1. Art and Diseño

Congreso bienal
RSME 2015
Universidad de Granada
Granada, 2 al 6 Febrero
Libro de resúmenes
de Conferencias
y Pósteres
Más información en http://rsme2015.ugr.es
email: [email protected]
Índice de conferencias y
pósteres
Índice de conferencias y pósteres . . . . . . . .
Conferencias plenarias . . . . . . . . . . . . . .
Ángel Castro . . . . . . . . . . . . . . . . . .
Fernando Codá Marques . . . . . . . . . . .
Luis V. Dieulefait . . . . . . . . . . . . . . .
Daniel Faraco Hurtado . . . . . . . . . . . .
María Angeles Gil Alvarez . . . . . . . . . .
Rafael Ortega . . . . . . . . . . . . . . . . .
David Pardo . . . . . . . . . . . . . . . . . .
María Pe Pereira . . . . . . . . . . . . . . . .
S01. Análisis armónico . . . . . . . . . . . . . .
Theresa C Anderson . . . . . . . . . . . . .
Arpad Benyi . . . . . . . . . . . . . . . . . .
Frédéric Bernicot . . . . . . . . . . . . . . .
Oleksandra Beznosova . . . . . . . . . . . .
Maria J. Carro . . . . . . . . . . . . . . . . .
Lucas Chaffee . . . . . . . . . . . . . . . . .
Leonardo Colzani . . . . . . . . . . . . . . .
David Cruz-Uribe . . . . . . . . . . . . . . .
W. Damián . . . . . . . . . . . . . . . . . . .
Luigi Fontana . . . . . . . . . . . . . . . . .
Jarod Hart . . . . . . . . . . . . . . . . . . .
Juha Kinnunen . . . . . . . . . . . . . . . .
Diego Maldonado . . . . . . . . . . . . . . .
Jose Maria Martell . . . . . . . . . . . . . . .
Francisco J. Martín-Reyes . . . . . . . . . .
Albert Mas . . . . . . . . . . . . . . . . . . .
Virginia Naibo . . . . . . . . . . . . . . . . .
Ioannis Parissis . . . . . . . . . . . . . . . .
Marco Peloso . . . . . . . . . . . . . . . . .
Salvador Rodriguez-Lopez . . . . . . . . . .
Javier Soria . . . . . . . . . . . . . . . . . . .
Lesley A. Ward . . . . . . . . . . . . . . . . .
Xinfeng Wu . . . . . . . . . . . . . . . . . . .
Qingying Xue . . . . . . . . . . . . . . . . .
S02. Análisis complejo y teoría de operadores
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Índice de conferencias y pósteres
Alexandru Aleman . . . . . . . . .
Oscar Blasco . . . . . . . . . . . . .
Aline Bonami . . . . . . . . . . . .
Jose Bonet . . . . . . . . . . . . . .
Carme Cascante . . . . . . . . . . .
Santiago Díaz Madrigal . . . . . .
María José González . . . . . . . .
Håkan Hedenmalm . . . . . . . . .
Yurii Lyubarskii . . . . . . . . . . .
Joaquim Ortega-Cerdà . . . . . . .
Jose Manuel Rodriguez . . . . . . .
Oliver Roth . . . . . . . . . . . . . .
Dragan Vukoti´c . . . . . . . . . . .
S03. Análisis funcional . . . . . . . . .
Xavier Barrachina Civera . . . . . .
Chiara Boiti . . . . . . . . . . . . .
Santiago Boza . . . . . . . . . . . .
J. Alejandro Chávez-Domínguez .
José E. Galé . . . . . . . . . . . . . .
Maria Angeles Japón Pineda . . . .
Vladimir Kadets . . . . . . . . . . .
Fernando Lledó . . . . . . . . . . .
Elisabetta Mangino . . . . . . . . .
Javier Merí . . . . . . . . . . . . . .
Marina Murillo Arcila . . . . . . . .
Matías Raja . . . . . . . . . . . . . .
Daniel Seco . . . . . . . . . . . . .
Juan B. Seoane-Sepúlveda . . . . .
Jesús Suárez . . . . . . . . . . . . .
S04. Análisis geométrico . . . . . . . .
J. Carlos Díaz-Ramos . . . . . . . .
Oscar Garcia-Prada . . . . . . . . .
Ana Hurtado . . . . . . . . . . . . .
Antonio Martínez . . . . . . . . . .
Vicente Palmer . . . . . . . . . . .
Daniel Peralta-Salas . . . . . . . .
David Ruiz . . . . . . . . . . . . . .
Francisco Torralbo . . . . . . . . .
S05. Análisis no lineal y EDP elípticas
Claudio Bonanno . . . . . . . . . .
Marco Ghimenti . . . . . . . . . . .
Tommaso Leonori . . . . . . . . . .
Ederson Moreira dos Santos . . . .
Filomena Pacella . . . . . . . . . .
Benedetta Pellacci . . . . . . . . .
Angela Pistoia . . . . . . . . . . . .
David Ruiz . . . . . . . . . . . . . .
Berardino Sciunzi . . . . . . . . . .
Gabriella Tarantello . . . . . . . . .
Susanna Terracini . . . . . . . . . .
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Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Índice de conferencias y pósteres
Gianmaria Verzini . . . . . . . . . . . . . . . . . . . . . . . . . . .
S06. Análisis numérico de EDP y modelización . . . . . . . . . . . .
Marta Benítez . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tomás Chacón . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
María Cruz Navarro . . . . . . . . . . . . . . . . . . . . . . . . . .
Rosa Donat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
José María Gallardo . . . . . . . . . . . . . . . . . . . . . . . . . .
Luca Gerardo-Giorda . . . . . . . . . . . . . . . . . . . . . . . . .
Heiko Gimperlein . . . . . . . . . . . . . . . . . . . . . . . . . . .
María González Taboada . . . . . . . . . . . . . . . . . . . . . . .
J.C. Jorge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Yvon Maday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Julia Novo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Francisco Pla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rodolfo Rodríguez . . . . . . . . . . . . . . . . . . . . . . . . . .
S07. Análisis numérico en la resolución de ecuaciones no lineales
Sergio Amat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alicia Cordero . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
José Antonio Ezquerro . . . . . . . . . . . . . . . . . . . . . . . .
José M. Gutiérrez . . . . . . . . . . . . . . . . . . . . . . . . . . .
José Luis Hueso . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ángel Alberto Magreñán . . . . . . . . . . . . . . . . . . . . . . .
Eulalia Martínez Molada . . . . . . . . . . . . . . . . . . . . . . .
Rosa M. Peris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Javier Segura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Juan R. Torregrosa . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jean-Claude Yakoubsohn . . . . . . . . . . . . . . . . . . . . . .
S08. Conocimiento profesional del profesor de matemáticas . . .
Lorenzo J. Blanco Nieto . . . . . . . . . . . . . . . . . . . . . . .
María Luz Callejo de la Vega . . . . . . . . . . . . . . . . . . . . .
Luis Carlos Contreras González . . . . . . . . . . . . . . . . . . .
Pablo Flores Martínez . . . . . . . . . . . . . . . . . . . . . . . .
Antonio Moreno Verdejo . . . . . . . . . . . . . . . . . . . . . . .
Gloria Sánchez-Matamoros García . . . . . . . . . . . . . . . . .
S09. Ecuaciones diferenciales y sistemas dinámicos . . . . . . . . .
Begoña Alarcón . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Juan Belmonte Beitia . . . . . . . . . . . . . . . . . . . . . . . . .
José Luis Bravo Trinidad . . . . . . . . . . . . . . . . . . . . . . .
Adriana Buic˘a . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alberto Cabada . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jose Angel Cid . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Carlos Escudero . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jaykov Foukzon . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jorge Galan-Vioque . . . . . . . . . . . . . . . . . . . . . . . . . .
Santiago Ibáñez . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eduardo Liz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rafael Obaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jesús Palacián . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Daniel Peralta-Salas . . . . . . . . . . . . . . . . . . . . . . . . .
Patricia Yanguas Sayas . . . . . . . . . . . . . . . . . . . . . . . .
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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43
44
44
45
46
46
46
47
48
48
48
49
50
50
51
53
53
53
54
54
55
56
56
57
57
57
58
59
59
59
60
60
60
61
62
62
62
62
63
64
64
65
65
66
67
67
68
68
69
70
Índice de conferencias y pósteres
Massimo Tarallo . . . . . . . . . . . . . . . . . . . . . . . . .
Joan Torregrosa . . . . . . . . . . . . . . . . . . . . . . . . .
Antonio J. Ureña . . . . . . . . . . . . . . . . . . . . . . . . .
S10. Espacios de aplicaciones y grupos de autoequivalencias
Urtzi Buijs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Federico Cantero . . . . . . . . . . . . . . . . . . . . . . . .
Ramón Flores . . . . . . . . . . . . . . . . . . . . . . . . . .
Juan Gonzalez-Meneses . . . . . . . . . . . . . . . . . . . .
Milagros Izquierdo . . . . . . . . . . . . . . . . . . . . . . .
Luis Paris . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Pitsch . . . . . . . . . . . . . . . . . . . . . . . . .
Sebastián Reyes-Carocca . . . . . . . . . . . . . . . . . . .
S11. Geometría algebraica . . . . . . . . . . . . . . . . . . . . .
Maria Alberich Carramiñana . . . . . . . . . . . . . . . . .
Leovigildo Alonso Tarrío . . . . . . . . . . . . . . . . . . . .
José Ignacio Burgos Gil . . . . . . . . . . . . . . . . . . . . .
Alberto Castaño Domínguez . . . . . . . . . . . . . . . . .
Guillermo Cortiñas . . . . . . . . . . . . . . . . . . . . . . .
Carlos D’Andrea . . . . . . . . . . . . . . . . . . . . . . . . .
José Ignacio Farrán Martín . . . . . . . . . . . . . . . . . . .
Ana Jeremías López . . . . . . . . . . . . . . . . . . . . . . .
José María Muñoz Porras . . . . . . . . . . . . . . . . . . . .
Fernando Pablos Romo . . . . . . . . . . . . . . . . . . . .
Orlando Villamayor . . . . . . . . . . . . . . . . . . . . . . .
S12. Geometría convexa e integral . . . . . . . . . . . . . . . .
David Alonso-Gutiérrez . . . . . . . . . . . . . . . . . . . .
Jesús Bastero . . . . . . . . . . . . . . . . . . . . . . . . . . .
Andreas Bernig . . . . . . . . . . . . . . . . . . . . . . . . .
Antonio Cañete . . . . . . . . . . . . . . . . . . . . . . . . .
Bernardo González Merino . . . . . . . . . . . . . . . . . .
Martin Henk . . . . . . . . . . . . . . . . . . . . . . . . . . .
María A. Hernández Cifre . . . . . . . . . . . . . . . . . . .
Carlos Hugo Jiménez G. . . . . . . . . . . . . . . . . . . . .
Monika Ludwig . . . . . . . . . . . . . . . . . . . . . . . . .
Antonio R. Martínez Fernández . . . . . . . . . . . . . . . .
José Pedro Moreno . . . . . . . . . . . . . . . . . . . . . . .
Manuel Ritoré . . . . . . . . . . . . . . . . . . . . . . . . . .
Camilo Sarmiento . . . . . . . . . . . . . . . . . . . . . . . .
Gil Solanes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Efstratios Vernadakis . . . . . . . . . . . . . . . . . . . . . .
Rafael Villa . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jesús Yepes Nicolás . . . . . . . . . . . . . . . . . . . . . . .
S13. Geometría diferencial y aplicaciones . . . . . . . . . . . .
Alfonso Carriazo . . . . . . . . . . . . . . . . . . . . . . . . .
Marco Castrillón López . . . . . . . . . . . . . . . . . . . .
Miguel Domínguez Vázquez . . . . . . . . . . . . . . . . .
Manuel Fernández-López . . . . . . . . . . . . . . . . . . .
Angel Ferrández . . . . . . . . . . . . . . . . . . . . . . . . .
Miguel Angel Javaloyes . . . . . . . . . . . . . . . . . . . . .
Marc Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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70
71
71
73
73
73
74
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75
75
75
77
77
77
78
78
79
79
79
80
80
81
81
82
82
82
82
83
83
84
84
85
85
86
86
87
87
87
88
88
89
90
90
90
91
91
91
92
93
Índice de conferencias y pósteres
Marco Rigoli . . . . . . . . . . . . . . . . . . . . . .
Alfonso Romero . . . . . . . . . . . . . . . . . . . .
José Ignacio Royo Prieto . . . . . . . . . . . . . . .
Luis Ugarte . . . . . . . . . . . . . . . . . . . . . . .
Cristina Vidal Castiñeira . . . . . . . . . . . . . . .
S14. Investigación operativa . . . . . . . . . . . . . . .
Víctor Blanco . . . . . . . . . . . . . . . . . . . . .
F. Javier Martin Campo . . . . . . . . . . . . . . . .
Federico Perea . . . . . . . . . . . . . . . . . . . . .
Diego Ponce . . . . . . . . . . . . . . . . . . . . . .
Miguel A. Pozo . . . . . . . . . . . . . . . . . . . . .
J. Tinguaro Rodríguez . . . . . . . . . . . . . . . .
Gregorio Tirado . . . . . . . . . . . . . . . . . . . .
S15. Matemática discreta . . . . . . . . . . . . . . . . .
Miguel Ángel Fiol . . . . . . . . . . . . . . . . . . .
Pedro A. García-Sánchez . . . . . . . . . . . . . . .
Delia Garijo . . . . . . . . . . . . . . . . . . . . . .
Justo Puerto . . . . . . . . . . . . . . . . . . . . . .
Eugenia Saorín Gómez . . . . . . . . . . . . . . . .
Oriol Serra . . . . . . . . . . . . . . . . . . . . . . .
Lluis Vena . . . . . . . . . . . . . . . . . . . . . . .
S16. Matemáticas de la teoría de la información . . .
María Bras-Amoros . . . . . . . . . . . . . . . . . .
Pino Caballero-Gil . . . . . . . . . . . . . . . . . .
Sara D. Cardell . . . . . . . . . . . . . . . . . . . . .
Joan-Josep Climent . . . . . . . . . . . . . . . . . .
Cristina Fernández-Córdoba . . . . . . . . . . . .
Jaime Gutierrez . . . . . . . . . . . . . . . . . . . .
Fernando Hernando . . . . . . . . . . . . . . . . .
F. J. Lobillo . . . . . . . . . . . . . . . . . . . . . . .
Irene Márquez Corbella . . . . . . . . . . . . . . .
Juan Jacobo Simón Pinero . . . . . . . . . . . . . .
Adriana Suárez Corona . . . . . . . . . . . . . . . .
Magda Valls . . . . . . . . . . . . . . . . . . . . . .
S17. Métodos categóricos en álgebra no conmutativa
Pere Ara . . . . . . . . . . . . . . . . . . . . . . . . .
Alessandro Ardizzoni . . . . . . . . . . . . . . . . .
V. V. Bavula . . . . . . . . . . . . . . . . . . . . . . .
Gabriella Böhm . . . . . . . . . . . . . . . . . . . .
Manuel Cortés Izurdiaga . . . . . . . . . . . . . . .
Juan Cuadra . . . . . . . . . . . . . . . . . . . . . .
Laiachi El Kaoutit . . . . . . . . . . . . . . . . . . .
Sergio Estrada . . . . . . . . . . . . . . . . . . . . .
Alberto Facchini . . . . . . . . . . . . . . . . . . . .
Xabier Garcia Martinez . . . . . . . . . . . . . . . .
Juan Ramón García Rozas . . . . . . . . . . . . . .
Tatiana Gateva-Ivanova . . . . . . . . . . . . . . .
Ramón González Rodríguez . . . . . . . . . . . . .
José Gómez-Torrecillas . . . . . . . . . . . . . . . .
Ivo Herzog . . . . . . . . . . . . . . . . . . . . . . .
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Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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112
113
113
113
113
114
114
115
115
116
116
117
118
118
119
119
120
Índice de conferencias y pósteres
7
Sergio R. López-Permouth . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Juan Antonio Lopez Ramos . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Manuel Saorín . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Feroz Siddique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Mercedes Siles Molina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Kornel Szlachanyi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Jan Trlifaj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Thomas Weigel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
S18. Modelización y predicción estocásticas . . . . . . . . . . . . . . . . . . 124
M. Carmen Aguilera-Morillo . . . . . . . . . . . . . . . . . . . . . . . . . 124
Antonio Jesús Barrera García . . . . . . . . . . . . . . . . . . . . . . . . . 124
Paula R. Bouzas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Victor Manuel Casero Alonso . . . . . . . . . . . . . . . . . . . . . . . . . 125
Manuel Febrero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Rosa María Fernández-Alcalá . . . . . . . . . . . . . . . . . . . . . . . . . 126
Rosa E. Lillo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Patricia Román Román . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Juan Eloy Ruiz-Castro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Javier Álvarez Liébana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
S19. Singularidades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Enrique Artal Bartolo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Antonio Campillo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Roi Docampo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Lorenzo Fantini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Carlos Galindo Pastor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Eugeny Gorsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Ignacio Luengo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Pedro Manchón . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Jorge Martín Morales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Irene Márquez Corbella . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Juan José Nuño Ballesteros . . . . . . . . . . . . . . . . . . . . . . . . . . 133
S20. Soluciones matemáticas e innovación en la industria . . . . . . . . . . 134
Maria Aguareles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Antonio Alonso Ayuso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Joaquim Bruna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Bartomeu Coll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Carmen-Ana Domínguez-Bravo . . . . . . . . . . . . . . . . . . . . . . . 136
Laureano Escudero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Adrián Galdrán . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Luca Gerardo-Giorda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
José Manuel González Vida . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Carlos Gorria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Gustavo Montero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Tim Myers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Carlos Parés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Lakhdar Remaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
S21. Teoría de aproximación y funciones especiales de la física matemática141
Renato Álvarez-Nodarse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Ruymán Cruz Barroso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Jaime Díaz de Bustamante . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
Índice de conferencias y pósteres
Manuel Mañas Baena . . . . . .
Sergio Medina Peralta . . . . . .
Luz Roncal Gómez . . . . . . . .
Vanesa Sánchez Canales . . . . .
Jesus Sánchez-Dehesa . . . . . .
Germán Sierra Rodero . . . . . .
Miguel Tierz Parra . . . . . . . . .
S22. Teoría de números . . . . . . . .
Sara Arias de Reyna Domínguez
Pedro A. García-Sánchez . . . . .
Enrique González Jiménez . . . .
Josep González-Rovira . . . . . .
Ma Ángeles Gómez Molleda . . .
Alvaro Lozano-Robledo . . . . .
Juan Carlos Peral Alonso . . . . .
Antonio Rojas León . . . . . . . .
Adrián Ubis Martínez . . . . . . .
Sesión de pósteres . . . . . . . . . . .
Antonio Bueno . . . . . . . . . .
Damián Castaño . . . . . . . . .
Óscar Ciaurri . . . . . . . . . . . .
Josep Domingo-Ferrer . . . . . .
Cristina Draper . . . . . . . . . .
Claudia Gallego . . . . . . . . . .
Juan Carlos García Ardila . . . . .
Jesús A. Laliena . . . . . . . . . .
Rafael López . . . . . . . . . . . .
José M. Manzano . . . . . . . . .
Misael E. Marriaga . . . . . . . .
Alexis Molino . . . . . . . . . . .
Juan José Moreno Balcázar . . .
M. Luisa Márquez García . . . .
M. Luisa Márquez García . . . .
Juan Núñez . . . . . . . . . . . . .
Juan Núñez . . . . . . . . . . . . .
Irene Ortiz . . . . . . . . . . . . .
Walter Andrés Ortiz Vargas . . .
Jose Manuel Rodríguez . . . . . .
Desirée Romero . . . . . . . . . .
Josu Sangroniz . . . . . . . . . . .
Changhwa Woo . . . . . . . . . .
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Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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142
142
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Conferencias plenarias
http://rsme2015.ugr.es/index.php?section=programa
Some recent results on fluid interface dynamics
Ángel Castro
Universidad Autónoma de Madrid
In this talk we will present some results concerning the evolution of an interface between two incompressible fluids with diferent characteristics. In particular
we will focus on the existence of singularities, global regular solutions and the structure of weak solutions. Different mediums will be considered corresponding with
different physical situations.
Minimal surfaces: variational aspects and applications
Fernando Codá Marques
Princeton University
[email protected]
Minimal surfaces are among the most natural objects in Differential Geometry,
and are fundamental tools in the solution of several important problems in mathematics. In this lecture we will describe the variational theory of minimal surfaces
and discuss recent applications to geometry and topology, as well as mention some
future directions in the field.
Cambio de base para GL(2) y otros casos de funtorialidad de Langlands
Luis V. Dieulefait
Universidad de Barcelona
[email protected]
Dentro del programa de Langlands, ocupan un papel importante las conjeturas
de funtorialidad, y de entre ellas, cuestiones como la siguiente: dadas formas modulares o automorfas con representaciones de Galois asociadas, ¿que operaciones
“elementales” a nivel de representaciones de Grupos (efectuadas sobre estas representaciones de Galois) tienen su contrapartida en el mundo modular/automorfo?
En esta charla nos centraremos en el caso de “cambio de base” (es decir, restricción
de la representación de Galois a un subgrupo de índice finito) para GL(2). En este
caso, he resuelto la conjetura por completo para el caso de formas modulares clásicas (y subgrupos correspondientes a cuerpos totalmente reales) y esto implica que a
una forma modular clásica le podemos asociar, sobre cualquier cuerpo F totalmente
real, una forma de Hilbert que es su “levantamiento”. Otros ejemplos que discutiremos brevemente son el caso de potencias simétricas y de productos tensoriales.
Las técnicas utilizadas en este tipo de resultados incluyen teoremas de “levantamiento automorfo” al estilo de Taylor-Wiles y técnicas de propagación.
10
Conferencias plenarias
11
Inverse Problems: A meeting point for quasiconformal mappings, oscillatory integrals and conformally invariant tensors
Daniel Faraco Hurtado
Universidad Autónoma de Madrid / ICMAT
[email protected]
I will give an overview of the classical Calderón Problem arising in impedance tomography as well as in the classical quantum scattering theory. Calderón problem
ask for the determination of the coefficient of a given P.D.E from the knowledge of its
solutions at the boundary of your domain. In particular I will focus on three results.
Quasiconformal machinary can be used to proved the stability of the planar problem with minimal regularity, Carleson problem on the convergence to initial data of
time dependent equation can be adapted to yield an optimal reconstruction procedure por planar quantum potentials and a delicated analysis of conformally invariant tensors allows to decide in which geometries (physically which anisotropies) the
problem can be solved.
Una metodología para el Análisis Estadístico de datos difusos
María Angeles Gil Alvarez
Universidad de Oviedo
[email protected]
El análisis estadístico de datos supone habitualmente que tales datos pueden
expresarse de forma exacta en escala numérica. Sin embargo, muchos datos del
mundo real son intrínsecamente imprecisos, especialmente cuando provienen de
apreciaciones o percepciones humanas. Los valores difusos suministran un modelo
que se ha empleado exhaustivamente para expresar dichos datos.
Al tratar de analizar los datos difusos desde una perspectiva estadística, surgen
dos obstáculos importantes:
- la falta de linealidad asociada a la consideración de la aritmética usual entre
valores difusos (que coincide nivel-a-nivel con la aritmética de conjuntos);
- la falta de modelos y de resultados límite para las distribuciones de los estadísticos basados en muestras de datos difusos.
Estos inconvenientes, pueden soslayarse en buena parte mediante el uso de
métricas adecuadas, el concepto de conjunto difuso aleatorio y algunos resultados
conocidos para elementos aleatorios con valores en espacios de Hilbert. Sobre la
base de estos se está desarrollando una metodología para el análisis de datos difusos, que va a exponerse sucintamente.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
Conferencias plenarias
12
Oscilaciones periódicas de un péndulo forzado: de la existencia a la estabilidad
Rafael Ortega
Universidad de Granada
[email protected]
Se considera la ecuación diferencial
x 00 + β sin x = f (t )
donde f (t ) es una función periódica. Se trata de un modelo simple que se usa con
frecuencia para ilustrar diversos métodos globales en Análisis No Lineal. La existencia de soluciones periódicas se suele obtener combinando herramientas que
provienen de la Topología y del Cálculo de Variaciones. El objetivo de esta charla es
mostrar que esas herramientas también son útiles en el estudio de la estabilidad de
las soluciones periódicas. La estabilidad se entiende en el sentido de Lyapunov.
Los resultados serán del tipo "Para casi toda función f (t ) que cumple. . . , hay
solución estable", por eso será importante la noción de conjunto de medida cero en
espacios de dimensión infinita.
Dimensionally adaptive methods for the simulation and inversión of electromagnetic geophysical measurements
David Pardo
Universidad del País Vasco UPV/EHU
[email protected]
Coautores: Shaaban Bakr, Carlos Torres-Verdín
A number of three dimensional (3D) simulators of geophysical logging measurements have been developed during the last two decades for oil-industry applications. These simulators have been successfully used to study and quantify different
physical effects occurring in 3D geometries. Despite such recent advances, there
are still many 3D effects for which reliable simulations are not available. Furthermore, in most of the existing results, only partial validations have been reported,
typically obtained by comparing solutions of simplified model problems against the
corresponding solutions calculated with a lower dimensional (2D or 1D) numerical method. The lack of 3D simulation results (as opposed to 2D results) is due
to major difficulties encountered when solving geometrically challenging problems.
Namely, for mesh-based methods (Finite Elements, Finite Differences, Boundary Elements, etc.), the size of the system of linear equations becomes excessively large to
be solved in real time.
When solving inverse geophysical problems (as opposed to forward simulation
problems), the cost of computations dramatically increase, making the use of 3D
simulators impractical. Often, even 2D simulators cannot be afforded due to their
elevated computational cost.
In this presentation, we first explain how oil companies record different types
of electromagnetic geophysical measurements. Then, we explain the main mathematical and computational difficulties associated to the simulation and inversion of
such measurements. Subsequently, we analyze a number of mathematical features
that a numerical method should possess in order to overcome the above challenges.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
Conferencias plenarias
13
Finally, we present a family of dimensionally adaptive methods that we are employing for solving such simulation and inversion problems.
One of the main objectives of this presentation is to raise the awareness and interest of the applied mathematics community on the topic, since its expertise is necessary in order to solve several mathematical problems that still remain open in the
area.
Additional info: http://sites.google.com/site/m2sigroup
Arcs spaces, birational geometry and singularity theory
María Pe Pereira
ICMAT
[email protected]
Given a singularity germ (X ,O), an arc is simply a curve parametrization that
passes through the origin in time 0. The space of arcs was introduced by J. Nash in
the 60Šs to understand the structure of the singularity in relation with their resolutions or more generally its birational geometry. He conjectured a precise relation in
the case of surfaces that was proved by myself and J. Fernandez de Bobadilla in 2011
and a more relax statement for the higher dimensional case. After counterexamples
in dimension greater than 3 by Ishii and Kollar in 2005 and in dimension 3 by de
Fernex in 2012, some precise positive answer was given in relation with the terminal
model with the singularity by de Fernex and Docampo in 2014.
Moreover arc spaces are the base for motivic integration (Kontsevich, Denef and
Loeser) and are a good tool to compute birational invariants (Mustata, Ein, Lazarsfeld , Ishii , de Fernez and others).
In this talk I will introduce arcs spaces, Nash problem and how they interact with
birational geometry and other classical problems in singularity theory.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
http://rsme2015.ugr.es/s01.php
A Framework for Calderon-Zygmund Singular Integral Operators on Spaces of
Homogeneous Type
Theresa C Anderson
Brown University
[email protected]
Coautores: Armen Vagharshakyan, Wendolin Damian
Weighted norm inequalities for singular integrals have received much attention
in recent years. Via local mean oscillation, Andrei Lerner was able to bound CalderonZygmund operators in norm by positive dyadic operators. I will focus on how the innovative techniques of Lerner, that I have worked to extend, have led to new results
in the area of sharp weighted norm inequalities for Calderon-Zygmund operators
and will also introduce how a reverse Holder extrapolation technique is useful in
proving two-weighted bounds.
Modulation spaces, Wiener randomization and PDEs
Arpad Benyi
Western Washington University, USA
[email protected]
Coautores: Tadahiro Oh (University of Edinburgh), Oana Pocovnicu (Princeton University)
We provide a brief introduction to the time-frequency analysis surrounding the
so-called modulation spaces and indicate how the intrinsically related Wiener randomization plays a special role in the well-posedness of non-linear PDEs.
Riesz transform and Sobolev Algebra through the heat semigroup
Frédéric Bernicot
CNRS - Université de Nantes
[email protected]
Coautores: Thierry Coulhon and Dorothee Frey
On a doubling Riemannian manifold, we are interested in Sobolev spaces, given
by Laplace operator. We give several new results about the L p -boundedness of the
Riesz transform, as well as for the algebra property (under the pointwise product)
for such Sobolev spaces. Our results aim to get around the use of any Poincaré inequality and improve some previous results. The Leibniz-type inequalities rely on
abstract paraproducts defined via functional calculus.
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On the two weighted boundedness of the dyadic paraproduct operator.
Oleksandra Beznosova
University of Alabama (Tuscaloosa)
[email protected]
Coautores: M.C. Pereyra, D. Chung, J.C. Moraes
We obtain sufficient conditions for the boundedness of the dyadic paraproduct
operator from L 2 (u) into L 2 (v). We also find several necessary conditions for such
boundedness and discuss connections with the bumped A 2 conjecture.
A method to arrive to weak type (1,1) estimates
Maria J. Carro
Universidad de Barcelona
[email protected]
Coautores: Carlos Domingo, Loukas Grafakos and Javier Soria
We shall present a variant of Rubio de FranciaŠs extrapolation theory that allows
us to obtain the weak type (1,1) boundedness of several operators in Harmonic Analysis such as the Bochner-Riesz operator at the critical index among many others.
Commutators of bilinear fractional integrals
Lucas Chaffee
University of Kansas
[email protected]
In this talk we will briefly discuss the history of commutators of singular integral
operators with point-wise multiplication, and we will conclude by showing new results characterizing BMO and CMO in terms of boundedness and compactness of
commutators with the bilinear fractional integral operator.
L(p) and Weak-L(p) estimates for the number of integer points in translated domains.
Leonardo Colzani
Università Milano Bicocca
[email protected]
Coautores: Luca Brandolini, Giacomo Gigante, Giancarlo Travaglini
¡ ¢
Revisiting and
extending
a recent result of M.Huxley, we estimate the L p Td
¡
¢
and W eak −L p Td norms of the discrepancy between the volume and the number
of integer points in translated domains.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
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Extrapolation in weighed variable Lebesgue spaces
David Cruz-Uribe
Trinity College
[email protected]
Coautores: Daniel Wang
The weighted variable Lebesgue spaces are a generalization of the classical weighted
L p spaces. Given an exponent function p(·) : Rn → [1, ∞], we define the variable A p(·)
class to be all weights such that
sup kwχQ kp(·) kw −1 χQ kp 0 (·) < ∞,
Q
where the supremum is taken over all cubes (or equivalently, all balls).
Earlier it was proved that if p(·) is log-Hölder continuous and 1 < p − ≤ p + <
∞, and if w ∈ A p(·) , then the Hardy-Littlewood maximal operator is bounded on
L p(·) (w).
Using this fact, we extend the theory of Rubio de Francia extrapolation to these
spaces. We prove weighted L p(·) estimates, off-diagonal estimates and limited range
extrapolation. These last results are new in the variable exponent setting even without weights.
As an application we show that a variety of classical operators from harmonic
analysis are bounded on the weighted variable Lebesgue spaces.
Recent advances on multilinear sharp weighted inequalities
W. Damián
Universidad de Sevilla
[email protected]
Coautores: Andrei K. Lerner (Bar-Ilan University) and Carlos Pérez (Universidad de
Sevilla)
In 2010, Tuomas Hytönen proved in full generality the well-known A 2 conjecture, which claimed that the sharp dependence of the L p (w) norm of a Calderón–
Zygmund operator T on the A 2 constant of the weight w might be linear. Shortly after, Andrei Lerner gave a alternative, simpler and beautiful proof based on the local
mean oscillation formula and the use of sparse operators. The versatility of Lerner’s
techniques are reflected in the development of the extensions of those linear results
to the multilinear scenario. In this talk we present a survey on the recent advances
in multilinear sharp weighted inequalities, as well as a glimpse of some interesting
proofs and open problems in the area.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
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Adams-Moser-Trudinger inequalities on Rn
Luigi Fontana
Universita’ di Milano-Bicocca, Milano, Italy
[email protected]
Coautores: Carlo Morpurgo University of Missouri, Columbia MO, USA
k,p
By 1988, the basic facts about the critical case of Sobolev imbeddings for W0 (Ω)
with Ω bounded, were essentially established. Unbounded domains and in particular R n itself pose several new problems that began to be investigated in the new century. Many papers on various aspects and special cases appeared. We will present
some new results from joint work with Carlo Morpurgo that settle the matter in a
fairly complete way
Hardy Space Estimates for Linear and Bilinear Calderón-Zygmund Operator
Jarod Hart
Wayne State University
[email protected]
Coautores: Guozhen Lu
In this joint work with Guozhen Lu, we find sufficient conditions for non-convolution
type linear and bilinear Calderón-Zygmund operators to be bounded on Hardy spaces.
For a linear operator T f , we show T is bounded on H p for 0 < p ≤ 1 under appropriate regularity and cancellation assumptions for T . Likewise for a bilinear operator T ( f 1 , f 2 ), we give sufficient regularity and cancellation conditions for T to be
bounded from H p 1 × H p 2 into H p for 0 < p 1 , p 2 , p ≤ 1. We formulate an approach
that uses Littlewood-Paley-Stein theory, without any direct use of atomic/molecular
characterizations of Hardy spaces or proof techniques involving atom to molecule
mapping properties. Furthermore, this approach naturally yields a pointwise estimate that is useful for analysis of singular integrals acting on distribution spaces.
The linear results are applied to prove that the Bony paraproduct, which was notably used by David and Journé in the proof of their T 1 theorem, is bounded on
H p for any 0 < p ≤ 1. The fundamental difficulty that arises in the bilinear Hardy
spaces estimates, that is not present in the linear setting, stems from the fact that
f 1 , f 2 ∈ H 1 does not imply f 1 · f 2 ∈ H 1/2 , i.e. the pointwise product operator ( f 1 , f 2 ) 7→
f 1 (x) f 2 (x) is not bounded from H 1 × H 1 into H 1/2 . The product structure of bilinear Calderón-Zygmund operators severely complicates analysis of operators on H p
when 0 < p ≤ 1, which stems from difficulties in understanding the oscillatory behavior of products of functions. Some Hardy space paraproduct boundedness properties for bilinear operators will also be discussed. In particular, one paraproduct
Π( f 1 , f 2 ) maps (and is bounded) from H p 1 × H p 2 into H p and resembles the product
operator, Π( f 1 , f 2 )(x) ≈ f 1 (x) f 2 (x), in the appropriate sense.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
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Parabolic weighted norm inequalities
Juha Kinnunen
Aalto University
[email protected]
Coautores: Olli Saari (Aalto University)
We discuss parabolic Muckenhoupt weights and functions of bounded mean oscillation (BMO) related to a doubly nonlinear parabolic partial differential equation. In the natural geometry for the doubly nonlinear equation the time variable
scales as the modulus of the space variable raised to a power. Consequently the
Euclidean balls and cubes have to be replaced by parabolic rectangles respecting
this scaling in all estimates. An extra challenge is given by the time lag appearing
in the estimates. The main result gives a full characterization of weak and strong
type weighted norm inequalities for parabolic forward in time maximal operators.
In addition, we give a Jones type factorization result for the parabolic Muckenhoupt
weights and a Coifman-Rochberg type characterization of the parabolic BMO through
parabolic Muckenhoupt weights and maximal functions. We also discuss connections and applications of the results to regularity of nonlinear parabolic partial differential equations. This is a joint work with Olli Saari at Aalto University.
Variational inequalities related to the Monge-Ampere equation
Diego Maldonado
Kansas State University
[email protected]
We will start with a description of geometric and measure-theoretic objects associated to certain convex functions in R n . These objects include a quasi-distance
and a Borel measure in R n which render a space of homogeneous type (i.e. a doubling quasi-metric space) associated to such convex functions. We will illustrate how
real-analysis techniques in this quasi-metric space can be applied to the regularity
theory of convex solutions u to the Monge-Ampere equation detD 2 u = f as well as
solutions v of the linearized Monge-Ampere equation L u (v) = g . Finally, we will
discuss recent developments regarding the existence of Sobolev and Poincare inequalities on these Monge-Ampere quasi-metric spaces and mention some of their
applications.
Beyond the Kato conjecture for degenerate elliptic operators
Jose Maria Martell
ICMAT
[email protected]
Coautores: David Cruz-Uribe, Cristian Rios
Consider the degenerate elliptic operator
¡
¢
L γ u(x) = −|x|γ div |x|−γ A(·) ∇u(·) (x),
x ∈ Rn ,
where A is a uniform elliptic matrix with complex bounded coefficients. In the nondegenerate or uniformly elliptic case (that is, when γ = 0) the solution to the Kato
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
19
conjecture by P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and Ph. Tchamitchian
leads to the L 2 -estimates
° 1/2 °
° °
°L f ° 2 ≈ °∇ f ° 2 .
0
L
L
When −n < γ < n, the “degeneracy” of L γ (i.e., w(x) = |x|−γ ) belongs to the class of
Muckenhout weights A 2 . Hence the Kato problem for degenerate operators studied
by C. Rios and D. Cruz-Uribe gives the L 2 (w)-estimates
° 1/2 °
° °
°L f ° 2 −γ ≈ °∇ f ° 2 −γ .
γ
L (|x| )
L (|x| )
In this talk we present a method that allows us to show that when 0 ≤ γ < 2 n/(n + 2)
the degenerate elliptic operator L γ satisfies L 2 -Kato estimates.
A counting problem in Ergodic Theory and Extrapolation for one-sided weights
Francisco J. Martín-Reyes
Universidad de Málaga
[email protected]
Coautores: María J. Carro (Universidad de Barcelona) and María Lorente (Universidad de Málaga)
The purpose of this talk is to present the following result: given a dynamical system (X , M , µ, τ) and 0 < q < 1, the Lorentz spaces L 1,q (µ) satisfy the so-called Return
Times Property for the Tail contrary to what happens in the case q = 1. In fact, we
consider a more general case than in previous results since we work with a σ-finite
measure µ and a transformation τ which is only Cesàro bounded. The proof uses
the extrapolation theory of Rubio de Francia for one-sided weights. These results
are of independent interest and can be applied to many other situations.
L p and weak-L 1 estimates for the variation of singular integrals on uniformly rectifiable measures
Albert Mas
Universitat Politècnica de Catalunya
[email protected]
Coautores: Xavier Tolsa
In this talk I will present a recent work in collaboration with Xavier Tolsa where
we extend some previous results concerning the variation for singular integrals with
odd kernel on Lipschitz graphs with small slope to the case of uniformly rectifiable
measures. Given a family of truncations of a singular integral operator over a meaµ
sure µ, namely T µ = {T² }²>0 , one defines the ρ-variation of T µ by
µ
Ã
(Vρ ◦ T ) f (x) := sup
X
{²m } m∈Z
µ
|T²m+1 f
µ
(x) − T²m
f (x)|
ρ
!1/ρ
,
where the supremum is taken over all decreasing sequences {²m }m∈Z ⊂ (0, ∞). We
will discuss the L p estimates for 1 < p < ∞ as well as the weak-L 1 endpoint case
when µ is uniformly rectifiable.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
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On the L ∞ ×L ∞ → B MO mapping property for certain bilinear pseudodifferential
operators
Virginia Naibo
Kansas State University
[email protected]
We will present boundedness results from L ∞ × L ∞ into B MO for bilinear pseudodifferential operators with symbols in a range of bilinear Hörmander classes of
critical order. These results are achieved by means of new continuity properties for
bilinear operators with symbols in certain classes and a new pointwise inequality
relating bilinear operators and maximal functions.
Weighted asymptotic estimates for maximal functions and embedding of Muckenhoupt weights
Ioannis Parissis
The University of the Basque Country and Ikerbasque
[email protected]
Coautores: Paul A. Hagelstein
I will discuss some asymptotic estimates for the level sets of the Hardy-Littlewood
maximal operator, applied to indicators of sets. We look at weights w such that we
have the asymptotic estimate
lim
sup
α→1− 0<w(E )<∞
w({x : M (χE )(x) > α})/w(E ) = 1.
It turns out that the property above characterizes the class of Muckenhoupt weights
A ∞ and this point of view provides us with a natural way to understand how A ∞
embeds into A p . These results extend to other differentiation bases such as the basis
of rectangles with sides parallel to the coordinate axes, giving us characterizations
of strong Muckenhoupt weights.
Analysis of Rumin’s complex on the Heisenberg group
Marco Peloso
Università degli Studi di Milano
[email protected]
Coautores: Marco Marchi
This work is concerned with the analysis of Rumin’s Laplacian ∆R on the Heisenberg group Hn . We obtain a decomposition of the space of Rumin’s forms with L 2 coefficients into invariant subspaces and describe the action of ∆R restricted to these
subspaces up to unitary equivalence. We prove that this decomposition provides a
L p decomposition of the space of Rumin’s forms.
We also prove a sharp Mihilin-Hörmander multiplier theorem for ∆R . Finally, we
discuss the L p -boundedness of the Riesz transforms defined by ∆R .
