustrate this estimation with anumerical experiment.
References[1] A. Dreves, F. Facchinei, A. Fischer, M. Herrich, A new error bound result for Generalized
Nash Equilibrium Problems and its algorithmic application, Comput Optim Appl, 2013
[2] F. Facchinei, A. Fischer, V. Piccialli, Generalized Nash equilibrium problems and Newtonmethods, Math. Program., Ser. B (2009) 163-194
[3] X. Hu, D. Ralph, Using EPECs to model bilevel games in restructured electricity marketswith locational prices. Operations research, 2007, vol. 55, no 5, p. 809-827.
[4] A. F. Izmailov, M. V. Solodov . On error bounds and Newton-type methods for general-ized Nash equilibrium problems. Computational Optimization and Applications, 1-18,2012
[5] B. Mordukhovich, Stability Theory for Parametric Generalized Equations and Varia-tional Inequalities Via Nonsmooth, Transactions of the American Mathematical Society,Vol. 343, No. 2 (Jun., 1994), pp. 609-657
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Boosting Topic Models for Text AnalysisMarcelo Mendoza
Abstract
Topic models have been of growing interest in the last decade. In particular, the tech-niques based on probabilistic latent variable models provide a solid theoretical base anda flexible framework that allow for the modeling of various kinds of documentary col-lections. These models consist of introducing a set of latent variables that allow one tocapture the relationships between terms, documents and other attributes of the docu-mentary collection that are not evidently manifested but can be modeled as unobservedrelationships. The flexibility of this modeling family allows one to incorporate relevantproperties of the text such as polysemy, making groups in sets of terms that describeconcepts, forming topics. The use of latent variables also allows one to make infer-ences about the presence of topics in each document. Topic models are fundamentallydivided into two broad approaches: the techniques resulting from Probabilistic LatentSemantic Analysis (PLSA) [1], which introduce latent variables without assuming dis-tribution priors, and the techniques based on Latent Dirichlet Allocation (LDA) [2],which assume distribution priors over topics and vocabulary by using a Dirichlet dis-tribution. Both approaches have strengths and weaknesses. On the one hand, PLSAfits the model by using the Expectation-Maximization (EM) algorithm [3] which is astandard method for the inference of parameters in latent variable models but tendsto over fit data limiting the generalization capability because it can only guaranteeconvergence to local optimums. On the other hand, LDA addresses this limitation byintroducing Dirichlet distribution priors on the vocabulary and on topic distributionsover documents, which corresponds to a Bayesian regularization over the input. Thisprocess allows for improvement in the generalization capability of the models, but itintroduces computational difficulties in the parameter estimation method, addressedusing Monte Carlo methods through Gibbs sampling. We propose to explore the useof regularization operators on the EM estimators of PLSA, which would allow for con-trol of the compromise between generalization and overfitting that is inherent in thelocal optimization methods. We introduce the eliteness versus background concept tomodel the production of text from two components. The idea is that when producingtext, the author selects words from the elite of the distributions of text associated witheach topic. However, there are words that are part of the natural language backgroundand do not correspond to specific terms from any topic but rather to common terms,transversal to the topics. To model this linguistic phenomenon, we propose to modifyPLSA, introducing sparsification over the latent variables associated with the termsand smoothing over a single latent variable that is capable of modeling the background.
Universidad Técnica Federico Santa María, e-mail: [email protected]
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References[1] Hofmann, T. (2001). Unsupervised learning by probabilistic latent semantic analysis.
Machine Learning, 42(2):177-196.
[2] Blei, D., Ng, A., Jordan, M. (2003). Latent Dirichlet Allocation. Journal of MachineLearning Research, 3(4-5):993-1022.
[3] Dempster, A., Laird, N., Rubin, D. (1977). Maximum likelihood from incomplete datavia the EM algorithm. Journal of the Royal Statistics Society, 39:1-38.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A Decomposition Method for Two-StageStochastic Programs with Risk-Averse Utilities
Tito Homem-de-Mello Sebastian Arpon Bernardo Pagnoncelli
Abstract
We discuss a decomposition method for two-stage stochastic programs with risk-averseutilities. More specifically, we consider problems of the form
minx
f (x) + Q(x)
subject to Ax = b (1)x ≥ 0,
where Q(x) = Eω [Q(x, ω)], ω is a random vector representing the uncertainty in theproblem and the function f (x) is convex. The second decision stage is represented byQ(x, ω), which has the following form:
Q(x, ω) = minzω
g(zω)
subject to Tωx+Wωzω = hω
zω ≥ 0
where g is a function representing a convex monotone utility.Suppose that ω has finite support with N points (if this is not the case, we can
consider a sample average approximation of the problem). By a proper re-arrangementof the variables, we can write the above problem in the format
minx,z
F (x) +G(z)
subject to Ax+ Bz = C. (2)
The advantage of reformulating the problem this way is that it fits the format of thewell-known Alternating Direction Method of Multipliers (ADMM) developed in theliterature. In this method constraint (2) is relaxed, creating the approximation
minx,z
F (x) +G(z) + ρ∥∥Ax+ Bz − C
∥∥2
and then an iterative procedure is applied whereby the primal and dual solutions areupdated alternatingly.
Universidad Adolfo Ibáñez, e-mail: [email protected]. El trabajo es financiado por el ProyectoFondecyt 1120244
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Some attractive features of the algorithm are its simplicity of implementation andits suitability for parallelization. Nevertheless, several questions arise when applyingthis approach to problem (1), both from the theoretical as well as from the compu-tational perspective. For example, it is important to establish conditions that ensurethe convergence of the algorithm. It is also important to provide guidelines to choosethe parameter ρ, as the practical performance of the algorithm appears to be sensitiveto the value of that parameter. In this talk we discuss these issues and present somenumerical results to illustrate the ideas.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
A primal-dual mix algorithm for convexnon-differentiable structured optimization in
Hilbert spacesCesare Molinari Juan Peypouquet
AbstractWe consider the following structured optimization problem:
min(x,y)∈X×Y
f (x) + g (y) : Ax+By = c ,
where
• X,Y, Z are Hilbert spaces;
• A,B are linear continuous operators and c an element in Z;
• f is a convex and differentiable function with Lipschitz-continuous gradient;
• g is a proper, convex and lower semi-continuous function (possibly non-differentiable).
Inspired by a work of Chen and Teboulle [1], the primal-dual iterative algorithm thatwe propose consists of four steps:
i) prediction of the Lagrange multiplier;
ii) gradient method on the Lagrangian in the differentiable variable;
iii) proximal-point algorithm in the non-differentiable variable;
iv) correction of the Lagrange multiplier.
Under qualification conditions for the solution and mild hypothesis on the step-size,we show the weak convergence of the sequence generated by the algorithm to an opti-mal pair. We investigate also the introduction of a fixed parameter in the prediction-correction steps: the aim is the acceleration of the dual process, in order to reach fasterthe correct Lagrange multiplier. Finally, we present some possible applications andnumerical experiments for optimal control of parabolic PDEs.
References[1] G. Chen, M. Teboulle, A proximal-based decomposition method for convex minimization prob-
lems, Mathematical Programming, Vol. 64, No. 1, pp. 81-101, 1994.
Universidad Técnica Federico Santa María, e-mail: [email protected] trabajo es financiado por el Conicyt, Proyecto Anillo ACT-1106
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Resultados sobre convexidad de la imagen defunciones cuadráticas
Felipe Opazo Lagos
Abstract
Los clásicos teoremas de Dines [1] (motivado por Finsler [2]) y de Brickman [3] tratansobre la convexidad de la imagen de dos funciones cuadráticas, cuyo dominio es unespacio Euclidiano o bien una esfera N-dimensional, con N ≥ 3, respectivamente.
Estos resultados dieron inicio al estudio de posibles generalizaciones (ver [4]), a lapar de aplicaciones a Optimización Cuadrática y otros temas (ver [5]). Las funcionescuadráticas en si mismas exhiben interesantes propiedades geométricas, lo que se tra-duce en una cierta ’convexidad oculta’ discutida en [4].
En esta charla se comentará sobre una investigación de 2014 [6] en que se caracterizóla convexidad de dos funciones cuadráticas no necesariamente homogéneas (en oposicióna [1]). También se presentarán resultados de una investigación en curso este año comoparte de una Memoria de Título, siempre sobre el tema de la convexidad y de lasfunciones cuadráticas.
References[1] L. Dines, On the mapping of quadratic forms, Bull. Amer. Math. Soc., 47 (1941), 494-498
[2] P. Finsler, Uber das Vorkommen definiter und semidefiniter Formen in Scharen quadratischerFormen, Comment. Math. Helv. 9 (1936/37), 188-192.
[3] L. Brickman, On the fields of values of a matrix, Proc. Amer. Math. Soc., 12 (1961), 61-66
[4] J.-B. Hiriart-Urruti and M. Torki, Permanently going back and forth between the ’quadraticWorld’ and the ’Convexity World’ in Optimization, Appl. Math. Optim. 45 (2002) 169-184.
[5] B. Polyak, Convexity of quadratic transformations and its use in Control and Optimization, J.Optim. Theory Appl., 99 (1998), 553-583.
[6] F. Flores-Bazán, F. Opazo Lagos, Joint-Range convexity for a pair of inhomogeneous quadraticfunctions and applications to QP, submited manuscript, arXiv:1508.01612.
Departamento de Ingeniería Matemática, Universidad de Concepción, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Generación de benchmark de fondos para elsistema de pensiones en Chile, un enfoque
basado en optimización estocásticaDaniel Espinoza G. Giorgiogiulio Parra De B.
