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II Congresso Brasileiro de Jovens
Pesquisadores em Matematica e Matematica
Aplicada
IMECC-Unicamp, 05 a 07 de dezembro de 2016.
Caderno de Resumos
Campinas-SP
Apoio financeiro: IMECC, Capes, CNPq, Fapesp, Unicamp/Faepex.
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COMISSAO ORGANIZADORA
Ademir Pastor FerreiraAnne Caroline Bronzi
Bianca Morelli R. CalsavaraBenilton de Sa Carvalho
Caio Lucidius Naberezny AzevedoDouglas Duarte Novaes
Eduardo Cardoso de AbreuFrancesco Matucci
Henrique N. Sa EarpJose Regis Azevedo Varao Filho
Juliana Marta Rodrigues de SouzaKelly Cristina Poldi
Lucio CentroneMariana Rodrigues Motta
Olivaine Santana de QueirozRicardo Miranda Martins
Simone Marchesi
COMITE CIENTIFICO
Alberto Saa (UNICAMP)Carlos Tomei (PUC - Rio de Janeiro)Carolina Bhering de Araujo (IMPA)
Daniel Pellegrino (UFPB)Eduardo Teixeira (UFC)
Ivan Shestakov (IME-USP)Jose Alberto Cuminato (ICMC-USP)
Jose Mario Martinez (UNICAMP)Marcos Jardim (UNICAMP)Maria Eulalia Vares (CBPF)Maria Jose Pacıfico (UFRJ)Nancy Lopes (UNICAMP)
Yuan Jin Yun (UFPR)
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Organizadores das sessoes tematicas
Algebra: Francesco Matucci (UNICAMP), Lucio Centrone (UNICAMP)
Algebraic Geometry: Abdelmoubine Amar Henni (UFSC), Emilio Franco Gomez (UNICAMP), EthanCotterill (UFF)
Analise Funcional, Espacos de Banach e Aplicacoes: Nacib Andre Gurgel e Albuquerque (UFPB),Pedro Levit Kaufmann (UNIFESP), Vinıcius Vieira Favaro (UFU)
Analise Numerica e Dinamica de Fluidos Computacional: Cassio Oishi (UNESP), Eduardo Abreu(UNICAMP), Fabricio Simeoni de Sousa (ICMC-USP), Pedro Peixoto (USP)
Current Trends in Partial Differential Equations and Harmonic Analysis: Edgard Pimentel(UFSCar), Tiago Picon (USP - Ribeirao Preto)
Probabilidade e Estatıstica: Caio Lucidius Naberezny Azevedo (Unicamp), Mariana Rodrigues Motta(Unicamp)
Folheacoes holomorfas e teoria de singularidades: Aurelio Menegon Neto (UFPB), Maurıcio Correa(UFMG), Michelle Ferreira Zanchetta Morgado (IBILCE/UNESP)
Geometry in Algebra and Algebra in Geometry: Cristian Ortiz (USP), Elizaveta Vishnyakova(USP), Kostiantyn Iusenko (USP)
Matematicas da Teoria da Informacao: Ana Lucila Sandoval Orozco (UCM - Espanha), RobertoAssis Machado (UNICAMP), Sara Dıaz Cardell (UNICAMP)
Nonlocal equations and applications: Juliana Pimentel (UFABC), Ricardo Alonso (PUC-RJ)
Otimizacao I: Lucas Pedroso (UFPR), Mael Sachine (UFPR)
Otimizacao II: Antoine Laurain (USP), Luıs Felipe Bueno (UNIFESP), Thadeu Alves Senne (UNI-FESP)
Sistemas Dinamicos: Gabriel Ponce (UNICAMP) , Hale Aytac (UFBA), Tiago de Carvalho (UNESP)
Teoria da Aproximacao: Mario Henrique de Castro (UFU), Thaıs Jordao (ICMC-USP)
Trends in Geometry and Topology: Andrew Clarke (UFRJ), Henrique Sa Earp (UNICAMP), SimonChiossi (UFF)
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Sumario
1 Programacao 5
2 Historico do Evento 6
3 Mesa Redonda: Mulheres nas ciencias matematicas: por que tao poucas? 7
4 Palestras Plenarias 8
5 Sessao tematica: Algebra 13
6 Sessao tematica: Algebraic Geometry 29
7 Sessao tematica: Analise Funcional, Espacos de Banach e Aplicacoes 43
8 Sessao tematica: Analise Numerica e dinamica de fluidos computacional 58
9 Sessao tematica: Current trends in partial differential equations and harmonic analy-sis 72
10 Sessao tematica: Probabilidade e Estatıstica 82
11 Sessao tematica: Folheacoes holomorfas e teoria de singularidades 97
12 Sessao tematica: Geometry in Algebra and Algebra in Geometry 105
13 Sessao tematica: Matematicas da Teoria da Informacao 115
14 Sessao tematica: Nonlocal equations and applications 130
15 Sessao tematica: Otimizacao 139
16 Sessao tematica: Sistemas Dinamicos 159
17 Sessao tematica: Teoria da Aproximacao 169
18 Sessao tematica: Trends in Geometry and Topology 181
19 Posters apresentados 194
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1 Programacao
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Abertura, Plenarias e Mesa Redonda: Centro de Convencoes da Unicamp.Assembleia, sessoes tematicas e encerramento: IMECC
Salas e datas das sessoes tematicas:
SESSÃO TEMÁTICA 05/12 06/12 07/12 ÁLGEBRA 124 -- 124 ALGEBRAIC GEOMETRY -- 223 223 ANÁLISE FUNCIONAL, ESPAÇOS DE BANACH E APLICAÇÕES
223 -- 221
ANÁLISE NUMÉRICA E DINÂMICA DE FLUIDOS COMPUTACIONAL
121 121 --
CURRENT TRENDS IN PARTIAL DIFFERENTIAL EQUATIONS AND HARMONIC ANALYSIS
151 -- 253
ESTATÍSTICA 325 325 -- FOLHEAÇÕES HOLOMORFAS E TEORIA DE SINGULARIDADES
222 324 --
GEOMETRY IN ALGEBRA AND ALGEBRA IN GEOMETRY
225 -- 225
MATEMÁTICAS DA TEORIA DA INFORMAÇÃO -- 151 151 NONLOCAL EQUATIONS AND APPLICATIONS -- 225 325 OTIMIZAÇÃO AUD AUD AUD SISTEMAS DINÂMICOS 323 323 -- TEORIA DA APROXIMAÇÃO 224 124 -- TRENDS IN GEOMETRY AND TOPOLOGY -- 224 224
AUD = AUDITÓRIO DO IMECC
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2 Historico do Evento
No Brasil devido as suas dimensoes continentais, muitos dos recem doutores que ingressam nas uni-versidades terminam por se distanciar de seus grupos de formacao. Em muitos casos, isto dificulta acontinuidade de suas pesquisas e, consequentemente, sua ascensao na carreira profissional.
Por esta razao, os jovens decidiram criar um marco apropriado para compartilhar vivencias e re-sultados de pesquisa, culminando na realizacao do I Congresso Brasileiro de Jovens Pesquisadores emMatematica Pura e Aplicada, entre os dias 10 a 12 de Dezembro de 2014, no IME-USP, em Sao Paulo.O evento contou com 221 palestrantes, distribuıdos em 15 sessoes tematicas de Matematica, MatematicaAplicada, Probabilidade e Estatıstica. Estiveram presentes pesquisadores de todas as regioes do Brasil ede mais de 20 paıses (palestrantes e ouvintes).
O evento colocou em posicao de destaque jovens pesquisadores produtivos em inıcio de carreira paradiscutir temas de vanguarda em diversas areas da matematica, bem como os problemas enfrentados noinıcio da carreira academica no paıs.
Este Caderno de Resumos contem resumos dos trabalhos apresentados no ”II Congresso Brasileirode Jovens Pesquisadores em Matematica Pura e Aplicada”, realizada no IMECC/Unicamp, em Campi-nas/SP, entre os dias 5 e 7 de dezembro de 2016. Esta edicao contou com 14 sessoes tematicas e cerca de350 participantes registrados.
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3 Mesa Redonda: Mulheres nas ciencias matematicas: por quetao poucas?
Com a participacao de:
Adriana Neumann de Oliveira (UFRGS)Cecılia Salgado Guimaraes da Silva (UFRJ)Nancy Lopes Garcia (UNICAMP)Sandra Augusta Santos (UNICAMP)
O objetivo desta mesa redonda e criar um espaco para reflexao sobre a insercao por que o numero demulheres nas ciencias exatas e tao baixo. Discutir as possıveis causas desta baixa insercao das mulheresnas ciencias exatas e os seus desdobramentos sao essenciais para que possamos discutir iniciativas quesejam capazes de reverter este quadro. Esperamos contar com a participacao de todos para que possamoster um debate rico, com troca de ideias e de pontos de vista.
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4 Palestras Plenarias
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Turbulencia hidrodinamica:
no cruzamento entre Matematica, Fısica e Engenharia
Alexei A. MailybaevIMPA, Rio de Janeiro
Abstract
O objetivo desta palestra e apresentar, de forma acessıvel, o problema de turbulenciahidrodinamica. Turbulencia premanece um dos mais importantes problemas em aberto daMatematica e da Fısica, embora sua solucao seja de extrema importancia para varias aplicacoes(predicao climatica, tecnologia aeroespacial, transporte, etc.). Em Matematica Pura, estaassociada ao um dos Sete Problemas do Milenio.
Darei uma visao geral de fenomenos naturais que continuam a desafiar a compreensaoteorica, e descreverei um espectro de problemas matematicos correlatos nas areas de sistemasdinamicos, equacoes diferenciais e probabilidade. Tambem mencionarei os principais desafiospara a analise numerica que surgem neste contexto.
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k-Modified Power Series Models
Katiane S. Conceicao
Department of Applied Mathematics and StatisticsInstitute of Mathematics and Computer Science
University of Sao PauloSao Paulo - Brazil
Abstract
In this work, we present a distribution family for count data called k-Modified Power Series, which is anextension of the Zero-Modified Power Series distribution family. This extension consists in modifying theprobability of observing k of each Power Series distribution and enabling the new k-modified distributionappropriately accommodate datasets which have any amount of observations k (for instance, k-inflated ork-deflated datasets). This work also describes the properties and particularities of the new distributionfamily for count data. The parameters of the distribution are estimated via maximum likelihood methodand the new family is used to analyze real datasets. We emphasize that the new distribution familycan accommodate sets of count data without any previous knowledge on the characteristic of k-inflation(-deflation) present in the dataset.
Keywords: Power Series Distribution; Inflation; Deflation; Count Data.
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Advances in Network Dynamics
Adilson E. MotterPhysics, Northwestern University
Abstract
An increasing number of complex systems are now modeled as networks of coupled dy-namical entities. Nonlinearity and high-dimensionality are hallmarks of the dynamics of suchnetworks but have generally been regarded as obstacles to control. In first part of this talk, Iwill discuss recent advances on mathematical and computational approaches to control high-dimensional nonlinear network dynamics under general constraints on the admissible inter-ventions. I will present applications to the stabilization of power grids, identification of newtherapeutic targets, mitigation of extinctions in food webs, and control of systemic failures insocio-economic systems.
In the second part of the talk, I will discuss fundamental new discoveries in the area ofnetwork synchronization, which has long served as a paradigm for behavioral uniformity thatcan emerge from network interactions. When the interacting entities are identical and theircoupling patterns are also identical, the complete synchronization of the entire network is thestate inheriting the system symmetry. As in other systems subject to symmetry breaking, suchsymmetric states are not always stable. In this presentation, I will report on the recent dis-covery of the converse of symmetry breaking—the scenario in which complete synchronizationis not stable for identically coupled identical oscillators but becomes stable when, and onlywhen, the oscillator parameters are judiciously tuned to nonidentical values, thereby breakingthe system symmetry to preserve the state symmetry.
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Symplectic dynamics: methods and results
Umberto L. HryniewiczUniversidade Federal do Rio de Janeiro (Brazil)
Ruhr-Universitat Bochum (Germany)[email protected] [email protected]
Abstract
Methods coming from symplectic topology and geometry have been recently causing abig impact in the study of Hamiltonian dynamical systems. Many of the results obtainedare applications of Floer theory, Gromov-Witten theory, Symplectic Field theory etc, andthese theories rely on careful study of solution sets of certain elliptic PDEs. In this talk I willpresent some of these modern tools, and explain how they are intimately related to Hamiltoniandynamics. Concrete examples of applications are certain types of forcing phenomena for Reebflows, which I obtained in collaboration with Pedro A. S. Salomao (USP) and Al Momin.
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5 Sessao tematica: Algebra
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Programação das palestras
Sessão especial: Álgebra.
Segunda-feira Sala 124
14:00 - 14:30 Ilir Snopche (UFRJ)Uncountably many non-commensurable finitely presented pro-p groups
14:30 - 15:00 Slobodan Tanusevski (UFRJ)Retracts of free groups
15:00 - 15:30 Cristina Acciarri (UnB)On profinite groups and the fixed points of coprime automorphisms
15:30 - 16:00 Café16:00 - 16:30 Martino Garonzi (UnB)
Desigualdades detectando propriedades de grupos finitos16:30 - 17:00 Raimundo Bastos (UnB)
Finiteness conditions for the non-abelian tensor product of groups17:00 - 17:30 Aline Pinto (UnB)
Some implications of Golod-Shafarevich inequality forsoluble pro-p groups of type FPm
17:30 - 18:00 Nicola Sambonet (USP)The partial multiplier of a group
Quarta-feira Sala 124
10:00 - 10:30 Diogo Diniz da Silva e Silva (UFCG)Graded identities of block-triangular matrices
10:30 - 11:00 Viviane Ribeiro Tomaz da Silva (UFMG)On Zp-graded identities and cocharacters of the Grassmann algebra
11:00 - 11:30 Manuela da Silva Souza (UFBA)Graded cocharacters of the Lie algebra W1
11:30–12:00 Fabrizio Martino (UNICAMP)Standard identities and matrices with superinvolutions
12:00–15:30 Almoço, plenária, etc.15:30–16:00 Thiago Castilho de Mello (UNIFESP)
Involutions gradings and identities on matrix algebras16:00–16:30 Javier Sánchez Serdà (USP)
A way of obtaining free group algebras inside division rings16:30–17:00 Angelo Calil Bianchi (UNIFESP)
Locally nilpotent derivations, some invariants and automorphism groups:a look at certain Danielewski surfaces
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Locally nilpotent derivations, some invariantsand
automorphism groups: a look at certain Danielewski surfaces.
Angelo Calil BianchiUNIFESP (Brazil)
Abstract
In this talk we discuss two important invariants related to locally nilpotent derivations:the ML-invariant and the Derksen invariant.To illustrate we describe the set of all locallynilpotent derivations of the quotient ring
C[X, Y, Z]/(f(X)Y '(X, Z)),
constructed from the defining equation f(X)Y = '(X, Z) of a generalized Danielewski surfacein C3 for a specific choice of polynomials f and '. As a consequence of this descriptionwe calculate the ML-invariant and the Derksen invariant of this ring anddetermine a set ofgenerators for the group of C-automorphisms of C[X, Y, Z]/(f(X)Y '(Z)) (alsofor a specificchoice of polynomials f and ').
Joint work with Marcelo Oliveira Veloso. This work waspartially supported by the CNPqgrant 462315/2014-2 and FAPESP grant 14/09310-5.
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On profinite groups and the fixed points of coprime
automorphisms
C. AcciarriUniversity of Brasilia (Brazil)[email protected]
Abstract
Many well-known results in the literature show that, if A is a finite group acting on a finitegroup G, then the structure of the centralizer CG(A) of A has influence over the structure ofG. The influence is especially strong if the action of A on G is coprime, that is if (|A|, |G|) = 1.
In this talk we will discuss results, in the realm of finite and profinite groups, that showthis phenomenon in the particular situation where, for any nontrivial element a in A, thecentralizers CG(a) consist of Engel elements.
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Graded identities of block-triangular matrices
D. DinizFederal University of Campina Grande (Brazil)
T. C. de MelloFederal University of Sao Paulo (Brazil)
Abstract
Let F be an infinite field and UT (d1, . . . , dn) be the algebra of upper block-triangularmatrices over F . In this presentation we describe a basis for the G-graded polynomial identitiesof UT (d1, . . . , dn), with an elementary grading induced by an n-tuple of elements of a groupG such that the neutral component corresponds to the diagonal of UT (d1, . . . , dn). In thecharacteristic zero case we also generalize these results for the algebra UT (d1, . . . , dn) Cwith a tensor product grading, where C is a color commutative algebra generating the varietyof all color commutative algebras.
17
Uncountably many non-commensurable finitely presented
pro-p groups
Ilir SnopcheUFRJ (Brazil)
Abstract
Let p be a prime. A pro-p group is the inverse limit of an inverse system of finite p-groups.Two groups are called commensurable if they have isomorphic subgroups of finite index.
In this talk I will show that for any positive integer m 3 there are uncountably manynon-commensurable finitely presented pro-p groups with minimal number of generators m (andminimal number of relations
m2
).
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A way of obtaining free group algebras inside division rings
Javier SanchezIME-USP (Brazil)
Abstract
In the mid eighties, L. Makar-Limanov conjectured the following:
• Let D be a division ring with center Z. If D is finitely generated (as a division ring) overZ and [D : Z] = 1, then D contains a noncommutative free Z-algebra.
In many of the examples for which the conjecture is known to be valid, the division ringD contains a (noncommutative) free group algebra over Z, not only a free Z-algebra. Notethat, in general, if X is the set of free generators of a free algebra inside a division ring, Xmay not be a set of free generators of a free group Z-algebra.
