ACTA DE RESUMENES LXXXIV Encuentro Anual

Transcripción

ACTA DE RESUMENES LXXXIV Encuentro Anual

ACTA DE RESUMENES
LXXXIV Encuentro Anual
Sociedad de Matemática de Chile
2015
Conferencias
y
Sesiones Invitadas
1
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.cl
Canadá 253 Departamento F. Providencia, Santiago. Chile.
2
Introducción
El Encuentro de la SOMACHI es un evento de carácter nacional que se realiza anualmente
y que tiene como uno de sus objetivos reunir por un par de días a los académicos, profesores
e investigadores de la disciplina para que intercambien sus conocimientos y experiencias.
Además en este encuentro la comunidad matemática chilena elige directiva y planifica iniciativas futuras.
Este año el Comité Científico ha estimado conveniente mostrar a la comunidad algunas
de 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 Leroy
P. Steele de la Sociedad Americana de Matemáticas 1999, Medalla de la Doble Hélice del
Laboratorio Cold Spring Harbor 2010, Premio Abel 2015. Para ello se han programado una
serie de Conferencias Plenarias dictadas por expertos en sus trabajos, que esperamos sean
de provecho para el desarrollo de nuestra matemática.
La SOMACHI también se ha preocupado fuertemente de fomentar la participación de
estudiantes de pre y post grado en las distintas sesiones de trabajo; esperamos que sea una
experiencia provechosa para los alumnos que han comenzado su formación matemática.
El Comité Organizador agradece el valioso aporte de los organizadores y de los coordinadores de sesión, sin cuya colaboración este encuentro no habría sido posible. En estas
actas se plasma un resumen de los trabajos presentados.
Comité Organizador
LXXXIV Encuentro
Pucón
Región de la Araucanía
26-28 Noviembre 2015
3
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)
4
Contents
Introducción
Conferencias Plenarias . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
16
Los trabajos de John Nash en Teoría de Juegos
Mario Bravo
17
Nash´s imbedding problem for Riemannian manifolds
Mauricio Godoy
18
Los trabajos geométricos de John Nash
Mark Spivakovsky
19
Diffusive models and their intrinsic regularity theories
Eduardo Teixeira
20
Análisis Funcional y Aplicaciones . . . . . . . . . . . . . . . . . . . .
22
Almost exponential decay for a system of Schrödinger equations
M.A. Astaburuaga
C. Fernández
23
Instability of eigenvalues for unitary perturbations
Víctor H. Cortés
24
Breather solutions of a 1D non-linear Shrödinger equation
Matías Courdurier
25
The topology of Chiral vector bundles: topological insulators of type AIII
Giuseppe De Nittis Kiyonori Gomi
26
Un modelo simple de una situación biológica
Manuel Elgueta
27
Exact β function and DGLAP-BFKL duality in a supersymmetric gauge
theory
28
Igor Kondrashuk
Teorema de Baillon para una ecuación integro differenciales
Juan C. Pozo Octavio Vera
30
On the fractional Schrödinger equation on a Hilbert space
Humberto Prado
31
A characterization of universally starlike functions
Andrew Bakan Stephan Ruscheweyh Luis Salinas
32
5
Aproximación de soluciones acotadas de ecuaciones diferenciales con argumento constante a trozos del tipo generalizado con impulsos
34
Manuel Pinto Ricardo Torres
Álgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Algunas propiedades del álgebra de símbolos pseudodiferenciales formales
Jarnishs Beltran Enrique G. Reyes
38
Módulos de tipo FP-infinito y la Categoría Estable de Módulos de un Anillo 39
Daniel Bravo
Macdonald polynomials in superspace and the 6 vertex model
Luc Lapointe
40
Teoría de representaciones graduada del álgebra de blob y el cálculo de
Soergel en dos colores
41
David Plaza
Groupoidal Gelfand Models in Group Theory
Jorge Soto Andrade
42
Algebraic structures on stellohedra and pterahedra
L. Berry S. Forcey M. Ronco P. Showers
43
Teoría de representaciones del álgebra de Yokonuma-Hecke y del álgebra de
braids and ties
44
Steen Ryom-Hansen
Análisis No Arquimediano . . . . . . . . . . . . . . . . . . . . . . . . .
C-álgebra de operadores lineales continuos definidos en c0 (I)
J. Aguayo
M. Nova J. Ojeda
46
47
Espacios y cuerpos residuales en espacios de tipo Hilbert sobre cuerpos con
valuaciones no-arquimedianas
48
Herminia Ochsenius Elena Olivos
Medida Espectral sobre Algebras de Operadores definidos en c0 (N)
J. Aguayo
M. Nova J. Ojeda
49
Espacios de Banach sobre cuerpos con valuación discreta
Elena Olivos
Herminia Ochsenius
50
An o-minimal approach to definability of functions in extensions of C
Javier Utreras
51
6
Análisis Numérico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Mathematical and numerical analysis for a nonlocal reaction diffusion system 53
V. Anaya
M. Bendahmane M. Langlais M. Sepúlveda
A kind of Mixed Finite Element Variational Formulation for a Parabolic
Problem
54
Mauricio Barrientos Karina Vilches
Discontinuos Galerkin scheme for Helmholtz equation
Tomás Barrios Rommel Bustinza
55
Hydraulic parameter estimation under non-saturated flow conditions in copper heap leaching
56
Emilio Cariaga Rubén Martínez Mauricio Sepúlveda
Torque free rigid body motion: an elliptic function primer
Roberto León Luis Salinas Claudio Torres
57
Analysis of an augmented pseudostress-based mixed formulation for a nonlinear 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
David Mora
59
Finite Volume Method Improved Sequential Solution Applied To Two-Dimensional
Convective-Diffusive Heat Transfer
60
Nelson O. Moraga Juan Jaime
On the analyses of three DG schemes for Stokes problem based on velocitypseudostress formulation
62
Tomás P. Barrios Rommel Bustinza Felipe Sánchez
Convergence of a finite volume scheme for a sorption-coagulation equation 63
Erwan Hingant Mauricio Sepúlveda
Numerical Analysis Of Non-Newtonian Flows In Liquid Foods, Solidification
Casting And Polymer Injection Molding
64
Nelson O. Moraga Edgardo J. Tabilo
Numerical Challenges for a Numerical Simulation of a Coupled Model for
Grain Growth in 2D
66
Claudio E. Torres Alejandro Sazo
7
On Nonlinearly and Linearly Implicit IMEX Runge-Kutta methods for a
class 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
Miguel Angel Alejo
71
Sobolev and Hardy-Sobolev type inequalities
Hernán Castro
72
On the control of the stabilized Kuramoto-Sivashinsky system by a single
force
73
Eduardo Cerpa
A Dirichlet problem involving the divergence operator
Gyula Csató
74
Solvability of fractional problems with supercritical drifts
Gonzalo Dávila
75
Formación de singularidades para el flujo de mapas armónicos
Juan Dávila
Solutions to a supercritical elliptic problem
Jorge Faya
76
77
Entire sign changing solutions with finite energy to the fractional Yamabe
equation
78
Danilo Garrido Monica Musso
Multiplicity of solutions for some semilinear problems involving nonlinearities
with zeros
80
Leonelo Iturriaga Jorge García-Melián
Kink dynamics in the φ4 model: asymptotic stability for odd perturbations
in the energy space
82
Michal Kowalczyk
Singularly perturbed PDEs and patterns with periodic profiles
Fethi Mahmoudi
83
Some results for a problem from Combustion
Alejandro Omón Arancibia
84
Desigualdades de restricción de Fourier: existencia y no existencia de máximos y cálculo de la mejor constante
86
René Quilodrán
8
Stationary harmonic functions whose Laplacian is a Radon measure
Rémy Rodiac
87
Fractional mean curvature flow
Mariel Sáez Enrico Valdinoci
88
Moderate solutions of semilinear elliptic equations with Hardy potential
PHUOC-TAI NGUYEN
89
0
On the C p -regularity conjecture
Eduardo Teixeira
90
Geometría . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Linear systems on IHS manifolds
Michela Artebani
92
Group actions on Riemann surfaces up to topological equivalence.
Antonio Behn
93
Acción de grupos en superficies y variedades abelianas
Angel Carocca
94
Sobre la existencia de acciones de grupos elementales en Superficies de Riemann
95
Mariela Carvacho
Dual families of Calabi-Yau varieties
Paola Comparin
96
On singular varieties with smooth subvarieties
M. R. Gonzalez-Dorrego
98
The 4-prims family.
Víctor González Aguilera
99
Gustavo Labbé Morales
Automorphisms of non-cyclic p-gonal surfaces
R. A. Hidalgo A. F. Costa
100
Lines on cubic hypersurfaces over finite fields
Antonio Laface
101
Authomorphims of graphs and Riemann surfaces
Alexander D. Mednykh
103
On Jacobian of circular graphs
Ilya A. Mednykh
104
Curvas de Tipo Fermat y sus Jacobianas
Jaime Pinto
105
9
Automorphisms Of Pseudo-Real Riemann Surfaces
S. Quispe R. A. Hidalgo
106
Superficies Algebraicas: Uniformización y Aritmeticidad
Sebastián Reyes-Carocca
107
Fixed points and rational representations of actions in abelian varieties
Rubí E. Rodríguez
108
Familias de Jacobianas completamente descomponibles y subvariedades especiales de Ag .
109
Anita M. Rojas
Small degree covers and reducible hyperplane sections
Andrea L. Tironi
111
Modelos Matemáticos de
Sistemas Biológicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Medidas de conservación ex situ de tipo impulsivo: Un enfoque metapoblacional a través del modelo clásico de Levins
113
Sandra Araya Crisóstomo Héctor Rojas-Castro
Consecuencias sobre la abundancia poblacional del Efecto Allee en hábitats
bajo fragmentación
115
Rodrigo Del Valle Fernando Córdova-Lepe
Control epidemiológico optimal por hospitalización impulsiva
M. Eugenia Solís Fernando Córdova-Lepe
116
Dinámica de la distribución genotípica bajo mortalidad diferenciada por
rasgos fenotípicos
117
Héctor Rojas-Castro Fernando Córdova-Lepe
Observaciones a la aproximación de L.A. Segel para las ecuaciones del Sistema
Ligando-Receptor
119
Fernando Córdova-Lepe
Neurodidactics: Analysis of Cellular Neural Network Models
Kuo-Shou Chiu
120
A vaccine-age structured model to study the effect of a pre-erythrocytic
vaccine on malaria prevalence
123
Katia Vogt Geisse Calistus Ngonghala Zhilan Feng
Modelación del cambio en la interacción de poblaciones biológicas. Estudio
de un caso
125
Marcelo E. Alberto et al.
10
Un Modelo Estocástico de Biorrectar de Autociclado
Ana Venegas
Ricardo Castro
Fernando Córdova
126
Dinámica de un modelo tritrófico con una respuesta funcional monotónica
no-diferenciable
127
Viviana Rivera Pablo Aguirre
Dinámica en el modelo de depredación de Holling-Tanner considerando interferencia entre los depredadores
128
Adrián Cecconato Eduardo González-Olivares
Modelling and stability analysis of a microalgal pond with nitrification
F. Mairet
H. Ramírez C
A. Rojas-Palma
130
Mathematical approach regarding the environmental effects upon trait diversity in a cell population.
132
Karina Vilches Ponce
On a nonlinear problem from catalysis: existence, multiplicity and qualitative
behaviour
133
Alejandro Omón Arancibia Gonzalo Robledo Veloso
A stochastic disease transmission in an epidemic model considering a hyperbolic incidence rate
135
A. Christen M. A. Maulén E. González-Olivares M. Curé
Sistemas Dinámicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Aspectos recientes de la Conjetura de Palis
Alma Armijo
138
Injectivity, Global and Almost Global Stability of Hurwitz Vector Fields.
Álvaro Castañeda Víctor Guíñez
139
Toeplitz and strong orbit equivalence
Maryam Hosseini
140
Dimensión de Hausdorff de los conjuntos de Borel-Bernstein
Felipe Pérez
141
Flexibility of some groups of homeomorphisms of the line
Cristobal Rivas
142
A linearization result for DEPCA systems
Manuel Pinto Gonzalo Robledo
143
Shearer’s inequality and the Infimum Rule
Pierre Paul Romagnoli
144
11
Polinomios de Fibonacci y Componentes Errantes
Eugenio Trucco
145
Matemática Discreta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Estudio de un modelo de evasión en el transporte público
Bastián Bahamondes Pizarro
147
Resource Augmentation Algorithm for Single Machine Scheduling with JobDependent Convex Cost
148
Rodrigo A. Carrasco
Avances recientes en la resolución exacta del problema de vendedor viajero149
Daniel Espinoza William Cook
Marcos Goycoolea
Quasirandom hypergraphs and subsets with small Fourier coefficients
Hiê.p Hàn
150
Efficient Implementation of Carathéodory’s Theorem for a Simple Scheduling
Polytope
151
Ruben Hoeksma
Maximum number of colourings without monochromatic Schur triples
Andrea Jimenez
152
On-line list coloring of random graphs
Dieter Mitsche
153
Computing income taxes under the new Chilean tax regime: Graphs, Markov
Chains and Algorithms.
154
Javiera Barrera
Eduardo Moreno
Sebastián Varas
Optimización . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Chance-constrained problems and rare events: an importance sampling approach
156
J. Barrera, T. Homem-de-Mello, E. Moreno
B. Pagnoncelli, G. Canessa
Nonsmooth Lur’e Dynamical Systems in Hilbert Spaces
Samir Adly Abderrahim Hantoute Ba Khiet Le
157
Stochastic Topology Design Optimization for Continuous Elastic Materials 158
Miguel Carrasco Benjamín Ivorra
Angel Manuel Ramos
Existence and approximation of generalized Lagrange multipliers for variational problems under uniform constraints on the gradient.
159
Felipe Alvarez Salvador Flores
12
Condiciones de Optimalidad en Problemas de Control Óptimo Discreto
C. Isoton M.A. Rojas-Medar V. Vivanco
L. dos Santos
161
Proximal Distances over Symmetric Cones
Julio López Erik Papa
163
Stability in Generalized Nash Equilibrium Problems with nonsmooth payoff
functions, application to Electricity market
164
Matthieu Maréchal Rafael Correa
Boosting Topic Models for Text Analysis
Marcelo Mendoza
166
A Decomposition Method for Two-Stage Stochastic Programs with RiskAverse Utilities
168
Tito Homem-de-Mello
Sebastian Arpon Bernardo Pagnoncelli
A primal-dual mix algorithm for convex non-differentiable structured optimization in Hilbert spaces
170
Cesare Molinari Juan Peypouquet
Resultados sobre convexidad de la imagen de funciones cuadráticas
Felipe Opazo Lagos
171
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 Viability
Theory
175
Héctor Ramírez
Optimal feedback synthesis and minimal time function for the bioremediation
of water resources with two patches
176
H. Ramírez C. A. Rapaport V. Riquelme
Dualidad en optimización vectorial
M. A. Rojas-Medar
L. Batista dos Santos
178
Camila Isoton
Comparision of MINC and MRMT configurations:
Effects of spatial structure and biomass diffusion
A. Rapaport H. Ramírez
A. Rojas-Palma J. de Dreuzzy
180
Optimization of the concentration changes in a chemostat with one species 182
Térence Bayen Jérôme Harmand Matthieu Sebbah
Problemas Inversos y Control de EDP . . . . . . . . . . . . . . . . 184
On the cost of null controllability of some linear partial differential equations185
Nicolás Carreño
13
On the control of the improved Boussinesq equation
Eduardo Cerpa
186
An Inverse Problem for the Helmhotz Equation in a Layered Media.
Matías Courdurier
187
Detection of Several Obstacles in a Stokes Flow: A mixed approach
Matías Godoy Campbell
188
Controllability of coupled systems with Schrödinger equations.
Alberto Mercado Saucedo
189
An ADER type scheme for evolving differential operators
G. Montecinos J. C. López R. Lecaros
J. Ortega E. F. Toro
190
Stability numbers to Timoshenko’s system with shear boundary dissipation192
Margareth Alves Jaime E. Muñoz Rivera Mauricio Sepúlveda
Teoría de Números . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Ramas y extensiones de cuerpos
L. Arenas-Carmona
194
Un retículo Hermiteano central
Ana Cecilia de la Maza Remo Moresi
195
Distribución asintótica de puntos de Hecke sobre Cp
Sebastián Herrero Miranda
196
Optimal bounds for Büchi’s problem in modular arithmetic
Pablo Sáez
Xavier Vidaux Maxim Vsemirnov
197
Una conexión entre la propiedad de Northcott y la indecidibilidad en anillos
de enteros totalmente reales
198
Xavier Vidaux
Estudio de Clases. Método Japonés
. . . . . . . . . . . . . . . . . 199
Promoviendo el desarrollo de habilidades del pensamiento matemático en
estudiantes del sistema escolar: Una experiencia en el complejo educacional
la Granja de Cajón
200
Pamela Alarcón Valeria Carrasco Ciro González
Teresa Sanhueza
Análisis de idoneidad didáctica del método japonés
desde un enfoque onto-semiótico de la instrucción matemática
Carlos Cabezas
Pedro Arteaga
202
14
Problemas incorrectos como medio para desarrollar aprendizaje profundo 203
Hugo Caerols Katia Vogt Geisse
El aprendizaje del cálculo diferencial bajo un diseño curricular Modular
Elías Irazoqui Becerra
204
Estudio de clases: hacia una alianza de la universidad con las escuelas
Soledad Estrella
Sergio Morales Raimundo Olfos
206
Avances y retrocesos en el Estudio de Clases (en el norte de Chile)
Eliseo Martínez Herrera.
207
Comunidades de Aprendizaje GEC
Soledad Estrella
Sergio Morales
Maria Tapia
208
Raimundo Olfos
La probabilidad en el aula de educación básica.
Un estudio de caso sobre los primeros elementos linguísticos
Claudia Vásquez Ortiz
210
Estudio de clases en didáctica de la matemática: proceso reflexivo de los
estudiantes de pedagogía en Educación Básica en la U. Santo Tomás
212
Pierina Zanocco Soto
15
Conferencias Plenarias
CONFERENCIAS
PLENARIAS
16
LXXXIV Encuentro Anual Sociedad de Matemática de Chile
26 – 28 de Noviembre 2015, Pucón
Los trabajos de John Nash en Teoría de Juegos
Mario Bravo
Abstract
A comienzos de los años 50, los trabajos de John Nash constituyeron avances fundamentales tanto en la teoría de juegos cooperativos como no-cooperativos. Las nociones
de equilibrio definidas por Nash y la variedad de interacciones a las que se aplican
cambiaron definitivamente la ciencia económica moderna. En esta charla daremos una
introducción a los aspectos principales del trabajo de Nash en este ámbito. Además,
discutiremos cómo estas ideas han impactado en lo más diversos campos de investigación.
Universidad de Santiago de Chile. e-mail: [email protected]
17
Nash´s imbedding problem for Riemannian
manifolds
Mauricio Godoy
Abstract
Imbedding problems are one of the most natural questions in the evolution of mathematics. When new concepts are defined, they are often concrete examples of a phenomenon;
afterward they are given abstract formulations that, in principle, generalize the particular situations studied before; and not too long after that, the question "does the
abstract definition include new examples ? " arises.
Nash´s imbedding is a fundamental result in Riemannian geometry that exemplifies the
difficulties that may appear when answering the last question. Employing approximation techniques coming from PDEs and geometric analysis, J. F. Nash proved in 1956
that any C k Riemannian manifold M (3 ≤ k ≤ ∞) can be C k isometrically imbedded
in RN , where N depends quadratically on the dimension of M , if M is compact, and
cubically if M is not compact.
In this talk I will give an overview of the problem and its historical background, a
sketch of Nash´s proof and some later results improving dimensions (e.g., Gromov´s
reduction to N = 5 for surfaces instead of Nash´s N = 17).
e-mail: [email protected]
18
Los trabajos geométricos de John Nash
Mark Spivakovsky
Abstract
En esta conferencia trataremos de resumir las principales contribuciones de John Nash a
la geometría, realizadas durante casi dos décadas entre 1950 y 1968. Nos concentraremos
sobre los siguientes resultados y construcciones:
• El teorema del encaje de una variedad diferencial como una componente conexa
de una variedad algebraica real.
• Dos versiones del célebre teorema de Nash de encaje. La primera versión, el teorema de Nash–Kuiper, dice que todo encaje débilmente contráctil de una variedad
C 1 de dimensión m en el espacio Euclideano de dimensión n > m puede ser
aproximado arbitráariamente bien par un encaje C 1 isométrico. En el teorema
original de Nash n era mayor o igual a m + 2, pero Kuiper mejoró la cota para
obtener n > m. La segunda versión, mucho más difícil de demostrar, también
publicada en los Annals of Mathematics, dice que para k entre 3 e infinito toda
variedad M de clase C k y dimensión m admite un encaje C k isométrico en
el espacio Euclideano de dimensión n, donde n ≤ m(3m+11)
si M es compacta
2
m(m+1)(3m+11)
si M no es compacta.
y n≤
2
La mayor parte de la conferencia será dedicada a las contribuciones de Nash en
los años sesenta del siglo XX:
• Explosión de Nash como un método para estudiar geometría y topología de singularidades y un método conjetural para construir una desingularización canónica
de variedades algebraicas (y analíticas) en característica cero.
• El problema de Nash sobre el espacio de arcos de una variedad algebraica singular.
19
Diffusive models and their intrinsic regularity
theories
Eduardo Teixeira
Abstract
Diffusive processes appear naturally in the mathematical formulation of a number of
models in pure and applied sciences, ranging from problems in physics, biology, economics, probability, differential geometry, etc. Among the most fundamental questions
pertaining to the mathematical treatment of such models, understanding their intrinsic
regularity theories has a central importance. In this talk I will present an overview
of the PDE theory for diffusive models and will discuss the regularity properties of
solutions of such equations.
20
SESIONES
INVITADAS
21
Análisis Funcional y Aplicaciones
Encargado de Sesión: M. A. Astaburuaga
Víctor H. Cortés
22
Almost exponential decay for a system of
Schrödinger equations
M.A. Astaburuaga
C. Fernández
Abstract
h1
Let h = h2 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. The
vector h is a generalized eigenvector of the Hamiltonian of the system H. This means
that Hh = k 2 h, k a complex number with Im k < 0, and h satisfies an outgoing condition at ∞. Since h is not square integrable we truncate h to an interval containing
the support of the perturbation and show that if the resonance k is close to the real
axis the survival probability of the truncated solution has an approximate exponential
behaviour 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’s
inequality, 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, Conference
U. of Alabam at Birmingham, 132-142, 1995.
Facultad de Matemáticas. Pontificia Universidad Católica de Chile
Fondecyt No. 1141120, ACT-1112 , email: [email protected]
23
Instability of eigenvalues for unitary
perturbations
Víctor H. Cortés
Abstract
In this paper we deal with the problem of instability of eigenvalues of a family of unitary
operators acting on a separable Hilbert space H by describing the corresponding Fermi
Golden Rule for a one parameter family {Uβ : |β| < β0 }. We apply it to prove that close
to an eigenvalue of U0 , embedded in the continuous spectra , there are not eigenvalues
of Uβ for β small.
Following [1], [2] we show that there is a close relation between this behavior and
the existence of a local commutator for the family Uβ .
This is a joint work with Astaburruaga and Bourget, Pontificia Universidad Católica
de 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 in
quantum 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 for
eigenvalues, J. Funct. Anal. 192 (2002), no. 2, 425-445.
Fondecyt 1120786, ACT-1112, e-mail: [email protected]
24
Breather solutions of a 1D non-linear
Shrödinger 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 the
non-linear Schrödinger equation
1
i∂t u = − ∂x2 u + V (x, u)
2
with a specific space dependent non-linear term V (x, u) and present an peculiar family
of breather solutions.
This is a joint work with Olivier Bourget and Claudio Fernandez, Pontificia Universidad Católica de Chile.
25
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 mathematical physics. In this talk I will describe a new approach to the classification of
topological quantum systems in class AIII which is based on the study of a new category of vector bundles. The objects of this category, the chiral vector bundles, are
pairs constituted by a complex vector bundle along with one of its automorphisms.
We provide a classification for the homotopy equivalence classes of these objects which
is based on the construction of a suitable classifying space. The computation of the
cohomology of the latter allows us to introduce a proper set of characteristic cohomology classes: Some of those just reproduce the ordinary Chern classes but there are
also new odd-dimensional classes which take care of the extra topological information
introduced by the chiral structure. Chiral vector bundles provide the proper geometric model for topological quantum systems in class AIII, namely for systems endowed
with a (pseudo-)symmetry of chiral type. The classification of the chiral vector bundles
over sphere and tori (explicitly computable up to dimension 4) recover the commonly
accepted classification for topological insulators of class AIII which is usually based on
the K-group K1. However, this new classification turns out to be even richer since it
takes 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]
Shinshu University, Nagano, Japan, e-mail: [email protected]
26
Un modelo simple de una situación biológica
Manuel 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 soluciones
estacionarias.
Fondecyt No. 1150028 , e-mail: [email protected]
27
Exact β function and DGLAP-BFKL duality in
a supersymmetric gauge theory
Igor Kondrashuk
Abstract
We consider a couple of integro-differential equations which can solved analytically. We
propose a new method based on a complex analysis to find the analytic solution. We
apply this result to a model for QCD dynamics in which DGLAP and BFKL equations
are a couple of the integro-differential equations of this type. The solution has been
found due to possibility to solve analytically the corresponding duality equations. This
duality relates DGLAP and BFKL kernels in a model of singlet parton evolution with a
dominant eigenvalue. The case of N = 1 supersymmetric gauge theory is considered in
detail because the running of a gauge coupling is known exactly. The complete structure
of the solution is determined by the properties of the Lambert function. This result is
obtained in collaboration with Gustavo Alvarez (Departamento de Fisica, Udec), Bernd
Kniehl (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 scattering
in 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 Nonabelian 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 in
Asymptotically Free Theories,” Phys. Lett. B 60 (1975) 50.
[6] E. A. Kuraev, L. N. Lipatov and V. S. Fadin, “Multi - Reggeon Processes in the YangMills 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 Nonabelian 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 Chromodynamics,” Sov. J. Nucl. Phys. 28 (1978) 822 [Yad. Fiz. 28 (1978) 1597].
[9] Y. L. Dokshitzer, “Calculation of the Structure Functions for Deep Inelastic Scattering
and 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. B
126 (1977) 298.
[11] G. Altarelli, R. D. Ball and S. Forte, “Singlet parton evolution at small x: A Theoretical
update,” 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, physicomathematico-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 the
Lambert 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-MannLow 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 QCD
running coupling,” JHEP 1112 (2011) 019 [arXiv:1110.2545 [hep-ph]].
29
Teorema de Baillon para una ecuación integro
differenciales
Juan C. Pozo
Octavio Vera
Abstract
Sea X un espacio de Banach, A : D(A) ⊆ X → X un operador lineal cerrado generador
de un C0 -semigrupo y g : R → R una función de variación acotada en [0, c] para todo
c > 0. En este trabajo mostramos que siempre existe una función continua f tal que el
problema