Abstract
Por ley en Chile, los fondos de pensión son administrados por entidades privadas con reg-ulaciones particulares denominadas “Administradora de Fondos de Pensiones” (AFP).Estas empresas administran los fondos en las cuentas de capitalización individual decada uno de sus afiliados, conformando en la actualidad un sistema con seis AFPs, 11millones de afiliados y US$ 112.5 billones americanos (109) en los fondos, un 43.6% delPIB. 1
La actual estructura de multifondos del sistema se obtuvo a través de un procesoiterativo que se inició el año 1981 con el Decreto Ley 3.500 del 6 de diciembre de 1980que reemplazó el antiguo sistema de repartos por uno basado en el ahorro obligatorioindividual. Inicialmente el sistema contó con un único fondo, conformado a diciembrede 2007 por un 27.47% en renta variable y 72.46% en renta fija. En mayo de 2000, entróen funcionamiento un fondo adicional, más conservador, compuesto sólo por renta fija.En febrero de 2002 se dicta la Ley 19.795 que introduce tres nuevos fondos, entregandola actual estructura de multifondos, conformado por cinco fondos designados por lasletras A,B,C,D y E, donde a través de la regulación en la elegibilidad, proporción ycobertura, se implementa los perfiles riesgo-retorno de los diferentes fondos, siendo elfondo A el más agresivo y el fondo E el más conservador.
Como es común en la administración activa de inversiones, los fondos de pensión decada AFP son evaluados contra el promedio del mes anterior de los demás participespara igual fondo, teniendo como restricción dura tener un tracking error menor al 3%en tres años. Esto entrega un fuerte incentivo al sistema a tomar comportamientode mandas, tal como expone [3] en el caso de Polonia y [4] para Chile por nombraralgunos, generando AFPs líderes del sistema y seguidores. En esta línea, el presentetrabajo entrega una visión y propuesta de benchmark para comparar la rentabilidadde los respectivos fondos y su promedio mensual, así como un mecanismo para evaluarmecanismos regulatorios basados en la administración del riesgo propiamente tal.
Departamento de Ingeniería Industrial, Universidad de Chile, e-mail: [email protected] de Ingeniería Industrial, Universidad de Chile, e-mail: [email protected]
1Cifras informadas por Superintendencia de pensiones al 7 de Septiembre de 2015, con PIB base del 2014
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Metodológicamente se resuelve un problema de optimización estocástica de portafo-lio, sobre un poliedro que incorpora los límites de inversión, restricciones de liquidezy concentración y, adicionalmente, permite diferenciar los perfiles de riesgo-retornode cada fondo, resolviendo como función objetivo diferentes propuestas de medidas deriesgo sobre la distribución de pérdidas del portafolio, con diferentes niveles de aversión,reconociendo el carácter estocástico del problema. En particular, se utiliza y resuelve deforma estadísticamente significativa la Medida de Riesgo Entrópica, no resuelta antesen problemas de este tipo.
Para la resolución de los problemas de optimización se incorpora la metodologíaSAA por sus siglas en inglés (Sample Average Aproximation), propuesta en [5] (y am-pliamente extendida y utilizada, por citar algunos: [6],[7]) que resuelve cada instanciaequivalente cierta del problema de optimización con realización particulares de las vari-ables aleatorias independientes e idénticamente distribuidas, repitiendo este procesouna cantidad significativa de veces, tratando los resultados como realizaciones aleato-rias, obteniendo con ellos, adicional a las soluciones, una medida de la convergenciaestocástica y significáncia estadística de la misma.
A fin de poder estudiar ciertas propiedades deseables, más allá de la coherencia [2],como es la consistencia intertemporal, se consideraron las siguientes medidas de riesgo:Valor esperado (Expected Value), Valor en riesgo condicional (Conditional Value atRisk (CVaR)), una combinación convexa de los dos anteriores (ECVaR) y la medida deriesgo Entrópica (Entropic Risk Measure).
Como conclusión y aporte metodológico, se logra resolver de forma estadísticamentesignificativa el problema de portafolios con la Medida de Riesgo Entrópica, siendo laúnica medida, como recientemente fue demostrada en [1], que cumple la consistencia enla aditividad, interpretable como una noción de consistencia intertemporal, propiedadesencial para lograr óptimos globales vía optimización local por rolling forward. Elenfoque de generación de cortes con ajuste iterativo de soportes, para la aproximación dela exponencial en la vecindad de la región óptima, muestra tener muy buenos resultados,siendo fácilmente generalizable a otras funciones no lineales.
References[1] Cominetti and Torrico, Alfredo. Additive consistency of risk measures and its application to
risk-averse routing in networksarXiv preprint arXiv:1312.4193, 2013.
[2] Artzner, Philippe and Delbaen, Freddy and Eber, Jean-Marc and Heath, David. CoherentMeasures of Risk1. Risk management: value at risk and beyond , pp.145, Cambridge UniversityPress (2002).
[3] Kominek, Zbigniew. Regulatory induced herding? Evidence from Polish pension funds. (2006)
[4] Stein, Roberto and Miranda, Pedro and Risco, Rodolfo. Herding in Chile: the case of equitytrading in the Chilean pension fund market, (2006) Estudios de Administración , volumen 18,(2011)
173
[5] Kleywegt, Anton J and Shapiro, Alexander and Homem-de-Mello, Tito The sample average ap-proximation method for stochastic discrete optimization, (2006) SIAM Journal on Optimization,V12, No.2,479–502 (2002) volumen 18, (2011)
[6] Verweij, Bram and Ahmed, Shabbir and Kleywegt, Anton J and Nemhauser, George andShapiro, Alexander.The sample average approximation method applied to stochastic routing problems: a computa-tional study, Computational Optimization and Applications, V.24, No.2-3, pp 289–333, 2003.
[7] Pagnoncelli, BK and Ahmed, Shapiro and Shapiro, ASample average approximation method for chance constrained programming: theory and appli-cations, Journal of optimization theory and applications, V.142, No.2, pp 399–416, 2009.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Assessing Fishery Management and RecoveryStrategies through Viability Theory
Héctor Ramírez
Abstract
In this talk we construct a theoretical framework that permits, first, to assess the sus-tainability of fishery management strategies and, second, to propose recovery plans foroverexploited fisheries. The proposed framework allows facing situations when severalconflicting objectives have to be accounted for. In a first application, stochastic via-bility ranks management strategies according to their probability to sustain economicand ecological outcomes over time. This approach is then extended to build stochasticsustainable production possibility frontiers representing the trade-offs between sustain-ability objectives at any risk level, given the current state of the fishery. We thus studythe viability of effort and quota strategies when catch and biomass levels have to besustained. In the second application, a suitable deterministic bioeconomic dynamic per-mits to simulate divers recovery plans and a related optimization problem allows findinga recovery plan which minimizes the total cost of the recovering process. Here the totalcost has been defined as the intertemporal sum of the differences between the objectiveof the community and the recovery strategy proposed by the model (both quantitiesare stated in term of catches), Finally, both approaches are applied and illustrated inChilean fisheries.
References[1] V. Martinet, J. Peña, M. de Lara, H. Ramírez: Risk and Sustainability: Assessing Fishery
Management Strategies, Environmental and Resource Economics, 2015, ISSN 0924-6460, pp.1–25.
[2] FIC-R BIP N 30110834: Regional Government of Valparaíso, Chile. Quantitative tools for asustainable rebuilding of the Chilean Hake, 2014.http://www.recuperemoslamerluza.cl/
[3] M. De Lara, P. Gajardo, and H. Ramírez: Viable states for monotone harvest models. Syst.Control Letters, 60:192-197, 2011.
∗Universidad de Chile, e-mail: [email protected]∗El trabajo es financiado por el proyecto BASAL (PFB03) Centro de Modelamiento Matemático de la
Universidad de Chile y equipo BIONATURE del centro CIRIC de Inria Chile.
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Optimal feedback synthesis and minimal timefunction for the bioremediation of water
resources with two patchesH. Ramírez C. A. Rapaport V. Riquelme
Abstract
We study the bioremediation, in minimal time, of a water resource or reservoir usinga single continuous bioreactor. The bioreactor is connected to two pumps, at differentlocations in the reservoir, that pump polluted water and inject back sufficiently cleanwater with the same flow rate. This leads to a minimal-time optimal control problemwhere the control variables are related to the inflow rates of both pumps. We obtaina non-convex problem for which it is not possible to directly prove the existence of itssolutions. We overcome this difficulty and fully solve the studied problem by applyingPontryagin’s principle to the associated generalized control problem. We also obtainexplicit bounds on its value function via Hamilton-Jacobi-Bellman techniques.
El trabajo es financiado por los Proyectos DYMECOS 2 INRIA Associated team, proyectoBIONATURE de CIRIC INRIA CHILE, CONICYT REDES 130067, CONICYT ACT 10336,FONDECYT 1110888, proyecto BASAL (Centro de Modelamiento Matemático, Universidad deChile), MathAmsud 15MATH-02, y Beca Doctorado Nacional Convocatoria 2013 folio 21130840CONICYT. Se agradece también al Departamento de Postgrado y Postítulo de la Vicerrectoría deAsuntos Académicos (Universidad de Chile) y al Institut Français (Embajada de Francia en Chile).
References[1] H. Ramírez C., A. Rapaport, V. Riquelme: Minimal-time bioremediation of water resources
with two patches, HAL-01154435, To appear in SIAM Journal on Control and Optimization.
[2] P. Gajardo, H. Ramírez C., A. Rapaport, V. Riquelme: Bioremediation of Natural WaterResources via Optimal Control Techniques In: Rubem P Mondaini. (ed): BIOMAT 2011, 178–190. BIOMAT consortium, Rio de Janeiro (2012).