Given a division ring D with a valuation , we obtain sucient conditions for the existenceof noncommutative free group algebras in D. These conditions involve the graded division ringgrad(D) associated to the filtration induced by the valuation.
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Graded cocharacters of the Lie algebra W1
Manuela da Silva SouzaUFBA (Brazil)
Abstract
The derivations of the polynomial algebra K[t] where K is a field of characteristic zero forma Lie algebra denoted by W1 and it is known as the Witt algebra. The set en = tn+1 d
dt:
n 1 is a basis of W1. The Lie algebra structure on the vector space W1 is given by themultiplication [ei, ej ] = (j i)ei+j . The algebra W1 is Z-graded,
W1 =M
i2ZLi
where Li is the subspace of L generated by ti+1 ddt
if i 1 and Li = 0 otherwise. In [1] theauthors describe the Z-graded identities of W1 exhibiting a infinite basis for these identities.
Fixed an integer s > 0 and g1, . . ., gs 2 Z in the support of the Z-grading of W1, thereexists a natural Sn1 · · ·Sns -action in the space of multilinear graded polynomials in whichthe first n1 variables are homogeneous of degree g1, next n2 variables are homogeneous ofdegree g2 and so on. Such action give rise to a definition of Z-graded cocharacters for W1. Inthis talk we describe this cocharacters. Joint work with Gildeane Almeida Duarte.
References
[1] Freitas, Jose A.; Koshlukov, Plamen; Krasilnikov, Alexei Z-graded identities of the Lie algebraW1. J. Algebra 427 (2015), 226–251.
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Desigualdades detectando propriedades de grupos finitos.
Martino Garonzi.
Abstract
Nessa apresentacao falarei de algumas desigualdades involvendo as ordens dos elementosde um grupo finito G associadas a propriedades estruturais de G. Por exemplo mostrarei queo numero dos subgrupos cıclicos de G e uma funcao das ordens dos elementos de G, se a ordemde G tem exatamente d divisores positivos entao G contem no mınimo d subgrupos cıclicos,e G contem exatamente d subgrupos cıclicos se e somente se G e ceclico. O trabalho e juntocom Massimiliano Patassini.
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The partial multiplier of a group
Nicola SambonetUSP (Brazil)
Abstract
It is becoming relatively common to encounter the word partial as a prefix for a classicterminology of group theory. This phenomenon is due to the increasing awareness about thealgebraic relevance of the partial symmetries, that is of the inverse semigroups, which implicitlyappear in various local/global principles from well-known mathematical topics. Henceforth,the partial multiplier classifies the partial projective representations of a group in the samemanner that the ordinary Schur multiplier classifies the projective representations. However,since the algebraic structures involved in a partial representation are not only groups but alsosemigroups, the new theory diverges from the original one and the partial multiplier resultsin a semilattice of abelian groups. This research completes the previous work on the subjectproviding an exhaustive description of the partial multiplier.
22
Finiteness conditions for the non-abelian tensor product of
groups
Raimundo de Araujo Bastos JuniorUnB (Brazil)
Abstract
Let G, H be groups. We denote by (G, H) a certain extension of the non-abelian tensorproduct GH by GH. We prove that if G and H are groups that act compatibly on eachother and such that the set of all tensors T(G, H) = g h : g 2 G, h 2 H is finite, thenthe non-abelian tensor product G H is finite. In the opposite direction we examine certainfiniteness conditions of G in terms of similar conditions for the tensor square G G. Thisis a joint work with Noraı Rocco (Universidade de Brasılia) e Irene Nakaoka (UniversidadeEstadual de Maringa).
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Retracts of free groups
Slobodan Tanushevski
Abstract
I will discuss some results on the geometry of retracts of free groups. This is joint workwith Ilir Snopce.
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Involutions gradings and identities on matrix algebras
Thiago Castilho de MelloUNIFESP (Brazil)
Abstract
Let G be an abelian group and F be an algebraically closed field of characteristic zero. Wesay that an F -algebra A is G-graded, if A decomposes as a direct sum
A = g2GAg
such that for all g, h 2 G, AgAh Agh.The possible gradings on matrix algebras, Mn(F ), are well known, as well as involutions
on such algebras. Bahturin, Shestakov and Zaicev (see [1] and [2]) described under certainhypothesis, the so called involution preserving gradings on the algebra A = Mn(F ), i.e.,gradings on the algebra Mn(F ), with involution satisfying (Ag) Ag, for all g 2 G. In thistalk, we present some recent results in collaboration with L. Fonseca, describing the involutionson the graded algebra A, satisfying 8g 2 G,
(Ag) Ag1 .
We also exhibit a basis of graded identities with involution of Mn(F ) endowed with an ele-mentary grading and with the transpose involution, where F is an infinite field of arbitrarycharacteristic, generalizing previous results obtained for a particular grading and for the fieldof complex numbers in [3].
References
[1] Yu. Bahturin, I.P. Shestakov, M.V.ZaicevGradings on simple Jordan and Lie algebras. Journalof Algebra 283 (2005) 849–868.
[2] Yu. Bahturin, M. V. Zaicev Involutions on graded matrix algebras, Journal of Algebra 315(2007), 527–540.
[3] D. Haile, M. Natapov, Graded polynomial identities for matrices with the transpose involution,J. Algebra 464 (2016), 175–197.
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On Zp-graded identities and cocharacters of the Grassmann
algebra
O. M. Di VincenzoUniversita della Basilicata (Italy)
P. KoshlukovUniversidade Estadual de Campinas (Brazil)
V. R. T. da SilvaUniversidade Federal de Minas Gerais (Brazil)
Abstract
Let p be an odd prime number, F a field of characteristic zero, and let E be the unitaryGrassmann algebra generated by the infinite-dimensional F -vector space L. In this talk wewill show some recent results about the Zp-graded identities and cocharacters of the algebraE endowed with any Zp-grading such that L is a homogeneous subspace.
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Some implications of Golod-Shafarevich inequality for soluble
pro-p groups of type FPm
Aline G. S. PintoUnB (Brazil)
Abstract
In the 90’s, John Wilson proved that Golod-Shafarevich inequality holds for finitely pre-sented soluble pro-p groups. Using this, he proved that, for finitely presented soluble pro-pgroups, every normal pro-p subgroup with quotient Zp is finitely generated.
In the context of pro-p groups, the properties of being finitely generated and finitely pre-sented can be translated as the homological properties FP1 and FP2, respectively. So, Wilson’sresult can be stated as, for soluble pro-p groups of type FP2, every normal pro-p subgroupwith quotient Zp is FP1. In a recent work jointly with D. Kochloukova, we generalize thisresult for a special class of soluble pro-p groups of type FPm. In this talk we will present thegeneralization we obtained and also some other implications of Golod-Shafarevich inequalityfor soluble pro-p groups of type FPm.
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Standard identities and matrices with superinvolutions
Fabrizio MartinoUnicamp (Brazil)
Abstract
A celebrated theorem of Amitsur and Levitski (see [1]) states that the standard polynomialSt2n is an identity of Mn(F ), the algebra of n n matrices over a field F . Moreover, ifcharF 6= 2, St2n is, up to a scalar, the only identity of minimal degree of Mn(F ).
The general question whether the Amitsur-Levitski theorem could be extended to a specialkind of matrices was consider by Kostant and then by Rowen. They proved some powerfulresults (see [3, 4, 5]) concerning symmetric and skew matrices with respect to the transposeor the symplectic involution.
In this talk we shall discuss analogous results in the setting of matrix algebras with super-involution (see [2]).
References
[1] A. S. Amitsur and J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1(1950), 449–463.
[2] A. Giambruno, A. Ioppolo and F. Martino, Standard polynomials and matrices with super-involutions, Linear Algebra Appl. 504 (2016), 272–291.
[3] B. Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory,J. Math. Mech. 7 (1958), 237–264.
[4] L. H. Rowen, Standard polynomials in matrix algebras, Trans. Amer. Math. Soc. 19 (1974),253–284.
[5] L. H. Rowen, A simple proof of Kostant’s theorem, and an analogue for the symplectic invo-lution, Contemp. Math. 13 (1982), 207–215.
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6 Sessao tematica: Algebraic Geometry
29
Programacao das palestras
Sessao especial: Geometrıa Algebrica.
Segunda-feira
Sem palestras programadas.
Terca-feira Sala 223
10:30 - 11:00 David FernandezNon-commutative bi-symplectic NQ-algebras and double Poisson algebras
11:00 - 11:30 Valeriano LanzaHilbert schemes of points of the total space of OP1(−n) as quiver varieties
11:30 - 12:00 Kostiantyn IusenkoSchofield’s theorem for posets
16:00 - 16:30 Alessia MandidiHyperpolygons and parabolic Higgs bundles
16:30 - 17:00 Roberto RubioThe Toledo invariant through the Toledo character, and maximal Higgs bundles
Quarta-feira Sala 223
10:30 - 11:00 Fabio FerrariCaracteres de Cheeger-Simons generalizados
11:00 - 11:30 Gregoire MenetOn the integral cohomology of complex varieties: examples and applications
11:30 - 12:00 Alex AbreuEstruturas enriquecidas sobre curvas algebricas e tropicais
15:30 - 16:00 Andre ContieroOn minimal generators of the ideal of a moduli space and the second cohomology module of a cotangent complex
16:00 - 16:30 Cecılia SalgadoTopologies for rational point sets I
16:30 - 17:00 Oliver LorscheidTopologies for rational point sets II
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Hyperpolygons and parabolic Higgs bundles
Alessia MandiniPUC-Rio (Brazil)
Abstract
Hyperpolygons spaces are a family of (finite dimensional, non-compact) hyperkaehler spaces,that can be obtained from coadjoint orbits by hyperkaehler reduction. In joint work with L.Godinho, we show that these space are di↵eomorphic (in fact, symplectomorphic) to certainfamilies of parabolic Higgs bundles. In this talk I will describe this relation and use it toanalyse the fixed points locus of a natural involution on the moduli space of parabolic Higgsbundles. I will show that each connected components of the fixed point locus of this involutionis identified with a moduli spaces of polygons in Minkowski 3-space.
This is based on joint works with Leonor Godinho and with Indranil Biswas, Carlos Flo-rentino and Leonor Godinho.
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Estruturas enriquecidas sobre curvas algbricas e tropicais
Alex AbreuUFF (Brazil)
Abstract
Nesta palestra definiremos estruturas enriquecidas sobre curvas tropicais, em analogia coma usual definio para curvas algbricas introduzida pela Main, e construiremos um espao de mod-uli para tais objetos. Como aplicao, daremos uma descrio trica do esquema que parametrizaestruturas enriquecidas sobre uma curva nodal fixada. Este um trabalho em colaborao comMarco Pacini.
32
On minimal generators of the ideal of a moduli space and the
second cohomology module of a cotangent complex
Andre ContieroUFMG (Brazil)
Abstract
The aim of this talk is to present an outline of a working in progress in which I describe acorrespondence between the generators, possibly minimal, of the ideal of the moduli space ofGorenstein curves with prescribed symmetric Weierstrass semigroup and the basis elementsof the negatively graded part of the second cohomology module of the cotangent complex ofa monomial Gorenstein curve.
33
Non-commutative bi-symplectic NQ-algebras and double
Poisson algebras
David FernandezUFRJ-IMPA (Brazil)
Abstract
The aim of this talk will be two-fold; firstly, it will be a friendly introduction to non-commutative algebraic geometry, which has its roots in the works of Kontsevich-Rosenberg,Crawley-Boevey-Etingof-Ginzburg and Van den Bergh. Its paradigm is the Kontsevich-Rosenbergprinciple, whereby a structure on an associative algebra has geometric meaning if it inducesstandard geometric structures on its representation schemes. Moreover, we will show thatcrucial Van den Bergh’s double Poisson algebras (the non-commutative analogues of Poissonmanifolds) can be regarded in terms of a non-commutative version of graded geometry. Jointwork with Luis lvarez-Cnsul.
34
Topologies for rational point sets I
Cecılia SalgadoUFRJ (Brazil)
Abstract
For certain topological rings R, like topological fields, valuation rings or the adele ring ofa global field, the set of R-rational points X(R) of a variety X over R comes equipped with atopology. This topology has been proven to be an important tool to study the variety X.
However, a definition of a topology for X(R) is missing for more general topological ringsR. In this talk, we introduce the fine topology for X(R), which is defined for any ring R,for any R-scheme X and for any topology on R. This definition reproduces the well-knowntopologies in the above mentioned cases.
35
Caracteres de Cheeger-Simons generalizados
Fabio Ferrari RufinoUFSCar (Brazil)
Abstract
Dada uma extensao diferencial de uma teoria cohomologica multiplicativa, mostraremos oemparelhamento natural entre o sub-grupo das classes planas e a teoria homologica dual. Issogeneraliza a holonomia de uma classe de cohomologia de Deligne plana. Em seguida, parageneralizar a holonomia de qualquer classe de Deligne, definiremos os caracteres de Cheeger-Simons generalizados. Trata-se de funes definidas a partir de uma adequada extensao diferen-cial dos ciclos, cujo valor sobre um bordo so depende da curvatura.
36
On the integral cohomology of complex varieties: examples
and applications
Gregoire MenetIMECC-Unicamp (Brazil)
Abstract
The integral cohomology of varieties endowed with the cup product becomes a lattice.This topological object turns out to be, in many circumstances, a very useful tool to studythe geometry of varieties. Torelli theorems or classification of automorphisms are among themost important examples of situations where the cup-product lattice provides geometricalinformation. After a general introduction on these concepts, we will specify our attention onthe generalized Kummer 4-fold K2(T ) and its quotient by a symplectic involution K2(T )/i.Following our main goal to provide a description of the integral cohomology of these twovarieties, we will also show some general techniques to calculate the integral cohomology ofquotient varieties.
37
Schofield’s theorem for posets
Kostiantyn IusenkoIME-USP (Brazil)
Abstract
Representations of partially ordered set (poset) appear in “moduli problems” in algebraicgeometry. For instance due to A. Klyachko any (stable) equivariant sheaf on toric variety givesrise to a certain family of filtrations of finite-dimensional vector spaces, and therefore to acertain (stable) representation of some poset. In 1991, Aidan Schofield gave a characterizationof Schurian roots for given quiver Q. This characterization is given in terms of the bilinearform associated with Q. The application of this result is a canonical choice of stability for givenSchurian representation of the quiver. We prove the direct analogue of Schofield’s theorem fora poset which has finite representation type, and for the critical posets of tame representationtype.
38
Codimension one holomorphic distributions on P3
Maurıcio CorreaUFMG (Brazil)
Abstract
We classify non-integrable codimension one holomorphic distributions on P3 with low de-gree. This is a joint work with Marcos Jardim and Omegar Calvo-Andrade.
39
The Toledo invariant through the Toledo character, and
maximal Higgs bundles
Roberto RubioIMPA (Brazil)
Abstract
We will first redefine the Toledo invariant for G-Higgs bundles over a compact Riemannsurface, with G a Hermitian group, by introducing the Toledo character. The proof of theMilnor-Wood inequality for this Toledo invariant, which relies on Jordan algebra theory, willtake us to rigidity results for maximal G-Higgs bundles. The main one is known as the Cayleycorrespondence. Namely, for G of tube type, by choosing a |Z(G)|-root of the canonical bundle,the moduli space of polystable maximal G-Higgs bundles is isomorphic to the moduli spaceof polystable K2-twisted H-Higgs bundles, where H is a non-compact dual of the maximalcompact subgroup H of G. (Joint work with O. Biquard and O. Garcia-Prada).
40
Topologies for rational point sets II
Oliver LorscheidIMPA (Brazil)
Abstract
It is a classical result that a complex variety is complete if and only if its set of C-rationalpoints is compact. This generalizes to a theorem for all local fields k, but one must take allfinite extensions K/k into account.
The fine topology allows us to ask for the validity of generalizations to other topologicalrings. We review the proof of the above theorem under this aspect, and explain how far wecan generalize the connection between completeness and compactness of k-varieties.
41
Hilbert schemes of points of the total space of OP1(n) as
quiver varieties
Valeriano LanzaIMECC-Unicamp (Brazil)
Abstract
In the paper Monads for framed sheaves on Hirzebruch surfaces (Adv. Geom. 15 (2015)55(76), Bartocci, Bruzzo, and Rava provide a description of moduli spaces of framed sheaves onHirzebruch surfaces in terms of monads. Specializing this description to the rank one case, weobtain ADHM data for Hilbert schemes of points of the total space of OP1(n). These lineardata can be interpreted also within the theory of quiver representations, so that our Hilbertschemes turn out to be isomorphic to suitable moduli spaces of semistable representations ofcertain ”framed” quivers.
42
7 Sessao tematica: Analise Funcional, Espacos de Banach eAplicacoes
43
Programacao das palestras
Sessao especial: Analise Funcional, Espacos de Banach e Aplicacoes.