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
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On the regularity of certain bilinear oscillatory integral operators
Salvador Rodriguez-Lopez
Imperial College London
[email protected]
Coautores: D. Rule and W. Staubach
In this talk, we will consider the regularity properties of certain bilinear oscillatory integral operators and related Fourier integral operators on (among others)
products of Banach and quasi-Banach Lebesgue spaces.
More specifically, we will establish the boundedness of operators of the type
Z Z
T ( f , g )(x) =
a(x, ξ, η) fb(ξ)gb(η)e i Φ(x,ξ,η) dξdη
where the amplitude a belongs to a bilinear Hörmander-type class and the phase
function Φ(x, ξ, η) = φ1 (x, ξ)+φ2 (x, η), with φ j satisfying suitable regularity and nondegeneracy conditions. These operators appear, for instance, in the study of the
regularity of the product of two solutions of wave-type equations.
Our investigation also yields the bilinear version of the classical theorem of A.
Seeger, C. Sogge and E. Stein concerning the L p boundedness of linear Fourier integral operators.
This is joint work with D. Rule (Linköping University) and W. Staubach (Uppsala
University).
Cross-section estimates on Sobolev spaces
Javier Soria
Universidad de Barcelona
[email protected]
Coautores: Nadia Clavero, Viktor Kolyada
We find the optimal domain and range spaces for Sobolev embeddings in terms
of rearrangement invariant mixed norms (involving the cross-sections of the functions), which extend well-known estimates by Gagliardo, Nirenberg and Fournier.
The study of these results lead us to consider further properties of iterated rearrangements, for which we establish sharp conditions in the case of classical Lebesgue
spaces.
Referencias
1. N. Clavero and J. Soria, Mixed norm spaces and rearrangement invariant estimates, J. Math. Anal. Appl. 419 (2014), no. 2, 878–903.
2. N. Clavero and J. Soria, Optimal rearrangement invariant Sobolev embeddings
in mixed norm spaces, preprint.
3. V. Kolyada and J. Soria, Mixed norms and iterated rearrangements, preprint.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
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Connections between continuous and dyadic function spaces on product spaces
of homogeneous type
Lesley A. Ward
School of Information Technology and Mathematical Sciences, University of South
Australia
[email protected]
The function spaces of harmonic analysis, such as BMO, VMO, H 1 , and the classes
of A p weights and reverse-Hölder weights, come in both continuous and dyadic
flavours. We know of two types of connections between the continuous and dyadic
versions of such a space. First, averaging procedures take us from the dyadic to the
continuous version. Second, the continuous version can be written as an intersection (for BMO, VMO, A p and R H p ), or a sum (for H 1 ), of finitely many dyadic versions. We present recent work that extends these connections from the Euclidean
world to the setting of spaces of homogeneous type (X , d , µ) in the sense of Coifman
and Weiss, in both the one-parameter and product situations. Our results build on
earlier work by Garnett, P. Jones, Pipher, Ward, Treil, Xiao, and Li [2,3,5,6,7]. This is
joint work with P. Chen, A. Kairema, J. Li, and M.C. Pereyra [1,4].
Referencias
1. P. Chen, J. Li and L.A. Ward, BMO from dyadic BMO via expectations on product spaces of homogeneous type, J. Funct. Anal. 265 (2013), 2420–2451.
2. J.B. Garnett and P.W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982),
no. 2, 351–371.
3. J. Li, J. Pipher, and L.A. Ward, Dyadic structure theorems for multiparameter
function spaces, Rev. Mat. Iberoamericana, to appear.
4. A. Kairema, J. Li, M.C. Pereyra and L.A. Ward, Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product
spaces of homogeneous type, in preparation.
5. J. Pipher and L.A. Ward, BMO from dyadic BMO on the bidisc, J. London Math.
Soc. 77 (2008), 524–544.
6. J. Pipher, L.A. Ward and X. Xiao, Geometric-arithmetic averaging of dyadic
weights, Rev. Mat. Iberoamericana 27 (2011), no. 3, 953–976.
7. S. Treil, H 1 and dyadic H 1 , in Linear and Complex Analysis: Dedicated to V. P.
Havin on the Occasion of His 75th Birthday (ed. S. Kislyakov, A. Alexandrov,
A. Baranov), Advances in the Mathematical Sciences 226 (2009), AMS, 179–
194.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S01. Análisis armónico
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Weighted Carleson Measure Spaces Associated with Different Homogeneities
Xinfeng Wu
University of Kansas
[email protected]
In this paper, we introduce the weighted Carleson measure spaces associated
with different homogeneities and prove the boundedness of composition of two
Calderón-Zygmund operators with different homogeneities on the weighted Carleson measure spaces. We also identify the dual spaces of the weighted Hardy spaces
with the weighted Carleson measure spaces.
On multilinear fractional strong maximal operator associated with rectangles and
multiple weights
Qingying Xue
School of Mathematical Sciences, Beijing Normal University, Beijing, 100875 P.R. China
[email protected]
Coautores: M. Cao and K. Yabuta
In this talk, the multilinear fractional strong maximal operator MR,α associated with rectangles and corresponding multiple type weights are introduced. Under the dyadic reverse doubling condition, a necessary and sufficient condition for
two-weight inequalities is given. As consequences, we first obtain a necessary and
sufficient condition for one-weight inequalities. Then, we give a new proof for the
weighted estimates of multilinear fractional maximal operator Mα associated with
cubes and multilinear fractional integral operator Iα , which is quite different and
simple from the proof known before.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S02. Análisis complejo y teoría de operadores
http://rsme2015.ugr.es/s02.php
Residual subspaces and spectral synthesis for differentiation on C ∞
Alexandru Aleman
Lund University
[email protected]
If I is an interval on the real line, the differentiation operator ddt restricted to its
invariant subspaces of C ∞ (I ) shows a fairly intricated behavior. The spectrum of
such a restriction, may be void, or equal to the whole complex plane, or consist of a
countable set of eigenvalues. Arbitrary invariant subspaces may contain a nontrivial
”residual part” where the spectrum of the restriction is void. The talk is focused on
two topics:
1) The structure of residual subspaces,
2) The appropriate spectral synthesis in invariant subspaces with a countable spectrum.
The material is based on earlier joint work with B. Korenblum and recent results with
A. Baranov and Y. Belov.
Averaging operators, Berezin transforms and atomic decomposition on weighted
Bergman spaces
Oscar Blasco
Universidad de Valencia
[email protected]
Coautores: Salvador Perez-Esteva
We study the class of weight
functions W in the unit disk for which the averaging
R
1
p
operators Ar φ(z) = |D(z,r
)| D(z,r ) φ(w)d A(w) are bounded on L (W ), where D(z, r )
is the disk centered at z and radius r in the hyperbolic metric. We also show the
atomic decompositions on the weighted Bergman spaces A p (W ) or Bergman-Herz
p
spaces A q (W ) for weights in the above class for which the Bergman projection is
p
continuous in L p (W ) or in the Herz spaces K q (W ).
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S02. Análisis complejo y teoría de operadores
25
Bergman projection and Bloch functions on symmetric domains of tube type
Aline Bonami
University of Orleans, France
[email protected]
I will mainly consider the tubes over the forward light cones and their bounded
realizations, the Lie balls. In both cases, one now knows the whole range of p for
which the Bergman projection extends into a bounded operator on L p , due to recent work of Bourgain and Demeter. It also makes sense for the Lie ball to ask for
estimates with loss, from L p to L q , with q < p and I will present some necessary and
some sufficient conditions. This translates in particular into integrability properties
for Bloch functions of the Lie balls, local integrability properties in the case of the
tube domains.
I will emphasize the differences with the same problems on the disc or the unit
ball.
This is a joint work with Gustavo Garrigós and Cyrille Nana.
Volterra operators on weighted Banach spaces of entire functions
Jose Bonet
Universitat Politecnica de Valencia
[email protected]
Coautores: Jari Taskinen (Helsinki, Finland)
We characterize boundedness, compactness and weak compactness of Volterra
operators Vg acting between different weighted Banach spaces H v∞ (C) and H v0 (C) of
entire functions with sup-norms in terms of the symbol g ; thus we complement recent work by Bassallote, Contreras, Hernández-Mancera, Martín and Paul for spaces
of holomorphic functions on the disc and by Constantin and Peláez for reflexive
weighted Fock spaces.
Holomorphic potentials and multipliers for Hardy-Sobolev spaces
Carme Cascante
Universitat de Barcelona
[email protected]
Coautores: Joan Fabrega and Joaquin M. Ortega
Our focus of interest comes out from the following fact in Rn : for a nonlinear
potential of a positive measure, it is enough to impose its boundedness to assure
that the potential is a pointwise multiplier of the Bessel space L s,p . We will check,
using different methods, an analogous result for non isotropic holomorphic potentials on the unit ball in Cn , B, showing that the bounded holomorphic potentials are
pointwise multipliers for the Hardy-Sobolev spaces. As a consequence, we construct
nontrivial examples of such multipliers and we give some applications.
p
We recall that if 1 ≤ p < ∞ and s ∈ R, the Hardy-Sobolev
space H s consists of the
X
holomorphic functions f on B such that if f =
f k is its homogeneous polynomial
k
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S02. Análisis complejo y teoría de operadores
expansion, and the fractional radial derivative is defined by (1 + R)s f :=
26
X
(1 + k)s f k ,
k
then
|| f ||H p := ||(1 + R)s f ||H p < ∞.
s
Referencias
1. Böe, B.: Construction of multipliers for Bessel potential spaces, unpublished.
2. Cascante, C.; Ortega, J.M.: Carleson Measures for weighted Hardy-Sobolev
spaces. Nagoya Math. J., 186, (2007), 29-68.
3. W.S. Cohn, I.E. Verbitsky.: Non-linear potential theory on the ball, with applications to exceptional and boundary interpolation sets, Michigan Math. J., 42,
(1995), 79-97.
4. Ortega, J.M.; Fabrega, J.: Multipliers in Hardy-Sobolev spaces. Int. Eq. Op. Th.,
55, (2006), 535-560.
Slopes in the theory of analytic semigroups
Santiago Díaz Madrigal
Universidad de Sevilla
[email protected]
Coautores: Manuel D. Contreras Márquez, Pavel Gumenyuk
Given a semigroup (ϕt ) of analytic self-maps of the unit disc D and fixed a point
z ∈ D, the function t ∈ [0, +∞) → ϕt (z) ∈ D can be seen as the trajectory of a certain
vector field. Indeed, most of the times these trajectories land at concrete points in
the circle. Dynamically thinking, this suggests the question of when these landings
hold with a definite slope.
In this paper, we give a panoramic view of this problem paying special attention
to several very recent developments which tell us that some kind of wild behaviour
is possible.
Quasiconformal mappings and the corona problem
María José González
Universidad de Cádiz
[email protected]
Coautores: José M. Enríquez Salamanca
We study the corona theorem for domains whose boundary lies in a smooth
curve. The idea is to transfer the problem via a quasiconformal mapping to a Denjoy
domain and use the results from Garnett and Jones. For that we will use a characterization of this type of curves that involves a Carleson condition on the dilatation
coefficient of the quasiconformal mapping.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S02. Análisis complejo y teoría de operadores
27
Weighted integrability of Polyharmonic functions
Håkan Hedenmalm
Royal Institute of Technology, Stockholm
[email protected]
In joint work with A. Borichev, we study the boundary behavior of polyharmonic
functions in the unit disk. We find that there exist functions that decay maximally
quickly and we determine the rate - in terms of p-means along concentric circles.
The results are expressed in terms of standard weighted area-L p spaces on the disk.
We then continue to analyze similar flatness of polyharmonic functions on other
domains.
Bandlimited Lipschitz functions
Yurii Lyubarskii
Norwegian University of Scinces and Technology
[email protected]
Coautores: Joaquim Ortega-Cerdà
We study the space of bandlimited Lipschitz functions in one variable. In particular we provide a geometrical description of interpolation and sampling sequences
in this space. We also give a description of trace of such functions to sequences of
critical density in terms of a cancellation condition.
Fekete points in complex manifolds
Joaquim Ortega-Cerdà
University of Barcelona
[email protected]
Coautores: Nir Lev, Bar-Ilan University.
I will present an overview of the distribution of Fekete points in several contexts.
These are points that minimize a certain energy and that appear naturally in problems ranging from electrostatics to approximation theory.
We will overview the results in different contexts, from the clasical weighted
Fekete points in the plane to more elaborated versions in complex Riemannian manifolds. We will explore how the conection to other sets of points can be enlightening
in their study and will provide some applications.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S02. Análisis complejo y teoría de operadores
28
Isoperimetric inequalities and quasi-isometries in Riemann surfaces
Jose Manuel Rodriguez
Universidad Carlos III de Madrid
[email protected]
Coautores: Alicia Cantón, Ana Granados, Ana Portilla
This work studies the stability of isoperimetric inequalities under quasi-isometries
between non-exceptional Riemann surfaces endowed with their Poincaré metrics.
This stability was proved by Kanai in the more general setting of Riemannian manifolds under the condition of positive injectivity radius. In the present work we prove
the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without any condition on their injectivity radii. It is also shown the stability of
any non-linear isoperimetric inequality.
The Schramm-Loewner equation for multiple slits
Oliver Roth
University of Würzburg
[email protected]
Coautores: Sebastian Schleissinger
We prove that any disjoint union of finitely many simple curves in the upper
half-plane can be generated in a unique way by the chordal multiple-slit Loewner
equation with constant weights.
Hilbert matrix as an operator on spaces of analytic functions
Dragan Vukoti´c
Universidad Autónoma de Madrid
[email protected]
Coautores: Ole F. Brevig, Karl-Mikael Perfekt, Kristian Seip, Aristomenis Siskakis
The Hilbert matrix is a prototype of a Hankel operator. Its norm and spectrum
have been studied on the sequence spaces `p . It can also be defined in a natural
way on Hardy and Bergman spaces of analytic functions in the unit disk. We briefly
review various earlier results in this setting.
The Hardy space of Dirichlet series has been studied extensively by a number
of authors in the recent years. Continuing a line of research started by Helson in his
last two papers, we produce a multiplicative version of the Hilbert matrix on a Hardy
space of Dirichlet series. This operator relates in a natural way to the Riemann ζfunction and turns out to have several properties analogous to those of the classical
Hilbert matrix.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S03. Análisis funcional
http://rsme2015.ugr.es/s03.php
Distributional chaos for the Forward and Backward Control traffic model
Xavier Barrachina Civera
Università degli Studi di Roma Tor Vergata
[email protected]
Coautores: J. Alberto Conejero, Marina Murillo-Arcila, Juan B. Seoane-Sepúlveda
The interest in car-following models has increased in the last years due to its connection with vehicle-to-vehicle communications and the development of driverless
cars. Some non-linear models, such as the Gazis-Herman-Rothery model, were already known to be chaotic (4). We consider the linear Forward and Backward Control traffic model for an infinite number of cars on a track. We show the existence
of solutions with a chaotic behavior by using some results of linear dynamics of C 0 semigroups. In contrast, we also analyze which initial configurations lead to stable
solutions.
References
1. A. A. Albanese, X. Barrachina, E. M. Mangino, and A. Peris. Distributional
chaos for strongly continuous semigroups of operators. Commun. Pure Appl.
Anal., 12(5):2069–2082, 2013.
2. X. Barrachina, J. A. Conejero, M. Murillo-Arcila, and J. B. Seoane-Sepúlveda.
Distributional chaos for the Forward and Backward Control traffic model. Preprint,
2014.
3. J. A. Conejero, M. Murillo-Arcila, and J. B. Seoane-Sepúlveda. Linear chaos for
the quick-thinking-driver model. Preprint, 2014.
4. D. C. Gazis, R. Herman, and R. W. Rothery. Nonlinear follow-the-leader models of traffic flow. Operations Res., 9(4):545–567, August 1961.
Wave front sets with respect to the iterates of an operator
Chiara Boiti
University of Ferrara, Italy
[email protected]
Coautores: David Jornet, Jordi Juan-Huguet
We characterize, in [3] and [2], the wave front set with respect to the iterates of a
linear partial differential operator with constant coefficients in the setting of ultradifferentiable functions of Beurling or Roumieu type, in the sense of Braun, Meise
and Taylor [4].
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S03. Análisis funcional
30
The properties of this new wave front set are analogous to those of the classical
wave front set, but it gives more precise information about the singularities compared to those obtained by the usual wave front set in the considered class of ultradifferentiable functions (cf. [1]).
Generalizing Theorems 8.1.4 and 8.4.14 of [5], we can construct a distribution
with prescribed wave front set with respect to the iterates of a hypoelliptic linear
p.d.o. with constant coefficients.
Moreover, in [2], we give some applications to the regularity of operators with
variable coefficients and constant strength.
Referencias
1. A.A. Albanese, D. Jornet, A. Oliaro, Quasianalytic wave front sets for solutions
of linear partial differential operators, Integr. Equ. Oper. Theory 66 (2010),
153–181.
2. C. Boiti, D. Jornet, A characterization of the wave front set defined by the iterates
of an operator with constant coefficients, in preparation
3. C. Boiti, D. Jornet, J. Juan-Huguet, Wave front sets with respect to the iterates
of an operator with constant coefficients, Abstr. Appl. Anal. 2014, Article ID
438716 (2014), pp. 1–17, http://dx.doi.org/10.1155/2014/438716
4. R.W. Braun, R. Meise, B.A. Taylor, Ultradifferentiable functions and Fourier
analysis, Result. Math. 17 (1990), 206–237.
5. L. Hörmander, The Analysis of Linear Partial Differential Operators I, SpringerVerlag, Berlin (1990).
Isometries in L 2 (X ), monotone functions and averaging operators
Santiago Boza
Universitat Politècnica de Catalunya
[email protected]
Coautores: Javier Soria (Universitat de Barcelona)
In this talk, we will study necessary and sufficient conditions to ensure that a
bounded operator T defined on the Hilbert space is indeed an isometry. We will
show that under some hypothesis is enough to restrict the operator to a smaller class
of functions, (if X = R+ , we can restrict ourselves to the cone of decreasing and positive functions). In the second part of the talk, we will study the existence of positive isometric averaging operators on `2 (Z, µ) and show that they are determined
by very subtle arithmetic conditions on the measure µ (even for very simple examples), contrary to what happens in the continuous case L 2 (R+ ), where any possible
average value is realized by a suitable positive isometry.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S03. Análisis funcional
31
Lipschitz tensor products and their duality
J. Alejandro Chávez-Domínguez
ICMAT and The University of Texas at Austin
[email protected]
Coautores: M. G. Cabrera-Padilla, A. Jiménez-Vargas, M. Villegas-Vallecillos
Inspired by classical ideas, we introduce the notion of a Lipschitz tensor product
between a metric space and a Banach space. We develop the basic theory of such
tensor products, which largely parallels that of tensor norms for Banach spaces. For
example, we define Lipschitz versions of the injective and projective tensor norms
and relate them to their Banach-space versions via Lipschitz-free spaces. We also
study the duality relationships between Lipschitz tensor products and ideals of Lipschitz maps, and in particular we obtain a Lipschitz version of the representation
theorem for maximal operator ideals
Geometric perspectives of reproducing kernels
José E. Galé
Departamento de Matemáticas, Universidad de Zaragoza
[email protected]
Coautores: D. Beltita, from the IMAR (Academy of Sciences), Bucharest.
There are quite diverse situations where one finds simultaneously reproducing
kernels of Hilbert spaces and differential geometry. On the basis of these examples,
it will be shown in this talk how a differential geometry theory emerges naturally
with every member of a wide class of reproducing kernels on Hermitian vector bundles. Properties of such a geometry of kernels like the existence of (Chern) covariant
derivatives compatible with complex and Hermitian structures and the positivity of
its curvature will be discussed.
Some connections between Renorming Theory and Fixed Point Theory
Maria Angeles Japón Pineda
Universidad de Sevilla
[email protected]
A Banach space (X , k · k) is said to satisfy the fixed point property (FPP) if every
nonexpansive mapping defined from a closed bounded convex subset of X into itself
has a fixed point. Recall that a mapping T : C → C is said to be nonexpansive if
kT x − T yk ≤ kx − yk for all x, y ∈ C . Notice that the nonexpansiveness is a metric
condition which may change if we replace the norm by an equivalent one.
It is well-known that every uniformly convex Banach space satisfies the FPP and
that the space (`1 , k · k1 ) fails to have this property. Nowadays, it is an open problem
whether every reflexive Banach space verifies the FPP and it was conjectured that
every Banach space with the FPP should be reflexive for a long time.
Fixed point theory for nonexpansive mappings and renorming theory definitively became connected after the publication of the following two results:
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S03. Análisis funcional
32
i) In 2009, T. Domínguez Benavides proved that every reflexive Banach X space
could be equipped with an equivalent norm such that (X , | · |) satisfies the FPP.
ii) In 2008, P. K. Lin proved that the Banach space `1 could be renormed to have
the FPP, answering in a negative way to the above conjecture.
Since then, some other authors have proved the existence of new renormings in
`1 with the FPP and the existence of some other non-reflexive Banach spaces that
are FPP-renormable.
We will talk about the recent advances in these topics and the open problems we
deal with.
Referencias
1. T. Domínguez Benavides, A renorming of some nonseparable Banach spaces
with the fixed point property, J. Math. Anal. Appl., 350(2) (2009), 525-530.
2. P. K. Lin, There is an equivalent norm on `1 that has the fixed point property.
Nonlinear Anal., 68 (8) (2008), 2303-2308.
3. C. A. Hernández-Linares and M. A. Japón. A renorming in some Banach spaces
with applications to fixed point theory. J. Funct. Anal. 258 (2010), 3452-3468.
4. C. A. Hernández-Linares, M. A. Japón, E. Llorens-Fuster. On the structure of
the set of equivalent norms on `1 with the fixed point property. J. Math. Anal.
App. 387 (2012), 645-54.
5. C. A. Hernández-Linares, M. A. Japón. Rays of equivalent norms with the fixed
point property in some nonreflexive Banach spaces. J. Nonlinear Convex Anal.
15, n. 2, (2014), 355-377.
Plasticity of the unit ball of a strictly convex Banach space
Vladimir Kadets
Kharkiv V.N.Karazin National University, Ukraine
[email protected]
Coautores: B. Cascales, J. Orihuela, E. J. Wingler
Let X be a strictly convex Banach space and B X be its closed unit ball. Then
every bijective non-expansive map F : B X → B X is an isometry.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S03. Análisis funcional
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Foelner C*-algebras and applications
Fernando Lledó
Dept. Mathematics, Universidad Carlos III Madrid and ICMAT
[email protected]
In this talk I will introduce the class of Foelner C*-algebras, which are defined in
terms of a net of unital completely positive maps from the algebra to matrices that
are asymptotically multiplicative in a weak sense. This class of C*-algebras include
the quasidiagonal ones.
I will then present several characterizations of Foelner C*-algebras in terms of
Foelner nets of projections or amenable traces and show some applications to spectral approximation problems. Finally, I will analyze uniform Roe algebras on discrete
metric spaces with bounded geometry under this perspective.
[Joint work with Pere Ara (UAB), Kang Li (U. Copenhagen) and Jianchao Wu (U.
Muenster)]
Frequent hypercyclic translation semigroups
Elisabetta Mangino
Dipartimento di Matematica e Fisica "E. De Giorgi" - Università del Salento - Lecce Italy
[email protected]
Coautores: Marina Murillo
Frequent hypercyclicity for translation C 0 -semigroups on weighted spaces of
continuous functions is investigated. The results are achieved by establishing an
analogy between frequent hypercyclicity for the translation semigroup and for weighted
pseudo-shifts and by characterizing frequent hypercyclic weighted pseudo-shifts
in spaces of vanishing sequences. Frequent hypercylic translation semigroups in
weighted L p -spaces are also characterized.
The Daugavet equation for Lipschitz operators
Javier Merí
Universidad de Granada
[email protected]
Coautores: Vladimir Kadets, Miguel Martín, and Dirk Werner
We show that introducing a reasonable substitute for the concept of slice for the
case of non-linear Lipschitz functionals one can transfer to the non-linear case several results about the Daugavet and the alternative Daugavet equations previously
known only for linear operators.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S03. Análisis funcional
34
Strong mixing measures and invariant sets in linear dynamics
Marina Murillo Arcila
Universitat Politècnica de València
[email protected]
Coautores: A. Peris
In this talk, we show that the Frequent Hypercyclicity Criterion for operators and
for C 0 -semigroups ensures the existence of invariant strongly mixing measures with
full support. Moreover, we provide several examples which illustrate these results.
We also study dynamical properties that are satisfied by autonomous and nonautonomous dynamical systems on certain invariant sets. Particular attention is
given to the case of positive operators and semigroups on lattices, and the (invariant) positive cone.
Bibliography:
1. M. Murillo-Arcila and A.Peris. Mixing properties for nonautonomous linear
dynamics and invariant sets. Appl. Math. Lett., 26 (2013), 215–218.
2. M. Murillo-Arcila and A.Peris. Strong mixing measures for linear operators
and frequent hypercyclicity. J. Math. Anal. Appl., 298 (2013), 462–465.
3. M. Murillo-Arcila and A.Peris. Strong mixing measures for C 0 -semigroups.
DOI:10.1007/s13398-014-0169-3. To appear in RACSAM.
4. M. Murillo-Arcila and A.Peris. Chaotic behavior on invariant sets of linear operators. DOI 10.1007/s00020-014-2188-z. To appear in Integral Equations and
Operator Theory.
Seeking the best modulus of uniform convexity
Matías Raja
Universidad de Murcia
[email protected]
The modulus of convexity of a norm k · k (given on a linear space) is the function
defined by
o
n
x+y
k : kxk = kyk = 1, kx − yk ≥ t
δk·k (t ) = inf 1 − k
2
for t ∈ [0, 2]. The norm is said uniformly convex if δk·k (t ) > 0 for t > 0. It is well
known that a Banach space has an equivalent uniformly convex norm if and only if
it is super-reflexive (James, Enflo). The modulus of convexity δk·k (t ) of any norm (on
p
any Banach space) is bounded from above by 1 − 1 − t 2 /4, which is the modulus of
convexity of the Hilbert space (Nörlander). This is usually interpreted as “no Banach
space is more convex than the Hilbert space”.
Among the different equivalent norms on a given super-reflexive space, it is natural to look for the one that makes δk·k (t ) as larger as possible. In a recent paper,
we have investigated this question from an asymptotic sense: to find a equivalent
norm on X such that the limit limt →0 δk·k (t ) = 0 converges in the feasible slowest
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S03. Análisis funcional
35
manner. Our results provide a quite satisfactory answer to this problem. As an application, we will deduce the classical Pisier’s renorming with power type modulus
of convexity: there is an equivalent norm such that δk·k (t ) ≥ c t p for some c > 0 and
p ≥ 2.
Cyclic polynomials in two variables
Daniel Seco
University of Warwick
[email protected]
Coautores: C. Beneteau, G. Knese, L. Kosinski, C. Liaw and A. Sola
Let f be a function in H , a Hilbert space of analytic functions over a fixed domain. The function f is called cyclic if the polynomials times f form a dense subspace of H . A classical theorem by Brown and Shields classifies cyclicity when f is a
polynomial in Dirichlet-type spaces over the unit disc. In this talk, we show the corresponding classifications of polynomials for Dirichlet-type spaces over the bidisc
(and depending on time, also for weighted Hardy spaces over the unit disc).
Some results and open questions on spaceability in function spaces
Juan B. Seoane-Sepúlveda
UCM and ICMAT
[email protected]
Coautores: J.A. Conejero, P. Enflo, V. Gurariy, G. A. Muñoz-Fernández, and M. MurilloArcila.
A subset M of a topological vector space X is called lineable (respectively, spaceable) in X if there exists an infinite dimensional linear space (respectively, infinite
dimensional closed linear space) Y ⊂ M ∪ {0}. In this lecture we shall present several new advances within this theory regarding several different classes of annulling
functions (functions having infinitely many zeros). We also discuss problems related
to these concepts for certain subsets of some important classes of Banach spaces
(such as C [0, 1] or Müntz spaces). We also propose several open questions in the
field and provide new directions of research.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S03. Análisis funcional
36
A Schur space which is not a uniform retract of its bidual
Jesús Suárez
Universidad de Extremadura
[email protected]
N.J. Kalton gave the first example of a Banach space X which is not a uniformly
continuous retraction of its bidual; such an X is an isometric L 1 -predual. We take
the construction of X a step further and provide a new example: a non-separable
L ∞ -space Z with the Schur property such that there is no uniformly continuous
retraction of Z ∗∗ onto Z .
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S04. Análisis geométrico
http://rsme2015.ugr.es/s04.php
Polar actions on symmetric spaces
J. Carlos Díaz-Ramos
University of Santiago de Compostela
[email protected]
An isometric action of a Lie group G on a Riemannian manifold M is said to be
polar if there is a submanifold Σ of M that intersects all the orbits of G and the orbits
of G and Σ are perpendicular at intersection points. The submanifold Σ is called a
section, and it is known to be totally geodesic.
Polar actions are a possible way to generalize polar, spherical or cylindrical coordinates in Euclidean spaces, and this is where its name comes from. They also generalize algebraic results such as the diagonalization of symmetric and other types of
matrices. For that reason, sections are also known as sets of cannonical forms.
In this talk I will review the main classification results for polar actions on symmetric spaces. Polar actions on symmetric spaces of compact type are more or less
well understood. However, the classification problem in the noncompact setting is
more difficult and many examples with no compact counterpart arise.
Gravitating vortices and Kaehler-Yang-Mills equations
Oscar Garcia-Prada
Instituto de Ciencias Matematicas - CSIC
[email protected]
Coautores: Luis Alvarez-Consul, Mario Garcia-Fernandez
After reviewing the vortex equations over a compact Riemann surface and their
relation with the Hermitian-Yang-Mills equations, we go on to introduce a system
of coupled equations for a Kaehler metric on a manifold and a hermitian metric
on a vector bundle. This system interpolates between the Hermitian-Yang-Mills
equations and the Donaldson-Tian-Yau problem for constant scalar curvature. We
explain how this relates to gravitating vortices and the theory of cosmic strings in
physics.
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S04. Análisis geométrico
38
Estimates of the first Dirichlet eigenvalue from exit time moment spectra
Ana Hurtado
Universidad de Granada
[email protected]
Coautores: Steen Markvorsen and Vicente Palmer
We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit
times from the ball. As an application of the model space theory we prove lower and
upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures.
Moreover, from this general setting we thereby obtain new generalizations of the
classical and celebrated results due to McKean and Cheung–Leung concerning the
fundamental tones of Cartan-Hadamard manifolds and the fundamental tones of
submanifolds with bounded mean curvature in hyperbolic spaces, respectively.
An Extension of Efimov’s Theorem
Antonio Martínez
Universidad de Granada
[email protected]
Coautores: José A. Gálvez and José L. Teruel
The classical Efimov theorem states that there is no C 2 -smoothly immersed complete surface in R3 with negative Gauss curvature uniformly separated from zero.
Here we analyze the case when the curvature of the complete surface is less that
−c 2 in a neighborhood of infinity, and prove the surface is topologically a finitely
punctured compact surface, the area is finite, and each puncture looks like cusps
extending to infinity, asymptotic to rays.
Curvature and tolopogy of submanifolds
Vicente Palmer
Universitat Jaume I, Castellón
[email protected]
Coautores: Vicent Gimeno
T. H. Colding and W. P. Minicozzi proved in 2008 that a complete embedded minimal surface with finite topology in R3 must be proper.
In this talk we consider the more general setting of a complete immersed manifold
in a Cartan-Hadamard manifold and we try to elucidate
if these hypotheses, (embeddedness, minimality, finiteness of the topology) can be
replaced by other hypothesis related with the volume or the metric.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S04. Análisis geométrico
39
Eigenfunctions with prescribed nodal sets
Daniel Peralta-Salas
ICMAT
[email protected]
I will show that given any separating hypersurface of a closed manifold, there
exists a Riemannian metric such that the nodal set of its first nontrivial LaplaceBeltrami eigenfunction is the aforementioned hypersurface. Applications to critical
points of low energy eigenfunctions and nodal sets of Dirichlet eigenfunctions will
be provided. This is based on joint work with A. Enciso (J. Differential Geom. in
press).
The Toda system on compact surfaces: a variational approach
David Ruiz
University of Granada
[email protected]
Coautores: L. Battaglia, A. Jevnikar and A. Malchiodi
This talk is devoted to the so-called Toda system on a compact surface Σ, which
is assumed to have total area equal to 1:
´
´
³
³

u2
u
 −∆u 1 = 2ρ 1 R h1 eu 1
− 1 − ρ 2 R hh2eeu2 dV − 1 ,
1 dV g
h
e
1
2
g
´
´
³ Σ
³ Σ
(1)
u
u1
 −∆u 2 = 2ρ 2 R h2 eu 2
R h 1 eu
−
1
−
ρ
−
1
.
1
2
1
h e dV
h e dV
Σ
2
g
Σ
1
g
Here ∆ is the Laplace-Beltrami operator, ρ 1 , ρ 2 ∈ R and h 1 , h 2 are smooth positive
functions. This system appears naturally in geometry and mathematical physics.
Solutions of (1) correspond to critical points of a certain energy functional, which
turns out to be unbounded from below. A minimization argument being impossible,
we approach the problem via min-max arguments.
Compact embedded minimal surfaces in S 2 xS 1
Francisco Torralbo
KU Leuven
[email protected]
Coautores: José M. Manzano and Julia Plehnert
We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in S2 ×S1 (r ), for arbitrary radius r .
We illustrate it by obtaining some periodic minimal surfaces in S2 × R via conjugate
constructions. The resulting surfaces can be seen as the analogy to the Schwarz Psurface in these homogeneous 3-manifolds.
Referencias
1. José M. Manzano, Julia Plehnert and Francisco Torralbo. Compact embedded
minimal surfaces in S2 × S1 . arXiv: 1311.2500 [math.DG].
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S05. Análisis no lineal y EDP elípticas
http://rsme2015.ugr.es/s05.php
Dynamics of solitons in nonlinear Schroedinger equations
Claudio Bonanno
Università di Pisa
[email protected]
We use variational and symplectic methods to study the particle-type behaviour
of a soliton solution for the nonlinear Schroedinger equation in presence of a singular external potential.
Sharp lower bounds for Coulomb energy
Marco Ghimenti
Università di Pisa
[email protected]
Coautores: Jacopo Bellazzini, Marco Ghimenti, Tohru Ozawa
We give an estimate on L p lower bounds for Coulomb energy for radially symmetric functions in the homogeneous fractional Sobolev space H˙ s (R 3 ) with 1/2<s<3/2.
In case 1/2 < s ≤ 1 we show that the lower bounds are sharp
Quasilinear elliptic equations with lower order terms
Tommaso Leonori
University of Granada
[email protected]
In this talk I want to discuss some problems related to equations of the type
−∆p u + H (x, u, ∇u) = 0
in Ω
where Ω is a bounded smooth subset of RN , ∆p u = div(|∇u|p−2 ∇u), p ≥ 2 H (x, s, ξ) :
Ω × R × RN → R is Carathéodory function on which suitable assumptions are made.
I deal with existence, uniqueness and some properties of solutions to this equation.
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S05. Análisis no lineal y EDP elípticas
41
On the symmetry of ground state and least energy nodal solutions of some problems of elliptic equations
Ederson Moreira dos Santos
Universidade de São Paulo - Brazil
[email protected]
In this talk, I will consider some problems involving elliptic equations, single
equations and Hamiltonian elliptic systems, set in a radially symmetric domain in
RN . I will present new results about the symmetry of least energy solutions as well
as least energy nodal solutions for these problems. In particular, I will show that the
approach addressed to the system case presents a unified variational treatment to
deal with Hamiltonian elliptic systems, fourth-order and second-order elliptic equations.
Morse index of sign changing solutions of semilinear elliptic equations
Filomena Pacella
Università di Roma Sapienza
[email protected]
We will discuss some estimates and related properties of the Morse index of sign
changing solutions of semilinear elliptic problems with general type of nonlinearities in symmetric bounded domains. Then, in the case of Lane-Emden problems will
present some recent results on the exact computation of the Morse index of nodal
radial solutions in the ball. These results have been obtained in collaboration with
F.De Marchis and I.Ianni.
Singularly perturbed elliptic problems with asymptotically linear nonlinearities.
Benedetta Pellacci
Università Napoli "Parthenope"
[email protected]
Coautores: Liliane Maia and Eugenio Montefusco
We consider a class of singularly perturbed elliptic problems with non-autonomous asymptotically linear nonlinearities. This kind of nonlinearities represents a
saturation effect observed in nonlinear optic studies. We investigate the existence
of nontrivial and nonnegative concentrating solutions.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S05. Análisis no lineal y EDP elípticas
42
Toda system: degree and blow-up
Angela Pistoia
La Sapienza Università di Roma
[email protected]
We prove existence of continua of solutions to a SU(3) Toda system which exhibite partial blow-up or asymmetric blow-up. The results have been obtained in
collaboration with Teresa D’Aprile and David Ruiz.
Standing waves for a Gauged Nonlinear Schrödinger Equation
David Ruiz
University of Granada
[email protected]
Coautores: Alessio Pomponio (Politecnico di Bari, Italy)
This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. At low energies, the Maxwell term can be
dropped, giving rise to the following problem, proposed by Jackiw & Pi in 1990.
i D 0 φ + (D 1 D 1 + D 2 D 2 )φ + |φ|p−1 φ = 0,
¯ 2 φ),
∂0 A 1 − ∂1 A 0 = Im(φD
¯ 1 φ),
∂0 A 2 − ∂2 A 0 = −Im(φD
1
2
∂1 A 2 − ∂2 A 1 = 2 |φ| .
(2)
Here D µ = ∂µ + i A µ denotes the covariant derivative (µ = 0, 1, 2).
The study of radially symmetric standig waves leads to a nonlinear stationary
Schrödinger equation involving a nonlocal term. This problem is the Euler-Lagrange
equation of a certain energy functional. In this talk we will be concerned with the
global behavior of such functional.
This is joint work with Alessio Pomponio (Politecnico di Bari, Italy).
Semilinear elliptic equations with singular nonlinearities
Berardino Sciunzi
UNICAL
[email protected]
I will start discussing some known and new results regarding existence and uniqueness to semilinear elliptic equations involving singular nonlinearities. Solutions
generally are not in H 1 in this setting but it is in any case possible to adapt the moving plane method, exploiting a suitable decomposition of the solutions.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S05. Análisis no lineal y EDP elípticas
43
The role of singular Liouville systems in the study of non-abelian Chern-Simons
vortices
Gabriella Tarantello
Università di Roma Tor Vergata
[email protected]
We describe recent results about the construction on non-abelian Chern-Simons
vortices of non-topological type in terms of entire solutions for a class of singular
Liouville systems in the plane.
On the eigenvalues of Aharonov-Bohm operators with varying poles
Susanna Terracini
Università di Torino
[email protected]
We consider a magnetic operator of Aharonov-Bohm type with Dirichlet boundary conditions in a planar domain. We analyse the behavior of its eigenvalues as the
singular pole moves in the domain. For any value of the circulation we prove that
the k-th magnetic eigenvalue converges to the k-th eigenvalue of the Laplacian as
the pole approaches the boundary. We show that the magnetic eigenvalues depend
in a smooth way on the position of the pole, as long as they remain simple. In case of
half-integer circulation, we show that the rate of convergence depends on the number of nodal lines of the corresponding magnetic eigenfunction.
Solutions with prescribed mass for nonlinear Schrödinger systems
Gianmaria Verzini
Politecnico di Milano
[email protected]
For a class of nonlinear Schrödinger equations and systems, we investigate the
existence and the orbital stability of standing waves having components with prescribed L 2 -mass. We provide a variational characterization of such solutions, which
gives information on the stability through of a condition of Grillakis-Shatah-Strauss
type.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S06. Análisis numérico de EDP y modelización
http://rsme2015.ugr.es/s06.php
Métodos puramente Lagrangianos y semi-Lagrangianos para modelos de la mecánica de medios continuos
Marta Benítez
Universidade da Coruña
[email protected]
Coautores: Alfredo Bermúdez (Universidade de Santiago de Compostela)
El objetivo de la comunicación es presentar nuevos métodos de características
puramente Lagrangianos y semi-Lagrangianos para la resolución numérica de diferentes problemas de convección. En concreto, se consideran problemas escalares
de convección-difusión, las ecuaciones de Navier-Stokes y problemas acoplados
fluido-estructura. Para la discretización espacial de los diferentes problemas se utilizan métodos de elementos finitos.
En primer lugar, introduciendo un cambio de variable bastante general, se obtiene una formulación unificada para un problema de convección-difusión escalar,
con la que es posible obtener a la vez métodos de características puramente Lagrangianos y semi-Lagrangianos (ver [1]). En particular, se presentan diferentes
métodos de características totalmente Lagrangianos y semi-Lagrangianos obtenidos
a partir de dicha formulación general. Uno de estos métodos es puramente Lagrangiano y de segundo orden en tiempo y ha sido analizado matemáticamente
y numéricamente en [2], [3] y [1]. Además, se muestran los resultados numéricos
obtenidos con diferentes métodos de características.
Aplicando estas ideas a las ecuaciones de Navier-Stokes, se obtiene una formulación general de dichas ecuaciones en función del desplazamiento, con la que es
posible obtener a la vez métodos de características puramente Lagrangianos y semiLagrangianos (ver [4]). Concretamente, se presentan dos métodos de características de segundo orden, uno Lagrangiano y otro semi-Lagrangiano, y un método
semi-Lagrangiano de primer orden. Además, se muestran los resultados numéricos obtenidos para ejemplos test académicos y para problemas muy extendidos en
la bibliografía. En particular, se consideran problemas de frontera libre. Los métodos puramente Lagrangianos son adecuados para la resolución numérica de estos
problemas, puesto que permiten resolver el problema sin necesidad de calcular y
mallar el dominio en cada paso de tiempo. Únicamente es necesario reinicializar la
transformación cuando el dominio de referencia presenta grandes deformaciones.