Departamento de IngenierÃŋa MatemÃątica y Centro de Modelamiento Matemático (UMI2807, CNRS),Universidad de Chile, Beauchef 851, Casilla 170-3, Santiago 3, Chile e-mail:[email protected]
MISTEA, UMR 729 INRA/Supagro, Montpellier, France, MODEMIC, INRA/Inria team, Sophia-Antipolis, France, e-mail: [email protected]
Departamento de IngenierÃŋa MatemÃątica y Centro de Modelamiento MatemÃątico (UMI 2807,CNRS) Universidad de Chile, Chile; MISTEA, UMR 729 INRA/Supagro, Montpellier, France MODEMIC,INRA/Inria team, Sophia-Antipolis, France, e-mail: [email protected]
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[3] P. Gajardo, J. Harmand, H. Ramírez C. A. Rapaport: Minimal time bioremediation of naturalwater resources, Automatica 47 (8), 1764–1769 (2011).
[4] P. Gajardo, H Ramírez C., A. Rapaport: Minimal time sequential batch reactors with boundedand impulse controls for one or more species, SIAM Journal on Control and Optimization,47(6) 2827–2856, (2008).
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Dualidad en optimización vectorialM. A. Rojas-Medar L. Batista dos Santos Camila Isoton
Abstract
El concepto de WD-invexidad ha sido recientemente introducido en problemas de pro-gramación no lineal con restricciones de desigualdad. Para tales problemas, un prob-lema dual fue propuesto y el siguiente resultado fue establecido: el problema dual esWD-invex si y solamente si el par de problemas primal-dual satisface la propiedad dedualidad débil. En este trabajo, discutimos la noción de WD-invexidad para problemasmultiobjetivos. Obtenemos resultados similares a [3].
La Teoría de la Dualidad es una herramienta fundamental en el análisis de problemasde optimización. Desde los años 80, muchos autores se han dedicado a la formulación dediferentes problemas duales y también a la relajación de las hipótesis de convexidad de lasfunciones involucradas, vea [1], [2], [4], [5] y las referencias ahí.
1 Formulación del problema y conceptos de soluciónConsideremos el siguiente problema de optimización vectorial:
Minimizar f(x) := (f1(x), · · · , fp(x))sujeto a:g(x) := (g1(x), · · · , gm(x)) 5 0x ∈ X
(P)
• Asumiremos que fj, gi : Rn→ R son diferenciables en el abierto X ⊂ Rn;
• F := x ∈ X : g(x) 5 0(6= ∅) es el conjunto factible;
• x ∈ F, I(x) := i : gi(x) = 0 son las restricciones activas en x.
Universidade Federal do Paraná, e-mail: [email protected] de Tarapacá, e-mail: [email protected] Federal do Paraná, e-mail: [email protected]
178
1.1 WD-invexidad I: dual de Wolfe:
Vamos a enunciar uno de los resultados obtenidos, para ello consideremos el siguiente prob-lema
Maximizar L(r, λ, u) := rTf(u) + λTg(u)sujeto a:rT∇f(u) + λT∇g(u) = 0λ = 0, u ∈ X
(WD)
Definición. Decimos que el problema (P) es WD-invex I si existe η : X ×X → Rn talque, para u ∈ X, x ∈ F una de las siguientes condiciones ocurre:
•f(x)− f(u)−∇f(u)η(x, u) = 0−g(u)−∇g(u)η(x, u) = 0
ó
•−∇f(u)η(x, u) > 0−∇g(u)η(x, u) = 0.
• Diremos que vale la propriedad de dualidad débil entre (P) y (WD) si para cada vectorr ≥ 0 fijado, se tiene
rTf(x) ≥ rTf(u) + λTg(u), ∀x ∈ F,∀(u, λ) ∈ D.
Denotaremos:F = x ∈ X : g(x) 5 0,D = (u, λ) ∈ X × Rm
+ : rT∇f(u) + λT∇g(u) = 0.
Teorema Vale dualidad débil entre (P) y (WD) si y solamente si el problema (P) esWD-invex I.
References[1] M. A. Hanson: On sufficiency of the Kuhn-Tucker conditions, JMAA, vol. 80, 545-550
(1981).
[2] D. H. Martin: The essence of invexity, JOTA, vol. 47, 65-76 (1985).
[3] V. I. Ivanov: Duality in nonlinear programming, Optim. Lett., 7, 1643-1658 (2013).
[4] R. Osuna-Gómez, A. Rufián-Lizana, P. Ruiz-Canales: Invex functions and generalizedconvexity in multiobjective programming, JOTA, vol. 98, 651-661 (1998).
[5] L.B. dos Santos, C. Isoton, M.A. Rojas-Medar, V.A. de Oliveira, WD-invexity in mul-tiobjective problems, Prepinter, 2015.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Comparision of MINC and MRMTconfigurations:
Effects of spatial structure and biomass diffusionA. Rapaport H. Ramírez A. Rojas-Palma J. de Dreuzzy
Abstract
In a paper in preparation we prove that under certain assumptions controllability, theMINC (Multiple Interacting Continua) and MRMT (Multirate Mass Transfer) config-urations can be input-output equivalents for certain parameters values, but withoutbiomass diffusion. In this presentation the idea is to extend the results to the case ofdiffusion of biomass (non-linear systems) by considering the simplest case, three reactorsconnected linear biomass growth. The existence and stability of a positive equilibriumwill be studied and two nonlinear optimization problems will be defined in steady state,which will serve to compare which of the two configurations is better in terms of theoutput flow.
References[1] D. Anderson , Compartmental Modeling and Tracer Kinetics, Lecture Notes in Biomathe-
matics, Vol. 50, Springer, 1983.
[2] J. Carrera, X. Sanchez-Vila, I. Benet, A. Medina, G. Galarza and J. Guimera, Onmatrix diffusion: formulations, solution methods and qualitative effects, Hydrogeology Journal,Vol. 6(1), pp. 178–190, 1998.
[3] L. Donado, X. Sanchez-Vila, M. Dentz, J. Carrera and D. Bolster, Multi-component reactive transport in multicontinuum media, Water Resources Research, Vol. 45(11),pp. 1–11, 2009.
UMR MISTEA, Montpellier; INRA-INRIA ’MODEMIC’ team, INRIA Sophia-Antipolis Méditerranée,France. e-mail: [email protected] , e-mail: [email protected]
e-mail: [email protected] Géosciences, Rennes, France, e-mail: [email protected] work was developed in the context of the DYMECOS 2 INRIA Associated team and of project
BIONATURE of CIRIC INRIA CHILE, and it was partially supported by CONICYT grant REDES 130067.The second and fourth authors were also supported by CONICYT-Chile under ACT project 10336, FONDE-CYT 1110888, BASAL project (Centro de Modelamiento Matemático, Universidad de Chile), CONICYTnational doctoral grant and CONICYT PAI/ Concurso Nacional Tesis de Doctorado en la Empresa, convo-catoria 2014, 781413008.
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[4] J.-R. de Dreuzy, A. Rapaport, T. Babey, J. Harmand; Influence of porosity struc-tures on mixing-induced reactivity at chemical equilibrium in mobile/immobile Multi-Rate MassTransfer (MRMT) and Multiple INteracting Continua (MINC) models, Water Resources Re-search, Vol. 49(12), pp. 8511–8530, 2013.
[5] G. de Marsily, Quantitative Hydrogeology: Groundwater Hydrology for Engineers, AcademicPress, Orlando, 1986.
[6] C. Fetter, Contaminant Hydrogeology, (2nd edition). Waveland Pr Inc., 2008.
[7] L. Farina and S. Rinaldi, Positive Linear Systems, Theory and Applications, Prentice Hall,2000.
[8] L. Gelhar, Stochastic Subsurface Hydrology Prentice Hall, Engelwood Cliffs, New Jersey,1993.
[9] L. Gelhar, C. Welty and R. Rhefeldt, A Critical Review of Data on Field-Scale Dis-persion in Aquifers. Water Resources Research, Vol. 28(7), pp. 1955–1974, 1992.
[10] R. Haggerty and S. Gorelick Multiple-rate mass transfer for modeling diffusion and sur-face reactions in media with pore-scale heterogeneity, Water Resources Research, Vol. 31(10),pp. 2383–2400, 1995.
[11] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1985.
[12] J. Jacquez and C. Simon Qualitative theory of compartmental systems, SIAM Review, Vol.35(1), pp. 43–79 , 1993.
[13] T. Kailath, Linear Systems, Prentice Hall, 1980.
[14] S. McKenna, L. Meigs and R. Haggerty, Tracer tests in a fractured dolomite 3. Double-porosity, multiple-rate mass transfer processes in convergent flow tracer tests Water ResourcesResearch, Vol. 37(5), pp. 1143–1154, 2001.
[15] K. Pruess and T. Narasimhan, A practical method for modeling fluid and heat-flow infractured porous-media, Society of Petroleum Engineers Journal, Vol. 25(1), pp. 14–26, 1985.
[16] C. Steefel, D. DePaolo and P. Lichtner, Reactive transport modeling: An essential tooland a new research approach for the Earth sciences, Earth and Planetary Science Letters, Vol.240(3-4), pp. 539–558., 2005.
[17] M. Vangenuchten and J. Wierenga, Mass-transfer studies in sorbing porous-media .1.Analytical solutions, Soil Science Society of America Journal, Vol. 40(4), pp. 473–480, 1976.
[18] , G. Walter and M. Contreras, Compartmental Modeling with Networks, Birkäuser, 1999.
[19] J. Warren, P. Root and M. Aime, The Behavior of Naturally Fractured Reservoirs, Societyof Petroleum Engineers Journal, Vol. 3(3), pp. 245–255, 1963.