Segunda-feira Sala 223
Chairman: Vinıcius14:00 - 14:30 Gustavo Araujo
Optimal estimates for summing multilinear operators14:30 - 15:00 Elisa R. Santos
Equacao de Daugavet e Equacao alternativa de Daugavet para polinomios15:00 - 15:30 Pedro Kaufmann
Produtos de espacos livres e aplicacoesChairman: Pedro16:00 - 16:30 Nacib G. Albuquerque
Holder’s inequality on mixed Lp spaces and classical inequalities16:30 - 17:00 Ariosvaldo Jatoba
O dual topologico do espaco dos polinomios (p, q)-nucleares17:00 - 17:30 Wilson Cuellar
Complex structures on Twisted Hilbert spaces17:30 - 18:00 Waleed Noor
A Bergman space approach to the Riemann Hypothesis
Quarta-feira Sala 221
Chairman: Nacib10:00 - 10:30 Ximena Mujica
Extending τ -summing and σ-nuclear operators and their related results10:30 - 11:00 Dana Bartosova
Algebra in the Samuel compactification11:00 - 11:30 Sergio Leon
Banach spaces of linear operators and homogeneous polynomialswithout the approximation property
Chairman: Vinıcius15:30 - 16:00 Leandro Candido
Copias complementadas de c0(ω1) em espacos de Banach C(Kn)16:00 - 16:30 Daniel Nunez-Alarcon
Optimal exponents for Hardy–Littlewood inequalities16:30 - 17:00 Cristobal Rodrıguez Porras
Combinatoria infinita e os automorfismos do espaco de Banach `∞/c0
44
O dual topologico do espaco dos polinomios (p, q)-nucleares
A. m. JatobaFederal University of Uberlandia (Brazil)
Abstract
Sejam E, F espacos de Banach, n 2 N, p, q 2 [1,1] tais que 1p
+ 1q 1 e 1
r: = 1
p+ 1
q 1.
Um polinmio n-homogneo P 2 P(nE; F ) dito (p; q)-nuclear se existem escalares (j)1j=1 2 `r,
polinmios (Pj)1j=1 2 `wq0(P(nE)) e vetores (yj)
1j=1 2 `wp0(F ) tais que
P (x) =1X
j=1
jPj yj(x) =1X
j=1
jPj(x)yj , (1)
para todo x 2 E. Denotamos o espaco vetorial de tais polinomios por PN(p,q)(nE; F ).
ChamandokPkN(p,q)
= infk(j)
1j=1kr · k(Pj)
1j=1kw,q0 · k(yj)
1j=1kw,p0
,
onde o nfimo tomado sobre todas as representaes de P como em (1), temos uma norma emPN(p,q)
(nE; F ). Definimos a transformada de Borel por:
:hPN(p,q)
(nE; F ), k · kN(p,q)
i0! L(P(nE), F 0)
7! ( ): P(nE) ! F 0
P 7! ( )(P ): F ! K( )(P )(y) = (P y)
onde P y: E ! F e definido por P y(x) = P (x) · y.Neste trabalho caracterizamos o dual topologico do espaco dos polinomios homogeneos
(p; q)-nucleares entre espaos de Banach via transformada de Borel.
45
Banach spaces of linear operators and homogeneous
polynomials without the approximation property
Sergio A. PerezUniversity of Campinas (Brazil)
Abstract
The approximation property was introduced by Grothendieck [6]. Enflo [4] gave the firstexample of a Banach space without the approximation property. Enflo’s counterexample is anartificially constructed Banach space. The first naturally defined Banach space without theapproximation property was given by Szankowski [7], who proved that the space L(`2; `2) ofcontinuous linear operators on `2 does not have the approximation property. Later Godefroyand Saphar [5] proved that, if LK(`2; `2) denotes the subspace of all compact members ofL(`2; `2), then the quotient L(`2; `2)/LK(`2; `2) does not have the approximation property.
Recently Dineen and Mujica [3] proved that if 1 < p q < 1, then L(`p; `q) does not havethe approximation property. They also proved that if 1 < p < 1, then the space P(n`p) ofcontinuous n-homogeneous polynomials on `p does not have the approximation property forevery n p.
In this event, by using the methods of Dineen and Mujica [3] and Godefroy and Saphar [5],we present many examples of Banach spaces of linear operators and homogeneous polynomialswhich do not have the approximation property, for example we show that if 1 < p q < 1,and E and F are closed infinite dimensional subspaces of `p and `q, respectively, then L(E; F )does not have the approximation property. This improves a result of Dineen and Mujica [3].We also show that if 1 < p q < 1, and E and F are closed infinite dimensional subspaces of`p and `q, respectively, then the quotient L(E; F )/LK(E; F ) does not have the approximationproperty. This improves a result of Godefroy and Saphar [5].
We present more examples of Banach spaces of linear operators without the approximationproperty. Our examples are Banach spaces of linear operators on Pelczynski’s universal spaceU1, on Orlicz sequence spaces `Mp , and on Lorentz sequence spaces d(w, p).
References
[1] Blasco, F. - Complementation of symmetric tensor products and polynomials. Studia Math,123, 165-173, 1997.
[2] Dineen, S. - Complex Analysis on Infinite Dimensional Spaces., Springer, 1999.
[3] Dineen, S. and Mujica, J. - Banach spaces of homogeneous polynomials without theapproximation property. Czechoslovak Math. J, 140, 367-374, 2015.
[4] Enflo, P. - A counterexample to the approximation problem in Banach spaces. Acta Math,130, 309-317, 1973.
[5] Godefroy, G. and Saphar, P.D. - Three-space problems for the approximation properties.Proc. Amer. Math. Soc, 105, 70-75, 1989.
[6] Grothendieck, A. - Produits Tensoriels Topologiques et Espaces Nuclaires. Mem. Amer.Math. Soc, 16, 140 pages, 1955.
[7] Szankowski, A. - B(H) does not have the approximation property. Acta Math , 147, 89-108,1981.
46
Combinatoria infinita e os automorfismos do espaco de
Banach `1/c0
Cristobal Rodrıguez PorrasUniversidade de Sao Paulo (Brazil)
Abstract
Apresentaremos um estudo sobre os automorfismos T : `1/c0 ! `1/c0 inspirado no con-hecimento acumulado durante decadas sobre os automorfismos da algebra de Boole (N)/Fin.Essa pista nos levou a considerar “fragmentos” de tais operadores T , ie, operadores da forma
TB,A : `1(A)/c0(A) ! `1(B)/c0(B),
onde A e B sao subconjuntos infinitos de N, e a nos perguntar sobre a possibilidade derepresenta-os mediante operadores lineares de `1(A) em `1(B), matrices infinitas, funcoescontinuas de B = B \ B em A ou funcoes bijetivas de B em A. Assim chegamos a umaanalise dos operadores sobre `1/c0 (nao necessariamente automorfismos) e a introducao denovas classes de tais operadores. Provamos que assumindo o Axioma da Coloracao Aberta(Open Coloring Axiom) e o Axioma de Martin, todo automorfismo sem “funis” ou sem “fontes”tem fragmentos (no sentido de acima) induzidos por uma bijecao. Com a Hipotese do Continuoconstruimos varios exemplos de automorfismos que mostram que este resultado e otimo emvarios sentidos.
47
Algebra in the Samuel compactification
Dana BartosovaUniversidade de Sao Paulo (Brazil)
Abstract
The Samuel compactification, or the greatest ambit, is an important compactification of atopological group for its dynamics. In the case of discrete groups, the Samuel compactificationcoincides with the Cech-Stone compactification and its algebra and combinatorics have beenextensively studied. We remind the Samuel compactification for automorphism groups in theultrafilter language and point out some di↵erences and similarities with the discrete case.We will then apply algebra and combinatorics to answer a problem of Ellis for the group ofpermutations of the integers. This is a joint work in progress with Andy Zucker (CarnegieMellon University).
48
Optimal exponents for Hardy–Littlewood inequalities
D. Nunez-AlarconUniversidade Federal de Pernambuco (Brazil)
Abstract
Let K be the real or complex scalar field. In 1934 Hardy and Littlewood proved severaltheorems on the summability of bilinear forms on `p `q (here, and henceforth, when p = 1we consider c0 instead of `1). For any function f we shall consider f(1) := lims!1 f(s) andfor any s 1 we denote the conjugate index of s by s, i.e., 1
s+ 1
s = 1.
For all p, q 2 (1,1], such that 1p
+ 1q
< 1, let us define
:=pq
pq p q.
From now on, (ek)1k=1 denotes the sequence of canonical vectors in `p. The followingtheorem is one of the classical Hardy–Littlewood inequalities for bilinear forms, see Bilinearforms bounded in space [p, q], G. Hardy and J. E. Littlewood, Quart. J. Math. 5 (1934),241–254:
Theorem 0.1. Let 1 < q 2 < p, with 1p
+ 1q
< 1. There is a constant Cp,q 1 such that
1X
j1,j2=1
|A(ej1 , ej2)|! 1
Cp,q kAk ,
for all continuous bilinear forms A : `p `q ! K.
The exponent in Theorem 0.1 was improved in An interpolation approach to Hardy–Littlewood inequalities for norms of operators on sequence spaces, B. Osikiewicz and A. Tonge,Linear Algebra Appl. 331 (2001), 1–9:
Theorem 0.2. Let 1 < q 2 < p, with 1p
+ 1q
< 1. If A : `p `q ! K is a continuous bilinearform, then 0
@1X
j1=1
1X
j2=1
|A(ej1 , ej2)|q!
q1A
1
kAk .
Hardy–Littlewood type inequalities were extensively investigated in recent years, but de-spite much progress there are still several open questions concerning the optimality of expo-nents and constants.
Our main results generalize Theorems 1.1 and 1.2 with optimal exponents and constants, tothe multilinear setting. This work is contained in the preprint: Optimal exponents for Hardy-Littlewood inequalities for m-linear operators, R. Aron, D. Nunez-Alarcon, D. Pellegrino, andD. M. Serrano-Rodrıguez, arXiv:1602.00178v2 [math.FA] 11 May 2016.
49
Equacao de Daugavet e Equacao alternativa de Daugavet
para polinomios
E. R. SantosUniversidade Federal de Uberlandia (Brazil)
Resumo
Seja X um espaco de Banach. Dizemos que X tem a propriedade de Daugavet (DP) setodo operador de posto um T : X ! X satisfaz a equacao
kId + Tk = 1 + kTk,
denominada equacao de Daugavet. E dizemos que X tem a propriedade alternativa de Daugavet(ADP) se todo operador de posto um T : X ! X satisfaz a equacao
maxkId + wTk : |w| = 1 = 1 + kTk,
conhecida como equacao alternativa de Daugavet. Exemplos de espacos de Banach com taispropriedades tem sido descobertos por diversos autores desde 1963.
Em 2007, o estudo da equacao de Daugavet e da equacao alternativa de Daugavet foiestendido para polinomios. Daı surgiram os conceitos de propriedade polinomial de Daugavete propriedade polinomial alternativa de Daugavet para um espaco de Banach X. Se todopolinomio de posto um P : X ! X satisfaz
kId + Pk = 1 + kPk,
dizemos que X tem a propriedade polinomial de Daugavet (PDP); e se todo polinomio de postoum P : X ! X satisfaz
maxkId + wPk : |w| = 1 = 1 + kPk,
dizemos que X tem a propriedade polinomial alternativa de Daugavet (APDP).Ainda nao e conhecido se a propriedade de Daugavet e a propriedade alternativa de Dau-
gavet implicam a propriedade polinomial de Daugavet e a propriedade polinomial alternativade Daugavet, respectivamente. Entretanto muitos resultados sobre (DP) e (ADP) podem sergeneralizados para (PDP) e (APDP). Neste trabalho, pretendemos dar um panorama de taisresultados.
50
Optimal estimates for summing multilinear operators
G. AraujoUniversidade Estadual da Paraıba (Brazil)
D. PellegrinoUniversidade Federal da Paraıba (Brazil)
Abstract
We show that given a positive integer m, a real number p 2 [2,1) and 1 s < p theset of non–multiple (r; s)–summing m–linear forms on `p · · · `p contains, except for thenull vector, a closed subspace of maximal dimension whenever r < 2ms
s+2mms. This result
is optimal since for r 2mss+2mms
all m–linear forms on `p · · · `p are multiple (r; s)–summing. In particular, among other results, we generalize a result related to cotype (from2010) due to Botelho, Michels and Pellegrino and we observe a new optimality componentof the Bohnenblust–Hille inequality. Moreover, we investigate the optimality of coincidenceresults for multiple summing operators in c0 and in the framework of absolutely summingmultilinear operator.
51
Copias complementadas de c0(!1) em espacos de Banach
C(Kn)
Leandro CandidoUniversidade Federal de Sao Paulo (Brazil)
Abstract
Dado um espaco compacto de Hausdor↵ K, denotamos por C(Kn) o espaco de Banach dasfuncoes contınuas da forma f : K n. . . K ! R, munido da norma do supremo, ou, de formaequivalente, o produto tensorial injetivo
N"C(K) de n copias de C(K) ou ainda o espaco de
Banach da funcoes contınuas a valores vetoriais C(K, C(K, C(K..., C(K)...). Nos estudamosa existencia de copias complementadas de c0(!1) em espacos C(Kn) sob a hipotese de c0(!1)ser isomorfo a um subespaco de C(K). Esta questao esta relacionada com um resultado de E.Saab e P. Saab, que X"Y contem uma copia complementada de c0 se algum dos espacos deBanach de dimensao infinita X, Y contem uma copia de c0, e tambem um resultado de E. M.Galego e J. Hagler, obtido sob a suposicao do Maximo de Martin, que se C(K) tem densidade!1 e C(K) tem uma copia de c0(!1), entao C(K K) contem uma copia complementada dec0(!1).
Nosso principal resultado, obtido sob o Princıpio | de Ostaszewski, afirma que para todon 2 N existe um espaco compacto de Hausdor↵ Kn de peso !1 tal que C(K) e um espacode Lindelof na topologia fraca, C(Kn) contem uma copia de c0(!1), C(Kn
n ) nao contem copiacomplementada de c0(!1) enquanto C(Kn+1
n ) contem uma copia complementada de c0(!1).Isso mostra que suposicoes adicionais de teoria dos conjuntos sao necessarias no resultado deE. Galego e J. Hagler.
52
Holder’s inequality on mixed Lp spaces and classical
inequalities
N. Gurgel AlbuquerqueUniversidade Federal da Paraıba (Brazil)
D. PellegrinoUniversidade Federal da Paraıba (Brazil)
Abstract
In the last years Holder’s inequality on mixed Lp spaces helped in provide deep and vastresults and applications. We present some of these consequences: an modern point of view ofthe classical Bohnenblust-Hille and Hardy-Littlewood multilinear inequalities and its recentdevelopments on summability of multilinear operators.
53
Produtos de espacos livres e aplicacoes
Pedro Levit KaufmannUniversidade Federal de Sao Paulo (Brazil)
Abstract
Nossa exposicao e sobre os chamados espacos livres sobre um espaco metrico. Estes espacosde Banach sao pre-duais isometricos naturais para espacos de funcoes lipschitzianas e codifi-cam importantes propriedades geometricas do espaco metrico original, em particular no queconcerne a transporte otimo. Apesar da definicao simples, muitas questoes elementares sobreos espacos livres permanecem sem resposta. Mostraremos que o espaco livre associado sobreum espaco de Banach X, denotado por F(X), e isomorfo a `1-soma de enumeraveis copiasde F(X). Como aplicacoes, deduzimos uma versao nao-linear do metodo de decomposicaode Pe lczynski, e identificamos o espaco livre sobre qualquer variedade riemanniana compactacom F(Rn), a menos de isomorfismo.
54
A Bergman space approahc to the Riemann Hypothesis
W. NoorUNICAMP (Brazil)
Abstract
The Nyman-Beurling criterion is a reformulation of the Riemann Hypothesis that askswhether a particular family of functions is complete in the Hilbert space L2([0, 1]). In thistalk, the objective is to show that much of Nyman-Beurling theory and the current researchactivity around it may be approached very naturally via Bergman space theory, i.e via aHilbert space of holomorphic functions on the open unit disk in C.
55
Complex structures on Twisted Hilbert spaces
Wilson Albeiro Cuellar CarreraUniversidade de Sao Paulo (Brazil)
V. FerencziUniversidade de Sao Paulo (Brazil)
J. CastilloUniversidad de Extremadura (Spain)
Y. MorenoUniversidad de Extremadura (Spain)
Abstract
We investigate complex structures on twisted Hilbert spaces, with special attention paidto the Kalton-Peck space. We show that given any twisted Hilbert space there is a complexstructure on the Hilbert space that cannot be extended to an operator on the twisted Hilbertspace.
56
Extending -summing and -nuclear operators and their
related results
Ximena MujicaDMat - UFPR/ FAMAT - UFU (Brazil)
Abstract
The concepts of -summing and -nuclear operators in Pietsch’s Operator Ideals (NorthHolland Publishing Company, Amsterdam, 1980), present a domination theorem ( -summingoperators), a factorization theorem (-nuclear operators) and a relation between the adjointof -nuclear operators and -summing operators. For p 1, the domination theorem has beenextended to (p)-summing and quasi-(p)-summing linear and multilinear operators. Thefactorization theorem was extended to (p)-nuclear linear and multilinear operators. As to arelation between -nuclear and -summing operators, we realized there is a relation betweenthe dual of (p)-nuclear operators [L(p)(E; F )]0 and (p)-summing operators L(p)(E
0; F 0),as long as F is a reflexive space, through the Borel transform, as well as a relation between thedual of (p)-nuclear operators [L(p)(E; F )]0 and quasi-(p)-summing operators L(p)(E
0; F 0),for any F . In both cases we must have E0
i with the bounded approximation property.
Joint work with G. Botelho. The author is supported by Grant CNPq PDS 115992/2015-4;DMat - UFPR, Brazil, [email protected].