Finalmente, se utilizan estas ideas para resolver numéricamente un problema
acoplado fluido-estructura. Concretamente, se obtiene una formulación del problema acoplado en función del desplazamiento y se propone un método de características de segundo orden en tiempo totalmente Lagrangiano combinado con un
método de elementos finitos para su resolución numérica. Además, se presentan
los resultados numéricos obtenidos para ejemplos test académicos y problemas de
la bibliografía. Estos nuevos métodos totalmente Lagrangianos en función del desplazamiento son convenientes para resolver problemas fluido-estructura, puesto
que no requieren el cálculo y el mallado del dominio en cada paso de tiempo y
además el acoplamiento en la frontera común resulta sencillo.
44
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45
Referencias
1. M. Benítez and A. Bermúdez, Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of convection-diffusion problems,
Int. J. Numer. Anal. Mod., vol. 11, (2014), 271–287.
2. M. Benítez and A. Bermúdez, Numerical Analysis of a second-order pure Lagrange-Galerkin method for convection-diffusion problems. Part I: time discretization, SIAM J. Numer. Anal., vol. 50, (2012), 858–882.
3. M. Benítez and A. Bermúdez, Numerical Analysis of a second-order pure Lagrange-Galerkin method for convection-diffusion problems. Part II: fully discretized scheme and numerical results, SIAM J. Numer. Anal., vol. 50, (2012),
2824–2844.
4. M. Benítez and A. Bermúdez, Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of Navier-Stokes equations (to appear in Appl. Numer. Math.).
Modelado de la turbulencia mediante métodos de Multiescala Variacional por
Proyección
Tomás Chacón
Universidad de Sevilla
[email protected]
En los últimos años se ha desarrollado una nueva clase de modelos de turbulencia, llamados de Multiescala Variacional (VMS). Se trata de modelos intrínsecamente discretos, construidos directamente a partir de la formulación variacional de
las Ecuaciones de Navier-Stokes, con ecuaciones para grandes y pequeñas escalas.
Esta estructura permite que el modelado de las escalas de sub-malla aparece únicamente en las ecuaciones de las pequeñas escalas, reduciendo el amortiguamiento
de las grandes escalas. En esta comunicación presentamos una clase especial de
métodos VMS en que la acción de las escalas de sub-malla sobre las pequeñas escalas resueltas se tiene en cuenta a través de operadores de proyección sobre una
malla más groser, con lo que en la práctica se puede hacer todo el cálculo con una
sola malla. Este modelo calcula con alta precisión flujos laminares regulares, e igualmente flujos turbulentos con precisión aceptable de los estadísticos de primer y segundo orden. Supone un buen compromiso entre economía de cálculo y precisión.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Thermoconvection to explain the formation and tilting of a dust devil eye
María Cruz Navarro
Universidad de Castilla-La Mancha
[email protected]
Coautores: Damián Castaño and Henar Herrero
Dust devils are columnar, ground-based whirlwinds, common in dry regions and
made by the dust picked up from the ground. Most authors emphasized the importance to dust devil formation of intense surface heating, which leads to high surface
air temperatures and superadiabatic lapse rates, but a crucial question remaining is
how they acquire rotation, how the eye is created and why dust devils tilt towards
the direction of motion. Convective cell circulations are the accepted theory to explain the source of angular momentum in dust devils. But the way these convective
cells generate vorticity is not understood. Here we show that vorticity in dust devils
is generated by a thermoconvective instability, we give a thermal explanation for the
morphology of the dust devil and demonstrate that tilting appears after a secondary
thermoconvective instability. The model consists of incomprensible Navier-Stokes
coupled to a heat equation under the Boussinesq approximation. The numerical
method is Chebyshev collocation.
Mathematical models for multiphase flow in vertical equilibrium and their numerical simulation
Rosa Donat
Universidad de Valencia
[email protected]
We will review certain models for two-phase and three-phase flow under vertical
equilibrium, and propose a general framework for multiphase flow that includes
them as particular cases. Sharp gradients are to be expected in the solutions to these
models, so that WENO schemes for the convective terms lead to robust codes. A
combination with IMEX-ODE solvers leads to reliable and cost-effective schemes
for the numerical simulation of these problems.
Efficient Osher-Solomon schemes for hyperbolic Systems
José María Gallardo
Universidad de Málaga
[email protected]
Coautores: M. J. Castro and A. Marquina
The Osher-Solomon scheme is a classical Riemann solver which enjoys a number of interesting features: it is nonlinear, complete, robust, entropy-satisfying,
smooth, etc. However, its practical implementation is rather cumbersome, computationally expensive, and applicable only to certain systems (compressible Euler
equations for ideal gases or shallow water equations, for example). In this work, a
new class of approximate Osher-Solomon schemes for the numerical approximation of general conservative and nonconservative hyperbolic systems is proposed.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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They are based on viscosity matrices obtained by polynomial or rational approximations to the Jacobian of the flux evaluated at some average states, and only require a
bound on the maximal characteristic speeds. These methods are easy to implement
and applicable to general hyperbolic systems, while at the same time they maintain
the good properties of the original Osher-Solomon solver. The numerical tests indicate that the schemes are robust, running stable and accurate with a satisfactory
time step restriction, and the computational cost is very advantageous with respect
to schemes using a complete spectral decomposition of the Jacobians.
Optimized partitioned procedures for the Stokes-Darcy coupled problem
Luca Gerardo-Giorda
BCAM – Basque Center for Applied Mathematics, Bilbao, Spain
[email protected]
Coautores: Marco Discacciati (Universitat Politècnica de Catalunya)
We consider a coupled Stokes-Darcy system for the filtration of an incompressible fluid through a porous medium. The model couples the solution of the Stokes
equation in the fluid region, with the solution of the Darcy equation in the porous
medium region through the surface separating the two physical domains. Partitioned procedures are modular algorithms commonly used for the solution of coupled multiphysics problems. They involve separate solvers for the different subproblems, that interact in an iterative framework through the exchange of suitable
transmission conditions at the multiphysics interface. In the framework of domain
decomposition methods, the Robin-type interface conditions introduced in [1] guarantee convergence in the absence of overlap between the different subregions. Following the ideas developed in [2] for Fluid-Structure Interaction problems, we optimize the performance of the corresponding algorithm, both in term of an iterative
solver and as a preconditioner for the fully coupled problem [3].
Referencias
1. M. Discacciati, A. Quarteroni and A. Valli 2007 Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal., Vol. 45
(3), pp. 2193-2213.
2. L. Gerardo-Giorda, F. Nobile, and C. Vergara 2010. Analysis and optimization
of Robin-Robin partitioned procedures in Fluid-Structure Interaction problems.
SIAM J. on Num. Anal., Vol. 48 (6), pp. 2091-2116.
3. M. Discacciati, L. Gerardo-Giorda 2014 Optimized Schwarz Methods for the
Stokes-Darcy coupling. In preparation.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Time-Domain BEM for Acoustic Problems
Heiko Gimperlein
Heriot Watt University, Edinburgh, UK
[email protected]
We consider the time-domain boundary element method for exterior Robin type
bvp’s for the wave equation. We apply a space-time Galerkin method, present a priori and a posteriori error estimates, and derive an h-adaptive algorithm in space and
time with mesh refinement driven by residual type error indicators. Numerical experiments are also given which underline our theoretical results. Special emphasis
is given to numerical simulations of the sound radiation of car tyres.
Stabilization and a posteriori error analysis of a mixed convection-diffusion
problem
María González Taboada
Universidade da Coruña
[email protected]
Coautores: Johan Jansson (BCAM) and Sergey Korotov (BCAM)
Convection-diffusion problems appear in a large number of applications, including the numerical simulation of incompressible fluid flows. As it is well-known,
the numerical solution of this type of problems is difficult when transport processes
are dominant. In this talk we will address the numerical approximation of convectiondiffusion problems in mixed form using an augmented mixed finite element method.
We will present a priori and a posteriori error estimates, paying special attention to
the role of the parameters of the problem. We will show numerical experiments that
support the theoretical results.
Métodos Runge-Kutta-Nyström de Pasos Fraccionarios: Evitando la reducción de
orden
J.C. Jorge
Departamento de Ingeniería Matemática e Informática, Universidad Pública
de Navarra (Spain)
[email protected]
En esta ponencia se presentan nuevos integradores temporales diseñados para
resolver eficientemente Problemas de Valor Inicial y de Contorno en los que la Ecuación o el sistema de Ecuaciones en Derivadas Parciales es de la forma
∂2 u
+ Au = f ;
∂t 2
(3)
donde A es un operador diferencial, típicamente de segundo o de cuarto orden, que
contiene las derivadas espaciales de la ecuación, y f es una función dato: dos ejem∂4
plos clásicos son la ecuación de Euler-Bernouilli (A =
) y la ecuación de ondas
∂x 4
(A = −∆). Veremos cómo tales integradores, combinados con una discretización
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espacial estándar y una partición del operador A adecuada, aportan ventajas computacionales importantes similares a las de los métodos clásicos de direcciones alternadas. Otros tipos de particiones llevan a métodos de tipo descomposición de
dominios muy eficientes. Prestaremos especial atención al fenómeno de la reducción de orden, típico de todos los métodos de un paso cuando se aplican a estos
problemas, que es especialmente importante cuando las condiciones de contorno
varían en el tiempo. Daremos una técnica sencilla de implementar, basada en modificar las condiciones de contorno asociadas a las etapas internas y analizaremos
la mejora de la consistencia proporcionada por dichas modificaciones. Finalmente
mostraremos algunos ensayos numéricos que ilustren el buen comportamiento de
los métodos propuestos.
New developments for the parareal in time algorithm
Yvon Maday
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France
[email protected]
The need for faster numerical simulations of complex phenomena, and the definition in this context of what a complex phenomenon is, is evolving in line with
the improvement of the platforms that are available for High Performance Computing. Indeed, what used to require hours or days of numerical simulations on large
computers can now be run in fractions of seconds on laptops. Nevertheless the understanding of real phenomena, the control and optimization of processes and the
monitoring of industrial problems propose new challenges where i) better accuracy,
ii) use of more involved mathematical models, iii) simulations on bigger object or
iv) on longer period of time for unsteady phenomena are required. The evolution of
the computing platforms helps in addressing bigger problems but is not sufficient.
Exascale systems which are achievable in the next 5-10 years will contain millions
of cores. In order to make efficient use of these systems, high-performance applications must have sufficient parallelism to support parallel execution across millions
of threads of execution. The development of more efficient and more highly parallel scalable solvers is therefore at the forefront of Exascale applications research
and development, in particular, the domain decomposition methods or task partitioning approaches reach their limits in their ability to use the entire computational
resource with the same efficiency as currently achieved on existing smaller systems.
Most simulations which are expected to deliver economic, societal and scientific
impact from Exascale systems contain time-stepping in some form and present-day
codes make little or no use of parallelism in the time domain; time stepping is currently treated as a serial process. For time dependent problems, either pure differential systems or coupled with partial differential equations, the time direction leads
to new families of algorithms that might allow to provide full efficiencies and speed
ups. The parareal (parallel in time) algorithm and the waveform relaxation methods
have been introduced to fill this gap and have the potential to extract very large additional parallelism from a wide range of time-stepping application codes. This is a
disruptive technology which will deliver performance speed-ups of between 10 and
100. By comparison, optimisations of current algorithms typically yield benefits in
the range of tens of percent, or at most a factor of 2-3 improvement.
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In this talk, we shall introduce the basics of the approach, taking care of the only
time direction. We shall present the efficiency that can be expected, the drawbacks
of the original approach and the way these can be circumvented. We shall then
present the way to combine this algorithm with other iterative procedures such as
algebraic linear or nonlinear solvers, domain decomposition methods or as control
problems. A current state of the art including the numerical analysis of these combined schemes will also be presented so as the challenges that need to be addressed
now.
Stabilization of finite element approximations to the Stokes and Oseen equations
Julia Novo
Universidad Autónoma de Madrid
[email protected]
When considering the numerical approximation of the Navier-Stokes equations
by means of mixed finite elements one can found two types on instabilities. On the
one hand, it is well known that the standard Galerkin finite element method suffers
from instabilities caused by the dominance of convection. On the other hand, stable
mixed finite element approximations to the Stokes and Navier-Stokes equations are
required to satisfy a discrete inf-sup condition. In this talk we study both kinds of
instabilities. In the first part of the talk, we consider the stabilization in the convection dominated regime by means of SUPG/grad-div stabilized methods using LBB
stable elements. We revise the existing literature pointing out some open questions.
In the second part of the talk, we consider non LBB stable elements and analyze
the so called pressure stabilized Petrov-Galerkin method for the continuous in time
discretization of the evolutionary Stokes equations. We show some recent advances
that avoid the so called instability of the discrete pressure for small time steps that
has been reported in the literature.
A POD reduced order model to calculate bifurcations in a Rayleigh-Bénard
convection problem
Francisco Pla
Universidad de Castilla-La Mancha
[email protected]
Coautores: Henar Herrero, José Manuel Vega
In this work, a flexible Galerkin method based on proper orthogonal decomposition (POD) is applied to a Rayleigh-Bénard convection problem in the limit of infinite Prandtl number using the Rayleigh number as a bifurcation parameter. The
fluid is confined between a lower solid plate and an upper non-deformable free surface, and the patterns are assumed to be periodic in the horizontal direction. Restricting to a half of a period, imposing symmetry conditions at the lateral boundaries (which breaks invariance under translations in the laterally infinite layer), and
using the Boussinesq approximation, the nondimensional continuity, momentum,
and energy conservation equations are considered.
This problem exhibits a horizontal reflection symmetry which is exact and also
an approximate vertical reflection symmetry, which is due to the large values of the
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S06. Análisis numérico de EDP y modelización
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Rayleigh number in the physically relevant regime. A reduced order model will be
presented with two main ingredients, namely (i) the symmetries are accounted for in
the calculation of the POD modes and (ii) advantage is taken of the property (already
tested in related bifurcation problems) that POD manifold resulting from snapshots
calculated in either Newton iterations or time-dependent runs for a particular value
of the Rayleigh number also contain good approximations of the steady states other
values of R. Using these and a basic continuation method on the reduced order
model, the bifurcation diagram is calculated at a fairly low computational cost.
Numerical approximation of Beltrami fields in a topologically non trivial domain
Rodolfo Rodríguez
Universidad de Concepción
[email protected]
Coautores: Eduardo Lara y Pablo Venegas
Vector fields H satisfying curl H = λ H with λ being a scalar field are called forcefree fields. This name arises from magnetohydrodynamics, since a magnetic field of
this kind induces a vanishing Lorentz force: F := J × B = curl H × (µH ). In 1958
Woltjer [4] showed that the lowest state of magnetic energy density within a closed
system is attained when λ is spatially constant. In such a case H is called a linear
force-free field and its determination is naturally related with the spectral problem
for the curl operator. This problem has a longstanding tradition in mathematical
physics. A large measure of the credit goes to Beltrami [1], who seems to be the first
who considered it in the context of fluid dynamics and electromagnetism. This is
the reason why the corresponding eigenfunctions are also called Beltrami fields.
A couple of numerical methods based on edge finite elements have been introduced and analyzed in a recent paper [3] on simply connected domains. This topological assumption is not just a technicality, since the eigenvalue problem for the
curl operator is ill-posed on multiply connected domains, in the sense that its spectrum is the whole complex plane as is shown in [5]. However, additional constraints
can be added to recover a well posed problem with a discrete spectrum [5,2]. We
choose as additional constraint a zero-flux condition of the curl on all the cutting
surfaces. We introduce a weak form of the corresponding problem, which is a convenient variation of one of the formulations studied in [3]. We prove well posedness,
spectral convergence and a priori error estimates and show how to modify the finite
element discretization from [3] to take care of the additional constraint. We also introduce a convenient variation of the other formulation from [3], which allows us to
compute more efficiently the eigenvalues. Finally, we report a numerical test which
allows us to assess the performance of the proposed methods.
Referencias
1. E. B ELTRAMI, Considerazioni idrodinamiche. Rend. Inst. Lombardo Acad. Sci.
Let., 22 (1889) 122–131. (English translation: Considerations on hydrodynamics, Int. J. Fusion Energy, 3 (1985) 53–57.)
2. R. H IPTMAIR , P.R. KOTIUGA AND S. T ORDEUX, Self-adjoint curl operators, Ann.
Mat. Pura Appl., 191 (2012) 431–457.
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3. R. R ODRÍGUEZ AND P. V ENEGAS, Numerical approximation of the spectrum of
the curl operator, Math. Comp., 83 (2014) 553–577.
4. L. W OLTJER, A theorem on force-free magnetic fields. Proc. Natl. Acad. Sci.
USA, 44 (1958) 489–491.
5. Z. Y OSHIDA AND Y. G IGA, Remarks on spectra of operator rot, Math. Z., 204
(1990) 235–245.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S07. Análisis numérico en la resolución de
ecuaciones no lineales
http://rsme2015.ugr.es/s07.php
A modified secant-type method for systems of nonlinear equations improving the
numerical stability
Sergio Amat
U.P. Cartagena
[email protected]
Coautores: M.A. Hernández-Verón, M.J. Rubio
A modification of the secant method for the approximation of nonlinear system
of equations is considered to improve the applicability of the secant method. This
modification changes the resolution of a linear system in each step, necessary to
apply the secant method, for several matrix multiplications. In this way, the numerical stability of the secant method can be improved. In addition, in this paper, we
prove that the modification considered keeps two important properties of the secant method, such as: does
p not use derivatives in its algorithm and has R-order of
convergence at least (1 + 5)/2.
On the weight function procedure for designing multidimensional iterative
schemes
Alicia Cordero
Universitat Politècnica de València
[email protected]
Coautores: Santiago Artidiello (Instituto Tecnológico de Santo Domingo) Juan R.
Torregrosa (Universitat Politècnica de València) María P. Vassileva (Instituto Tecnológico de Santo Domingo)
We present two classes of iterative methods whose order of convergence are four
and five, respectively, for solving systems of nonlinear equations, by using the technique of weight functions in each step (see [1] and the references therein). Moreover,
we show an extension to higher order, adding only one functional evaluation of the
vectorial nonlinear function. We perform some numerical tests to compare the proposed methods with other schemes in the literature (see [2-4])and check their effectiveness on specific nonlinear problems. Moreover, some real basins of attraction
are analyzed by using the software designed in [5] in order to check the relation between the order of convergence and the set of convergent starting points.
References
1. M.S. Petkovic, B. Neta, L.D. Petkovic, J. Dzunic, Multipoint methods for solving
nonlinear equations, Academic Press, 2013.
2. P. Jarratt, Some fourth order multipoint iterative methods for solving equations, Mathematics of Computation, 20 (1966) 434–437.
53
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3. J. R. Sharma, R. K. Guna, R. Sharma, An efficient fourth order weighted-Newton
method for systems of nonlinear equations, Numerical Algorithms, 62 (2013)
307–323.
4. M.P. Vassileva, Métodos iterativos eficientes para la resolución de sistemas no
lineales, Tesis Doctoral, Universidad Politécnica de Valencia, 2011.
5. F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters
planes of iterative families and methods, The Scientific World Journal Volume
2013, Article ID 780153, 11 pages.
A variation of the Lipschitz condition for the semilocal convergence of Newton’s
method
Miguel Ángel Hernández-Verón
University of La Rioja
[email protected]
Coautores: José Antonio Ezquerro
Newton’s method is the most used iterative method to solve nonlinear operator
equations F (x) = 0 in a Banach space. In this work, we focus our attention of the
analysis of the semilocal convergence of Newton’s method. Two types of conditions
are then needed: conditions on the operator involved F and conditions on the starting point. The usual usual condition required to the operator F is that F 0 is Lipschitz
continuous in the domain where F is defined, what affects significantly the value
of the Lipschitz constant. Our main aim is then to relax this dependence, so that
we can improve the domain of starting points for Newton’s method. For this, we
introduce a variation of the Lipschitz condition for the operator F 0 .
Multidimensional generalization of iterative methods and applications
José M. Gutiérrez
Departamento de Matemáticas y Computación, Universidad de La Rioja, Logroño,
España
[email protected]
Coautores: Miquel Grau-Sánchez, Miquel Noguera (Universidad Politécnica de Catalunya)
In this work we we extend to the multidimensional case some iterative methods that are known in their scalar version. We have considered here methods with
fourth-order local convergence, some of them containing derivatives and other without derivatives. We analyze the efficiency of these four new algorithms and conclude
which ones are the most efficient. We illustrate these results with some numerical
examples and applications. In particular, we have used these methods in the resolution of the systems arising from a Hammerstein’s integral equation. Finally, we
compare the methods introduced here with other known fourth-order methods for
solving nonlinear systems of equations. The numerical examples considered in this
paper allow us to introduce some new numerical tools, such as a modified stopping
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S07. Análisis numérico en la resolución de ecuaciones no lineales
55
criterium, a computational order of convergence and an adaptive arithmetic that
minimizes the elapsed time in comparison with the use of a fixed arithmetic.
Dynamical Study while Searching Equilibrium Solutions in N-body Problem
José Luis Hueso
Universidaad Politécnica de Valencia
[email protected]
Coautores: D. A. Budzko E. Martínez C. Teruel
The dynamics of different iterative methods for the solution of nonlinear equations has been widely studied (see, for example [1] and the references therein) by
analyzing the properties of the rational functions in the complex plane that arise
when applying the method to polynomials of certain degree. Here we deal with an
interesting real application of celestial mechanics that produces a 2 ×2 system of algebraic equations in the real 2-dimensional plane. Our idea is to apply the tools of
the theory of complex dynamics to this case.
In the classical N-body problem, the search of equilibrium solutions is a very intricate problem itself, because the number of real solutions increases very fast while
the number of interacting bodies N grows and at present nobody knows even how
to find the number of all equilibrium solutions (or central configurations) for arbitrary N. The equations that determine equilibrium solutions in N-body problem
are nonlinear algebraic equations and usually contain some geometric or dynamic
parameters.
In this paper we study the equilibrium solutions in the restricted four-body problem [2]. The corresponding system
³
´
´
p
¢³
¡p
( 3x − y) 1 − (x 2 +y1 2 )3/2 + µ1 3(x − 1) + y 1 − ((x−1)21+y 2 )3/2 = 0
´
´
³
¢³
¡p
1 p
2y 1 − (x 2 +y1 2 )3/2 + µ2 3(x − 1) + y 1 −
2
2 3/2 = 0
((x−1/2) +(y− 3/2) )
has from 8 to 10 solutions, depending on two mass parameters µ1 and µ2 [2].
By applying Newton’s method to the solution of this system, we find that the attraction basins of the roots are very irregular, chaotic and full of noise, especially if
parameters µ1 and µ2 are close to zero (the most interesting case). This means that
the problem of choosing initial estimations for the iterative method is very sensitive
and needs careful consideration, because the next step is the stability analysis of every solutions under any values of parameters. We have confronted the numerical
solutions with the solutions obtained in the form of power series in order to determine if the last ones are suitable as starting points for the Newton’s iterations.
Referencias
1. J. L. Varona, Graphic and numerical comparison between iterative methods,
Math. Intelligencer, 24, pp. 37–46 (2002).
2. Budzko, D.A. and Prokopenya, A.N. Symbolic-numerical analysis of equilibrium solutions in a restricted four-body problem. Programming and Computer Software, Vol. 36:2, pp. 68–74 (2010).
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S07. Análisis numérico en la resolución de ecuaciones no lineales
56
Purely iterative algorithms for Newton’s maps
Ángel Alberto Magreñán
Universidad Internacional de La Rioja
[email protected]
Coautores: Sergio Amat Plata, Sonia Busquier, Gerardo Honorato
In this talk, the behavior of a family of purely iterative algorithms for Newton’s
map is studied. Some anomalies, such us convergence to extraneous fixed points
or different cycles, are found by means of studying the dynamical behavior of the
family applied to quadratic polynomials. Parameter spaces using the Convergence
Plane are shown and the study of the stability of the fixed points is presented. Dynamical planes for members with good and bad dynamical behavior are also provided.
Applying iterative methods in the Splitting technique for solving partial
differential equations
Eulalia Martínez Molada
Universitat Politècnica de València
[email protected]
Coautores: Jurgen Geiser, José L. Hueso
In this paper we propose to use Newton’s method and higher order iterative
methods to the splitting technique to solve nonlinear differential equations and
time-dependent partial differential equations. We deal with two splitting schemes:
Non-iterative splitting schemes and iterative ones, whose convergence study can be
found in [1]. Finally, we present some numerical results. In the first example we
apply the results to the Bernoulli’s ordinary differential equation and the second example is a mixture of a convection-diffusion and Burger’s equation. [2]
Referencias
1. J. Kanney, C. Miller, and C.T. Kelley, Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems, Advances
in Water Resources 26 (2003) 247-261.
2. J. Geiser, Discretisation Methods with embedded analytical solutions for convection dominated transport in porous media, Proc. NA&A ’04, Lecture Notes
in Computer Science, Vol.3401, Springer, Berlin, 2005, pp. 288-295.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Métodos iterativos de alto orden tipo Kurchatov
Rosa M. Peris
Universitat de València
[email protected]
Coautores: Vicente F. Candela
Presentaremos una familia de métodos iterativos de tercer orden para resolver
ecuaciones no-lineales libre de derivadas. Estos métodos están basados en los métodos clásicos tipo Halley-Chebyshev de tercer orden, aproximando las derivadas de
la función mediante diferencias tipo Kurchatov. Cada método de la familia necesita
tres evaluaciones de la función en cada iteración y demostraremos su convergencia
cúbica. Para finalizar mostraremos algunos ejemplos numéricos de resolución de
ecuaciones con raíces simples donde estos métodos funcionan y los compararemos
con los métodos clásicos de tercer orden que usan primera y segunda derivada y con
los métodos de tercer orden tipo Steffensen libres de derivadas.
On the numerical inversion of cumulative distribution functions
Javier Segura
Universidad de Cantabria
[email protected]
Rx
The inversion of cumulative distribution functions F (x) = x0 f (t )d t (x ∈ [x 0 , x 1 ],
F (x 1 ) = 1), where f (x) is the probability density function, is an important problem
with many applications. The inversion of the equation F (x) = p usually requires
accurate starting values in order to ensure fast convergence for standard iterative
methods; this is particularly true at the tails of the distribution (p close to 0 or 1).
The reason is that these cumulative distributions F (x) have a sigmoidal shape sometimes with very flat tails. We chose the particular examples of the gamma distribution ( f (t ) = t a−1 e −t /Γ(a), x 0 = 0, x 1 = +∞) and the beta distribution ( f (t ) =
t a−1 (1 − t )b−1 /B (a, b), B (a, b) = Γ(a)Γ(b)/Γ(a + b), x 0 = 0, x 1 = 1) in order to illustrate two different but complementary approaches to the problem: the derivation
of accurate starting values by analytical methods and the construction of methods
specially tailored for this type of sigmoid functions. In particular we obtain a fourth
order method with good global convergence properties.
A biparametric family for solving nonlinear problems
Juan R. Torregrosa
Universitat Politècnica de València
[email protected]
Coautores: Alicia Cordero (Universitat Politècnica de València) Javier G. Maimó (Instituto Tecnológico de Santo Domingo) María P. Vassileva (Instituto Tecnológico de
Santo Domingo)
In this paper, by using a generalization of Ostrowski’ and Chun’s methods two
optimal bi-parametric families of predictor-corrector iterative schemes, with order
of convergence 4 for solving nonlinear equations, are presented. The predictor of
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S07. Análisis numérico en la resolución de ecuaciones no lineales
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the first family is Newton’s method, and the first step of the second class is Steffensen’s scheme. This second family is derivative-free and is designed by using the
idea described in [1]. One of them is extended to the multidimensional case.
Some numerical tests are performed to compare the proposed methods with the
original Ostrowski’ [2] and Chun’s [3] methods and also with Jarratt’s scheme [4], to
confirm the theoretical results.
References
1. A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions
with no derivatives of any order of convergence, Journal of Comp. and Appl.
˝
Mathematics, 275 (2015) 502U515.
2. R. King, A family of fourth order methods for nonlinear equations, SIAM Journal Numer. Anal, 10 (1973) 876–879.
3. C. Chun, Construction of Newton-like iterative methods for solving nonlinear
equations, Numerical Mathematics, 104 (2006) 297–315.
4. P. Jarratt, Some fourth order multipoint methods for solving equations, Mathematics and Computation, 20 (1966) 434–437.
The Newton’s method in the singular case
Jean-Claude Yakoubsohn
Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France
[email protected]
Coautores: Gregorio Malajovich Instituto de Matemática, Universidade Federal do
Rio de Janeiro Brasil
We define a such type Newton method named "lower rank approximation" (lra)
Newton method for polynomial systems in the singular case, i.e., when the jacobian
matrix of the system has not a constant rank in a ball containing the solution set.
We give two kinds of results concerning the numerical analysis of the lra Newton
method. We first prove a γ-theorem which explain the quadratic at the neighborhood of a singular variety. We next state an α-theorem which prove the existence
of singular solution from a punctual criterion and its fast approximation by the lra
Newton sequence. These two results complete in a certain sense the numerical analysis of Newton’s method where γ-theorem and α-theorem are known in the various
cases where the Jacobian matrix of the polynomial system of constant rank : for example, in the regular case, in the overdetermined case and underdetermined case.
Referencias
1. Jean-Pierre Dedieu, Points Fixes, Zéros et la Méthode de Newton, Springer
Verlag, Mathématiques et Applications 54, 2006.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S08. Conocimiento profesional del profesor de
matemáticas
http://rsme2015.ugr.es/s08.php
La evaluación en Matemáticas: una necesidad y un problema
Lorenzo J. Blanco Nieto
Universidad de Extremadura
[email protected]
Coautores: Janeth A. Cárdenas Lizarazo
Asumimos tres premisas: i. la evaluación es uno de los organizadores del currículo y como tal debe estar integrada en el proceso de enseñanza y aprendizaje, ii.
La Resolución de Problemas es un contenido específico que los estudiantes deben
aprender, iii. La resolución de problemas debe ser objeto de evaluación. En nuestra investigación nos hemos planteado: Caracterizar las concepciones y prácticas
de evaluación de los profesores de secundaria y bachillerato al evaluar la resolución
de problemas en matemáticas. Para ello hemos aplicado dos cuestionarios que nos
permiten describir sus concepciones, hemos analizado los instrumentos de evaluación que nos han facilitado y que desarrollan en sus prácticas en el aula y, finalmente, hemos mantenido entrevistas que nos han permitido profundizar sobre los
resultados obtenidos. Los resultados obtenidos y las contradicciones que aparecen
al considerar conjuntamente todos los instrumentos de investigación nos indican la
necesidad de profundizar acerca de la evaluación en matemáticas y la inclusión en
los programas de formación inicial y permanente del profesorado la evaluación en
matemáticas como uno de los contenidos específicos.
El problema de identificar y caracterizar el conocimiento del profesor de
matemáticas
María Luz Callejo de la Vega
Universidad de Alicante
[email protected]
Una competencia del profesor de matemáticas es analizar, interpretar y valorar
las respuestas de los estudiantes para tomar decisiones sobre el proceso de enseñanza-aprendizaje. Algunas investigaciones sobre el desarrollo profesional se han
centrado en identificar el conocimiento necesario para desarrollar esta competencia en dominios matemáticos específicos. En esta comunicación caracterizamos el
conocimiento del profesor en el dominio del pensamiento algebraico.
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S08. Conocimiento profesional del profesor de matemáticas
60
MTSK: un modelo analítico para el estudio del conocimiento del profesor de
matemáticas.
Luis Carlos Contreras González
Universidad de Huelva
[email protected]
Coautores: María Cinta Muñoz-Catalán
Nuestro trabajo parte de distintos marcos teóricos que han intentado caracterizar el conocimiento del profesor de matemáticas. Asi, se describen aquellos modelos de conocimiento del profesor que han sido fundamentales para la elaboración de
nuestro modelo y se detallan los aspectos que lo han originado. Luego se presenta
este modelo (Mathematics Teacher’s Specialised Knowledge -MTSK) y se muestran
ejemplos de elementos de los diferentes subdominios del MTSK referidos a un núcleo conceptual concreto. Finalmente, se presentan algunas reflexiones.
El conocimiento profesional del profesor de matemáticas
Pablo Flores Martínez
Universidad de Granada
[email protected]
La necesaria profesionalización de los docentes de matemáticas tiene que basarse en identificar y caracterizar un cuerpo específico de conocimiento del profesor
de matemáticas. En esta sesión especial se dan a conocer algunas líneas de investigación en torno a este tema, distinguiendo dos problemas, la identificación del
conocimiento y su caracterización. Se abordan desde la perspectiva investigadora
(primeras sesiones), y desde la práctica (quinta sesión). Finalmente se abre un debate sobre estas dos cuestiones generales, empleando como puntos de apoyo los
aportes realizados en cada sesión.
Práctica del profesor de matemáticas
Antonio Moreno Verdejo
Departamento de Didáctica de la Matemática
[email protected]
El conocimiento profesional forma de una conciencia y cultura profesional del
profesor de matemáticas que resulta difícil hacer explícita.
Los profesores expertos disponen de conocimientos distintos a los que poseen
los profesores noveles. Estos conocimientos distintos están estructurados de modo
adecuado a las exigencias del entorno y se muestran en parte a través del dominio de
procedimientos y en parte a través de rutinas pero no habitualmente en conocimientos reproducibles.
Esta dificultad provoca a veces un distanciamiento entre las herramientas facilitadas en la formación inicial del profesorado de matemáticas y los requerimientos
de la propia práctica docente.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S08. Conocimiento profesional del profesor de matemáticas
61
Del análisis de la comprensión de los estudiantes a la formación de profesores
Gloria Sánchez-Matamoros García
Departamento Didáctica de las matemáticas Facultad Ciencias de la Educación,
Universidad de Sevilla
[email protected]
La formación de profesores de matemáticas en la actualidad debe apoyarse en
un cuerpo de conocimientos generando a partir de investigaciones realizadas en
el campo de Didáctica de las Matemáticas. Las investigaciones sobre la comprensión y el desarrollo de dicha comprensión de un determinado concepto matemático
aportan conocimiento científico. Trasladar este conocimiento científico a los programas de formación de profesores conllevan cierta transferencia del conocimiento.
Esta transferencia del conocimiento hay que entenderla en el sentido de que dichas
investigaciones aportan información para el diseño de módulos de formación de
estudiantes para profesor. En este trabajo se presentarán algunos ejemplos de este
proceso de transferencia usando el concepto de derivada como ejemplo.
Palabras claves: desarrollo de la comprensión, formación de profesores, competencia docente, mirar profesionalmente, derivada.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S09. Ecuaciones diferenciales y sistemas dinámicos
http://rsme2015.ugr.es/s09.php
Rotation Numbers for Planar Attractors of Equivariant Homeomorphisms
Begoña Alarcón
Universidade Federal Fluminense
[email protected]
Given an integer n > 1, we consider Zn -equivariant and orientation preserving
homeomorphisms of the plane with an asymptotically stable fixed point at the origin. We present examples without periodic points and having some complicated
dynamical features. The key is a preliminary construction of Zn -equivariant Denjoy
maps of the circle.
References
1. B. Alarcón. Rotation numbers for planar attractors of equivariant homeomorphisms. Topological Methods in Nonlinear Analysis, 42 n.2, 327-343, 2013.
Dynamics and optimal control of chemotherapy for low grade gliomas
Juan Belmonte Beitia
UCLM
[email protected]
Coautores: Clara Rojas, Victor M. Pérez-García
We discuss the optimization of chemotherapy treatment for low-grade gliomas
using a mathematical model. We analyze the dynamics of the model, study the stability of the solutions and characterize the optimal controls on drug distribution,
using different strategies, including quadratic and linear controls. We establish the
existence of the optimal control, and solve for the control in both the quadratic and
linear case
Centros y cíclos límite para familis de ecuaciones de Abel
José Luis Bravo Trinidad
Universidad de Extremadura
[email protected]
Coautores: M.J. Álvarez, M. Fernández, R. Prohens
Consideremos la familia de ecuaciones de abel
Ã
!
Ã
!
n
m
X
X
x0 =
a i A i (t ) x 2 +
b i B i (t ) x 3 ,
i =1
i =1
donde A i , B i son monomios trigonométricos fijados y (a 1 , . . . , a n , b 1 , . . . , b m ) ∈ Rn+m
parámetros. Para un valor concreto de los parámetros, decimos que la ecuación
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S09. Ecuaciones diferenciales y sistemas dinámicos
63
tiene un centro si toda solución acotada es periódica y, si no es un centro, denominamos ciclo límite a una solución periódica aislada en el conjunto de las soluciones
periódicas
En este contexto, planteamos dos problemas:
1. Caracterizar los A i , B i tales que la ecuación tenga un centro para todo valor
de los parámetros
2. Caracterizar los A i , B i tales que exista un ciclo límite para algún valor de los
parámetros
Mostraremos algunos resultados parciales a estos problemas.
Referencias
1. M.J. Álvarez, J.L. Bravo, M. Fernández, R. Prohens, Centers and limit cycles for
a family of Abel equations, in preparation.
2. M.J. Álvarez, J.L. Bravo, M. Fernández, R. Prohens, Existence of non-trivial
limit cycles in Abel equations with symmetries, Nonlinear Anal. TMA. 84 (2013)
18–28.
3. M.A.M. Alwash, N.G. Lloyd, Nonautonomous equations related to polynomial
two dimensional systems, Proc. Roy. Soc.Edinburgh, 105A (1987) 129–152.
4. A. Cima, A. Gasull, F. Maosas, A simple solution of some composition conjectures for Abel equations, J. Math. Anal. Appl. 398 (2013), 477–486.
La persistencia de un punto de equilibrio como solución periódica en sistemas
forzados
Adriana Buic˘a
Universitatea Babes-Bolyai, Cluj-Napoca, Romania
[email protected]
Coautores: Rafael Ortega, Universidad de Granada
En dimensión dos, obtenemos una caracterización del hecho de que un punto
de equilibrio aislado de un sistema autónomo persiste como solución T-periódica
en sistemas forzados T-periódicos. Además, presentamos resultados en dimensión
arbitraria.
Referencias
1. A. Buic˘a, R. Ortega, Persistence of equilibria as periodic solutions of forced
systems, J. Differential Equations 252 (2012), 2210–2221.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S09. Ecuaciones diferenciales y sistemas dinámicos
64
Problemas con reflexión y oscilación de soluciones de ecuaciones phi-laplacianas
Alberto Cabada
Departamento de Análisis Matemático, Universidad de Santiago de Compostela.
[email protected]
Coautores: Adrián Tojo, Departamento de Análisis Matemático, Universidad de Santiago de Compostela.
En un resultado reciente los autores han probado la equivalencia de un problema de primer orden con argumento reflejado y una determinada ecuación diferencial phi-laplaciana de segundo orden.
Con este fin, estudiamos en este trabajo, la existencia y oscilación de las soluciones de ecuaciones diferenciales phi-laplacianas, prestando especial atención al
cálculo explícito del período. Los resultados obtenidos nos permiten obtener condiciones suficientes que garantizan la existencia de solución periódica del problema
con reflexión.
Soluciones positivas para un problema de frontera periódico relacionado con el
fenómeno de Liebau
Jose Angel Cid
Universidad de Vigo
[email protected]
Coautores: G. Infante, M. Tvrdy y M. Zima
En esta charla presentaremos condiciones suficientes para la existencia y no existencia de soluciones positivas del problema de frontera periódico
x 00 (t ) + ax 0 (t ) = r (t )x α (t ) − s(t )x β (t ),
x(0) = x(T ),
t ∈ [0, T ],
x 0 (0) = x 0 (T ),
siendo a > 0, r, s ∈ C [0, T ] y 0 < α < β < 1. Nuestros resultados generalizan algunos
de los obtenidos en [3], donde se estudió un problema singular relacionado con el
fenómeno de Liebau [1,2,5].
Este trabajo ha sido realizado en colaboración con G. Infante, M. Tvrdý y M.
Zima, [4].
Referencias
1. G. Liebau, Über ein ventilloses Pumpprinzip, Naturwissenschaften 41 (1954),
327.
2. G. Propst, Pumping effects in models of periodically forced flow configurations, Physica D 217 (2006), 193-201.
3. J. A. Cid, G. Propst y M. Tvrdý, On the pumping effect in a pipe/tank flow configuration with friction, Physica D 273-274 (2014), 28-33.
4. J. A. Cid, G. Infante, M. Tvrdý y M. Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. Appl. 423 (2015),
1546-1556.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S09. Ecuaciones diferenciales y sistemas dinámicos
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5. P. J. Torres, Mathematical models with singularities, aparecerá en Atlantis Briefs,
Springer.
Dynamics of an infinite dimensional gradient flow of fourth order
Carlos Escudero
Universidad Autónoma de Madrid
[email protected]
This talk will review a series of recent results obtained for a parabolic fourth order partial differential equation with a second order quadratic nonlinearity. We will
summarize the existence and multiplicity of stationary solutions as well as the existence and uniqueness of solutions for the full evolution problem. Moreover we
study the evolution problem as an infinite dimensional dynamical system and partially establish the different dynamical outputs. This talk is based on several joint
works with R. Hakl, F. Gazzola, I. Peral and P. J. Torres.