[20] M. Willmann, J. Carrera, X. Sanchez-Vila, O. Silva and M. Dentz, Coupling ofmass transfer and reactive transport for nonlinear reactions in heterogeneous media, WaterResources Research, Vol. 46(7), pp. 1–15, 2010.
[21] B. Zinn, L. Meigs, C. Harvey, R. Haggerty, W. Peplinski and C. von Schwerin,Experimental visualization of solute transport and mass transfer processes in two-dimensionalconductivity fields with connected regions of high conductivity, Environmental Science & Tech-nology, Vol. 38(14), pp. 3916–3926, 2004.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Optimization of the concentration changes in achemostat with one species
Térence Bayen Jérôme Harmand Matthieu Sebbah
Abstract
In this work, we study the problem of driving in minimal time a system describing achemostat model to a target point. More precisely, we consider the usual chemostatmodel of a single biomass and a single substrate given by the following equations (see[4]):
X = µ(S)X − uX,S = −µ(S)X + u(Sin − S),
(1)
where X, resp. S, represents the biomass concentration, resp. the substrate concen-tration, Sin the input substrate concentration, u the dilution rate and µ the growthfunction of the biomass.
Given an initial condition (X0, S0) and a target point (X,S), we are interested inreaching (X,S) from (X0, S0) in minimal time under variations of the dilution rate u,that is, studying the value function v defines as follows:
v(X0, S0) := infu(·)
t(u) s.t. Xu(t(u)) = X and Su(t(u)) = S, (2)
where t(u) is the first time such that Xu(t(u)) = X and Su(t(u)) = S.When µ is of Monod-type, i.e. of the form
µ(s) = µS
S + k,
for some µ > 0 and k > 0, the problem has been studied in [2], where it is shown thatthe control (the dilution rate) is necessarily bang-bang.
In this work, we consider a growth function of Haldane type, i.e. of the form
µ(s) = µS
S2
ki+ S + ks
,
for some µ > 0, ki > 0 and ks > 0. Using the Pontryagin Maximum Principle ([3]) andgeometric control theory ([1]), we show that in some cases the control is not necessarilybang-bang but might admit frame curves such as singular locus and switching curve.
Térence Bayen, e-mail: [email protected]érôme Harmand, e-mail: [email protected] Sebbah, e-mail: [email protected]
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References[1] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Mani-
folds, vol. 43, Springer-Verlag, Berlin, 2004.
[2] G. D’Ans, P. Kokotovic, D. Gottlieb, Time-Optimal Control for a Model ofBacterial Growth, J. Optim. Theory and Applications, vol. 7, 1, 1971.
[3] L.S. Pontryagin, V.G. Boltyanskiy, R.V. Gamkrelidze, E.F. Mishchenko,Mathematical theory of optimal processes, The Macmillan Company, 1964.
[4] H.L. Smith and P. Waltman, The theory of the chemostat, Dynamics of microbialcompetition, Cambridge University Press, 1995.
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Problemas Inversos y Control de EDP
Encargado de Sesión : Rodrigo Lecaros
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On the cost of null controllability of some linearpartial differential equations
Nicolás Carreño
Abstract
In this talk, we will present some results concerning the cost of null controllability oftwo linear equations posed on a bounded interval. First, we consider a linear KdVequation with a boundary control on the left extreme of the interval. We prove that, asthe dispersion coefficient goes to zero, the size of the control that drives the solution tothe null state increases exponentially for every control time. Then, we consider a linearfourth-order equation with two boundary controls. In this case, we show that the size ofthe controls explode as the diffusion coefficient vanishes if the control time is small. Onthe other hand, if the control time is large enough, the controls are uniformly boundedwith respect to the diffusion coefficient and, furthermore, their norms decrease to zeroexponentially.
References[1] N. Carreño, S. Guerrero. On the non-uniform null controllability of a linear KdV equation.
Asymptot. Anal. 94 (2015), no. 1-2, 33–69.
[2] N. Carreño, P. Guzmán. On the cost of null controllability of a linear fourth order parabolicequation. Preprint.
Universidad Técnica Federico Santa María, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
On the control of the improved Boussinesqequation
Eduardo Cerpa
Abstract
The so called ÒbadÓ Boussinesq equation was introduced to describe the flow of shallowwater waves with small amplitude. This equation can be approached by the ImprovedBoussinessq equation. In this talk we are interested in the control properties of thissystem. First, we consider a boundary control and prove that the system is approx-imately controllable but not exactly controllable. Second, we introduce an internalcontrol supported on a moving region and prove that the system is exactly controllable.The main tools we use are spectral analysis and the Moment Theory.
References[1] E. Cerpa, E. Crépeau, On the control of the improved Boussinesq equation, under review.
Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile, e-mail:[email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
An Inverse Problem for the Helmhotz Equationin a Layered Media.
Matías Courdurier
Abstract
An important element in the propagation of waves on a stratified media is thepropagation of the wave along waveguides. For acoustic waves in 2D, in a time harmonicregime, on an infinite medium with a single layer of width 2h, the propagation of thewave is described by the solution of the Helmholtz equation
∆u+ k2n2(x, z)u = f, (x, y) ∈ R2,
with a picewise constant index of refraction
n(x, z) =
n0 |x| < h
ncl |x| > h,
and where the right radiation conditions are described in [1]. In this talk we will proposean inverse problem in this setting and we will present the progress made in the studyof such inverse problem using the Green’s function provided in [2].
References[1] Ciraolo G., Magnanini R., A radiation condition for uniqueness in a wave propagation problem
for 2-D open waveguides, Math. Methods in the Applied Sciences 32 (10) (2009), 1183-1206
[2] Magnanini R., Santosa F. , Wave propagation in a 2-D optical waveguide, SIAM J. Appl. Math.,61 (2001) 1237-1252.
[3] Wilcox, C. H. Sound Propagation in Stratified Fluids. Applied Mathematical Sciences. Vol. 50.New York: Springer-Verlag. (1984).
Pontificia Universidad Católica de Chile. e-mail: [email protected] work with Eric Bonnetier, Université Joseph Fourier Axel Osses Universidad de Chile. and
Faouzi Triki. Université Joseph Fourier.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Detection of Several Obstacles in a Stokes Flow:A mixed approachMatías Godoy Campbell
Abstract
We consider the inverse problem of detecting the location and shape of several objectsimmersed in a fluid flowing in a larger bounded domain Ω from boundary measurements.The fluid is governed by the steady-state Stokes equations.
For this goal we consider a Kohn-Vogelius cost type function. This functional penal-izes erroneous configurations for the considered system and even more, its minimizationwith respect of all possible admissible configurations is equivalent to the resolution ofour inverse problem.
In order to determine numerically the number and relative position of the objects,we perform a topological sensitivity analysis of the considered functional, obtaining anasymptotic expansion which leads to the expression of the so-called topological gradientof the cost function.
As a complementary task, we compute the shape derivative of the cost function,which allows to improve the shape of the detected objects via our primary topologicalmethod.
Then, we present some numerical simulations of this mixed approach which combinesthe topological and geometrical shape optimization methods.
We finally discuss, briefly, the possibilities when there exists an inaccessible regionof the boundary for the measurements which leads to a data completion problem.
This is a joint work with Fabien Caubet (IMT, France) and Carlos Conca (U. deChile).
References[1] F. Caubet, C. Conca and M. Godoy: On the detection of several obstacles in 2D Stokes flow:
Topological sensitivity and combination with shape derivatives, to appear in Inverse Problemsand Imaging.
[2] F. Caubet and M. Dambrine: Localization of small obstacles in Stokes flow, Inverse Problems,28(10) (2012).
Departamento de Ingeniería Matemática, Universidad de Chile, Institut de Mathématiques de Toulouse,Université Paul Sabatier. e-mail: [email protected]
This project has been partially supported by ECOS-CONICYT Grant C13E05, PFBasal-01, PFBasal-03and Fondecyt Grant No.1140773 and also by the Grant CONICYT-PCHA/Doctorado Nacional/2012.
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Controllability of coupled systems withSchrödinger equations.
Alberto Mercado Saucedo
Abstract
In this talk we present some control problems regarding systems of coupled partialdifferential equations, when one of them is a linear Schrödinger equation. We addressthe problem of controlling the system using less controls than equations. We presenttwo controllability problems: 1) A system with two N -dimensional linear Schrödingerequations with a control supported in a region not satisfying the classical geometricalcontrol condition (see [1]); and 2) The problem of internal null controllability of asystem coupling a Schrödinger and a linear Korteweg - de Vries equation (see [2]).
References[1] M. López-García, A. Mercado, L. de Teresa. Null controllability of a cascade system of
Schrödinger equations. Submitted.
[2] F.D. Araruna, E. Cerpa, A. Mercado, M.C. Santos. Internal null controllability of a linearSchrödinger-KdV system on a bounded interval. J. Differential Equations 260 (2016) 653Ð687.