57
8 Sessao tematica: Analise Numerica e dinamica de fluidoscomputacional
58
Programacao das palestras
Sessao Tematica: Analise Numerica e Dinamica de Fluidos Computacional
Segunda-feira
14:00 - 14:30 Lucas Catao / IMECCUniqueness in critical Besov-Morrey spaces for Navier-Stokes Equations
14:30 - 15:00 Alexei Mailybaev / IMPAToward analytic theory of the Rayleigh-Taylor instability: lessons from a toy model
15:00 - 15:30 Juan Galvis / Univ. Nacional ColombiaMultiscale discretizations and iterative methods for high-contrast elliptic problems
16:00 - 16:30 Pablo Javier Blanco / LNCCVariational Derivation of a Multi-Scale Fluid Flow Model for Permeable Media
16:30 - 17:00 Pedro Peixoto / IME-USPBeyond spatial scalability limitations with a massively parallel methodfor linear oscillatory problems
17:00 - 17:30 Elias Gudino / UNESPNon-Fickian mass transport applied to coronary drug eluting stents
Terca-feira
14:00 - 14:30 Giuseppe Romanazzi / IMECCModelos Multiescala e Homogeneizado das primeiras fases deCarcinogenese no Colon
14:30 - 15:00 John Perez / ITM-IMECCA high-order non-oscillatory Lagrangian-Eulerian scheme for hyperbolic conservation laws
15:00 - 15:30 Wanderson Lambert / UNIFALSistemas hiperbolicos usados para modelar fenomenos de fluxo em meiosporosos: modelos bifasicos
16:00 - 16:30 Juliana Castanon Xavier/ UTFPRAnalise de Estabilidade e Convergencia de um Metodo EspectralTotalmente Discreto para Sistemas de Boussinesq
16:30 - 17:00 Jardel Vieira / IMECCComputing numerical solutions of the pseudo-parabolicBuckley-Leverett equation with dynamic capillary pressure
17:00 - 17:30 Andre Pierro de Camargo / UFABCInterpoladores de Floater-Hormann estendidos: uma alternativaa interpolacao trigonometrica
59
Variational Derivation of a Multi-Scale Fluid Flow Model for
Permeable Media
P. J. Blancoa, R. A. Feijoooa, and A. Clausseb
aLaboratorio Nacional de Computacao Cientıfica (Brazil)b Universidad del Centro de la Provincia de Buenos Aires (Argentina)
Abstract
Many two-phase systems consisting of ferrofluid droplets immersed in a Newtonian fluidhave a great potential in important technological applications such as the design new materials(Me↵ord et al. 2007) and drug delivery (Alexiou et al. 2002, p. 233; Voltairas et al. 2002).The understanding of droplet deformation and the coupled motion inside and outside thedroplet is important for the ecient control of ferrofluid droplets by an external magneticfield (Afkhami et al. 2008). In this work, the governing equations are the static Maxwellequations and the equations for the conservation of momentum and incompressibility. Tosolve these equations, we employ an adaptive interface tracking method to achieve full three-dimensional time-dependent two-phase simulations In which one of the phases is a ferrofluiddroplet. An unstructured mesh with geodesic triangles represents the surface of the droplet. Toour knowledge, this is the first full three-dimensional time-dependent two-phase flow simulationof such a system. A series of numerical studies is conducted to examine the response of asheared ferrofluid droplet to an applied magnetic field and to the capillary e↵ects influencingthe dynamics, deformation, and orientation of the droplet.
60
Computing numerical solutions for a pseudo-parabolic
Buckley-Leverett model with dynamic capillary
pressure in one and two dimensions
E. Abreua and J. Vieiraa,∗
a Department of Applied Mathematics,Institute of Mathematics, Statistics and Scientific Computing
University of Campinas (Brazil)
∗Presenting Author: [email protected]
Abstract
In this work, we present and discuss a novel techinique for numerically solving a partialdifferential pseudo-parabolic Buckley-Leverett-like model, which models incompressible twophase flow in porous media taking into account dynamic effects in the capillary pressure.First, we briefly discuss two numerical schemes based on the operator splitting techniquefor one-dimensional models. Our numerical experiments show that the standard splitting,widely used for numerically solving parabolic problems, may fail when applied to pseudo-parabolic models. As an illustration, we give an example for this case. Next, we present anew operator splitting scheme based on a dispersive-like character that obtains qualitativelycorrect numerical solutions. Then, we discuss an unsplit efficient numerical modelling, locallyconservative by construction, for one dimensional problems. This framework is based ona fully coupled space-time mixed hybrid finite element discretization approach in order toaccount for the delicate balance between the numerical approximations of the hyperbolic fluxand the pseudo-parabolic term, but linked to a natural dispersive-like character of the fullpseudo-parabolic equation. We compare our numerical results with approximate solutionsconstructed with state-of-the-art numerical methods introduced in the specialized literature,in order to establish that we are computing the expected qualitative behaviour of the solutions.We were also able to extend the novel fully coupled mixed hybrid finite element approach fortwo-dimensional problems.
61
Interpoladores de Floater-Hormann estendidos: uma
alternativa a interpolacao trigonometrica
Andre Pierro de CamargoUniversidade Federal do ABC (Brasil)
Resumo
Em 2007, Floater e Hormann introduziram a seguinte formula racional rd(x,x,y) parainterpolar o vetor y = (y0, y1, . . . , yn) nos nos x = (x0, x1, . . . , xn), a qual nao possui polosreais e reproduz exatamente polinomios de grau menor ou igual a d
rd(x,x,y) :=
ndPi=0
i(x,x)pi(x,x,y)
ndPi=0
i(x,x)
,
onde pi(x,x,y) e o unico polinomio de grau menor ou igual a d que interpola os dados(xi, yi), . . . , (xi+d, yi+d) e as funcoes peso i(x,x) sao definidas por
i(x,x) := (1)i
(xxi)...(xxi+d), i = 0, 1, . . . , n d.
No entanto, esses interpoladores sao mal condicionados (e consequente numericamente instaveis)para interpolacao em nos igualmente espacados quando o parametro d, o qual define a ordemde aproximacao do metodo, e grande. Os interpoladores de Floater-Hormann estendidos re-presentam uma tentativa pouco efetiva de melhorar o condicionamento dos interpoladores deFloater-Hormann para nos igualmentes espacados. Contudo, recentemente constatamos que osinterpoladores de Floater-Hormann estendidos podem ser adaptados para interpolacao estavelde funcoes periodicas em nos igualmente espacados. O metodo resultante e mais estavel doque interpolacao trigonometrica quando os nos de interpolacao nao sao exatamente igualmenteespacados. Nessa conversa discutiremos esses resultados com maior detalhe.
62
Uniqueness in critical Besov-Morrey spaces for Navier-Stokes
Equations
Lucas C. F. Ferreira, Jhean E. Perez-LopezDepartamento de Matematica - Unicamp (Brazil)
Abstract
We show the bi-continuity of the bilinear term associated to the mild formulation of theNavier-Stokes equations in critical Besov-weak-Morrey (BWM) spaces. Our bilinear estimateinvolves only the norm of the natural persistence space and does not employ auxiliary norms(like e.g. Kato time-weighted norms). As a corollary, we obtain the uniqueness of mildsolutions in the class of continuous functions from [0, T ) to critical Besov-Morrey-spaces whichis linked to the existence-class of Kozono-Yamazaki CPDE (1994). For our purposes, we needto show interpolation properties, heat semigroup estimates, and a characterization of preduals(of BWM-spaces) that are Besov type spaces based on Lorentz-block ones. Another ingredientis a product estimate in our framework.
63
Non-Fickian mass transport applied tocoronary drug eluting stents
E. GudinoDepartment of Mathematics and Computer Science, Universidade Estadual Paulista, Brazil.
C. OishiDepartment of Mathematics and Computer Science, Universidade Estadual Paulista, Brazil.
A. SequeiraDepartment of Mathematics and CEMAT-IST, Universidade de Lisboa, Portugal.
Abstract
A stent is an expandable metallic tube used to open an artery that has been narroweddue to the presence of an atherosclerotic plaque. In this talk we propose a fully coupledmodel for the description of blood flow and mass transport by a coronary drug elutingstent. We consider the steady state Navier-Stokes equations to describe the blood flowin the lumenal region coupled with the Darcy equations to describe plasma filtration inthe arterial wall. The drug release model is governed by advection, non-Fickian di↵usionand dissolution. A numerical solution of the problem, using a stabilized finite elementmethod in combination with a Nitsche type penalty method for the coupling betweendi↵erent local sub-problems is proposed. Numerical experiments to show the eciency ofthe methods will also be presented.
64
Multiscale discretizations and iterative methods for
high-contrast elliptic problems
Juan GalvisUniversidad Nacional de Colombia - Bogota
Abstract
In this talk, we review recent results in the numerical solution of elliptic equations withhigh-contrast multiscale coecients. We present an overview of the Generalized MultiscaleFinite Element Method (GMsFEM) that was recently introduced to ecient compute solutionsof porous media flow problems posed on multiscale and heterogeneous media with high-contrastin the properties values. We mention several applications and present with some more detail anapplication to two-phase fluid flow. Numerical examples are o↵ered to validate the performanceof the method in this particular application.
65
Beyond spatial scalability limitations with a massively
parallel method for linear oscillatory problems
Martin Schreiber (University of Exeter-UK)Pedro S. Peixoto (Universidade de Sao Paulo-Brasil)Terry Haut (Los Alamos National Laboratory-USA)
Beth Wingate (University of Exeter-UK)
Presenting Author: [email protected]
Abstract
This talk presents, discusses and analyses a massively parallel-in-time high order solverfor linear oscillatory PDEs, which is a key numerical component for evolving weather, ocean,climate and seismic models. The time parallelization in this solver allows us to significantlyexceed the computing resources used by parallelization-in-space methods and results in a cor-respondingly significantly reduced wall-clock time. One of the major diculties of achievingExascale performance for weather prediction is that the strong scaling limit – the parallelperformance for a fixed problem size with an increasing number of processors – saturates. Amain avenue to circumvent this problem is to introduce new numerical techniques that takeadvantage of time parallelism. In this paper we use a time-parallel approximation that retainsthe frequency information of oscillatory problems. This approximation is based on (a) refor-mulating the original problem into a large set of independent terms using an accurate rationalapproximation of an exponential integrator and (b) solving each of these terms independent ofeach other which can now be accomplished on a large number of HPC resources. Our resultsare conducted on up to 3586 cores for problem sizes with a parallelization-in-space scalabilitylimited already on a single node. We gain significant reductions in the time-to-solution of118.3 for spectral methods and 1503.0 for finite-di↵erence methods with the parallelization-in-time approach. A developed and validated performance model gives the scalability limitationsa-priory for this new approach and allows us to extrapolate the performance method towardslarge-scale system. This work has the potential to contribute as a basic building block ofparallelization-in-time approaches, with possible major implications in applied areas modellingoscillatory dominated problems.
66
Sistemas hiperbolicos usados para modelar fenomenos de
fluxo em meios porosos: modelos bifasicos
Wanderson LambertUniversidade Federal de Alfenas - ICT (Brazil)
Abstract
Apresentaremos nessa palestra alguns modelos 1-D usados para o estudo de fenomenos defluxo em meios porosos de fluidos bifasicos. Como aplicacao discutiremos os modelos quımicose termicos usados em recuperacao melhorada de petroleo.
Tais modelos tem a forma generica descrita pelas equacoes que podem ser dadas na forma:
@G(V )
@t+@uF (V )
@x= Q(V ), (1)
no qual as incognitas sao V = V (x, t) : R R+ ! Rn, u = u(x, t) : R R+ ! R+;G = (G1(V ), · · · , Gn+1(V ))T : ! Rn+1 e F = (F1(V ), · · · , Fn+1(V ))T : ! Rn+1 sao asfuncoes vetoriais do termo de acumulacao e fluxo e Q = (Q1(V ), · · · , Qn+1(V ))T : ! Rn+1
e o termo de fonte.Para o problema bifasico, se desconsiderarmos o fenomeno de adsorcao, cada Gi e Fi, para
i = 1, 2, · · · , n + 1, pode ser escrita como:
Gi = s1Ci1(y) + s2Ci2(y) Fi = f1Ci1(y) + f2Ci2(y), (2)
no qual y sao especies quımicas no modelo e s1 e s2 sao a saturacao de cada uma das fases 1e 2 e f1 e f2 sao os respectivos fluxos fracionarios.
Em geral usamos uma equacao de restricao, tal que, s1 + s2 = 1. Dessa forma a incognitaV e o vetor (s1, y).
Nosso interesse e para o caso em que o sistema (1) e hiperbolico ou hiperbolico-parabolico.O termo de fonte, Q(V ), muda o seu comportamento dependendo de quais fenomenos o
sistema esta modelando.Quando desconsideramos quaisquer transferencias entre especies quımicas ou mudancas de
fase, Q(V ) = 0, e o modelo esta na forma conservativa. Por outro lado, por exemplo, paraproblemas fısicos no qual consideramos mudancas de fase, elas sao bastante rapidas e taistermos exibem o comportamento de relaxacao, que e um grande desafio tanto do ponto devista teorico quanto numerico.
Discutiremos a teoria utilizada para o estudo do comportamento de tais modelos, bemcomo varias questoes em aberto que sao motivo de pesquisa do ponto de vista teorico e comimplicacoes inclusive numericas.
67
Toward analytic theory of the Rayleigh-Taylor instability:
lessons from a toy model
Alexei A. MailybaevIMPA, Rio de Janeiro
Abstract
The Rayleigh–Taylor instability occurs at an interface between two fluids of di↵erent den-sities, when heavier fluid is placed above the lighter one. Propagation of a disturbance fromsmaller to larger scales due to nonlinear interaction generates a very complex turbulent dynam-ics in a growing mixing layer. Occurrence of the RT instability is abundant in nature, whichincludes astrophysical, geological and atmospheric phenomena, as well as various technologicalapplications such as combustion.
In this work we suggest that a turbulent phase of the Rayleigh-Taylor instability can beexplained as a universal stochastic wave traveling with constant speed in a properly renor-malized system. This wave, originating from ordinary deterministic chaos in a renormalizedtime, has two constant limiting states at both sides. The theoretical analysis is confirmed withextensive numerical simulations for a new shell model of convection turbulence.
68
A high-order non-oscillatory Lagrangian-Eulerian scheme for
hyperbolic conservation laws
E. Abreua and J. Perezb
aUniversity of Campinas (Brazil)bITM-Institucion Universitaria (Colombia)
Abstract
In this work, we present a formal construction of an accurate, high-resolution, solver fornumerically solving nonlinear hyperbolic conservation law problems using a novel Lagrangian-Eulerian framework. The hyperbolic conservation law is written in a space-time divergenceform so that inherent conservation properties of the problem are reflected in the numericalscheme, in which leads to first order conservative three-point numerical schemes. In orderto increase the order and but keeping good properties of the resulting scheme, such as TotalVariation Non-Increasing and monotonicity, we apply the Harten theory, yielding a new explicitsecond-order scheme (for the smooth regions) and with very good resolution of weak solutions(shocks). Numerical calculations were performed on several test cases for which analyticalresults are made available in the literature. The results show significant evidence we arecomputing qualitatively correct solutions with high-resolution.
69
Modelos Multiescala e Homogeneizado das primeiras fases de
Carcinogenese no Colon
G. RomanazziIMECC, Universidade Estadual de Campinas (Brazil)
Abstract
O cancer colorretal (CRC) e um dos tumores malignos mais difusos no mundo [WorldCancer Report, IARC Press, 2014].Embora este cancer tem uma taxa de mortalidade alta epossıvel reduzir os seus efeitos mediante uma diagnose precoz, ate anos antes que o cancerseja mortal. Devido a este particularidade diferentes grupos cientıficos, dos genetistas aosmatematicos e informaticos, sao interessados ao estudo da sua morfogenese nos estados ini-ciais, e em particular no desenvolvimento de modelos matematicos e das relativas simulacoesnumericas para puder analisar e prever o desenvolvimento do CRC e ajudar assim os medicosna sua diagnose [IET Systems Biology, 7, 3, pp.57-73, 2013].
O colon e um sıtio do corpo propenso a aparencia do cancer pela sua contınua renovacaoque conta um elevado numero de divisoes de celulas por dia. A parte interior do colon (epitelio)e formada de milhoes de pequenas cavidades chamadas criptas. E globalmente reconhecidoque as mutacoes geneticas das celulas nas criptas sao os principais responsaveis do inıcio doprocesso de carcinogenese.
Uma primeira manifestacao da carcinogenese observavel atraves as imagens obtidas durantea colonoscopia e a presena de Aberrant Crypt Foci (ACF) que e um conjunto de criptasabnormais adicentes caraterizados da um comportamento atıpico das celulas.
Nesta palestra, um modelo multiescala periodico e um modelo homogeneizado que de-screvem a formacao dos ACF sao apresentados [SIAM J. Appl. Math., 76, 3, pp.1152-1177,2016]. Estes modelos sao construidos a partir das dinamicas, ao nıvel microescala, das celulasdentro das criptas com o objetivo de simular e prever a evolucao do ACF ao nıvel macroescala,que e o nıvel observavel durante a colonoscopia. Uma analise e comparacao entre os dois mod-elos e discutida. Resultados numericos mostram a convergencia do modelo periodico para omodelo homogeneizado.
70
Analise de Estabilidade e Convergencia de um Metodo
Espectral Totalmente Discreto para Sistemas de Boussinesq
J. C. XavierUniversidade Tecnologica Federal do Parana (Brasil)
M. A. Rincon, D. G. Alfaro VigoUniversidade Federal do Rio de Janeiro (Brasil)
D. E. AmundsenCarleton University (Canada)
Abstract
Neste trabalho, vamos apresentar a analise de estabilidade e de convergencia da famılia desistemas de Boussinesq dada por
t + ux + (u)x + auxxx bxxt = 0,ut + x + uux + cxxx duxxt = 0,
(1)
que foi obtida e analisada como um modelo assintotico obtido a partir das equacoes de Eulerpara ondas de pequena amplitude e grande comprimento de onda (Bona J., Chen M., and SautJ.C., Boussinesq equations and other systems for smalle-amplitude long waves in nonlineardispersive media: I. Derivation and linear theory, J. Nonlinear Sci., 12, 283 - 318, 2002). Osparametros (a, b, c, d) devem satisfazer
a =1
2
2 1
3
, b =
1
2
2 1
3
(1 ), c =
1
2
1 2
µ, d =
1
2
1 2
(1 µ)
com , µ 2 R e 0 1.Nosso objetivo e analisar a influencia de diferentes escolhas dos parametros (a, b, c, d) na
eficiencia e precisao de um esquema numerico. Para isto, aplicamos conjuntamente o metodoespectral de colocacao de Fourier na discretizacao da variavel espacial, e o metodo de RungeKutta de quarta ordem aplicado na discretizacao da variavel temporal.