Z t

0
u (t) = Au(t) +
g(t − s)Au(s)ds + f (t), t ∈ [0, τ ],
(1)
0

u(0) = x ∈ X,
no admite solución a menos que A sea un operador acotado o X contenga un subespacio
cerrado isomorfo a c0 .
Considerando g ≡ 0, este problema ha sido abordado en varios trabajos [1, 2, 3] y
es 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 and
semigroups 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 equation, Proc. Amer. Math. Soc. 82 (3) (1981), 425–430.
Pozo, Proyecto FONDECYT 3140103, e-mail: [email protected]
Vera, Proyecto FONDECYT 1121120, e-mail: [email protected]
30
On the fractional Schrödinger equation on a
Hilbert space
Humberto Prado
Abstract
Let A be a given self-adjoint operator on a Hilbert space H. Then by an application
of the spectral theorem we prove existence and uniqueness of strong solutions for the
∂αu
(t) = (−i)α Au(t) with the initial condition
linear fractional Schrödinger equation
∂tα
u(0) = u0 , in which 0 < α < 1. We show existence of an operator solution family
{Uα (t)}t≥0 . Furthermore, the solution is defined as uα (t) = Eα ((−it)α A)u0 in which
uα (t) is obtained by means of the functional calculus defined by the the Mittag-Leffler
function.
Additionally we show that the operators {Uα (t)}t≥0 have a continuous dependence
on the parameter α. Moreover at the limit when α → 1 we obtain that U1 (t) is equals
to 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, and
J. Trujillo from University of La Laguna, Tenerife.
Departamento de Matemáticas y Ciencias de la Computación. Universidad de Santiago de Chile
FONDECYT grant No. 1130554, e-mail: [email protected]
31
A characterization of universally starlike
functions
Andrew Bakan
Stephan Ruscheweyh
Luis Salinas
Abstract
This communication deals with extensions of the classical Polya-Schoenberg conjecture
about the convolution invariance (Hadamard product invariance) of convex univalent
functions in the unit disk. There is a continuous passage of function classes convex (and
similar classes starlike, pre-starlike, etc.) in the unit disk to those in the slit domain
Λ := C \ [1, ∞), and it has been shown in the papers [1, 3, 4, 5] that the theory of
convex –resp. starlike, pre-starlike, etc.– functions analytic in Λ (understood in the
right sense) is formally very similar to the one in D.
A function F with F (0) = 0, F 0 (0) = 1 and analytic in Λ is called universally
starlike if it maps every circular subdomain of Λ containing the origin univalently onto
a starlike domain with respect to the origin. Here, the word universal comes from
the fact that these functions represent the universal multipliers (with respect to the
Hadamard product) of the classes of starlike analytic functions in arbitrary circular
subdomains of Λ. A universally starlike function f is a Pick function which can be
represented in either of the next two forms:
(Z
) Z
1
z
f (z) = z exp
log
dµ(t) =
dρ(t) , z ∈ Λ ,
1 − tz
[0,1]
[0,1] 1 − tz
where µ and ρ are probability measures on [0, 1]. Note that there is a one-to-one relation
between f and µ (a.e.) but this is not the case between f and ρ. An interesting question
is to identify those functions ρ which describe a universally starlike f (and those which
don’t). The purpose of this note is to communicate a recent solution of this problem
obtained by Bakan, Ruscheweyh and Salinas [2].
References
[1] A. Bakan, St. Ruscheweyh, L. Salinas, Universal convexity and universal starlikeness of polylogarithms. 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, 97074
Würzburg, 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 of
Analysis 15 (2007), 247–254
[4] S. Ruscheweyh, L. Salinas and T. Sugawa, Completely monotone sequences and universally
prestarlike functions. Israel J. Math. Vol. 171 Nr. 1 (2009), 285–304.
[5] St. Ruscheweyh, L. Salinas Universally Prestarlike Functions as Convolution Multipliers. Mathematische Zeitschrift 263 (3) (2009), 607–617.
33
Aproximación de soluciones acotadas de
ecuaciones diferenciales con argumento
constante a trozos del tipo generalizado con
impulsos
Manuel Pinto
Ricardo Torres
Abstract
En [1], M.U.Akhmet consideró la ecuación
x0 (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 a
Trozos del Tipo Generalizado (DEPCAG), las cuales poseen soluciones continuas, a
pesar de que γ(t) no lo sea. En los extremos de los intervalos de constancia, estas
ecuaciones generan una ley recursiva, la cual da origen a una ecuación discreta. Es
por esto que estas ecuaciones corresponden al tipo híbridas, ya que combinan tanto
propiedades de ecuaciones discretas como de continuas.
En una DEPCAG, al no considerar la continuidad en los extremos de los intervalos
Ik = [tk , tk+1 ); es decir, al considerar una condición de salto en dichos puntos, se
da origen a las Ecuaciones Diferenciales con Argumento Constante a Trozos del tipo
Generalizado con Impulsos. (IDEPCAG)
x0 (t) = f (t, x (t) , x (γ (t))) ,
∆x|t=tk = Qk x t−
,
k
t 6= tk
t = tk .
(1)
En cada intervalo Ik se satisface la ecuación diferencial ordinaria
x0 (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(tk ) ,
M. Pinto agradece el apoyo del proyecto FONDECYT 1120709, e-mail: [email protected]
R. Torres agradece el apoyo del proyecto FONDECYT 1120709, e-mail: [email protected]
34
donde asumiremos que el límite lateral izquierdo
x(t−
k ) = lim x(t)
t→tk
t<tk
existe ∀tk con k ∈ N y x t+
k = x (tk ) está definido por
−
x (tk ) = x(t−
)
+
Q
x(t
)
,
k
k
k
(Ver ([1, 12, 14])
En esta ocasión, estableceremos condiciones para producir aproximaciones de soluciones de sistemas impulsivos del tipo CNN (Cellular Neural Networks), por medio de
soluciones IDEPCAG utilizando un argumento constante a trozos que aproxime a la
identidad. Es decir, se aproximará el sistema semilineal impulsivo
yi0 (t) = −ai (t)yi (t) + Hi (t, y(t)),
∆yi =
−qi,k yi (t−
k)
+
t 6= tk
Ii,k (yi (t−
k )),
t = tk
y (t0 ) = y0 ,
t = t0
(2)
donde i = 1, 2, . . . , m, y k ∈ N, mediante el sistema IDEPCAG
zi0 (t) = −ai (t)zi (t) + Hi (t, z(γ(t))),
−
−
∆zi |t=γ(tk ) = −qi,k zi (γ(tk ) ) + Ii,k (zi (γ(tk ) )),
z (ζ0 ) = z0 ,
t 6= γ(tk )
t = γ(tk )
(3)
ζ0 = γ(t0 )
con i = 1, 2, . . . , m, k ∈ N, donde
Hi (t, y(t)) =
m
X
bij (t)fj (yj (t)) + ci (t),
j=1
tal que qi,k 6= 1, ai (t), qi,k > 0∀t ∈ R+ , ∀i ∈ [1, m] y
t
γ(t) =
δ, k ∈ N, δ > 0.
δ
(Ver [3, 5, 9, 10, 11]). Se demostrará que al considerar tal función γ (t) (la cual converge
uniformemente a la identidad cuando δ → 0), además de condiciones del tipo Lipschitz
sobre Hi (t, y(t)) y de estabilidad del sistema impulsivo lineal asociado a (2),
Z t
X
ai (u)du +
ln(1 + qi,k ) ≥ σ(t − s), σ > 0, ∀i ∈ [1, m],
s
s≤tk <t
entonces
sup |y(t) − z(t)| → 0,
δ→0
t∈[t0 ,∞)
y
sup |y(tr ) − z(ζr )| → 0,
δ → 0.
t∈[t0 ,∞)
Este método fue propuesto por I.Györi en [7], para la ecuación escalar
x0 (t) = f (t, x(t), x(t − τ ))
con
f (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 caso
acotado, 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 constant
argument. Nonlinear Analysis: Real World Applications. 11:2584-2593. (2010).
[4] K.L.Cooke, I.Györi. Numerical approximation of the solutions of delay differential equations
on an infinite interval using piecewise constant arguments. Computer and Mathematics with
applications, 28(1-3):81-92. (1994).
[5] A.Coronel, M.Pinto, R.Torres. Exponential periodic attractor of impulsive Hopfield-type neural
network system with piecewise constant argument. En preparación.
[6] L.González. Aproximación de soluciones casi periódicas de ecuaciones diferenciales mediante
argumento 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 piecewise
constant arguments. Internat. J. Math. Sci. 14(1). pp. 111-126. (1991).
[8] I.Györi. On numerical approximation using differential equations with piecewise-constant arguments. Periodica Mathematica Hungarica, (56)-1, pp 55-69 (2008).
[9] J.J.Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc.Nat.Acad.Sci. U.S.A. 79 (1982).
[10] M.Pinto, G. Robledo. Existence and stability of almost periodic solutions in impulsive neural
network models. Applied Mathematics and Computation, 217(8):4167-4177, (2010).
[11] M.Pinto, R.Torres. Approximation of bounded solutions of an impulsive differential system using
piecewise 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 con
impulsos. Tesis de Magister. Facultad de Ciencias. Universidad de Chile. (2015).
[14] J. Wiener. Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993).
36
Álgebra
Encargado de Sesión: Nicolás Libedinsky
37
Algunas propiedades del álgebra de símbolos
pseudodiferenciales formales
Jarnishs Beltran
Enrique G. Reyes
Abstract
Se presentará el álgebra de Lie ΨDO de símbolos pseudodiferenciales formales en una y
varias variables. Se mostrará la construcción de una derivación exterior que induce una
extension central del álgebra de Lie ΨDO. También se presentará el álgebra de símbolos pseudodiferenciales formales torcidos y se estudiarán sus extensiones centrales. Se
construyen jerarquías de extensiones centrales de álgebras de Lie de dimensión infinita,
generalizando de esta manera trabajos previos de Khesin [2]. Se construirán triples
de Manin para estas álgebras, y se observará (motivado por de Bajo et al [3]) que
la construcción de estas álgebras de símbolos pseudodiferenciales fomales proporciona
ejemplos 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 Advances 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 the
derivative operator,âĂİ International Mathematics Research Notices, no. 1, pp. 1âĂŞ5,
1992
[3] Bajo, I., Benayadi, S. and Medina, A. âĂĲ Symplectic structures on quadratic Lie
algebrasâĂİ, Journal of Algebra, 316(1), 174-188, 2007
Centro de Investigacion en Complejidad Social,
Universidad
[email protected]
Universidad de Santiago de Chile , e-mail: :[email protected]
del
Desarrollo
e-mail:
38
Módulos de tipo FP-infinito y la Categoría
Estable 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 por
FP ∞ a la clase de estos módulos.
En esta charla, hablaremos de las propiedades de FP ∞ , en particular, como esta
clase de módulos se comporta en el contexto de anillos Noetherianos [Lam01], coherentes
[Gla89] y finalmente para cualquier anillo en general [Bie81]. Mostraremos como esta
clase de módulos nos permite definir una generalización de la categoría estable de
módulos [BGH14].
A lo largo de la charla algunos resultados sobre a la clase de módulos módulos
finitamente n-presentados, denotados por FP n , 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 general
ring. preprint, arXiv:1405.5768v1, 2014.
[BP15] D. Bravo and M. Perez.
arXiv:1510.08966, 2015.
Finiteness conditions and cotorsion pairs.
preprint,
[Gla89] S. Glaz. Commutative Coherent Rings. Lecture Notes in Mathematics. Springer
Berlin Heidelberg, 1989.
[Lam01] T. Y. Lam. A First Course in Noncommutative Rings. Graduate Texts in Mathematics. Springer New York, 2001.
Universidad Austral de Chile
39
Macdonald polynomials in superspace and the 6
vertex model
Luc Lapointe
Abstract
The Macdonald polynomials in superspace are symmetric polynomials involving commuting and anticommuting variables that generalize the Macdonald polynomials. We
will describe how the combinatorics of the Macdonald polynomials extends to superspace. We will focus in particular on how the partition function of the 6 vertex model
arises in the Pieri rules for the Macdonald polynomials in superspace.
Proyecto Fondecyt #1130696, e-mail: [email protected]
40
Teoría de representaciones graduada del álgebra
de blob y el cálculo de Soergel en dos colores
David Plaza
Abstract
En esta charla estudiaremos la teoría de representaciones graduada del álgebra de blob
bn (q, m). En primer lugar, explicaremos como podemos equipar a bn (q, m) con la estructura de álgebra celular graduada [PR14]. Usando esta construcción podemos definir
y 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 grupo
diedral infinito. Si el tiempo lo permite, formularemos una conjetura que relaciona el
álgebra de blob con la categoría de bimódulos de Soergel en dos colores, que explicaría
a nivel categórico la coincidencia combinatoria entre los números de descomposición de
bn (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,
41
Groupoidal Gelfand Models in Group Theory
Jorge Soto Andrade
Abstract
We have conjectured [2] that for any finite group of Lie type G some canonical Gset may be found from which a Gelfand Model [4, 5, 2, 1] for G may be obtained by
geometric induction from a linear character of the associated motion groupoid. This
conjecture has been proved for dihedral groups, the symmetric groups [3]and the projective general linear group of rank 2. The case of the general linear group is work in
progress. This construction via groupoids works even for groups that have no involution
model (like GL(2, q)) in the sense of Bump and Ginzburg [5]. We have also conjectured
that a Gelfand Model of a (finite) group G always lies in the Green ring Green(G) of
G (generated by all permutation representations of G) and that it may be realised as a
top cohomology space of some G-set for a suitable equivariant cohomology theory. We
present some recent counter examples [2] and discuss the domain of validity and variants of these conjectures. We conjecture moreover that not only Gelfand Models but
all symmetric functions of the irreducible representations of G lie in Green(G). Then
the irreducible representations of G would appear as the roots in the representation
ring R(G) of G of a polynomial equation with coefficients in Green(G), whose degree
is the number of conjugacy classes of G. This is joint work with Anne-Marie Aubert
and 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 Gelfand
Models, 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 278 (2004) 294Ð313
¯
Supported by Fondecyt Project 1140510, e-mail: [email protected]
42
Algebraic structures on stellohedra and
pterahedra
L. Berry S. Forcey
M. Ronco
P. Showers
Abstract
Stellohedra and pterahedra are families of polytopes, which may be obtained by the
contraction 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 polytopes
K(Gn ) associated to graphs which are suspensions of other finite graphs. The stellohedron of dimension n is the polytope K(Stn ) associated to the suspension of the trivial
graph with n nodes, while the pterahedra of dimension n is the polytope associated to
the suspension of the line graph with n nodes.
We study the simplicial complex of faces of the stellohedron and the permutohedron
in terms of differents types of planar trees , and describe an extension of the Tamari
order in this case.
Moreover, we defined associative products of degree −1 on the vector spaces spanned
by the nodes of both families of polytopes. These notion uses the Hopf algebra structure
on 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 structures
on the spaces spanned by the faces of Carr and Devadoss’s polytopes of the suspensions
of 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 Solomon
descent algebra, J. of Algebra 177 (3) (1995) 967-982.
[3] A. Tonks, Relating the associahedron and the permutohedron, in Operads: Proceedings of Renaissance 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]
Fondecyt Regular 1130939, e-mail: [email protected]
NSA Grant H98230-14-0121, e-mail: [email protected]
43
Teoría de representaciones del álgebra de
Yokonuma-Hecke y del álgebra de braids and
ties
Steen 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 representaciones de YHn y del álgebra de ’braids and ties’ En introducida por Aicardi y
Juyumaya. 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 y
P. Martin. Sin embargo, en general la relación todavía no está clara. Construimos
módulos espacios tensoriales para ambas álgebras y demostramos que son fieles. Esto
nos permite construir un isomorphismo concreto entre YHn y el álgebra modificada de
Ariki-Koike, introducida por Shoji. Tambien obtenemos de esta manera la presentacion
usada 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 for
classical knots and links from the Yokonuma-Hecke algebras, arXiv:1505.06666.
[5] M. Chlouveraki, S. Lambropoulou, The Yokonuma-Hecke algebras and the HOMFLYPT polynomial, 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 algebra
of 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 new
generators 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. Algebra
Comb., 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
46
C-álgebra de operadores lineales continuos
definidos en c0(I)
J. Aguayo
M. Nova
J. Ojeda
Abstract
Sea I un conjunto arbitrario de índices. Se estudia una clase de operadores lineales
continuos definidos sobre
c0 (I) := (xi )i∈I : xi ∈ K, limxi = 0 ,
i∈I
la cual resulta ser una C-álgebra de Banach conmutativa con unidad. Se muestra que
bajo ciertas condiciones, esta álgebra es isométricamente isomorfa a un espacio de funciones 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]
Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción.
Este trabajo es parcialmente financiado por Proyecto VRID N◦ 214.014.038-1.0IN
47
Espacios y cuerpos residuales en espacios de tipo
Hilbert sobre cuerpos con valuaciones
no-arquimedianas
Herminia Ochsenius
Elena Olivos
Abstract
Sea K un cuerpo completo en una valuación no arquimediana de rango infinito. Un
espacio tipo Hilbert según la norma (NHS) es un K-espacio de Banach de tipo contable
en el cual todo subespacio cerrado admite un complemento ortogonal en el sentido de
la norma. Cuando esta norma proviene de una forma bilineal simétrica, se habla de
espacios tipo Hilbert según la forma (FHS), que son espacios ortomodulares. La teoría
de operadores en estos espacios usa como una herramienta importante la reducción a
espacios residuales sobre cuerpos residuales. Se expondrá su construcción y ejemplos
de teoremas sobre operadores que surgen de su aplicación, correspondientes a trabajos
conjuntos con H. Keller, E. Olivos y W. Schikhof.
Depto. de Matemática y Est. Universidad de La Frontera. e-mail: [email protected]
Agradecimientos a Proyecto DIUFRO DI15-0043
48
Medida Espectral sobre Algebras de Operadores
definidos en c0(N)
J. Aguayo
M. Nova
J. Ojeda
Abstract
En un contexto no-arquimediano, se estudia el espacio Gelfand de ciertas álgebras de
operadores lineales definidos sobre el espacio c0 (N), las cuales resultan ser C-álgebras
de Banach conmutativas con unidad. Se muestra que bajo ciertas condiciones, estas
álgebras son isometricamente isomorfas a un espacio de funciones continuas definidas
sobre conjunto compacto particular. La isometría entre álgebras antes mencionada
preserva elementos idempotentes y permite definir la medida asociada, la cual es conocida como medida espectral. Se muestra, además, que cada elemento del álgebra de
Banach descrita en este estudio, puede ser representada como la integral de alguna
función 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 operators on Free Banach Spaces of countable type over the complex Levi-Civita field, Journal of
Mathematical Physics, Vol. 54(2), 2013.
[3] V. Berkovich, Spectral Theory and analytic geometry over non-archimedean fields, Mathematical 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 y Física Aplicadas, Universidad Católica de la Santísima Concepción.
Este trabajo es parcialmente financiado por Proyecto VRID N◦ 214.014.038-1.0IN.
49
Espacios de Banach sobre cuerpos con valuación
discreta
Elena Olivos
Herminia Ochsenius
Abstract
Si el grupo de valores G de un cuerpo con valuación no arquimediana K es cíclico, el
G-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 la
acción g(b, g 0 ) 7→ (b, gg 0 ) y orden antilexicográfico. Como consecuencia, el espacio E
es un NHS si y solo si B es bien ordenado. En este trabajo estudiamos las propiedades
de espacios uno-ortogonal, rigidos y que contienen c0 cuando el cuerpo tiene valuación
discreta.
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 by
J. 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).
Agradecimientos Proyecto DIUFRO DI15-0043
50
An o-minimal approach to definability of
functions in extensions of C
Javier Utreras
Abstract
In 1984, A. Pillay and C. Steinhorn introduced the concept of an o-minimal structure to
obtain analytical properties of R, and some other related real closed fields, in a limited
tame topological setting. In 2001, Y. Peterzil and S. Starchenko obtained analogues of
many 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 functions
definable in this setting are, almost everywhere, obtained from a given set of functions
via standard complex analytic operations (Schwarz reflection, differentiation, implicit
definition and composition).
We will present a generalization of Wilkie’s result to non-Archimedean extensions
of C, using the fact that under this tame topological setting many standard results
from analysis may still work, after giving some consideration to the parameters of the
functions.
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, School
of Mathematics, The University of Manchester, 2014.
[4] A. J. Wilkie, Some local definability theory for holomorphic functions. Model Theory with
Applications to Algebra and Analysis, Vol 1 (2008) LMS Lecture Note Series 349, CUP, 197213.
51
Análisis Numérico
Encargado de Sesión: Mauricio Sepúlveda
52
Mathematical and numerical analysis for a
nonlocal 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 disease between two spatially distributed host populations having non-coincident spatial
domains with nonlocal and cross-diffusion, the epidemic disease transmission occurring
through a contaminated environment. The mobility of each class is assumed to be influenced by the gradient of the other classes. We address the questions of existence of
weak solutions by using a regularization method. Moreover, we propose a finite volume
scheme and proved the well-posedness, nonnegativity and convergence of the discrete
solution. The convergence proof is based on deriving a series of a priori estimates and
by using a general Lp compactness criterion. Finally, the numerical scheme is illustrated
by 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 in
Applied Sciences, 21, (2011), 307–344.
[2] M. Bendahmane and M. Langlais, A reaction-diffusion system with cross-diffusion modelling 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 of
Numerical 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]
CI2 MA and DIM, Universidad de Concepción, Concepción, Chile, e-mail: [email protected]
53
A kind of Mixed Finite Element Variational
Formulation for a Parabolic Problem
Mauricio Barrientos
Karina Vilches
Abstract
In this work we study a parabolic problem arised from Keller-Segel model for chemotaxis. A new formulation of the system of partial differential equations is obtained
by the introduction of a new variable, which have a mixed variational formulation
structure. At this point, the applicability of adaptive moving meshes theory is carried
out with the purpose to obtain a cheap and better description of the behavior of the
particles 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 parabolic
partial differential equations. Comput. & Fluids, vol. 46 (2011), 353–361.
[4] A. Morrocco, Numerical Simulation of Chemotactic Bacteria Aggregation via Mixed Finite
Elements, ESAIM: M2AN., vol. 37 (4) (2003) 617–630.
PUCV, Pontificia Universidad Católica de Valparaíso, Chile. e-mail: [email protected].
Partially supported by Proyecto DI Regular 037.438/2015.
Departamento de Ingeniería Matemática, Universidad de Chile, Chile. e-mail: [email protected]
54
Discontinuos Galerkin scheme for Helmholtz
equation
Tomás Barrios
Rommel Bustinza
Abstract
In this talk, in order to describe the different phenomena, we first apply the local
discontinuous Galerkin (LDG for short) method to solve a Helmholtz problem in a
bounded domain. We establish existence, uniqueness as well as optimal rate of convergence, for meshes such that its meshsize is small enough. Additionaly, with the aim
to obtain more flexibility in the elections of the discrete subspaces pairs, we analysed
an stabilized mixed discontinuous formulation for this problem. The procedure of the
stabilization is through of the addition of an appropriate Galerkin least squares term to
the mixed formulation. Following the same ideas than in the first part, we also prove
the well posedness of this scheme and the optimal convergence are guaranteed for h
small enough, too. Finally, several numerical experiments confirming the theoretical
properties 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 augmented
discontinuous Galerkin formulation. IMA Journal of Numerical Analysis, vol 30, 4, pp. 9871008, (2010).
[3] T.P. Barrios and R. Bustinza: An a posteriori error analysis of an augmented discontinuous 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 method
for solving boundary value problems for the Helmholtz equation: A priori and a posteriori
error analyses. Preprint 2013-13, Departamento de Ingeniería Matemática, Universidad de
Concepción, (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]
CI2 MA and DIM, Universidad de Concepción, Concepción, Chile. e-mail: [email protected]
55
Hydraulic parameter estimation under
non-saturated flow conditions in copper heap
leaching
Emilio Cariaga
Rubén Martínez
Mauricio Sepúlveda
Abstract
The mathematical modeling of the unsaturated flow problem requires the simultaneous
resolution of two problems: the Richards equation and the estimation of the hydraulic
parameters involved in hydraulic conductivity and in the retention curve. Various
techniques have been applied to both problems in a wide range of situations. In this
article, a novel implementation of the processing techniques involved in copper heap
leaching is presented. Specifically, the impact of the used numerical method and the
selection 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 from
the Chilean copper mining industry and related works.
References
[1] E.Cariaga, R.Martínez, M.Sepúlveda, Hydraulic parameter estimation under nonsaturated 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]
UACh, Universidad Austral de Chile, Chile, e-mail: [email protected]
56
Torque free rigid body motion: an elliptic
function primer
Roberto León
Luis Salinas
Claudio Torres
Abstract
An analysis of the motion for a rigid box is presented when a force or torque is not