Departamento de Matemática, Universidad Técnica Federico Santa María.e-mail:[email protected]
Partially supported by FONDECYT 1120610, Basal CMM U. de Chile and ANILLO ACT1106
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An ADER type scheme for evolving differentialoperators
G. Montecinos J. C. López R. LecarosJ. Ortega E. F. Toro
Abstract
In this work we propose a numerical strategy to solve partial differential equations inwhich the evolution of differential terms, including mixed time and spatial derivatives,is addressed. Examples of this type of problems are those derived from the water wavesequations. Like the Saint-Venant type equations and the Shallow water equations.Motivated by the work of Zambra et al. [6], we propose a one-step finite volumeevolution of differential operators. In [6], a globally implicit strategy to solve theRichard equation, was proposed. In the present work a locally implicit formulationis investigated and the operator to be evolved may contain differential terms. In thepresent methodology we follow the ADER philosophy first put forward by Toro et al.[4, 5]. This methodology is based on two steps: i) a polynomial reconstruction ofthe data; ii) solutions to Generalized Riemann Problems (GRP), which allow us toevaluate numerical fluxes and source terms. For a review of GRP solvers, see [1, 3].We implement here the GRP solver proposed by Dumbser et al. [2]. This solver usesthe Discontinuous Galerkin method to construct a local predictor inside of each cells.It is well known that this solver is a suitable method to deal with problems containingstiff source terms, reconciling stability and accuracy. Additionally, the present methodallows the use of a CFL-type condition and thus the high-order of accuracy is achievedin space and time. We show theoretically, the convergence of the present scheme.Furthermore, we carried out an empirical convergence rate assessment, in order toillustrate the high-order of accuracy.
Key words: Water-wave equations, ADER schemes, high-order of accuracy.
References[1] C. E. Castro and E. F. Toro. Solvers for the high–order Riemann problem for hyperbolic
balance laws. Journal of Computational Physics, 227:2481–2513, 2008.
CMM Universidad de Chile,DIM Universidad de Chile e-mail: [email protected] , e-mail:[email protected], e-mail: [email protected], e-mail: [email protected]
DICAM University of Trento, e-mail: [email protected]
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[2] M. Dumbser, C. Enaux, and E. F. Toro. Finite volume schemes of very high order of accuracyfor stiff hyperbolic balance laws. Journal of Computational Physics, 227(8):3971–4001, 2008.
[3] G. Montecinos, C. E. Castro, M. Dumbser, and E. F. Toro. Comparison of solvers for the gen-eralized Riemann problem for hyperbolic systems with source terms. Journal of ComputationalPhysics, 231:6472–6494, 2012.
[4] E. F. Toro, R. C. Millington, and L. A. M. Nejad. Towards very high–order Godunov schemes.In Godunov Methods: Theory and Applications. Edited Review, E. F. Toro (Editor), pages905–937. Kluwer Academic/Plenum Publishers, 2001.
[5] E. F. Toro and V. A. Titarev. Solution of the generalised Riemann problem for advection–reaction equations. Proceedings of the Royal Society of London A, 458:271–281, 2002.
[6] C. E. Zambra, M. Dumbser, E. F. Toro, and N. O. Moraga. A novel numerical method ofhigh-order accuracy for flow in unsaturated porous media. International Journal for NumericalMethods in Engineering, 89(2):227–240, 2012.
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Stability numbers to Timoshenko’s system withshear boundary dissipation
Margareth Alves Jaime E. Muñoz Rivera Mauricio Sepúlveda
Abstract
In this paper we consider a Timoshenko’s model with only one boundary dissipation,effective over the shear force. We introduce two numbers χ0 which depends on thedifference of the wave speed and χ1 that depends on the size of the interval. Thisnumbers will describe the asymptotic behavior of the system. That is, we prove strongstability if and only if χ1 is not a rational multiple of π2. If additionally χ0 = 0 andχ1 < 1/2, then the corresponding semigroup is exponentially stable.
References[1] D. Almeida Júnior, M. L. Santos and J. E. Muñoz Rivera, Stability to 1-D thermoelas-
tic Timoshenko beam acting on shear force, Z. Angew. Math. Phys., 65 (2014), pp. 1233–1249.
[2] M. Alves, J. Muñoz-Rivera, M. Sepúlveda, O. Vera and M. Zegarra, The asymptoticbehaviour of the linear transmission problem in viscoelasticity, Math. Nachr., 287 (2014), pp.483–497.
[3] M. Alves, J. Muñoz-Rivera, M. Sepúlveda and O. Vera, Exponential and the lack ofexponential stability in transmission problems with localized Kelvin-Voigt dissipation, SIAM J.Appl. Math., 74 (2014), pp. 354–365 .
Departamento de Matemática. Universidade Federal de Viçosa. Viçosa. 36570-000. MG. Brasil, e-mail:[email protected]
National Laboratory for Scientific ComputationRua Getulio Vargas 333, Quitadinha-Petrópolis 25651-070, Rio de Janeiro, RJ, Brasil, e-mail:[email protected]
CI2MA and DIM, Universidad de Concepción, Concepción, Chile, e-mail: [email protected]
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Teoría de Números
Encargado de Sesión : Amalia Pizarro
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Ramas y extensiones de cuerposL. Arenas-Carmona
Abstract
El árbol de Bruhat-Tits es un grafo cuyos vértices son los órdenes maximales del álgebraM2(K) dondeK es un cuerpo local. Dos de tales órdenes son vecinos si en algún sistemade coordenadas tienen la forma
D1 =
(OK OKOK OK
), D2 =
(OK π−1OKπOK OK
),
donde π es un parámetro uniformizante de K. En trabajos anteriores (ver [5]), el autordefinió la rama de un suborden H como el mayor subgrafo cuyos vértices correspondena órdenes que contienen a H. En este trabajo estudiamos el comportamiento de estasramas bajo extensiones de cuerpos y aplicamos esta idea para calcular los invariantesde una rama para un órden dado explícitamente en términos de sus generadores.
References[1] L. Arenas-Carmona, Applications of spinor class fields: embeddings of orders and quater-
nionic lattices, Ann. Inst. Fourier 53 (2003), 2021–2038.
[2] L. Arenas-Carmona, Representation fields for commutative orders, Annales de l’institutFourier 62 \2 (2012), 807-819.
[3] L. Arenas-Carmona, Maximal selectivity for orders in fields, J. Number T. 132, (2012),2748-2755.
[4] L. Arenas-Carmona, Representation fields for cyclic orders. Acta Arith. 156 (2012), 143-156.
[5] L. Arenas-Carmona, Eichler orders, trees and representation fields. Int. J. Number Theory,9 (2013), 1725-1741.
[6] M. Arenas, L. Arenas-Carmona, and J. Contreras, On optimal embeddings and trees,Preprint.
[7] J. Brzezinski, On embedding numbers into quaternion orders. Comm. Math. Helvetici 66(1991), 302-318.
Fac. Cs., Univ. de Chile, e-mail: [email protected] by Fondecyt No 1140533. En conjunto con M. Arenas y C. Bravo.
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Un retículo Hermiteano centralAna Cecilia de la Maza Remo Moresi
AbstractDefiniciones previas
Un retículo Hermiteano es un algebra (L, 0, 1, · , + , ⊥, b) tal que
i) (L, 0, 1, · ,+) es un retículo modular con cotas universales 0 , 1.ii) ⊥ : L→ L es una operación unitaria con 1⊥ = 0 y
x ≤ (x⊥y)⊥ ∀ x, y ∈ L; (1)
iii) b ∈ L es una operación nula con
xx⊥ ≤ b ∀ x ∈ L. (2)
Un ejemplo canónico de retículo Hermiteano indexado está dado por el retículo de lossubespacios L(E) de un espacio Hermiteano (E, φ) de dimensión menor o igual a ℵ0,sobre algún algebra de divisón k, junto con la relación de ortogonalidad inducida porla forma φ. El rol de b lo juega el subespacio formado por los vectores de traza valuadaE∗, subespacio que adquiere importancia cuando la caracterŊística no es 2. .
En esta charla se describirá el retículo hermiteano generado por a con las condiciones
a ≤ b⊥ ∧ bb⊥ = 0, (3)
junto con dos condiciones que hacen este retŠculo finito.
References[DM1] A.C. de la Maza, R.Moresi, On modular lattices generated by chains, Algebra Universalis 54
(2005), 475-488.
[DM2] A.C. de la Maza, R.Moresi, Hermitean (semi) lattices and Rolf ’s lattice, Algebra Universalis66 (2011), 49-62.
[G1] H. Gross, Quadratic forms in infinite dimensional vector spaces, Birkäuser, Boston, 1979.
[G2] H. Gross, Lattices and infinite-dimensional forms. “The lattice method”, Order 4 (1987),233-256.
[KKW] H. A. Keller, U.-M. Künzi, M. Wild (eds), Orthogonal geometry in infinite dimensionalvector spaces, Heft 53, Bayreuther Mathematische Schriften, Bayreuth, 1998.
[R] H. L. Rolf, The free lattice generated by a set of chains, Pacific J. Math. 8 (1958), 585-595.
Departamento de Matemática y Estadística, Universidad de la Frontera,Temuco, Chile email :[email protected]
Cerfim, cp 1132, 6601 Locarno, Switzerland. email : [email protected]
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Distribución asintótica de puntos de Hecke sobreCp
Sebastián Herrero Miranda
Abstract
Sea p un número primo, Cp la completación de una clausura algebraica de Qp y Ell(Cp)el espacio de moduli de curvas elípticas sobre Cp (módulo isomorfismo sobre Cp). DadaE ∈ Ell(Cp) y n ∈ N definimos los puntos de Hecke de orden n asociados a E comolos puntos E′ ∈ Ell(Cp) que admiten una isogenia E → E′ de grado n. Esto equivale atener E′ = E/C donde C es un subgrupo de E de cardinalidad n. Con estos puntos deHecke podemos construir el divisor
Tn(E) =⊕
C≤E,#C=n
E/C
sobre Ell(Cp). Nosotros estamos interesados en describir la distribución de Tn(E)cuando n tiende a infinito. El caso clásico sobre C es bien conocido: los puntos deHecke se equidistribuyen respecto a una medida natural en Ell(C), la medida hiper-bólica. En particular, la distribución asintótica de dichos puntos es independiente delpunto inicial E ∈ Ell(C). Nuestro resultado principal es una descripción de la distribu-ción asintótica de Tn(E) cuando E ∈ Ell(Cp) bajo ciertas condiciones sobre el tipo dereducción de E módulo M, el ideal maximal del anillo de enteros de Cp, y sobre lanorma p-ádica de n.Esta presentación se basa en un trabajo en colaboración con Ricardo Menares (PUCV)y Juan Rivera Letelier (PUC - U. of Rochester).