A analise de convergencia relacionada com a discretizacao da variavel espacial do problema(1), indica qual a ordem de precisao no espaco em que podemos obter solucoes aproximadasdesse problema. Essa ordem de precisao esta fortemente relacionada com a regularidade dassolucoes desse sistema.
Por outro lado, a analise da discretizacao da variavel temporal indica a eficiencia em quepodemos obter solucoes aproximadas do problema (1). Essa informacao nao esta diretamenterelacionada com a regularidade espacial da solucao, mas sim com propriedades dispersivas ede estabilidade das equacoes totalmente discretizadas.
Apos essa analise numerica, somos capazes de identificar quais escolhas particulares dosparmetros (a, b, c, d) geram os mais precisos e eficientes esquemas numericos. Para os casos emque a condicao de estabilidade do problema linear e dada por t Cx`, ` = 0, 1, e obtidoum resultado sobre a convergencia dos metodos numericos mencionados anteriormente.
Experimentos numericos sao fornecidos com o objetivo de verificar a estabilidade dassolucoes do problema em cada regiao de parametros, e confirmar a ordem de precisao e con-vergencia das solucoes aproximadas do problema nao linear.
71
9 Sessao tematica: Current trends in partial differentialequations and harmonic analysis
72
Programacao das palestras
Sessao especial: Current Trends in Partial Differential Equations and Harmonic Analysis
Segunda-feira Sala 151
14:00 - 14:30 Olivaine Santana de QueirozTBA
14:30 - 15:00 Damiao Junio Arajo
Towards the Cp′-regularity conjecture
15:00 - 15:30 Eder Ritis AragaoSobre a dicotomia exponencial em espacos de Frechet
15:30 - 16:00 COFFEE BREAK16:00 - 17:00 Emanuel Carneiro
Recent progress on the regularity of maximal operators17:00 - 17:30 Diego Marcon de Farias
Weakly coupled system of Hamilton-Jacobi Equations17:30 - 18:00 Luiz Roberto Hartmann Junior
Zeta-determinants of Sturm-Liouville operators with quadratic potentials at infinity
Quarta-feira Sala 253
10:00 - 10:30 Jean Carlo Pech MoraesHaar Basis on spaces of homogeneous type
10:30 - 11:00 Gustavo da Silva AraujoSome recent applications of Holder’s inequality
11:00 - 11:30 Mateus Costa de SousaUm estudo sobre os estremizantes para desigualdades de restricao no hiperboloide
11:30 - 12:00 Joao Vıtor da SilvaRegularity estimates for fully nonlinear elliptic equations with free boundaries
73
Some recent applications of Holder’s inequality
Gustavo AraujoUniversidade Estadual da Paraıba (Brazil)
Daniel PellegrinoUniversidade Federal da Paraıba (Brazil)
Abstract
When Leonard James Rogers (1862-1933) and Otto Holder (1859-1937) discovered, inde-pendently, the famous inequality that holds Holder’s name, they could have never imaginedthat they had just started a “revolution” in Functional Analysis (since its appearance in1888, Holder’s inequality has played a fundamental role in Mathematical Analysis and maybe considered a milestone in Mathematics). As it happens to almost every important resultin mathematics, several extensions and generalizations of Holder’s inequality have appearedalong the time. One of the extensions is the variant of Holder’s inequality for mixed Lp spaces,which appeared in 1961, in the seminal work of A. Benedek and R. Panzone [Duke Math. J.28 (1961), 301–324]. In this talk we show how this variant of Holder’s inequality (althoughwell-known in PDEs; the paper of A. Benedek and R. Panzone [op. cit.] has more than 100citations, mainly related to PDEs) was essentially overlooked in Functional/Complex Analysisand has had a crucial (and in some sense unexpected) influence in very recent advances indi↵erent fields of Mathematics. Some of these recent advances have been appearing since 2012and include the theory of Dirichlet series, the famous Bohr radius problem, certain classicalinequalities (such as Bohnenblust–Hille or Hardy–Littlewood), and Mathematical Physics.
74
Towards the Cp0-regularity conjecture
Damiao AraujoUniversidade da Integrao Internacional da Lusofonia Afro-Brasileira
Abstract
We establish a new oscillation estimate for solutions of nonlinear partial di↵erential equa-tions of degenerate elliptic type, which yields a precise control on the growth rate of solutionsnear their set of critical points. We then apply this new tool in the investigation of a longstand-ing conjecture which inquires whether solutions of the degenerate p-Poisson equation with abounded source are locally of class C1,1/p1. This is a joint work with Eduardo Teixeira andJos Miguel Urbano
75
Recent progress on the regularity of maximal operators
Emanuel CarneiroInstituto Nacional de Matemtica Pura e Aplicada (IMPA)
Abstract
This talk will be a brief survey of recent results on the regularity of maximal operators act-ing on Sobolev and BV functions. Most of such problems can be posed either in a continuousor in a discrete setting. We will present some results for the classical Hardy-Littlewood max-imal operator, for operators of convolution type (associated to elliptic and parabolic PDE’s)and also for fractional maximal operators. The talk will be accessible to a broad audience.
76
Haar Basis on spaces of homogeneous type
Jean Carlo MoraesUniversidade Federal do Rio Grande do Sul (UFRGS)
Abstract
In this talk, we show how to generalize the Haar basis for L2(R) to general spaces ofhomogeneous type (X, v,), a positive doubling measure and v a weight. We then showthat this generalization is indeed a complete orthonormal basis for L2(vd).
We also present a constructive proof of a collection dyadic cubes in a space of homogeneoustype (SHT) which are “honest,” i.e., that there are no more than two children per generation.We will also show how these cubes can be used to prove a generalization of D. Chung’s GoodBellman Function in SHT.
We will use these constructions to estimate the L2(d) norm of the t-Haar multipliers (T v )t
associated to a weight v 2 Ad2(d). The estimates that we will present are sharp in the sense
that they cannot be improved and be expected to hold for all , since any improvement willnot hold if is the Lebesgue measure.
77
Regularity estimates for fully nonlinear elliptic equations
with free boundaries
Joao Vitor da SilvaUniversidad de Buenos Aires (Argentina)
Abstract
In this Lecture we will discuss about regularity estimates for second order fully nonlinearelliptic equations of degenerate/singular type with strong absorption as following
F (x, Du, D2u) = 0(x)uµ+(x)inB1,
where 0 µ < + 1 ( > 1 is the homogeneity degree of the gradient) and 0 is a functionbounded away from zero and infinity. This model permits the formation of dead core zones,i.e., regions where non-negative solutions vanish identically. We shall establish improved C
estimates along free boundary points @u > 0\B1, where the sharp value of the exponent isgiven explicitly by
= + 2
+ 1 µ.
Similar regularity estimates will be also obtained for the gradient and Hessian of such solutions.We shall prove a Non-degeneracy result of solutions, and some consequences will be discussed.
78
Zeta-determinants of Sturm-Liouville operators with
quadratic potentials at infinity
Luiz HartmannUniversidade Federal de Sao Carlos
Abstract
Consider a Sturm-Liouville operators on [a,1), a > 0, with potentials that are growing atmost quadratically at infinity. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solution adaptedto the boundary conditions. This is a joint work with Matthias Lesch (Bonn University) andBoris Vertman (Mnster University).
79
An optimization problem in heat conduction for the
fractional Laplacian
D. MarconUniversidade Federal do Rio Grande do Sul (Brazil)
R. TeymurazyanUniversidade de Coimbra (Portugal)
Abstract
We study a free boundary optimization problem in heat conduction, ruled by the fractionalLaplacian operator, with a lower temperature bound and a volume constraint. We prove theexistence and the Holder regularity of solutions, we derive geometric properties of solutionsand of the corresponding exterior and interior free boundaries.
80
Um estudo sobre os estremizantes para desigualdades de
restricao no hiperboloide
Mateus Costa de SousaInstituto Nacional de Matemtica Pura e Aplicada (IMPA)
Abstract
Neste palestra apresentaremos alguns resultados de existencia, assim como alguns de naoexistencia, de extremizantes para estimativas de restricao da transformada de Fourier emhiperboloides. Nossas tecnicas passam por um estudo das simetrias de Lorentz do problema,a existencia de certos princıpios de concentracao e compacidade e a conexao com estimativasde Strichartz para equacao de Klein-Gordon.
81
10 Sessao tematica: Probabilidade e Estatıstica
82
Programação das palestras
Sessão especial: Probabilidade e Estatística
Segunda-feira Sala 325
14h00 - 14h30 Rodrigo Lambert (UFU)On dependent Bernoulli sequences with memory lapses
14h30 - 15h00 Tatiana Benaglia (Unicamp)A Generalized Non-Parametric Genotype Caller using an EM-like Algorithm
16h00 - 16h30 Rafael Izbicki (UFSCAR)Converting High-dimensional Regression to High-dimensional ConditionalDensity-Estimation
16h30 - 17h00 Afrânio Vieira (UFSCAR)Bayesian Binary Overdispersed Longitudinal model via Marginalized Likelihood
17h00 - 17h30 Christophe Gallesco (Unicamp)Uma introdução às desigualdades de concentração
17h30 - 18h00 Victor Fossaluza (USP)Sequential allocation methods in clinical trials
Tera-feira Sala 325
10h30 - 11h00 Rafael Maia (Unicamp)A Discrete Mixed Competing Risk Model for Discrete Time with Masked Causes
11h00 - 11h30 Camila Borelli (UFJF)Contribuição Matemática para Acelerar o Algoritmo EM sob a Classede Distribuições Misturas de Escala Skew-Normal
11h30 - 12h00 Marcos Prates (UFMG)Where geography lives? A projection approach for spatial confounding
16h00 - 16h30 Sandro Gallo (UFSCAR)Frogs on homogeneous trees and renewal theory
16h30 - 17h00 Mariana Curi (USP)Aprendizagem de Máquina: a Estatística sob nova direção?
17h00 - 17h30 Guilherme Ludwig (Unicamp)Fusão de dados de sensores móveis e estáticos com aplicaçõesem mapeamento de risco no ambiente de trabalho
17h30 - 18h00 Benillton Carvalho (Unicamp)The Importance of Analytics on the Brazilian Initiative on Precision Medicine
83
Tecnicas para obter desigualdades de concentracao: uma
introducao.
Christophe F. GallescoUniversity of Campinas (Brazil)
Abstract
Por “desigualdade de concentracao”, entendemos uma cota superior para o desvio de umafuncao de variaveis aleatorias independentes em relacao a sua esperan ca. Nesta palestra,faremos uma breve apresentacao de algumas das principais tecnicas usadas para obter taisdesigualdades.
84
Contribuicao Matematica para Acelerar o Algoritmo EM sob
a Classe de Distribuicoes Misturas de Escala Skew-Normal
Camila Borelli ZellerUniversidade Federal de Juiz de Fora (Brasil)
Clecio da Silva FerreiraUniversidade Federal de Juiz de Fora (Brasil)
Graciliano Marcio Santos LouredoUniversidade Federal de Juiz de Fora (Brasil)
Resumo
O algoritmo EM (Dempster et al., 1977) e suas extensoes, denominadas algoritmos tipo-EM, saoferramentas muito populares no calculo iterativo de estimativas de maxima verossimilhanca em umavariedade de modelos estatısticos na fısica, na medicina, nas ciencias sociais, dentre outras areas.Alem de serem conceitualmente atrativos, os algoritmos tipo-EM sao simples para implementare numericamente estaveis, pois convergem monotonicamente em termos da log-verossimilhanca.Uma crıtica frequente, entretanto, e a convergencia lenta em algumas situacoes. Portanto, tem sidopropostas varias aceleracoes para melhorar o algoritmo (Lange, 1995 e Liu et al., 1998). Nenhumdos metodos de aceleracao propostos sao teoricamente dominantes e comparacoes experimentaissao escassas. Nessa linha de pesquisa, desenvolvemos alguns resultados adicionais para a classe dedistribuicoes misturas de escala skew-normal (Branco & Dey, 2001). Especificamente, propomosum metodo simples para acelerar o algoritmo EM sob a classe das distribuicoes assimetricas.Finalmente, com o intuito de examinar o desempenho (e a utilidade) da metodologia proposta,alguns estudos de simulacao sao apresentados.
85
Aprendizagem de Maquina: a Estatıstica sob nova direcao?
Mariana CuriUniversity of Sao Paulo (Brazil)
Abstract
Aprendizagem de maquina e um campo da area de Ciencias de Computacao que estudaalgoritmos que “aprendem” e fazem previsao a partir de dados. Modelagem de dados e previsaotem tambem papel central na Estatıstica. Nao coincidentemente, ha muito em comum entreesses algoritmos e os modelos estatısticos. Mas por que este nao e apenas mais um caso em queuma area da ciencia utiliza metodos estatısticos para a realizacao de suas pesquisas e solucaode seus problemas?
86
Bayesian Binary Overdispersed Longitudinal model viaMarginalized Likelihood
Afrânio M. C. VieiraUniversidade Federal de São Carlos
Resumo
Many experiments and observational studies produce binary (or dichotomous) responses.Some of these studies have a typical longitudinal structure. Once these kind of data structureexpress naturally correlation between the repeated measures of a subject, usually, marginalmodels based on Generalized Estimating equations (GEE) or subject-specific models like Ge-neralized Linear Mixed Models (GLMM) perform well accommodating the serial correlation orclustered correlated data. However, when both sources of correlation express themselves an ex-tended latent structure is demanded. Molemberghs at al. (2010) have proposed an extendedclass of GLMM with multiplicative conjugated random effects. On a frequentist likelihoodestimation, the optimization phase includes numerical integration methods like adaptativeGaussian quadrature. In this work will be presented on a Bayesian perspective the use of aanalytical marginalized likelihood, what reduces the computational work. An application onToenail Onychomicosis patology will be show as well the comparison of the both approaches.
87
The Importance of Analytics on the Brazilian Initiative on
Precision Medicine
B. S. CarvalhoUniversity of Campinas (Brazil)
Abstract
Precision Medicine has emerged recently as a concept in which scientific knowledge andtechnology come together to provide the foundation for the 21st century medicine. It embracestranslational research, genomic medicine and personalized medicine and proposes a new levelof integration of data in order to improve health care.
BIPMed is currently based on a recently launched software platform, built following theguidelines and principles prosed by of the Global Alliance for Genomics and Health, andobserving the responsible and ethical sharing of genomic and clinical data. This platform isthe first of its kind in Latin America and aims to o↵er public access to genomic and phenotypicdata. It is intended to be used by clinicians and scientists all over the world, to share andobtain information about various aspects of genomic medicine and human health, as well as tosupport dissemination and training in the areas of human molecular genetics, computationalbiology and others. Thus, BIPMed is already playing an essential role in the development ofprecision medicine, allowing for significant advances in health care and research.
88
Fusao de dados de sensores moveis e estaticos com aplicacoes emmapeamento de risco no ambiente de trabalho
Guilherme Ludwig (Unicamp)
Avancos tecnologicos tem permitido a coletas de dados personalizada e emlarga escala para fatores de risco no ambiente de trabalho. A analise estatsticadesses dados, no entanto, carece de ferramentas que considerem as especifici-dades da area. Nossos objetivos sao (1) desenvolver uma nova metodologia deestatstica espaco-temporal que combine dados provenientes de sensores moveise estaticos, e permita a construcao de Mapas de Risco de trabalho dinamicos,e (2) comparar o novo mtodo com as praticas atuais na indstria, discutir asvantagens de mapas dinamicos e seu impacto na avaliacao de riscos de trabalhoe nas diretrizes de regulacoes de risco. A metodologia sera ilustrada com umestudo de caso conduzido em uma fabrica de construcao de motores, onde ointeresse encontra-se em limitar a exposicao dos trabalhadores a ruıdos de altaintensidade.
1
89
Where geography lives? A projection approach for spatial
confounding
M. O. Prates, R. M. AssuncaoUniversidade Federal de Minas Gerais (Brazil)
E. C. RodriguesUniversidade Federal de Ouro Preto (Brazil)
Abstract
Spatial confounding between the spatial random e↵ects and fixed e↵ects covariates hasbeen recently discovered and showed that it may bring misleading interpretation to the modelresults. Solutions to alleviate this problem are based on decomposing the spatial random e↵ectand fitting a restricted spatial regression. In this paper, we propose a di↵erent approach: atransformation of the geographic space to ensure that the unobserved spatial random e↵ectadded to the regression is orthogonal to the fixed e↵ects covariates. Our approach, namedSPOCK, has the additional benefit of providing a fast and simple computational method toestimate the parameters. Furthermore, it does not constrain the distribution class assumedfor the spatial error term. A simulation study and a real data analysis are presented to betterunderstand the advantages of the new method in comparison with the existing ones.