applied. The fundamental equations describing the motion of the rigid box are the well
known equations from classical mechanics [1],
X
F = ma,
X
·
MG = HG
where the sum of all external forces acting on the body is equal to mass by the instantaneous acceleration of the center of mass G, and the sum of torques applied to the
body 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
·
·
I22 ω 2 = (I33 − I11 ) ω3 ω1 ,
·
I33 ω 3 = (I11 − I22 ) ω1 ω2 ,
I11 ω 1 = (I22 − I33 ) ω2 ω3 ,
where I11 , I22 and I33 are the principal moments of inertia, and ω1 , ω2 and ω3 are the
instantaneous angular velocity of the rigid body.
Solving these set of differential equations, it is obtained as solutions, the set of Jacobi
elliptic functions sn, cn and dn [2] for each rotational velocity of the rigid body, and it
will depend on the initial conditions which Jacobi elliptic function correspond to each
rotational velocity. An algorithm to obtain these functions is presented with the aim
to get an analytical solution. A comparison between analytical and numerical solution
is 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 Andres
Bello, e-mail: [email protected] , [email protected] , [email protected]
57
Analysis of an augmented pseudostress-based
mixed formulation for a nonlinear Brinkman
model of porous media flow
Gabriel N. Gatica
Luis F. Gatica
Filánder A. Sequeira
Abstract
In this work we introduce and analyze an augmented mixed finite element method for
the two-dimensional nonlinear Brinkman model of porous media flow with mixed boundary conditions. More precisely, we extend a previous approach for the respective linear
model to the present nonlinear case, and employ a dual-mixed formulation in which
the main unknowns are given by the gradient of the velocity and the pseudostress. In
this way, and similarly as before, the original velocity and pressure unknowns are easily
recovered through a simple postprocessing. In addition, since the Neumann boundary
condition becomes essential, we impose it in a weak sense, which yields the introduction
of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange
multiplier. We apply known results from nonlinear functional analysis to prove that
the corresponding continuous and discrete schemes are well-posed. In particular, a feasible choice of finite element subspaces is given by Raviart-Thomas elements of order
k ≥ 0 for the pseudostress, piecewise polynomials of degree ≤ k for the gradient of the
velocity, and continuous piecewise polynomials of degree ≤ k + 1 for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator
for this problem. Finally, several numerical results illustrating the performance and the
robustness of the method, confirming the theoretical properties of the estimator, and
showing 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 analyses
of 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.
CI2 MA 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 la
Santísima Concepción, Concepción, Chile, e-mail: [email protected]
Escuela de Matemática, Universidad Nacional de Costa Rica, Heredia, Costa Rica, e-mail:
[email protected]
58
A mixed FEM for a vorticity based formulation
of the Brinkman problem
David Mora
Abstract
In this talk, we develop a mixed finite element method for the Brinkman equations
formulated in terms of velocity, vorticity and pressure. By employing an extension
of the Babuska-Brezzi theory, it is proved that the resulting continuous and discrete
variational formulations are well-posed. In particular, we show that Raviart-Thomas
elements of order k ≥ 0 for the approximation of the velocity field, piecewise continuous
polynomials of degree k + 1 for the vorticity, and piecewise polynomials of degree k for
the pressure, yield unique solvability of the discrete problem. We establish a priori
error estimates in the natural norms. Finally, we report several numerical experiments
illustrating the behavior of the proposed scheme and confirming our theoretical results.
We will report on results obtained in collaboration with V. Anaya, R. Oyarzúa and
R. 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 the
vorticity-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 CI2 MA, Universidad de
Concepción, Concepción, Chile e-mail: [email protected]
59
Finite Volume Method Improved Sequential
Solution Applied To Two-Dimensional
Convective-Diffusive Heat Transfer
Nelson O. Moraga
Juan Jaime
Abstract
The numerical solution of fluid mechanics with convective heat transfer problems in
most of the industrial applications is obtained by the Finite Volume Method, FVM [1].
One node per volume and suitable interpolation functions for convective and diffusion
terms of the governing partial differential equations are used in the FVM. The solution
of the discretized mathematical model is obtained by a sequential-iterative-implicitprocedure to solve in a segregated way the continuity, linear momentum and energy
non-linear coupled PDEâĂŹs [2]. The purpose of this paper is to describe improvements in the efficiency of the numerical solution, estimated in terms of reduction in
the 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 heat
convective cooling, with and without solidification, inside inner rectangular cavities and
in the annular space between horizontal concentric cylindrical containers. Results of
the evolution of velocity, temperature and liquid-solid interface position obtained by
the classical sequential algorithm Semi-Implicit Method for Pressure Linked Equations,
SIMPLE [4] and by the improved PSIMPLER algorithm, are presented and discussed
for each problem. The omission of the corrections of the velocity components of SIMPLE, avoided in PSIMPLER and the use to a second prediction-correction step in the
proposed iterative algorithm allows the use of higher values for the under-relaxation coefficients 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 analyzed
and 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 Its
Applications), Springer, 2015.
Departamento de Ingeniería Mecánica, Universidad de La Serena, Chile, e-mail: [email protected]
Doctorado en Ingeniería en Alimentos y Bioprocesos, Universidad de La Serena, Chile
El trabajo es financiado por CONICYT-Chile en Proyecto FONDECYT 1140074
60
[2] H. K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics: The
Finite Volume Method, Longman Scientific Technical, 2007.
[3] N. Moraga, S. RamÃŋrez, M. Godoy, P. Ticchione, Study of convective non-Newtonian alloy
solidification 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 transfer
in three-dimensional parabolic flows, International Journal Heat Mass Transfer, 15: 1787-1806,
1972
61
On the analyses of three DG schemes for Stokes
problem based on velocity-pseudostress
formulation
Tomás P. Barrios
Rommel Bustinza
Felipe Sánchez
Abstract
In this talk we first discuss the well posedness of a modified LDG scheme of Stokes
problem, considering a velocity-pseudostress formulation. The difficulty here relies on
the fact that the application of classical Babuška–Brezzi theory is not easy, so we proceed in a non standard way. For uniqueness we apply a discrete version of Fredholm’s
alternative theorem, while the a priori error analysis is done introducing suitable projections of exact solution. As a result, we prove that the method is convergent, and under
suitable regularity assumption on the exact solution, the optimal rate of convergence
is guaranteed. Next, we propose a second discrete formulation, by adding a div-div
stabilization term, which helps to guarantee its well posedness as well as the a priori
error estimates. These are done by application of standard theory. Finally, with the
aim of having more freedom to choose the approximation spaces, we propose a third DG
scheme, 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 us
to redefine one of the parameters that define the so called numerical fluxes, in contrast
with 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. 9871008, (2010).
[2] B. Cockburn, G. Kanschat, D. Schötzau and C. Schwab: Local discontinuous Galerkin
method 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]
CI2 MA and DIM, Universidad de Concepción, Concepción, Chile. e-mail: [email protected]
DIM, Universidad de Concepción, Concepción, Chile, e-mail: [email protected]
62
Convergence of a finite volume scheme for a
sorption-coagulation equation
Erwan Hingant
Mauricio Sepúlveda
Abstract
This work is devoted to the derivation and the mathematical study of a new model
for water-soluble polymers and metal ions interactions, which are used in chemistry for
their wide range of applications. First, we motivate and derive a model that describes
the evolution of the configurational distribution of polymers. One of the novelty resides in the configuration variables which consider both, the size of the polymers and
the quantity of metal ions they captured through sorption. The model consists in a
non-linear transport equation with a quadratic source term, the coagulation. Then, we
prove the existence of solutions for all time to the problem thanks to classical fixed
point theory. Next, we reformulate the coagulation operator under a conservative form
which allows to write a finite volume scheme. The sequence of approximated solutions
is proved to be convergent (toward a solution to the problem) thanks to a L1 −weak stability principle. Finally, we illustrate the behaviour of the solutions using this numerical
scheme 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, Discrete
Contin. Dyn. S. - B, vol. 17, 3, pp. 775-799, 2012.
[2] H. Hingant, M. Sepúlveda Derivation and mathematical study of a sorption-coagulation
equation, Nonlinearity, vol. 28, 10, pp. 3623âĂŞ3661, 2015
[3] B. Rivas, E.D. Pereira, M. Palencia, J., Sánchez, Water-soluble functional polymers
in conjunction with membranes to remove pollutant ions from aqueous solutions, Progress in
Polymer Science vol. 36, 2, pp. 294-322, 2011.
UAMat, Universidad Federal de Campina Grande, Brazil, e-mail: [email protected]
63
Numerical Analysis Of Non-Newtonian Flows In
Liquid Foods, Solidification Casting And
Polymer Injection Molding
Nelson O. Moraga
Edgardo J. Tabilo
Abstract
The objective of this work is to describe the numerical analysis and solution of fluid
mechanics and convective heat transfer in three industrial processes. The cases investigated and the numerical methods are: the Finite Element Method in the fabrication of
yoghurt 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 industry [4,5]. Most of the fluids in liquid foods and in molten metals, alloys and polymers
are non-Newtonians. In these applications the relation between shear stresses and deformation rate introduces additional non-linear terms in the momentum equations that
must be added to the non-linearity caused by the convective terms. The applications
of non-Newtonian flows studied in this presentation are: the rotational movement of
thixotropic yoghurt in a cylindrical container, the melting of pseudo-plastic Al-1.7%Si
alloy inside a square mold and injection and three-dimensional solidification of CrossWLF polyester in a mold cooled by water. In each case the procedure starts from the
construction of the mathematical models in terms of a system of PDEÂťs, followed by
the 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 of
the dependent variables and of the effective viscosity are analyzed. Finally, a discussion
of 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 / nonNewtonian liquid solidification for various cavity configurations and rheological models, International Journal of Heat and Mass Transfer, 54: 5116-5125, 2011.
Departamento de Ingeniería Mecánica, Universidad de La Serena, Chile, e-mail: [email protected]
Doctorado en Ingeniería en Alimentos y Bioprocesos, Universidad de La Serena, Chile
El 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 in
mold, 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 the
thermal 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 moulding
with a new Control Volume FEM / Volume of Fluid, International Journal for Numerical
Methods in Fluids, 71: 1509-1523, 2013.
65
Numerical Challenges for a Numerical
Simulation of a Coupled Model for Grain
Growth in 2D
Claudio E. Torres
Alejandro Sazo
Abstract
Mathematical analysis of numerical algorithms that model the evolution of grain structure of polycrystalline materials in 2D has gained interest over the past years [1,2]. This
interest has been driven by disagreement between experimental data and simulated data.
To create simulated data, a number of assumptions and simplifications are made. From
the simplifications made, two family of approaches appear: curvature driven model and
vertex model. Both of them have been greatly studied but unfortunately neither of
them has been able to fully predict correctly experimental data. In this direction, we
propose a new coupled model were we connect both approaches.
A grain structure can be defined as a collection of non-overlapping grains that
completely cover a bounded periodic/non-periodic 2D domain. Grains are of polygonal
shapes with curved sides, such that two grains meet on a grain boundary and grain
boundaries meet at grain boundary junction (also known as triple junctions).
The mathematical models are based on the total grain boundary energy:
XZ 1
E(t) =
γ(∆αk ) klk (s, t)kds,
k
0
where γ is the grain boundary energy, ∆αk is the misorientation of grains meeting at
dξ~k (s, t) ~k
grain boundary k, lk =
, ξ (s, t) is a parametric curve of the grain boundary
ds
k. The evolution of the grain structure is modeled such that dissipation of the energy
d
is enforced, i.e.
E(t) < 0. Thus,
dt
K
X
d
E(t) = −
dt
k=1
Z
0
1
M
3
X
X
∂
(T(k) ) · v(k) ds +
vm ·
T(m,l)
∂s
m=1
l=1
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
Z
1
∂ (k)
T (s, t) φi (s) ds , i 6= {1, N }
0 ∂s
Z 1
3
X
(m,l)
(m,l)
ẋm (t) = λ
−T
+
T
(s, t) φ1,m (s) ds
(k)
ẋi (t) = µ
l=1
0
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, Numerical Analysis of the Vertex Models for Simulating Grain Boundary Networks, SIAM Journal
on Applied Mathematics, Vol. 75 Nr. 2 (2015), 762-786.
[2] Kinderlehrer, David and Livshits, Irene and Ta’asan, Shlomo, A Variational Approach to Modeling 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 Mathematics, 2000.
67
On Nonlinearly and Linearly Implicit IMEX
Runge-Kutta methods for a class of degenerate
convection-diffusion problems
Luis-Miguel Villada Osorio
Abstract
Multi-species kinematic flow models with strongly degenerate diffusive corrections give
rise to systems of nonlinear convection-diffusion equations of arbitrary size. Aplications
to these systems include models of polydisperse sedimentation and multi-class traffic
flow. Implicit-explicit (IMEX) Runge-Kutta (RK) methods [1] are suitable for the solution of these convection-diffusion problems since the stability restrictions, coming from
the explicitly treated convective part, are much less severe than those that would be
deduced from an explicit treatment of the diffusive term. These schemes usually combine an explicit Runge-Kutta scheme for the time integration of the convective part
with a diagonally implicit one for the diffusive part. In [4], a nonlinear implicit IMEXRK scheme of this type is proposed, where the nonlinear and non-smooth systems of
algebraic equations arising in the implicit treatment of the degenerate diffusive part
are solved by smoothing of the diffusion coefficients combined with a Newton-Raphson
method with line search. A particularly efficient variant of these schemes, so-called
linearly implicit IMEX-RK schemes [3], arise from discretizing the diffusion terms in a
way that more carefully distinguishes between stiff and nonstiff dependence, such that
in each time step only a linear system needs to be solved. In this talk a serie of examples of polydisperse sedimentation [2] and multi-class traffic flow [5] it is demonstrated
that these linearly implicit IMEX-RK schemes approximate the same solutions as the
nonlinearly 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 (Universidad 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 timedependent 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 systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math., 64 (2003),
41–80.
CI2 MA & GIMNAP-Departamento de Matemática,
[email protected]
Universidad del Bío-Bío,
Chile e-mail:
68
[3] S. Boscarino, R. Bürger, P. Mulet, G. Russo and L.M. Villada. Linearly implicit IMEX RungeKutta 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 applied
to 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-WhithamRichards 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
70
Stability of mKdV breathers in the energy space
and numerical results
Miguel Angel Alejo
Abstract
In this talk I will show some recent results about the H 1 stability of breather solutions
of mKdV. I will also present Bäcklund transformations for the mKdV and I will show
some numerical results about the study of the spectra of linearized operators.
71
Sobolev and Hardy-Sobolev type inequalities
Hernán Castro
Abstract
In this talk we ask ourself about the validity of the following family of inequalities of
the type
Z
1/q
Z
1/p
|u(x)|q v(x)dx
≤C
|∇u(x)|p w(x)dx
,
Ω
Ω
where v and w are weights functions and Ω is some subset of RN , usually Ω = RN ,
N
Ω = RN
+ (the half-space), or Ω = (R )+ (the positive cone).
There are some famous weights that have been known and vastly used for a long
time, for instance v(x) = w(x) = 1 (Sobolev inequality), v(x) = |x|−q , w(x) = 1 (Hardy
inequality), 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
+|A|
.
ai ≥ 0 for all p ≥ 1. They show that if |A| = a1 + a2 + . . . + aN , then p∗ = NN+|A|−p
Our goal is to extend the result of Cabré and Ros-Oton to obtain a Hardy-Sobolev
type inequality for monomial weights, that is to consider weights of the form
v(x) = xb11 · . . . · xbNN
and w(x) = xa11 · . . . · xaNN .
IMAFI, Universidad de Talca, e-mail: [email protected]
72
On the control of the stabilized
Kuramoto-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 the
domain and acting only on either the KS equation [1] or the heat equation [2]. The
local null-controllability of the system is proven. The proof is based on a Carleman
estimate for the linearized system around the origin. A local inversion theorem is
applied to get the result for the nonlinear system.
References
[1] E. Cerpa, A. Mercado, A. Pazoto, Null controllability of the stabilized KuramotoSivashinsky 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 system
by a single control acting on the heat equation, under review.
Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile, e-mail:
[email protected]
73
A Dirichlet problem involving the divergence
operator
Gyula Csató
Abstract
Consider the following classical and broadly treated problem: Given a function f on
Ω ⊂ Rn , find a vector field u such that
(
div u = f in Ω
u=0
on ∂Ω.
R
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 + ha, ui = f in Ω
u=0
on ∂Ω,
where a is a given vector field and h , i is the scalar product. What is now the necessary
and sufficient condition for solvability? What is the expected regularity result? The
answer is easy, if a is of the special form a = grad A. We present some results and
conjectures about the general case. This is joint work with B. Dacorogna appearing in
the 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
Solvability of fractional problems with
supercritical drifts
Gonzalo Dávila
Abstract
We study the existence of viscosity solutions u : RN → R of quasilinear integrodifferential equations given by the following model
(−∆)s u + g(|Du|) = f
in Ω,
subject to the exterior Dirichlet condition
u = ϕ in Ωc ,
with f ∈ C(Ω̄), ϕ ∈ C(Ωc ) and bounded. Here g ∈ C 1 (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 supercritical
regime.
We prove that under suitable conditions there are viscosity solutions attaining the
boundary data in the classical sense. As a by product of our proof we also get boundary
regularity for the viscosity solution.
This is a joint work with Alexander Quaas (UTFSM) and Erwin Topp (UTFSM).
Universidad Santa María
75
Formación de singularidades para el flujo de
mapas armónicos
Juan Dávila
Abstract
Estudiamos formación de singularidades en tiempo finito para el flujo de mapas
armónicos de un dominio en el plano a valores en la esfera 2-dimensional. Encontramos
el perfil y la tasa de reviente para ciertas condiciones iniciales, y la estabilidad de
este fenómeno. Anteriormente van den Berg, Hulshof y King (2003) habían encontrado
formalmente la tasa de reviente, y Raphael y Schweyer (2013) probaron estos resultados
en 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 CMM
Millenium Nucleus CAPDE NC130017, e-mail: [email protected]
76
Solutions to a supercritical elliptic problem
Jorge Faya
Abstract
We consider the supercritical problem
−∆v = |v|p−2 v
in Θ,
v = 0 on ∂Θ,
in a bounded smooth domain Θ ⊂ RN , N ≥ 3, for p >
reduce this problem to a critical problem of the form
4
−∆u = Q(x)|u| n−2 u
in Ω,
u=0
2N
N −2 .
on ∂Ω,
(1)
In some cases one can
(2)
in a domain Ω ⊂ RN of lower dimension by means of a Hopf map and some symmetry
considerations. 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 joint
work with Mónica Clapp and Angela Pistoia.
CMM, Universidad de Chile
77
Entire sign changing solutions with finite energy
to the fractional Yamabe equation
Danilo Garrido
Monica Musso
Abstract
We are interested in the existence of finite-energy sign-changing solutions to the fractional Yamabe type equation in Rn ,
(−∆)s u = γ |u|p−1 u in Rn
(1)
where n ≥ 3, p is the fractional critical Sobolev exponent p = n+2s
n−2s and γ > 0 is a
constant chosen for normalization purposes. For any s ∈ (0, 1), (−∆)s is the nonlocal
operator defined as
Z
Z
u(x) − u(y)
u(x) − u(y)
s
(−∆) (x) = c(n, s) P.V.
dy = c(n, s) lim
dy,
n+2s
n+2s
+
→0
Rn |x − y|
Rn \B(x,) |x − y|
n
where P.V. stands for the principal value and c(n, s) = π −(2s+ 2 )
Γ( n
+s)
2
Γ(−s) .
Finite energy sign-changing solutions to (1), are poorly understood. The purpose of
this paper is to give a first example of finite-energy sign-changing solutions to (1), in all
dimensions n ≥ 3, and for s ∈ ( 21 , 1): we build a solution to Equation (1) which looks
like the soliton U surrounded by k negative copies U properly scaled and distributed
along the vertices of a regular polygon with radius 1. Our main result is the following
Theorem
2jπi
Let n ≥ 3 and s ∈ ( 21 , 1). Write Rn = C × Rn−2 and let ξjk = (e k , 0), j = 1, . . . , k.
Then for any sufficiently large k there is a finite energy solution to Problem (1) of the
form
k
X
− n−2s
uk (x) = U (x) −
µk 2 U µ−1
k (x − ξj ) + o(1),
j=1
where
µk =
[2
n−2s
2
P∞
j=1 j
k2
2s−n ]−1
(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 nonlocal
equations. Proc. Amer. Math. Soc., 141 (2013), no. 11, 3865ÃćâĆňâĂĲ-3870.
[6] E. Di Nezza, G. Palatucci, E. Valdinoci. HitchhikerÃćâĆňâĎćs guide to the fractional Sobolev
spaces.Bull. Sci. Math., 136 (5), (2012) 521ÃćâĆňâĂĲ-573.
[7] M. del Pino, M. Musso, F. Pacard, A. Pistoia. Large Energy Entire Solutions for the Yamabe
Equation. Journal of Differential Equations 251, (2011), no. 9, 2568–2597.
[8] M. del Pino, M. Musso, F. Pacard. A. Pistoia. Torus action on S n and sign changing solutions
for 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 equation
with 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- Sobolev
inequality. In Spectral theory, function spaces and inequalities, volume 219 of Oper. Theory
Adv. 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. Analysis 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. of
Math. (2), 118 (2), (1983) 349ÃćâĆňâĂĲ- 374.
79
Multiplicity of solutions for some semilinear
problems involving nonlinearities with zeros
Leonelo Iturriaga
Jorge García-Melián
Abstract
In this talk we consider the semilinear elliptic problem
−∆u = λf (u) in Ω
u=0
on ∂Ω
where f is a nonnegative, locally Lipschitz continuous function with r positive zeros, Ω
is a smooth bounded domain and λ > 0 is a parameter. We show that for large enough
λ 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 on
the sub and supersolutions method and topological degree arguments, together with
the use of a new Liouville theorem which is an extension of recent results to this type
of 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 operators 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 equations
via 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 elliptic
equations, Comm. Pure Appl. Math. 34 (1981), 525–598.
Departamento de Matemática, Universidad Técnica Federico Santa María.
[email protected]
Departamento de Análisis Matemático, Universidad de La Laguna.
C/. Astrofísico Francisco Sánchez s/n, 38271 – La Laguna, SPAIN
and
Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica,
Molecular y Fotónica, Facultad de Física, Universidad de La Laguna, e-mail: [email protected]
e-mail:
80
[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 critical
and 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-Laplacian
involving 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’s
Operator, Comm. Partial Differential Equations 31 (2006), 987–1003.
81
Kink dynamics in the φ4 model: asymptotic
stability for odd perturbations in the energy
space
Michal 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 by H(x) = tanh(x/ 2),
is an explicit stationary solution of this model. From a result of Henry, Perez and
Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability
of the kink for odd perturbations in the energy space. The proof is based on Virialtype estimates partly inspired from previous works of Martel and Merle on asymptotic
stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein
in the case of general nonlinear Klein-Gordon equations with potential: the interactions
of the so-called internal oscillation mode with the radiation, and the different rates of
decay of these two components of the solution in large time.
82
Singularly perturbed PDEs and patterns with
periodic profiles
Fethi Mahmoudi
Abstract
We consider a class of singularly perturbed equations in planar domains: as the singular
perturbation parameter tends to zero, we exhibit a family of solutions concentrating