Large deviations principles of Non-Freidlin-Wentzell type
Jaykov Foukzon
Israel Institute of Technology
<[email protected]>
Generalized Large deviation principles (SLDP) was developed for ColombeauIto SDE with a random coefficients. We is significantly expand the classical theory of
large deviations for randomly perturbed dynamical systems developed by Freidlin
and Wentzell.Using SLDP approach, jumps phenomena, in financial markets, also
is considered. Jumps phenomena, in financial markets is explained from the first
principles, without any reference to Poisson jump process. In contrast with a phenomenological approach we explain such jumps phenomena from the first principles, without any reference to Poisson jump process.
http://arxiv.org/abs/0803.2072
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S09. Ecuaciones diferenciales y sistemas dinámicos
66
Global Continuation of symmetric periodic orbits in the Sitnikov Problem.
Jorge Galan-Vioque
Universidad de Sevilla
[email protected]
Coautores: D. Nuñez and A. Rivera
The Sitnikov problem is a special case of restricted 3-body problems where the
two primaries with equal masses are moving in a circular or an elliptic orbit of the 2body problem, and the infinitesimal mass is moving on the straight line orthogonal
to the plane of motion of the primaries which passes through their center of mass.
In [1] Llibre and Ortega studied analytically making use of the global continuation theorem the families of symmetric periodic orbits of the elliptic Sitnikov problem for non necessarily small values of the eccentricity e, and showed that some
periodic orbits for e = 0 can be continued to all values of e in [0, 1). In [2] Ortega
and Rivera analyzed the bifurcation of solution from the center of mass which is an
equilibrium of the problem. There are also numerical studies by Belbruno et al [3] a
nd Jiménez-Lara and Escalona Buendía [4] describing the families of periodic orbits
for almost all values of the eccentricity.
In this work we concentrate on the stability and bifurcation behavior of the families of symmetric periodic orbits that are born at the circular problem or emanate
form the equilibrium solution and provide complementary information to the existing results. We present a combination of analytical estimates of the eccentricity
intervals of ellipticity and numerical results based on a continuation technique developed for conservative and symmetric systems [5,6].
Referencias
1. J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem, SIAM J. Applied Dynamical Systems., 7 (2008), 561-576
2. R. Ortega and A. Rivera, Global bifurcations from the center of mass in the
Sitnikov problem. Discrete and Continuous Dynamical Systems, Series B 14,
719-732 (2010).
3. E. Belbruno, J. Llibre, and M. Ollé, On the families of periodic orbits which
bifurcate from the circular Sitnikov motions, Celestial Mech. Dynam. Astronom., 60 (1994), 99-129.
4. L. Jiménez-Lara and A. Escalona-Buendía Symmetries and bifurcations in the
Sitnikov problem. Celestial Mech. Dynam. Astronom., 79 (2001), 97-117.
5. Muñoz-Almaraz, F.-J., E. Freire, J. Galaán, E. Doedel and A. Vanderbauwhede
(2003) Continuation of periodic orbits in conservative and Hamiltonian systems. Physica D 181, 138.
6. Galán, J., Munñoz-Almaraz, F. J., Freire, E., Doedel, E. and Vanderbauwhede,
A.: 2002, Stability and bifurcations of the figure-8 solution of the three-body
problem, Physical Review Letters 88, 241101-4.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Caos en cadenas alimenticias de tres especies
Santiago Ibáñez
Universidad de Oviedo
[email protected]
Coautores: Pawel Pilarczyk
Los mecanismos subyacentes a la génesis de oscilaciones en interacciones depredador-presa fueron explicados por Lotka y Volterra y, a partir de sus trabajos, el
estudio de cadenas alimenticias con dos especies se convirtió en uno de los principales intereses de la Ecología Teórica. Hace dos décadas Hastings y Powel [3] observaron que las cadenas tróficas de tres especies podían exhibir comportamientos
caóticos. Desde entonces numerosos estudios han tenido como objetivo comprender el origen de tales comportamientos.
Nosotros proponemos el uso de una herramienta novedosa, basada en resultados de teoría de bifurcación local, para probar la existencia de dinámicas caóticas.
Siendo más precisos, explicaremos como cierto tipo de singularidades pueden jugar el papel de centros organizadores de tales procesos y aplicaremos el método a
algunos modelos bien conocidos de cadenas tróficas de tres especies (ver [1,5). Los
resultados que se presentan están recogidos en [4].
El método ya ha sido aplicado con éxito a un modelo de reacciones químicas
acopladas (ver [2]) y con este trabajo queremos ilustrar el amplio rango de situaciones en las que es susceptible de ser utilizado.
Referencias
1. B. Blassius, A. Huppert, L. Stone, Complex dynamics and phase synchronization in spatially extended ecological systems, Nature 399 (1999) 354–359.
2. F. Drubi, S. Ibáñez, J. A. Rodríguez, Coupling leads to chaos, J. Differential
Equations 239 (2) (2007) 371–385.
3. A. Hastings, T. Powell, Chaos in a three-species food chain, Ecology 72 (1991)
896–903.
4. S. Ibáñez, P. Pilarczyk, Nilpotent equilibria and chaos in tri-trophic food chains
(preprint).
5. L. Stone and D. He, Chaotic oscillations and cycles in multi-trophic ecological
systems, J. Theor. Biol. 248 (2007) 383–390.
Dinámica de modelos de población con efecto Allee
Eduardo Liz
Universidad de Vigo
[email protected]
El efecto Allee fue descrito en los años 30 para indicar que en algunas especies
las densidades de población demasiado bajas reducen la aptitud de los individuos
para sobrevivir. En el caso de un efecto Allee fuerte, hay un umbral para la densidad
de población por debajo del cual hay un gran riesgo de extinción. El efecto Allee ha
sido objeto de extensivo estudio en los últimos años por su importancia en la gestión
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S09. Ecuaciones diferenciales y sistemas dinámicos
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de recursos naturales. En este trabajo consideramos una población cuya dinámica
está gobernada por una ecuación diferencial con retardo y analizamos la influencia del efecto Allee en conexión con otros parámetros del modelo, como el tamaño
del retardo y la intensidad de la mortalidad (generalmente debida a la presencia de
depredadores o a la acción humana en forma de caza, pesca o control de plagas).
Basados en un modelo discreto asociado, en nuestros principales resultados
aportamos condiciones suficientes para la persistencia de la especie y la región de
atracción de un equilibrio positivo. Además, analizamos los cambios en la dinámica
cuando se escoge la intensidad de mortalidad como parámetro de bifurcación y
mostramos que la consideración de un retardo no sólo da lugar a oscilaciones en el
tamaño de la población, sino que puede interactuar con el efecto Allee para provocar la extinción o la permanencia de la especie, dependiendo de las condiciones
iniciales.
La charla está basada en un artículo conjunto con Alfonso Ruiz Herrera.
Referencias
1. E. Liz, A. Ruiz-Herrera, Delayed population models with Allee effects and exploitation, Math. Biosci. Eng. 12 (2015), in press.
Sistemas hamiltonianos débilmente disconjugados. Aplicaciones en teoría
de control.
Rafael Obaya
Universidad de Valladolid
[email protected]
Introducimos los sistemas hamiltonianos débilmente disconjugados y analizamos
sus propiedades cualitativas más importantes. Utilizamos técnicas de dinámica noautónoma para probar que dichos sistemas son fundamentales para resolver problemas de control cuadráticos no autónomos con horizonte infinito.
KAM Tori for Near-Rectilinear Motions in the Spatial Three-Body Problem
Jesús Palacián
Departamento de Ingeniería Matemática e Informática
[email protected]
Coautores: Flora Sayas y Patricia Yanguas
We deal with the spatial three-body problem in the various regimes where the
Hamiltonian is split as the sum of two Keplerian systems plus a small perturbation.
This is a region of the phase space T ∗ R6 where the perturbation is small [2], the
so called perturbing region (Qε,n ). In particular we prove the existence of quasiperiodic motions where the inner particles describe bounded near-rectilinear trajectories whereas the outer particle follows an orbit lying near the invariable plane.
These motions fill in five-dimensional invariant tori. Moreover, the inner particles
move in orbits either near an axis perpendicular to the invariable plane or near the
invariable plane.
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S09. Ecuaciones diferenciales y sistemas dinámicos
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By averaging over the mean anomalies, truncating higher-order terms and using singular reduction theory we get a one-degree-of-freedom Hamiltonian system
defined in a singular reduced space, the so called orbit space. In [2] we analyse the
relative equilibria and bifurcations and in [3] we reconstruct the invariant tori corresponding to motions of non-rectilinear type. Three of the relative equilibria of
the reduced Hamiltonian in the orbit space are elliptic points that correspond to
near-rectilinear motions of the inner bodies and these are the ones we study in the
present paper. We carry out the reconstruction of the KAM 5-tori surrounding the
three equilibria. By means of our reduction process we regularise the double inner collisions and this allows us to build sets of action-angle coordinates needed to
apply KAM theory. The motions we deal with admit different combinations, for instance, the outer particle may move in a near-circular orbit or the invariable plane
may coincide with the horizontal plane. This leads to various situations that have to
be analysed in different intermediate reduced spaces. We achieve our study by considering all possible cases, constructing an adequate set of coordinates and computing the corresponding torsion in each case. Hence, our analysis is global and we
characterise properly all type of bounded motions of the three particles (excluding
triple collisions). In order to achieve the existence of the quasi-periodic motions we
use a theorem by Han, Li and Yi [1] that allows us to handle the high-order degeneracy of the Hamiltonians involved in the process. The application of this theorem is
not straightforward as one needs to bring the Hamiltonian to normal form through
successive changes of symplectic coordinates and these transformations are rather
cumbersome.
This work is part of the second author’s PhD thesis.
Referencias
(1) Y. H AN , Y. L I , AND Y. Y I, Invariant tori in Hamiltonian systems with high order
proper degeneracy, Ann. Henri Poincaré, 10 (2010), pp. 1419–1436.
(2) J. F. PALACIÁN , F. S AYAS , AND P. YANGUAS, Regular and singular reductions in
the spatial three-body problem, Qual. Theory Dyn. Syst., 12 (2013), pp. 143–
182.
(3) J. F. PALACIÁN , F. S AYAS , AND P. YANGUAS, Flow reconstruction and invariant
tori in the spatial three-body problem, submitted.
Existence of stationary knotted vortex tubes
Daniel Peralta-Salas
ICMAT
[email protected]
I will show the existence of knotted and linked invariant tori of arbitrary topology
for steady solutions to the incompressible Euler equation in R3 . The problem of the
existence of steady knotted vortex tubes can be traced back to Lord Kelvin. This is
based on joint work with A. Enciso (Acta Math. in press).
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S09. Ecuaciones diferenciales y sistemas dinámicos
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Singular reduction for resonant Hamiltonian systems
Patricia Yanguas Sayas
Departamento de Ingeniería Matemática e Informática, Edificio Encinas, Campus de
Arrosadía, 31006 Pamplona, Navarra, Spain
[email protected]
Coautores: Ken R. Meyer, Jesús F. Palacián Subiela
The problem we discuss in this presentation has been a testing ground of many
different methods for analysing Hamiltonian systems and in particular finding periodic solutions and their stability. Here we illustrate the use of singular reduction on
this classic problem. Singular reduction lowers the dimension of the problem under
study; so, given that our first test problem is a two degrees of freedom Hamiltonian
system in R4 , it will be reduced to a Hamiltonian system of one degree of freedom on
a two-dimensional real algebraic surface called an orbifold. The two-dimensionality
leads itself to a graphical representation with a better geometric insight on the flow
of the system.
The planar restricted three body problem is considered as a benchmark for the
last 85 years, and in particular there has been a bunch of works, mainly of numerical type, to obtain the periodic solutions and related invariant manifolds around
the equilibrium points L 4 and L 5 . We shall illustrate how reduction theory is used
to establish rigorously the existence and stability of these solutions, as well as the
different types of bifurcations. We shall also mention how our theoretical approach
can be combined with normal forms in order to obtain good initial conditions to
approximate the periodic solutions and their associated manifolds.
Finally we shall jump to n degrees of freedom, and in particular to the three degrees of freedom case where the polynomial invariants needed in the singular reduction theory are computed using an algorithm based on integer programming. After
computing these invariants we use Gröbner bases theory and the division algorithm
for multivariate polynomials to deal with the equations of motion in terms of the
invariants. We shall apply the theory with the aim of finding some periodic solutions in resonant Hamiltonian systems of three degrees of freedom with semisimple
linear part.
Favard condition and recurrent solutions of almost periodic equations
Massimo Tarallo
Universita’ degli Studi di Milano - Italia
[email protected]
Coautores: Juan Campos
It is well known that linear almost periodic equations may not have almost periodic solutions, even when bounded solutions are known to exist. This pathology is
related to the failure of the so–called Favard separation condition which, to some extent, is a kind of natural divide between nice and ugly equations. On the other hand,
all the known examples of pathological equations admit solutions with a recurrence
property weaker than almost periodicity: the so–called almost automorphy. During
the talk, I will introduce the notion of almost automorphy and present a joint work
with J. Campos, showing that such solutions exist in every almost periodic linear
equations with bounded solutions. The result depends on a careful analysis of the
way Favard separation condition breaks down.
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Center, weak-focus and ciclicity problems for planar systems with few monomials
Joan Torregrosa
Universitat Autònoma de Barcelona
[email protected]
The center-focus problem consists in distinguishing whether a monodromic singular point is a center or a focus. For singular points with imaginary eigenvalues,
usually called nondegenerate singular points, this problem was already solved by
Poincaré and Lyapunov, see [1]. The solution consists in computing several quantities called commonly the Poincaré–Lyapunov constants, and study whether they are
zero or not.
Despite the existence of many methods, the solution of the center-focus problem for simple families, like for instance the complete cubic systems or the quartic
systems with homogeneous nonlinearities, has resisted all the attempts. For this
reason, we propose to push on this question in another direction. We study this
problem for a natural family of differential systems with few free parameters but arbitrary degree.
We consider planar systems with a linear center at the origin that in complex
coordinates the nonlinearity terms are formed by the sum of few monomials. For
some families in this class, we study the center problem, the maximum order of a
weak-focus and the ciclicity problem. Several centers inside this family are done.
The list includes a new class of Darboux centers that are also persistent centers. We
study if the given list is exhaustive or not. We show that for each natural number p
there are differential equations of this type having at least p limit cycles. Moreover,
for a particular case which has homogeneous nonlinearities, we show examples with
several limit cycles and give a condition that ensures uniqueness and hyperbolicity
of the limit cycle.
The talk will be a review of the results [2,3].
Referencias
1. A. Gasull, J. Giné, J. Torregrosa. Center problem for systems with two monomial
nonlinearities. Preprint. 2014.
2. A. Gasull, C. Li, J. Torregrosa. Limit cycles for 3-monomial differential equations. Preprint. 2014.
3. A. M. Lyapunov. The general problem of the stability of motion. Taylor & Francis, Ltd., London, 1992. Translated from Edouard Davaux’s French translation
(1907) of the 1892 Russian original and edited by A. T. Fuller. Reprint of Internat. J. Control 55 (1992), no. 3.
The higher-dimensional Poincaré-Birkhoff theorem for Hamiltonial systems
Antonio J. Ureña
Universidad de Granada
[email protected]
Coautores: Alessandro Fonda (Università degli Studi di Trieste)
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S09. Ecuaciones diferenciales y sistemas dinámicos
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We propose a higher dimensional generalization of the Poincaré-Birkhoff Theorem which applies to Poincaré time maps of Hamiltonian systems. The maps under
consideration are neither required to be close to the identity nor to have a monotone twist. The annulus is replaced by the product of an N-dimensional torus and
the interior of an embedded sphere in the N -dimensional euclidean space; on the
other hand, the classical boundary twist condition is replaced by an avoiding rays
condition.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S10. Espacios de aplicaciones y grupos de
autoequivalencias
http://rsme2015.ugr.es/s10.php
Espacios de aplicaciones y teoría de homotopía racional de Quillen generalizada
de la deformación desde un punto de vista homotópico
Urtzi Buijs
Universidad de Málaga
[email protected]
Coautores: Aniceto Murillo
Comenzaremos presentando el principio de Deligne según el cual cada funtor
de deformación está gobernado por un álgebra de Lie graduada diferencial (LDG).
Por otra parte, describiremos cómo estas estructuras algebraicas modelan el tipo
de homotopía racional de los espacios de aplicaciones continuas.
Uniendo ambos enfoques bajo el nexo común de las LDG’s, detallaremos cómo
cada funtor de deformación puede ser “realizado geométricamente” como el tipo de
homotopía de un cierto espacio.
On homological stability for configuration spaces on closed manifolds
Federico Cantero
WWU Münster
[email protected]
Coautores: Martin Palmer
We introduce a new map between configuration spaces of points in a background
manifold —the replication map— and prove that it is a homology isomorphism in a
range with certain coefficients. This is particularly of interest when the background
manifold is closed, in which case the classical stabilisation map does not exist.
We then establish conditions on the manifold and on the coefficients under
which homological stability holds for configuration spaces on closed manifolds.
These conditions are sharp when the background manifold is a two-dimensional
sphere, the classical counterexample in the field. For field coefficients this extends
results of Church (2012) and Randal-Williams (2013) to the case of odd characteristic, and for p-local coefficients it improves results of Bendersky and Miller (2014).
Referencias
1. Church (2012), Homological stability for configuration spaces of manifolds,
Invent. Math. 188 (2): 465–504.
2. Randal-Williams (2013), Homological stability for unordered configuration spaces, Q. J. Math 64(1): 303-326.
3. Bendersky and Miller (2014), Localisation and homological stability for configuration spaces, Q. J. Math 65(3): 807–815
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Colímites homotópicos de espacios nilpotentes
Ramón Flores
Departamento de Matemáticas, Universidad Autónoma de Madrid
[email protected]
Coautores: Wojciech Chacholski, Emmanuel Farjoun, Jerome Scherer.
En esta charla describiremos una versión modificada de la clásica torre de Bousfield-Kan, y la utilizaremos para estudiar la A-homotopía de espacios nilpotentes.
En particular, probamos que las aproximaciones celulares de secciones de Postnikov
nilpotentes producen de nuevo secciones de Postnikov, y que un resultado análogo
puede obtenerse para espacios clasificadores de grupos nilpotentes, y de p-grupos
finitos en particular. Concluiremos con algunas consecuencias sobre aciclidad y
asfericidad de espacios nilpotentes.
Inyecciones geométricas de grupos de trenzas
Juan Gonzalez-Meneses
Universidad de Sevilla
[email protected]
Los grupos de trenzas pueden verse como grupos de automorfismos de superficies, salvo isotopía. Una inyección de una superficie en otra induce, en la mayoría
de los casos, una inyección de sus grupos de trenzas, que se conoce como inyección
geométrica.
En esta charla describiremos las inyecciones geométricas de los grupos de trenzas, demostrando que estas inyecciones no fusionan clases de conjugación. Veremos además que este resultado, válido para discos punteados, no se puede generalizar a otro tipo de superficies.
On the Connectivity of Branch Loci in Spaces of Fuchsian, NEC and
Schottky Groups
Milagros Izquierdo
Linköping University, Linköping, Sweden
[email protected]
Coautores: Antonio, F. Costa, Rubén A. Hidalgo and Ana M. Porto
Moduli spaces of Riemann surfaces of genus g , Klein surfaces of genus g and k
boundary components and handlebodies of genus g har orbifold structure where
the branch loci consist of such surfaces or handlebodies admitting automorphisms,
other than the identity.
We will see that while the branch locus in the moduli space of Riemann surfaces
of genus g is disconnected with many connected components, with a few exceptions
for low genera, the branch locus in the moduli space of orientable Klein surfaces
with boundary is connected.
The branch locus of Schottky space is connected for odd genera, it consist of
two connected components for genus two and it consist of at most two connected
components for even genera greater or equal to four.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S10. Espacios de aplicaciones y grupos de autoequivalencias
75
Subgrupos parabólicos en los grupos Artin y geodésicas
Luis Paris
Université de Bourgogne, France
[email protected]
Esta presentación se basa en una mezcla de dos trabajos, el primero en colaboración con Eddy Godelle, y el segundo en colaboración con Ruth Charney.
Un grupo de Artin A es un grupo abstracto que se define por una presentación
con un conjunto de generadores S y un conjunto de relaciones de la forma st s · · · =
t st · · · , donde la palabra de la izquierda y la de la derecha tienen la misma longitud. Los grupos de trenzas son ejemplos relevantes de tales grupos. El subgrupo A T
de A generado por un subconjunto T de S se llama subgrupo parabólico de A. Es
un hecho no trivial que un tal subgrupo es a su vez un grupo de Artin. Un ejemplo
de subgrupo parabólico es el grupo trenzas B m inmerso en forma estándar en B n ,
donde m ≤ n. No hay en general ninguna retracción A → A T a la inmersión A T ,→ A
que sea un homomorfismo, pero se puede describir un retracción conjuntista “natural” similar a un homomorfismo. Por ejemplo, la retracción de B n en B m se define
borrando las n − m ultimas cuerdas de una trenza con los puntos de salida y llegada
que ya no esten conectados. En esta presentación, describiremos esta retracción,
daremos una interpretación en términos de espacios celulares, y mostraremos una
aplicación a la convexidad de los subgrupos parabólicos.
Dupont-Guichardet-Wigner quasi-morphisms on mapping class groups
Wolfgang Pitsch
Universidad Autónoma de Barcelona
[email protected]
Coautores: Luis Funar (CNRS-Université Joseph Fourier)
In this talk we will present the construction of Dupont-Guichardet-Wigner quasimorphisms on the universal central extension of the mapping class groups. They
araise as pull-backs of the quasi-morphisms on pseudo-unitary groups via the quantum representations of mapping class groups. Their interest lies in the fact that on
the one hand, by deep results of Burger and Iozzi, they classify Zariski dense representations on pseudo-unitary groups and on the other hand they allow to control
the kernel of the quantum representations of mapping class groups.
On the Arithmeticity of Kodaira Fibrations
Sebastián Reyes-Carocca
Universidad Autónoma de Madrid
[email protected]
Coautores: Gabino González-Diez
A Kodaira fibration consists of a non-singular compact complex surface S, a
compact Riemann surface C and a surjective holomorphic map f : S → C everywhere of maximal rank such that the fibers are connected and not mutually isomorphic Riemann surfaces. In this talk, we study this kind of fibrations and we show that
whether or not the algebraic surface S is arithmetic (i.e. it is defined over a number
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S10. Espacios de aplicaciones y grupos de autoequivalencias
76
field) depends only on the biholomorphic class of its universal cover. This in turn
speaks of the diversity of universal covers in the world of complex surfaces in contrast with the uniformity of the one-dimensional case. In fact, we construct a very
explicit collection of Kodaira fibrations that gives rise to uncountably many mutually non-biholomorphic universal covers.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S11. Geometría algebraica
http://rsme2015.ugr.es/s11.php
Jumping numbers on rational surface singularities
Maria Alberich Carramiñana
Universitat Politècnica de Catalunya
[email protected]
Coautores: J. Álvarez Montaner, F. Dachs-Cadefau and V. González Alonso
We will present some new results on the computation of jumping numbers with
their multiplicities of any ideal sheaf around a rational surface singularity. In the
case of m-primary ideals (where m is the maximal ideal of the local ring of the surface at the rational point singularity), when these invariants are encoded all together
in a Poincaré series form, we will give a sort of rational expression for it.
Homología de Hochschild bivariante en esquemas
Leovigildo Alonso Tarrío
Departamento de Álxebra, Facultade de Matemáticas, Universidade de Santiago de
Compostela, E-15782 Santiago de Compostela, Spain
[email protected]
Coautores: Ana Jeremías López, Joseph Lipman
La homología de Hochschild en geometría algebraica se ha estudiado en el caso
especial de variedades lisas sobre un cuerpo de característica 0. La consideración de
las transformaciones de Fourier-Mukai por Caldararu, en conexión con esta teoría
cohomológica y su relación con el formalismo de Hodge-De Rham (a través del
isomorfismo HKR) muestra el interés de una exploración general sobre bases arbitrarias.
En la charla mostraremos cómo construir una teoría bivariante en el sentido de
Fulton y McPherson cuya homología coincide con la homología de Hochschild usual
en esquemas. Asimismo definiremos y expondremos las propiedades de la clase
fundamental en este contexto. La clase fundamental generaliza construcciones clásicas como la relación entre n-formas diferenciales y el haz dualizante y está relacionada con invariantes clásicos en cohomología. Veremos cómo es compatible con
cambio de base étale y posee la propiedad de transitividad que hace de la teoría bivariante de Hochschild una teoría orientada. Además, existe una segunda teoría
dual que coincide con la anterior cuando ambos esquemas son lisos.
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Essential minimum and equidistribution of small points on toric varieties
José Ignacio Burgos Gil
ICMAT (CSIC)
[email protected]
Coautores: Patrice Philippon, Martín Sombra
The toric dictionary between geometric properties of polarized toric varieties
and combinatorial properties of lattice polytopes can be extended to arithmetic
properties by the introduction of the roof function. In this talk we will show how
the roof function determines two arithmetic properties: the essential and absolute
minima of a toric variety and the equidistribution property of Galois orbits of small
points.
Cohomología de Gauss-Manin de familias de Dwork
Alberto Castaño Domínguez
Departamento de Álgebra e Instituto de Matemáticas (IMUS), Universidad de Sevilla
[email protected]
Una familia de Dwork es una deformación uniparamétrica monomial de una
hipersuperficie de Fermat (cf. [1, s. 2]). Fueron introducidas por Bernard Dwork
en los sesenta para comprender el efecto de una deformación en la función zeta de
una hipersuperficie sobre un cuerpo finito. En los últimos años ha aumentado el
interés en ellas gracias al descubrimiento de sus conexiones con otros problemas
provenientes de la geometría algebraica, la teoría de números o la física teórica.
La parte invariante de la cohomología de Gauss-Manin de dichas familias bajo
cierta acción de un grupo como en [1, s. 3] se ha estudiado en detalle por su conexión con las sumas de Kloosterman. Siguiendo el desarrollo de las cohomologías de
Weil, estos trabajos usan técnicas de cohomología `-ádica, análisis p-ádico o geometría diferencial compleja.
En esta charla presentaré el resultado análogo en el caso complejo algebraico,
conseguido gracias a la teoría de D-módulos, mencionando sus diferencias (y virtudes) con respecto a los anteriores y, si el tiempo lo permite, su posible extensión a
otras familias o al resto de autoespacios de la cohomología.
Referencias
1. N. M. Katz, Another look at the Dwork family, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol II, 89-126, Progr. Math., 270, Birkhäuser
Boston, Inc., Boston, MA (2009).
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Cohomología de Monsky-Washnitzer y homología cíclica.
Guillermo Cortiñas
Universidad de Buenos Aires
[email protected]
Coautores: Joachim Cuntz
La cohomología de Monsky-Washnitzer es una variante de la cohomología de de
Rham para variedades suaves sobre un cuerpo finito. En la charla mostraremos que
esta cohomología puede interpretarse en términos de homología cíclica periódica.
Este resultado es parte de un proyecto conjunto con Joachim Cuntz, actualmente
en desarrollo, en el que nos proponemos encontrar la “buena definición" de la homología cíclica periódica para álgebras sobre un cuerpo característica positiva.
Quantitative equidistribution of algebraic points in the N-dimensional torus
Carlos D’Andrea
Universitat de Barcelona
[email protected]
Coautores: Marta Narvaez Clauss, Martin Sombra
Bilu’s classical equidistribution theorem establishes that, given a generic sequence of points in the algebraic torus {αn }n∈N ⊂ (˘aQ× )N of small height, the sequence of
their Galois orbits equidistributes uniformly around the compact torus (S 1 )N . We
present a quantitative version of this result, obtained by reducing the problem via
projections to the one-dimensional case and reconstruction using techniques of
Fourier analysis.
Geometría Algebraica y códigos de evaluación
José Ignacio Farrán Martín
Universidad de Valladolid
[email protected]
Los códigos correctores de errores están presentes en múltiples aplicaciones de
la vida real, como los códigos de barras o los dispositivos ópticos de almacenamiento
de datos (CD/DVD). En las últimas décadas han surgido diversas construcciones de
códigos a partir de objetos de la Geometría Algebraica, como los códigos AG (a partir de curvas algebraicas), los códigos tóricos (a partir de variedades tóricas), o los
códigos diferenciales (a partir de formas diferenciales algebraicas), entre otras. En
el primer caso, los códigos AG han conseguido sobrepasar, desde el punto de vista
asintótico, la llamada cota de Gilbert-Varshamov.
Todas estas construcciones tienen un punto en común: la idea de evaluar funciones, extraídas de cierto espacio vectorial de dimensión finita, en un conjunto
finito de puntos de un determinado objeto geométrico. Eligiendo estos objetos
de manera apropiada, pueden estimarse los parámetros de los códigos obtenidos,
utilizando técnicas y resultados más o menos clásicos de la Geometría Algebraica.
Asimismo, el problema práctico de su construcción efectiva e implementación (codificación y decodificación) se relacionan con interesantes problemas de Geometría
Algebraica Computacional.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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En esta conferencia se presentarán diversas construcciones de códigos de evaluación, analizando los puntos clave de cada construcción, tanto desde el punto de
vista teórico como computacional, poniendo de manifiesto en cada caso las técnicas geométricas subyacentes.
Haces cuasi coherentes sobre pilas geométricas
Ana Jeremías López
Departamento de Álxebra, Facultade de Matemáticas, Universidade de Santiago de
Compostela, E-15782 Santiago de Compostela, Spain
[email protected]
Coautores: L. Alonso Tarrío, M. Pérez Rodríguez y M. J. Vale
El concepto de pila algebraica (algebraic stack) generaliza el de esquema y tiene
una gran relevancia en problemas de moduli, donde los objetos que se pretenden
clasificar pueden presentar automorfismos, lo que prohibe la existencia de un espacio fino de parámetros. De modo análogo al diccionario Álgebra — Geometría que
relaciona los esquemas afines y los anillos conmutativos, se tiene un diccionario
similar que relaciona algebroides de Hopf con pilas algebraicas cuasi-compactas y
semiseparadas (es decir, pilas geométricas) dotadas de una presentación fielmente
plana por un esquema afín. De hecho, un algebroide de Hopf es precisamente el
dual de un grupoide interno en la categoría de esquemas afines. Discutiremos cómo
describir haces cuasi coherentes sobre este tipo de pilas algebraicas en términos
de comódulos y veremos que la categoría de tales haces es abeliana y tiene buenas propiedades. La categoría derivada de haces cuasi coherentes sobre una pila
geométrica es monoidal cerrada. Además, bajo la condición de existencia de resoluciones globales esta categoría satisface los axiomas de categoría de homotopía
estable en el sentido de Hovey, Palmieri y Strickland.
Ecuaciones KP, conexiones con Física, Aritmética y Geometría
José María Muñoz Porras
Universidad de Salamanca
[email protected]
La charla está basada en resultados de varios trabajos conjuntos con E. Gómez
González, F. Pablos Romo y F. Plaza Martín. El principal objetivo de estos trabajos
es ofrecer una formulación algebraica de la teoría de solitones y aplicar sus resultados al estudio de problemas clásicos de moduli de curvas y fibrados vectoriales.
Asimismo se ofrece una nueva interpretación de las leyes de reciprocidad para curvas algebraicas.
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Conmensurabilidad y Ley General de Reciprocidad en Geometría Algebraica
Fernando Pablos Romo
Dpto. de Matemáticas, Universidad de Salamanca, Plaza de la Merced, 1-4, 37008,
Salamanca
[email protected]
El objetivo de la charla es ofrecer una teoría general de leyes de reciprocidad para
símbolos en espacios vectoriales arbitrarios (a partir de la noción de conmensurabilidad introducida por J. Tate en [1]), y mostrar que leyes de reciprocidad clásicas en
Geometría Algebraica son casos particulares de esta teoría (suma de valoraciones en
una curva completa, Teorema de los Residuos, Ley de Reciprocidad de Weil o Ley de
Reciprocidad para el Residuo Normado de Hilbert). Además, varias leyes de reciprocidad introducidas en los últimos años por D. V. Osipov, A. N. Parshin, I. Horozov, I.
Horozov - M. Kerr o D. Hernández Serrano - F. Pablos Romo, también pueden ser
deducidas de la expresión general.
Referencias
1. Tate, J., Residues of Differentials on Curves, Ann. Scient. Éc. Norm. Sup., 4a
série 1, (1968) 149-159.
On the blow up at equimultiple centers, and simplification of singularities.
Orlando Villamayor
Universidad Autónoma de Madrid
[email protected]
Fix a perfect field k and a variety X , or, more generally, a pure dimensional
scheme of finite type over k. Let d be the dimension of X , and fix a singular point
x ∈ X of multiplicity e. A local simplification of x ∈ X is a proper birational map,
X ← X 1 so that x 1 ∈ X 1 has multiplicity < e at any x 1 mapping to x. We don’t know
of the existence of simplifications but we show that one con construct inclusions in
a complete intersection, say X ⊂ X 0 , so that any local simplification of x ∈ X induces
one of x ∈ X 0 and vice versa. We use this to prove our main result: The existence of a
local simplification of x ∈ X is guaranteed by induction on the dimension d , unless
(C X ,x )r ed is regular. Here C X ,x denotes the tangent cone of x ∈ X , and (C X ,x )r ed the
underlying reduced scheme.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S12. Geometría convexa e integral
http://rsme2015.ugr.es/s12.php
Desigualdades de Rogers-Shephard y cuerpos de convolución
David Alonso-Gutiérrez
Universitat Jaume I
[email protected]
Coautores: Bernardo González, C. Hugo Jiménez, Rafael Villa
We prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal
relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in
all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study
of some geometrical properties of these new sets.
Three related conjectures for log-concave probabilities: KLS, thin shell width and
the slicing problem
Jesús Bastero
Universidad de Zaragoza
[email protected]
In this lecture the main known results on the Kannan-Lovász-Simonovits spectral gap, the thin shell width conjectures and their relations with the slicing problem
will be surveyed. Also the contributions of the author with David Alonso on the hyperplane projections of the `np balls will be presented.
Integral geometry of transitive group actions
Andreas Bernig
Goethe-Universität Frankfurt
[email protected]
I will give an overview over recent results on kinematic formulas for groups that
act transitively on the sphere bundle of an affine space. The most interesting case is
that of hermitian integral geometry, where a complete set of kinematic formulas was
recently worked out in collaboration with Joseph Fu (University of Georgia) using
Alesker’s theory of valuations, representation theory and geometric measure theory.
Some other cases (quaternionic integral geometry and some exceptional groups like
G 2 ) will be briefly mentioned.
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Divisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter
Antonio Cañete
Universidad de Sevilla
[email protected]
In this talk we shall study an optimization problem involving the diameter functional. More precisely, fix k ∈ N, k ≥ 3, and consider a k-rotationally symmetric
planar convex body C . The question we shall focus on is: which is the division of
C into k connected subsets minimizing the maximum relative diameter? We recall
that the maximum relative diameter is the maximum of the diameters of the k subsets determined by the division. We shall see that the so-called standard k-partition,
consisting of k inradius segments symmetrically placed, is a minimizing division for
k ≤ 6, but not when k ≥ 7.
Moreover, for each k ∈ N, k ≥ 3, we shall characterize the optimal body for this
problem (that is, the set with the division attaining the lowest value for the maximum
relative diameter functional). We will finish our talk making some comments for the
case k = ∞.
This is part of a joint work with Uwe Schnell (University of Applied Sciences Zittau/Görlitz) and Salvador Segura (Universidad de Alicante).
Referencias
1. A. Cañete, C. Miori, S. Segura, Trisections of a 3-rotationally symmetric planar
convex body minimizing the maximum relative diameter, Journal of Mathematical Analysis and Applications, 418 (2014), 1030–1046.
2. A. Cañete, U. Schnell, S. Segura, Subdivisions of k-rotationally symmetric planar convex bodies minimizing the maximum relative diameter, preprint 2014.
On complete systems of inequalities
Bernardo González Merino
C/Madrid, no 1, 2o A, 30003, Murcia
[email protected]
Coautores: René Brandenberg
For many years mathematicians have studied the behavior of two or more geometric functionals by means of inequalities relating them as well as extremal sets
satisfying their equality conditions.
Each new inequality obtained is interesting on its own, but it is also possible to
ask if a finite collection of inequalities concerning several geometric magnitudes is
large enough to determine the existence of the convex set. Such a collection is called
a complete system of inequalities: a system of inequalities relating all the geometric
functionals such that for any set of numbers satisfying those conditions, a convex
set with these values of the characteristics exists.
Historically Blaschke (1) and Santaló (4) were the first mathematicians studying
these problems. In the last years Hernández Cifre and Segura Gomis (see, for instance, (2,3)) showed new insights to some of the classical problems.
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In the talk we will discuss these results and recent development in the topic.
In particular, we will discuss a complete system of inequalities for n-dimensional
convex sets.
Referencias
(1) W. Blaschke, Eine Frage Über Konvexe Körper, Jahresber. Deutsch. Math. Ver.,
25 (1916), 121–125.
(2) M. A. Hernández Cifre, Is there a planar convex set with given width, diameter
and inradius?, Amer. Math. Monthly, 107 (2000), 893– 900.
(3) M. A. Hernández Cifre, S. Segura Gomis, The missing boundaries of the Santaló diagrams for the cases (d , w, R) and (w, R, r ), Discrete Comp. Geom., 23
(2000), 381–388.
(4) L. Santaló, Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa planas, Math. Notae, 17 (1961), 82–104.
Cone-volume measure of convex bodies
Martin Henk
Technische Universität Berlin
[email protected]
Coautores: Károly J. Böröczky
We show that the cone-volume measure of a convex body with centroid at the
origin satisfies the subspace concentration condition. This implies, among others, a
conjectured best possible inequality for the U-functional of a convex body. For both
results we provide stronger versions in the sense of stability inequalities.
Extensions of Minkowski’s theorem on successive minima
María A. Hernández Cifre
Universidad de Murcia
[email protected]
Coautores: Martin Henk and Matthias Henze
Let K be a 0-symmetric convex body, i.e., a compact and convex set satisfying
that K = −K , in the n-dimensional Euclidean space Rn , and let Zn denote the integer lattice. The well-known Minkowski 2nd theorem in the Geometry of Numbers
provides optimal upper and lower bounds for the volume of K in terms of its successive minima:
n
n
Y
2
2
1 Y
≤ vol(K ) ≤
;
n
n
n! i =1 λi (K , Z )
i =1 λi (K , Z )
©
ª
here, λi (K , Zn ) = min λ > 0 : dim(λK ∩ Zn ) ≥ i is the i -th successive minimum
of K with respect to the integer lattice, 1 ≤ i ≤ n, and vol(K ) denotes the volume
(Lebesgue measure) of K .
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In this talk we will make a brief historical tour on this inequality and its generalizations, and then we will show new analogs of the theorem from two different
points of view: either relaxing the symmetry condition, assuming for instance that
the centroid of the body lies at the origin, or replacing the volume functional by the
surface area.
Characterization of dual mixed volumes via polymeasures
Carlos Hugo Jiménez G.
Universidad Federal de Minas Gerais
[email protected]
Coautores: Ignacio Villanueva
We present a proof of a characterization of the dual mixed volume in terms of
functional properties of the polynomial associated to it. To do this, we use tools
from the theory of multilinear operators on spaces of continuos functions. Along
the way we reprove, with these same techniques, a recently found characterization
of the dual mixed volume.
Valuations on lattice polytopes
Monika Ludwig
TU Wien
[email protected]
Coautores: Károly J. Böröczky (Central European University, Budapest, and Alfréd
Rényi Institute of Mathematics)
Lattice polytopes are convex hulls of finitely many points with integer coordinates in Rn . The classification of real-valued invariant valuations on lattice polytopes by Betke & Kneser is classical (and will be discussed during the talk). Building
on this, a complete classification is established of Minkowski valuations on lattice
polytopes that intertwine the special linear group over the integers and are translation invariant. In the contravariant case, the only such valuations are multiples
of projection bodies. In the equivariant case, the only such valuations are generalized difference bodies combined with multiples of the newly defined discrete Steiner
point.
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The isodiametric problem and other inequalities in the constant curvature
2-spaces
Antonio R. Martínez Fernández
Universidad de Murcia
[email protected]
Coautores: María Ángeles Hernández Cifre
If K is a planar convex body, i.e., a compact convex set in the plane, with area
A(K ), perimeter p(K ) and diameter D(K ), the well-known isodiametric inequality
states that
πD(K )2 ≥ 4A(K ),
with equality if and only if K is a circle. The isodiametric inequality can be obtained
as a consequence of the famous isoperimetric inequality, p(K )2 ≥ 4πA(K ), and the
classical Rosenthal-Szasz’s theorem p(K ) ≤ πD(K ). The isoperimetric problem has
its analog on the sphere S2κ and the hyperbolic plane H2κ with curvature κ ≷ 0: if K
is the region bounded by a convex curve on the sphere (respectively, the hyperbolic
plane), then
p(K )2 ≥ 4πA(K ) − κA(K )2 ,
with equality only for the geodesic discs.
In this talk we will recall the few classical inequalities of the Euclidean plane
which have been translated into the sphere and the hyperbolic space (for instance,
Jung’s inequality or Bonnesen’s inequality), showing next several new inequalities
for centrally symmetric convex bodies in the 2-dimensional spaces of constant curvature κ (which have their analog in the plane). For instance, we show the relation
between the perimeter and the diameter of a symmetric convex body (RosenthalSzasz inequality) which, together with the well-known spherical/hyperbolic isoperimetric inequality, allows to solve the corresponding isodiametric problem.