Este trabajo fue financiado por beca CONICYT Doctorado Nacional 21130412, e-mail:[email protected]
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Optimal bounds for Büchi’s problem in modulararithmetic
Pablo Sáez Xavier Vidaux Maxim Vsemirnov
Abstract
We study the second order analogue of the problem of finding optimal lower and upperbounds for the length of sequences of squares in arithmetic progression modulo a prime,and some connections with the computational problem of finding a quadratic non-residue modulo a prime. More precisely, we work modulo an integer and our objects ofstudy are those sequences of squares whose second difference is an invertible constant.The main results of our work is a number of exact formulae that allow to reducethe problem to prime moduli. We observe several phenomena which are supported byextensive numerical computations. We also discuss the case where the leading coefficientof the second degree polynomial defining the sequence of squares is non invertible.
References[1] H. Pasten, T. Pheidas, X. Vidaux, A survey on Büchi’s problem : new presen-
tations and open problems, Proceedings of the Hausdorff Institute of Mathemat-ics, Zapiski POMI 377, 111-140, Steklov Institute of Mathematics. Published onlinehttp://www.pdmi.ras.ru/znsl/2010/v377.html (2010).
[2] P. Sáez, X. Vidaux, M. Vsemirnov, Optimal bounds for Büchi’s problem in modular arithmetic,Journal of Number Theory 149, 368-403 (2015).
Independent, Casilla 64, San Pedro de la Paz, e-mail: [email protected] de Matemática, Universidad de Concepción, Casilla 160-C, Concepción-3, e-mail:
[email protected]. Petersburg Department of V. A. Stelkov Institute of Mathematics, 27 Fontanka, St. Petersburg,
191023, and St. Petersburg State University, Department of Mathematics and Mechanics, 28 Universityprospekt, St. Petersburg, 198504, Russia, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Una conexión entre la propiedad de Northcott yla indecidibilidad en anillos de enteros
totalmente realesXavier Vidaux
Abstract
Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción-3, e-mail:[email protected]
198
Estudio de Clases. Método Japonés
Encargado de Sesión: Carlos Cabezas
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Promoviendo el desarrollo de habilidades delpensamiento matemático en estudiantes del
sistema escolar: Una experiencia en el complejoeducacional la Granja de Cajón
Pamela Alarcón Valeria Carrasco Ciro GonzálezTeresa Sanhueza
Abstract
En este trabajo, se presentan los principales lineamientos de desarrollo de un proyectointerno de Pasantías Docentes en Centros Educativos, aprobado a mediados del pre-sente año en la Facultad de Educación de la UC Temuco.
El objetivo del Proyecto es “fortalecer el vínculo entre la Universidad Católica deTemuco y uno de los Centros Educativos que colaboran en la formación inicial de profe-sores, a través de la realización actividades conjuntas que fomenten en los estudiantes delestablecimiento educativo el desarrollo de las habilidades del pensamiento matemático”.
Conformado el equipo de trabajo de la Carrera de Pedagogía Media en Matemática,se propone coordinar entre este equipo y los profesores de matemática del Centro Ed-ucativo las necesidades de ayuda técnica de acuerdo a la realidad escolar en la cualse encuentran inmersos, de modo que a través del Estudio de Clases, se planifiquene implementen acciones que contribuyan al fortalecimiento profesional de los profe-sores de matemática del Centro Educativo, en relación al desarrollo de habilidades delpensamiento matemático (como la resolución de problemas, las habilidades de comu-nicación y argumentación, las de representación, y las de modelamiento matemático),implementar de manera conjunta estrategias didácticas que permitan a los estudiantesel desarrollo de estas habilidades que propicien un impacto positivo en sus aprendiza-jes, y analizar las implementaciones realizadas con el propósito de mejorar las prácticaseducativas a nivel de aula.
El presente trabajo pretende presentar el día de las jornadas los avances en el procesode ayuda técnica inicial que se le dio a las profesores, del proceso de recogida de lasnecesidades en función del tratamiento de las habilidades del pensamiento matemático,de la planificación de estrategias innovadoras, de la observación y la posterior reflexiónsobre la práctica docente ( Estudio de Clase: un caso).
200
References[1] Bruner, 1961, “The act of discovery”, Harvard Educational Review.
[2] George Polya, a partir de 1945, con su libro “How to solve it”
[3] Isoda, M.; Arcavi, A.; Mena, A., 2007, “El Estudio de Clases japonés en matemáticas”, EditorialUniversitaria Valparaíso, Valparaíso.
[4] Ministerio de Educación, 2013, “Bases Curriculares de Matemáticas vigentes”, Decreto 614 de2013.
[5] NCTM, 2003, pp 64: “Principios y Estándares para la educación matemática”.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Análisis de idoneidad didáctica del métodojaponés
desde un enfoque onto-semiótico de lainstrucción matemática
Carlos Cabezas Pedro Arteaga
Abstract
Esta ponencia, se enfoca en el análisis didáctico del método Estudio de Clases, en sucomponente "Ejecución de la Clase" y en la metodología para su desarrollo conocidacomo "Método Japonés". Se estudian las diversas componentes de una clase típicaorientada por un plan de clases, teniendo como herramienta de análisis, el sistemade indicadores empíricos que desarrollan la noción de idoneidad didáctica, introducidaen el marco del enfoque onto-semiótico del conocimiento y la instrucción matemática(Godino J., 2013). Es una característica del método japonés, la confección de un plande clases que contemple los diversos aspectos tanto disciplinares como didácticos y otroselementos que intervienen en el proceso de enseñanza y aprendizaje de la matemática.El plan de clases es concebido como un plan orientador de la clase que especifica ladirección en que se llevarán las diferentes interacciones, en todo lo que involucra elproceso de aula de modo que se logren los objetivos de aprendizaje propuestos por elprofesor. Un aporte de este trabajo se verifica en la comprobación empírica, con baseen el enfoque onto-semiótico de la instrucción matemática, de la idoneidad didáctica delmétodo japonés y en el análisis desde esta perspectiva, del plan de clases, que aportaantecedentes, orientaciones y criterios para la confección de otros planes de clase quecubran todos los aspectos involucrados en los planes ya tradicionales propuestos por elmétodo japonés.
References[1] Godino J. Indicadores de la idoneidad didáctica de procesos de enseãnza y aprendizaje de
las matemáticas. Cuadernos de Investigación y Formación en Educación Matemática .Año 8.Número 11. pp 111-132. Costa Rica. 2013.
[2] Isoda, M. & Olfos, R. (2010). El enfoque de Resolución de problemas en la enseñanza de lamatemática a partir del Estudio de Clases. Ediciones Universitarias de Valparaíso, P. Univer-sidad Católica de Valparaíso.
Universidad Católica del Maule, e-mail: [email protected] de Granada, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Problemas incorrectos como medio paradesarrollar aprendizaje profundo
Hugo Caerols Katia Vogt Geisse
Abstract
Uno de los objetivos fundamentales de quienes dirigimos los procesos de Enseñanza-Aprendizaje en matemáticas y también en otras disciplinas, es conseguir en nuestrosestudiantes un aprendizaje profundo de los temas de estudio. Este aprendizaje dependede varios factores, tales como el interés del estudiante por el tema, sus estrategias deaprendizaje, la dinámica de las clases, el tiempo del que dispone el estudiante para elcurso y del tipo de problemas a los que se enfrenta.
La mayoría de los problemas que proponemos a nuestros estudiantes, y que aparecenen los actuales textos de estudio, buscan una aplicación principalmente de maneradirecta de algún resultado y los menos están dirigidos al análisis de la materia. En estacharla presentaremos algunos resultados sobre cómo han reaccionado nuestros alumnosal verse enfrentados de manera dirigida a problemas incorrectos.
Encontrar y formular este tipo de problemas no es una tarea sencilla. Mostraremosun problema de modelamiento que hemos presentado a los alumnos de seis secciones delcurso de Ecuaciones Diferenciales Ordinarias dictado en la Universidad Adolfo Ibáñez.El problema de por si es sencillo de entender y bastante interesante. También pre-sentaremos un análisis de los resultados obtenidos por los alumnos, quienes trabajaronen grupo siguiendo las indicaciones de discutir sus resultados y analizar las distintasformas que encontraron de enfrentar el problema. Mostraremos cuánto se acercaron losestudiantes a una visión profunda del planteamiento y análisis del modelo asociado alproblema.
References[1] Arias, A. V., Cabanach, R. G., Pérez, J. C. N., Riveiro, J. M. S., Aguín, I. P., Martínez, S. R.
(2000). Enfoques de aprendizaje en estudiantes universitarios. Psicothema, 12(3), 368-375.
[2] Entwistle, N. Approaches to studying and levels of understanding: The influences of teachingand assessment. En J. C. Smart y W. G. Tierney (Eds.), Higher education: Handbook of theoryand research. New York: Springer, 2000.
Facultad de Ingeniería y Ciencias Universidad Adolfo Ibáñez, e-mail: [email protected] , e-mail:[email protected]
Agradecimientos al Plan de Mejoramiento Institucional UAI 1303
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
El aprendizaje del cálculo diferencial bajo undiseño curricular Modular
Elías Irazoqui Becerra
Abstract
Se presenta una experiencia de aula, donde se aplica un diseño curricular modularpara propiciar el aprendizaje del cálculo diferencial, en estudiantes de primer año deIngeniería en Alimentos de la Universidad del Bío-Bío, Campus Chillán, Chile.