at the boundary with asymptotically periodic profile. As solutions with uniform profile at the boundary were known to exist, the result here reflects the phenomenon of
Turing’s instability, which triggers formation of inhomogeneous structures from more
homogeneous ones.
CMM and Universidad de Chile [email protected]
83
Some results for a problem from Combustion
Abstract
This work presents results concerning the Perturbed Gelfand Problem, given by
∂v
= ∆v + λ ev/(1+ v)
∂t
v(0, x) = v0 (x) in Ω,
in ]0, T ) × Ω,
(1)
v = 0 on ]0, τ ) × ∂ Ω,
and an Arrhenius problem of the type
∂v
= ∆v + λ (1 + v)q ev/(1+ v)
∂t
v(0, x) = v0 (x) in Ω,
in ]0, T ) × Ω,
(2)
v = 0 on ]0, τ ) × ∂ Ω
with q > 0 and q < 0. The sign of q, including zero, depends on the character of the
chemical reaction under modelling, as both (1) and (2) are motivated by Combustion
Theory, see [5] or [11]. In particular, when & 0, it is recovered the classical Gelfand
Problem.
Within the results to present there are multiplicity ones in the case of the
steady problems, and blow-up results for the evolution problem. Numerical results
will be also presented as part of the study, in particular bifurcation diagrams and also
computation 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 Mathematical 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]
84
[5] J. Buckmaster, G.S.S. Ludford: Theory of Laminar Flames; Cambridge Monographs on Mechanics and Mathematics, Cambridge University Press (1982).
[6] Y. Du, Y. Lou: Proof os a conjecture for the perturbed Gelfand Equation from Combustion
theory; J. Diff. Equations, vol. 173 (2001), pp. 213-230.
[7] Y. Du: Exact multiplicity and S-shape bifurcation curve for some semilinear elliptic problems
from combustion theory; SIAM J. Math. Anal., vol. 32-4 (2000), pp. 707-733.
[8] G. Gavallas: Nonlinear Differential Equations of Chemically Reacting Systems; Springer Tracks
in 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. SchidtLainé, CNRS-France (1996).
[10] A. Kapila: Asymptotic Treatment of Chemically Reacting Systems; Applied Mathemtical Series
PITMAN (1983).
[11] A. Liñán, F. Williams: Fundamental Aspects of Combustion; The Oxford Engineering Science
Series, 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.
85
Desigualdades de restricción de Fourier:
existencia y no existencia de máximos y cálculo
de la mejor constante
René Quilodrán
Abstract
La transformada de Fourier de una función integrable en Rd es una función continua que
tiende a cero en infinito y por lo tanto tiene sentido restringirla a una variedad S, como
la esfera, el paraboloide, etc., mientras que para una función de L2 la transformada de
Fourier 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 , para
p ∈ [1, 2], más precisamente, si existe una desigualdad
kfˆS kLq ≤ Ckf kLp .
Hablaremos de esta desigualdad, su relación con desigualdades de Strichartz para EDPs
clásicas, de la mejor constante C y de la existencia y no existencia de funciones que
maximizan kfˆS kLq /kf kLp .
Universidad de los Lagos, e-mail: [email protected]
86
Stationary harmonic functions whose Laplacian
is a Radon measure
Rémy Rodiac
Abstract
Let Ω ⊂ R2 be a bounded set. In this talk I will study local regularity properties of
Radon measures µ and functions h ∈ H 1 (Ω) such that
∆h = µ
(1)
ωh := (∂x h)2 − (∂y h)2 − 2i(∂x h)(∂y h)
(2)
and the quantity
is holomorphic in Ω. This problem is related to limiting vorticities measures of GinzburgLandau system as shown by Sandier and Serfaty. It is also linked to vorticity measure
of the time independent Euler system in fluid mechanics and to limiting vorticity measures of system of point vortices. I will prove that, locally, near almost every point of
the domain, h can be written as h = |H| for some smooth harmonic function H. In
particular we deduce that the measure ∆h is concentrated on lines, which are sets of
zeros of harmonic functions.
87
Fractional mean curvature flow
Mariel 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 fractional
mean curvature and analyze their behavior under suitable assumptions. I will discuss
in more depth the evolution of graphical hyper-surfaces, which is an important model
in the local case.
This is joint work with Enrico Valdinoci
Pontificia Universidad Católica
[email protected]
Weierstrass
Institut
für
[email protected]
de
Chile,
partially
Angewandte
funded
Analysis
by
und
Fondecyt
1150014
Stochastike,
e-mail:
e-mail:
88
Moderate solutions of semilinear elliptic
equations with Hardy potential
PHUOC-TAI NGUYEN
Abstract
Let Ω be a bounded smooth domain in RN . We study positive solutions of equation
(E) −Lµ u + uq = 0 in Ω where Lµ = ∆ + δµ2 , 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. If
µ < CH (Ω) (the Hardy constant for Ω) every positive Lµ - harmonic functions can be
represented 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 introduce
a notion of normalized boundary trace by which we obtain a complete classification of
the positive moderate solutions of (E) in the subcritical case, 1 < q < qµ,c . (The critical
value depends only on N and µ.) For q ≥ qµ,c we show that there exists no moderate
solution with an isolated singularity on the boundary.
Pontificia Universidad Catolica de Chile, e-mail: [email protected]
89
0
On the C p -regularity conjecture
Eduardo Teixeira
Abstract
C 1,α regularity estimate for solutions of the p-Poisson equation with bounded source,
∆p u = f (x) ∈ L∞ , has been well established since early 1980’s. Determining the
optimal (universal) Hölder exponent α has been a tantalizing question since then, with
major implications to the PDE theory and its applications. The explicit example,
0
0
∆p (|x|p ) = cte sets the ground for what has been termed the C p -regularity conjecture:
The optimal Hölder continuity exponent for the gradient of a function whose p-laplacian
1
is bounded is p−1
, at least when p > 2. In this talk I will discuss a recent proof of
0
the C p -regularity conjecture in dimension two. This is a joint work with Araujo and
Urbano.
Universidade Federal do Ceara, Brasil, e-mail: [email protected]
90
Geometría
Encargado de Sesión : Maximiliano Leyton
91
Linear systems on IHS manifolds
Michela Artebani
Abstract
A classical result in the theory of surfaces is that any complex K3 surface S which has
an ample and base point free class h ∈ H 2 (S, Z) with h2 = 2 is a double cover of the
projective plane branched along a smooth sextic curve [5]. By a result of Beauville
and Fujiki [1, 3] the cohomology group H 2 (X, Z) of a holomorphic symplectic manifold
(or IHS) has a lattice structure given by a non-degenerate quadratic form qX which
generalizes the intersection pairing for K3 surfaces. It is thus natural to ask for a
geometric characterization of IHS manifolds carrying a class h ∈ H 2 (X, Z) with qX (h) =
2. In [4] O’Grady conjectured that general IHS should behave as K3 surfaces, more
precisely that if X is a generic deformation of the Hilbert scheme S [r] for some K3
surface S, equipped with an ample class h with qX (h) = 2, then h is base point free
and the associated morphism φh : X → PN is a double cover.
In a joint work in progress with Samuel Boissière and Alessandra Sarti we are
exploring this problem in case X = S [2] , where S is a projective K3 surface of Picard
number one. A nice example by Beauville shows that in this case O’Grady conjecture
is false [2]. In this talk I will give a short introduction to IHS manifolds and to these
problems 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 and
analytic manifolds (Katata, 1982), 1983, 1–26.
[3] Fujiki, A., On the de Rham cohomology group of a compact Kähler symplectic manifold, Algebraic geometry, Sendai, 1985 (Advanced Studies in Pure Mathematics 10), ed. T. Oda, North
Holland, 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]
92
Group actions on Riemann surfaces up to
topological equivalence.
Antonio Behn
Abstract
Let G be a finite group acting via holomorphic maps on a compact Riemann surface
X 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 having
product 1 and gi of order mi . Conversely, orbits by the action of Aut(G) × Br on the
set of generating vectors corresponding to a fixed signature give rise to topologically
non-equivalent actions, called Hurwitz equivalence classes. To produce families of coverings, and to study their images in different moduli spaces, is of interest for many
problems in algebraic geometry. Therefore to compute Hurwitz equivalence classes is
an experimental tool that is required in different settings.
Working with Anita Rojas and Camila Muñoz we developed scripts in SAGE that
compute the orbits of generating vectors for a given group and signature. These scripts
were part of Camila’s Master Thesis. We show several applications, including a list of
all 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), 409426. https://sites.google.com/a/u.uchile.cl/polygons/
[2] A. Behn, C. Muñoz and A. M. Rojas, Classification of topologically non-equivalent actions
using generating vectors, a SAGE Package.
Partially supported by Fondecyt Grant 1140507, e-mail: [email protected]
93
Acción de grupos en superficies y variedades
abelianas
Angel Carocca
Abstract
En esta charla abordaremos diferentes aspectos de acciones de grupos finitos en superficies de Riemann y variedades abelianas, con interés en las representaciones lineales
inducidas por tales acciones. Presentaremos algunos resultados sobre la acción lineal
inducida en el espacio de Riemann-Roch.
Universidad de La Frontera, Temuco, Chile. Parcialmente financiado por ACT1415 y FONDECYT
94
Sobre la existencia de acciones de grupos
elementales en Superficies de Riemann
Mariela Carvacho
Abstract
Estudiamos acciones de Zkp , el p−grupo elemental abelidano de rango k, en superficies
de Riemann compactas de género g > 1. Por simplicidad tomamos p = 2 y asumimos
que la acción tiene R ≥ 3 puntos de ramificación sobre la esfera. En esta charla
mostraremos condiciones numeéricas necesarias y suficientes para construir vectores
generadores asegurando la existencia de tales acciones.
Este es un trabajo en conjunto con Anthony Weaver.
95
Dual families of Calabi-Yau varieties
Paola 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 hypersurfaces in Q-Fano toric varieties with canonical singularities. This is based on the
definition of good pairs of polytopes. We also show how this duality generalizes both
Batyrev and Berglund-Hübsch-Krawitz (BHK) constructions of mirror Calabi-Yau varieties.
Among the constructions of mirror pairs of Calabi-Yau varieties, we consider the duality
presented 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. The
family 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 be
repeated 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(∆) and
F(∆∗ ) satisfy the topological mirror test:
n−p,q
hp,q
(X ∗ )
st (X) = hst
where n is the dimension of X and X ∗ and hp,q
st 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-Yau
variety is defined by a transverse polynomial W in P(w) = P(w0 , . . . , wn ) and a symplectic
group G. The number of monomials in W is chosen equal to the number of variables (Delsarte
type) so that W can be taken of the form
W =
n Y
n
X
a
xj ij ,
aij ∈ Z≥0
i=0 i=0
and this defines a square matrix AW = (aij ) of exponents which is assumed invertible over
Q. Then one takes
Pn the quotient of {W = 0}/G. Under the assumption that the total degree
d of W is d = i=0 wi , 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 the
topological mirror test on Hodge numbers.
In [1] we present a duality of Calabi-Yau hypersurfaces in Q-Fano toric varieties with
canonical singularities (w.c.s) which generalizes the two constructions shown above. The
key 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-Yau
varieties 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 contained
in the anticanonical polytope of X, let F(∆) be the family of anticanonical hypersurfaces
whose 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 CalabiYau. Similarly, the general element in F(∆∗2 ) is a Calabi-Yau hypersurface in X∆∗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 we
allows 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 ). The
same can be done for the trasponsed pair (ATW , GT ), obtaining the pair (∆T1 , ∆T2 ). Theorem
2 shows that the duality of good pairs is a generalization of the generalized version of the
BHK construction.
Theorem 2. In the previous setting, the pair (∆1 , ∆2 ) is a good pair and (∆T1 , ∆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 varieties, 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.
97
On singular varieties with smooth subvarieties
M. R. Gonzalez-Dorrego
Abstract
Let k an algebraically closed field, char k = 0. Let Z be a reduced irreducible nonsingular subvariety of a normal n-fold X with certain type of singularities, such that Z
intersects Sing(X). We study the singularities of X through which Z passes.
Universidad Autónoma de Madrid, e-mail: [email protected]
98
The 4-prims family.
Víctor González Aguilera
Gustavo Labbé Morales
Abstract
cg be the set of stable
Let Mg be the moduli space of smooth curves of genus g and M
cg can be endowed with a structure of projective complex variety and
curves. The set M
contains Mg as a dense open variety.
The 4-prism is a stable graph thus it determine a terminal stable curve (or a noded
c5 , therefore it is a limit of a 1-dimensional family of smooth
Riemann surface) in M
algebraic curves (or Riemann surfaces) in M5 . In this short note we give a Fuchsian
description and an algebraic description of this family. We also describe the others
c5 − M5 .
points of this family that belong to M
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 dimensional
families of Riemann surfaces. Preprint 2015.
[3] Deligne, P. and Mumford, D. The irreducibility of the space of curves. Publications
Mathématiques de L’I.H.E.S. 36 , 75-109, 1965.
99
Automorphisms of non-cyclic p-gonal surfaces
R. A. Hidalgo
A. F. Costa
Abstract
In this talk I will be concerned with holomorphic automorphisms of a non-cyclic p-gonal
Riemann surface S of genus g > (p − 1)2 , where p ≥ 3 is a prime integer. We will see
that the order of any automorphism is bounded above by 2(g + p − 1) and that this
maximal order is attained for infinitely many genera. This generalizes the similar result
for the particular case p = 3 recently obtained by Costa-Izquierdo in [2]. Moreover, we
also observe that the full group of holomorphic automorphisms of S is either the trivial
group 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 case
that S admits a holomorphic automorphism of order 2(g + p − 1), then its full group
of automorphisms is the cyclic group generated by it and every p-gonal map of S is
necessarily simple.
As a first consequence, we note that each pair (S, π), where S is a Riemann surface
of 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 surface
S is different from a non-trivial cyclic group, then S can be also be defined over its field
of moduli.
A second consequence is that every dessin d’enfant of prime degree is definable over
its 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]
100
Lines on cubic hypersurfaces over finite fields
Antonio Laface
Abstract
Let Fq be the finite field with q elements and let X be a smooth n-dimensional cubic
hypersurface of Pn+1
Fq . We address the problem of deciding if X contains a line defined
over the base field Fq . If n = 2 and there exists a non-cube a ∈ Fq , then the surface of
equation x31 + x32 + x33 + 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 Fano
scheme of lines of X. Using a recent formula of Galkin–Shinder [GS] which relates the
number of Fq -points on F (X) with the number of Fq - and Fq2 -points on X we find
the zeta function of F (X). Then we make use of Weil conjectures [D1, D2] to show
that X always contains Fq -lines when q ≥ 11 and n = 3 or q ≥ 5 and n = 4. Using
Magma [BC], we produce examples of smooth cubic threefolds containing no lines for
q ∈ {2, 3, 4, 5} leaving only the cases where q ∈ {7, 8, 9} open, at least when X is
smooth.
If X is a threefold which admits mild singularities, i.e. one singular point of type
A1 or of type A2 , the geometry of its Fano scheme F (X) is closely related to that of
a 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 when
q ≥ 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. of
Math. 95 (1972), 281–356.
[DLR] Debarre, O., Laface, A., Roulleau, X., Lines on cubic hypersurfaces over finite fields eprint
arXiv:1510.05803
Proyecto Fondecyt regular n. 1150732, e-mail: [email protected]
101
[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 hypersurface, 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, in
Algebraic and arithmetic structures of moduli spaces, Sapporo 2007, 27–45, Adv. Stud. Pure
Math. 58, Math. Soc. Japan, Tokyo, 2010.
102
Authomorphims of graphs and Riemann surfaces
Alexander D. Mednykh
Abstract
We give a short survey of old and new results about automorphism groups and branched
coverings 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) and
others. The branched covering of graphs are also known as harmonic maps or vertically
holomorphic maps of graphs. The main idea of the present talk is to create a parallel between classical results on branched covering of Riemann surfaces and those for
graphs. We introduce the notion of harmonic action on a graph and discuss the Hurwitz type theorems for the groups acting harmonically. These results can be regarded
as 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 automorphism
group acting on a Riemann surface.
We present discrete versions of theorems by Wiman (1895), Oikawa (1956) and
Arakawa (2000), which sharpen the Hurwitz upper bound for various classes of groups
acting on a Riemann surface of given genus.
Then we define a hyperelliptic graph as two fold branched covering of a tree and
a γ- hyperelliptic graph as two fold branched covering of a graph of genus γ. A few
discrete versions of the well-known results on γ-hyperelliptic Riemann surface will be
given.
Sobolev Institute of Mathematics, Novosibirsk, and Institute of Mathematics, Siberian Federal University, Krasnoyarsk, e-mail: [email protected]
103
On Jacobian of circular graphs
Ilya 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 satisfying
the first and the second Kirchhoff rules. One also can define a circulant graph as
the 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 and
Moebius ladder graph. We propose a new method to find the structure of Jacobians
for a large subfamily of circulant graphs.
Sobolev Institute of Mathematics, Novosibirsk, and Institute of Mathematics, Siberian Federal University, Krasnoyarsk, e-mail: [email protected]
104
Curvas de Tipo Fermat y sus Jacobianas
Jaime Pinto
Abstract
En esta charla se hablará sobre cierto tipo de superficies de Riemann compactas con
acciones de grupos de la forma G = Zn o Zn , donde n es una potencia de primo
impar. Mencionaremos una representación de estas superficies como curvas planas
afines cíclicas n-gonales, y estudiaremos la acción de G en sus variedades Jacobianas
asociadas, mostrando las representaciones irreducibes complejas de G para determinar
una 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 Journal
of Mathematics 157 (2007) ,103-122.
U. de Chile. e-mail: [email protected]
105
Automorphisms Of Pseudo-Real Riemann
Surfaces
S. Quispe
R. A. Hidalgo
Abstract
Let S be a pseudo-real Riemann surface of genus g with conformal automorphism group
Aut(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 subgroup
K of Aut(S) isomorphic to H and with S/K of the same signature as S/H it holds that
H = K), then the number of cone points of S/H of the same order is even, and the
cocient group Aut(S)/H is either trivial or cyclic. This result is applied to pseudo-real
hyperbolic generalized Fermat curve of type (k, n) [2, 3, 4], also this results generalizes
the result obtains by E. Bujalance and A. Costa for Cyclic p-gonal pseudo-real Riemann
surfaces 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 Algebra
321, 2009, 1643–1660.
[3] R. A. Hidalgo. Non-hyperelliptic Riemann surfaces with real field of moduli but not definable
over the reals. Archiv der Mathematik 93 (2009), 219–222.
[4] R. A. Hidalgo, A. Kontogeorgis, M. Leyton and P. Paramantzoglou. Automorphisms of the
Generalized 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]
Partially supported by Project FONDECYT 1150003, e-mail: [email protected]
106
Superficies Algebraicas: Uniformización y
Aritmeticidad
Sebastián Reyes-Carocca
Abstract
Una variedad algebraica (compleja, proyectiva y lisa) X es aritmética si existen polinomios homogéneos con coeficientes algebraicos definiendo una variedad isomorfa a X.
El teorema de Belyi [1] caracteriza todas las curvas algebraicas (equivalentemente, superficies de Riemann compactas) que son aritméticas. Para el caso dos-dimensional,
si X es una fibración de Kodaira entonces la aritmeticidad está completamente determinada por (la clase de isomorfía) de su cobertor universal [2]. Generalizando este
hecho, en esta charla veremos cómo para una superficie algebraica la propiedad de ser
aritmética puede ser distinguida en el cobertor universal, cuando ésta corresponde al
espacio total de una familia de superficies de Riemann de tipo finito. Luego aplicaremos
lo anterior para proveer una caracterización de aritmeticidad para cualquier superficie
algebraica de tipo general en términos de los cobertores universales de sus abiertos de
Zariski.
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 is
determined 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 La
Frontera, e-mail: [email protected]
107
Fixed points and rational representations of
actions in abelian varieties
Rubí E. Rodríguez
Abstract
In this talk we will present some known and new results on the relations between fixed
points of an endomorphism of an abelian variety and its rational representation.
We will also discuss the natural generalization to the rational representation of a
group 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]
108
Familias de Jacobianas completamente
descomponibles 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 vector
generador 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 las
correspondientes variedades Jacobianas JX. Para referencias vea [2] o [5]. Por otro
lado, el grupo simpléctico Sp(2g, Z) actúa en el semiespacio superior de Siegel Hg , y
Ag = Sp(2g, Z) \ Hg es un espacio analítico complejo que parametriza las variedades
abelianas 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, y
en 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 las
cuales Z(G, m, θ) es una subvariedad especial. Su criterio es como sigue, si la dimensión
de HG
g es igual a la dimensión de J (G, m, θ), la cual es r − 3, entonces Z(G, m, θ) es
una 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 Tg0 .
Adicionalmente, en [3, Question 6.6] los autores preguntan por subvariedades especiales de Tg de dimensión positiva, tales que la variedad abeliana correspondiente al
punto genérico es isógeno a un producto de curvas elípticas; esto es, completamente
descomponible.
En [4] encontramos varios ejemplos de variedades Jacobianas completamente descomponibles, incluyendo varias familias. Tenemos así un amplio escenario donde buscar
subvariedades especiales de Ag : Dado un par (m, θ) para un G fijo, usando [1] se puede
calcular la dimensión de HG
g , aunque es computacionalmente caro, además se conoce
la dimensión de la familia J (G, m, θ). Si ambas coinciden, la clausura de dicha familia
tendrá las propiedades deseadas.
En esta charla explicaremos los conceptos involucrados en estas preguntas y mostraremos ejemplos de subvariedades especiales correspondientes a familias de variedades
Jacobianas completamente descomponibles, estas fueron desarrolladas en trabajo conjunto con Jennifer Paulhus de Grinnell College y se utiliza la técnica de descomposición
según el álgebra de grupo.
Fondecyt Regular 1140507, e-mail: [email protected]
109
References
[1] A. Behn, R. E. Rodríguez, A.M. Rojas, ‘Adapted Hyperbolic Polygons and Symplectic Representations 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. Cambridge University Press, 1996.
110
Small degree covers and reducible hyperplane
sections
Andrea L. Tironi
Abstract
Let X be a smooth complex projective variety with dim X ≥ 4 and let L be an ample
line 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 ≤ 3
for every i = 1, ..., r. We classify pairs (X, L) as above under the assumption that Yi is a
very 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]
111
Modelos Matemáticos de
Sistemas Biológicos
Encargado de Sesión : Fernando Córdova
112
Medidas de conservación ex situ de tipo
impulsivo: Un enfoque metapoblacional a través
del modelo clásico de Levins
Sandra Araya Crisóstomo
Héctor Rojas-Castro
Abstract
La Biología de Metapoblaciones estudia los efectos de la dinámica local de poblaciones sobre la persistencia regional de una especie y, por lo tanto, ha alcanzado su
máxima aplicación en la Biología de la conservación (Hanski y Simberloff, 1977), ver
[2].
Consideremos una metapoblación, que como tal, se encuentra distribuida en fragmentos o parches, ver [3], [5]. Suponiendo que las tasas de colonización y de extinción
son parámetros constantes que no se ven afectados por las propiedades que determinado
parche pueda tener, entonces la dinámica de la metapoblación puede modelarse a través
del modelo planteado por Richard Levins en su trabajo “Some demographic and genetic
consequences of environmental heterogeneity for biological control", ver [4]:
u(t)
0
u (t) = m u(t) 1 −
− g u(t),
(1)
v
donde u corresponde a la cantidad de parches ocupados, v al total de parches que es
posible 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 una
metapoblación que está extinguiéndose (m < g) y que con el propósito de evitar que
la población total se extinga completamente, se aplica como medida de conservación la
colonización artificial de K parches cada τ unidades de tiempo, es decir, cada cierta
cantidad de tiempo (fija y determinada con anterioridad) se colonizarán K parches
vacíos 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 expresarse
mediante el sistema de ecuaciones diferenciales impulsivo:

si t 6= k τ,
 u(t) = m u 1 − uv − g u,
(2)

+
u(t ) = u(t) + K,
si t = k τ.
A partir del sistema (2), se obtienen y presentan resultados analíticos y de simulación, a fin de establecer la dinámica de largo plazo.
Universidad Católica del Maule, e-mail: [email protected]
113
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 Heterogeneity 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.
114
Consecuencias sobre la abundancia poblacional
del Efecto Allee en hábitats bajo fragmentación
Rodrigo Del Valle
Abstract
En este trabajo deducimos un modelo de crecimiento poblacional de una especie
el cual considera una fragmentación progresivo de su hábitat [3, 5, 6], proceso que
finalmente 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 cápita
de 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 especie
en cuestión. Nuestro interés está en estudiar las consecuencias sobre la dinámica de
la abundancie de la concurrencia simultánea de Efecto Allee y la fragmentación, esta
última 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 habitat
fragmentation in progress, Proceeding of CMMSE 2014,Vol II, 429-440, 2014.
[3] Fahrig L. Efects of habitat fragmentation on biodiversity, Annual Review of Ecology, Evolutions
and 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 know
for sure?, Ecography, 22, 225-232, 1999.
[6] K.W. Herbener, S.J. Tavener & N.T. Hobbs: The distinct efects of habitat fragmentation on
population size, Theor. Ecol., 5, 73-82, 2012.
Universidad Católica del Maule, Proyecto Interno 434171 (2015-2016). e-mail: [email protected]
115
Control epidemiológico optimal por
hospitalización impulsiva
M. Eugenia Solís
Abstract
Este trabajo formula una problemática de control optimal para la erradicación de
una enfermedad contagiosa desde un sistema productivo de animales (granja). Se considera 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–ésima
hospitalización) de los animales infecciosos hacia sitios de aislamiento y cuidados especiales, para en un tiempo posterior retornar al grupo susceptible. El problema central
es encontrar la sucesión {ρk } que minimiza las pérdidas. El contexto matemático es la
maximización de un funcional de una variable ligada a una ecuación diferencial impulsiva.
References
[1] Diekmann and Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, Chichester. (2000).
[2] Shulaw and Bowman On-Farm Biosecurity: Traffic Control and Sanitation. Extension FactSheet, 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).
116
Dinámica de la distribución genotípica bajo
mortalidad diferenciada por rasgos fenotípicos
Héctor Rojas-Castro
Abstract
Este trabajo considera una población mendeliana, cerrada, dipliode y panmíctica, la
cual admite un modelo de crecimiento malthusiano. Si A y a son los dos posibles alelos
para un locus determinado, compartimentamos la población en tres tipos de individuos
según 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 (·) la
abundancia 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 (·) y
fa (·), como la proporción que se observa de un alelo específico respecto al conjunto de los
que pueden ocupar un locus determinado en la población. Entoces, fA (t) = x(t)+y(t)/2
y 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 cierta
porción es capturada. La cantidad de individuos capturados por unidad de tiempo es
distinta en cada una de las subpoblaciones genotípicas, pues expresan una característica
fenotípica que facilita o dificulta la captura. En [5] se describen ejemplos en que la
captura de algún recurso pesquero ha implicado una variación en la diversidad genética
de la población. Por ejemplo, se comenta el caso del Coregonus lavaretus en el lago
Femund de Noruega, ver [4], que presenta tres formas distintas, y una estrategia puntual
de pesca aumentó la capturabilidad de los individuos de una de estas tres formas, lo
que ocacionó una disminución de estos individuos y afectó la variabilidad genética de
la población total.
Consideramos captura impulsiva, ver [2, 3], esto es, ocurre en una secuencia creciente
de instantes {tk } igualmente espaciados, i.e., tk+1 − tk = τ , cierto τ > 0, para cada
k ≥ 0. Además, en cada momento de captura y para cada subpoblación genotípica lo
cosechado por unidad de biomasa es proporcional al esfuerzo de pesca ejercido (E), es
decir, la Hipótesis de Schaefer, pero con capturabilidad diferenciada qi , i ∈ {1, 2, 3}.
117
En resumen, tenemos el modelo diferencial impulsivo siguiente:


0 (t ) = α f 2 (t)P (t) − µP
PAA

AA (t),

A


0 (t ) = 2α f (t)f (t)P (t) − µP

P
(t),
t 6= tk ,

a
A
Aa
Aa



0
2
Pa a (t ) = α fa (t)P (t) − µPa a
(t).
+ ) = (1 − q E)P
P
(t
(t),


1
AA
AA



 PA a (t+ ) = (1 − q2 E)PA a (t),
t = tk ,



Pa a (t+ ) = (1 − q3 E)Pa a (t),
(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 la
dinámica de largo plazo del sistema (1), los cuales son interpretados en el contexto
especificado de diversidad genética final.
References
[1] Clark, C. (1990). Mathematical Bioeconomics: The Optimal Management of Renewable Resources. John Wiley and Sons.
[2] Córdova, F., Pinto, M. (2002). Mathematical Bioeconomics. Explotation of resources and
preservation. Cubo Mat. Educ., 239, 49–54.
[3] Córdova, F., Del Valle, R., Robledo, G. (2012). A pulse fishery model with closures as
function 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 polymorphic
whitefish (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 impacts of fishing. FAO Fisheries Technical Paper, 344, 53p. Disponible en:
http://www.fao.org/docrep/003/V4865E/V4865E00.HTM
118
Observaciones a la aproximación de L.A. Segel
para las ecuaciones del Sistema
Ligando-Receptor
Abstract
Todos los organismos a diferentes escalas de organización presentan sistemas sensoriales bastantes comparables. Estos son sistemas que responden a señales, pero que
presentan adaptación cuando están expuestos a estos estímulos por un tiempo prolongado, esto es, las respuestas del sistema deben decrecer en magnitud o simplemente no
existir.
En este trabajo realizamos algunas observaciones al modelo de adaptación exacta
de Segel et al [1] y pretendemos una versión impulsiva del mismo. Nuestra hipótesis es
que la aproximación con esta técnica matemática facilita el análisis del modelo.
En nuestro sistema, tipo Receptor-Ligando, concurren dos estados libres, un receptor R y su covalente D, junto con sus correlativos estados ligados RL y DL. Las
reacciones que representan la asociación-disociación del ligando L (verticales), se consideran comparativamente más rápidas que las reacciones de conversión entre estados
libres o entre estados ligados (horizontales), con ambas definidas por leyes de acción de
masas simples. La naturaleza híbrida está en que: (a) la impulsividad (pulso) se incorpora 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 celulares. Procesos biofísicos complejos. Simposio sobre complejidad biológica. Julio A. Hernández
and 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 to
show adaptative response over a widw range of levels of stimulus. J. Theor. Biol. 229: 45-57.
Universidad Católica del Maule. e-mail: [email protected]
119
Neurodidactics: Analysis of Cellular Neural
Network Models
Kuo-Shou Chiu
Abstract
Cellular neural networks [9,10] have been extensively studied in past years and found
many applications in different areas such as pattern recognition, associative memory,
and combinatorial optimization. Such applications heavily depend on the dynamical
behaviors. It is well known that studies on neural dynamical systems not only involve
discussion of stability property, but also involve other dynamics behaviors such as periodic solution, bifurcation, chaos et al. Recently, the properties of periodic oscillatory
solutions are of great interest because one has been found many network such as human brain are often in periodic oscillatory or even chaotic state. Many results for
the existence of their periodic solutions and the exponential convergence properties for
cellular 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. However, many real world systems and natural processes cannot be categorized into one of
them. They display characteristics both continuous and discrete styles. For instance,
some biological neural networks in biology, bursting rhythm models in pathology and
optimal control models in economics are characterized by abrupt changes of state. These
are the familiar impulsive phenomena.
Other examples can also be found in information science, electronics, automatic
control systems, computer networking, artificial intelligence, robotics, and telecommunications, etc. Such a kind of phenomena, in which sudden and sharp changes often
occur in a continuous process, which cannot be well described by pure continuous or
pure discrete models. Therefore, it is important and, in effect, necessary to study a
new type of neural networks - neural networks with piecewise constant arguments as an appropriate description of these phenomena. See, for instance, Refs. [1,6] and
references cited therein.
Differential equations with piecewise constant arguments (briefly DEPCA) arise in
an attempt to extend the theory of functional differential equations with continuous
arguments to differential equations with discontinuous arguments. This task is of considerable applied interest, because DEPCA can be seen in many of the phenomena in
the real world. These phenomena may often be modeled by piecewise constant systems
with corresponding differential equations containing piecewise constant arguments.
Partially supported by FIE 11-14 DIUMCE, Departamento de Matemática, Facultad de Ciencias Básicas, 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. Fix
a real sequence ti , i ∈ Z, such that ti < ti+1 for all i ∈ Z, ti → ±∞ as i → ±∞. Let
γ : R → R be a step function given by γ(t) = ti for t ∈ Ii = [ti , ti+1 ) and consider the
DEPCA with this general γ. In this case we speak of DEPCA of general type, in short
DEPCAG. 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 neural
networks with a general piecewise constant argument and impulse effects (IDEPCAG):
ẋi (t) = −ai xi (t) +
n
X
{bij fj (xj (t)) + cij gj (xj (γ(t)))} + di
j=1
where n denotes the number of neurons in the network, xi (t) corresponds to the state
of the i–th unit at time t, fj (xj (t)) and gj (xj (γ(t))) denote, respectively, the measures
of activation to its incoming potentials of the unit j at time t. Moreover, ai > 0 and
bij , cij , Ii are positive real numbers; bij denotes the synaptic connection weight of the
unit j on the unit i at time t, cij denotes the synaptic connection weight of the unit j
on 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 analysis
techniques, some new sufficient conditions are obtained ensuring existence, uniqueness
and global exponential stability of periodic solution of nonautonomous cellular neural
networks with piecewise constant argument of generalized type. The results given here
extend and improve the earlier publications. Moreover, connection with the neurodidactics 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., International
Journal of Systems Science, 31 (2000) 1313-1316.
[3] Kuo-Shou Chiu and M. Pinto, Periodic solutions of differential equations with a general piecewise 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 for
differential 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 piecewise 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 a
general piecewise constant argument. In preparation.
[7] S. Xu, Y. Chu and J. Lu, An analysis of global asymptotic stability of delayed cellular neural
networks, 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.
121
[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 recurrent
neural networks with delays, Physics Letters A, 298 (2002) 393-404.
[12] S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time
cellular neural networks with delays, Applied Mathematics and Computation, 135(2003), 1738.
[13] M. Pinto, Asymptotic equivalence of nonlinear and quasilinear differential equations with piecewise constant arguments, Mathematical and Computer Modelling, 49 (2009), 1750-1758.
[14] M. Pinto, Cauchy and Green matrices type and stability in alternately advanced and delayed
differential 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 networks
with time-varying delays, Physics Letters A, 352 (2006) 371-379.
122
A vaccine-age structured model to study the
effect of a pre-erythrocytic vaccine on malaria
prevalence
Katia Vogt Geisse
Calistus Ngonghala
Zhilan Feng
Abstract
A deterministic compartmental malaria model will be presented to study the effects
of a pre-erythrocytic vaccine on malaria dynamics in Africa. Malaria is a parasitic
disease transmitted to humans through bites of infected mosquitoes and represents an
important international health problem. The model divides the human population into
susceptible, infected and vaccinated individuals, separating the latter into two vaccinated classes, the first consisting of individuals that received initial vaccination doses
and the second of those who received an additional booster dose. All epidemiological
classes vary with chronological time and vaccinated individuals in both compartments
are additionally structured by vaccine-age. A vaccine-age dependent transition rate
between vaccinated classes makes it possible to model a minimum vaccine-age required
for receiving the booster dose. The mosquito population is divided into two compartments: susceptible and infected. The model consists of a coupled system of ordinary
differential equations and first order partial differential equations. An expression for
the control reproduction number R will be derived. The local and global stability of
the disease free equilibrium, conditions for existence of endemic equilibria, as well as
the dynamics of the system under different vaccination policies will be discussed. In
particular, the model exhibits backward bifurcation dynamics, indicating that R = 1
is 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,” Malaria
Journal, vol. 10, no. 202, 2011.
Universidad Adolfo Ibáñez, e-mail: [email protected]
Department of Global Health and Social Medicine, Harvard Medical
[email protected]
Department of Mathematics, Purdue University, e-mail: [email protected]
School,
e-mail:
123
[3] C. M. Kribs-Zaleta and J. X. Velasco-Hernández, “A simple vaccination model with multiple
endemic states,” Mathematical biosciences, vol. 164, no. 2, pp. 183–201, 2000.
[4] P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic 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 epidemiology,” Springer, 2011.
124
Modelación del cambio en la interacción de
poblaciones 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 Hernández
 [1] y Hernández ny Barradas
o [2].
 x0 (t) = r1 x(t) 1 − x(t)
K1
o
n
2
 y 0 (t) = r2 y(t) 1 − y(t) + bx(t)−x2 (t) x(t)
K2
1+cx (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 estadísticamente el tipo de interacción entre ambas por Regresión de Poisson. Los datos son
también utilizados para la estimación de los parámetros del modelo matemático en el
caso 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 interaction 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 and
applications, 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]
125
Un Modelo Estocástico de Biorrectar de
Autociclado
Ana Venegas
Ricardo Castro
Fernando Córdova
Abstract
En el presente trabajo se analizará un modelo de biorreactor en donde la tasa de consumo es perturbada por un ruido estocástico. El modelo estudiado corresponde a uno
de autociclado con una acción de vaciado - llenado rápida, lo anterior cuando el sustrato
alcanza un cierto nivel poblacional predeterminado. Las preguntas matemáticas se centrarán en demostrar existencia y unicidad del proceso solución del modelo y también
en 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. Biotechnology
and Bioengineering 1993; 42:1175-1180.
[4] Samoilenko AM, Perestyuk NA. Impulsive Differential Equations. World Scientific: River Edge
NJ, 1995.
[5] Smith RJ, Wolkowicz GSK. Analysis of a model of the nutrient driven self-cycling fermentation
process. 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 impulsivedependent times. In BIOMAT 2006, International Symposium on Mathematical and Computational 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]
Universidad Tecnológica Metropolitana [email protected]
Universidad Católica del Maule e-mail: [email protected]
126
Dinámica de un modelo tritrófico con una
respuesta 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 y
3, mientras que la especie 2 es depredador (específico) de la especie 3, siguiendo una
respuesta funcional no diferenciable del tipo Holling II [2].
El modelo está dado por un sistema de ecuaciones diferenciales ordinarias, el cual
se estudia cualitativamente: se demuestra la unicidad de soluciones en el sistema no
diferenciable, la existencia de una bifurcación de Hopf [4] y una bifurcación de Hopf
Generalizada [3] . Respaldados por la teoría de bifurcaciones explicaremos los cambios
topológicos del modelo, los cuales representan situaciones o fenómenos concretos en la
dinámica poblacional [1]. Los resultados analíticos son complementados con un análisis
numérico 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 in
Population 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 Bifurcation
of Vector Fields. Second Edition, 1985.
Programa de Incentivos a
[email protected]
la
Iniciación
Científica
(PIIC),
DGIP
USM.
e-mail:
127
Dinámica en el modelo de depredación de
Holling-Tanner considerando interferencia entre
los depredadores
Adrián 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 tipo
logístico [7, 9] y que la capacidad de soporte de la especie depredadora es proporcional
al tamaño poblacional de las presas Asimismo, se asume para el modelo propuesto
que el depredador es especialista y que su respuestas funcional es hiperbólica, un tipo
particular de las respuestas funcionales del tipo Holling II [9]
La característica distintiva que el presente desarrollo pretende modelizar está asociada a la competencia intraespecífica de los individuos de la especie depredadora al
momento de atacar una presa, fenómeno que es referido como interferencia entre los
depredadores [5].
El modelo es descrito por el siguiente sistema de ecuaciones diferenciales autónomas
del tipo Kolmogorov [4]:
dx
qx p
x
=r 1− K
x − x+a
y
dt
Xµ :
dy
y
=
s
1
−
y
dt
nx
donde, x = x (t) e y = y (t) representan los tamaños poblacionales de las presas
y los depredadores respectivamente, en función del tiempo, para t ≥ 0 (medidos en
densidad, 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 topológicamente equivalente al original [6], determinando una región de invarianza, probando el
acotamiento de las soluciones y determinando condiciones para la existencia de puntos
de equilibrio al interior del primer cuadrante, y que el sistema puede presentar una, dos
o 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.
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.) BIOMAT
2010 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-prey
model 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 Applied
Mathematics 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 density
dependent 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 complexities
in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Applied
Mathematical 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 Applied
Mathematics 59 (1999) 1867-1878.
[9] P. Turchin, Complex population dynamics. A theoretical/empirical synthesis, Mongraphs in Population Biology 35 Princeton University Press 2003.
129
Modelling and stability analysis of a microalgal
pond 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 represents the dynamics of these two species (microalgae and nitrifying bacteria) in competition for nitrogen (present as ammonium and nitrate produced by nitrification) in
a continuous process. The existence and local stability of system equilibria is studied.
Reduction by conservation principle, perturbed systems and Lyapunov methods are
used to provide sufficient conditions for the global asymptotic stability of the system
equilibria. Finally, we illustrate our analysis with a case study, showing which operating
conditions (dilution rate and pond depth) can promote the presence of nitrifiers with
microalgae.
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 2nd
IWA/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 University
Press, 1990.
Inria BIOCORE, BP93, 06902 Sophia-Antipolis Cedex, France email:[email protected].
[email protected].
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 la
Empresa, convocatoria 2014, 781413008 (A. Rojas-Palma), the BASAL Project (Centro de Modelamiento
Matemático, 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 perfectly 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 Two
Perfectly 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 chemostat
with 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 of
chemostat 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 population
biology. Taiwanese J. Math., 9(2):151–173, 2005.
[10] S. B. Hsu, K. S. Cheng, and S. P. Hubbell. Exploitative Competition of Microorganisms for
Two 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. Pearson
Education, Limited, 2013.
131
Mathematical approach regarding the
environmental effects upon trait diversity in a
cell population.
Karina Vilches Ponce
Abstract
The environmental effects in phenotype and trait diversity were explained extensively
by ecologists, biologists and zoologists obtained different versions of its real influence in
the evolutionary and natural selection. We propose to study the asymptotic behavior of
a no local transport equation that models the environmental influence on trait evolution
u
perhaps over time in some populations using a viscosity and the anzats e . Concluding
in a mathematical sense: If there is a stable environment then the trait variability in all
populations 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 integral
equations: a general convergence result.
[3] Calsina.A, Cuadrado. S. Small mutation rate and evolutionary stable strategies in infinite
dimensional 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 structured 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: An
illuminating 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, physiologically structured models. Journal of Theoretical Biology (2009).
[8] Lorz. A, Mirrahimi. S and Perthame. B Dirac mass dynamics in a multidimensional nonlocal
parabolic equation. (2010)
132
On a nonlinear problem from catalysis:
existence, multiplicity and qualitative behaviour
Gonzalo Robledo Veloso
Abstract
This work studies a model of chemical reactor with an Arrhenius dependence on temperature, this is a nonlinear ordinary system of two variables: one which corresponds
to the normalized concentration of the reacting species, and the temperature. The
system also has a non trivial dependence on the parameters involved: the Damköhler
number, the activation energy and the heat release. Different behaviour is identified in
term 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 in
a neighborhood of the steady solution. One of the main improvements of in all previous mentioned topics is the fact that the presented analysis is extended for the case
of the activation energy strictly positive, which since the formulation of the problem
in reference [5] up to recent references like [6] is neglected, as in all the literature is
assumed the hypothesis that the activation energy is zero. Numerical test showing the
behaviour 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 runaway
boundaries of heterogeneous reactor models; Ind. Eng. Chem. Res., vol. 36 (1997), pp. 32303241.
[5] A. Poore: A model equation arising from chemical reactor theory Arch. Rat. Mech. Anal., vol.
52-4 (1973), pp. 358-388.
Gonzalo Robledo Veloso, e-mail: [email protected]
133
[6] M.Z. Solórzano, W.H. Ray: Multiplicity and stability of chemical reactors with evaporating
cooling 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-bed
reactors; Ind. Eng. Chem. Res., vol. 45 (2006), pp. 7057-7066.
134
A stochastic disease transmission in an epidemic
model 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 model
proposed by Roberts and Saha [?], considering a hyperbolic type nonlinear incidence
rate. According to our knowledge this incidence rate has not been previously used for
this type of epidemic models. Although this kind of rate has receive more attention in
last year for being more realistic. Assuming the proportion of infected population varies
with 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 the
solution of this equation (deterministic model) is found when time tends to infinity.
Then, the asymptotic behaviour of a stochastic fluctuation due to the environmental
variation in the coefficient of disease transmission is studied. So, a stochastic differential
equation (SDE) is obtained. In the last results obtained the existence of an unique
solution is proved. Moreover, the SDE is analysed through the associated FokkerPlanck equation to obtain the probability density function (its invariant probability
distribution) when the proportion of the infected population reaches steady state. An
explicit expression for invariant measure is found together with some interesting features
about it. The long time behaviour of deterministic and stochastic models are compared
in simulations.
References
[1] C. A. Braumann and C. Carlos, Allee effects models in randomly varying environments, Proceedings of the 13th International Conference on Computational and Mathematical Methods
in Science and Engineering CMMSE 2013, pp. 304-307.
[2] S. Busenberg and K. L. Cooke, Vertically Transmitted Diseases, Biomathematics 23, SpringerVerlag, Berlin, 1993.
[3] V. Capasso and D. Bakstein, An introduction to continuous-time stochastic processes (Second
edition). 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].
Universidad de Valparaíso, e-mail: [email protected]
135
[5] P. Das, D. Mukherjee and A. K. Sarkar, Study of an S-I epidemic model with nonlinear incidence 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 AIDS
transmission, 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 with
treatment 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 behaviour
of a logistic epidemic model with two stochastic perturbations, Mathematical Problems in
Engineering Article ID 904383 Volume 2009 7 pages.
[9] E. González-Olivares, B. González-Yañez, J. Mena-Lorca and R. Ramos-Jiliberto, Modelling
the Allee effect: are the different mathematical forms proposed equivalents? In R. Mondaini
(Ed.) Proceedings of the 2006 International Symposium on Mathematical and Computational
Biology, 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 SIS
epidemic 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 and
treatment, Mathematical Biosciences 238 1 (2012) 12-20.
[14] P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments. 1994. Springer.
[15] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear 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 models
with 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 without
immunity, Journal of Mathematical Biology 32 (1994) 809-834.
136
Sistemas Dinámicos
Encargado de Sesión : Irene Inoquio
137
Aspectos recientes de la Conjetura de Palis
Alma Armijo
Abstract
En los años 90 Palis conjeturó que se puede clasificar los difeomorfismos con la
topología C r . En difeomorfismos hiperbólicos y como son los que no lo son. Muchos
matemáticos han tratado de darle respuesta a esta conjetura, vamos a ver los avances
realizados por varios autores y en particular el trabajo realizado en mi tesis de doctorado.
Universidad de Santiago, e-mail: [email protected]
138
Injectivity, Global and Almost Global Stability
of 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 Conjecture. Furthermore, we construct a family of almost Hurwitz vector fields such that the
origin is almost globally asymptotically stable by using the novel concept of density
functions, and we give a family of the Hurwitz vector fields, perturbing the previous
family, 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 Mathematics, 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 the
Polynomial 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 to
Diff. 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]
139
Toeplitz and strong orbit equivalence
Maryam Hosseini
Abstract
We show that for any unital Dimension group, (G, u), of rank bigger than one with
non-cyclic rational subgroup and for any countable subgroup of the unit circle, Λ, there
exists a Toeplitz system (X, T ) with an invariant measure µ such that K 0 (X, T ) = G
and SPµ (T ) = Λ. This is a joint work with M. Isabel Cortéz at university of Santiago
and Thierry Giordano at university of Ottawa.
Universidad de Santiago, e-mail: [email protected]
140
Dimensión de Hausdorff de los conjuntos de
Borel-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) ≥ B n i.o. }, siendo an (x) el n-ésimo término de la expansión en fracciones contínuas de x, tiene medida de Lebesgue 0. En [3], Wang y Wu probaron que la
función que codifica la dimensión de Hausdorff de E(B), D : B 7→ dimH E(B) definida
en (1, ∞) es contínua, decreciente, y satisface limB→1 D(B) = 1 y limB→∞ D(B) = 1/2.
Utilizando los métodos del Formalismo Termodinámico, es posible mejorar la regularidad 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 fraction
expansions. Advances in Mathematics, 218(5), 1319-1339.
141
Flexibility of some groups of homeomorphisms
of the line
Cristobal Rivas
Abstract
There are many result stating rigidity phenomena all across dynamical sistems. In
this talk, I will explain a recent result obtained in collaboration with Juan Alonso and
Joaquin Brum (U. de la Republica) stating that any action by homeomorphism on the
line of the fundamental group of a closed surface is flexible. By this we mean that the
action considered can be approximated in a natural topology by another action which is
not semi-conjugated to the original one. This is somehow surprising since for the case
of actions on the circle, these groups admits action that are rigid : any other action
close 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-orderings
of the groups considered.
References
[1] K. Mann, Space of surface group representation. Invent. Math. 201 (2015).
Partially supported by
[email protected]
Proyecto
Anillo
1103
and
FONDECYT
1150691,
e-mail:
142
A linearization result for DEPCA systems
Manuel Pinto
Gonzalo Robledo
Abstract
We generalize the topological equivalence results obtained by Palmer [2] and Shi &
Xiong [4] to the systems of differential equations with piecewise constant arguments
γ(·):
ẋ(t) = A(t)x(t) + A0 (t)x(γ(t)) + f (t, x(t), x(γ(t))),
(1)
and
ẏ(t) = A(t)y(t) + A0 (t)y(γ(t)),
(2)
that is, we obtain conditions ensuring the existence of a homeomorphism between the
solutions of the systems (1) and (2).
The function t 7→ γ(t) is defined with the help of two sequences {ti }i∈Z and {ζi }i∈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 considered: a) we will assume that (2) admits a property of exponential dichotomy recently
introduced 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 delayed
differential 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]
Universidad de Chile – Departamento de Matemáticas, e-mail: [email protected]
143
Shearer’s inequality and the Infimum Rule
Pierre Paul Romagnoli
Abstract
We review subbadditivity properties of Shannon entropy, in particular, from the
Shearer’s inequality we derive the infimum ruleİ for actions of amenable groups. We
briefly discuss applicability of the infimum formula to actions of other groups. Then
we pass to topological entropy of a cover. We prove Shearer’s inequality for disjoint
covers and give counterexamples otherwise. We also prove that, for actions of amenable
groups, the supremum over all open covers of the infimum fomulaİ gives correct value
of 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
J. Amer. Math. Soc. 23 (2010), 217–245.
[3] P.
Burton,
Naive
entropy
http://arxiv.org/pdf/1503.06360v1.pdf
of
for
actions
of
dynamical
countable
systems,
sofic
groups,
preprint,
[4] T. Downarowicz, Entropy in dynamical systems, Cambridge University Press, New Mathematical
Monographs 18, Cambridge 2011.
[5] E. Lindenstrauss, Pointwise theorems for amenable groups, Electronic Research Announcements
of 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 amenable
groups 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]
The author acknowledges the support of Programa Basal PFB 03, CMM, Universidad de Chile
Joint 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
144
Polinomios de Fibonacci y Componentes
Errantes
Eugenio Trucco
Abstract
En el caso de una función racional
R : P1C −→ PC1
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 (no
pre-periódicas) fue respondida por D. Sullivan en 1985 utilizando técnicas de geometría
cuasi-conforme. Él demuestra que el conjunto de Fatou de una función racional no tiene
componentes errantes. En el estudio de la dinámica de funciones racionales sobre un
cuerpo no arquimediano K es más conveniente estudiar la acción de la función racional
en la línea proyectiva de Berkovich asociada al cuerpo K. Esto es, estudiar
1,an
R : P1,an
K −→ PK
El resultado de Sullivan sobre la no existencia de componentes errantes no es cierto en
el caso de funciones racionales sobre cuerpos no arquimedianos. Existen ejemplos de
funciones racionales que presentan una componente errante en su conjunto de Fatou.
Todos los ejemplos conocidos están relacionados a un fenómeno llamado ramificación
salvaje, una propiedad que no ocurre en los números complejos. Por lo anterior, se cree
que la ramificación salvaje es necesaria para la existencia de componentes errantes. En
esta charla estudiaremos una combinatoria relacionada con los números de Fibonacci
para así comprender esta relación.
Universidad Austral de Chile, e-mail: [email protected]
145
Matemática Discreta
Encargado de Sesión : José Soto
146
Estudio de un modelo de evasión en el
transporte 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→
PZ+ , y una distribución 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 sobre
uno 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 la
distribución p. El evasor escogerá un camino entre s y t y lo recorrerá de manera que
si atraviesa el arco donde se encuentra el policía, será multado con una infracción de
costo M > 0 y continuará su viaje ya sea por el camino previamente escogido (versión
no adaptativa) o por un shortest path (versión adaptativa).
Mientras que el evasor enfrenta el problema de minimizar el costo esperado del
camino que escoge, el policía busca establecer una distribución de probabilidad sobre
los arcos de manera de hacer lo más alto posible el costo del evasor, o bien maximizar
la multa esperada que recauda. Se presentan resultados algorítmicos concernientes a la
versión no adaptativa del problema del evasor y de complejidad referentes al problema
del policía.
Universidad de Chile, e-mail: [email protected]
147
Resource Augmentation Algorithm for Single
Machine Scheduling with Job-Dependent
Convex Cost
Rodrigo A. Carrasco
Abstract
In this work we combine resource augmentation and alpha-point scheduling techniques
to compute approximate solutions for a general family of scheduling problems: each
job has a convex non-decreasing cost function applied to its completion time and the
goal is to compute a schedule that minimizes the total cost subject to precedence
constraints. We show that our algorithm is a O(1)-speed 1-approximation algorithm
and our numerical experiments show that the speed-scaling ratio needed is actually
close to 1.
Faculty of engineering and Sciences, Universidad Adolfo Ibañez, e-mail: [email protected]
148
Avances recientes en la resolución exacta del
problema de vendedor viajero
Daniel Espinoza
William Cook
Marcos Goycoolea
Abstract
El problema del vendedor viajero (TSP) es uno de los problemas más clasicos en optimización combinatorial [?]; y uno de los problemas más usados para probar algoritmos
de optimización, heurísticas y algoritmos de aproximación.
El TSP también ha sido uno de los ejemplos donde más logros ha cosechado la teoría
poliedral .
En esta charla recorreremos brevemente algunos de los hitos más importantes en la
resolución práctica de instancias del TSP, con un énfasis en los avances poliedrales; asi
como también últimos desarrollos en desigualdades válidas y algoritmos de separación
que explotan planaridad.
Universidad de Chile. e-mail: [email protected]
University of Waterloo, e-mail: [email protected]
Universidad Adolfo Ibañez, e-mail: [email protected]
Se han omitido las referencias por no ser compatibles con el formato.
149
Quasirandom hypergraphs and subsets with
small Fourier coefficients
Hiê.p Hàn
Abstract
Quasi-randomness forms a central theme in modern discrete mathematics. Informally
speaking quasi-random properties are deterministic properties which are characteristic
for random objects. Such exist for various discrete structures such as graphs, hypergraphs, subsets of abelian groups etc. Being interesting for their own sake, their study
has also revealed many connections between various areas of mathematics such as graph
and hypergraph theory, number theory, geometry and also to algorithms and complexity. In this talk we give an introduction into the topic of quasirandomness, putting
emphasis on linear quasirandom hypergraphs and subsets of integers with small nontrvial Fourier coefficients.
Joint work with Elad Aigner-Horev.
150
Efficient Implementation of Carathéodory’s
Theorem for a Simple Scheduling Polytope
Ruben Hoeksma
Abstract