Some geometry of convex bodies in C(K)
José Pedro Moreno
Departamento de Matemáticas, UAM
[email protected]
Coautores: R. Schneider
In this talk we are concerned with some problems related to vector addition and
diametric completion procedures of convex bodies in C (K ) spaces. The results follow from a systematic investigation of generalized order intervals and intersections
of balls. We will present some characterization of the underlying compact Hausdorff
space as a Stonean space in terms of some properties of convex bodies.
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Isoperimetric inequalities in unbounded convex bodies
Manuel Ritoré
Universidad de Granada
[email protected]
Coautores: Gian Paolo Leonardi, Efstratios Vernadakis
We consider the relative isoperimetric problem in unbounded convex domains
in Euclidean space and extend some of the results already proven in the Ph. D. Thesis
of E. Vernadakis, such as the (strict) convexity of the isoperimetric profile. This is
joint work in progress with Gian Paolo Leonardi and Efstratios Vernadakis.
Dyck path triangulations and extendability
Camilo Sarmiento
Otto-von-Guericke-Universität Magdeburg
[email protected]
Coautores: Cesar Ceballos y Arnau Padrol
In this talk, we introduce the Dyck path triangulation of the cartesian product of
two simplices: its maximal simplices are given by Dyck paths along with their orbit
under a cyclic action. The construction also naturally produces triangulations of the
product of two simplices consisting of rational Dyck paths. Our study of the Dyck
path triangulation is motivated by an extendability problem for certain kinds of partial triangulations of the product of two simplices. We present a complete solution
to this extendability problem and, with an explicit construction of non-extendable
partial triangulations, we prove that our characterization of extendability is optimal.
Time permitting, we will briefly mention interesting interpretations of our results in
the language of tropical oriented matroids, which are analogous to classical results
in oriented matroid theory. The content of the talk is based on joint work with C.
Ceballos and A. Padrol.
Integral Geometry of Curved Spaces
Gil Solanes
Universitat Autònoma de Barcelona
[email protected]
The kinematic formulas of Blaschke, Santaló, Federer and Chern are fundamental results in the Integral Geometry of Euclidean space. The generalization of these
formulas to the sphere is also classical and has important applications. It is known
since the 90’s that similar formulas exist in other ambient spaces, but only recently
they could be found explicitly in some cases. In the talk I will present both the classical and the new results and also the algebraic approach that made the latter possible.
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The isoperimetric profile of a bounded convex body
Efstratios Vernadakis
Universidad de Granada
[email protected]
Coautores: Manuel Ritoré
In this talk we shall consider the problem of minimizing the relative perimeter
under a volume constraint in the interior of a bounded convex body, i.e., a compact
convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we
shall prove the equivalence between Hausdorff and Lipschitz convergence, the continuity of the isoperimetric profile with respect to the Hausdorff distance and the
convergence in Hausdorff distance of sequences of isoperimetric regions and their
free boundaries. We shall also describe the behavior of the isoperimetric profile for
small volume, and the behavior of isoperimetric regions for small volume.
Convolution bodies and volume inequalities
Rafael Villa
Universidad de Sevilla
[email protected]
Coautores: David Alonso-Gutiérrez, Bernardo González-Merino, C. Hugo Jiménez.
A quantitative version of Minkowski sum, involving the proportional measure of
the intersections, gives the following subset of A + B
A +θ B = {x ∈ A + B : |A ∩ (x − B )| ≤ θM (A, B )}
for θ ∈ [0, 1], whenever M (A, B ) := sup x∈A+B |A∩(x −B )| is finite. This set is called the
θ-convolution set of A and B. This set has been widely studied for symmetric convex
bodies in the literature; the term convolution body was first introduced by Tsolomitis. However, our notation differs from the one used there, in order to emphasize the
connection with the standard Minkowski sum.
Our purpose is to find volume estimates, from above and below, of the θ-convolution of two sets, finding a quantitative version of the classical Brunn-Minkowski
inequality. This study leads us to get some classical geometric inequalities, such as
Rogers-Shephard or Zhang, as well as other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies.
The extension of θ-convolution to more than two sets is also given.
References
1. D. Alonso-Gutiérrez, C. H. Jiménez, R. Villa, Brunn-Minkowski and Zhang inequalities for convolution bodies, Adv. in Math., 238 (2013), 50–69.
2. D. Alonso-Gutiérrez, B. González Merino, C. H. Jiménez, Volume inequalities
for the i-th-convolution bodies, to appear in J. Math. Anal. Appl.
(arXiv:1312.6005)
3. D. Alonso-Gutiérrez, B. González Merino, C. H. Jiménez, R. Villa, Rogers-Shephard inequality for log-concave functions, Preprint. (arXiv:1410.2556)
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On Brunn-Minkowski type inequalities
Jesús Yepes Nicolás
Universidad de Murcia
[email protected]
Coautores: María A. Hernández Cifre, Eugenia Saorín Gómez.
Brunn-Minkowski’s inequality establishes that the n-th root of the volume of two
convex bodies K , E ⊂ Rn is a concave function, and assuming that both sets have a
projection onto a hyperplane with the same measure (or a common maximal volume section through parallel hyperplanes to a given one), it was proved that the
volume itself is concave, namely,
vol(λK + (1 − λ)E ) > λvol(K ) + (1 − λ)vol(E ),
for all λ ∈ [0, 1].
(4)
In this talk we will show, on the one hand, that under the above-mentioned projection/section assumption, if (4) holds with equality for some λ ∈ (0, 1), then (up to
degenerated convex bodies) K may be specifically recovered via K = L + E , with L
being a segment. We will also discuss that this extra assumption is needed in order to obtain such a characterization, even in the more general case in which (4)
holds with equality for all λ ∈ [0, 1]. This problem is connected with a conjecture
relating the roots of the Steiner polynomial of a pair of convex bodies to their relative inradius. We will show some counterexamples for the general case as well as a
counterexample to a conjecture by Matheron on inner parallel bodies.
On the other hand, we will show that the expected result of concavity for the kth root of the volume (cf. (4)), when a common projection onto an (n − k)-plane
(or a common maximal volume (n − k)-section) is assumed, is not true, by explicitly
giving (a family of ) convex bodies providing a counterexample for this statement.
Nevertheless, other Brunn-Minkowski type inequalities can be derived under these
hypotheses.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S13. Geometría diferencial y aplicaciones
http://rsme2015.ugr.es/s13.php
Normal approximations of regular curves and surfaces
Alfonso Carriazo
Universidad de Sevilla
[email protected]
Coautores: M. Carmen Márquez and Hassan Ugail
Bézier curves and surfaces are very useful tools in Geometric Modeling, with
many applications to fields such as computer graphics.
In this talk, we will offer a new method to provide approximations of regular
plane and spatial curves by Bézier curves. As an application, we will also develop a
method to approximate regular surfaces by Bézier ones. We will illustrate our methods by showing many examples and some animations.
Referencias
1. A. Carriazo, M. C. Márquez and H. Ugail. Normal approximations of regular
curves and surfaces. Submitted.
Pseudo-Riemannian Homogeneous Manifolds
Marco Castrillón López
ICMAT - UCM
[email protected]
Undoubtedly, isometries play an essential role in pseudo-riemannian geometry.
This fact makes homogenous spaces be a specially interesting instance in the realm
of smooth manifolds.One of the most succesfull toos to tackle the study of homogeneity is the so-called homogeneous structure tensors. The goal of this talk is to
present the applications of these tensors for Lorentzian manifolds by giving a review of some main results obtained by them. In particular, the connection of these
tensors with the notion of plane waves will be explored, in the real, complex and
quaternionic cases.
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Two construction methods of isoparametric hypersurfaces in noncompact symmetric spaces
Miguel Domínguez Vázquez
Instituto de Matematica Pura e Aplicada (IMPA), Brasil
[email protected]
A hypersurface in a Riemannian manifold is isoparametric if it and its nearby
equidistant hypersurfaces have constant mean curvature. The study of these objects
goes back to Levi-Civita, Segre and Cartan in the thirties. It turns out that most of
the known examples are homogeneous, that is, orbits of isometric actions on the
ambient manifold.
In this talk I will explain two methods to construct isoparametric hypersurfaces
in symmetric spaces of noncompact type. Both techniques rely on the algebraic
structure that underlies symmetric space of noncompact type. The first method,
proposed in joint work with J. Carlos Díaz-Ramos, allows to construct many inhomogeneous examples in the rank one symmetric spaces of nonconstant curvature.
The second one, which is based on the so-called horospherical decomposition of
the symmetric space, allows to enlarge examples from a rank one symmetric space
to a higher rank symmetric space.
On gradient Ricci solitons with constant scalar curvature
Manuel Fernández-López
IES María Sarmiento, Viveiro, Lugo
[email protected]
Coautores: Eduardo García-Río
Ricci solitons are fixed points of the Ricci flow on the space of Riemannian metrics modulo diffeomorphims and scalings. Also, they often arise as limits of dilations
of singularities in the Ricci flow. Moreover, they constitute a natural generalization
of Einstein manifolds.
The talk will be divided into two parts. In the first one we introduce the concept
of Ricci soliton and present some known results. In particular, we will give several
classification results under adequate geometric hypothesis (locally conformal flatness, harmonic Weyl tensor, ...). In the second part our recent results on gradient
Ricci solitons with constant scalar curvature will be presented.
Extremal curves of the total curvature in homogeneous 3-spaces
Angel Ferrández
Universidad de Murcia
[email protected]
Coautores: Manuel Barros, Universidad de Granada, and Oscar J. Garay, Universidad
del País Vasco
We obtain the space of extremals in homogeneous 3-spaces whose isometry
group has dimension four, also known as rotationally symmetric homogeneous 3spaces. Most of the geometry in these spaces is governed by the existence of a unit
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Killing vector field, ξ, sometimes called the Reeb vector field, which turns the homogeneous 3-space into the source of a Riemannian submersion whose target space is
a surface with constant curvature. We will show that a curve is an extremal of the
total curvature energy if and only if ξ lies into either the rectifying plane or the osculating plane along that curve. Then, we prove that every rotationally symmetric homogeneous 3-space, except H2 × R, admits a real one-parameter class of extremals
with horizontal normal (Lancret helices). The whole family of extremals is completed with a second class made up of those curves with horizontal binormal. In
contrast with the first class, it appears in any rotationally symmetric space, with no
exception, and it can be modulated in the space of real valued functions. We also
work out geometric algorithms to solve the so called solving natural equations for
extremals, allowing us to determine them explicitly in many cases. Furthermore, we
solve the closed curve problem by showing the existence of two families of closed
extremals.
Spacetimes and Finsler metrics
Miguel Angel Javaloyes
Universidad de Murcia
[email protected]
Let us recall that a pseudo-Finsler metric in a manifold M is a smooth function
L : A ⊂ T M \ {0} → R, such that A is a conic subset, L is positive homogeneous of
degree 2, namely, L(λv) = λ2 L(v) for every λ > 0 and v ∈ A, and its fundamental
tensor defined as
1 ∂2
L(v + su + t w)|t =s=0
g v (u, w) =
2 ∂t ∂s
is non-degenerate at every v ∈ A.
In our talk, we will review the relation between spacetimes and Finsler metrics.
This relation can happen in several ways. We will focus on two cases:
(i) the Fermat metric associated to a stationary spacetime [1] or more generally
the wind Finsler metric associated to a standard spacetime endowed with a Killing
vector field not necessarily timelike [2]. We will describe how this metric controls
the causality of the spacetime. Observe that the Fermat metric has positive definite
fundamental tensor.
(ii) we will discuss the definition of Finsler spacetime, namely, a manifod endowed with a pseudo-Finsler metric L : A ⊂ T M \ {0} → (0, +∞), which extends as
zero to the boundary of A [3]. We will review some causality notions and results for
Finsler spacetimes. In particular, we will explain how Penrose singularity theorem
can be extended to Finsler spacetimes [4].
References
1. E. C APONIO, M. A. J AVALOYES , AND M. S ÁNCHEZ, On the interplay between
Lorentzian causality and Finsler metrics of Randers type, Rev. Mat. Iberoamericana, 27 (2011), pp. 919–952.
2. E. C APONIO, M. A. J AVALOYES , AND M. S ÁNCHEZ, Wind Finslerian structures:
from Zermelo’s navigation to the causality of spacetimes,
arXiv:1407.5494 [math.DG]
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3. M. A. J AVALOYES AND M. S ÁNCHEZ, Finsler metrics and relativistic spacetimes,
Int. J. Geom. Methods Mod. Phys., 11 (2014), p. 1460032 (15 pages).
4. A. A AZAMI , M. A. J AVALOYES, Penrose’s singularity theorem in Finsler spacetimes, arXiv:1410.7595 [math.DG]
Hypersurface data in pseudo-riemannian manifolds, constraint equations,
energy-momentum map and applications to shells
Marc Mars
Instituto de Física Fundamental y Matemáticas, Universidad de Salamanca
[email protected]
The notions of metric hypersurface data and hypersurface data are introduced
and their basic properties, such as the relationship with the geometry of hypersurfaces embedded in pseudo-riemannian manifolds, are described. These concepts
allow for a unified description of hypersurfaces of arbitrary causal character and
generalizes the standard hypersurface data for non-degenerate hypersurfaces, in
particular concerning the notion of constraint equations. The concept of energymomentum map for hypersurface data is introduced and its connection to the Israel
equations for shells of constant causal character in General Relativity is presented.
Some geometric applications of two new forms of the weak maximum principle
Marco Rigoli
Universita degli Studi di Milano
[email protected]
Coautores: Luis J. Alias and Juliana F.R. Miranda
In this talk we introduce two equivalent new forms of the weak maximum principle and we show how they can be used to obtain information on some geometric
problems, for instance, related to the geometry of hypersurfaces in a product manifold. Other applications can be given to Yamabe type PDS’s and to "a priori" estimates.
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Complete maximal hypersurfaces in spatially parabolic Generalized RobertsonWalker spacetimes
Alfonso Romero
Departamento de Geometria y Topologia, Universidad de Granada
[email protected]
A Generalized Robertson-Walker (GRW) spacetime such that the universal Riemannian covering of the fiber is parabolic (thus so is the fiber) is said to be spatially parabolic. Spatially parabolic GRW spacetimes extend to spatially closed GRW
spacetimes from the point of view of the geometric-analysis of the fiber. On the
contrary to spatially closed GRW spacetimes, these spacetimes could be compatible with certain cosmological principle, and they can be used for modeling open
relativistic universes. A complete spacelike hypersurface in a spatially parabolic
GRW spacetime inherits the parabol- icity, whenever some boundedness assumptions on the restriction of the warping function to the spacelike hypersurface and
on the hyperbolic angle between the unit normal vector field and a certain timelike
vector field are assumed. Conversely, the existence of a simply connected parabolic
spacelike hypersurface, under the previous assumptions, in a GRW spacetime also
leads to its spatial parabolicity. All the complete maximal hypersurfaces in a spatially parabolic GRW spacetime are determined in several cases. As an application,
all the entire solutions of the maximal hypersurface equation on a parabolic Riemannian manifold are found, solving new Calabi-Bernstein problems.
Flujos Riemannianos Tensos
José Ignacio Royo Prieto
Universidad del País Vasco UPV/EHU
[email protected]
Coautores: Hiraku Nozawa, Ritsumeikan University (Japón)
En este trabajo, demostramos que toda foliación de dimensión 1 sobre una variedad diferenciable, no necesariamente compacta, es fuertemente tensa; es decir,
que admite una métrica para la cual la forma de curvatura media es básica y cerrada.
Este resultado (cf. 3.) es una generalización parcial de un conocido teorema de
D. Domínguez (cf. 1.) para variedades compactas . Como aplicación, obtenemos,
para el caso de variedades no compactas, varios resultados ya conocidos para el
caso de flujos riemannianos sobre variedades compactas, como sucesiones de tipo
Gysin (cf. 4.) y la caracterización de la minimalidad de las hojas en términos de la
cohomología básica (cf. 2.).
Referencias
1. Domínguez D., Finiteness and tenseness theorems for Riemannian foliations,
˝
Amer. J. Math. 120 (1998) 1237-U1276.
2. X. Masa, Duality and minimality in Riemannian Foliations, Comment. Math.
Helv. 67 (1992), 17–27.
3. H. Nozawa, J.I.Royo Prieto, Tenseness of Riemannian flows, aparecerá en Annales de l’Institut Fourier, 64 (2014)
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4. J.I.Royo Prieto, The Euler Class for Riemannian Flows, C.R.Acad. Sci. Paris
332, Série I (2001) 45–50.
Hermitian geometry on solvmanifolds
Luis Ugarte
Universidad de Zaragoza
[email protected]
Coautores: Anna Fino, Antonio Otal
In this talk we consider solvmanifolds endowed with invariant complex structures for which their canonical bundle is holomorphically trivial. This class is a
natural extension of the class of complex nilmanifolds. In dimension 6, the moduli space of such complex structures can be obtained for each solvmanifold. This
allows to study the existence of important classes of Hermitian metrics, and we will
focus on strong Kähler with torsion, balanced and generalized Gauduchon metrics.
The behaviour of some properties under holomorphic deformations of the complex
structure can also be studied on this class of solvmanifolds.
Hipersuperficies con dos curvaturas principales en CP 2 y CH 2
Cristina Vidal Castiñeira
Universidad de Santiago de Compostela
[email protected]
El siguiente problema fue propuesto por Niebergall y Ryan [1] en los 90:
“Hay hipersuperficies en CP 2 o CH 2 que tengan ≤ 2 curvaturas principales además de los ejemplos estándar?”
En esta charla mostraré que existen ejemplos no estándar y presentaré la clasificación de hipersuperficies reales con dos curvaturas principales no constantes en
los planos proyectivo e hiperbólico complejo [2]. Resulta que cada hipersuperficie de ese tipo es foliada por superficies Lagrangianas llanas equidistantes con curvatura media paralela, o lo que es equivalente, por órbitas principales de una acción
polar de comohogeneidad dos.
Referencias
1. R. Niebergall, P. J. Ryan: Real Hypersurfaces in Complex Space Forms, Tight
and Taut Submanifolds, MSRI Publications, Volume 32, 1997.
2. J. C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Real hypersurfaces with two principal curvatures in complex projective and hyperbolic
planes. arXiv:1310.0357v1 [Math.DG]
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S14. Investigación operativa
http://rsme2015.ugr.es/s14.php
Localización Continua: Momentos, Sumas de Cuadrados y Programación
Semidefinida
Víctor Blanco
Universidad de Granada
[email protected]
La localización continua nace con el problema de Fermat (siglo XVII) que consiste encontrar las coordenadas del plano cuya suma de las distancias (euclídeas)
a tres puntos dados fuera lo más pequeña posible. Weber (1909) generaliza este
problema a la localización de fábricas que deben servir a un conjunto de clientes,
y cuya minimización de las distancias de servicio (ponderadas) a estas representa
un objetivo económico. El problema de Fermat-Weber es complejo de resolver en
general (para cualquier número de clientes o puntos de demanda, para cualquier
tipo de distancia y en cualquier dimensión). Aún más si en vez de localizar un sólo
servicio queremos localizar un conjunto de éstos o utilizamos objetivos tipo mediana ordenada, cuyo uso ha sido extendido en los últimos años. En esta charla
presentaré algunos resultados recientes en los que transformaciones adecuadas de
distintos problemas de localización contínua permiten reformular éstos como problemas de optimización sobre el cono de segundo orden (SOCP), que son resolubles
en tiempo polinomial. Además, en el caso en el que el/los servicio/s a localizar
esté/n restingido/s a pertenecer a regiones semialgebraicas compactas, el problema
de los momentos (Lasserre, 2006) permite resolver hasta el grado de aproximación
deseado el problema de localización resolviendo problemas de programación semidefinida (SDP).
Resolución de conflictos aéreos vía optimización multiobjetivo entera mixta no
lineal
F. Javier Martin Campo
Universidad Complutense de Madrid
[email protected]
Coautores: Antonio Alonso Ayuso, Laureano F. Escudero
Debido al incremento de la demanda de transporte aéreo, cada vez son más las
técnicas usadas para la mejora de la gestión de tráfico aéreo. Un problema abierto
es la detección y resolución de conflictos aéreos, cuyo objetivo es proporcionar un
plan de vuelo para cada aeronave considerada, de tal modo que no se violen las
distancias mínimas establecidas de separación entre aeronaves. Para la consecución de tal fin, se puede modificar el plan de vuelo inicial de un avión mediante
tres maniobras: cambios de velocidad, giro y altura. Sin embargo, existen relaciones
de preferencia entre ellas. En términos generales de costes, las preferencias vienen
dadas en el siguiente orden: cambios de altura, giro y velocidad. Sin embargo, en
términos de confort de los pasajeros, el orden de preferencia es precisamente el opuesto. En este trabajo proponemos tres métodos multiobjetivo para comparar estas
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dos situaciones: optimización por metas, compromiso en un paso con norma l −1 y
compromiso en dos pasos con normas l − ∞ y l − 1. En cualquier caso, dado el tipo
de construcción geométrica de las condiciones a satisfacer en cualquier solución
factible, el problema se aborda mediante un modelo de optimización entera mixta
no lineal. En esta ponencia, después de presentar una panorámica general de la optimización multiobjetivo, proponemos una metodología para resolver el problema
en cuestión y los principales resultados numéricos obtenidos en una extensa experiencia computacional realizada para estimar el tiempo de respuesta y la calidad de
la solución a ofrecer para resolver el problema.
Exact and Heuristic Approaches for The Unrelated Parallel Machine Scheduling
problem with additional Resources
Federico Perea
Instituto Tecnológico de Informática, Universitat Politècnica de València
[email protected]
Coautores: Rubén Ruiz
The unrelated parallel machine scheduling problem (UPMS) consists of processing a set of jobs in a set of available machines. A typical objective in these problems
is the minimization of the largest job completion time, also known as makespan. In
other words, the objective is to find the assignment of jobs to machines so that the
latest job being processed finishes as soon as possible. The machines are parallel
because they can process jobs simultaneously, and unrelated because job processing times need not be the same for all machines. Examples of real situations in which
the UPMS arises are production systems in which two or more tasks need to be done
and we do not have to wait for the end of the processing of one task to start the processing of another one. Ever since the first papers related to this topic, in the 1950’s,
the interest of the scientific community on the UPMS has not stopped increasing.
In this word we assume that the during the length of time in which a job is processed, besides one of the available machines, a discrete amount of a scarce renewable processing resource which depends on the job and the machine are needed, for
instance operators. Resources are:
• renewable because after the processing of the job is finished, the needed resources are again available for other jobs.
• discrete because a discrete amount of them need to be assigned to job-machine
pairs.
• processing because they are needed while the job is processed.
The resulting problem is called the Unspecified Unrelated Parallel Machine Scheduling problem with additional Resources (UUPMSR). The problem is unspecified because there is no pre-fixed job-machine assignment. We study the dynamic version
of the problem, meaning that the allocation of resources to machines need not be
fixed for the whole processing time. The static version has been already proposed
and studied in the literature.
In this research, an integer linear programming (ILP) program is introduced, as
well as three iterative processes based on such ILP program. Because the ILP based
approaches cannot handle real-size problems, the constructive phase of a GRASP
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algorithm is proposed. All these approaches are tested and compared over a large
set of randomly generated instances.
Nuevos enfoques del problema discreto de la mediana ordenada
Diego Ponce
Universidad de Sevilla
[email protected]
Coautores: Martine Labbé (Université Libre de Bruxelles) Justo Puerto (Universidad
de Sevilla)
El problema discreto de la mediana ordenada, conocido por su acrónimo en inglés DOMP, permite estudiar los problemas de localización discreta a través de una
única formulación. En esta charla vamos a ver las nuevas formulaciones que han
surgido basándose en similitudes con algunos de los conocidos como scheduling
problems, las cuales nos han reportado grandes avances en el estudio poliédrico del
problema. También introduciremos una formulación cuyas variables están basadas
en un número exponencial de conjuntos, lo que nos llevará a aplicar la técnicas de
generación de columnas y Branch & Price en nuestro problema, explicando en detalle este proceso.
Referencias
1. Boland N., Domínguez-Marín P., Nickel S. and Puerto J. (2006). Exact procedures for solving the discrete ordered median problem, Computers & Operations Research, 33:3270-3300.
2. Gamrath G. (2010). Generic Branch-Cut-and-Price, PhD thesis, Technischen
Universität Berlin.
3. Marín A., Nickel S., Puerto J. and Velten S. (2009). A flexible model and efficient
solution strategies for discrete location problems, Discrete Applied Mathematics, 157:1128-1145.
4. Nickel S. (2000). Discrete ordered weber problems, In Operations Research
Proceedings, 71-76.
5. Puerto J. (2007). A new formulation of the capacitated discrete ordered median problems with {0,1}-Assignment, Operations Research Proceedings, 165170.
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Multiobjective combinatorial optimization with ordering. Applications to spanning trees, perfect matchings and shortest paths.
Miguel A. Pozo
Universidad de Sevilla
[email protected]
Coautores: Elena Fernández, Miguel A. Pozo, Justo Puerto
Multiobjective combinatorial optimization deals with problems considering more
than one viewpoint or scenario. The problem of aggregating multiple criteria to
obtain a globalizing objective function is of special interest when the number of
Pareto solutions becomes considerably large or when a single, meaningful solution
is required. Ordered Weighted Average or Ordered Median operators are very useful when preferential information is available and objectives are comparable since
they assign importance weights not to specific objectives but to their sorted values.
In this paper, Ordered Weighted Average optimization problems are studied from
a modeling point of view. Alternative integer programming formulations for such
problems are presented and their respective domains studied and compared. In addition, their associated polyhedra are studied and some families of facets and new
families of valid inequalities presented. The proposed formulations are particularized for two well-known combinatorial optimization problems, namely, minimum
spanning trees, shortest path and minimum cost perfect matching, and the results
of computational experiments presented and analyzed. These results indicate that
the new formulations reinforced with appropriate constraints can be effective for
efficiently solving medium to large size instances.
SEDD: Sistema para la Evaluación y el Diagnóstico de Desastres
J. Tinguaro Rodríguez
Universidad Complutense de Madrid
[email protected]
Coautores: Grupo de Investigación “DEC-HUMLOG: Decision Aid Models and Humanitarian Logistics”, Instituto de Matemática Interdisciplinar
En la gestión de desastres y emergencias, es crucial una correcta valoración inicial de las posibles consecuencias de los fenómenos adversos. Esta permite entonces tomar decisiones estratégicas adecuadas de cara al diseño de operaciones de
respuesta y ayuda a la población afectada. No obstante, para elaborar esa valoración
es preciso confrontar diversos tipos de incertidumbre, que surgen en un contexto en
el que la información suele ser incompleta, imprecisa y poco fiable, y en el que es
además necesario tomar decisiones con urgencia. En este trabajo, estos factores son
tratados mediante una combinación de herramientas de lógica borrosa, estadística
e inteligencia artificial que posibilitan el desarrollo de un sistema de ayuda a la decisión, SEDD (Sistema para la Evaluación y el Diagnóstico de Desastres), particularmente diseñado para adaptarse a los requisitos y las posibilidades de los países en
desarrollo y las organizaciones no gubernamentales habitualmente implicadas en
la confección y la implementación de operaciones de respuesta a desastres.
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Modelos multicriterio de ayuda a la decisión para problemas de reparto de ayuda
humanitaria
Gregorio Tirado
Universidad Complutense de Madrid
[email protected]
Coautores: Grupo de Investigación “DEC-HUMLOG: Decision Aid Models and Humanitarian Logistics”, Instituto de Matemática Interdisciplinar
Tras el estallido de un desastre, natural o provocado por el hombre, el uso eficiente de la ayuda disponible y la atención urgente de las necesidades principales
de la población afectada requieren una planificación logística adecuada y adaptada
a las condiciones de la situación de emergencia existente. En este contexto surge la
denominada “logística humanitaria”, que se caracteriza principalmente por la poca
disponibilidad de información fiable y precisa, la necesidad de una respuesta urgente y la existencia de recursos muy escasos. En este trabajo se presentan varios
modelos de decisión multicriterio para planificar operaciones de distribución de
ayuda humanitaria, considerando factores como la equidad en el reparto, la atención prioritaria a poblaciones especialmente vulnerables, la fiabilidad del itinerario
o su seguridad, que en este contexto tienen una especial relevancia, junto a otros
atributos como el coste y el tiempo, habituales en la logística empresarial.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S15. Matemática discreta
http://rsme2015.ugr.es/s15.php
El Teorema del Exceso Espectral: Variaciones y Aplicaciones
Miguel Ángel Fiol
Universitat Politècnica de Catalunya
[email protected]
Los grafos distancia-regulares aparecen a menudo en el estudio de estructuras
matemáticas con un alto grado de simetría y/o regularidad. Un ejemplo bien conocido de tales grafos son los esqueletos de los sólidos platónicos. Desde que fueron
propuestos por N. Biggs a principios de los 70, los grafos distancia-regulares han
sido objeto de un intenso estudio que incluye numerosas caracterizaciones, tanto
de carácter combinatorio como algebraico. Como ejemplo del primer caso, P. Rowlinson demostró que un grafo es distancia-regular si, y sólo si, el número de caminos
de una longitud dada entre dos vértices sólo depende de la distancia entre dichos
vértices. Esta charla versa sobre una caracterización casi-espectral de dichos grafos,
debida al conferenciante y E. Garriga, conocida en la literatura como el ‘teorema del
exceso espectral’. Este resultado afirma que un grafo es distancia-regular si, y sólo
si, su exceso espectral (una cantidad calculable a partir de su matriz de adyacencia)
es igual a su exceso medio (el número medio de vértices a distancia máxima de cada
vértice). Desde su aparición, este teorema ha dado lugar a diversas variaciones, concernientes tanto a familias concretas de grafos distancia-regulares, como a otras estructuras combinatorias más generales, como son los códigos completamente regulares y los esquemas de asociación p-polinomiales. Asimismo, el teorema del exceso
espectral ha sido clave en la demostración de otros resultados importantes como,
por ejemplo, la obtención por E.R van Dam y J. Koolen de la primera familia infinita
de grafos distancia-regulares que no son vértice transitivos.
Hiper Lformas
Pedro A. García-Sánchez
Universidad de Granada
[email protected]
Coautores: Francisco Aguiló-Gost, David Llena
Dado un semigrupo numérico de dimensión de inmersión tres, podemos asociar
al conjunto de Apéry del generador más grande un diagrama de distancias mínimas.
Éstos tienen forma de L y teselan el plano. Es por ello que se han venido llamando
Lformas en la literatura, siendo utilizados para el estudio tanto de semigrupos como
de factorizaciones en éstos. En el caso de dimensión de inmersión tres, sólo hay
como mucho dos posibles Lformas asociadas a un semigrupo numérico.
De igual forma podemos definir, asociado al conjunto de Apéry del mayor generador minimal de un semigrupo de dimensión de inmersión cuatro, un diagrama
de distancias mínimas. En este caso es una figura tridimensional, y es por eso que
les llamamos hyper Lformas. A diferencia del caso de dimensión de inmersión tres,
existen familias que tienen un número arbitrario de hyper Lformas asociadas.
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Generating strongly polynomial sequences of graphs
Delia Garijo
Universidad de Sevilla
[email protected]
Coautores: Andrew Goodall and Jaroslav Nešetˇril
De la Harpe and Jaeger [3] provided a method of generating strongly polynomial graph sequences, which were defined as sequences of graphs (Hk ) with k ∈
N such that the number of homomorphisms of every graph G to Hk , denoted by
hom(G, Hk ), is polynomial in k for every k ∈ N. A well-known example is the sequence of cliques (K k ), where hom(G, K k ) is the value of the chromatic polynomial
of G at k for each k ∈ N. On the other hand, we established [2] for which edgeweighted graphs H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial T (G; x, y), the Averbouch-Godlin-Makowsky polynomial ξG (x, y, z) [1], and the Tittmann-Averbouch-Makowsky polynomial QG (x, y)
[4]. The edge-weighted graphs H obtained for the three polynomials take the form of
a sequence of graphs indexed by a multivariate parameter. This motivates determining in general which sequences of graphs (Hk ) indexed by a multivariate parameter
k = (k 1 , . . . , k h ), h ≥ 1, have the property that for all graphs G, hom(G, Hk ) is the value
of a multivariate graph polynomial p(G; x 1 , . . . , x h ) at k.
In this talk, we will describe a new method to generate strongly polynomial sequences of graphs (Hk ) determining a polynomial p(G; x 1 , . . . , x h ). Whilst the chromatic polynomial, the Tutte polynomial and the Averbouch–Godlin–Makowsky polynomial can be obtained from strongly polynomial sequences of graphs by adaptation of the techniques used in [3], the Tittmann–Averbouch–Makowsky polynomial
cannot be thus obtained. Our construction includes this polynomial and, in fact,
we formulate our results in a more general context, using tree models for graphs,
so that the above-mentioned polynomials are obtained from strongly polynomial
sequences generated from cotrees for cographs.
(1) I. Averbouch, B. Godlin, J.A. Makowsky, A most general edge elimination polynomial, in: H. Broersma, T. Erlebach, T. Friedetzky, D. Paulusma (eds.), GraphTheoretic Concepts in Computer Science, 34th International Workshop, WG2008,
Durham, UK, June/July 2008, Lect. Notes Comput. Sci. 5344 (2008), 31–42
(2) D. Garijo, A.J. Goodall, J. Nešetˇril, Distinguishing graphs by left and right homomorphism profiles, European J. Combin. 32 (2011), 1025–1053
(3) P. de la Harpe, F. Jaeger, Chromatic invariants for finite graphs: theme and
polynomial variations, Lin. Algebra Appl. 226–228 (1995), 687–722
(4) P. Tittmann, I. Averbouch, J.A. Makowsky, The enumeration of vertex-induced
subgraphs with respect to the number of components, European J. Combin.
32:7 (2011), 954–974
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Political districting, clustering and partitioning graphs into centered connected
components in a nutshell
Justo Puerto
Instituto de Matemáticas de la Universidad de Sevilla
[email protected]
Partitioning a graph into connected components is a typical problem arising in
different areas, such as clustering, image processing, parallel computing, pattern
recognition, territorial districting and others. Many optimization problems can be
defined in these contexts, with different constraints and objectives. Several applications require a fixed number of components in the partition and/or capacity constraints on the components. In some cases there is a subset of vertices corresponding to centers and the problem requires that each component of the partition contains exactly one center. In cluster analysis and classification a dissimilarity criterion
is defined for each pair of units and in an optimal partition the units belonging to
the same cluster must be as similar as possible, or, equivalently, the units belonging
to different clusters must be as dissimilar as possible. In a large number of applications a weight is associated to each subset of vertices and an objective function
based on these weights must be maximized or minimized according to the specific
meaning of the weights. The objective function may represent, for example, a cost
for the assignment of the vertices to the components of the partition, or it can be
formulated in order to balance the weights of the components as much as possible
(equipartition problems).
The most studied graph partitioning problems are those with a cardinality constraint on the number of components. These problems are NP-hard on arbitrary
graphs, but in many cases they can be solved in polynomial time on trees (see, e.g.,
[1]). Polynomial time algorithms are also known for the max-split clustering problem on ladder graphs [4], and, more generally, on outerplanar graphs [5], as well as,
for equipartition problems on ladder graphs [2]. For some equipartition problems
pseudo-polynomial algorithms on series parallel graphs and an FPTAS on interval
graphs were presented in [3] and [6], respectively.
In this talk, we consider a connected graph G = (V, E ), and distinguish a subset S of vertices in V that are assumed to be centers, while the vertices in U = V \ S
are units. Under this framework we address different problems related to the partition of the graph, analyze their complexity, provide polynomial algorithms whenever possible and relate those problems to different applications in clustering and
political districting.
Referencias
1. Apollonio, N., Lari, I., Puerto, J., Ricca, F., and Simeone, B. (2008). Polynomial
algorithms for partitioning a tree into single-center subtrees to minimize flat
service costs. Networks, 51, 78–89.
2. Becker, R.I., Lari, I., Lucertini, M., and Simeone, B. (2001). A Polynomial-Time
Algorithm for Max-Min Partitioning of Ladders. Theory of Computing Systems,
34, 353–374.
3. Ito, T., Zhou, X., and Nishizeki, T. (2006). Partitioning a graph of bounded treewidth to connected subgraphs of almost uniform size. Journal of Discrete Algorithms, 4, 142–154.
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4. Lari, I. (2002). Connected Maximum Split Clustering of Ladder Graphs, in W.
Gaul, G. Ritter eds, Proc. of the 24th Annual Conference of the Gesellschaft für
Klassifikation, Studies in Classification, Data Analysis, and Knowledge Organization 2002, 107–114.
5. Lari, I., Ricca, F., and Scozzari, A. (2002). The forest wrapping problem on
outerplanar graphs. Lecture Notes in Computer Science, 2573, 345–354.
6. Wu, B.Y. (2012). Fully Polynomial-Time Approximation Schemes for the MaxMin Connected Partition Problem on Interval Graphs. Discrete Mathematics,
Algorithms and Applications, 4, doi: 10.1142/S179383091250005X.
Some geometrical aspects of difference bodies
Eugenia Saorín Gómez
Otto-von-Guericke Universität Magdeburg
[email protected]
Coautores: Judit Abardia
For n even, P ⊆ Rn a polytope and C ⊆ R2 a convex polygon with edge lengths l i
and unit normals to the edges αi ∈ S 1 ⊂ C, i = 1, · · · , N , the polytope
D C P :=
N
X
l i αi P
1
is called the complex difference body of P , with respect to C .
For a general planar convex body C and an arbitrary convex body K in an even
dimensional vector space, the complex difference body D C K was introduced by J.
Abardia in 2012 in the framework of the theory of valuations.
In this talk we will introduce this construction and study geometrical properties
and inequalities satisfied by the complex difference body.
In particular, we will prove that D C K is a polytope if and only if both C ⊆ R2
and K ⊆ Rn are polytopes. We prove further, that its dimension depends on the
position of K and characterize the bodies for which the complex difference body is
a Euclidean ball.
Dimensional versions of sumset inequalities
Oriol Serra
Universitat Politècnica de Catalunya
[email protected]
The small sumset problem asks for upper bounds on the cardinalities of sumsets
in terms of the cardinality of the summands in an abelian group. These inequalities
play a central role in additive combinatorics. One of the classical inequalities of
this kind is given by the theorem of Kneser. Hou, Leng and Xian gave an analog of
Kneser’s theorem in separable extensions of fields, where dimensions of subspaces
play the role of cardinalities. One of the nice features of this dimension version is
that it gives the classical one as a Corollary.
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We will discuss a dimension version of the simplest inverse theorem in additive
combinatorics, the theorem of Vosper. In contrast with Kneser’s theorem, this version can be essentially proved by extending the so–called isoperimetric method to
the dimension setting, opening the way to other dimension analogues of results in
additive combinatorics. Moreover it can be seen as a genuine generalization of the
theorem of Vosper. The proof includes the nonexistence of maximum distance separating codes in a space of bilinear forms with respect to a natural metric, a result
which has interest of itself.
A removal lemma for linear configurations and its applications
Lluis Vena
University of Toronto
[email protected]
Szemrédi’s Theorem from 1975 states that any set of integers, with positive upper density, contains a non-trivial, arbitrarily long arithmetic progression. In 1978
Furstenberg and Katznelson showed a multidimensional version of it; any set X of
positive upper density in the t -dimensional integer lattice contains t +1 points forming a simplex.
One of the results that can be used to prove the above existential theorems is the
so-called removal lemma for hypergraphs. This result roughly says that, if a large
(hyper)graph K does not have many copies of a given (hyper)graph H , then K can
be made free of copies of H by deleting a small number of edges.
In this talk we present a translation of the removal lemma to the arithmetic setting. This removal lemma states that, given a group G and some subset X of G, if a
linear configuration system Ax = 0 does not have many solutions with x i in X , then
we can obtain a new set X 0 where the system Ax = 0 has no solution if the variables
x i take values in X 0 . The set X 0 has been obtained from X by removing a few of its
elements.
Such an arithmetic removal lemma gives a unified approach to finding linear
configurations in dense sets of finitely generated abelian groups, thus showing Szemerédi’s theorem and its multidimensional version. Moreover, it can be used to
provide results for counting sets in finitely generated abelian groups free of linear
configurations.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S16. Matemáticas de la teoría de la información
http://rsme2015.ugr.es/s16.php
Semigroup ideals and generalized Hamming weights
María Bras-Amoros
Universitat Rovira i Virgili
[email protected]
A sharp upper bound for the maximum integer not belonging to an ideal of a
numerical semigroup is given and the ideals attaining this bound are characterized.
Then, the result is used, through the so-called Feng-Rao numbers, to bound the generalized Hamming weights of algebraic-geometry codes.
Árboles, Hash y Revocación
Pino Caballero-Gil
Universidad de La Laguna
[email protected]
Coautores: Francisco Martín-Fernández, Cándido Caballero-Gil
Los árboles constituyen uno de los tipos de grafos con mayor número de aplicaciones prácticas, y con mayor variedad ya que los hay binarios, k-arios, balanceados,
completos, perfectos, etc. Por otra parte, las funciones hash son también utilizadas
en distintos campos, entre los que destaca la protección de la integridad, donde se
requiere en particular el uso de funciones hash criptográficas. Una de las funciones
hash criptográficas más interesantes actualmente es la escogida en como estándar
en 2012, bautizada como SHA-3. Por último, la revocación de claves es uno de los
problemas más difíciles de resolver en criptografía, principalmente debido al problema de escalabilidad que implica y sobre todo cuando se utilizan infraestructuras
de clave pública o PKI (Public Key Infrastructure). Este trabajo analiza las relaciones
entre los conceptos de árboles y funciones hash en la búsqueda de soluciones eficientes y seguras al problema de la revocación.