90
A Discrete Mixed Competing Risk Model for Discrete Time
with Masked Causes
R. P. Maia,University of Campinas (Brazil)
C. G. B. Demetrio,University of Sao Paulo (Brazil)
R. Labouriau,Aarhus University (Denmark)
Abstract
The problem of competing risks arises in time-to-event analysis when the subjects observedmay experience one, and only one, among a set of possible competing events. For instance,in longevity studies often the interest lies in modeling the time until death of a group of in-dividuals, that might die of di↵erent specific causes. The time to death of an individual issaid to be cause-masked, or simply masked, when it is observed the time until death but notnot the specific cause of death. This work will study some techniques, based on suitable vari-ants of the EM-algorithm, to perform statistical inference in a competing risk scenario in thepresence of partial masking and right censoring. The goal is to extent a class of multivariateproportional hazard models for competing risks containing suitably defined gaussian randomcomponents to characterize the quantitative genetic determination of longevity in large scaleanimal production systems proposed by Maia et. al (2014). Moreover, it is provided a simula-tion study evaluating the performance of the proposed procedures with respect the estimationof variance-covariance components. An application with real dataset is also presented.
91
Converting High-dimensional Regression to High-dimensional
Conditional Density-Estimation
Rafael IzbickiUniversity of So Carlos (Brazil)
Ann B. LeeCarnegie Mellon University (USA)
Abstract
There is a growing demand for nonparametric conditional density estimators (CDEs) infields such as astronomy and economics. In cosmology, for example, one can dramaticallyimprove estimates of the parameters that dictate the evolution of the Universe by workingwith full conditional densities instead of regression (i.e., conditional mean) estimates. Moregenerally, standard regression falls short in any prediction problem where the distributionof the response is more complex with multi-modality, asymmetry or heteroscedastic noise.Nevertheless, much of the work on high-dimensional inference concerns regression and classifi-cation only. Here we propose FlexCode, a fully nonparametric approach to conditional densityestimation that reformulates CDE as a non-parametric orthogonal series problem where theexpansion coecients are estimated by regression. By taking such an approach, one can ef-ficiently estimate conditional densities in high dimensions by drawing upon the success inhigh-dimensional regression. Depending on the choice of regression procedure, our methodcan adapt to a variety of challenging high-dimensional settings with, for example, a largenumber of irrelevant components or nonlinear manifold structure in the data. We study thetheoretical and empirical performance of our proposed method, and we compare our approachwith traditional conditional density estimators on simulated as well real-world data.
92
On dependent Bernoulli sequences with memory lapses
M. Gonzalez-NavarreteIME-USP / GSSI - Italy
R. LambertFederal University of Uberlandia
Abstract
We propose an approach to construct Benoulli sequences combining dependence and inde-pendence periods. We call the trials Xi, i 1 as Bernoulli sequence with random dependence(BSRD). The dependence on a function of the past (Si = X1 + · · · + Xi) is activated by acollection of Bernoulli trials Yi, i 1, called memory switch sequence. We characterize theproperty of memory lapses, which are intervals of consecutive independent trials in BSRD. Westudy properties and discuss several applications of BSRD. The main result includes classicallimit theorems for a class of linear BSRD.
93
Sandro Gallo
Tıtulo: Frogs on homogeneous trees and renewal theory
Resumo: Suppose in every site of homogeneous tree. Initially, only the frog at the root isactive all the others are inactive. Each active particle starts a random walk with geometriclifetime of parameter 1 p. When an active particle visits a site containing an inactiveparticle, the later is activated. A slight modification of the dynamic yields the self-avoidingfrog model considered by Lebensztayn, Machado and Martinez (2008). In both models,there exists a critical parameter pc which separates between almost-sure extinction of theactive frogs and positive probability of survival. In the proofs of the literature, whenderiving the upper bound, a delicate recursive formula appears. The main objective ofthe present talk is to explain a new approach based on renewal theory. While our methodyields the exact same bound for the original frog model, it gives a slight improvement forthe self-avoiding model. This is work in progress with Pablo Rodriguez.
94
A Generalized Non-Parametric Genotype Caller using an
EM-like Algorithm
Tatiana BenagliaUNICAMP - Department of Statistics (Brazil)
Abstract
Current biomedical researches make constant use of high-throughput technologies for geno-typing. Microarrays and, more currently, sequencing experiments provide information at mil-lions of variant sites, which are later investigated for association with a phenotype of interest.Prior to this association assessment, genotyping algorithms quantify the allele-specific abun-dance and classify subjects into genotype clusters. The series of statistical procedures involvedin the aforementioned quantification and classification steps is complex and includes stronghypotheses on the distribution of the data. Models for microarray data assume that intensitysignals follow an Exponential distribution combined with Gaussian error, while count datafrom sequencing are seen as results of a Negative-Binomial processes.
We propose an alternative genotype caller that does not rely on assumptions regardingthe distribution of the data. Our method uses a non-parametric approach combined with anadapted Expectation-Maximization (EM-like) algorithm to determine the membership likeli-hoods of each subject at every loci. Following this strategy, we use the maximum posteriorprobability to accurately perform genotype calling.
95
Sequential allocation methods in clinical trials
V. Fossaluza, J. B. Diniz, C. A. B. Pareira, S. WechslerUniversidade de Sao Paulo (Brazil)
Abstract
Randomized clinical trials are the gold standard for testing ecacy of treatment interven-tions. However, although randomization protects against deliberately biased samples, it doesnot guarantee random imbalances will not occur. Methods of intentional allocation that canovercome such deficiency of randomization have been developed, but are less frequently appliedthan randomization. Initially, we introduce a fictitious case example to revise and discuss thereasons of researchers? resistance to intentionally allocate instead of simply randomizing. Wethen introduce a real case example to evaluate the performance of an intentional protocol forallocation based on compositional data balance. A real case of allocation of 50 patients in twoarms was compared with an optimal allocation of global instead of sequential arrivals. Perfor-mance was measured by a weighted average of Aitchison distances, between arms, of prognosticfactors. To compare the intentional allocation with simple random allocation 50,000 arrivalorderings of 50 patients were simulated. To each one of the orders, both kinds of allocationsinto two arms were considered. Intentional allocation performed as well as optimal allocationin the case considered. In addition, out of the 50,000 simulated orders, 61% of them performedbetter with intentional allocation than random allocation. Hence, we conclude that intentionalallocation should be encouraged in the design of future interventional clinical trials as a wayto prevent unbalanced samples. Our sequential method is a viable alternative to overcometechnical diculties for study designs that require sequential inclusion of patients as it doesnot require prior knowledge about the total sample composition.
96
11 Sessao tematica: Folheacoes holomorfas e teoria desingularidades
97
Programacao das palestras
Sessao especial: Folheacoes Holomorfas e Teoria de Singularidades
Segunda-feira Sala 222
14:00 - 15:00 Arturo Fernandez PerezFolheacoes do 2 tipo via ındices de folheacoes.
16:00 - 17:00 Fernando LourencoUma efetiva maneira de calcular resıduos de folheacoes.
Terca-feira Sala 324
10:30 - 11:30 Luis Renato G. DiasDetection of the bifurcation values of real polynomial maps.
16:00 - 17:00 Thiago FassarellaConnections and moduli spaces.
17:00 - 18:00 Edson SampaioRegularidade Lipschitz e a multiplicidade de conjuntos analıticos
98
Folheacoes do 2 tipo via ındices de folheacoes.
A. Fernandez PerezUniversidade Federal de Minas Gerais (Brazil)
Abstract
Estudamos germes de folheacoes do 2 tipo em (C2, 0), isto e, folheacoes holomorfas singu-lares que nao admitem sela-nos tangentes na sua reducao de singularidades. Usando os ındicesde Baum-Bott, Camacho-Sad, GSV e o excesso polar, daremos uma caracterizacao destas fol-heacoes, tanto no caso dicrıtico e nao-dicrıtico. Os resultados sao parte de uma trabalho emconjunto com Rogerio Mol.
99
Regularidade Lipschitz e a multiplicidade de conjuntos
analıticos
J. Edson SampaioUniversidade Federal do Ceara (Brasil)
Abstract
Nesta palestra, apresentaremos um estudo sobre regularidade Lipschitz de conjuntos analı-ticos tanto reais quanto complexos. Nos mostraremos que um conjunto analıtico complexo quee uma variedade Lipschitz necessariamente e suave. Entretanto, mostraremos que isto nemsempre ocorre no caso real. Alem disso, nos daremos uma resposta parcial para a conjecturade Zariski para a multiplicidade, mostraremos que a multiplicidade de superfıcies analıticascomplexas em C3 e um invariante bi-Lipchitz.
100
Topological classification of finitely determined map germs
with non isolated zeros.
E. B. BatistaUniversidade Federal de Sao Carlos (Brazil)
J. J. Nuno-BallesterosUniversity of Valencia (Spain)
J. C. F. CostaUniversidade Estadual Paulista (Brazil)
Abstract
The topological type of a finitely determined map germ f : (R3, 0) ! (R2, 0) is givenby the so-called link of f. The link of f is obtained by taking a small enough representativef : U R3 ! (R2 and the intersection of its image with a small enough sphere S1
centered atthe origin in R2 . As a consequence of Fukudas theorem, two finitely determined map germsf, g : (R3, 0) ! (R2, 0) are topologically equivalent if their associated links are topologicallyequivalent. Inspired by the works of Arnold, Maksymenko and Prishlyak we introduce anadapted version of the Reeb graph that turns out to be a complete topological invariant forthe links of finitely determined map germs with non isolated zeros.
101
Uma efetiva maneira de calcular resıduos de folheacoes.
F. LourencoUniversidade Federal de Lavras (Brazil)
Abstract
Nessa palestra consideraremos folheacoes holomorfas F de codimensao k numa variedadecomplexa. Nesse contexto, apresentaremos uma maneira efetiva de calcular resıduos para taisfolheacoes quando a dimensao do conjunto singular da folheacao tem dimensao um a menos doque a dimensao das folhas. Nesse trabalho nao utilizamos da hipotese de regularidade usadapor Baum-Bott em [1], nem da assuncao de nao degenerecencia de Vishik em [2].
Bibliography
[1] Baum, P. and Bott, R. Singularities of holomorphic foliations, J. Di↵erential Geom.7(1972), 279-342.
[2] Vishik, S. M. Singularities of analytic foliations and characteristic classes, FunctionalAnal. Appl. 7 (1973) 1-15.
[3] Correa, M. and Fernandez Prez A. Absolutely k-convex domains and holomorphic foli-ations on homogeneous manifolds, Preprint arXiv: 1403.4286v5.
[4] Brunella, M. and Perrone, C. Exceptional singularities of codimension one holomorphicfoliations, Publicacions Matematiques 55 (2011), 295-312.
102
Detection of the bifurcation values of real polynomial maps.
L. R. G. DiasUniversidade Federal de Uberlandia (Brazil)
Abstract
The bifurcation locus of a polynomial map f : Rn ! Rp, n p, is the smallest subsetB(f) Rp such that f is a locally trivial C1-fibration over Rp \ B(f). In this talk, wedescribe an e↵ective estimation of the “nontrivial” part of the bifurcation locus of a polynomialmapping f : Rn ! Rp.
Connections and moduli spaces.T. Fassarella
Universidade Federal de Uberlandia (Brazil)
Abstract
In this talk we will investigate the geometry of moduli spaces of rank 2 logarithmic connec-tions on an elliptic curve minus 2 points with fixed spectral data. This is a work in progresswith Frank Loray.
Estudamos germes de folheacoes do 2 tipo em (C2, 0), isto , folheacoes holomorfas singularesque no admitem sela-nos tangentes na sua reducao de singularidades. Usando os ıindices de Baum-Bott, Camacho-Sad, GSV e o excesso polar, daremos uma caracterizao destas folheaes, tanto nocaso dicrtico e no-dicrtico. Os resultados so parte de uma trabalho em conjunto com Rogrio Mol.
103
Connections and moduli spaces.
T. FassarellaUniversidade Federal Fluminense (Brazil)
Abstract
In this talk we will investigate the geometry of moduli spaces of rank 2 logarithmic connec-tions on an elliptic curve minus 2 points with fixed spectral data. This is a work in progresswith Frank Loray.
104
12 Sessao tematica: Geometry in Algebra and Algebra inGeometry
105
Programacao das palestras
Sessao especial: Geometry in Algebra and Algebra in Geometry.
Segunda-feira Sala 225
16:00 - 16:40 Alejandro CabreraDescribing local Lie groupoids in terms of infinitesimal data
16:40 - 17:20 Alexandre QuesneyDeformation of homotopy actions and the Swiss Cheese operad
17:20 - 18:00 Luiz Enrique RamirezPolyhedra associated with weight modules
Terca-feira Sala 225
Sem palestras
Quarta-feira
10:00 - 10:40 Oliver LorscheidQuiver Grassmannians of extended Dynkin type D
10:40 - 11:20 Elizaveta VishnyakovaLie supergroups and complex homogeneous supermanifolds
11:20 - 12:00 Ivan StruchinerRigidity Phenomena for Lie Groups
15:30 - 16:10 Artem LopatinInvariantes ortogonais indecomposıveis de matrizes
16:15 - 16:55 Pedram HekmatiA topological application of T-duality
106
Describing local Lie groupoids in terms of infinitesimal data
Alejandro CabreraUniversidade Federal do Rio de Janeiro (Brazil)
Abstract
In this talk, I will review our recent joint work with M.A. Salazar and I. Marcut aboutthe structure of local Lie groupoids. In particular, our first main result is that a Lie algebroidtogether with a suitable spray vector field fully determines the structure maps of the associatedlocal Lie groupoid integrating it. The second main result is an explicit characterization ofgroupoid morphisms integrating given Lie algebroid ones. Finally, there is a surprisingly longlist of applications, including Poisson, Dirac and Jacobi geometries that I will elaborate on.
107
A topological application of T-duality
Pedram HekmatiIMPA (Brazil)
Abstract
T-duality is a discrete symmetry that was discovered by physicists in the context of stringtheory, but has now matured into a precise mathematical statement. In this talk I will give abrief review of the mathematical formulation of T-duality and explain how it can be used togive a new, surprisingly simple proof of Hodgkin’s famous theorem on the K-theory of compactsimply connected Lie groups. This is joint work with David Baraglia.
108
Invariantes ortogonais indecomposıveis de matrizes
Artem LopatinUNICAMP (Brazil)
artem [email protected]
Abstract
Trabalhamos sobre um corpo F infinito da caracterıstica char(F) 2. Consideramos aalgebra RO(n) dos invariantes de d matrizes n n ao respeito da acao pelo grupo O(n) pelaconjugacao. Essa algebra e gerada pelos coeficientes dos polinomios caracterısticos de todosprodutos das matrizes genericas e matrizes genericas transpostas.
Nos provamos que no caso 0 < char(F) n o limite do grau maximo dos invariantesindecomposıveis e infinito, quando d esta tendendo a infinito. Este resultado e bem conhecidono caso da acao de GL(n). Por outro lado, para caracterıstica di↵erente este fenomeno nao everdadeiro. Nos consideramos o mesmo problema para matrizes simetricos e skew-simetricos.Demonstramos tambem que RO(n) nao e uma algebra livre no caso d > 1 e no caso d = 1 en > 9.
109
Quiver Grassmannians of extended Dynkin type D
Oliver LorscheidIMPA (Brazil)
Abstract
Since cluster algebras have been introduced in the early 2000’s, they receive growing at-tention from researchers in representation theory and other other areas of mathematics andphysics. A fundamental problem in this subject are explicit descriptions of cluster algebras interms of generators and relations. While cluster algebras of extended Dynkin type A are well-understood, there are only partial results for type D. In collaboration with Thorsten Weist,we have established formulas for the Euler characteristics of the associated quiver Grassman-nians, which leads to explicit formulas for the cluster variables in terms of the initial clustervariables.
In this talk, we will give an overview over the methods that entered the lengthy proof ofthese formulas. The central result of this work is that all quiver Grassmannians of extendedDynkin type admit decompositions into ane spaces.
110
Deformation of homotopy actions and the Swiss Cheese
operad
Alexandre QuesneyICMC-USP (Brazil)
Abstract
We begin by exposing a few classical facts on operads, focusing our attention on the En
operads and the Swiss Cheese operad. Next, we speak about deformations of homotopy actionsencoded by the (chain operad of the) Swiss Cheese operad.
111
Polyhedra associated with weight modules
Luis Enrique RamirezUFABC (Brazil)
Abstract
In this talk I will present some known results about polyhedra associated with weightmodules over complex finite dimensional semisimple Lie algebras, in particular characterizationof modules such that its associated polyhedrom is a polytope. Also is proposed a general settingand consider analogous questions for the case of Gelfand-Tsetlin gl(n) modules.
112
Rigidity Phenomena for Lie Groups
Ivan StruchinerIME-USP (Brazil)
Abstract
In this talk I will describe how to detect rigid structures in the context of Lie groupsand their homomorphisms. I will explain the relevant cohomology theories which controlthe associated deformation problems and a Moser type argument that can be used to proverigidity. Parts of the talk will be based on joint work with Cristian Cardenas (IME-USP), andother parts are from a joint work with Joao Nuno Mestre (University of Coimbra) and MariusCrainic (Utrecht University).
113
Lie supergroups and complex homogeneous supermanifolds
Elizaveta VishnyakovaIME-USP (Brazil)
Abstract
Our talk is devoted to Lie supergroups and homogeneous supermanifolds. The main struc-ture theorem in the theory of real supergroups was proven by B. Kostant. It is an equivalenceof the category of real Lie supergroups and the category of real Harish-Chandra pairs. Fromthis theorem it follows that a Lie supergroup depends only on the underlying Lie group andits Lie superalgebra with certain compatibility conditions. Further the structure sheaf of aLie supergroup and the supergroup morphisms can be explicitly described in terms of thecorresponding Lie superalgebra and underlying Lie supergroup. We will discuss another proofof this result that works also in the complex-analytic and algebraic cases.