In a fundamental paper in polyhedral combinatorics, Queyranne describes the complete
facial structure of a classical object in combinatorial optimization, the single machine
scheduling polytope. In the same paper, he answers essentially all relevant algorithmic
questions with respect to optimization and separation. In this talk, motivated by
recent 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 point
in the scheduling polytope as a convex combination of the vertices of the polytope. We
here give a combinatorial O(n2 ) time algorithm, where n is the number of jobs of the
scheduling 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 by
Grötschel, Lovász and Schrijver, applied to one of the subpolytopes of the barycentric
subdivision. We thereby also shed new light on an algorithm recently proposed for a
special case, namely the permutahedron.
151
Maximum number of colourings without
monochromatic Schur triples
Andrea Jimenez
Abstract
We study subsets of finite abelian groups that maximize the number of r-colourings
without 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 achieved
only by largest sum-free sets. For r > 3 this phenomenon does not persist and the
problem becomes harder. We resolve the problem for abelian groups of even order and
r = 4, 5.
Joint work with Hiê.p Hàn.
Universidad de Valparaiso, e-mail: [email protected]
152
On-line list coloring of random graphs
Dieter 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 surely
asymptotic 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)2 n2/3 . For sparser graphs, we
are slightly less successful; we show that if d > (log n)2+epsilon for some > 0, then the
on-line choice number is larger than the chromatic number by at most a multiplicative
factor of C, where C ∈ [2, 4], depending on the range of d. Also, for d = O(1), the
on-line choice number is by at most a multiplicative constant factor larger than the
chromatic number.
Joint work with Alan Frieze, Xavier Pérez-Giménez and Pawel Prałat.
CMM, Universidad de Chile, e-mail: [email protected]
153
Computing income taxes under the new Chilean
tax 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, Chilean
corporations will attribute all their incomes to their shareholders and these will be taxed
on the incomes attributed to them. This change required a new tool for computing
the taxable income of taxpayers accurately and efficiently. In this work, we show the
mathematical conceptualization and the solution to the problem, proving that there
is only one way to distribute incomes to taxpayers. Moreover, using the theory of
Absorbing Markov Chains, we define a mathematical model for computing the taxable
incomes of each taxpayer. In addition, we implement the mathematical model creating
an algorithm based on the properties of Absorbing Markov Chains and the Tarjan’s
strongly connected components algorithm. This allows us to compute the solution
accurately and with efficient use of computational resources.
Faculty of engineering and Sciences, Universidad Adolfo Ibañez
Faculty of engineering and Sciences, Universidad Adolfo Ibañez
154
Optimización
Encargado de Sesión : Luis Briceño
155
Chance-constrained problems and rare events:
an importance sampling approach
J. Barrera, T. Homem-de-Mello, E. Moreno
B. Pagnoncelli, G. Canessa
Abstract
We study chance-constrained problems in which the constraints involve the probability
of a rare event. We discuss the relevance of such problems and show that the existing
sampling-based algorithms cannot be applied directly in this case, since they require an
impractical number of samples to yield reasonable solutions. Using a Sample Average
Approximation (SAA) approach combined with importance sampling (IS) techniques,
we show how variance can be reduced uniformly over a suitable approximation of the
feasibility 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 prove
asymptotic convergence of the combined SAA-IS approach. We apply our methodology
to a telecommunications problem, find IS distributions that satisfy the conditions laid
out for uniform variance reduction in that context and present numerical results to
illustrate the ideas.
Universidad Adolfo Ibáñez, e-mail: [email protected]
156
Nonsmooth Lur’e Dynamical Systems in Hilbert
Spaces
Samir Adly
Abderrahim Hantoute
Ba Khiet Le
Abstract
In this paper, we study the well-posedness and stability analysis of set-valued Lur’e
dynamical systems in infinite-dimensional Hilbert spaces. The existence and uniqueness
results are established under the so-called passivity condition. Our approach uses a
regularization procedure for the term involving the maximal monotone operator. The
Lyapunov stability as well as the invariance properties are considered in detail. On
the other hand, this work generalizes the classical composition between a maximal
monotone 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 nonsmooth
multivalued 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]
El trabajo es financiado por el Proyecto Fondecyt Postdoctorado 3150332
157
Stochastic Topology Design Optimization for
Continuous Elastic Materials
Miguel Carrasco
Benjamín Ivorra
Angel Manuel Ramos
Abstract
In this work, we develop a stochastic model for topology optimization. We find robust structures that minimize the compliance for a given main load having a stochastic
behavior. We propose a model that takes into account the expected value of the compliance and its variance. We show that, similarly to the case of truss structures, these
values can be computed with an equivalent deterministic approach and the stochastic model can be transformed into a nonlinear programming problem, reducing the
complexity of this kind of problems. First, we obtain an explicit expression (at the continuous level) of the expected compliance and its variance, then we consider a numerical
discretization (by using a finite element method) of this expression and finally we use
an optimization algorithm. This approach allows solving design problems which include
point, surface or volume loads with dependent or independent perturbations. We check
the capacity of our formulation to generate structures that are robust to main loads and
their 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 solutions
of the expected-compliance and variance weight coefficients, the laws used to describe
the random loads, the variance of the perturbations and the dependence/independence
of the perturbations. Finally, the results are compared with similar ones found in the
literature for a different modeling approach.
El trabajo es financiado por el Proyecto FONDECYT 1130905
Universidad de los Andes, Facultad de Ingeniería y Ciencias Aplicadas, e-mail: [email protected]
Universidad Complutense de madrid,
Departamento de Matemt́ica Aplicada,
e-mail:
[email protected]
Universidad Complutense de madrid, Departamento de Matemt́ica Aplicada, e-mail: [email protected]
158
Existence and approximation of generalized
Lagrange multipliers for variational problems
under uniform constraints on the gradient.
Felipe Alvarez
Salvador Flores
Abstract
In this work we present existence results concerning generalized, measure-valued Lagrange multipliers for variational problems with uniform constraints on the gradient of
the type |∇u| ≤ 1 c.t.p. Our main technique is power-penalty, therefore our results hold
under very mild hypothesis. In particular, we do not require constraint qualification
conditions.
We study a class of constrained Dirichlet problems from the calculus of variations of the
type
inf{J(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 multipliers
associated to the uniform constraint on the gradient.
We approximate the problem by a sequence of unconstrained problems penalizing the
uniform constraint by a power term. Next we address the existence and approximation of
Lagrange multipliers for the uniform constraint on the gradient. The underlying rationale
bears some resemblances to certain methods for showing existence of Lagrange multipliers
without recourse to separation theorems, such as the Fritz – John optimality conditions in
nonlinear programming.
In [1] we proved the following. Theorem
Let T (x, ξ) = |ξ|, and g ∈ C 2 (Ω̄) be such that k∇gk∞,Ω < 1/2. Let {up }p≥p1 be a sequence
of solutions to the penalized problem. Let u∞ be a cluster point of {up }p≥p1 for the topology
of C(Ω). Under appropiate conditions on f , there exists a nonnegative Radon measure
multiplier 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 D0 (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−1 }p≥p1 converges to λ in the bitting sense.
In this talk we shall discuss this result and its connection with similar ones from [2] obtained
using infinite dimensional duality. As time permits, we shall also discuss the connections
with overdetermined boundary value problems and its implications for the numerical analysis
of 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 Mathematical
Analysis 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.
160
Condiciones de Optimalidad en Problemas de
Control 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 control
óptimo, formulado en tiempo discreto, no lineal. Para esto, extendemos la noción de
KT-invexidad para problemas de programación matemática a problemas de control óptimo discreto. Se demuestra que las condiciones necesarias de optimalidad establecidas
por 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 lineal:
N
−1
X
Minimizar Φ(x, u) =
f (xi , ui , i)
i=0
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
x(i) 7→ xi
u : [0, N − 1] → Rm
(2)
u(i) 7→ ui
(3)
xi es la variable de estado; ui es el parámetro de control; [0, N ] es el intervalo discreto
de 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 la
trayectoria correspondiente.
Universidade Federal do Paraná, Curitiba, Brasil, e-mail: [email protected] , e-mail:
[email protected]
Universidad de Tarapacá, Arica, Chile, e-mail: [email protected]
Universidad Católica de la SantíÂŋsima Concepción,Concepción, Chile, e-mail: [email protected]
El trabajo es financiado por CAPES-PDSE, Fondecyt: 1120260, Dirección de Investigación, Universidad
Católica de la Santísima Concepción, proyecto DIN 05/2015
161
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 Estabilización Económica [8].
El estudio de las condiciones de optimalidad es un tema importante en Análisis
Variacional, en Optimización y también en Control Óptimo. Existen muchos artículos
en la literatura que visan el estudio de las condiciones de optimalidad para estos problemas; véase [4]. Nuestro énfasis en este trabajo es presentar condiciones de optimalidad
para el problema de control óptimo discreto (1), para ello usaremos algunos resultados
probados en [5].
Los resultados obtenidos se basan en la teoría desarrollada por [2] y generalizada
en [1]. Esto se hace a través de una conveniente reinterpretación del problema (PCD)
como un problema de programación matemática, al cual aplicamos el formalismo de
Dubovitskii-Milyutin [3]. El concepto de KT-invexidad, que fue introducido por Martin
[6] para problemas de programación matemática y posteriormente generalizado por
Osuna-Gómez et al. [7] para problemas multiobjetivos, no sólo es interesante para
la obtención de las condiciones suficientes de optimalidad; también nos proporciona
una completa caracterización de optimalidad. En este trabajo, se demuestra que un
problema de control óptimo discreto es KT-invex si y sólo si todo proceso admisible
que 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 el
Principio 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 from
the Russian by Ron Hardin. John Wiley-Sons, New York-Toronto, Ont.; Israel Program for
Scientific 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 Mathematical
Systems, Vol. 67. Springer-Verlag, Berlin-New York, 1972. iv+136 pp.
[4] Hilscher, R.; Zeidan, V. Discrete optimal control: second order optimality conditions. In honour
of 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 constraints, 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 convexity 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
162
Proximal Distances over Symmetric Cones
Julio 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 two
class of functions of real-valued satisfying some assumptions. Then, we show that its
corresponding spectrally defined function defines a proximal distance. In addition, we
present several examples and properties of this distance. The properties are useful for
the 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 secondorder 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 algebras, Mathematics Operations Research, 33(2):421-445, 2008.
Universidad Diego Portales, e-mail: [email protected]
Universidad Federal de Rio de Janeiro, e-mail: [email protected]
163
Stability in Generalized Nash Equilibrium
Problems with nonsmooth payoff functions,
application to Electricity market
Matthieu Maréchal
Rafael Correa
Abstract
This talk deals with the calmness of a Generalized Nash Equilibrium Problem with
non-differentiable data. The approach consists in obtaining some error bound property
for the KKT system associated to the Generalized Nash Equilibrium Problem, and
returning to the primal problem thanks to the Slater constraint qualification. We
apply these results to study a smoothing method applied to the electricity market.
We have considered a perturbed Generalized Nash Equilibrium Problem (GNEP), which
1
p
consists in finding an element x̄ ∈ Rn = Rn × · · · × Rn satisfying:
∀ν ∈ {1, · · · , p} ,
x̄ν ∈ arg
min
−ν
Xν (x̄
,u)
θν (·, x̄−ν , u)
(1)
where {1, · · · , p} denotes the set of players, xν is the strategy played by player ν, x−ν is the
vector containing the strategy played by all players excepted player ν, Xν (x−ν , u) is the set
of admissible strategies for player ν depending on x−ν and on the parameter u. The function
θν (xν , x−ν , u) is the loss function for player ν depending on a parameter u ∈ U . We suppose
that the set Xν (x−ν , u) is described by a inequality system, that is
ν
Xν (x−ν , u) = {xν ∈ Rn
: g ν (xν , x−ν , u) ≤ 0}.
We denote S(u) the solution set of problem (1). This talk deals with the regularity properties
of S, more precisely gives some sufficient conditions in order to ensure the calmness of S
when the loss functions θν are not supposed to be differentiable.
Definition
We say that S is calm at a point (ū, x̄), where x̄ ∈ S(ū), if there exist constants r, L > 0
such that, for all x ∈ B(x̄, r), for all u ∈ B(ū, r),
S(u) ∩ B(x̄, r) ⊂ S(ū) + B(0, Lku − ūk)
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] trabajo
es financiado por el Proyecto Fondecyt 3130596
164
The sufficient conditions for the calmness of S are obtained from the study of the KKT system
of 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 the
function 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 revenue
function solving the following program
P (a−i , b−i )
max Ri (ai , a−i ) = ai qi (ai , a−i ) − Ci (qi (ai , a−i )).
Ai ≤ai ≤Ai
where ai is the price strategy of the producer i, a−i is the price strategy of the other producers 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 can
replace the revenue function Ri by a smooth approximation R̃i .
In this part we use the stability study about GNEP in order to derive an estimation of the
distance between the solutions of the Nash Equilibrium Problem which models the electricity
market and the solution of the smooth approximation, and illustrate this estimation with a
numerical 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 Newton
methods, Math. Program., Ser. B (2009) 163-194
[3] X. Hu, D. Ralph, Using EPECs to model bilevel games in restructured electricity markets
with 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 generalized Nash equilibrium problems. Computational Optimization and Applications, 1-18,
2012
[5] B. Mordukhovich, Stability Theory for Parametric Generalized Equations and Variational Inequalities Via Nonsmooth, Transactions of the American Mathematical Society,
Vol. 343, No. 2 (Jun., 1994), pp. 609-657
165
Boosting Topic Models for Text Analysis
Marcelo Mendoza
Abstract
Topic models have been of growing interest in the last decade. In particular, the techniques based on probabilistic latent variable models provide a solid theoretical base and
a flexible framework that allow for the modeling of various kinds of documentary collections. These models consist of introducing a set of latent variables that allow one to
capture the relationships between terms, documents and other attributes of the documentary collection that are not evidently manifested but can be modeled as unobserved
relationships. The flexibility of this modeling family allows one to incorporate relevant
properties of the text such as polysemy, making groups in sets of terms that describe
concepts, forming topics. The use of latent variables also allows one to make inferences about the presence of topics in each document. Topic models are fundamentally
divided into two broad approaches: the techniques resulting from Probabilistic Latent
Semantic Analysis (PLSA) [1], which introduce latent variables without assuming distribution priors, and the techniques based on Latent Dirichlet Allocation (LDA) [2],
which assume distribution priors over topics and vocabulary by using a Dirichlet distribution. Both approaches have strengths and weaknesses. On the one hand, PLSA
fits the model by using the Expectation-Maximization (EM) algorithm [3] which is a
standard method for the inference of parameters in latent variable models but tends
to over fit data limiting the generalization capability because it can only guarantee
convergence to local optimums. On the other hand, LDA addresses this limitation by
introducing Dirichlet distribution priors on the vocabulary and on topic distributions
over documents, which corresponds to a Bayesian regularization over the input. This
process allows for improvement in the generalization capability of the models, but it
introduces computational difficulties in the parameter estimation method, addressed
using Monte Carlo methods through Gibbs sampling. We propose to explore the use
of regularization operators on the EM estimators of PLSA, which would allow for control of the compromise between generalization and overfitting that is inherent in the
local optimization methods. We introduce the eliteness versus background concept to
model the production of text from two components. The idea is that when producing
text, the author selects words from the elite of the distributions of text associated with
each topic. However, there are words that are part of the natural language background
and 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 modify
PLSA, introducing sparsification over the latent variables associated with the terms
and smoothing over a single latent variable that is capable of modeling the background.
Universidad Técnica Federico Santa María, e-mail: [email protected]
166
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 Machine
Learning Research, 3(4-5):993-1022.
[3] Dempster, A., Laird, N., Rubin, D. (1977). Maximum likelihood from incomplete data
via the EM algorithm. Journal of the Royal Statistics Society, 39:1-38.
167
A Decomposition Method for Two-Stage
Stochastic 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-averse
utilities. More specifically, we consider problems of the form
min
x
subject to
f (x) + Q(x)
Ax = b
(1)
x ≥ 0,
where Q(x) = Eω [Q(x, ω)], ω is a random vector representing the uncertainty in the
problem and the function f (x) is convex. The second decision stage is represented by
Q(x, ω), which has the following form:
Q(x, ω) = min
zω
subject to
g(zω )
Tω x + Wω zω = 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-arrangement
of the variables, we can write the above problem in the format
min
x̄,z̄
subject to
F (x̄) + G(z̄)
Āx̄ + B̄ z̄ = C̄.
(2)
The advantage of reformulating the problem this way is that it fits the format of the
well-known Alternating Direction Method of Multipliers (ADMM) developed in the
literature. In this method constraint (2) is relaxed, creating the approximation
2
min
F (x̄) + G(z̄) + ρ Āx̄ + B̄ z̄ − C̄ x̄,z̄
and then an iterative procedure is applied whereby the primal and dual solutions are
updated alternatingly.
Universidad Adolfo Ibáñez, e-mail: [email protected]. El trabajo es financiado por el Proyecto
Fondecyt 1120244
168
Some attractive features of the algorithm are its simplicity of implementation and
its suitability for parallelization. Nevertheless, several questions arise when applying
this approach to problem (1), both from the theoretical as well as from the computational perspective. For example, it is important to establish conditions that ensure
the convergence of the algorithm. It is also important to provide guidelines to choose
the parameter ρ, as the practical performance of the algorithm appears to be sensitive
to the value of that parameter. In this talk we discuss these issues and present some
numerical results to illustrate the ideas.
169
A primal-dual mix algorithm for convex
non-differentiable structured optimization in
Hilbert spaces
Cesare Molinari
Juan Peypouquet
Abstract
We consider the following structured optimization problem:
min
{ f (x) + g (y) : Ax + By = c } ,
(x,y)∈X×Y
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 that
we 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 optimal pair. We investigate also the introduction of a fixed parameter in the predictioncorrection steps: the aim is the acceleration of the dual process, in order to reach faster
the correct Lagrange multiplier. Finally, we present some possible applications and
numerical experiments for optimal control of parabolic PDEs.
References
[1] G. Chen, M. Teboulle, A proximal-based decomposition method for convex minimization problems, Mathematical Programming, Vol. 64, No. 1, pp. 81-101, 1994.
El trabajo es financiado por el Conicyt, Proyecto Anillo ACT-1106
170
Resultados sobre convexidad de la imagen de
funciones cuadráticas
Felipe Opazo Lagos
Abstract
Los clásicos teoremas de Dines [1] (motivado por Finsler [2]) y de Brickman [3] tratan
sobre la convexidad de la imagen de dos funciones cuadráticas, cuyo dominio es un
espacio Euclidiano o bien una esfera N-dimensional, con N ≥ 3, respectivamente.
Estos resultados dieron inicio al estudio de posibles generalizaciones (ver [4]), a la
par de aplicaciones a Optimización Cuadrática y otros temas (ver [5]). Las funciones
cuadráticas en si mismas exhiben interesantes propiedades geométricas, lo que se traduce en una cierta ’convexidad oculta’ discutida en [4].
En esta charla se comentará sobre una investigación de 2014 [6] en que se caracterizó
la convexidad de dos funciones cuadráticas no necesariamente homogéneas (en oposición
a [1]). También se presentarán resultados de una investigación en curso este año como
parte de una Memoria de Título, siempre sobre el tema de la convexidad y de las
funciones 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 quadratischer
Formen, 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 ’quadratic
World’ 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 quadratic
functions and applications to QP, submited manuscript, arXiv:1508.01612.
Departamento de Ingeniería Matemática, Universidad de Concepción, e-mail: [email protected]
171
Generación de benchmark de fondos para el
sistema de pensiones en Chile, un enfoque
basado en optimización estocástica
Daniel Espinoza G.
Giorgiogiulio Parra De B.
Abstract
Por ley en Chile, los fondos de pensión son administrados por entidades privadas con regulaciones particulares denominadas “Administradora de Fondos de Pensiones” (AFP).
Estas empresas administran los fondos en las cuentas de capitalización individual de
cada uno de sus afiliados, conformando en la actualidad un sistema con seis AFPs, 11
millones de afiliados y US$ 112.5 billones americanos (109 ) en los fondos, un 43.6% del
PIB. 1
La actual estructura de multifondos del sistema se obtuvo a través de un proceso
iterativo que se inició el año 1981 con el Decreto Ley 3.500 del 6 de diciembre de 1980
que reemplazó el antiguo sistema de repartos por uno basado en el ahorro obligatorio
individual. Inicialmente el sistema contó con un único fondo, conformado a diciembre
de 2007 por un 27.47% en renta variable y 72.46% en renta fija. En mayo de 2000, entró
en 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, entregando
la actual estructura de multifondos, conformado por cinco fondos designados por las
letras A,B,C,D y E, donde a través de la regulación en la elegibilidad, proporción y
cobertura, se implementa los perfiles riesgo-retorno de los diferentes fondos, siendo el
fondo A el más agresivo y el fondo E el más conservador.
Como es común en la administración activa de inversiones, los fondos de pensión de
cada AFP son evaluados contra el promedio del mes anterior de los demás participes
para 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 comportamiento
de mandas, tal como expone [3] en el caso de Polonia y [4] para Chile por nombrar
algunos, generando AFPs líderes del sistema y seguidores. En esta línea, el presente
trabajo entrega una visión y propuesta de benchmark para comparar la rentabilidad
de los respectivos fondos y su promedio mensual, así como un mecanismo para evaluar
mecanismos regulatorios basados en la administración del riesgo propiamente tal.
Departamento de Ingeniería Industrial, Universidad de Chile, e-mail: [email protected]
Departamento de Ingeniería Industrial, Universidad de Chile, e-mail: [email protected]
1
Cifras informadas por Superintendencia de pensiones al 7 de Septiembre de 2015, con PIB base del 2014
172
Metodológicamente se resuelve un problema de optimización estocástica de portafolio, sobre un poliedro que incorpora los límites de inversión, restricciones de liquidez
y concentración y, adicionalmente, permite diferenciar los perfiles de riesgo-retorno
de cada fondo, resolviendo como función objetivo diferentes propuestas de medidas de
riesgo sobre la distribución de pérdidas del portafolio, con diferentes niveles de aversión,
reconociendo el carácter estocástico del problema. En particular, se utiliza y resuelve de
forma estadísticamente significativa la Medida de Riesgo Entrópica, no resuelta antes
en problemas de este tipo.
Para la resolución de los problemas de optimización se incorpora la metodología
SAA por sus siglas en inglés (Sample Average Aproximation), propuesta en [5] (y ampliamente extendida y utilizada, por citar algunos: [6],[7]) que resuelve cada instancia
equivalente cierta del problema de optimización con realización particulares de las variables aleatorias independientes e idénticamente distribuidas, repitiendo este proceso
una cantidad significativa de veces, tratando los resultados como realizaciones aleatorias, obteniendo con ellos, adicional a las soluciones, una medida de la convergencia
estocástica 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 at
Risk (CVaR)), una combinación convexa de los dos anteriores (ECVaR) y la medida de
riesgo Entrópica (Entropic Risk Measure).
Como conclusión y aporte metodológico, se logra resolver de forma estadísticamente
significativa el problema de portafolios con la Medida de Riesgo Entrópica, siendo la
única medida, como recientemente fue demostrada en [1], que cumple la consistencia en
la aditividad, interpretable como una noción de consistencia intertemporal, propiedad
esencial para lograr óptimos globales vía optimización local por rolling forward. El
enfoque de generación de cortes con ajuste iterativo de soportes, para la aproximación de
la 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 networks
arXiv preprint arXiv:1312.4193, 2013.
[2] Artzner, Philippe and Delbaen, Freddy and Eber, Jean-Marc and Heath, David. Coherent
Measures of Risk1. Risk management: value at risk and beyond , pp.145, Cambridge University
Press (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 equity
trading 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 approximation 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 and
Shapiro, Alexander.
The sample average approximation method applied to stochastic routing problems: a computational study, Computational Optimization and Applications, V.24, No.2-3, pp 289–333, 2003.
[7] Pagnoncelli, BK and Ahmed, Shapiro and Shapiro, A
Sample average approximation method for chance constrained programming: theory and applications, Journal of optimization theory and applications, V.142, No.2, pp 399–416, 2009.
174
Assessing Fishery Management and Recovery
Strategies through Viability Theory
Héctor Ramírez
Abstract
In this talk we construct a theoretical framework that permits, first, to assess the sustainability of fishery management strategies and, second, to propose recovery plans for
overexploited fisheries. The proposed framework allows facing situations when several
conflicting objectives have to be accounted for. In a first application, stochastic viability ranks management strategies according to their probability to sustain economic
and ecological outcomes over time. This approach is then extended to build stochastic
sustainable production possibility frontiers representing the trade-offs between sustainability objectives at any risk level, given the current state of the fishery. We thus study
the viability of effort and quota strategies when catch and biomass levels have to be
sustained. In the second application, a suitable deterministic bioeconomic dynamic permits to simulate divers recovery plans and a related optimization problem allows finding
a recovery plan which minimizes the total cost of the recovering process. Here the total
cost has been defined as the intertemporal sum of the differences between the objective
of the community and the recovery strategy proposed by the model (both quantities
are stated in term of catches), Finally, both approaches are applied and illustrated in
Chilean 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 a
sustainable 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.
∗
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.
∗
175
Optimal feedback synthesis and minimal time
function for the bioremediation of water
resources with two patches
H. Ramírez C.
A. Rapaport
V. Riquelme
Abstract
We study the bioremediation, in minimal time, of a water resource or reservoir using
a single continuous bioreactor. The bioreactor is connected to two pumps, at different
locations in the reservoir, that pump polluted water and inject back sufficiently clean
water with the same flow rate. This leads to a minimal-time optimal control problem
where the control variables are related to the inflow rates of both pumps. We obtain
a non-convex problem for which it is not possible to directly prove the existence of its
solutions. We overcome this difficulty and fully solve the studied problem by applying
Pontryagin’s principle to the associated generalized control problem. We also obtain
explicit bounds on its value function via Hamilton-Jacobi-Bellman techniques.
El trabajo es financiado por los Proyectos DYMECOS 2 INRIA Associated team, proyecto
BIONATURE de CIRIC INRIA CHILE, CONICYT REDES 130067, CONICYT ACT 10336,
FONDECYT 1110888, proyecto BASAL (Centro de Modelamiento Matemático, Universidad de
Chile), MathAmsud 15MATH-02, y Beca Doctorado Nacional Convocatoria 2013 folio 21130840
CONICYT. Se agradece también al Departamento de Postgrado y Postítulo de la Vicerrectoría de
Asuntos 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 Water
Resources 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 (UMI
2807, 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, SophiaAntipolis, 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]
176
[3] P. Gajardo, J. Harmand, H. Ramírez C. A. Rapaport: Minimal time bioremediation of natural
water resources, Automatica 47 (8), 1764–1769 (2011).
[4] P. Gajardo, H Ramírez C., A. Rapaport: Minimal time sequential batch reactors with bounded
and impulse controls for one or more species, SIAM Journal on Control and Optimization,
47(6) 2827–2856, (2008).
177
Dualidad en optimización vectorial
M. A. Rojas-Medar
L. Batista dos Santos
Camila Isoton
Abstract
El concepto de WD-invexidad ha sido recientemente introducido en problemas de programación no lineal con restricciones de desigualdad. Para tales problemas, un problema dual fue propuesto y el siguiente resultado fue establecido: el problema dual es
WD-invex si y solamente si el par de problemas primal-dual satisface la propiedad de
dualidad débil. En este trabajo, discutimos la noción de WD-invexidad para problemas
multiobjetivos. Obtenemos resultados similares a [3].
La Teoría de la Dualidad es una herramienta fundamental en el análisis de problemas
de optimización. Desde los años 80, muchos autores se han dedicado a la formulación de
diferentes problemas duales y también a la relajación de las hipótesis de convexidad de las
funciones involucradas, vea [1], [2], [4], [5] y las referencias ahí.
1
Formulación del problema y conceptos de solución
Consideremos el siguiente problema de optimización vectorial:
Minimizar f (x) := (f1 (x), · · · , fp (x))
sujeto a:
g(x) := (g1 (x), · · · , gm (x)) 5 0
x∈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]
Universidad de Tarapacá, e-mail: [email protected]
Universidade 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 problema