Los sistemas de revocación tradicionales, como las listas de revocación de certificados o CRLs (Certificate Revocation Lists) y el protocolo OCSP (Online Certificate Status Protocol) requieren, respectivamente, un gran ancho de banda y una
alta carga computacional. Existe una propuesta diferentes para la mejora de dichos
sistemas, basada en una estructura de datos autenticada o ADS (Authenticated Data
Structure), similar a una CRL, pero que permite que los usuarios puedan hacer consultas sobre posibles revocaciones a servidores inseguros. En particular, la estructura de datos utilizada [1] es un árbol hash basado en los árboles hash de Merkle
[2], llamado árbol de revocación de certificados o CRT (Certificate Revocation Tree),
en el que la raíz está firmada por la Autoridad de Certificación o CA (Certificate Authority), y las hojas son los certificados revocados ordenados por número de serie.
Así, cuando un usuario desea validar un certificado, envía una consulta a un servidor
representante de la CA, de forma que cualquiera de estos servidores inseguros puede
proporcionar una prueba convincente de que el certificado está (o no) en el CRT.
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Además, el problema de la revocación no solo se presenta en PKIs basadas en certificados, sino que también se da por ejemplo cuando se utiliza criptografía basada en
identidad. En estos casos, de nuevo la combinación de árboles y funciones hash permite afrontar el problema mediante árboles de revocación. La mecánica de uso de
árboles de revocación requiere un análisis específico en cada uno de los casos, para
intentar optimizar todas las operaciones implicadas: consulta, inserción y borrado
de nodos, actualización, almacenamiento, envío, etc. En esta charla se presenta un
estudio de distintos sistemas de revocación basados en árboles hash para diferentes
circunstancias y utilizando diferentes herramientas.
Referencias
1. P. Kocher, On Certificate Revocation and Validation, Financial Cryptography
(FC98), Lecture Notes in Computer Science 1465, Springer-Verlag, pp. 172177, 1998.
2. R. Merkle, A certified digital signature, Advances in Cryptology (CRYPTO89).
Lecture Notes in Computer Science 435, Springer-Verlag, pp. 234-246, 1989.
Linealización del generador auto-shrinking a través de autómatas celulares
Sara D. Cardell
Universidad de Alicante
[email protected]
Coautores: Amparo Fúster-Sabater (Consejo Superior de Investigaciones Científicas)
Algunos autómatas celulares de una dimensión generan exactamente las mismas PN-secuencias que un LFSR de longitud máxima. Por lo tanto, un autómata
celular puede ser considerado como un generador alternativo a estos LFSR [1]. Además,
algunos generadores de secuencias cifrantes pueden ser modelizados como estructuras lineales basadas en autómatas celulares lineales [1,2]. En este trabajo, intentamos modelizar el generador auto-shrinking usando la regla 102.
El generador auto-shrinking fue diseñado por Meier y Staffelbach [3]. Es muy
fácil de implementar dado que usa un solo LFSR cuya PN-secuencia {a i } es decimada por sí misma. La regla de decimación es bastante simple; dados dos bits consecutivos de la PN-secuencia (a 2i , a 2i +1 ), con i = 0, 1, 2, . . ., un bit s j de la secuencia
auto-shrinking {s j } será igual a a 2i +1 si a 2i es uno. A su vez, a 2i +1 es descartado si
a 2i es cero.
Un autómata celular (CA) es un modelo compuesto por n celdas cuyo contenido
(binario en nuestro caso) se actualiza siguiendo una ley de transición de estados que
determina el estado de cada celda en función del estado actual de esa celda y de las
celdas adyacentes [4]. Los autómatas celulares considerados en este trabajo son
lineales (sólo se consideran operaciones XOR), regulares (todas las celdas siguen
la misma ley) y nulos (se consideran celdas con contenido nulo adyacentes a las
celdas de los extremos). En nuestro caso, sólo consideramos la ley 102. Cuando k
es 3, esta ley viene dada por la expresión: x it +1 = x it + x it+1 . Dado un LFSR de longitud máxima, existe un CA lineal que genera la secuencia auto-shrinking obtenida
a través de este registro, utilizando la ley 102. La longitud de dicho CA es igual a la
complejidad lineal de esta secuencia.
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Por ejemplo, dado el polinomio primitivo p(x) = 1+x 3 +x 4 y el estado inicial 1000
la secuencia auto-shrinking sería 01101001 . . .. El polinomio característico de esta
secuencia es p 5 (x) = (1 + x)5 y, así, la complejidad lineal es 5. En la siguiente tabla
se ofrece un ejemplo de CA de una dimensión y longitud 5 que genera la secuencia
auto-shrinking:
102
102
102
102
102
0
1
1
0
1
0
0
1
1
0
1
1
1
0
1
1
1
1
0
0
1
1
0
0
0
1
0
1
0
1
0
1
1
1
1
1
1
1
1
1
El esfuerzo realizado por parte de los criptógrafos por introducir decimación
para romper la linealidad de las secuencias generadas por los LFSR ha sido inútil, ya
que la secuencia auto-shrinking puede ser modelizada como una secuencia de salida de una estructura lineal basada en autómatas celulares que usan, en este caso, la
ley 102. Esto implica que estas secuencias son sensibles de sufrir una criptoanálisis
que se aproveche de esta linealidad. Nuestro trabajo se basa en analizar la familia
de autómatas celulares lineales, regulares, nulos, de una dimensión que describen
el comportamiento del generador auto-shrinking, diseñado como no lineal en su
origen. Además, en [2] los autores modelizaron este generador usando una familia
de autómatas celulares híbridos basados las leyes 150 y 90. Por lo tanto, podemos realizar un estudio de las ventajas y desventajas de una familia sobre otra y utilizar ambas familias para llevar a cabo un criptoanálisis sobre la secuencia auto-shrinking.
Agradecimientos: Este trabajo ha sido financiado parcialmente por los proyectos MTM2011-24858 y TIN2011-25452 del Ministerio de Ciencia e Innovación del
Gobierno de España. El trabajo de la primera autora ha sido financiado por una
beca postdoctoral de la Generalitat Valenciana con referencia APOSTD/2013/081.
Referencias
1. Fúster-Sabater, A., Caballero-Gil, P.: Linear solutions for cryptographic nonlinear sequence generators. Physics Letters A. 369, 432–437 (2007).
2. Fúster-Sabater, A., Pazo-Robles, M. E., Caballero-Gil, P.: A simple linearization
of the self-shrinking generator by means of cellular automata. Neural Networks. 23(3), 461–464 (2010).
3. Meier, W., Staffelbach, O.:The self-shrinking generator. In: Advances in Cryptology, EUROCRYPT 1994. LNCS. 950, 205–214. Springer-Verlag (1994).
4. Das, A. K., Ganguly, A, Dasgupta, A, Bhawmik, S., Chaudhuri, P. P.: Efficient
characterisation of cellular automata. IEE Proceedings E: Computers and Digital Techniques. 137(1), 81–87 (1990).
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Representación entrada-estado-salida de códigos convolucionales concatenados
Joan-Josep Climent
Universitat d’Alacant
[email protected]
Coautores: Victoria Herranz (Universidad Miguel Hernández de Elche), Carmen Perea
(Universidad Miguel Hernández de Elche)
En este trabajo caracterizamos dos modelos de códigos convolucionales desde la
perspectiva de la teoría de sistemas lineales. Concretamente, introducimos una representación entrada-estado-salida de dichos modelos y estudiamos las condiciones
para obtener una representación minimal del código convolucional concatenado.
También introducimos condiciones para que el código convolucional concatenado sea observable y presentamos una cota inferior de su distancia libre en función
de las distancias libres de los códigos convolucionales constituyentes.
β
β
α
Double cyclic codes over the rings Zα
2 × Z2 and Z2 × Z4
Cristina Fernández-Córdoba
Universitat autònoma de Barcelona
[email protected]
Coautores: Joaquim Borges Ayats, Roger Ten-Valls
β
Consider the rings R 1 and R 2 , such that R 1 is an R 2 -module, and C ⊂ R 1α × R 2 an
additive code. The code C is a double cyclic code if the set of coordinates can be
partitioned into two subsets, the set of coordinates in R 1 and the set of coordinates
in R 2 , such that any cyclic shift of the coordinates of both subsets leaves invariant
the code. The code can be identified as submodules of the R 2 [x]-module R 1 [x]/(x α −
1) × R 2 [x]/(x β − 1). We define two cases. First, when the code C is binary, that is
R 1 = R 2 = Z2 , which is called Z2 -double cyclic. The second case is when R 1 = Z2 and
R 2 = Z4 , that is the code is a Z2 Z4 -additive code, and it is called Z2 Z4 -cyclic. In both
cases, we determine the structure of these double cyclic codes giving their generator
polynomials. We also determine the related polynomial representation of its duals
in terms of the generator polynomials.
HIMMO: A Lightweight, Fully Collusion Resistant Key Pre-Distribution Scheme
Jaime Gutierrez
Universidad de Cantabria
[email protected]
Coautores: Oscar Garcia-Morchon; Domingo Gomez; Ronald Rietman; Ludo Tolhuizen
Public-key cryptography addresses key distribution and agreement in a very elegant way by allowing any pair of nodes to generate a common secret without sharing any information beforehand. In the alternative approach of key pre-distribution
schemes (KPS), a trusted-third party (TTP) securely provides each node with a (nodedependent) secret function allowing pairs of nodes to agree on a common key in a
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non-interactive way, which is a big advantage in delay critical applications. However, no known key KPS is simultaneously secure and efficient. This paper proposes HIMMO, a KPS which relies on the recently introduced Hiding Information
(HI) and Mixing Modular Operations (MMO) problems. Our security analysis shows
that HIMMO is fully collusion resistant for appropriate parameter choices. HIMMO
is also lightweight for these parameters and thus makes non-interactive key establishment feasible even in very large networks. Additionally, the identity-based nature of HIMMO enables implicit certification and verification of credentials, as well
as secure broadcast by the TTP. HIMMO can a also accommodate multiple TTPs so
that no single TTP knows the keys shared between nodes. All these features make
HIMMO a very promising candidate to enable more efficient security protocols.
Quantum codes from evaluation
Fernando Hernando
Universidad Jaume I
[email protected]
Coautores: Carlos Galindo and Diego Ruano
Stabilizer codes obtained via the CSS code construction and the Steane’s enlargement of subfield-subcodes and matrix-product codes coming from generalized
Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes with
good quantum parameters are supplied, in particular, some binary codes of lengths
127 and 128 improve the parameters of the codes in http://www.codetables.de.
Cyclic convolutional codes over separable extensions.
F. J. Lobillo
Departamento de Algebra, Universidad de Granada
[email protected]
Coautores: J. Gómez-Torrecillas, G. Navarro
We show that, under mild conditions of separability, an ideal code is a direct
summand of an Ore extension and, consequently, it is generated by an idempotent
element. We also design an algorithm for computing one of these idempotents.
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Nuevo enfoque del problema de la equivalencia de códigos lineales
Irene Márquez Corbella
INRIA Rocquencourt - Equipe SECRET - (France)
[email protected]
Coautores: Natalia Dück, Edgar Martínez Moro
En [3] se muestra cómo se puede asociar a cualquier código lineal un ideal binomial que contiene toda la información del código en los exponentes de cada binomio. Esta correspondencia se ha demostrado que es interesante para resolver
varios problemas considerados difíciles en teoría de códigos, como por ejemplo la
descodificación completa, la búsqueda del conjunto de palabras de soporte minimal o el cálculo de la capacidad de corrección. Todas estas aplicaciones requieren
el cálculo de bases de Gröbner respecto de un orden graduado.
En esta charla introduciremos una nueva invariante de un código lineal, que
hemos llamado el Gröbner Fan graduado. Esta estructura se define como la colección geométrica del conjunto de Bases de Gröbner graduadas del ideal asociado
al código. Veremos cómo calcular de forma eficiente esta estructura, y para ello
mostraremos cómo adaptar el software TIGERS (Toric Gröbner bases Enumeration
by Reverse Search) desarrollado por R. R. Thomas [1]. Además descubriremos la
aplicación de esta estructura al problema de equivalencia de códigos, tal como hemos
planteado en [2].
Referencias
1. B. Huber and R. R. Thomas. Computing Gröbner fans of toric ideals. Experimental Mathematics, 9(3):321-331, 2000. linear code. In 4th International
Castle Meeting on Coding Theory and and Applications (4ICMCTA). CIM-MS
Series by Springer-Verlag, 2014.
2. I. Márquez-Corbella, E. Martínez-Moro and E. Suárez-Canedo. On the ideal
associated to a linear code. arXiv: 1206.5124, 2014.
Cyclic and BCH Codes whose Minimum Distance Equals their Maximum BCH bound
Juan Jacobo Simón Pinero
Universidad de Murcia
[email protected]
Coautores: José Joaquín Bernal Buitrago and Diana H. Bueno Carreño
We study the family of cyclic codes such that its minimum distance reaches the
maximum of its BCH bounds. We also study a way to construct cyclic codes with
that property by means of computations of some divisors of a polynomial of the
form x n − 1. We apply our results to the study of those BCH codes C , with designed
distance δ that have minimum distance d (C ) = δ. Finally, we present some examples
of new binary BCH satisfying that condition. To do this, we make use of two related
tools: the discrete Fourier transform and the notion of apparent distance of a code,
originally defined for multivariate abelian codes.
Note:Partially supported by MINECO (Ministerio de Economía y Competitividad), (Fondo Europeo de Desarrollo Regional) project MTM2012-35240, Programa
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Hispano Brasileño de Cooperación Universitaria PHB2012-0135, and Fundación Séneca
of Murcia.
Cifrado Inverso
Adriana Suárez Corona
Universidad de León
[email protected]
Coautores: David Naccache, Rainer Steinwandt y Moti Yung
El cifrado de clave pública inverso (Reverse Public-Key Encryption (RPKE)) es un
modo de operación que utiliza un esquema de cifrado de clave pública en el que no
se puede distinguir qué clave ha sido utilizada para cifrar el mensaje. El mensaje a
cifrar en modo inverso determina la clave a usar.
RPKE permite una forma alternativa de enviar información, en el caso de que
se rompa el esquema de cifrado, pero se siga manteniendo la privacidad de la clave.
Las construcciones propuestas anteriormente [NSY09] son poco eficientes para
enviar mensajes de más de un bit de longitud en el modo inverso. En esta charla,
veremos cómo se pueden construir esquemas que permitan el envío de mensajes de
mayor tamaño de forma más eficiente, utilizando esquemas de Anomymous Broadcast Encryption (ANOBE).
Un uso simultáneo de los modos de operación tradicional e inverso permite enviar dos mensajes independientes en un mismo texto cifrado, o de forma alternativa,
proporciona un canal esteganográfico dentro del criptosistema.
NSY09 David Naccache, Rainer Steinwandt, and Moti Yung. Reverse Public Key Encryption. In BIOSIG 2009 Proceedings, Lecture Notes in Informatics, pages
155–169. GI, Springer, 2009.
Constructing credential-based E-voting systems from offline E-coin protocols
Magda Valls
Universitat de Lleida
[email protected]
Coautores: Víctor Mateu, Francesc Sebé
Mu and Varadharajan proposed a remote voting paradigm in which participants
receive a blindly signed voting credential that permits them to cast a vote anonymously. If some participant tries to cheat by submitting more than one vote, her
anonymity will be lifted. In the last years, several proposals following this paradigm,
including Mu and Varadharajan, have been shown to be cryptographically weak. In
this paper we first show that a recent proposal by Baseri et al. is also weak. After
that, we give a general construction that, employing an offline e-coin protocol as a
building block, provides an anonymous voting system following the aforementioned
paradigm.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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http://rsme2015.ugr.es/s17.php
Isomorphisms of graph algebras and groupoids
Pere Ara
Departament de Matemàtiques, Universitat Autònoma de Barcelona
[email protected]
Abrams and Tomforde have shown that any ∗-algebra isomorphism L C (E ) →
L C (F ) between complex Leavitt path algebras can be extended to a ∗-isomorphism
C ∗ (E ) → C ∗ (F ) between the corresponding graph C*-algebras. Not all the ∗-isomorphisms between graph C*-algebras restrict to the corresponding Leavitt path algebras, but it is an open question whether the existence of a ∗-isomorphism between C ∗ (E ) and C ∗ (F ) implies the existence of a ∗-isomorphism between L C (E )
and L C (F ). Using the theory of groupoids, we are able to show that the answer is positive, even for the Leavitt path algebras over an arbitrary field, if the ∗-isomorphism
from C ∗ (E ) to C ∗ (F ) respects a certain natural commutative C ∗ -subalgebra.
A functorial approach to Lie Theory
Alessandro Ardizzoni
University of Torino
[email protected]
Coautores: C. Menini (University of Ferrara)
Hom-Lie algebras, Lie color algebras, Lie superalgebras and other type of generalized Lie algebras are recovered by means of an iterated construction, known as
monadic decomposition of functors, which is based on Eilenberg-Moore categories.
This talk is mainly based on the work [A. Ardizzoni and C. Menini, Milnor-Moore
Categories and Monadic Decomposition, preprint. (arXiv:1401.2037)]
Weakly left localizable rings and their characterizations
V. V. Bavula
University of Sheffield
[email protected]
We introduce a new class of rings - the class of weakly left localizable rings - and
give its characterization.
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Multiplier bialgebras in braided monoidal categories
Gabriella Böhm
Wigner RCP, Budapest
[email protected]
Coautores: Stephen Lack (Macquarie University, Sydney)
A bialgebra A — over a field or, more generally, in any braided monoidal category
— can equivalently be described without referring separately to the multiplication
µ : A ⊗ A → A and the comultiplication ∆ : A → A ⊗ A; just in terms of the unit, the
counit and the so-called fusion morphism [1]
∆⊗id
id⊗µ
A ⊗ A −→ A ⊗ A ⊗ A −→ A ⊗ A.
This treatment has the advantage of applicability also in the absence of a unit and
a proper comultiplication; as Van Daele’s approach to multiplier Hopf algebras [2]
shows.
Based on the use of counital (but no longer unital) fusion morphisms, we propose a definition of multiplier bialgebras in arbitrary braided monoidal categories.
The categories of appropriately defined modules and comodules are shown to possess monoidal structures admitting strict monoidal forgetful functors to the base
category. These features are explained by the structures carried by the functor induced by tensoring with a multiplier bialgebra.
The talk is based on [3].
Referencias
1. S. Lack and R. Street, Skew monoidales, skew warpings and quantum cate˝
gories, Theory Appl. Categ. 26 (2012), 385U402.
2. A. Van Daele, Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994),
˝
no. 2, 917U932.
3. G. Böhm and S. Lack, Multiplier bialgebras in braided monoidal categories, J.
Algebra, in press. arXiv:1405.4668
Rooted rings with several objects
Manuel Cortés Izurdiaga
Departamento de Matemáticas, Universidad de Almería
[email protected]
Coautores: Blas Torrecillas Jover
A ring with several objects is a functor category (C, Ab) where C is a small preadditive category and Ab is the category of abelian groups. The objective of the talk is
to introduce the notion of rooted ring with several objects and to characterize, over
such rings, some classes of modules.
Since each ring with enough idempotents is equivalent to the functor category
over a small preadditive category, our results allow us to characterize these classes
of modules over rings with enough idempotents.
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Semisimple Hopf actions on Weyl algebras
Juan Cuadra
University of Almería, Dpt. Mathematics, 04120 Almería, Spain
[email protected]
In this talk we will discuss the following problem: under which conditions the
action of a finite-dimensional Hopf algebra on an algebra factors through that of a
group algebra? This question was studied for the first time by Chan et al. in [1] and
[2]. Several positive results were obtained there for semisimple Hopf algebras acting
faithfully (plus some other hypotheses) on various particular algebras, including the
Weyl algebra.
Etingof and Walton proved in [4] the following general theorem. Let H be a
semisimple and cosemisimple Hopf algebra over an algebraically closed field k. If
H acts faithfully on a commutative domain over k, then H is a group algebra. As
an application, they established a similar result for the Weyl algebra by using the
associated graded algebra, whenever the H -action preserves the standard filtration.
The aim of this talk is to show a different strategy for the proof of this result,
which allow us to remove the condition on the H -action. The techniques used include reduction modulo a prime number and an extension of Etingof-Walton’s Theorem to division algebras. The results that will be presented appear in the joint work
[3] with Pavel Etingof and Chelsea Walton (Massachusetts Institute of Technology).
References
1. K. Chan, C. Walton, Y.H. Wang and J.J. Zhang, Hopf actions on filtered regular
algebras. J. Algebra 397 (2014), 68-90.
2. K. Chan, C. Walton and J.J. Zhang, Hopf actions and Nakayama automorphisms.
J. Algebra 409 (2014), 26-53.
3. J. Cuadra, P. Etingof and C. Walton, Semisimple Hopf actions on Weyl algebras.
ArXiv:1409.1644.
4. P. Etingof and C. Walton, Semisimple Hopf actions on commutative domains.
Adv. Math. 251 (2014), 47–61.
Morita Theory for Commutative Hopf Algebroids, and Hovey-Strickland conjecture.
Laiachi El Kaoutit
Universidad de Granada
[email protected]
Coautores: Niels Kowalzig
A commutative Hopf algebroid is a presheaf of groupoids on affine schemes (a
pre-stack or préchamp for the fpqc topology). Its category of (right) comodules is
monoidally equivalent to the category of equivariant quasi-coherent sheaves, that
is, the category of representations of the associated prestack.
Two (flat) commutative Hopf algebroids are said to be Morita equivalent, if their
categories of (right) comodules are equivalent as symmetric monoidal categories. A
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morphism between two Hopf algebroids is a weak equivalence when the associated
induction functor (which is by definition a symmetric monoidal functor) induces
an equivalence of categories of comodules. Thus, two weakly equivalent Hopf algebroids are evidently Morita equivalent. The converse was conjectured by Hovey
and Strickland in 2005. Precisely, they conjectured that two Morita equivalent Hopf
algebroids are connected by a chain (of zig-zag type) of weak equivalences.
The main motivation of this research is two folds: First is to answer positively
to the above conjecture. Second to give a parallel theory (the algebraic counterpart)
to the Lie groupoid Morita theory developed by Moerdijk-Mrˇcun, for (flat) commutative Hopf algebroids.
In this talk and if time allows, we will expose all the machinery of Morita theory for (flat) commutative Hopf algebroids. Explicitly, we introduce the notion of
principal bi-bundle in the Hopf algebroids context, and show that two Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only
if there exists a principal bi-bundle connecting them. This will give a positive answer to the above conjecture by explicitly constructing a two stage zig-zag of weak
equivalences. Next, we gather (flat) Hopf algebroids and principal bundles along
with their morphisms in a bicategory. We show that the 2-functor assigning to each
morphism of Hopf algebroids its associated trivial bundle, transforms weak equivalences into invertible 1-cells. Finally, we exhibit this 2-functor as a universal solution for 2-functors which send weak equivalences to invertible 1-cells; establishing
by this a kind of calculus of fractions in the 2-category of (flat) Hopf algebroids with
respect to weak equivalences in the sense of Pronk.
This is a joint work with Niels Kowalzig, based on arXiv.math:1407.7461v1.
On pure exact structures
Sergio Estrada
Universidad de Murcia
[email protected]
Coautores: James Gillespie and Sinem Odabasi
Let A be closed symmetric monoidal Grothendieck category. We define the pure
derived category with respect to the monoidal structure via a relative injective model
category structure on the category C(A ) of unbounded chain complexes in A . Then
we will focus on applications of this general setting in concrete categories.
Direct products of modules whose endomorphism rings have at most two maximal ideals
Alberto Facchini
Università di Padova
[email protected]
Coautores: Adel Alahmadi
Let R be a ring, Mod-R the category of all right R-modules and C a full subcategory of Mod-R whose class of objects Ob(C ) consists of indecomposable right
R-modules. A completely prime ideal P of C consists of a subgroup P (A, B ) of the
additive abelian group HomR (A, B ) for every pair of objects A, B ∈ Ob(C ) such that:
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(1) for every A, B,C ∈ Ob(C ), every f : A → B and every g : B → C , one has that
g f ∈ P (A,C ) if and only if either f ∈ P (A, B ) or g ∈ P (B,C ); and (2) P (A, A) is a
proper subgroup of HomR (A, A) for every object A ∈ Ob(C ). If A, B are objects of
C , we will say that A and B belong to the same P class, and write [A]P = [B ]P , if
there exist right R-module morphisms f : A → B and g : B → A with f ∉ P (A, B ) or
g ∉ P (B, A). The full subcategory C of Mod-R is said to satisfy (DSP) if whenever
A, B,C , D are right R-modules with A ⊕ B ∼
= C ⊕ D and A, B,C ∈ Ob(C ), then also
D ∈ Ob(C ).
Sample result:
Theorem. Let C be a full subcategory of Mod-R in which all objects are indecomposable right R-modules and let P , Q be two completely prime ideals of C with
the property that, for every A ∈ Ob(C ), f : A → A is an automorphism if and only if
f ∉ P (A, A) ∪ Q(A, A). Assume that C satisfies Condition (DSP). Let { A i | i ∈ I } and
{ B j | j ∈ J } be two families of objects of C . Assume that there exist two bijections
σ, τ : I → J such that [A i ]P = [B σ(i ) ]P and [A i ]Q = [B τ(i ) ]Q for every i ∈ I . Then the
Q
Q
R-modules i ∈I A i and j ∈J B j are isomorphic.
Referencias
1. A. Alahmadi and A. Facchini, Direct Products of Modules Whose Endomorphism Rings have at Most Two Maximal Ideals, to appear 2015.
Universal central extensions of Lie–Rinehart algebras
Xabier Garcia Martinez
University of Santiago de Compostela
[email protected]
Coautores: Jose Luis Castiglioni, Manuel Ladra
In this work we study central extensions of Lie–Rinehart algebras. They do an
algebraic codification of Lie algebroids. The concept of Lie–Rinehart A-algebra generalizes the concept of Lie A-algebra and A-module and the main example of Lie–
Rinehart algebra is the set DerK (A) of all K -derivations of A.
We study central extensions of Lie–Rinehart algebras an we prove that if L is A2
projective then the second cohomology group HRin
(L, I ) classifies central extensions
of L by I . Then we build a non-abelian tensor product of Lie–Rinehart algebras extending the non-abelian tensor product of Lie algebras and we obtain the existence
of the universal central extension when the Lie–Rinehart algebra is perfect and we
characterize it with the non-abelian tensor product.
Referencias
1. G. J. Ellis. A nonabelian tensor product of Lie algebras. Glasgow Math. J.,
33(1):101–120, 1991.
2. J. Huebschmann. Poisson cohomology and quantization. J. Reine Angew.
Math., 408:57–113, 1990.
3. G. S. Rinehart. Differential forms on general commutative algebras. Trans.
Amer. Math. Soc., 108:195–222, 1963.
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Connections between relative Gorenstein dimensions and Auslander and
Bass classes
Juan Ramón García Rozas
Catedrático de Álgebra. Departamento de Matemáticas. Universidad de Almería
[email protected]
Coautores: Driss Bennis, Luis Oyonarte
It is well known the relation between the class of modules with finite Gorenstein
projective (resp. injective) dimension and the Auslander (resp. Bass) class when
these mentioned classes are built with a dualizing module over noetherian n-perfect
rings. Basically, the results are necessary conditions to ensure that both classes coincide. In this note we try to extend, and sometimes improve, some of these results by
weakening the condition of being dualizing. As an application, we are able to study
when the Auslander (Bass) class is covering (resp. enveloping).
Associative algebras and Lie algebras defined by Lyndon words
Tatiana Gateva-Ivanova
American University in Bulgaria and IMI BAS
[email protected]
We study the class C(X ,W ) of associative graded k-algebras A generated by X
and with a fixed obstructions set W consisting of Lyndon words in the alphabet X .
Important examples are the monomial algebras A = k〈X 〉/(W ), where W is an antichain of Lyndon words of arbitrary cardinality and the enveloping algebra U g of
any X -generated Lie k-algebra g. We prove that all algebras A in C(X ,W ) share the
same Poincaré-Birkhoff-Witt type k-basis built out of the so called Lyndon atoms N
(determined uniquely by W ) but, in general, N may be infinite. We prove that A has
polynomial growth if and only if the set of Lyndon atoms N is finite. In this case
α
α α
A has a k-basis N = {l 1 1 l 2 2 · · · l d d | αi ≥ 0, 1 ≤ i ≤ d }, where N = {l 1 , · · · , l d }. Surprisingly, in the case when A has polynomial growth its global dimension does not
depend on the shape of its defining relations but only on the set of obstructions W .
We prove that if A has polynomial growth of degree d then A has global dimension d
and is standard finitely presented, with d − 1 ≤ |W | ≤ d (d − 1)/2. We study when the
set of standard bracketings [W ] = {[w] | w ∈ W } is a Groebner-Shirshov Lie basis. We
use our general results to classify the Artin-Schelter regular algebras A generated by
two elements, with defining relations [W ] and global dimension ≤ 7. We give an extremal class of monomial algebras, the Fibonacci-Lyndon algebras, F n , with global
dimension n and polynomial growth, and show that the algebra F 6 of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin-Schelter regular
algebra.
Referencias
1. T. G ATEVA -I VANOVA , G. F LOYSTAD, Monomial algebras defined by Lyndon words,
J. Algebra 403 (2014), 470–496.
2. T. G ATEVA -I VANOVA, Quadratic algebras, Yang-Baxter equation, and Artin- Schelter regularity, Adv. in Math. 230 (2012), 2152-2175.
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Cleft and Galois extensions for weak Hopf quasigroups
Ramón González Rodríguez
Universidade de Vigo. Departamento de Matemática Aplicada II
[email protected]
The aim of this talk is to present the notions of cleft and Galois (with normal basis) extension associated to a weak Hopf quasigroup. We show that, under suitable
conditions, both notions are equivalent. As a particular instance we recover the results for (weak) Hopf algebras proved in [1] and [4]. Moreover, taking into account
that weak Hopf quasigroups generalize the notion of Hopf quasigroup, we obtain
the definitions of cleft and Galois (with normal basis) extension associated to a Hopf
quasigroup and we get the equivalence betwen these extensions in this setting. The
results that will be presented are part of a joint work with J.N. Alonso and J.M. Fernández (see [2] and [3]).
Referencias
1. J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, A. B. Rodríguez Raposo, Weak C-cleft extensions and weak Galois extensions, J. Algebra 299 (2006), 276-293.
2. J.N. Alonso Álvarez, J.M. Fernández Vilaboa y R. González Rodríguez, R., Weak
Hopf quasigroups. math.QA, arXiv:1410.2180 (2014)
3. J.N. Alonso Álvarez, J.M. Fernández Vilaboa y R. González Rodríguez, R., Cleft
and Galois extensions associated to a weak Hopf quasigroup (2014).
math.QA, arXiv:1412.1622
4. H. F. Kreimer, M. Takeuchi, Hopf algebras and Galois extensions of an algebra,
Indiana Univ. Math. J. 30 (1981), 675-691.
Weak Multiplier Bialgebras
José Gómez-Torrecillas
Universidad de Granada
[email protected]
Coautores: Gabriella Böhm and Esperanza López-Centella
A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the ‘base algebras’) are shown to carry coseparable co-Frobenius coalgebra
structures. Appropriate modules over a weak multiplier bialgebra are shown to
constitute a monoidal category via the (co)module tensor product over the base algebra. The relation to Van Daele and Wang’s (regular and arbitrary) weak multiplier
Hopf algebra is discussed.
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Some Examples of Tilting and Cotilting Modules
Ivo Herzog
The Ohio State University at Lima
[email protected]
Coautores: Silvana Bazzoni, Jan Šaroch, and Jan Trlifaj
Let (T , F ) be a torsion pair in a Grothendieck category G . This torsion pair induces a t -structure in the derived category D(G ), whose heart H t is an abelian category. The question of whether the abelian category H t is itself a Grothendieck
category has been considered by Parra and Saorin [2], who found necessary and
sufficient conditions. If R denotes an associative ring, we show that a necessary
condition for a torsion pair in the Grothendieck category R-Mod of left R-modules
to satisfy these conditions is that both classes T and F be (finitely) axiomatizable in the language of left R-modules. Using this characterization, we verify that
for some classes of rings, every torsion pair in R-Mod induces a t -structure with a
Grothendieck heart. To find a counterexample, we consider the representation theory of the ring R considered by Dubrovin and Puninski in [2]. We show that R has
a 1-tilting module and a 2-tilting module, neither of which are equivalent to finitely
generated modules.
Referencias
1. Parra, C., and Saorin, M., Direct limits in the heart of a t -structure,
arXiv:1311.6166v2.
2. Dubrovin, N., and Puninski, G., Classifying projective modules over some semilocal rings, J of Alg and Its Appl 6(5), Oct 2007.
Rings whose cyclic modules have minimal injectivity domain: preliminary report.
Sergio R. López-Permouth
Ohio University
[email protected]
Coautores: Noyan Er and Nguyen Khanh Tung
A module M is said to be poor if it is injective relative to only semisimple modules.
We consider rings for which every non-zero cyclic right R-module is poor. A
non-semisimple ring R with that property is shown to be an indecomposable ring
satisfying the following properties: the singular submodule Z (R R ) is essential in R R ,
every noetherian right R-module is artinian, and every ideal of R is either below the
prime radical or above the Jacobson radical of R. We also show that a right noetherian ring R whose non-zero cyclics are poor is isomorphic to a matrix ring over a
non-uniserial local right artinian ring. Thus, in particular, if a commutative noetherian ring R satisfies that property then R is isomorphic to a direct product of fields.
We point out relations between these notions and other families of rings characterized in similar ways; in particular, we study connections between this problem
and rings without a middle class (those rings for which every module is either injective or poor.)
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Monoid actions providing public key cryptosystems
Juan Antonio Lopez Ramos
Department of Mathematics, University of Almeria, 04120 Almeria, Spain
[email protected]
Coautores: Joan-Josep Climent, Leandro Tortosa
The aim of this work is to introduce a framework to get new public key cryptosystems in non-commutative settings. Inspired by the cryptographical applications of the group over an elliptic curve and the necessity of enlarging the keylength
of the traditional methods used for public key cryptography due to recent advances
in cryptanalysis techniques and computations capabilities, there has been an increasing interest on finding new public key cryptosystems based on different problems to those on classical Number Theory. We define a general protocol for key exchange based on the so-called Decomposition Problem, i.e., given a monoid G and
x, y ∈ G, find g , h ∈ G such that x = g yh, extending a recently cryptanalyzed protocol
on a commutative ring of matrices and providing alternatives on non-commutative
rings. We give an example where most of its elements are non-invertible. Then
we define an ElGamal type public key cryptosystem whose cryptanalysis shows to
be equivalent to break the precedent key exchange protocol and thus its fortress is
based on the difficulty to solve the Decomposition Problem.
Locally coherent categories as hearts of t-structures
Manuel Saorín
Universidad de Murcia
[email protected]
We will show that if R is a commutative Noetherian ring and t is any t-structure
in its unbounded derived category D(R) which restricts to the bounded derived category (of finitely generated R-modules) D b (R), then the heart H of the t-structure
is a locally coherent Grothendieck category.
On endomorphism rings of Σ-injective modules
Feroz Siddique
Saint Louis University
[email protected]
Coautores: Ashish K Srivastava
Wolfson [1] and Zelinsky [2] showed that every linear transformation of a vector
space V over a division ring D is the sum of two invertible linear transformations
except when V is one-dimensional over Z2 . Khurana and Srivastava [3] extended this
by proving that that every element of a right self-injective ring R is the sum of two
units if and only if R has no factor ring isomorphic to Z2 . We study a variation of this
problem and show that if M is a Σ-injective module such that each homomorphism
of M is a sum of two commutating automorphisms then M is directly finite. Further
more, in a joint work with A.K. Srivastava [4], we show that if R is a right self-injective
ring then for each element a ∈ R there exists a unit u ∈ R such that both a + u and
a − u are units if and only if R has no factor ring isomorphic to Z2 or Z3 .
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Morita equivalences from a ring-theoretical point of view
Mercedes Siles Molina
Universidad de Málaga
[email protected]
Coautores: José Félix Solanilla Hernández
One of the aims of this talk will be to show that if two idempotent rings R and S
are Morita equivalent then for every von Neumann regular element a ∈ R the local
algebra of R at a, R a , is isomorphic to Mn (S)u for some natural n and some idempotent u in Mn (S). The converse of this result is not true in general, as we will see,
although it will be valid for σ-unital rings having a σ-unit consisting of von Neumann regular elements. A consequence is that, for idempotent rings, a property is
Morita invariant if it is invariant under taking local algebras at von Neumann regular
elements and under taking matrices.
We will use our results to check the Morita invariance of certain ring properties:
locally left/right artinian/noetherian, categorically left/right artinian, being an I 0 ring and being properly purely infinite), and of certain graph properties for Leavitt
path algebras as Condition (L), Condition (K) and cofinality. And also to provide
another proof of the fact that a graph with an uncountable emitter does not admit a
desingularization.
Skew monoidal categories and monoids
Kornel Szlachanyi
Wigner Research Centre for Physics, Budapest
[email protected]
The recently introduced skew monoidal categories offer a new approach to bialgebroids and Hopf algebroids. After reviewing the motivating example of skew monoidal structures on the category of one-sided modules over a ring we discuss the
one object case, i.e., skew monoidal monoids. We give several equivalent descriptions of skew monoidal monoids as monoids with extra structure. The symmetry of
(co)module categories of skew monoidal monoids is a dual pair of source-regular
bialgebroids. These bialgebroids have the salient feature of being submonoids of
their own base.
Referencias
1. K. Szlachanyi, Skew-monoidal categories and bialgebroids, Advances in Mathematics 231, 1694-1730 (2012)
2. K. Szlachanyi, Skew monoidal categories beyond bialgebroids,
http://maths-temp.swan.ac.uk/staff/tb/LMS-Workshop,
slides of a talk at the LMS workshop "Categorical and Homological Methods
in Hopf Algebras", Swansea, UK (2013)
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Very flat and locally very flat modules
Jan Trlifaj
Univerzita Karlova, MFF, Praha
[email protected]
Coautores: Alexander Slavik
In [1], L. Positselski introduced very flat and contraadjusted modules in his investigation of contraherent cosheaves over schemes. Here, we concentrate on the
‘affine case’, that is, we describe the structure of very flat modules over certain classes
of commutative noetherian rings. We then use it to study approximation properties
of locally very flat modules. Our guiding principle is the analogy between projective and flat Mittag-Leffler modules on one hand, and very flat and locally very flat
modules on the other (based on joint work with A. Slavik, [2]).
Referencias
1. L. Positselski, Contraherent cosheaves, preprint, arXiv:1209.2995v4.
2. A. Slavik and J. Trlifaj, Very flat and locally very flat modules, preprint.
Cohomology of cohomological Mackey functors
Thomas Weigel
Universita’ di Milano-Bicocca
[email protected]
Coautores: Blas Torrecillas, Claudio Quadrelli
Mackey functors have been introduced by A. Dress around 40 years ago. The
category of cohomological Mackey functors of a finite group G coincides with the category of contravariant additive functors of the category of finitely generated Z[G]permutation modules.
In this short talk I would like to present several old and new results concerning
the structure and cohomology theory of cohomological Mackey functors for a finite
group G. The first classical theorem which should be mentioned in this context is
Hilbert’s theorem 94. It states that for a finite cyclic Galois extension of number
fields L/K the order of the capitulation kernel is divisible by |L : K |. We will see
that this statement is an easy consequence of the general features of cohomological
Mackey functors for cyclic groups. Indeed, it also provides an interpretation if the
multiplication arising in this context. This short talk is based on joint work with B.
Torrecillas and C. Quadrelli.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S18. Modelización y predicción estocásticas
http://rsme2015.ugr.es/s18.php
Clasificación multinomial de datos funcionales mediante regresión penalizada de
mínimos cuadrados parciales
M. Carmen Aguilera-Morillo
Dpto. de Estadística. Universidad Carlos III de Madrid
[email protected]
Coautores: Ana M. Aguilera (Universidad de Granada) y Mariano J. Valderrama (Universidad de Granada)
El objetivo de este trabajo es clasificar un conjunto de datos funcionales de acuerdo a una variable de respuesta multinomial. Para conseguirlo se propone una metodología basada en análisis siscriminante lineal (LDA) de la variable de respuesta
multinomial sobre un conjunto óptimo de componentes de mínimos cuadrados
parciales funcionales (FPLS). Con objeto de mejorar la estimación de las componentes PLS y la capacidad de clasificación del modelo, se introducirán además, distintas formas de estimación penalizada del modelo de regresión FPLS. La precisión
de las estimaciones obtenidas será evaluada son datos simulados y datos espectrales
del campo de la quimiometría.