Es sabido que el aprendizaje del cálculo diferencial presenta dificultades, muchosinvestigadores en esta materia así lo han reportado a través del tiempo: Artigue [1],Ortega y Sierra [5], Tall [8], Hitt [4], Salinas y AlanÃŋs [6], Salinas et al. [7] y Rincónet al. [9]. Ahora bien, no cabe ninguna duda que el aprendizaje del cálculo diferenciales un tópico importante para el estudiante si se piensa en las materias afines que siguena ésta, como lo son el cálculo integral, las ecuaciones diferenciales y el cálculo en variasvariables.
Cuando se habla de aprendizaje, en general, puede ser importante atender a las consid-eraciones que Biggs [2] hace sobre él al referirse a: aprendizaje superficial y aprendizajeprofundo. Por otro lado, las propuestas en materia del aprendizaje del cálculo, queponen el acento en la derivada, son variadas. Algunas de ellas ponen el acento en eluso de los recursos informáticos, otras en la modelización etc. En nuestro caso, y bajoun diseño curricular modular se pone énfasis en la resolución de actividades didácticasen los cuatro temas centrales que aborda cualquier curso de cálculo de primer año deuniversidad, como son: las funciones, límites y continuidad, derivadas y aplicaciones deéstas.
Por último, por un diseño curricular modular se entiende una disposición de los con-tenidos en dos módulos de trabajo, que son unidades de contenidos precisas y queimplican aprobar el primer módulo para realizar el segundo, de no ser así se ha derepetir el módulo nuevamente. La nota final del curso resulta del promedio obtenido enambos módulos de trabajo. En esta ocasión presentamos los resultados de una experi-encia de aula con estudiantes de Ingeniería en Alimentos, de la Universidad del Bío-Bío,Campus Chillán, donde se aplicó el diseño curricular modular.
References[1] Artigue, M. (1995). La enseñanza de los principios del cálculo: problemas epistemológicos, cog-
nitivos y didácticos. Ingeniería Didáctica en educación Matemática. México: Grupo EditorialIberoamericana.
Depto Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, e-mail: [email protected]
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[2] Biggs, J. (2010). Calidad del aprendizaje universitario. Cuarta Edición. Madrid: NARCEA,S.A. de Ediciones.
[3] Engler, A. (2011). >Es posible innovar en la enseñanza del cálculo diferencial? trabajamos conla derivada. Lestón, P. (Ed.) Acta Latinoamericana de Matemática Educativa, Vol. 24. México:Colegio Mexicano de Matemática Educativa A. C. y Comité Latinoamericano de MatemáticaEducativa A. C.
[4] Hitt, F. (2003). Las dificultades en el aprendizaje del Cálculo. Recuperado de:http://uqam.academia.edu/FERNANDOHITT/Papers (fecha de consulta: 20 de octubre de2013).
[5] Ortega, T. y Sierra, M. (1998). El concepto de derivada: algunas indicaciones para su en-señanza. Revista interuniversitaria de formación de profesores. NÂř 32, pp. 87- 115.
[6] Salinas, P., y Alanís, J. (2009). Hacia un nuevo paradigma en la enseñanza del cálculo dentro deuna institución educativa. Revista Latinoamericana de Investigación en Matemática Educativa,12(3), 355-382.
[7] Salinas P.; Alanís, A,; Pulido, R. ;, Santos, F.; Escobedo, J. y Garza J. (2010). Elementosdel Cálculo. Reconstrucción conceptual para el aprendizaje y la enseñanza. México: EditorialTrillas.
[8] Tall, D. (1996). Advanced Mathematical Thinking and the Computer. Proceedings of the 20thUniversity Mathematics Teaching Conference, Shell Centre, Nottingham, 8, p. 1-8.
[9] Rincón, E., Cienfuegos, D., Galván, D. y Fabela, M. (2014). El aprendizaje activo como es-trategia didáctica para la enseñanza del cálculo. Lestón, P. (Ed.). Acta Latinoamericana deMatemática Educativa, Vol. 27. México, DF: Colegio Mexicano de Matemática Educativa A.C. y Comité Latinoamericano de Matemática Educativa A. C.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Estudio de clases: hacia una alianza de launiversidad con las escuelas
Soledad Estrella Sergio Morales Raimundo Olfos
Abstract
Esta presentación tiene por objetivo compartir la experiencia del Instituto de Mate-mática de la PUCV y dar cuenta de los esfuerzos realizados durante los últimos 8 añospara promover la práctica del Estudio de Clases en distintos establecimientos escolaresde la región de Valparaíso.
Favorecer la conformación de Grupos de Estudio de Clases constituye una opor-tunidad de vinculación estratégica entre la Universidad y las instituciones educativas,que ayuda a promover la investigación acción en las escuelas en pro del desarrollode conocimientos y habilidades docentes para el mejoramiento de la enseñanza de lamatemática y estadística. Al mismo tiempo abre espacios de investigación para launiversidad que permitirían robustecer la base de conocimiento en torno al desarrolloprofesional docente, inicial y continuo, a la indagación en procesos de aprendizaje deconceptos matemáticos y estadísticos en alumnos, y a las dinámicas que surgen en lascomunidades de aprendizajes en torno a grupos de Estudio de Clases.
En la presentación se dará cuenta del proceso y las acciones llevadas a cabo porlos académicos del Instituto de Matemática de la PUCV para promover el Estudio deClases como una estrategia de desarrollo profesional docente efectiva.
References[1] Estrella (2015) Estudio de clases para el mejoramiento de la enseñanza de la estadística en Chile.
En: A. Salcedo (Ed.), Educación Estadística en América Latina: Tendencias y Perspectivas.(pp. 167 âĂŞ- 192).
[2] Isoda, M. & Olfos, R. (2010). El enfoque de Resolución de problemas en la enseñanza de lamatemática a partir del Estudio de Clases. Ediciones Universitarias de Valparaíso, P. Univer-sidad Católica de Valparaíso.
[3] Olfos, R., Estrella, S., & Morales, S. (2014). What can we learn from natural disasters toprevent loss of life in the future? In Lessons learned from across the world Prek-8. NCTM,National Council of Teachers of Mathematics. VA: NCTM.
Pontificia Universidad Católica de Valparaíso, e-mail: [email protected] Superior de Comercio de Valparíso, e-mail: [email protected] Universidad Católica de Valparaíso, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Avances y retrocesos en el Estudio de Clases (enel norte de Chile)
Eliseo Martínez Herrera.
Abstract
Se explican algunas buenas experiencias en el Estudio de Clases en una parte de Chile.Sus pequeños logros y se proponen causas de su gran retroceso. Entre éstas, a juicionuestro, por el exceso de currículo nulo en gran parte de nuestros programas curricularesdel Sistema K-12 (y también en el sistema universitario).
En efecto, por lo general, las preguntas abiertas, en el inicio de una propuesta para unestudio de clases van en contradicción con las propuestas ya fuertemente establecidasen los programas, en los libros, y en general en la inercia conductista, formulista einagotable que caracteriza a nuestra actual enseñanza de la matemática.
Hoy, en la actualidad, existe un postítulo de cobertura nacional, para los profesoresdel sistema K-8, conducido por el MINEDUC, en que hay cierta esperanza en intro-ducir fuertemente el Estudio de Clases en virtud de la propia exigencia del programa.Esperanzas de un avance.
References[1] Olfos, R. Lesson Study in Chile: The lesson of a collaboration Program. Paper presented at the
Fourth APEC - Tsukuba International Conference: Innovation of Mathematics Teaching andLearning through Lesson Study - Connection between Assessment and Subject Matter, February17 - 21. Tokyo, Japan. 2011
[2] Isoda, M., Mena-Lorca-Lorca, A. Arcavi, A. El Estudio de Clases japonés en Matemáticas.Ediciones Universitarias de Valparaíso. 2008
[3] Isoda M, OLfos R. El estudiode clases y las demandas curriculares: La enseñanza de la multi-plicación. Ediciones Universitarias de Valparaíso. 2009
[4] Manríquez L., Honores P. Currículum para enseñanza pre-básica y básica. Edición Universidadde Antofagasta, Vicerrectoría Académica, Dirección de Docencia. 2006
Universidad de Antofagasta, e-mail: [email protected]
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Comunidades de Aprendizaje GECSoledad Estrella Sergio Morales Raimundo Olfos
Maria Tapia
Abstract
La presentación tiene como propósito motivar a los participantes a apoyar la creacióny desarrollo de comunidades de aprendizaje GEC (Grupos de Estudio de Clases) enestablecimientos o grupos de establecimientos de su Región.
La conformación de Grupos de Estudio de Clases en Chile constituye una opor-tunidad para promover el desarrollo de la matemática en el país, mejorar la calidadde los aprendizajes escolares, favorecer el desarrollo profesional docente, aportar a lainnovación curricular y robustecer la relación Universidad y establecimientos Escolares.
Las políticas actuales del MINEDUC favorecen la conformación de Comunidadesde Aprendizaje proveyendo horas de dedicación exclusiva a los profesores para su de-sarrollo profesional y preparación de la enseñanza. El MINEDUC, a través del CPEIPestá desarrollando un plan que establece la creación de “comités locales de desarrolloprofesional docente” integrados por representantes regionales o provinciales del Ministe-rio de Educación, sostenedores de establecimientos municipales y consejeros regionalesprovinciales.