Maximizar L(r, λ, u) := rT f (u) + λT g(u) 


sujeto a:
(WD)
rT ∇f (u) + λT ∇g(u) = 0



λ = 0, u ∈ X
Definición. Decimos que el problema (P) es WD-invex I si existe η : X × X → Rn tal
que, 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 vector
r ≥ 0 fijado, se tiene
rT f (x) ≥ rT f (u) + λT g(u), ∀x ∈ F, ∀(u, λ) ∈ D.
Denotaremos:
F = {x ∈ X : g(x) 5 0},
T
T
D = {(u, λ) ∈ X × Rm
+ : r ∇f (u) + λ ∇g(u) = 0}.
Teorema Vale dualidad débil entre (P) y (WD) si y solamente si el problema (P) es
WD-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 generalized
convexity 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 multiobjective problems, Prepinter, 2015.
179
Comparision of MINC and MRMT
configurations:
Effects of spatial structure and biomass diffusion
A. Rapaport
H. Ramírez
A. Rojas-Palma
J. de Dreuzzy
Abstract
In a paper in preparation we prove that under certain assumptions controllability, the
MINC (Multiple Interacting Continua) and MRMT (Multirate Mass Transfer) configurations can be input-output equivalents for certain parameters values, but without
biomass diffusion. In this presentation the idea is to extend the results to the case of
diffusion of biomass (non-linear systems) by considering the simplest case, three reactors
connected linear biomass growth. The existence and stability of a positive equilibrium
will 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 the
output flow.
References
[1] D. Anderson , Compartmental Modeling and Tracer Kinetics, Lecture Notes in Biomathematics, Vol. 50, Springer, 1983.
[2] J. Carrera, X. Sanchez-Vila, I. Benet, A. Medina, G. Galarza and J. Guimera, On
matrix 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, Multicomponent 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]
UMR Géosciences, Rennes, France, e-mail: [email protected]
This 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, FONDECYT 1110888, BASAL project (Centro de Modelamiento Matemático, Universidad de Chile), CONICYT
national doctoral grant and CONICYT PAI/ Concurso Nacional Tesis de Doctorado en la Empresa, convocatoria 2014, 781413008.
180
[4] J.-R. de Dreuzy, A. Rapaport, T. Babey, J. Harmand; Influence of porosity structures on mixing-induced reactivity at chemical equilibrium in mobile/immobile Multi-Rate Mass
Transfer (MRMT) and Multiple INteracting Continua (MINC) models, Water Resources Research, Vol. 49(12), pp. 8511–8530, 2013.
[5] G. de Marsily, Quantitative Hydrogeology: Groundwater Hydrology for Engineers, Academic
Press, 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 Dispersion 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 surface 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. Doubleporosity, multiple-rate mass transfer processes in convergent flow tracer tests Water Resources
Research, Vol. 37(5), pp. 1143–1154, 2001.
[15] K. Pruess and T. Narasimhan, A practical method for modeling fluid and heat-flow in
fractured 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 tool
and 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, Society
of Petroleum Engineers Journal, Vol. 3(3), pp. 245–255, 1963.
[20] M. Willmann, J. Carrera, X. Sanchez-Vila, O. Silva and M. Dentz, Coupling of
mass transfer and reactive transport for nonlinear reactions in heterogeneous media, Water
Resources 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-dimensional
conductivity fields with connected regions of high conductivity, Environmental Science & Technology, Vol. 38(14), pp. 3916–3926, 2004.
181
Optimization of the concentration changes in a
chemostat 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 a
chemostat model to a target point. More precisely, we consider the usual chemostat
model of a single biomass and a single substrate given by the following equations (see
[4]):
Ẋ = µ(S)X − uX,
(1)
Ṡ = −µ(S)X + u(Sin − S),
where X, resp. S, represents the biomass concentration, resp. the substrate concentration, Sin the input substrate concentration, u the dilution rate and µ the growth
function of the biomass.
Given an initial condition (X0 , S0 ) and a target point (X, S), we are interested in
reaching (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 ) := inf t(u) s.t. Xu (t(u)) = X and Su (t(u)) = S,
(2)
u(·)
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) = µ
S
,
S+k
for some µ > 0 and k > 0, the problem has been studied in [2], where it is shown that
the 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) = µ S 2
ki
S
+ S + ks
,
for some µ > 0, ki > 0 and ks > 0. Using the Pontryagin Maximum Principle ([3]) and
geometric control theory ([1]), we show that in some cases the control is not necessarily
bang-bang but might admit frame curves such as singular locus and switching curve.
Térence Bayen, e-mail: [email protected]
Jérôme Harmand, e-mail: [email protected]
Matthieu Sebbah, e-mail: [email protected]
182
References
[1] U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, vol. 43, Springer-Verlag, Berlin, 2004.
[2] G. D’Ans, P. Kokotovic, D. Gottlieb, Time-Optimal Control for a Model of
Bacterial 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 microbial
competition, Cambridge University Press, 1995.
183
Problemas Inversos y Control de EDP
Encargado de Sesión : Rodrigo Lecaros
184
On the cost of null controllability of some linear
partial differential equations
Nicolás Carreño
Abstract
In this talk, we will present some results concerning the cost of null controllability of
two linear equations posed on a bounded interval. First, we consider a linear KdV
equation with a boundary control on the left extreme of the interval. We prove that, as
the dispersion coefficient goes to zero, the size of the control that drives the solution to
the null state increases exponentially for every control time. Then, we consider a linear
fourth-order equation with two boundary controls. In this case, we show that the size of
the controls explode as the diffusion coefficient vanishes if the control time is small. On
the other hand, if the control time is large enough, the controls are uniformly bounded
with respect to the diffusion coefficient and, furthermore, their norms decrease to zero
exponentially.
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 parabolic
equation. Preprint.
185
On the control of the improved Boussinesq
equation
Eduardo Cerpa
Abstract
The so called ÒbadÓ Boussinesq equation was introduced to describe the flow of shallow
water waves with small amplitude. This equation can be approached by the Improved
Boussinessq equation. In this talk we are interested in the control properties of this
system. First, we consider a boundary control and prove that the system is approximately controllable but not exactly controllable. Second, we introduce an internal
control 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]
186
An Inverse Problem for the Helmhotz Equation
in a Layered Media.
Matías Courdurier
Abstract
An important element in the propagation of waves on a stratified media is the
propagation of the wave along waveguides. For acoustic waves in 2D, in a time harmonic
regime, on an infinite medium with a single layer of width 2h, the propagation of the
wave is described by the solution of the Helmholtz equation
∆u + k 2 n2 (x, z)u = f, (x, y) ∈ R2 ,
with a picewise constant index of refraction
(
n0
n(x, z) =
ncl
|x| < h
|x| > h,
and where the right radiation conditions are described in [1]. In this talk we will propose
an inverse problem in this setting and we will present the progress made in the study
of 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]
Joint work with Eric Bonnetier, Université Joseph Fourier Axel Osses Universidad de Chile. and
Faouzi Triki. Université Joseph Fourier.
187
Detection of Several Obstacles in a Stokes Flow:
A mixed approach
Matías Godoy Campbell
Abstract
We consider the inverse problem of detecting the location and shape of several objects
immersed 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 penalizes erroneous configurations for the considered system and even more, its minimization
with respect of all possible admissible configurations is equivalent to the resolution of
our 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 an
asymptotic expansion which leads to the expression of the so-called topological gradient
of 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 topological
method.
Then, we present some numerical simulations of this mixed approach which combines
the topological and geometrical shape optimization methods.
We finally discuss, briefly, the possibilities when there exists an inaccessible region
of 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. de
Chile).
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 Problems
and 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-03
and Fondecyt Grant No.1140773 and also by the Grant CONICYT-PCHA/Doctorado Nacional/2012.
188
Controllability of coupled systems with
Schrödinger equations.
Alberto Mercado Saucedo
Abstract
In this talk we present some control problems regarding systems of coupled partial
differential equations, when one of them is a linear Schrödinger equation. We address
the problem of controlling the system using less controls than equations. We present
two controllability problems: 1) A system with two N -dimensional linear Schrödinger
equations with a control supported in a region not satisfying the classical geometrical
control condition (see [1]); and 2) The problem of internal null controllability of a
system 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 linear
Schrödinger-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
189
An ADER type scheme for evolving differential
operators
G. Montecinos J. C. López R. Lecaros
J. Ortega E. F. Toro
Abstract
In this work we propose a numerical strategy to solve partial differential equations in
which 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 waves
equations. 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 volume
evolution of differential operators. In [6], a globally implicit strategy to solve the
Richard equation, was proposed. In the present work a locally implicit formulation
is investigated and the operator to be evolved may contain differential terms. In the
present 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 of
the data; ii) solutions to Generalized Riemann Problems (GRP), which allow us to
evaluate 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 uses
the 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 containing
stiff source terms, reconciling stability and accuracy. Additionally, the present method
allows the use of a CFL-type condition and thus the high-order of accuracy is achieved
in space and time. We show theoretically, the convergence of the present scheme.
Furthermore, we carried out an empirical convergence rate assessment, in order to
illustrate 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]
190
[2] M. Dumbser, C. Enaux, and E. F. Toro. Finite volume schemes of very high order of accuracy
for 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 generalized Riemann problem for hyperbolic systems with source terms. Journal of Computational
Physics, 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), pages
905–937. 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 of
high-order accuracy for flow in unsaturated porous media. International Journal for Numerical
Methods in Engineering, 89(2):227–240, 2012.
191
Stability numbers to Timoshenko’s system with
shear 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 the
difference of the wave speed and χ1 that depends on the size of the interval. This
numbers will describe the asymptotic behavior of the system. That is, we prove strong
stability if and only if χ1 is not a rational multiple of π 2 . If additionally χ0 = 0 and
χ1 < 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 thermoelastic 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 asymptotic
behaviour 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 of
exponential 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 Computation
Rua Getulio Vargas 333, Quitadinha-Petrópolis 25651-070, Rio de Janeiro, RJ, Brasil, e-mail:
[email protected]
192
Teoría de Números
Encargado de Sesión : Amalia Pizarro
193
Ramas y extensiones de cuerpos
L. Arenas-Carmona
Abstract
El árbol de Bruhat-Tits es un grafo cuyos vértices son los órdenes maximales del álgebra
M2 (K) donde K es un cuerpo local. Dos de tales órdenes son vecinos si en algún sistema
de coordenadas tienen la forma
OK π −1 OK
OK OK
,
D2 =
,
D1 =
πOK
OK
OK OK
donde π es un parámetro uniformizante de K. En trabajos anteriores (ver [5]), el autor
definió la rama de un suborden H como el mayor subgrafo cuyos vértices corresponden
a órdenes que contienen a H. En este trabajo estudiamos el comportamiento de estas
ramas bajo extensiones de cuerpos y aplicamos esta idea para calcular los invariantes
de 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 quaternionic lattices, Ann. Inst. Fourier 53 (2003), 2021–2038.
[2] L. Arenas-Carmona, Representation fields for commutative orders, Annales de l’institut
Fourier 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), 143156.
[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]
Supported by Fondecyt No 1140533. En conjunto con M. Arenas y C. Bravo.
194
Un retículo Hermiteano central
Ana Cecilia de la Maza
Remo Moresi
Abstract
Definiciones 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)
∀ x ∈ L.
(2)
iii) b ∈ L es una operación nula con
xx⊥ ≤ b
Un ejemplo canónico de retículo Hermiteano indexado está dado por el retículo de los
subespacios 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 por
la forma φ. El rol de b lo juega el subespacio formado por los vectores de traza valuada
E ∗ , 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 Universalis
66 (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 dimensional
vector 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]
:
195
Distribución asintótica de puntos de Hecke sobre
Cp
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 ). Dada
E ∈ Ell(Cp ) y n ∈ N definimos los puntos de Hecke de orden n asociados a E como
los puntos E 0 ∈ Ell(Cp ) que admiten una isogenia E → E 0 de grado n. Esto equivale a
tener E 0 = E/C donde C es un subgrupo de E de cardinalidad n. Con estos puntos de
Hecke podemos construir el divisor
M
Tn (E) =
E/C
C≤E,#C=n
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 de
Hecke se equidistribuyen respecto a una medida natural en Ell(C), la medida hiperbólica. En particular, la distribución asintótica de dichos puntos es independiente del
punto inicial E ∈ Ell(C). Nuestro resultado principal es una descripción de la distribución asintótica de Tn (E) cuando E ∈ Ell(Cp ) bajo ciertas condiciones sobre el tipo de
reducción de E módulo M, el ideal maximal del anillo de enteros de Cp , y sobre la
norma 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
[email protected]
por
beca
CONICYT
Doctorado
Nacional
21130412,
e-mail:
196
Optimal bounds for Büchi’s problem in modular
arithmetic
Pablo Sáez
Xavier Vidaux
Maxim Vsemirnov
Abstract
We study the second order analogue of the problem of finding optimal lower and upper
bounds for the length of sequences of squares in arithmetic progression modulo a prime,
and some connections with the computational problem of finding a quadratic nonresidue modulo a prime. More precisely, we work modulo an integer and our objects of
study 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 reduce
the problem to prime moduli. We observe several phenomena which are supported by
extensive numerical computations. We also discuss the case where the leading coefficient
of the second degree polynomial defining the sequence of squares is non invertible.
References
[1] H. Pasten, T. Pheidas, X. Vidaux, A
tations and open problems, Proceedings
ics, Zapiski POMI 377, 111-140, Steklov
http://www.pdmi.ras.ru/znsl/2010/v377.html
survey on Büchi’s problem : new presenof the Hausdorff Institute of MathematInstitute of Mathematics. Published online
(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]
Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción-3, e-mail:
[email protected]
St. 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 University
prospekt, St. Petersburg, 198504, Russia, e-mail: [email protected]
197
Una conexión entre la propiedad de Northcott y
la indecidibilidad en anillos de enteros
totalmente reales
Xavier 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
199
Promoviendo el desarrollo de habilidades del
pensamiento matemático en estudiantes del
sistema escolar: Una experiencia en el complejo
educacional la Granja de Cajón
Pamela Alarcón
Valeria Carrasco Ciro González
Teresa Sanhueza
Abstract
En este trabajo, se presentan los principales lineamientos de desarrollo de un proyecto
interno de Pasantías Docentes en Centros Educativos, aprobado a mediados del presente año en la Facultad de Educación de la UC Temuco.
El objetivo del Proyecto es “fortalecer el vínculo entre la Universidad Católica de
Temuco y uno de los Centros Educativos que colaboran en la formación inicial de profesores, a través de la realización actividades conjuntas que fomenten en los estudiantes del
establecimiento 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 Educativo las necesidades de ayuda técnica de acuerdo a la realidad escolar en la cual
se encuentran inmersos, de modo que a través del Estudio de Clases, se planifiquen
e implementen acciones que contribuyan al fortalecimiento profesional de los profesores de matemática del Centro Educativo, en relación al desarrollo de habilidades del
pensamiento matemático (como la resolución de problemas, las habilidades de comunicación y argumentación, las de representación, y las de modelamiento matemático),
implementar de manera conjunta estrategias didácticas que permitan a los estudiantes
el desarrollo de estas habilidades que propicien un impacto positivo en sus aprendizajes, y analizar las implementaciones realizadas con el propósito de mejorar las prácticas
educativas a nivel de aula.
El presente trabajo pretende presentar el día de las jornadas los avances en el proceso
de ayuda técnica inicial que se le dio a las profesores, del proceso de recogida de las
necesidades en función del tratamiento de las habilidades del pensamiento matemático,
de la planificación de estrategias innovadoras, de la observación y la posterior reflexión
sobre 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”, Editorial
Universitaria Valparaíso, Valparaíso.
[4] Ministerio de Educación, 2013, “Bases Curriculares de Matemáticas vigentes”, Decreto 614 de
2013.
[5] NCTM, 2003, pp 64: “Principios y Estándares para la educación matemática”.
201
Análisis de idoneidad didáctica del método
japonés
desde un enfoque onto-semiótico de la
instrucción matemática
Carlos Cabezas
Pedro Arteaga
Abstract
Esta ponencia, se enfoca en el análisis didáctico del método Estudio de Clases, en su
componente "Ejecución de la Clase" y en la metodología para su desarrollo conocida
como "Método Japonés". Se estudian las diversas componentes de una clase típica
orientada por un plan de clases, teniendo como herramienta de análisis, el sistema
de indicadores empíricos que desarrollan la noción de idoneidad didáctica, introducida
en 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 plan
de clases que contemple los diversos aspectos tanto disciplinares como didácticos y otros
elementos 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 la
dirección en que se llevarán las diferentes interacciones, en todo lo que involucra el
proceso de aula de modo que se logren los objetivos de aprendizaje propuestos por el
profesor. Un aporte de este trabajo se verifica en la comprobación empírica, con base
en el enfoque onto-semiótico de la instrucción matemática, de la idoneidad didáctica del
método japonés y en el análisis desde esta perspectiva, del plan de clases, que aporta
antecedentes, orientaciones y criterios para la confección de otros planes de clase que
cubran todos los aspectos involucrados en los planes ya tradicionales propuestos por el
método 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 la
matemática a partir del Estudio de Clases. Ediciones Universitarias de Valparaíso, P. Universidad Católica de Valparaíso.
Universidad de Granada, e-mail: [email protected]
202
Problemas incorrectos como medio para
desarrollar aprendizaje profundo
Hugo Caerols
Katia Vogt Geisse
Abstract
Uno de los objetivos fundamentales de quienes dirigimos los procesos de EnseñanzaAprendizaje en matemáticas y también en otras disciplinas, es conseguir en nuestros
estudiantes un aprendizaje profundo de los temas de estudio. Este aprendizaje depende
de varios factores, tales como el interés del estudiante por el tema, sus estrategias de
aprendizaje, la dinámica de las clases, el tiempo del que dispone el estudiante para el
curso y del tipo de problemas a los que se enfrenta.
La mayoría de los problemas que proponemos a nuestros estudiantes, y que aparecen
en los actuales textos de estudio, buscan una aplicación principalmente de manera
directa de algún resultado y los menos están dirigidos al análisis de la materia. En esta
charla presentaremos algunos resultados sobre cómo han reaccionado nuestros alumnos
al verse enfrentados de manera dirigida a problemas incorrectos.
Encontrar y formular este tipo de problemas no es una tarea sencilla. Mostraremos
un problema de modelamiento que hemos presentado a los alumnos de seis secciones del
curso de Ecuaciones Diferenciales Ordinarias dictado en la Universidad Adolfo Ibáñez.
El problema de por si es sencillo de entender y bastante interesante. También presentaremos un análisis de los resultados obtenidos por los alumnos, quienes trabajaron
en grupo siguiendo las indicaciones de discutir sus resultados y analizar las distintas
formas que encontraron de enfrentar el problema. Mostraremos cuánto se acercaron los
estudiantes a una visión profunda del planteamiento y análisis del modelo asociado al
problema.
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 teaching
and assessment. En J. C. Smart y W. G. Tierney (Eds.), Higher education: Handbook of theory
and 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
203
El aprendizaje del cálculo diferencial bajo un
diseño curricular Modular
Elías Irazoqui Becerra
Abstract
Se presenta una experiencia de aula, donde se aplica un diseño curricular modular
para propiciar el aprendizaje del cálculo diferencial, en estudiantes de primer año de
Ingeniería en Alimentos de la Universidad del Bío-Bío, Campus Chillán, Chile.
Es sabido que el aprendizaje del cálculo diferencial presenta dificultades, muchos
investigadores 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 Rincón
et al. [9]. Ahora bien, no cabe ninguna duda que el aprendizaje del cálculo diferencial
es un tópico importante para el estudiante si se piensa en las materias afines que siguen
a ésta, como lo son el cálculo integral, las ecuaciones diferenciales y el cálculo en varias
variables.
Cuando se habla de aprendizaje, en general, puede ser importante atender a las consideraciones que Biggs [2] hace sobre él al referirse a: aprendizaje superficial y aprendizaje
profundo. Por otro lado, las propuestas en materia del aprendizaje del cálculo, que
ponen el acento en la derivada, son variadas. Algunas de ellas ponen el acento en el
uso de los recursos informáticos, otras en la modelización etc. En nuestro caso, y bajo
un diseño curricular modular se pone énfasis en la resolución de actividades didácticas
en los cuatro temas centrales que aborda cualquier curso de cálculo de primer año de
universidad, como son: las funciones, límites y continuidad, derivadas y aplicaciones de
éstas.
Por último, por un diseño curricular modular se entiende una disposición de los contenidos en dos módulos de trabajo, que son unidades de contenidos precisas y que
implican aprobar el primer módulo para realizar el segundo, de no ser así se ha de
repetir el módulo nuevamente. La nota final del curso resulta del promedio obtenido en
ambos módulos de trabajo. En esta ocasión presentamos los resultados de una experiencia 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, cognitivos y didácticos. Ingeniería Didáctica en educación Matemática. México: Grupo Editorial
Iberoamericana.
Depto Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, e-mail: [email protected]
204
[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 con
la derivada. Lestón, P. (Ed.) Acta Latinoamericana de Matemática Educativa, Vol. 24. México:
Colegio Mexicano de Matemática Educativa A. C. y Comité Latinoamericano de Matemática
Educativa 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 de
2013).
[5] Ortega, T. y Sierra, M. (1998). El concepto de derivada: algunas indicaciones para su enseñanza. 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 de
una 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). Elementos
del Cálculo. Reconstrucción conceptual para el aprendizaje y la enseñanza. México: Editorial
Trillas.
[8] Tall, D. (1996). Advanced Mathematical Thinking and the Computer. Proceedings of the 20th
University 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 estrategia didáctica para la enseñanza del cálculo. Lestón, P. (Ed.). Acta Latinoamericana de
Matemática Educativa, Vol. 27. México, DF: Colegio Mexicano de Matemática Educativa A.
C. y Comité Latinoamericano de Matemática Educativa A. C.
205
Estudio de clases: hacia una alianza de la
universidad con las escuelas
Soledad Estrella
Sergio Morales
Raimundo Olfos
Abstract
Esta presentación tiene por objetivo compartir la experiencia del Instituto de Matemática de la PUCV y dar cuenta de los esfuerzos realizados durante los últimos 8 años
para promover la práctica del Estudio de Clases en distintos establecimientos escolares
de la región de Valparaíso.
Favorecer la conformación de Grupos de Estudio de Clases constituye una oportunidad de vinculación estratégica entre la Universidad y las instituciones educativas,
que ayuda a promover la investigación acción en las escuelas en pro del desarrollo
de conocimientos y habilidades docentes para el mejoramiento de la enseñanza de la
matemática y estadística. Al mismo tiempo abre espacios de investigación para la
universidad que permitirían robustecer la base de conocimiento en torno al desarrollo
profesional docente, inicial y continuo, a la indagación en procesos de aprendizaje de
conceptos matemáticos y estadísticos en alumnos, y a las dinámicas que surgen en las
comunidades de aprendizajes en torno a grupos de Estudio de Clases.
En la presentación se dará cuenta del proceso y las acciones llevadas a cabo por
los académicos del Instituto de Matemática de la PUCV para promover el Estudio de
Clases 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 la
[3] Olfos, R., Estrella, S., & Morales, S. (2014). What can we learn from natural disasters to
prevent 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]
Instituto Superior de Comercio de Valparíso, e-mail: [email protected]
Pontificia Universidad Católica de Valparaíso, e-mail: [email protected]
206
Avances y retrocesos en el Estudio de Clases (en
el 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 juicio
nuestro, por el exceso de currículo nulo en gran parte de nuestros programas curriculares
del Sistema K-12 (y también en el sistema universitario).
En efecto, por lo general, las preguntas abiertas, en el inicio de una propuesta para un
estudio de clases van en contradicción con las propuestas ya fuertemente establecidas
en los programas, en los libros, y en general en la inercia conductista, formulista e
inagotable que caracteriza a nuestra actual enseñanza de la matemática.
Hoy, en la actualidad, existe un postítulo de cobertura nacional, para los profesores
del sistema K-8, conducido por el MINEDUC, en que hay cierta esperanza en introducir 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 and
Learning through Lesson Study - Connection between Assessment and Subject Matter, February
17 - 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 multiplicación. Ediciones Universitarias de Valparaíso. 2009
[4] Manríquez L., Honores P. Currículum para enseñanza pre-básica y básica. Edición Universidad
de Antofagasta, Vicerrectoría Académica, Dirección de Docencia. 2006
Universidad de Antofagasta, e-mail: [email protected]
207
Comunidades de Aprendizaje GEC
Soledad Estrella
Sergio Morales
Maria Tapia
Raimundo Olfos
Abstract
La presentación tiene como propósito motivar a los participantes a apoyar la creación
y desarrollo de comunidades de aprendizaje GEC (Grupos de Estudio de Clases) en
establecimientos o grupos de establecimientos de su Región.
La conformación de Grupos de Estudio de Clases en Chile constituye una oportunidad para promover el desarrollo de la matemática en el país, mejorar la calidad
de los aprendizajes escolares, favorecer el desarrollo profesional docente, aportar a la
innovación curricular y robustecer la relación Universidad y establecimientos Escolares.
Las políticas actuales del MINEDUC favorecen la conformación de Comunidades
de Aprendizaje proveyendo horas de dedicación exclusiva a los profesores para su desarrollo profesional y preparación de la enseñanza. El MINEDUC, a través del CPEIP
está desarrollando un plan que establece la creación de “comités locales de desarrollo
profesional docente” integrados por representantes regionales o provinciales del Ministerio de Educación, sostenedores de establecimientos municipales y consejeros regionales
provinciales.
A continuación se ofrecen dos ejemplos de grupos de estudio de clases vigentes en
la Región de Valparaíso. El primer caso corresponde a un grupo de profesores de
matemáticas del Liceo INSUCO de Valparaíso, quienes realizan sus actividades del
año 2013 a la fecha. Estos profesores generan desarrollo profesional al interior de su
Institución. 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 de
educación general básica y educadoras diferenciales entre escuelas, pertenecientes a la
Comuna de La Calera. Este grupo se constituyó en el año 2014. Durante la presentación
también 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 sus
localidades 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]
Escuela Palestina de La Calera, e-mail: [email protected]
208
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 la
[3] Olfos, R., Estrella, S., Morales, S. (2014). Open lessons impact statistics teaching teachersâĂŹ
beliefs. 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.
209
La probabilidad en el aula de educación básica.
Un estudio de caso sobre los primeros elementos
linguísticos
Claudia Vásquez Ortiz
Abstract
En las últimas décadas se observa una fuerte tendencia por incorporar la probabilidad en
los currículos de Educación Primaria, con el objeto de promover que los alumnos aprendan conocimientos probabilísticos que les sirvan de base para la recogida, descripción e
interpretación de datos. En definitiva, se trata de ofrecerles herramientas que faciliten
la toma de decisiones en situaciones en las que la incertidumbre es relevante, para que
progresivamente sean ciudadanos bien informados y consumidores inteligentes. Es en
este contexto que la probabilidad "proporciona una excelente oportunidad para mostrar
a los estudiantes cómo matematizar, cómo aplicar la matemática para resolver problemas reales" (Godino, Batanero y Cañizares, 1997, p.12). Por tanto, surge la necesidad
de educar a los estudiantes en esta área desde temprana edad, para así, contar con
ciudadanos alfabetizados probabilísticamente "capaces de hacer frente a una amplia
gama de situaciones del mundo real que implican la interpretación o la generación de
mensajes probabilísticos, así como la toma de decisiones" (Gal, 2005, p.40).
En este sentido, el National Council of Teachers of Mathematics incluyó a "Datos y
Azar" como área temática en Curriculum and Evaluation Standard for School Mathematics (NCTM, 1989), reforzando esta iniciativa en Principles and Standard for School
Mathematics (NCTM, 2000), que contemplan que los programas de enseñanza deberían
capacitar a los alumnos para aprender conocimientos relacionados con el análisis de
datos y la probabilidad a partir del nivel Pre-K (tres años). Esta tendencia, como
decíamos, se ha reflejado en los currículos de matemáticas de muchos países, entre ellos
Chile, que han incorporado la probabilidad en Educación Primaria para promover un
enfoque experimental que proporcione una experiencia estocástica desde las primeras
edades (Mineduc, 2012). Producto de esta necesidad científica, profesional y social,
Chile ha incluido en las actuales Bases Curriculares (2012) el estudio de la probabilidad
a lo largo de todo el currículo escolar, con el propósito de que "todos los estudiantes se
inicien en temas relacionados con las probabilidades" (Mineduc, 2012, p. 5), y de este
modo cumplir con parte de los objetivos generales propuestos en la Ley General de Educación (2009) para la Educación Básica, referidos explícitamente a "que los educandos
desarrollen los conocimientos, habilidades y actitudes que les permitan: pensar en forma
reflexiva, evaluando y utilizando información y conocimientos, de manera sistemática
y metódica, para la formulación de proyectos y resolución de problemas; comprender y
utilizar conceptos y procedimientos matemáticos básicos en la resolución de problemas
Pontificia Universidad Católica de Chile, e-mail: [email protected]
210
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 probabilidad, de manera informal en los primeros niveles, donde el lenguaje probabilístico
asociado a situaciones problemáticas centradas en los juicios que emiten los estudiantes
con base en sus propias experiencias, juega un rol fundamental, pues es a partir del
lenguaje informal y cotidiano que los estudiantes desarrollarán, paulatinamente, un razonamiento más abstracto y cuantitativo, que permitirá el tránsito entre los diversos
significados de la probabilidad en el contexto de la matemática escolar, para así alcanzar
la construcción de un conocimiento probabilístico de un nivel de abstracción mayor.
En este trabajo se presenta un estudio que contempla el análisis del proceso de
enseñanza de la probabilidad y, en concreto, cómo el profesor utiliza una multiplicidad
de términos, expresiones orales y escritas, símbolos y representaciones (tablas y gráficos)
cuando enseña probabilidad a estudiantes de educación primaria que no han recibido
instrucción previa sobre el tema con el propósito de que éstos aprendan gradualmente
la 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 in
school: 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 of
Teachers of Mathematics. 2000
[5] Mineduc . Bases Curriculares 2012: Educación Básica Matemática. Santiago de Chile: Unidad
de Curriculum y Evaluación. 2012
[6] Mineduc . Ley General de Educación. Santiago: MINEDUC, www.mineduc.cl. 2009
211
Estudio de clases en didáctica de la matemática:
proceso reflexivo de los estudiantes de pedagogía
en Educación Básica en la U. Santo Tomás
Pierina Zanocco Soto
Abstract
Esta ponencia, la cual se centra en la estrategia Estudio de clases, es parte de la investigación "Generación de ambientes reflexivos y decisiones pedagógicas fundamentadas, en
la Didáctica de la Matemática: Estudio de casos y Estudio de clases" focalizándose en
las asignaturas Didáctica de la Matemática I y II, trabajando con 30 estudiantes de la
carrera Educación Básica, aplicando la estrategia mencionada. Se privilegia potenciar
la generación de espacios reflexivos y toma de decisiones pedagógicas fundamentadas
en la enseñanza de la Matemática. Investigaciones destacan la importancia que tiene
la preparación de profesores en la planificación de clases (Liping Ma, 2010) y en el trabajo 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 en
la 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; presentación pública de sus clases; retroalimentación de sus pares; autocrítica a posteriori
de sus planificaciones; formulación del plan final. Las evidencias están relacionadas con
logros en habilidades de pensamiento mencionadas y en los avances en sus formas de
planificar.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 Improving
Education 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 Matemática
a partir del Estudio de Clases. Valparaíso: Ediciones Universitarias de Valparaíso. 2009
[4] Liping Ma. Conocimiento y enseñanza de las matemáticas elementales. La comprensión de las
matemáticas fundamentales que tienen los profesores en China y los EEUU. Santiago: Ediciones
Academia Chilena de Ciencias. 2010
Escuela
de
Educación,
[email protected]
Universidad
Santo
Tomás,
Sede
Santiago,Chile,
e-mail:
212

ACTA DE RESUMENES LXXXIV Encuentro Anual

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