Procesos de difusión hiperbolásticos. Estimación mediante nuevos algoritmos
metaheurísticos
Antonio Jesús Barrera García
Universidad de Granada
[email protected]
Coautores: Patricia Román Román (Universidad de Granada) y Francisco Torres
Ruiz (Universidad de Granada)
Las curvas hiperbolásticas representan un avance importante en el estudio matemático de fenómenos dinámicos, en particular fenómenos de crecimiento, en los
últimos años. Su flexibilidad y versatilidad las sitúan por delante de curvas clásicas
ampliamente usadas como las curvas Gompertz o Weibull, permitiéndoles además
abarcar un amplio espectro de campos de investigación, siendo la Biomedicina su
principal marco de aplicación. Sin embargo, su uso original queda restringido al
ámbito determinista. Por esto se hace necesaria una extensión al plano estocástico,
considerando para ello la evolución hacia un proceso de difusión hiperbolástico que
pretenda abarcar en su desarrollo aquellos elementos aleatorios presentes en cada
fenómeno y cuya influencia no puede ser ignorada. En este proceso de extensión al
campo estocástico, se introducen nuevos elementos matemáticos que incrementan
la complejidad a la hora de poder estimar los modelos considerados. Para considerar estos problemas se emplean nuevos algoritmos metaheurísticos, como el algoritmo Firefly, capaces de obtener soluciones con un bajo coste computacional y que,
junto al desarrollo teórico en el campo de los procesos estocásticos, configuran una
solución completa ante los nuevos enfoques que demandan los problemas actuales.
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Inferencia en procesos de Cox mediante Análisis de Datos Funcionales
Paula R. Bouzas
Dpto. de Estadística e I.O. Universidad de Granada
[email protected]
Coautores: Nuria Ruiz-Fuentes (Universidad de Jaén)
La literatura ofrece diferentes métodos de inferencia para procesos de recuento
y en particular para procesos de Cox. La mayoría de esos métodos se centran en la
estimación más que en la predicción, y a su vez, asumen una cierta estructura estocástica de la intensidad del proceso. En este trabajo se revisa una alternativa hasta
los resultados más recientes. El proceso de Cox tiene como intensidad un proceso
estocástico en si mismo y caracteriza al primero; de ahí la importancia de su estudio. El Análisis de Datos Funcionales permite modelizar un proceso estocástico a
partir de sus observaciones. Por lo tanto, la intensidad es subceptible de ser abordada mediante esta técnica modelizándose sin más asunciones y, por consiguiente,
el propio proceso de Cox. Desde esta perspectiva, es posible hacer inferencia a partir solamente de funciones muestrales. Además de estimar y predecir los procesos
intensidad y media del de Cox, tambén se hace con otros estadísticos de recuento o
tiempo. Este método de inferencia se amplía incluso a generalizaciones del proceso
de Cox como el compuesto, el multicanal o en el tiempo-espacio. Aún más, permite construir tests de bondad de ajuste para contrastar la intensidad de éstos. Los
procesos de Cox modelizan gran cantidad de fenómenos reales (recuento de operaciones financieras, emisiones de isótopos, puntos de cambio de una magnitud, etc.)
por lo que su aplicación es muy extensa.
Optimal designs subject to cost constraints in simultaneous equations models
Victor Manuel Casero Alonso
Departamento de Matemáticas. Universidad de Castilla-La Mancha
[email protected]
Coautores: Jesús López Fidalgo (Universidad de Castilla-La Mancha)
A procedure based on a multiplicative algorithm for computing optimal experimental designs subject to cost constraints in SE models is presented. A convex criterion function based on a usual criterion function and an appropriate cost function is considered. A specific L-optimal design problem and a numerical example
are taken from Conlisk (1979) to compare the procedure. The problem would need
integer non-linear programming to obtain exact designs. To avoid this he solves a
continuous non-linear programming problem and then he rounds-off the number
of replicates of each experiment. The procedure provided here reduces dramatically
the computational efforts computing optimal approximate designs. It is based on
a specific formulation of the asymptotic covariance matrix of the full-information
maximum likelihood estimators, which simplifies the calculations. The design obtained for estimating the structural parameters of the numerical example by this
procedure is not only easier to compute, but also more efficient than the design
provided by Conlisk.
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Extendiendo el clasificador-DD en el contexto funcional
Manuel Febrero
Dpto. Estadística e I.O. Universidad de Santiago de Compostela
[email protected]
Coautores: Juan Cuesta (Universidad de Cantabria) y Manuel Oviedo (Universidad
de Santiago de Compostela)
El gráfico- DD es una herramienta basada en profundidad que fue introducida
por Liu et al. (1999) para la comparación gráfica de dos grupos o distribuciones
multivariantes. Un gráfico-DD es un gráfico bidimensional donde se dibuja el par
(D1(x);D2(x)),siendo Di(x), la profundidad del elemento x respecto al grupo i-ésimo.
Usando este gráfico, Li et al. (2012) desarrollaron un clasificador no paramétrico
mejor que el principio de máxima profundidad pero con algunas limitaciones. Este
trabajo tiene dos objetivos: primero extender el clasificador-DD para más de dos
grupos, segundo, aplicar métodos de clasificación multivariante de fácil interpretación a los gráficos-DD para obtener diagnósticos útiles para el proceso de clasificación. A lo largo de este trabajo se revisarán varias nociones de profundidad
para datos funcionales al mismo tiempo que se aplica a la nueva propuesta llamada clasificador-DDh. El trabajo se completa con un estudio de simulación y la
aplicación a varios conjuntos clásicos de la literatura de clasificación con datos funcionales.
Estimación Gaussiana basada en un procesamiento cuaternión ampliamente
lineal
Rosa María Fernández-Alcalá
Dpto. de Estadística e I.O. Universidad de Jaén
[email protected]
Coautores: Jesús Navarro-Moreno (Universidad de Jaén) y Juan Carlos Ruiz-Molina
(Universidad de Jaén)
Se presenta una solución al problema clásico de estimación Gaussiana bajo una
formulación continuo-discreta, que se basa en un modelo cuaternión que incorpora tanto la información proporcionada por la señal como la correspondiente a
la señal cuadrada. Específicamente, dado un proceso observación complejo observado en presencia de ruido propio blanco Gaussiano aditivo, se define un nuevo
proceso observación cuaternión formado por las partes real e imaginaria del proceso observación original y de su cuadrado. Entonces, aplicando un procesamiento
cuaternión ampliamente lineal (CAL) y la teoría de los desarrollos en serie de procesos estocásticos, se obtiene un estimador subóptimo expresado como la suma del
estimador complejo ampliamente lineal y una función compleja, que mejora a la
solución basada en el procesamiento complejo ampliamente lineal. Finalmente, se
propone un algoritmo recursivo para el cálculo del estimador CAL propuesto y su
error asociado. A modo de aplicación, se incluye un ejemplo de simulación.
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Modelización de fallos mediante el proceso MAP no estacionario
Rosa E. Lillo
Dpto. de Estadística. Universidad Carlos III de Madrid
[email protected]
Coautores: Joanna Rodríguez (Universidad Carlos III de Madrid) y Pepa RamírezCobo (Universidad de Cádiz)
El proceso de llegadas estacionario Markovian Arrival Processes (MAP) ha sido
utilizado en la literatura en la modelización de tiempos de llegadas que no se pueden
asumir ni independientes, ni exponenciales, lo cual ocurre con bastante frecuencia en la sucesión de fallos de componentes de sistemas de diferente índole. Sin
embargo, cuando se analizan datos en el contexto de fallos se puede observar que
asumir que los tiempos entre-fallos son idénticamente distribuidos no es correcto
en todas las situaciones. En este sentido, el trabajo que se presenta analiza las
propiedades del proceso MAP no estacionario que permite tiempos entre-llegadas
no igualmente distribuidos. En concreto, se obtiene su forma canónica y sus bondades en la modelización de fallos cuando se observan varias muestras independientes, pues permite obtener de forma matricial las cantidades de interés asociadas
a la fiabilidad del sistema. La inferencia del proceso también se ha estudiado y se
ilustra con un conjunto de datos reales que recoge fallos de componentes eléctricas.
Modelización estocástica del crecimiento tumoral en presencia de terapias. Problemas de tiempos de primer paso
Patricia Román Román
Dpto. de Estadística e I.O. Universidad de Granada
[email protected]
Coautores: Francisco Torres Ruiz (Universidad de Granada)
Se aborda el problema de la modelización del crecimiento de tumores mediante
procesos de difusión. A partir de procesos de difusión conocidos que modelizan el
crecimiento tumoral, se consideran procesos modificados mediante funciones temporales afectando a su tendencia que describen el efecto de terapias. Dichas terapias
pueden afectar tanto al crecimiento como a la muerte celular por lo que las funciones temporales introducidas pueden afectar a los distintos parámetros del proceso. Se plantean procedimientos para el ajuste de tales funciones temporales y se
proponen metodologías que permiten deducir la naturaleza (o, al menos, el efecto
prevalente) de una terapia en estudios experimentales. El estudio de problemas de
tiempo de primer paso permitirá comparar la efectividad de diversos tratamientos
a través del estudio de índices estocásticos alternativos al TGI (Tumor Growth Inhibition) y TGD (Tumor Growth Delay).
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D-MAPs en la modelización de un sistema multi-estados de fiabilidad con mantenimiento preventivo
Juan Eloy Ruiz-Castro
Dpto. de Estadística e I.O. Universidad de Granada
[email protected]
El mantenimiento preventivo juega un papel muy importante en el campo de la
fiabilidad. Uno de los objetivos en la modelización de sistemas de fiabilidad es el
de evitar los fallos del mismo o retrasar su aparición. En muchas ocasiones el fallo
de un sistema puede ocasionar daños materiales y personales que pueden ser evitados mediante mantenimiento preventivo. Por otro lado, para aumentar la fiabilidad
de un sistema se consideran unidades dispuestas en redundancia. En este trabajo
se analiza un sistema multi-estados complejo de fiabilidad con componentes en redundancia activa que evoluciona en tiempo discreto. El sistema está compuesto por
la unidad principal (online) y resto en redundancia activa. En cualquier tiempo las
unidades pueden sufrir fallos reparables debido al desgaste de las mismas. Además,
el desgaste de la unidad principal puede sufrir fallos no reparables que pueden ocurrir desde cualquier estado. Cuando ocurre un fallo reparable la unidad va al canal
de reparación donde hay un reparador para realizar un mantenimiento correctivo.
El tiempo de reparación correctivo depende de si la unidad falló siendo la principal o una en reserva. En el caso de que el fallo sea no reparable dicha unidad es
reemplazada inmediatamente por otra igual y nueva. De forma aleatoria se realizan
inspecciones sobre la unidad principal, de tal manera que si se observan daños considerables, susceptibles a fallo, entonces la unidad pasa a canal de reparación para
que se le realice mantenimiento preventivo. Se ha modelizado el sistema mediante
procesos markovianos y procesos de llegadas markovianas en tiempo discreto (DMAP), obteniendo las medidas y resultados asociados de forma algorítmica a través
de expresiones algebraico matriciales.
Multivariate autoregressive Hilbertian prediction
Javier Álvarez Liébana
Dpto. de Estadística e I.O. Universidad de Granada
[email protected]
Coautores: M. Dolores Ruiz-Medina (Universidad de Granada)
This paper deals with the problem of functional prediction in the framework of
multivariate autoregressive Hilbertian processes. The asymptotic properties of the
formulated plug-in predictor in a multivariate infinite dimensional framework are
analyzed. Specically, well-known results from the operator algebra and, in particular, noncommutative algebra theory are applied in the derivation of these asymptotic properties, considering a multivariate spectral functional decomposition of the
covariance and autocorrelation operator matrices. A simulation is carried out to illustrate the results obtained in the context of Gaussian surfaces with temporal autoregressive interaction.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S19. Singularidades
http://rsme2015.ugr.es/s19.php
Monodromy theorem for Artin kernels.
Enrique Artal Bartolo
Universidad de Zaragoza
[email protected]
Given a simplicial graph, its associated right-angled Artin group is a group generated by the vertices and such that two vertices commute if they are joined by an
edge; the kernel of an epimorphism onto the infinite cyclic group (non vanishing on
the vertices) is called an Artin kernel and its homology is in a natural way a module
over a ring of Laurent polynomials. We study the module structure of this homology
which can be interpreted as the homology of the Milnor fiber of a monomial function on a highly singular space for which a version of the Monodromy Theorem can
be stated. It is joint work in progress with J.I. Cogolludo and D. Matei.
Teoremas de Cayley-Bacharach para ecuaciones diferenciales algebraicas
Antonio Campillo
Universidad de Valladolid
[email protected]
El teorema de Cayley-Bacharach, en su versión más clásica, garantiza que si
una cúbica plana pasa por ocho de los puntos base de un haz de cúbicas entonces
pasa también por el noveno. Para foliaciones por curvas algebraicas sobre espacios proyectivos, también se tiene que si una foliación de grado dado tiene entre
sus puntos singulares un cierto conjunto finito de puntos, entonces tiene que tener
también otros relacionados con los de dicho conjunto como puntos singulares. Este
tipo de resultados, obtenidos en colaboración con Jorge Olivares, pueden entenderse como versiones del teorema de Cayley-Bacharach para ecuaciones diferenciales algebraicas.
Valoraciones terminales y el problema de Nash
Roi Docampo
IMPA, Rio de Janeiro
[email protected]
El espacio de arcos de una variedad parametriza gérmenes de curvas formales
dentro de la variedad. En un preprint muy influyente que comenzó a circular en los
años 60, Nash propone estudiar singularidades mirando al subconjunto del espacio
de arcos correspondiente a los arcos que tocan el lugar singular. Este subconjunto
se descompone en un número finito de componentes (las familias de Nash), y el
problema de Nash consiste en encontrar una interpretación geométrica para estas
familias.
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Se puede asociar a cada familia de Nash una valoración esencial. Esto es lo
que se conoce como la aplicación de Nash, y es natural preguntar si es una biyección. Este acabó por ser el caso en dimensión dos (por un teorema de Fernández de
Bobadilla y Pe Pereira), pero existen contraejemplos en dimensión superior.
En esta charla presentaré progresos recientes en el estudio del problema de Nash
en dimensión superior. En colaboración con Tommaso de Fernex, demostramos
que toda valoración terminal pertenece a la imagen de la aplicación de Nash. Las
valoraciones terminales se definen en el sentido del programa Mori, como aquellas valoraciones dadas por los divisores excepcionales en un modelo minimal sobre
la singularidad. En dimensión dos obtenemos una nueva prueba del teorema de
Fernández de Bobadilla y Pe Pereira.
Links of surface singularities: a valuative approach
Lorenzo Fantini
Ecole Polytechnique (Paris)
[email protected]
After discussing some examples of Berkovich spaces, we construct a non-archimedan
model for the link of the singularities of an algebraic variety. We study the structure
of those spaces in the case of surfaces, deducing a characterization of log essential
valuations, i.e. those valuations whose center on every log resolution of a given surface is a divisor.
Families of Plane Valuations at Infinity having good algebraic and geometric properties.
Carlos Galindo Pastor
Universitat Jaume I
[email protected]
I will introduce the concept of plane valuation at infinity. On the one hand, I
will give families of plane valuations at infinity of each of the types of the classification of plane valuations. I will prove that these families satisfy an analogous to
the Abhyankar-Moh semigroup theorem. On the other hand, I will explain the good
behavior that present global objects as the Cox ring and the cone of curves of surfaces defined by certain divisorial plane valuations at infinity. These valuations are
defined on the fraction field of the polynomial ring in two variables k[x, y] and have
either positive or nonnegative sign on k[x, y]. I will characterize these valuations
and I will show that they need not to be related with curves with one place at infinity. Results in this talk have been obtained in collaboration with F. Monserrat.
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Singular curves and knot invariants
Eugeny Gorsky
Columbia University
[email protected]
The intersection of an algebraic plane curve singularity with a small 3-sphere
is an algebraic link. It turns out that the topological invariants of this link (such
as Alexander or HOMFLY polynomials) can be expressed in terms of the geometry
of some natural moduli spaces associated with a curve. I will review these relations
(due to Campillo, Delgado, Gusein-Zade, Oblomkov, Shende and others) and explain
the recent computation of the Heegaard-Floer link homology, "categorifying" the
Alexander polynomial. The talk is based on joint work with Andras Nemethi.
Pares jacobianos
Ignacio Luengo
Universidad Complutense
[email protected]
Un par jacobiano en el plano es un par de polinomios f , g tal que su jacobiano
Jac( f , g ) = 1 y φ = ( f , g ) no es un automorfismo. Describiremos como calcular el
arbol de divisores comunes de f y g en el infinito y
Z
χ( f −1 (t ))d χ
K
a partir de dicho arbol. Este cálculo da una condicion fuerte que permite demostrar
en ciertos casos que no exiten pares jacobianos.
Una introducción a las homologías de nudos
Pedro Manchón
Universidad Politécnica de Madrid
[email protected]
En esta charla haré un pequeño recorrido por las homologías de Floer y Khovanov para nudos, presentando algunos aspectos e ideas combinatoriales. No está
previsto presentar resultados nuevos.
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Bloques de Jordan de singularidades de Yomdin-Lê de superficie.
Jorge Martín Morales
Centro Universitario de la Defensa, Zaragoza
[email protected]
El objetivo de la charla es describir los bloques de Jordan de singularidades de
Yomdin de superficie en función de su cono tangente. Las principales herramientas
e ingredientes son una generalización de la fórmula de A’Campo y la sucesión espectral de Steenbrink en el contexto de las Q-resoluciones, es decir, el espacio ambiente,
en lugar de ser liso, puede contiene singularidades cocientes abelianas.
Utilización de códigos geométricos en Criptografía
Irene Márquez Corbella
École Polytechnique, Palaiseau, France
[email protected]
En esta charla se presenta un ataque polinomial contra el criptosistema de McEliece
basado en códigos geométricos o bien en subcódigos de un código geométrico [2,3].
Este ataque permite recuperar un algoritmo de decodificación para las claves propuestas en menos de O (n 4 ) operaciones sobre Fq .
En 1978 McEliece [6] introduce el primer criptosistema de clave pública basado
en la teoría de códigos. La seguridad de este esquema se basa en la complejidad
del problema de decodificación de un código aleatorio. Es, por tanto, un candidato
interesante para la criptografía post-cuántica o, dicho de otra forma, para los criptosistemas que resisten ataques frente un hipotético ordenador cuántico. Además, la
rapidez de cifrado y descifrado es mayor que la que presentan los esquemas basados
en el problema de la factorización o en el problema del logaritmo discreto. Su principal inconveniente es, sin embargo, que requiere la utilización de claves de gran
tamaño.
En su artículo original, McEliece propone utilizar códigos Goppa binarios, esta
familia sigue siendo segura. En los años posteriores, otras familias de códigos han
sido propuestas para este esquema buscando reducir el tamaño de las claves. La
principal exigencia es que la familia tenga un algoritmo de decodificación rápido
y que permita corregir un gran número de errores. Por ejemplo (y esta lista está
lejos de ser exhaustiva), los códigos Reed-Solomon Generalizados fueron sugeridos
en [8], sus subcódigos en [1] y los códigos Reed-Muller binarios en [9]. Todos estos
sistemas han sido atacados en tiempo polinomial o sub-exponencial [10, 7 11].
Los códigos geométricos son códigos de evaluación de funciones racionales sobre curvas algebraicas. Se trata de códigos casi-optimales (tienen una gran capacidad de corrección) y, además, se conocen algoritmos de decodificación eficaces. Estas propiedades convierten a estos códigos en una alternativa interesante para el
esquema de McEliece, Janwa y Moreno los introducen con fines criptográficos en
[5]. En 2008, Faure y Minder [4] plantean un ataque estructural de este criptosistema si los códigos se construyen utilizando curvas de género g ≤ 2. Sin embargo su
aproximación no es generalizable a curvas de género superior.
Referencias
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1. T. Berger and P. Loidreau. How to mask the structure of codes for a cryptographic use. Des. Codes Cryptogr., 35:63–79, 2005.
2. A. Couvreur, I. Márquez-Corbella, and R. Pellikaan. Cryptanalysis of PublicKey cryptosystems that use Subcoces of Aalgebraic Geometry Codes. In 4th
International Castle Meeting on Coding Theory and Applications (4ICMCTA),
CIM-MS Series by Springer-Verlag. 2014.
3. A. Couvreur, I. Márquez-Corbella, and R. Pellikaan. A polynomial time attack
against algebraic geometry code based public key cryptosystems. In ISIT 2014,
IEEE Information Theory Society. 2014.
4. C. Faure and L. Minder. Cryptanalysis of the McEliece cryptosystem over hyperelliptic codes. In ACCT 2008, pages 99–107, 2008.
5. H. Janwa and O. Moreno. McEliece public cryptosystem using algebraicgeometric codes. Des. Codes Cryptogr., 8:293–307, 1996.
6. R. J. McEliece. A public-key cryptosystem based on algebraic coding theory.
DSN Progress Report, 42–44:114–116, 1978.
7. L. Minder and A. Shokrollahi. Cryptanalysis of the Sidelnikov cryptosystem.
In EUROCRYPT 2007, volume 4515 of Lecture Notes in Comput. Sci., pages
347–360. Springer-Verlag Berlin Heidelberg, 2007.
8. H. Niederreiter. Knapsack-type cryptosystems and algebraic coding theory.
Problems of Control and Information Theory, 15(2):159–166, 1986.
9. V. Sidelnikov. A public-key cryptosytem based on Reed-Muller codes. Discrete
Math. Appl., 4(3):191–207, 1994.
10. V. M. Sidelnikov and S. O. Shestakov. On the insecurity of cryptosystems based
on generalized Reed-Solomon codes. Discrete Math. Appl., 2:439–444, 1992.
11. C. Wieschebrink. Cryptanalysis of the Niederreiter public key scheme based
on GRS subcodes. In Post-Quantum Cryptography, volume 6061 of Lecture
Notes in Comput. Sci., pages 61–72. Springer-Verlag Berlin Heidelberg, 2010.
Equisingularity of families of isolated determinantal singularities.
Juan José Nuño Ballesteros
Universitat de València
[email protected]
We study the topological triviality and the Whitney equisingularity of a family of
isolated determinantal singularities. On one hand, we give a Lê-Ramanujam type
theorem for this kind of singularities by using the vanishing Euler characteristic. On
the other hand, we extend the results of Teissier and Gaffney about the Whitney
equisingularity of hypersurfaces and complete intersections, respectively, in terms
of the constancy of the polar multiplicities.
(Joint work with B. Oréfice-Okamoto and J.N. Tomazella)
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S20. Soluciones matemáticas e innovación
en la industria
http://rsme2015.ugr.es/s20.php
Computation of asymptotic formulas for the maximum voltage drop in on-chip
power distribution networks
Maria Aguareles
Departament d’Informàtica, Matemàtica Aplicada i Estadística. Universitat de
Girona.
[email protected]
Integrated circuits have nowadays a very high density of semiconductor devices
and this requires a good supply voltage across the whole chip. This is a very challenging problem from both the theoretical and technological point of view. The
chip designers must control from the very preliminary stages of the design that the
voltage drop in the power distribution network does not exceed the maximum allowed by the chip components. The chip design process is very tedious and long
and at the very late stages of it the designers use software tools that provide precise
estimations of such voltage drop throughout the chip. However, these simulations
should be just a mere check that the chip specifications are met since failing to pass
this check would imply an unaffordable waste of time and resources. In this talk
we shall present a model, in terms of a partial differential equation, for the most
used power distribution network of microchips that enables the computation of a
closed formula for the maximum voltage drop which could be used by the designers
to guarantee, from the very beginning the correctness of their design. We shall also
comment on other possible configurations for the power network which would lead
to lower power losses.
Optimization applied to the management of air traffic
Antonio Alonso Ayuso
Risk, Time & Optimization (RiTo) and Dpto. Estadística e Investigación Operativa,
Universidad Rey Juan Carlos.
[email protected]
Coautores: Laureano F. escudero, Javier Martín-Campo
En este trabajo se presentarán dos modelos para la gestión del tráfico aéreo. El
problema consiste en encontrar una planificación para los vuelos de una red de
aeropuertos de forma que no se supere la capacidad de los distintos elementos del
sistema (aeropuertos y sectores aéreos). Para ello, el modelo pude asignar retrasos en tierra y/o aire a parte de los vuelos, uso de rutas alternativas e, incluso, cancelación de alguno de los vuelos. A partir del modelo de optimización combinatoria propuesto por Bertsimas y Stock en 1998, son varias las extensiones que se han
ido presentando en la literatura, con modelos de optimización combinatoria muy
feries que permiten obtener soluciones a problemas reales en tiempos computacionales razonables. En este trabajo presentaremos un modelo matemático basado
en problemas de flujos compatibles a coste mínimo que permite relajar algunas de
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las hipótesis más restrictivas que se imponían en los modelos anteriores, con lo que
mejora su aplicabilidad.
Pseudocodes for optical encoders
Joaquim Bruna
Centre de Recerca Matemática (CRM) y Servicio de Consultoría y Transferencia (SCT),
Universitat Autònoma de Barcelona
[email protected]
Los encoders ópticos son dispositivos de alta precisión que permiten determinar
la posición y velocidad de un eje giratorio en todo momento, con una determinada
resolución N, cuyo valor puede llegar a ser de 10000, es decir se miden diezmilésimas de grado en la posición. Para ello se utilizan dos señales ópticas basadas en
laser, una señal de referencia Z y una segunda señal A que referida a A da la posición y mide revoluciones. La señal Z ha de ser mucho mas intensa que la A. El cómo
generar Z acaba siendo matemáticamente un problema de teoría de códigos, concretamente el de la generación de señales binarias cuya función de autocorrelación
no circular sea óptima en un cierto sentido. En la presentación se explicará principalmente el contexto y desarrollo matemático del problema que, sorprendentemente, muestra que la solución aportada a la empresa és la óptima teórica.
A Mathematical Model for an Application to Satellite Images
Bartomeu Coll
TAMI (Tratamiento y Analisis Matematico de Imagenes), Depto. Ciencias Matematicas e Informatica. Universitat de les Illes Balears.
[email protected]
Coautores: J. Duran, A. Buades, C. Sbert
Many Earth observation satellites provide continuously growing quantities of remote sensing images useful for a wide range of tasks. Most satellites decouple the
acquisition of a panchromatic (grayscale) image at high spatial resolution from the
acquisition of a multispectral image at lower spatial resolution. The pansharpening problem refers to the fusion process of inferring a high-resolution multispectral
image from a high-resolution panchromatic image and a low-resolution multispectral one. We present a functional that incorporates a nonlocal regularization term
and two fidelity terms, one describing the relation between the panchromatic and
the high-resolution spectral channels and the other one preserving the colors from
the low-resolution modality. This model is applied on real images from the satellite
Pléiades thanks to a joint project with CNES, the French spatial agency.
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Heuristic Optimization to design Solar Power Tower systems
Carmen-Ana Domínguez-Bravo
Instituto Universitario de Investigación de Matemáticas (IMUS) Universidad de Sevilla
[email protected]
Coautores: Emilio Carrizosa, Enrique Fernández-Cara, Manuel Quero
The design of solar power tower systems involves, among others, the heliostat
field design (heliostats number, location and size) and the receiver design (number,
size, position, aperture tilt, etc.). The different problems under consideration have
been written as large-scale nonlinear nonconvex mathematical optimization problems. Under a contract financed by Abengoa Solar NT, the research team has designed a prototype implementing new algorithms to address the above mentioned
issues. Due to the large dimensionality of the problem (thousands of variables), the
time-consuming evaluation of the objective function (given by a blackbox procedure) and the highly nonconvex shape of the feasible region, heuristics have been
developed. As output of the project, designs with higher efficiency than those reported in the literature are being obtained, and unexplored challenges so far have
been answered as well.
Stochastic optimization for risk management in energy generation capacity and
transmission expansion problem
Laureano Escudero
Universidad Rey Juan Carlos, Mostoles, Madrid.
[email protected]
Coautores: M.A. Garín, M. Merino, G. Pérez
One of the great and difficult problems that EU is facing today consists of the
estimation of timing for clean power generation technologies and electricity free
transmission expansion network at a pan-European level in a long term (e.g., 30
years time horizon). EU has established aggressive pollutant emission reduction
targets: a 20
Progressive Visibility Recovery on Mammographic Images
Adrián Galdrán
Tecnalia Research & Innovation and the University of the Basque Country (UPV/EHU)
[email protected]
With the introduction of digital mammography, image processing and computer
vision algorithms are becoming standard in the field of mammographic image analysis, to support diagnosis and early cancer detection. In this context, image enhancement methods are regularly applied, in order to improve the visibility of abnormalities within the breast, as well as to serve as a pre-processing step to ease
posterior tasks such as segmentation, detection or classification.
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In this talk, we present recent the latest advances in the Computer Vision group
of Tecnalia to design an iterative visibility recovery algorithm, able to gradually increase quality of a mammogram. Contrarily to other one-step methods, our approach ensures that potentially critical image characteristic are preserved and available to visual examination, since starting from the original image, subsequent improved versions of it remain accessible to the radiographer that can easily appreciate
the evolution of the enhancement process.
Uncertainty Quantification and its role in designing efficient simulations for biomedical applications
Luca Gerardo-Giorda
Basque Center for Applied Mathematics (BCAM), Bilbao.
[email protected]
Mathematical modeling and Scientific computing have been closely interacting
with medical science for the last 25 years at least. The possibility of performing a
virtual surgery on a patient-specific geometry is fundamental to have additional,
noninvasive, insights at both diagnostic and prognostic levels. Numerical models
and fast dedicated solvers already exist and allow in-silico exploration of the mechanisms underlying the pathologies of interest at the cost of large-scale simulations,
and CFD (Computational Fluid Dynamics) and electrophysiological simulations on
real geometries extracted from medical imaging have become an important supporting tool in advanced clinical practice. In particular, CFD simulations are today
a standard step in the decision process for both cardiovascular surgeons treating
diseases such as cerebral aneurisms, arterial stenosis, and bypass design, and nasal
surgeons that need to assess the performance of a given morphology of the nose.
However, such techniques suffer from a series of limitations that question their reliability in view of their direct application to the bedside in clinical routine. In particular, such methodologies are based on deterministic models that do not fully take into
account the huge variability associated with such complex problems. The reconstruction of the geometry of interest using MRI or CT scan is affected by errors, the
coefficients of the models are known only through a range of admissible values, and
the boundary conditions may significantly vary in time, entailing meaningless simulations. In order to overcome such limitations, Uncertainty Quantification (UQ)
can be used: UQ is a broad term encompassing a variety of methodologies whose
common goal is the assessment of the effect that input uncertainties have on the response output of interest. In this talk I will highlight the impact of UQ in designing
reliable simulations.
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New trends in TEWS: Establishing the HySEA-tsunami model for the Italian TEWS
José Manuel González Vida
Dpto. Matemática Aplicada. Universidad de Málaga.
[email protected]
Coautores: M. J. Castro, J. Macías, M. de la Asunción, C. Parés, I. Molinari, D. Melini,
F. Romano, R. Tonini, S. Lorito, A. Piatanesi
The current state of the art for the propagation phase in TEWS (Tsunami Early
Warning Systems) relies on databases of pre-computed elementary tsunami scenarios and on the linearity of the propagation of tsunami waves in deep waters. The
inundation phase sometimes is computed in reduced coastal domains or estimated
using empirical formulations, both methods using as input the result of the precomputed propagation phase. Up to date no TEWS around the world performs
real time simulations to reproduce, in much faster than real time, observed submarine tsunamigenic earthquakes effects in coastal areas. This strategy of direct
real time computation, that could seem unfeasible a decade ago, it is now foreseeable thanks to the astonishingly recent increase in the computational power and
band wide evolution of modern GPUs. The INGV in collaboration with the EDANYA
Group (University of Málaga) are proposing a new paradigm in TEWS by using a
FTRT (Faster Than Real Time) Tsunami Simulation approach to be implemented
in the Italian TEWS, namely the Centro Allerta Maremoti (CAT), which will be preoperational starting from 1 October 2014, in the 24/7 seismic monitoring room at
INGV. Tsunami-HySEA model, developed by EDANYA Group, implements in the
same code the three phases of an earthquake generated tsunami: generation, propagation and coastal inundation. The generation step implements the classical Okada
model. Once initial conditions are generated, both propagation and inundation
steps are computed using an efficient GPU-based model which uses mixed finitedifference and finite volume schemes. At the same time, this model has been implemented in nested meshes with different resolution and multi-GPU environment,
which allows much faster than real time simulations. The challenge set by the Italian
TEWS is to be able to compute the generation, propagation and a first inundation
stage of a tsunami generated in the Mediterranean Sea, in computation time below
few minutes for the whole basin.
Optimizing the supply of weighted customer-oriented services: blood transfusions items on sale, and tourist attractions
Carlos Gorria
The University of the Basque Country (UPV/EHU)
[email protected]
Coautores: M. Lezaun, F. J. López
In this work there are shown three cases of optimization models developed by
our research group in collaboration with a healthcare institution and with two private companies. The first one concerns an inventory problem related to the optimal management of blood stock and transfusions. The other two concern statistical
tools that adapt leisure, entertainment and tourist services to the customerŠs preferences. The degree of accuracy reached by the solution is highly influenced by the
correct choice of the model, the availability of resources of the service provider and
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S20. Soluciones matemáticas e innovación en la industria
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the demand forecast. Several mathematical tools are used in order to get a good fit
between the predicted results and the real observations.
3-D Wind Fields Simulation over Complex Terrain
Gustavo Montero
University Institute for Intelligent Systems and Numerical Applications in Engineering, SIANI, University of Las Palmas de Gran Canaria
[email protected]
Coautores: A. Oliver, E. Rodríguez, J. Ramírez, J.I. López, M. Brovka, J.M. Escobar, F.
Díaz, G.V. Socorro, R. Montenegro, J.M. Cascón
A new methodology for wind field simulation or forecasting over complex terrain is introduced. The idea is to use wind measurements or predictions of the
HARMONIE mesoscale model as the input data for an adaptive finite element mass
consistent wind model. The method has been recently implemented in the freelyavailable Wind3D code. A description of the HARMONIE Non-Hydrostatic Dynamics can be found in. HARMONIE provides wind prediction with a maximum resolution about 1 Km that is refined by the finite element model in a local scale (about a
few meters). An interface between both models is implemented such that the initial
wind field approximation is obtained by a suitable interpolation of the HARMONIE
results. The final model approximation is adjusted to this interpolated field verifying
incompressibility and tangency to terrain. In addition, measured data can be considered to improve the reliability of the simulations. An automatic tetrahedral mesh
generator, based on the meccano method, is applied to adapt the three-dimensional
discretization to complex terrains. This method combines several former procedures: a mapping from the meccano boundary to the solid surface, a 3-D local refinement algorithm and a simultaneous mesh untangling and smoothing. The key
of the method lies in defining a one-to-one volumetric transformation between the
parametric domain (a simple cuboid in this case) and the physical domain. The
main characteristic of the whole framework is a minimal user intervention. The final goal is to validate our model in several realistic applications in Gran Canaria
Island, Spain. For this purpose, genetic algorithms are used to obtain the optimal
model parameter values. These wind simulations can also be used for air pollution
modeling.
Mathematics at the nanoscale
Tim Myers
Centre de Recerca Matematica (CRM) and Dept. de Matematica Aplicada I, Universitat Politecnica de Catalunya, Barcelona.
[email protected]
In this talk I will briefly discuss a number of applications of mathematics to research topics in nanotechnology.
1. Phase change: The mathematical description of the change of phase of a substance, for example from liquid to solid, is well established. However, in certain
situations the standard formulations break down. I will describe our recent work
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on the melting of nanoparticles which includes two important effects that are normally neglected: the decrease in melt temperature due to the high value of surface
tension induced stress and the density variation through the phase change. 2. Heat
transfer of nanofluids: A nanofluid is a fluid containing nanoparticles. It has been
suggested that nanofluids can remove significantly more heat than standard fluids
and so may be able to solve the imminent crisis of heat removal in modern electronic
devices. We will develop a time-dependent model for the heat flow to show that the
ŠanomalousŠ increase in the thermal conductivity of the fluid is actually captured
by a simple mathematical model. We will then use boundary layer theory to show
how perhaps the most popular model for nanofluid flow, which has been used by
numerous authors to explain enhanced heat transfer, in fact predicts the opposite
result. 3. Enhanced flow in carbon nanotubes: Carbon nanotubes are viewed as
one of the most exciting new materials with applications in electronics, optics, materials science and architecture. One unusual property s that liquid flows through
nanotubes have been observed up to five orders of magnitude faster than predicted
by classical fluid dynamics. I will describe a model for fluid flow in a CNT and show
that the theoretical limit is closer to 50 times the classical value. This result is in
keeping with recent experimental and molecular dynamics papers. Along the way
we will see a physical interpretation of the Navier slip length for atomically smooth
surfaces.
Presentación de la Red Española Matemática-Industria math-in
Carlos Parés
Ecuaciones Diferenciales, Análisis Numérico y Aplicaciones (EDANYA) Universidad
de Málaga and Member of the management board of math-in
[email protected]
Esta presentación inaugural comenzará con un saludo y bienvenida a los participantes y asistentes, informándoles del ámbito y propósitos de la sesión. Seguidamente un miembro de la junta directiva de la red de matemática industria “math-in”
hará una breve exposición de la composición, funcionamiento, historia y objetivos
de la asociación, de los grupos que la componen y del tipo de sectores industriales
que más demandan la cooperación con grupos de investigación de matemáticas.
Mesh adaptation and fluid dynamic simulation for industrial applications
Lakhdar Remaki
BCAM- Basque center for applied mathematics, Bilbao, Spain
[email protected]
The talk will be on the fluid dynamic simulation by the discretization of NavierStokes equations by a cell-centered finite volume method. A mesh adaptivity technique will be presented focusing on a shock-filtering PDE-based model that allows
better capturing of multiple-shocks. The talk will end with some applications to
challenging industrial problems.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S21. Teoría de aproximación y funciones especiales
de la física matemática
http://rsme2015.ugr.es/s21.php
Una generalización del concepto de pares coherentes al caso discreto y sus aplicaciones
Renato Álvarez-Nodarse
Universidad de Sevilla
[email protected]
Coautores: J. Petronilho, N. C. Pinzón-Cortés y R. Sevinik-Adıgüzel
Sean dos sucesiones de polinomios {P n (x)}n≥0 y {Q n (x)}n≥0 ortogonales con respecto a los funcionales lineales regulares U y V , respectivamente. Diremos que el
par (U , V ) es un par (M , N )-coherente de orden (m, k) si las familias {P n (x)}n≥0 y
{Q n (x)}n≥0 satisface la siguiente relación de estructura:
ΣiM=0 a i ,n D m P n+m−i (x) = ΣiN=0 b i ,n D k Q n+k−i (x),
n ≥ 0,
donde M , N , m, k ∈ N ∪ {0}, a i ,n i b i ,n son números complejos tales que a M ,n 6= 0
para n ≥ M , b N ,n 6= 0 para n ≥ N , y a i ,n = b i ,n = 0 para i > n, siendo D j el operador
derivada de orden j .
En esta charla, vamos a mostrar como la teoría de los pares coherentes “continuous” se puede extender al caso discreto. Es decir, cuando cambiamos el operador
p(x+ω)−p(x)
o el operador
derivada d /d x por el operador diferencias finitas D ω p(x) =
ω
p(q x)−p(x)
q-derivada de Jackson D q p(x) = (q−1)x (ω ∈ C\{0}, q ∈ C\{0, 1}), respectivamente.
Revisaremos algunos aspectos de la teoría de polinomios clásicos discretos y
enumeraremos varios problemas abiertos relacionados con el problema de la coherencia discreta. Finalmente mostraremos una aplicación a los polinomios de Sobolev discretos.
On para-orthogonal polynomials
Ruymán Cruz Barroso
Universidad de La Laguna
[email protected]
Coautores: Kenier Castillo, Francisco Perdomo Pío
The aim of this talk is to present some new results on para-orthogonal polynomials (on the unit circle). First, the role played by the free parameter that characterizes
these polynomials will be discussed, in particular, in the construction of quadrature
formulas on the unit circle. Secondly, the recursive computation and analogous to
the classical Favard and Geronimus-Wendroff theorems will be stated. The second
part of this talk is a part of a joint work with K. Castillo and F. Perdomo-Pío.
Referencias
1. K. Castillo, R. Cruz-Barroso and F. Perdomo-Pío, Para-orthogonality from a
new viewpoint, submitted 2014.
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Algunos resultados sobre la interpolación de Hermite en el intervalo acotado
Jaime Díaz de Bustamante
Universidad de Vigo
[email protected]
Coautores: Elías Berriochoa, Alicia Cachafeiro
En esta charla se presentan algunos resultados sobre la interpolación de Hermite
en el intervalo acotado con nodos de Chebyshev-Lobato. Se obtienen fórmulas explícitas para los coeficientes de la segunda fórmula baricéntrica. Además, se estudia
su convergencia y otros problemas relacionados.
Polinomios ortogonales multivariantes y sistemas integrables
Manuel Mañas Baena
Universidad Complutense de Madrid
[email protected]
Coautores: Gerardo Araznibarreta
En esta charla abordaremos como la factorización de Gauss-Borel de la matriz de
momentos es útil en la descripción de polinomios ortogonales multivariantes. Consideramos tanto polinomios con medidas soportados en el espacio euclidiano como
polinomios de Laurent en el toro unitario. Obtenemos fórmulas de recurrencia y de
Christoffel-Darboux. También presentamos flujos integrables discretos y continuos
con los correspondientes elementos de la teoría de integrabilidad como funciones
de onda, pares de Lax, ecuaciones de curvatura nula, ecuaciones bilineales y matrices quasi-tau. En particular para el caso discreto tratamos las transformaciones de
Darboux elemental y su iteración. Todos estos elementos son expresados en términos de quasi-determinantes.
Uniform convergence of Hermite-Padé approximants for different systems of
Markov type functions
Sergio Medina Peralta
Universidad Carlos III de Madrid
[email protected]
This talk deals with simultaneous rational approximation. In particular we study
type I and type II Hermite-Padé approximants of analytic and meromorphic functions of Markov type. In the literature one can find a number of results on the convergence of type II Hermite-Padé approximants, in this talk we present recent result
about the convergence of type I Hermite-Padé approximants to a Nikishin system
which has been perturbed by rational functions. This kind of problem was first study
by A.A Gonchar in 1975 for the usual Padé approximantion.