Further, we will consider a very important class of complex supermanifolds, non-splitsupermanifolds. A supermanifold is called split if it is isomorphic to a vector bundle with apurely even base and purely odd fiber. In the smooth category, all supermanifolds are knownto be split (although non-canonically). Another example of a split supermanifold is any Liesupergroup. The second question is how to find out whether a given complex homogeneoussupermanifold is split or non-split.
114
13 Sessao tematica: Matematicas da Teoria da Informacao
115
Programacao das palestras
Sessao especial: Matematicas da Teoria da Informacao.
Terca-feira (6 Dez) Sala 151
10:30 - 11:00 Diego NappConvolutional codes in random linear network coding
11:00 - 11:30 Diana H. Bueno-CarrenoStrong apparent distance of abelian codes
11:30 - 12:00 Ana L. Sandoval OrozcoEfficient Algorithms for Searching Multiple Burst-ErrorCorrecting Cyclic and Shortened Cyclic Codes
16:00 - 16:30 Gustavo T. BastosClassification and Multilevel Construction of Geometrically Uniform Subspace Codes
16:30 - 17:00 Cintya W. O. BeneditoAlgebraic Lattices from Maximal Quaternion Orders
17:00 - 17:30 Grasiele C. JorgeUnimodular lattices obtained via Z[ζ2rp + ζ−1
2rp] for p = 3 and p = 517:30 - 18:00 Carina Alves
Space-Time Block Code Designs Based on Quadratic Field Extension
Quarta-feira (7 Dez) Sala 151
10:30 - 11:00 Cesar H. Rodrıguez-GaravitoAnalysis of measurement concerning the extrinsic parameters in stereo vision
11:00 - 11:30 Jose MarquezCalculating Packets on a Linear Network Coding Model
11:30 - 12:00 Daniel M. CaldasAnalysis and Extraction of Behavioral and Static Features for Malware Profiling
16:00 - 16:30 Maiara BollaufConstruction C and applications in information theory
16:30 - 17:00 Christiane B. RodriguesThe space of binary asymmetric channels: properties and geometry
116
Space-Time Block Code Designs Based on Quadratic Field
Extension
C. AlvesSao Paulo State University - UNESP, Rio Claro (Brazil)
E.M. Moro, A.A. de AndradeSao Paulo State University - UNESP, Sao Jose do Rio Preto (Brazil)
Abstract
High rate and large diversity product (or coding advantage, or coding gain, or minimumproduct distance) are two of the most important criteria often used for good space-time codedesigns. In recent lattice-based space-time code designs, more attention is paid to the highrate criterion but less to the large diversity product criterion. By employing some algebraicnumber theory, in this work, we consider these two criteria together for quadratic space-timecode designs. The optimal design of the quadratic space-time block codes in terms of a diversityproduct criterion is also presented.
117
Classification and Multilevel Construction of Geometrically
Uniform Subspace Codes
G. T. Bastos,Federal University of Sao Joao del Rei (Brazil)
R. Palazzo Jr.State University of Campinas (Brazil)
Abstract
In this work we characterize all geometrically uniform subspace codes in the graassmannianGq(n, k). These codes are obtained from the metric space defined by the set of all vectorsubspaces of Fn
q , denoted by Pq(n), with the well known subspace distance (dS), introducedby Koetter and Kschischang in “Coding for Errors and Erasures in Random Network Coding”(IEEE Transactions on Information Theory, vol 54, 3579 - 3591, 2008).
In classical coding theory, geometrically uniform codes were broadly taken into consider-ation in the 90´s, due to their richer geometric and algebraic structures. On the other hand,subspace codes can be seen as a generalization of linear block codes for each codeword insubspace coding is a vector subspace of Fn
q . Hence, by using the concept of orbit codes, whichare subspace codes generated by the action of a subgroup of the general linear group GLn (Fq)on a k-dimensional subspace, we classify all geometrically uniform subspace codes.
In addition to this, by adapting the multilevel constructions of Ungerboeck (Channel Cod-ing with Multilevel/Phase Signals, IEEE Transactions on Information Theory, vol 28, 55 -67, 1982) and Calderbank (Multilevel Codes and Multistage Decoding, IEEE Transactions onCommunications, vol 37, 222 - 229, 1989) to subspace codes we generalize Nobrega’s proposalwhich is based on the definition of an L-level partition originally proposed by Calderbank.Basically, consider a chain of partitions of a subspace code C, such that at the last level thecomplete code partition is given by each subspace (codeword). In each level, a codeword froma classical code is used to label each subset. Hence, every subspace of C can be labeled byclassical codewords, resulting from each path in the partition chain as the juxtaposition oflabels necessary to reach this codeword. The codes obtained from this construction are calledL-codes, where L is the number of partitions. Therefore, from the geometric structure ofgeometrically uniform subspace codes, the multilevel subspace coding constructions by use oforbit codes are derived. Such constructions are nested and the partitions are easily obtained,since we know both its structure and the group acting on any codeword of the considered orbitcode. Furthermore, if the code admits L levels, then it is possible to reduce the number ofnecessary computations to obtain the minimum distance of the L-code, using the geometricaluniformity of their partitions.
118
Algebraic Lattices from Maximal Quaternion Orders
C. W. O. Benedito,Sao Paulo State University (UNESP), Campus of Sao Joao da Boa Vista (Brazil)
C. Alves,Sao Paulo State University (UNESP), Campus Rio Claro (Brazil)
N. G. Brasil Jr., S. I. R. CostaUniversity of Campinas (Brazil)
Resumo
Our main goal is to present an algebraic construction of lattices in dimensions 4n by using amaximal quaternion order in a quaternion algebra. It has been shown that algebraic lattices,i.e., lattices constructed via the canonical embedding of an algebraic number field, providean ecient tool for designing lattice codes (E. Bayer-Fluckiger et al., 2004). More recently,lattices and quaternion algebras have been connected. In (F.-T. Tu and Y. Yang, 2012), ideallattices from a definite quaternion algebra was constructed, in (C. Alves and J-C. Belfiore,2015), the E8-lattice was presented by using quaternion algebras over an imaginary quadraticfield and in (C. Q. Queiroz et al., 2015), hyperbolic lattices via maximal quaternion orders wasobtained. In this work we show that we can define algebraic lattices from quaternion algebrasin the same way that we define algebraic lattices from number fields.
Let K be a totally real number field with degree n and A be a definite quaternion algebraover K with a maximal order M. If I is an ideal in M and ↵ is a totally positive element inK, then = (I,↵) denote the lattice associated to the quadratic form TrK/Q(↵Trd(xy)).
Definicao 0.1 Let K be a totally real number field with degree n, ↵ a totally positive elementof K and let A be a definite quaternion algebra over K. If x = x1 + x2i + x3j + x4k 2 A then↵A : A ! R4n is an embedding defined by
↵A(x) = (p
21(↵)1(x1), · · · ,p
2n(↵)n(x1), · · · ,p
21(↵)1(x4), · · · ,p
2n(↵)n(x4)),
where 1, · · · ,n are the n-embeddings of K in R.
Teorema 0.1 Let K be a totally real number field with degree n, ↵ a totally positive elementof K and let A be a definite quaternion algebra over K. If I M is a right ideal of a maximalquaternion order M of A, then ↵A(I) is a lattice with volume
V ol(↵A(I)) = (N(↵)dK)2 (N (det1s,s04(trd(vsvs0)))1/2 ,
where N(↵) is the norm of ↵, dK is the discriminat of K and v1, v2, v3, v4 is a OK-basis ofI.
By using the Definition 0.1 and the Theorem 0.1, we can construct algebraic lattices ofdimensions 4n. As examples, we construct algebraic lattices of dimensions 4, 8, 12, 16 and 24that are rotated versions of the densest lattices D4, E8, K12, BW16 and 24, respectively.
119
Construction C and applications in information theory
M. Bollauf, S. I. R. CostaUniversity of Campinas (Brazil)
R.ZamirTel Aviv University (Israel)
Keywords: lattice constructions, geometrically uniform constellations, multistage decoding
Abstract
When we think in lattice codes we immediately remember the classic constructions A andD [Conway e Sloane 1999]. However, there exist some constructions that produce nonlat-tice constellations in general, which is the case of Construction C, defined by the followingexpression as a superposition of L codes [Forney 1988]:
C = C1 + 2C2 + . . . + 2L1CL + 2LZn, (1)
where C1, . . . , CL Fn2 are not necessarily linear or nested. We can notice that Construction
C generates lattices in some specific cases, for example, if the codes are linear and L = 1 itfalls into Construction A and if the linear codes are nested and the chain of codes is closedunder Schur product, such construction coincides with Construction D [Kositwattanarerk eOggier 2014]. Even without usually presenting a lattice structure, Construction C has someinteresting properties, such as: if the underlying codes are linear, the minimum distance isthe same for all points in the constellation, although we cannot say the same with respectto the kissing number. The case when L = 2 is special, because it is geometrically uniform(i.e., given two points there exists an isometry that maps one point to another and preservesconstellation), however for L 3 this property does not hold [Bollauf e Zamir 2016]. Regardingto applications, we found the main potential in information theory, where we know that whenwe apply multistage decoding in Construction C, it achieves high SNR capacity in AWGNchannel [Forney, Trott e Chung 2000]. Another application that we are interested in are thehybrid solutions, where we use a lattice and a nonlattice (via Construction C) to propose analternative to network coding and interference alignment [Zamir e Shamai 2002].
120
Strong apparent distance of abelian codes
Diana H. Bueno-Carreno,Departamento de Ciencias Naturales y Matematicas
Pontificia Universidad Javeriana Seccional Cali, Colombia
Jose Joaquın Bernal, Juan Jacobo SimonDepartamento de Matematicas, Universidad de Murcia, Spain.
[email protected], [email protected], [email protected]
Abstract
The BCH bound is the oldest lower bound for the minimum distance of a cyclic code. Thestudy of this bound and its generalizations are classical topics, which includes the study of thevery well-known family of BCH codes. In 1970, P. Camion extended the notion of BCH boundto the family of abelian codes by introducing the apparent distance of polynomials. Camionshowed that the minimum value of the apparent distance of certain polynomials associatedto codewords is less than or equal to the minimum distance of the code. The mentionedminimum value is known as the apparent distance of an abelian code. We strengthen thenotion of apparent distance by introducing the notion of Strong apparent distance; then, wepresent an algorithm to compute the strong apparent distance of an abelian code, based onsome manipulations of hypermatrices associated to its generating idempotent
121
Analysis and Extraction of Behavioral and Static Features for
Malware Profiling
D. M. Caldas, R. de O. Albuquerque, F. E. G. de DeusUniversidade de Brasılia (UNB)
Departamento de Engenharia EletricaCampus Universitario Darcy Ribeiro, Asa Norte - 70910-900 - Brasılia - DF - Brasil
[email protected], [email protected], [email protected]
Abstract
Most criminal actions on the internet are committed using hostile software, commonlycalled malware. Malware are programs designed to infiltrated a computer system in an un-lawful way, in order to cause damage, alter data or steal information. Virus, Worms, Trojanhorses and others are some examples of malware.
Most of the research done today in the malware analysis area is focused on detectingmalicious program, as a way to prevent and avoid attacks. Profiling and classification canbe very useful when the time is short, and the consequences are critical. As reported byBroadhurst et al., e↵orts made by governments and law enforcement agencies around theworld has helped to fight cybercrime.
This paper discusses the development of a methodology to recognition and extraction ofkey features of binary malware files. The attributes will be used to elect malware profiles, andclassify malware samples within one of the designated profiles.
The proposed methodology is based on a preliminary study of malware samples captured bya honeypot, the separation of these malware groups (profiles), the identification of features thatbest allow the classification of these samples in one of the di↵erent profiles, and the discussionabout the best way to extract and use these features to achieve the desired profiling.
We combine behavioral and static features of the malware samples, to be used in theclassification between the di↵erent profiles. The static features were extracted directly from thebinary file. To extract the behavioral characteristics, a VirtualBox virtual machine installedwith Windows XP operating system was used. Each sample is individually executed in a cleaninstance of the virtual machine, and was studied using the tools indicated.
All malware samples were executed and had their features extracted according to theproposed Methodology. Some of the features analyzed have been revealed promising to classifymalware samples in the designated profiles. Next steps in this work includes increasing thenumber of samples analyzed, enhancing and refining the list of features, and the developmentof tools for automated extraction and classification.
122
Unimodular lattices obtained via Z[2rp + 12rp] for p = 3 and
p = 5
Grasiele C. JorgeFederal University of Sao Paulo (Brazil)
Agnaldo J. FerrariSao Paulo State University (Brazil)
Abstract
A lattice is a discrete additive subgroup of Rn. Equivalently, Rn is a lattice i↵ thereare linearly independent vectors v1, . . . , vm 2 Rn such that =
Pmi=1 aivi; ai 2 Z, 8i
.
Let K be a number field of degree n, OK its ring of integers and ↵ 2 OK a totally real,totally positive element. Recently, Eva Bayer-Fluckiger introduced a twisted embedding ↵ :K ! Rn such that if I OK is a free Z-module of rank n, then ↵(I) is a lattice in Rn.These lattices are called here algebraic lattices. Special algebraic lattice constructions can beused to obtain certain lattice parameters such as diversity and minimum product distance,which are usually dicult to calculate for general lattices in Rn and which are importantparameters related to the signal transmission error probability over Rayleigh fading channels.
In this work we construct families of unimodular algebraic lattices obtained via the twistedembedding applied to the ring of the integers of Q(2rp + 1
2rp) for p = 3 and p = 5 and
calculate their minimum product distances. The lattices obtained are direct sum of one Z8
with 2r5(p 1) 1 E8’s.
This work was partially supported by FAPESP, grants 2013/25977-7, 2014/14449-2 and 2015/17167-0.
123
Calculating Packets on a Linear Network Coding Model
J. Marquez, I. Gutierrez, J. TeheranUniversidad del Norte (Colombia)
jmarquez, isgutierr, [email protected]
Abstract
This paper presents a simple model of how the transmission and reception of packets ona network that applies the Multicast Linear Network Coding model, defined on a finite fieldFq sized q = 2m(m 1) should be performed. The proposal aims to show a procedure tocalculate in the intermediate nodes packets corresponding to link processes and in sink nodespackets sized m-bits that were sent by the source node simultaneously.
The present model is defined on the basis of a G = (V, E) graph, where V represents theset of vertices or nodes (routers in a network), and E represents the set of links or directedarcs that make up the graph (they are wired or wireless network links). Within the graph Ga s 2 V source node is defined, from which the multicast transmission originates towards aT E set of sink nodes. This algorithm shows how packets belonging to Fq field sent fromthe source node s can reach the T set of multiple multicast destinations.
In order to maximize the bandwidth in compliance with Max-flow and Min-cut theorem,in the work a data network (butterfly network) is defined, represented by a graph G = (V, E)with source node s, set of sink nodes T E and a bandwidth on each link of one bit per timeunit. Similarly, this network is shown in other works to explain the use of a linear combinationto process the input bits to a node to generate a resulting bit to be sent by the links to thefollowing nodes. Our objective is to work with each one of the processes as bit flows withinthe finite field Fq size q = 2m(m 1). That is, each packet is considered as a one-dimensionalarray of size m-bits.
The model is free of delay, since each bit flows for a link l is transmitted after source nodeof l has received the corresponding symbols in each one of the input processes. These symbolstransmitted in a link l are a scalar linear function in the finite field F2. This function is formedby local coding coecients defined in each source node of l. In order to form output symbolsfor each sink node t, it is necessary to build a scalar linear combination of its input symbolsusing decoding coecients. The global coding coecients are functions of the local codingoperations in the network nodes.
Two algorithms are proposed:The first one, generates at each intermediate node and the source node, packets defined at
the finite field Fq, with q = 2m, working with each one of the bits making up the incomingpackets in the order they appear and applying global coding coecients previously found.
This algorithm returns at each node all packets formed by applying a function on each bitj of each incoming packet to node u. Packet Pl formed by m-bits is returned at the end. Forsource node packets to be evaluated, the increased graph G0 is taken, which includes the nodes0 and packets corresponding to the source processes Pi are the initial inputs.
Packets are processed at each node u and then are forwarded to the appropriate destinationnodes d(l), where l 2 |O(u)|. The complexity of this algorithm is on the order of |V 0T | |O(u)| m |I(o(l))|.
The second algorithm generates each one of the packets sent from the source node in eachone of the sink nodes. This algorithm generates packets generated at the source node in eachsink node. With the execution of this algorithm it is possible to obtain at each sink nodepackets sent from each source node, i.e. P(t,i) = Pi. There are r output links at each sinknode, corresponding to the packets generated. The complexity of this algorithm is on theorder of |T | r m |I(t)|.
The importance of this work is to visualize, at the level of each one of the links of thenetwork (graph G) packets belonging to the finite field Fq with q = 2m. In addition, thealgorithms proposed exploit the formulation set for the case where m = 1 given to establishthe groundwork for the Multicast Linear Network Coding. These algorithms ratify that nodes(routers) constituted under the paradigm of Network Coding must process the incoming infor-mation and then forward it. That is, the routers do not work only with the store and forwardparadigm.