A continuación se ofrecen dos ejemplos de grupos de estudio de clases vigentes enla Región de Valparaíso. El primer caso corresponde a un grupo de profesores dematemáticas del Liceo INSUCO de Valparaíso, quienes realizan sus actividades delaño 2013 a la fecha. Estos profesores generan desarrollo profesional al interior de suInstitución. En la presentación se profundizará acerca de la organización, experiencia,productos y alcances del grupo. El segundo caso se refiere a un grupo de profesores deeducación general básica y educadoras diferenciales entre escuelas, pertenecientes a laComuna de La Calera. Este grupo se constituyó en el año 2014. Durante la presentacióntambién se relatará su iniciación, productos y proyecciones.
Un producto destacable de estos grupos constituye la creación de clases novedosas,pertinentes y efectivas. Varias de los cuales han sido compartidas con profesores en suslocalidades a través de clases públicas.
Pontificia Universidad Católica de Valparaíso, e-mail: [email protected],[email protected]
Instituto Superior de Comercio de Valparíso, e-mail: [email protected] Palestina de La Calera, e-mail: [email protected]
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References[1] Estrella (2015) Estudio de clases para el mejoramiento de la enseñanza de la estadística en Chile.
En: A. Salcedo (Ed.), Educación Estadística en América Latina: Tendencias y Perspectivas.(pp. 167 âĂŞ- 192).
[2] Isoda, M. , Olfos, R. (2010). El enfoque de Resolución de problemas en la enseñanza de lamatemática a partir del Estudio de Clases. Ediciones Universitarias de Valparaíso, P. Univer-sidad Católica de Valparaíso.
[3] Olfos, R., Estrella, S., Morales, S. (2014). Open lessons impact statistics teaching teachersâĂŹbeliefs. In K. Makar, B. de Sousa, & R. Gould (Eds.), Sustainability in statistics education.Proceedings of the Ninth International Conference on Teaching Statistics, Flagstaff, Arizona,USA. Voorburg, The Netherlands: International Statistical Institute.
[4] Olfos, R., Estrella, S., Morales, S. (2015). Clase pública de un estudio de clases de estadística:Una instancia de cambio de creencias en los profesores. Revista Electrónica Educare, 19(3),1–17.
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
La probabilidad en el aula de educación básica.Un estudio de caso sobre los primeros elementos
linguísticosClaudia Vásquez Ortiz
Abstract
En las últimas décadas se observa una fuerte tendencia por incorporar la probabilidad enlos currículos de Educación Primaria, con el objeto de promover que los alumnos apren-dan conocimientos probabilísticos que les sirvan de base para la recogida, descripción einterpretación de datos. En definitiva, se trata de ofrecerles herramientas que facilitenla toma de decisiones en situaciones en las que la incertidumbre es relevante, para queprogresivamente sean ciudadanos bien informados y consumidores inteligentes. Es eneste contexto que la probabilidad "proporciona una excelente oportunidad para mostrara los estudiantes cómo matematizar, cómo aplicar la matemática para resolver proble-mas reales" (Godino, Batanero y Cañizares, 1997, p.12). Por tanto, surge la necesidadde educar a los estudiantes en esta área desde temprana edad, para así, contar conciudadanos alfabetizados probabilísticamente "capaces de hacer frente a una ampliagama de situaciones del mundo real que implican la interpretación o la generación demensajes probabilísticos, así como la toma de decisiones" (Gal, 2005, p.40).
En este sentido, el National Council of Teachers of Mathematics incluyó a "Datos yAzar" como área temática en Curriculum and Evaluation Standard for School Mathe-matics (NCTM, 1989), reforzando esta iniciativa en Principles and Standard for SchoolMathematics (NCTM, 2000), que contemplan que los programas de enseñanza deberíancapacitar a los alumnos para aprender conocimientos relacionados con el análisis dedatos y la probabilidad a partir del nivel Pre-K (tres años). Esta tendencia, comodecíamos, se ha reflejado en los currículos de matemáticas de muchos países, entre ellosChile, que han incorporado la probabilidad en Educación Primaria para promover unenfoque experimental que proporcione una experiencia estocástica desde las primerasedades (Mineduc, 2012). Producto de esta necesidad científica, profesional y social,Chile ha incluido en las actuales Bases Curriculares (2012) el estudio de la probabilidada lo largo de todo el currículo escolar, con el propósito de que "todos los estudiantes seinicien en temas relacionados con las probabilidades" (Mineduc, 2012, p. 5), y de estemodo cumplir con parte de los objetivos generales propuestos en la Ley General de Ed-ucación (2009) para la Educación Básica, referidos explícitamente a "que los educandosdesarrollen los conocimientos, habilidades y actitudes que les permitan: pensar en formareflexiva, evaluando y utilizando información y conocimientos, de manera sistemáticay metódica, para la formulación de proyectos y resolución de problemas; comprender yutilizar conceptos y procedimientos matemáticos básicos en la resolución de problemas
Pontificia Universidad Católica de Chile, e-mail: [email protected]
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cotidianos, y apreciar el aporte de la matemática para entender y actuar en el mundo"(LGE, 2009, artículo 29, p. 10).
Para el logro de lo anterior, es fundamental el desarrollo de nociones básicas de prob-abilidad, de manera informal en los primeros niveles, donde el lenguaje probabilísticoasociado a situaciones problemáticas centradas en los juicios que emiten los estudiantescon base en sus propias experiencias, juega un rol fundamental, pues es a partir dellenguaje informal y cotidiano que los estudiantes desarrollarán, paulatinamente, un ra-zonamiento más abstracto y cuantitativo, que permitirá el tránsito entre los diversossignificados de la probabilidad en el contexto de la matemática escolar, para así alcanzarla construcción de un conocimiento probabilístico de un nivel de abstracción mayor.
En este trabajo se presenta un estudio que contempla el análisis del proceso deenseñanza de la probabilidad y, en concreto, cómo el profesor utiliza una multiplicidadde términos, expresiones orales y escritas, símbolos y representaciones (tablas y gráficos)cuando enseña probabilidad a estudiantes de educación primaria que no han recibidoinstrucción previa sobre el tema con el propósito de que éstos aprendan gradualmentela noción de probabilidad y adquieran el respectivo lenguaje probabilístico asociado.
References[1] Godino, J. D., Batanero, C. y Cañizares, M. J. Azar y Probabilidad. Fundamentos didácticos y
propuestas curriculares. Madrid: Síntesis. 1997
[2] Gal, I., Towards ’probability literacy’ for all citizens. In G. Jones (ed.), Exploring probability inschool: Challenges for teaching and learning (pp. 43-71). Kluwer Academic Publishers. 2005
[3] NCTM. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
[4] NCTM. Principles and standards for school mathematics. Reston, Va.: The National Council ofTeachers of Mathematics. 2000
[5] Mineduc . Bases Curriculares 2012: Educación Básica Matemática. Santiago de Chile: Unidadde Curriculum y Evaluación. 2012
[6] Mineduc . Ley General de Educación. Santiago: MINEDUC, www.mineduc.cl. 2009
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LXXXIV Encuentro Anual Sociedad de Matemática de Chile26 – 28 de Noviembre 2015, Pucón
Estudio de clases en didáctica de la matemática:proceso reflexivo de los estudiantes de pedagogía
en Educación Básica en la U. Santo TomásPierina Zanocco Soto
AbstractEsta ponencia, la cual se centra en la estrategia Estudio de clases, es parte de la investi-gación "Generación de ambientes reflexivos y decisiones pedagógicas fundamentadas, enla Didáctica de la Matemática: Estudio de casos y Estudio de clases" focalizándose enlas asignaturas Didáctica de la Matemática I y II, trabajando con 30 estudiantes de lacarrera Educación Básica, aplicando la estrategia mencionada. Se privilegia potenciarla generación de espacios reflexivos y toma de decisiones pedagógicas fundamentadasen la enseñanza de la Matemática. Investigaciones destacan la importancia que tienela preparación de profesores en la planificación de clases (Liping Ma, 2010) y en el tra-bajo colaborativo y reflexivo para mejorar sus prácticas pedagógicas (Hiebert y Stigler,1999). Esta estrategia se aplica principalmente en la formación continua de profesores,el aporte de este proyecto se centra en la formación inicial.
Se presentarán evidencias del impacto que la estrategia Estudio de Clases tiene enla formación inicial de profesores en las habilidades de pensamiento reflexivo y crítico,a través de: procesos de análisis a priori de sus planes de clases implementados; pre-sentación pública de sus clases; retroalimentación de sus pares; autocrítica a posterioride sus planificaciones; formulación del plan final. Las evidencias están relacionadas conlogros en habilidades de pensamiento mencionadas y en los avances en sus formas deplanificar.Se presentan los resultados obtenidos y el modelo aplicado.
References[1] Arcavi, A., Isoda, M y Mena, A. El Estudio de Clases Japonés en Matemáticas. Valparaíso:
Ediciones Universitarias de Valparaíso. 2008
[2] Hiebert, J., Stigler, J. The Teaching Gap: Best Ideas from the World’s Teachers for ImprovingEducation in the Classroom. Nueva York: The Free Press. 1999
[3] Isoda, M. y Olfos, R. El enfoque de Resolución de Problemas. En la enseñanza de la Matemáticaa partir del Estudio de Clases. Valparaíso: Ediciones Universitarias de Valparaíso. 2009
[4] Liping Ma. Conocimiento y enseñanza de las matemáticas elementales. La comprensión de lasmatemáticas fundamentales que tienen los profesores en China y los EEUU. Santiago: EdicionesAcademia Chilena de Ciencias. 2010
Escuela de Educación, Universidad Santo Tomás, Sede Santiago,Chile, e-mail:[email protected]
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