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Special functions in a discrete Laplacian
Luz Roncal Gómez
Universidad de La Rioja
[email protected]
Coautores: Ó. Ciaurri, T. A. Gillespie, P. R. Stinga, J. L. Torrea, J. L. Varona.
It is known that the fundamental solution to
u t (n, t ) = ∆d u(n, t ) := u(n + 1, t ) − 2u(n, t ) + u(n − 1, t ),
n ∈ Z,
t > 0,
with u(n, 0) = δnm for every fixed m ∈ Z, is given by u(n, t ) = e −2t I n−m (2t ) (see [3]
and [4]), where I k (t ) is the Bessel function of imaginary argument. Consequently,
the heat semigroup is given by the formal series
Wt f (n) = Σm∈Z e −2t I n−m (2t ) f (m).
This function is the solution to the discrete heat equation with initial data { f (n)}n∈Z .
Other second order differential operators and the associated discrete heat kernels
arise when dealing with equations connected with physics, see [4,5].
By using semigroup theory, the formula for Wt f (n) will allow us to tackle problems in two different contexts.
• On one hand, we will define and analize some classical operators of the Harmonic Analysis associated with the discrete Laplacian ∆d , such as maximal
operators, square functions, and Riesz transforms [1].
• On the other hand we will be able to define the fractional powers of the discrete Laplacian on a mesh of size h and then to show the convergence to the
fractional Laplacian on the whole space in the discrete supremum norm as
h → 0 [2].
Since the discrete heat semigroup is given in terms of modified Bessel functions,
the careful and exhaustive use of some properties and facts about these functions is
crucial to get the results.
The work is in collaboration with Ó. Ciaurri, T. A. Gillespie, P. R. Stinga, J. L. Torrea
and J. L. Varona.
Referencias
1. Ó. Ciaurri, T. A. Gillespie, L. Roncal, J. L. Torrea and J. L. Varona, Harmonic
analysis associated with a discrete Laplacian, arXiv:1401.2091, to appear in J.
Anal. Math..
2. Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, The discrete
fractional Laplacian, preprint (2014).
3. F. A. Grünbaum, “The bispectral problem: an overview”, in Special functions
2000: current perspective and future directions (Tempe, AZ), 129–140, NATO
Sci. Ser. II Math. Phys. Chem. 30, Kluwer Acad. Publ., Dordrecht, 2001.
4. F. A. Grünbaum and P. Iliev, Heat kernel expansions on the integers, Math.
Phys. Anal. Geom. 5 (2002), no. 2, 183–200.
5. P. Iliev, Heat kernel expansions on the integers and the Toda lattice hierarchy,
Selecta Math. (N. S.) 13 (2007), no. 3, 497–530.
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Some interesting examples of discrete families of orthogonal matrix polynomials
Vanesa Sánchez Canales
Universidad de Sevilla
[email protected]
Coautores: Antonio Durán Guardeño
In this talk we show several examples of discrete orthogonal matrix polynomials in arbitrary size defined by a matrix Rodrigues’ formula. These examples satisfy
some interesting properties: they are eigenfunctions of a second order difference
operator, their differences are orthogonal. We also show that these properties are
equivalent in the scalar case but not in the matrix one.
Ortogonalidad polinómica: entropía, complejidad y entrelazamiento
Jesus Sánchez-Dehesa
Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada
[email protected]
La ortogonalidad polinómica de tipo Shohat-Favard (también llamada, a veces,
clásica o hipergeométrica) no solo ha jugado un papel fundamental en el desarrollo
de la teoría de funciones especiales, sino que ha sido determinante en numerosos
problemas científicos. En particular, ha permitido calcular analíticamente las soluciones exactas de la ecuación de movimiento mecano-cuántica no-relativista (i.e.,
ecuación de Schrödinger) de un conjunto reducido de sistemas físicos realistas, que
incluye el hidrógeno. Ello ha posibilitado recientemente la determinación de las
medidas teórico-informacionales de tales sistemas en términos de funcionales polinómicos de tipo entrópico y de complejidad, cuyo significado y cálculo matemáticos serán tratados en esta charla. Además se discutirá la necesidad de otros tipos
de ortogonalidad polinómica (e.g., matricial, multivariada) para poder explicar los
efectos relativistas y de entrelazamiento en los sistemas cuánticos.
Una realización espectral de los ceros de Riemann
Germán Sierra Rodero
Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid
[email protected]
Polya y Hilbert conjeturaron entorno a 1910 que la hipótesis de Riemann podría
ser demostrada si la parte imaginaria de los ceros no triviales de la función zeta de
Riemann fueran los autovalores de un Hamiltoniano. Existen numerosos indicios de
la existencia de dicho Hamiltoniano procedentes de la Teoría de Matrices Aleatorias
y del Caos Cuántico. En la charla mostraremos cómo los ceros de Riemann aparecen
en el espectro de un fermión de Dirac en un espacio tiempo de Rindler y sometido
a un potencial construído a partir de los números primos. Este resultado confirma
la conjetura de Polya y Hilbert y podría llevar a una demostración de la hipótesis de
Riemann.
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Mordell integrals and supersymmetric gauge theory
Miguel Tierz Parra
Universidad Complutense de Madrid
[email protected]
We show how to compute observables of Chern-Simons theories with supersymmetric matter, using Mordell integrals. We also explain how these results follow from
the random matrix model description of the gauge theory and the ensuing orthogonal polynomial solution.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
S22. Teoría de números
http://rsme2015.ugr.es/s22.php
Curvas de género 3 y realizaciones de GSp6 (F` ) como grupo de Galois sobre Q
Sara Arias de Reyna Domínguez
University of Luxembourg
[email protected]
Coautores: Cécile Armana, Valentijn Karemaker, Marusia Rebolledo, Lara Thomas,
Núria Vila
Dada una curva C de género n, definida sobre el cuerpo Q de los números racionales y un número primo `, la acción del grupo de Galois absoluto G Q sobre los
puntos de `-torsion de la variedad Jacobiana J (C ) asociada a C proporciona una
representación de Galois
ρ ` : G Q → GSp2n (F` ),
que a su vez nos proporciona una realización de la imágen de ρ ` como grupo de
Galois sobre Q.
En esta charla consideramos el siguiente problema para dimensión n = 3: dado
un primo `, construir explícitamente una curva C de género 3 sobre Q tal que la
imagen de ρ ` coincida con GSp6 (F` ).
Nuevos métodos para el cálculo de invariantes de factorización no única
Pedro A. García-Sánchez
Universidad de Granada
[email protected]
Recientemente el estudio de las factorizaciones en dominios de integridad se ha
trasladado al ambiente de los monoides (pues podemos prescindir de la suma). Así
términos como dominio de factorización única y dominios de factorización media
(todas las factorizaciones de un elemento tienen igual longitud) pasan a estudiarse
en monoides cancelativos (e incluso no cancelativos o no conmutativos).
Invariantes como la elasticidad, catenariedad, amansamiento y w-primalidad
se tratan desde el punto de vista de las presentaciones del monoide. Podemos así
usar grafos asociados a elementos, programación lineal entera e ideales binomiales
para el cálculo de estos invariantes. Daremos ejemplos de cálculos hechos con GAP,
Normaliz y Singular.
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Torsión de curvas elípticas racionales sobre cuerpos de números
Enrique González-Jiménez
Universidad Autónoma de Madrid
[email protected]
Coautores: Filip Najman, José M. Tornero
Dada una curva elíptica definida sobre el cuerpo de los racionales. Estudiamos
la relación entre el subgrupo de torsión sobre los racionales y el subgrupo de torsión
sobre un cuerpo de números de grado d.
En esta charla, se presentan varios trabajos conjuntos que abarcan el caso cuadrático (d=2, junto con J.M. Tornero [1,2]) y el caso cúbico (d=3, junto con F. Najman
y J.M. Tornero [3]).
Referencias
1. E. González-Jiménez, J.M. Tornero. "Torsion of rational elliptic curves over
quadratic fields". Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM
108 (2014), 923-934.
2. E. González-Jiménez, J.M. Tornero. "Torsion of rational elliptic curves over
quadratic fields II". arXiv: 1411.3468.
3. E. González-Jiménez, F. Najman, J.M. Tornero. "Torsion of rational elliptic
curves over cubic fields". arXiv: 1411.3467.
Pierre Fermat y su último Teorema
Josep González Rovira
Universitat Politècnica de Catalunya
[email protected]
Es una exposición histórica, que basada en la correspondencia de Fermat, intenta mostrar las técnicas y herramientas matemáticas que él utilizaba para sus demostraciones. Finalmente, la charla se centra en su último Teorema.
Norma relativa y cuerpos intermedios en extensiones de cuerpos numéricos
Ma Ángeles Gómez Molleda
Universidad de Málaga
[email protected]
Daremos una demostración de la existencia, para cada extensión de cuerpos
numéricos K ⊂ M , de un elemento primitivo α tal que todo cuerpo intermedio L
está generado sobre K por la norma relativa NLM (α). Veremos cómo calcularlo y su
aplicación en algunos casos.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Cuerpos de definición de puntos de torsión en curvas elípticas con ramificación
mínima
Alvaro Lozano-Robledo
University of Connecticut y Universidad Autonoma de Madrid (visita)
[email protected]
Sea E una curva elíptica definida sobre Q, sea p un número primo, y sea n ≥
1. Es bien sabido que el cuerpo de definición de la p n -torsión Q(E [p n ]) de una
curva elíptica E contiene las p n -esimas raices de la unidad. Por tanto, la extensión de Galois Q(E [p n ])/Q se ramifica sobre el primo p, y el índice de ramificación
e(p, Q(E [p n ])/Q) de un ideal primo ℘ de Q(E [p n ]) sobre p es divisible por ϕ(p n ). El
objectivo de esta charla es construir curvas elípticas E /Q tales que e(p, Q(E [p n ])/Q)
es precisamente ϕ(p n ), y tales que el grupo de Galois de Q(E [p n ])/Q sea tan grande
como es posible, es decir isomorfo a GL(2, Z/p n Z).
Elliptic curves and Diophantine triples
Juan Carlos Peral Alonso
UPV/EHU
[email protected]
Coautores: Andrej Dujella (Zagreb University)
A set {a 1 , a 2 , . . . , a m } of m non-zero integers (rationals) is called a (rational) Diophantine m-tuple if a i · a j + 1 is a perfect square for all 1 ≤ i < j ≤ m. In this presentation, we consider elliptic curves of the form
y 2 = (ax + 1)(bx + 1)(c x + 1),
where {a, b, c} is a rational Diophantine triple. We say that this elliptic curve is induced by the Diophantine triple {a, b, c}.
By Mazur’s theorem, there are at most four possibilities for the torsion group of
such curves, namely, Z/2Z × Z/2Z, Z/2Z × Z/4Z, Z/2Z × Z/6Z and Z/2Z × Z/8Z.
In this presentation (joint work with Andrej Dujella), we study the rank of elliptic curves induced by Diophantine triples with torsion Z/2Z × Z/4Z. The previous
records for this torsion group were rank 8 over Q and rank ≥ 3 over Q(t ).
We have found new examples of such curves over Q with rank 8 and one example
with rank 9, and a parametric family of elliptic curves with torsion group Z/2Z ×
Z/4Z and with rank ≥ 4. Moveover, we will prove that its generic rank is equal to 4
and find the generators of the Mordell-Weil group by using a recent result of I. Gusi´c,
P. Tadi´c.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Descomposición de jacobianos de curvas sobre cuerpos finitos
Antonio Rojas León
Universidad de Sevilla
[email protected]
Coautores: Omran Ahmadi, Gary McGuire
Consideramos el problema de cuándo el jacobiano de una curva definida sobre un cuerpo finito tiene un factor isógeno al jacobiano de otra, lo que se refleja
en una relación de divisibilidad entre sus polinomios L. Existen varios ejemplos ya
conocidos, provenientes de relaciones geométricas entre las curvas (por ejemplo, la
existencia de un morfismo finito entre ellas). En esta charla repasaremos estos resultados anteriores y describiremos un nuevo ejemplo relacionado con una conjetura
sobre ciertas sumas exponenciales.
Primos y órbitas unipotentes
Adrián Ubis Martínez
Universidad Autónoma de Madrid
[email protected]
Coautores: P. Sarnak
Hablaré sobre una versión cuantitativa del Teorema de Ratner para el flujo horocíclico xu n , n ∈ N, en la superficie modular y su aplicación al estudio de la distribución de los puntos xu p , con p primo.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
Sesión de pósteres
http://rsme2015.ugr.es/posters.php
Superficies de traslación de tipo lineal Weingarten en el espacio euclídeo
Antonio Bueno
Departamento de Geometría y Topología, Universidad de Granada
[email protected]
Coautores: Rafael López
Consideramos superficies en el espacio euclídeo R3 que satisfacen una relación
de tipo aH + bK = c, donde a, b, y c son números reales y H y K son la curvatura
media y la curvatura de Gauss, respectivamente. Estas superficies son llamadas en
la literatura superficies lineales de Weingarten [3,5,6]. Esta familia de superficies
generaliza a las de H constante (b = 0) y K constante (a = 0).
En este trabajo estudiamos superficies de traslación, es decir, aquéllas que locamente se escriben de la forma z = f (x) + g (y), donde (x, y, z) son las coordenadas
cartesianas de R3 . Las superficies de traslación con K o H constantes fueron clasificadas en [4] y son un plano, un cilindro generalizado o la superficie de Scherk.
El resultado que probamos, proporcionando una demostración significativamente
más simple que la que aparece en [2], es el siguiente [1]:
Teorema. Una superficie lineal de Weingarten de traslación en R3 es una superficie con K constante o H constante. En particular, la superficie es congruente con un
plano, una superficie mínima de Scherk, o un cilindro generalizado.
Con pequeñas modificaciones, este teorema se extiende al espacio de LorentzMinkowski.
Referencias
1. A. Bueno, R. López, Translation surfaces of linear Weingarten type,
arXiv:1410.2510 (2014).
2. F. Dillen, W. Goemans, I. Van de Woestyne, Translation surfaces of Weingarten
type in 3-space. Bull. Transilvania Univ. Brasov (Ser. III), 50 (2008), 109–122.
3. A. Gálvez, A. Martínez, F. Milán, Linear Weingarten surfaces in R 3 , Monatsh.
Math. 138 (2003), 133–144.
4. H. Liu, Translation surfaces with constant mean curvature in 3-dimensional
spaces, J. Geom. 64 (1999), 141–149.
5. R. López, On linear Weingarten surfaces. Int. J. Math 19 (2008), 439–448.
6. H. Rosenberg, R. Sa Earp, The geometry of properly embedded special surfaces in R 3 ; e.g., surfaces satisfying aH + bK = 1, where a and b are positive,
Duke Math. J. 73 (1994), 291–306.
150
Sesión de pósteres
151
Vórtices termoconvectivos secundarios en un anillo cilíndrico con calentamiento
no homogéneo por debajo
Damián Castaño
Universidad de Castilla - La Mancha. Departamento de Matemáticas. Facultad de
CC. y TT. Químicas, Ciudad Real, España
[email protected]
Coautores: María Cruz Navarro, Henar Herrero
La importancia de los procesos termoconvectivos en la formación e intensidad
de fenómenos meteorológicos como torbellinos o huracanes es bien conocida [1].
Los torbellinos se forman más fácilmente en presencia de grandes gradientes de
temperatura horizontal, y la evolución de la intensidad en huracanes depende, entre otros factores, del intercambio de calor con la superficie del océano que se encuentra justo debajo del ojo [2]. Estos fenómenos atmosféricos tienen una estructura vortical común caracterizada por un movimiento primario en espiral alrededor
del ojo. En numerosas ocasiones se observa la aparición de vórtices secundarios
embebidos en la circulación primaria [3]. Estos vórtices secundarios siguen esencialmente trayectorias circulares concéntricas en torno al centro del torbellino.
En Ref. [4], se prueba que bajo ciertas condiciones térmicas (incluyendo gradientes verticales y horizontales de temperatura) y condiciones geométricas (relación
de aspecto) pueden generarse numéricamente vórtices a través de una inestabilidad
termoconvectiva en un problema de Rayleigh-Bénard en un anillo cilíndrico con calentamiento no homogéneo por debajo, y con un flujo lateral de entrada/salida. Estos vórtices son estados estacionarios axisimétricos caracterizados por el giro alrededor del cilindro interior.
En el póster, se mostrará cómo estas estructuras vorticales se desestabilizan a
través de una bifurcación secundaria, así como la forma de dicha perturbación creciente, que nos dará información sobre el estado que se estabiliza tras la bifurcación.
Hemos observado que la perturbación creciente lleva a la formación de vórtices secundarios embebidos en la circulación primaria. El resultado es relevante ya que
de una manera sencilla (inestabilidades termoconvectivas) se explican las observaciones de campo en torbellinos [3].
Referencias
1. N. O. Rennó, M. L. Burkett, and M. P. Larkin, A simple thermodynamical theory
for dust devils, J. Atmos. Sci 55 (1998), 3244-3252
2. K. S. Emanuel, Thermodynamic control of hurricane intensity, Nature 401
(1999), 665-669
3. P. C. Sinclair, The lower structure of dust devil, J. Atmos. Sci 30 (1973), 15991619
4. M. C. Navarro and H. Herrero, Vortex generation by a convective instability in
a cylindrical annulus, Physica D 240 (2011), 1181-1188
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Boundedness of operators in Morrey spaces
Óscar Ciaurri
Departamento de Matemáticas y Computación, Universidad de La Rioja
[email protected]
Coautores: Alberto Arenas and Edgar Labarga
In the last few years some classical harmonic analysis operators have been analyzed in Morrey spaces. This poster contains two new results connected with this
topic. On the one hand, we study the boundedness of the partial sum operator related to Fourier–Jacobi expansions. As a consequence we characterize the convergence of these series in Morrey spaces. On the other hand, we focus on the multiplier of the interval [0, 1] for the Hankel transform of order α ≥ −1/2. In this case we
consider Morrey spaces with the measure d µα (x) = x 2α+1 d x.
Coutilidad: protocolos autocumplidos (self-enforcing) sin mecanismos de coordinación
Josep Domingo-Ferrer
Universitat Rovira i Virgili
[email protected]
Coautores: Jordi Soria-Comas, Oana Ciobotaru
La realización de una tarea entre un conjunto de pares (peers) requiere el uso de
algún protocolo que regule las interacciones entre ellos. Si los pares son racionales,
pueden intentar subvertir el protocolo para su propio beneficio, en un intento de
alcanzar un resultado que les proporcione más utilidad. Revisitamos los conceptos clásicos de protocolos autocumplidos (self-enforcing) implementados mediante
conceptos existentes de solución de teoría de juegos.
Seguidamente, describimos sus inconvenientes en aplicaciones del mundo real
y proponemos un nuevo tipo de protocolos autocumplidos, llamados protocolos
coútiles. Estos protocolos representan un concepto de solución que puede implementarse sin mecanismo de coordinación alguno en situaciones en que el concepto
clásico de protocolo autocumplido requiere un mecanismo de coordinación.
Los protocolos coútiles son claramente ventajosos en sistemas descentralizados
de pares racionales, a causa de su eficiencia y equidad. Ilustramos la aplicación de
protocolos coútiles a las tecnologías de la información, en concreto a preservar la
privacidad de los perfiles de consulta de los usuarios de motores de búsqueda y/o
bases de datos. La privacidad del perfil se mide usando teoría de la información y se
emplea como función de utilidad de los usuarios.
Referencias
1. J. Domingo-Ferrer, J. Soria-Comas and O. Ciobotaru (2015) “Co-utility: selfenforcing protocols without coordination mechanisms”, en Proc. of the 2015
IEEE Intl. Conference on Industrial Engineering and Operations Management,
IEEE, en prensa.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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153
Simetrías en las formas reales de e6
Cristina Draper
Universidad de Málaga
[email protected]
Coautores: Valerio Guido
Una G-graduación de un álgebra, para G grupo abeliano, es una descomposición
del álgebra en suma directa de subespacios indizada en G de modo compatible con
el producto. En el caso de álgebras complejas, dichas graduaciones están en correspondencia con subgrupos diagonalizables del grupo de automorfismos del álgebra,
lo que ha permitido clasificar todas las graduaciones finas (que no pueden partirse
más) de las álgebras de Lie simples complejas finito-dimensionales. La monografía
[4], recientemente publicada por la AMS, recoge dicha clasificación, con excepción
de los casos e7 y e8 , de los que sólo aparece una conjetura (para más información,
consultar [2]).
Esta técnica no puede emplearse en el caso de álgebras reales, cuyas simetrías
son bastante más desconocidas. En este póster describiremos una amplia colección
de graduaciones finas en las cinco formas reales del álgebra de Lie excepcional e6 : 9
en la forma real split, 9 en e6,2 , 6 en e6,−14 , 4 en e6,−26 y 2 en el caso compacto. Es muy
probable que dispongamos de la totalidad de graduaciones finas salvo equivalencia.
Algunos de los precedentes en los que se apoya este trabajo son [3], en el que se
clasifican las 14 graduaciones finas del álgebra compleja e6 (clasificación que también aparece en [4]), y [1], en el que se estudian las graduaciones de las formas reales
de las álgebras de Lie excepcionales de dimensión inferior a la de e6 .
Los resultados que presentamos en este póster son parte de la tesis doctoral [5],
recientemente defendida en la Universidad del Salento y cuyos resultados están aún
sin publicar.
Referencias
1. A.J. Calderón, C. Draper and C. Martín. Gradings on the real forms of g2 and
f4 . J. Math. Phys. 51(5) (2010), 053516, 21 pp.
2. C. Draper and A. Elduque. Fine gradings on the simple Lie algebras of type E .
Note Mat. 34 (2014), no. 1, 53–86.
3. C. Draper and A. Viruel. Fine gradings on e6 . arXiv:1207.6690v1. To appear in
Pub. Mat.
4. A. Elduque and M. Kochetov. Gradings on Simple Lie Algebras. Mathematical
Surveys and Monographs, Vol 189. Amer. Math. Soc. 2013.
5. V. Guido. Gradings on e6 . Ph.D.Thesis. Dottorato di Ricerca in Matematica,
Università del Salento, 2013-2014.
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Gröbner bases for modules on skew P BW extensions.
Claudia Gallego
Universidad Nacional de Colombia, sede Bogotá
[email protected]
We present the Buchberger’s algorithm for computing Gröbner bases of modules
defined on a new class of noncommutative rings: the skew P BW extensions, introduced by us in [3], as a generalization of the PBW extensions established by Bell and
Goodearl in [1]. Further, we show some elementary applications of this, such as:
membership problem, syzygy module, presentation of a module, kernel and image
of a homomorphism.
Referencias
1. Bell, A. and Goodearl, K., Uniform rank over differential operator rings and
Poincaré-Birkhoff-Witt extensons, Pacific Journal of Mathematics, 131(1), 1988,
13-37.
2. Bueso, J., Gómez-Torrecillas, J. and Verschoren, A., Algorithmic Methods in
noncommutative Algebra: Applications to Quantum Groups, Kluwer,2003.
3. Gallego, C. and Lezama, O., Gröbner bases for ideals of skew P BW extensions,
Communications in Algebra, 39, 2011, 50-75.
4. Gallego, C., Gröbner bases for bijective skew P BW extensions, Preprint.
5. Lezama, O. and Reyes, M.A., Some homological properties of skew PBW extensions, Communications in Algebra, 42, 2014, 1200-1230
Multiple Geronimus transformation
Juan Carlos García Ardila
Universidad Carlos III de Madrid
[email protected]
Coautores: Maxim Derevyagin, Francisco Marcellán
We consider multiple Geronimus transformations and show that they lead to discrete (non-diagonal) Sobolev type inner products. Moreover, it is shown that every
discrete Sobolev inner product can be obtained as a multiple Geronimus transformation. A connection with Geronimus spectral transformations for matrix orthogonal polynomials is also considered.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Álgebras de Lie cuadráticas nilpotentes
Jesús A. Laliena
Universidad de La Rioja
[email protected]
Coautores: Pilar Benito, Daniel de la Concepción
Una forma bilineal simétrica, B , sobre un álgebra de Lie, L, se dice invariante si
verifica:
B ([x, y], z) = B (x, [y, z])
La forma Killing es una forma invariante sobre cualquier álgebra de Lie semisimple.
Un álgebra de Lie nilpotente se dice cuadrática si está dotada de una forma invariante no degenerada.
Las álgebras de Lie cuadráticas nilpotentes están relacionadas con la Física y la
Geometría Riemanniana. Han sido investigadas por diversos autores. Así, por ejemplo, las álgebras de Lie nilptentes cuadráticas de dimensión hasta 7 fueron clasificadas en 1987 por G. Favre y I. J. Santharoubane; y I. Kath clasificó hasta las de dimensión 10 en 2007. G. Ovando probó en 2012 que hay álgebras de Lie cuadráticas
2-nilpotentes de cualquier dimensión, salvo 2 y 4.
En este póster se presenta una equivalencia entre la categoría de las álgebras de
Lie cuadráticas t -nilpotentes con d generadores y una categoría cuyos objetos son
las formas bilineales simétricas invariantes sobre el álgebra de Lie libre t -nilpotente
con d generadores, nd ,t .
Con esta equivalencia se muestra que la clasificación, salvo isomorfismo, de las
álgebras de Lie cuadráticas t -nilpotentes con d generadores tiene relación con la
determinación de las órbitas de una acción del grupo de los automorfismos de nd ,t
sobre las formas bilineales invariantes de nd ,t .
Convexidad de las soluciones espaciales de la ecuación de curvatura media
constante en el espacio de Lorentz-Minkowski
Rafael López
Departamento de Geometría y Topología. Universidad de Granada
[email protected]
Coautores: Alma L. Albujer (Universidad de Córdoba), Magdalena Caballero (Universidad de Córdoba)
Dado un número real H ∈ R y un dominio Ω ⊂ R2 , la solución del problema de
Dirichlet
!
Ã
Du
= 2H , |Du| < 1 en Ω
div p
1 − |Du|2
u = 0 en ∂Ω
representa una superficie espacial con curvatura media constante H en el espacio
de Lorentz-Minkowski R3 y con frontera ∂Ω ([2,3]). Consideramos el problema de
convexidad de la solución en el caso que Ω sea un dominio compacto y convexo.
Se dice que la curva ∂Ω tiene frontera pseudo-elíptica si interseca a lo más en cinco
puntos a cualquier rama de cualquier hipérbola. En este trabajo probamos ([1])
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Teorema. Sea Σ un grafo compacto espacial en R3 con curvatura media constante
H 6= 0. Si la frontera ∂Σ es una curva plana que es pseudo-elíptica, entonces Σ tiene
curvatura de Gauss negativa en todos sus puntos interiores. En particular, Σ es una
superficie convexa.
La demostración sigue ideas de Chen-Huang de comparación de un grafo euclídeo de curvatura media constante con semi-cilindros euclídeos ([4]). En nuestro
caso, donde el espacio ambiente es lorentziano, utilizamos cilindros hiperbólicos,
los cuales tienen la particularidad de ser grafos sobre todo R2 . La condición sobre la
curva frontera no se puede eliminar, y mostramos un ejemplo de un grafo espacial
definido sobre un dominio convexo que tiene curvatura media constante pero no es
estrictamente convexo.
Referencias
1. A. Albujer, M. Caballero, R. López, Convexity of the solutions to the constant
mean curvature spacelike surface equation in the Lorentz-Minkowski space,
aparecerá en J. Differential Equations.
2. L.J. Alías, R. López, J.A. Pastor, Compact spacelike surfaces with constant mean
curvature in the Lorentz-Minkowski 3-space, Tohoku Math. J. (2) 50 (4) (1998)
491–501.
3. R. Bartnik, L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1) (1982/1983) 131–152.
4. J. T. Chen, W. H. Huang, Convexity of capillary surfaces in the outer space,
Invent. Math. 67 (1982) 253–259.
A Calabi-type correspondence for the prescribed mean curvature equation
José M. Manzano
Politecnico di Torino
[email protected]
Coautores: Hojoo Lee
A Killing submersion is a Riemannian submersion Π : E → M from an orientable
3-manifold E to a surface M , such that the fibres of the submersion are the integral
curves of a unit Killing vector field. If E is endowed with a Riemannian metric, then
Π is called Riemannian, whereas Π is called Lorentzian when the Killing vector field
is timelike in E. In any of the two cases, there exists a natural geometric function in
M , called bundle curvature, which encodes completely the geometry and topology
of E (see [2]).
In 1970 Calabi proved a remarkable correspondence between minimal surfaces
in the Euclidean space R3 and maximal spacelike surfaces in the Minkowski space
L3 . Using the fact that the bundle curvature and the mean curvature of a surface
in a Killing submersion admit divergence-type equations (at least locally when the
surface is transversal to the Killing vector field), we are able to generalize the aforementioned Calabi’s correspondence to a correspondence between
(a) Mean curvature H graphs in Riemannian Killing submersions over some surface with bundle curvature τ.
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(b) Mean curvature τ spacelike graphs in Lorentzian Killing submersions over the
same surface with bundle curvature H .
Here τ and H are arbitrary smooth functions defined on the base M , which leads to a
quite general result with applications, among other, to the existence of solutions for
the prescribed mean curvature equation in R3 or to the non-existence of complete
spacelike surfaces in a large class of spacetimes (see [1]).
1. Lee, H., Manzano, J.M., Generalized Calabi’s correspondence and complete
spacelike surfaces (2013, arXiv:1301.7241).
2. Manzano, J.M., On the classification of Killing submersions and their isometries. Pacific Journal of Mathematics, 270 (2014), no. 2, 367-392.
Matrix Pearson equations for bivariate Koornwinder weights
Misael E. Marriaga
Universidad Carlos III de Madrid
[email protected]
Coautores: Francisco Marcellán, Teresa E. Pérez, Miguel Piñar
We consider Koornwinder’s method for constructing orthogonal polynomials in
two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with
these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.
Continua of solutions for quasilinear elliptic problems
Alexis Molino
Universidad de Granada
[email protected]
Coautores: Lourdes Moreno-Mérida
We study the existence of positive solutions for some quasilinear elliptic equations, having lower order terms with quadratic growth in the gradient and singularities. In particular, we consider the problem
u ∈ H01 (Ω) : −∆u + µ(x)g (u) |∇u|2 = λu p + f 0 (x)
where Ω is a smooth bounded and open subset of RN , N ≥ 3. The functions µ ∈
L ∞ (Ω), g ∈ C 1 ((0, +∞)) and f 0 ∈ L q (Ω) for some q > N /2 are nonnegative and nontrivial.
Using topological methods we obtain the existence of an unbounded continuum
of solutions and we improve the results obtained in [1, 2].
The authors present the results of a joint work with José Carmona (see [3]).
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Referencias
1. D. Arcoya, J. Carmona, P.J. Martínez-Aparicio, Bifurcation for Quasilinear Elliptic Singular BVP, Communications in Partial Differential Equations 36, 670692 (2011).
2. L. Boccardo, L. Orsina, M. A. Porzio, Existence results for quasilinear elliptic
and parabolic problems with quadratic gradient terms and sources, Adv. Calc.
Var. 4, no. 4, 397-419 (2011).
3. J. Carmona, A. Molino, L. Moreno-Mérida, Existence of a continuum of solutions for a quasilinear elliptic singular problem, Preprint.
Asymptotics for a nonstandard family of discrete orthogonal polynomials
Juan José Moreno Balcázar
Departamento de Matemáticas, Universidad de Almería
[email protected]
We provide a Mehler-Heine type formula for a nonstandard family of discrete orthogonal polynomials. Concretely, we consider the ∆-Meixner-Sobolev polynomials
which are orthogonal with respect to an inner product involving the Pascal distribution and the forward difference operator. Consequences on the zeros of these
polynomials are analyzed and illustrated numerically.
This research was partially supported by Junta de Andalucía, Research Group
FQM-0229 (belonging to Campus of International Excellence CEI-MAR) and Ministerio de Ciencia e Innovación of Spain–European Regional Development Fund,
grant MTM2011-28952-C02-01.
1. ∆–Meixner–Sobolev orthogonal polynomials: Mehler–Heine type formula and
zeros, J.J. Moreno-Balcázar, J. Comput. Appl. Math., in press, 2015.
Virtual dynamic models. Application to the teaching and investigation on architectural forms
M. Luisa Márquez García
University of Granada (Spain)
[email protected]
Coautores: Ángel H. Delgado-Olmos
The explanations of the bodies analysis topics as well as of their sections and
intersections among they are usually some themes that have a great difficulty to assimilate for the students.
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On the other hand the construction of problems on these topics also creates difficulties to the professor, especially for inserting the problem appropriately in positions and measures that are compatible with the limits of the format used for its
resolution.
In this work we treat about these topics by means of the making of 3D scale models that allow the professor the developing of these themes and to build presentations with virtual scale models, susceptible of movements for to show the solutions
as well as to make the adequate analysis.
Static Proportions: The Leon Cathedral
M. Luisa Márquez García
University of Granada
[email protected]
Coautores: C. Valverde, M.L. Márquez-Garcia, M. Pasadas
The 2:3 static proportion does not have a great relevance in a specific historic
period but it appears with certain force in some moments in history.
Although this proportion appears in some rooms of buildings of different architectonic styles, where acquires greater importance is in the Ancient Egypt and in the
Gothic.
In the Ancient Egypt appears on the temples, particularly shaping the room dedicated to the God of the temple, that is, the most important room of the building.
In the Gothic period appears in the headers of the Spain cathedrals, specifically
in the León cathedral it acquires a great importance since it appears in a trapeze
shape and moreover it determines all the spaces of the cathedral header.
In this work we study some properties of the 2:3 static proportion and its using
in the Gothic architecture.
Invariant functions of filiform Lie algebras
Juan Núñez
Dpto de Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla.
[email protected]
Coautores: José María Escobar (Dpto de Geometría y Topología. Universidad de
Sevilla) y Pedro Pérez-Fernández (Dpto. de Física Aplicada III, Escuela Técnica Superior de Ingeniería. Universidad de Sevilla.)
The invariant functions ψ and ϕ were introduced in 2007 by Hrivnák and Novotný [Petr Novotný, Jirí Hrivnák, On (α, β, γ)-derivations of Lie algebras and corresponding invariant functions, Journal of Geometry and Physics 58:2 (2008), 208-217]
as a tool to get advances in the knowledge of Lie algebras, particularly in the study of
contractions. As a particular type of Lie algebras, filiform Lie algebras, which are the
most structured subclass of nilpotent Lie algebras, were introduced by M. Vergne in
the late 60’s of the past century [Vergne, M., Cohomologie des algèbres de Lie nilpotentes, Application à l’étude de la variété des algebres de Lie nilpotentes. Bull. Soc.
Math. France 98 (1970), 81-116]. The structure of these algebras allows us to use
and study them easier than other Lie algebras, hence its importance. In this poster,
although by using a different procedure, we particularize the study of the previously
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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cited invariant functions of Lie algebras in general to the case of filiform Lie algebras. In fact, we confirm some results by the previously mentioned authors and we
deal with the filiform case in dimensions 3, 4 and 5.
Isotopisms of Lie algebras
Juan Núñez
Dpto de Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla.
[email protected]
Coautores: Óscar J. Falcón (dpto. de Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla) y Raúl M. Falcón (dpto de Matemática Aplicada I, Facultad de Informática e Ingeniería. Universidad de Sevilla).
The distribution of algebras into equivalence classes is usually done according to
the concept of isomorphism. However, such a distribution can also be done into isotopism classes. The concept of isotopism was explicitly introduced in 1942 by Abraham Adrian Albert [A. A. Albert, Non-Associative Algebras: I. Fundamental Concepts
and Isotopy, Annals of Mathematics, Second Series 43 (1942), no. 4, 685–707] to classify non-associative algebras. In this poster we deal with the study of isotopisms of
Lie algebras. The reasons for using both criteria, isotopisms and isomorphisms, to
classify Lie algebras is due to that classifications by isotopisms are different from
those by isomorphisms, which involves obtaining new information about these algebras. On a sake of example, we indicate some recent results obtained by ourselves,
which are related to the distribution into isomorphism and isotopism classes of filiform Lie algebras over finite fields. In the poster, a very brief survey about the theory
of isotopisms of algebras and quasigroups is also included.
First stability eigenvalue of compact CMC surfaces
Irene Ortiz
Universidad de Murcia
[email protected]
Coautores: Miguel A. Meroño
We find out upper bounds for the first eigenvalue of the stability operator for
compact constant mean curvature (CMC) orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the strong stability of such surfaces
is studied. We also characterize constant mean curvature Hopf tori as the only ones
attaining the bound in certain cases.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
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Orden de eliminación de variables para el cálculo de las bases de Gröbner
Walter Andrés Ortiz Vargas
Universidad del Tolima, Ibagué,Colombia
[email protected]
The study of Gröbner Basis on the ring of polynomials in several variables,
K[x 1 , x 2 , . . . , x n ] (K field) begins with the concept of reduction, which allows for the
multivariate algorithm division; the Gröbner basis describing the ideals of the ring
of polynomials,so that they can make effective calculations using the Buchberger
algorithm.To find these bases is important to establish order in the variables (lex,
deglex, degrevlex) and the monomials. Is another way to calculate using the variables elimination method. Which allows finding the intersection of two ideal, so is
also generalized to more than two ideal
Referencias
1. W. W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, Graduate
Studies in Mathematics, vol. 3, American Mathematical Society, RI, 1994.
2. T. Becker and V. Weisfening, Gröbner Bases: A Computational Aproach to Conmutative Algebra, Springer Verlag, Berlin and New York, 1993.
3. W.Ortiz. Orden de eliminación para el cálculo de las bases de Gröbner, tesis
Lic en Matemáticas. Universidad del Tolima.2012
Markov-type inequalities and duality in weighted Sobolev spaces
Jose Manuel Rodríguez
Universidad Carlos III de Madrid
[email protected]
Coautores: Francisco Marcellán, Yamilet Quintana
The aim of this work is to provide Markov-type inequalities in the setting of
weighted Sobolev spaces when the considered weights are generalized classical
weights. Also, as results of independent interest, we prove some basic facts about
Sobolev spaces with respect to measures: separability, reflexivity, uniform convexity
and duality.
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Estudio comparativo de métodos de obtención de estimaciones máximo
verosímiles de los parámetros del proceso Gompertz-Lognormal
Desirée Romero
Universidad de Granada
[email protected]
Coautores: Rico, Nuria (Universidad de Granada) G-Arenas, Maribel (Universidad
de Granada)
Para ajustar un proceso de difusión a unos datos muestrales se necesita estimar
los parámetros del mismo. Entre los métodos de estimación, el método de máxima verosimilitud presenta la ventaja de que permite estimar las funciones de los
parámetros del modelo de forma directa a partir de las estimaciones de los mismos, pero tiene el inconveniente de que la función a maximizar puede plantear algunas dificultades. El sistema de ecuaciones normales no siempre es directamente
resoluble, y en ocasiones es necesario recurrir a métodos numéricos para obtener
una solución. Dichos métodos pueden presentar problemas de dependencia de
la solución inicial, lo cual puede evitarse combinándolo con otros métodos metaheurísticos. Otra opción es usar algoritmos bio-inspirados para la optimización directa de la verosimilitud. En este trabajo se estudian y comparan varios de estos
métodos para resolver el problema de la estimación de los parámetros del proceso
de difusión Gompertz-lognormal, el cual puede utilizarse para modelizar datos con
tendencia exponencial, Gompertz o una combinación de ambas. Finalmente, para
el estudio comparativo se han simulados datos del proceso y se han considerado
diversas medidas de errores.
The GC-content of a family of cyclic codes
Josu Sangroniz
Universidad del País Vasco, UPV/EHU
[email protected]
Coautores: Luis Martínez
In this work we study some properties of a family of cyclic codes defined over
the finite field with q r , r > 1, elements that include the quadratic-residue codes over
the field with q 2 elements. Most notably, we can count how many codewords have
a fixed number of coordinates in the subfield with q elements. Our results can be
used to find large DNA-codes with fixed GC-content (that is, codes over an alphabet
with four letters, say A,G,C , T , all of whose words have the same number of letters
G or C ).
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015
Sesión de pósteres
163
Real hypersurfaces in complex two-plane Grassmannians with GTW connections
Changhwa Woo
Department of Mathematics, Kyungpook National University
[email protected]
Coautores: Juan de Dios Pérez and Young Jin Suh
In this talk, we will give some non-existence properties for Hopf real hypersurfaces in complex two-plane Grassmannians with certain geometric conditions. First,
real hypersurfaces in complex two-plane Grassmannians with generalized TanakaWebster recurrent shape operator A will be talked in detail. Next, harmonic curvature with generalized Tanaka-Webster connection for Hopf hypersurfaces in complex two-plane Grassmannians and its related topics will be given.
Referencias
1. J. Berndt, Riemannian geometry of complex two-plane Grassmannians, Rend.
Sem. Mat. Univ. Politec. Torino, 55 (1997), 19-83.
2. J. Berndt and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatsh. Math., 127 (1999), 1–14.
3. N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math., 20 (1976), 89-102.
4. S. Tanno, Variational problems on contact Riemannian manifolds, Trans. AMS
314 (1989), 349-379.
5. S.M. Webster, Peudo-Hermitian structures on a real hypersurface. J. Diff. Geom.,
13 (1978), 25-41.
Congreso bienal de la RSME. Granada, 2–6 de febrero de 2015