124
Convolutional codes in random linear network coding
Diego NappDepartamento de Matematica, Universidade de Aveiro, Portugal
Abstract
A cornerstone of our digital society stems from the explosive growth of multimedia tracin general and video in particular. Video already accounts for over 50% of the internet tractoday and mobile video trac is expected to grow by a factor of more than 20 in the next fiveyears. This massive volume of data has resulted in a strong demand for implementing highlyecient approaches for video transmission. Error correcting codes for reliable communicationhave been studied for several decades. However, ideal coding techniques for video streamingare fundamentally di↵erent from the classical error correction codes. In order to be optimizedthey must operate under lowlatency, sequential encoding and decoding constrains, and assuch they must inherently have a convolutional structure. Such unique constrains lead tofascinating new open problems in the design of error correction codes. In this talk we aim tolook from a system theoretical perspective at these problems. In particular we propose theuse of convolutional codes in this context.
125
The space of binary asymmetric channels:
properties and geometry
C. B. Rodrigues, M. FirerUniversity of Campinas (Brazil)
Abstract
In Coding and Information Theory, the most commom model considers the vector space Fn2
over the binary field F2, the Binary Symmetric Channel (BSC) and the Hamming distance.In this work, we consider a n-dimensional Binary Asymmetric Channel (BAC), denoted
by BACn(p, q), for some constants p, q 2 (0, 12]. The channel BACn(p, q) is a memoryless
channel, i.e. , given x = (x1, . . . , xn) and y = (y1, . . . , yn), with x, y 2 Fn2 , we have P(y|x) =Qn
i=1 P(yi|xi), where P(yi|xi) = p, if yi = 1 and xi = 0 and P(yi|xi) = q, if yi = 0 and xi = 1.For 1 n 2 N, we denote by Pn the space of all n-dimensional BACs. Given a channel
BACn(p, q) 2 Pn consider the Hamming cube as a bi-oriented weighted graph G = (Fn2 , E),
where xy 2 E is an edge of E if, and only if, dH(x, y) = 1 and dH denotes the Hammingdistance. The weight of an oriented edge xy 2 E is log2(
1p1) or log2(
1q1) if wH(y)wH(x) =
1 or wH(y) wH(x) = 1, respectively. An oriented path in G, with exactly r edges, joiningthe vertexes x and y is a sequence X = (xi)
ri=0, where x0 = x, xr = y and xixi+1 2 E. Its
lenght is the sum of the weights of the r edges. If X is a path, with minimum lenght, amongall the paths that connecting the vertexes x and y then its lenght is denoted by n
p,q(x, y). Thefunction n
p,q : Fn2Fn
2 ! R+ is an additive quasi metric (M.M. Deza, E. Deza, Encyclopedia ofdistances, second ed., Springer, 2013). For each (p, q) 2 (0, 1
2]2, the distance n
p,q is associatedto the corresponding channel BACn(p, q) in the sense that
np,q(x, y) < n
p,q(x, z) , Pnp,q(z|x) < Pn
p,q(y|x), (1)
for all x, y, z 2 Fn2 .
From the relation (1), it is possible to identify the equivalence (BACn(p, q) BACn(p0, q0))between two channels BACn(p, q), BACn(p0, q0) 2 Pn. An equivalence class determined by apair (p, q) is denoted by [p : q] and
[p : q] = [p0 : q0] , np,q(x, y) < n
p,q(x, z) , np0,q0(x, y) < n
p0,q0(x, z),
for all x, y, z 2 Fn2 and p, q, p0, q0 2 (0, 1
2]. The set of all equivalence classes is denoted by
Pn/.The first problem solved in this work, from the coding viewpoint, is a simple description
of the equivalence classes of Pn/, considering the transformation
(p, q) =log2
1
p 1, log2
1
q 1
.
The set of BACs may be parametrized by a pair of non negative real numbers and, oncen is fixed, (R+)2 is partitioned into |Pn/| convex cones such that each one corresponds to aunique equivalence class of BACs.
Next, we describe the (quasi-) metric balls and show how the error correction depends onthe equivalence class.
This work is supported by grants 140368/2015-9 and 2013/25977-7, CNPq and FAPESP,respectively.
126
Analysis of measurement concerning the extrinsic parameters
in stereo vision
1,2C. H. Rodrıguez-Garavito, 2A. de la Escalera and 2J.M. Armingol1Universidad de La Salle Bogota Colombia.
2Universidad Carlos III de Madrid.
[email protected], 100286133,escalera,[email protected]
Abstract
The stability of the extrinsic parameters is essential to preserve visual consistency betweencaptured images from a scene, where a change of perspective is done. Therefore, It is importantto analyse and assess the reliability in parameters estimation that define the camera position.
This extrinsic analysis is performed on each simultaneously captured image pair, referenceimage and complementary image, each one taken from di↵erent spatial positions, separated acertain distance, called base line.
A first step of the analysis is the correction of lens aberrations, consecuence of manufactur-ing errors in the optics of each camera. The following step is the image alignment by settingthe projected plane of the reference image in coplanar position with the projected plane ofthe complementary image in order to match each line of both images. This process is calledrectification and is applied by means of a linear transformation that rotates and moves thecomplementary image to achieve the alignment in the visual information projected on eachimage sensor. Subsequently, each pixel from reference image is detected on the complementaryimage in the same line position and the row di↵erence between detections, called disparity,is used for estimating (by triangulation) the 3D position P3D = [X, Y, Z] corresponding toreference pixel in the world frame, assuming the pin-hole model for the projection of a sceneon an optical sensor, obtaining a three-dimensional representation of the scene or PC (PointCloud).
The analysis continues to search in the PC for the road plane, corresponding with themost populated plane, ec. (1). This plane can be found through parametric estimationalgorithms such as: RANSAC, Random Sample and Consensus, proposed by Fischler andBolles in 1981 and LMedS Least-Median-of-Squares (Rousseeuw, 1984), among others. Theoutput parameters for these methods are [a, b, c, d], o ~n and h, unitary vector perpendicularto the plane detected, and the distance from plane to PC reference system, respectively.
(~p) := ax + by + cz + d = 0 := ~n · ~p = h (1)
Therefore, ~n define orientation angles, pitch (↵) and roll () of the camera by relativeposition, likewise, h establish the height sensor corresponding to z coordinate in a referencesystem m attached to the ground plane.
Now, a practical test is possible over a dynamic reference sequence, taken from the KITTI(Karlsruhe Institute of Technology and Toyota Technological Institute at Chicago) repository.The experiment compare the behaviour of four parametric estimation methods constrainedto uniform sampling reduction for di↵erent resolutions (number of points taken from thePC). The aim of the experiment is to find the best trade-o↵ algorithm between accuracy andcomputational cost.
Parameter: Height (mm) Pitch () Roll () Time (ms)
Accuracy 14.7 0.29 1.83 5
Table 1: RANSAC performance for the KITTI sequence “Road 26092011 0015”
The method with the best accuracy-time is the RANSAC estimator with reduction byuniform sampling of 2000 points. The accuracy achieved with the method is shown in table 1.
This work was auspices by the Universidad de La Salle Bogota Colombia, COLCIENCIAS and UniversidadCarlos III de Madrid, UC3M.
127
Ecient Algorithms for Searching Multiple Burst-Error
Correcting Cyclic and Shortened Cyclic Codes
A. L. Sandoval Orozco, L. J. Garcıa Villalba, M. BlaumDepartment of Software Engineering and Artificial Intelligence (DISIA)
Faculty of Computer Science and EngineeringUniversidad Complutense de Madrid (UCM)
asandoval, javiergv, [email protected]
Abstract
Research in Error Correcting Codes has been concerned primarily with coding techniquesfor channels on which transmission errors occur independently in digit positions (i.e., eachtransmitted digit is a↵ected independently by noise); however, there are communication chan-nels that are a↵ected by disturbances that cause transmission errors to cluster into burst. Ingeneral, codes for correcting random errors are not ecient for correcting burst errors, so it isdesirable to design codes specifically for correcting burst errors, namely, burst-error-correctingcodes.
In other words, in many communication channels, due to physical/mechanical failures,noise disturbances tend to occur in the form of bursts. Channels modeled by Gilbert andElliot fall into this category. Examples of such channels are wireless, magnetic recording, andrecently, flash channels, that tend to su↵er from degradation of data in which their reliabilitydeteriorates significantly over time to a degree that compromises data integrity. Hence, errorsare often clustered into bursts. With higher transmission rates or higher storage densitiesthis may even be more so in the future. Under these circumstances it is not ecient to userandom-error-correcting codes. Codes capable of correcting multiple burst are ecient toolsof error control for such channels.
Under these circumstances it is not ecient to use random error-correcting codes. Codescapable of correcting multiple bursts are ecient tools of error control for such channels.
The problem of correcting bursts of errors is a dicult one. In the literature on errorcorrection codes there has been a good deal of attention given to the correction of a (single)burst of errors.
The problem of treating multiple bursts of errors is still more complicated and has re-ceived correspondingly less attention. In practice, Reed-Solomon codes, either interleaved ornot, are used for correcting multiple bursts. There are good reasons for this. One of them isthat, although good burst-correcting codes have been found by computer search, there are noknown general constructions giving cyclic codes that approach the Reiger bound. Interleavingof Reed-Solomon codes on the other hand, provides a burst-correcting code whose redundancy,asymptotically, approaches the Reiger bound. The longer the burst we want to correct, themore ecient interleaving of Reed-Solomon codes is. The second reason for choosing interleav-ing of Reed-Solomon codes, and probably the most important one, is that, by increasing theerror-correcting capability of the individual RS codes, we can correct multiple bursts. In brief,Reed-Solomon codes are often employed due to their “Maximum Distance Separable” prop-erty and moreover to the existence of ecient algorithms to correct up to half the minimumdistance of random errors.
This work focuses on the search for such codes. Some new bounds are presented for multipleburst-correcting codes that extend the previously known bounds, such as the Reiger and theGallager bounds. It is shown that both the new bounds coincide for block codes.
First, we have found new (lower) bounds for multiple burst-correcting codes. In the caseof the Gallager bound for single burst-correcting codes, we have extended it by generalizingthe definition of guard space to multiple bursts. In the case of block codes, we have presenteda straightforward extension of the Reiger bound followed by a more involved generalization.
We compared the new bounds to the existing values of the well known volume bound.For large values of the length of the code n, keeping b and db fixed, the volume bound is thebest, while for small values of n the most ecient bound is the extended Reiger bound. Thegeneralized Reiger bound improves the others for intermediate values of n.
128
Then, we have extended the concept of binary random error-correcting codes MDS codesto binary burst-correcting MDS codes. Although the former are trivial, this is not the casefor burst-correction. Theorem gives a kind of converse to the well known property that theonly binary MDS codes of distance greater than 2 are repetition codes. Since the only knownexamples of [n, k, db] burst-correcting MDS codes with k > b involve codes with minimumburst distance db = 3, an open question is the existence of burst-correcting MDS codes withdb 4. Theorem gives a negative answer for large enough db.
It is of interest to find multiple burst-correcting codes that are optimal in terms of redun-dancy. Such codes may be either of block or of convolutional type. However, in this Thesiswe have concentrated on the search of codes of block type. Moreover, we have searched forcodes that are either cyclic or shortened cyclic. There are two reasons for this: one is thata (shortened) cyclic code depends only on its generator polynomial. A general block codedepends on its parity-check matrix, making the search on all possible parity-check matricesintractable. A second reason is that (shortened) cyclic codes are easy to encode and decode.
Meanwhile, the known cyclic codes are designed, in general, to correct only one burst. Theproblem of finding the best (shortened) cyclic burst-correcting codes is a dicult one, evenin the case of single burst-correcting codes. The best known family of cyclic single burst-correcting codes is given by the Fire codes. However, for specific parameters and codes ofrelatively short length n, some of the best burst-correcting codes were found by computersearch (for large values of n, the search may become intractable), often improving the pa-rameters of (shortened) Fire codes. Many of the results of such searches can be found in thetables given in the literature. The above considerations imply that it is of interest have e-cient search algorithms that can extend the known optimal (shortened) cyclic burst-correctingcodes. We provide two algorithms in this Thesis.
Both algorithms are based on syndrome computation for each possible generator polyno-mial. The search algorithms follow the method developed in algorithm for determining if acode is single burst correcting. Given a generator polynomial g(x) in binary form, for a givenlength code, it tests whether the (shortened) cyclic code generated by g(x) can correct twobursts or not. The degree of g(x) corresponds to the number of parity bits of the code, i.e.,n k. In order to see if there exists an [n, k] (shortened) cyclic code capable of correcting twobursts, we need to check all possible generator polynomials of degree nk. If we find one thatgives an [n, k] double burst-correcting code, we stop the search. If there is none, then we tryto find an [n, k 1] code using the same procedure, and so on, until we determine the largestpossible value of k. These algorithms developed are optimal. The eciency of the algorithmsstems from the fact that no repeated syndromes are computed. It is shown how to achievethis goal using Gray codes.
129
14 Sessao tematica: Nonlocal equations and applications
130
Programacao das palestras
Sessao especial: Nonlocal Equations and Applications
Terca-feira Sala 225
10:30 - 11:10 Bruno AndradeEquacoes de Volterra com aplicacoes em modelos parabolicos com memoria
11:10 - 11:50 Matheus SantosGradient flow approach to bidimentional Euler equation
16:00 - 16:40 Julian HaddadAffine inequalities using convex analysis
16:40 - 17:20 Rafael BarostichiLocal well posedness for the generalized Camassa-Holm equation in spaces of analytic functions
17:20 - 18:00 Giovana SiracusaExistence results for fractional integro-differential inclusions with state-dependent delay
Quarta-feira Sala 325
10:00 - 10:40 Diego MarconAn optimization problem in heat conduction for the fractional Laplacian
10:40 - 11:20 Silvia Sastre-GomezStability results of nonlocal diffusion problems
11:20 - 12:00 Fabiana TravessiniTBA
15:30 - 16:10 Jesus RosadoTBA
131
An optimization problem in heat conduction for the
fractional Laplacian
D. MarconUniversidade Federal do Rio Grande do Sul (Brazil)
R. TeymurazyanUniversidade de Coimbra (Portugal)
Abstract
We study a free boundary optimization problem in heat conduction, ruled by the fractionalLaplacian operator, with a lower temperature bound and a volume constraint. We prove theexistence and the Holder regularity of solutions, we derive geometric properties of solutionsand of the corresponding exterior and interior free boundaries.
132
Equacoes de Volterra com aplicacoes em modelos parabolicos
com memoria
Bruno de AndradeUniversidade Federal de Sergipe
Abstract
Nesta palestra apresentaremos alguns de nossos resultados recentes (Math. Ann. 2016 eMath. Nachr. 2016) sobre existencia local, regularidade e dependencia continua com respeitoaos dados iniciais de solucoes brandas -regulares para o problema
u0 = Au +
Z t
0
g(t s, u(s))ds + f(t, u(t)), t 0, (1)
u(0) = u0 2 D(A), (2)
onde A : D(A) X0 ! X0 e um operador linear tal que A e uma operador setorial, X0 e umespaco de Banach e as funoes g e f satistazem certas condicoes. Mostraremos tambem aplicacoesde tais resultados em equacoes diferenciais parciais com memoria.
133
Existence results for fractional integro-di↵erential inclusions
with state-dependent delay
Giovana SiracusaUniversidade Federal de Sergipe
Claudio Cuevas & Hernan R. Henrı[email protected]
Abstract
In this work we are concerned with a class of abstract fractional integro-di↵erential inclu-sions with state-dependent and infinite delay. Our approach is based on the existence of aresolvent operator for the homogeneous equation. We establish the existence of mild solutionsusing both contractive and condensing maps. Finally, an application to the theory of di↵usionproblems in materials with memory is given.
134
Ane inequalities using convex analysis
Julian HaddadUniversidade Federal de Minas Gerais (Brazil)
M. Montenegro, H. [email protected]
Abstract
We show that the Lp Busemann-Petty centroid inequality, an inequality about convexsets, provides an elementary and powerful tool to the study of certain sharp ane functionalinequalities. Our theorems show the geometric nature behind them. In constrast with classi-cal sharp functional inequalities, the ane ones have extremal functions satisfying non-localpartial di↵erential equations.
135
Gradient flow approach to bidimentional Euler equation
M. C. SantosUniversidade Federal do Rio Grande do Sul
L. C. F. FerreiraUniversidade Estadual de Campinas
Abstract
The vorticity formulation of the two-dimensional incompressible Euler equation in R2 isan example of Hamiltonian flow. We will show that there are solutions to this equation whichalso admit a gradient flow formulation associated to a new entropy functional. Using optimaltransport techniques, these solutions are obtained through a minimizing movement scheme inthe Wasserstein space of probability measures.
136
Local well posedness for the generalized Camassa-Holm
equation in spaces of analytic functions
R. F. Barostichi and G. PetronilhoFederal University of Sao Carlos (Brazil)
A. HimonasUniversity of Notre Dame (USA)
Abstract
In this lecture, we will present the problem of wave propagation, providing a historicaloverview of its development, with special emphasis on the so-called Camassa-Holm equation,stressing its historical importance in this context. We will conclude by presenting recentadvances in the theory of well posedness of some generalizations of the Camassa-Holm equationin spaces of analytic functions on both the line and the torus.
137
Stability results of nonlocal diffusion problems
Silvia Sastre-GomezUniversidade Federal de Pernambuco (Brazil)
Anibal Rodriguez-BernalUniversidad Complutense de Madrid (Spain)
Abstract
Let RN be bounded, X = Lp(), with 1 p 1 and consider the nonlocal diffusion problem
ut(x, t) = Ku(x, t) h0(x)u(x, t) +f(u(x, t)) x 2 , t 0u(x, 0) = u0(x) x 2 ,
(1)
where u0 2 X , the function f defined as f : R ! R, with f 2 C2(R) is globally Lipschitz, andrepresents the nonlinear reaction term, and the nonlocal diffusion operator denoted by K, is given byKu(x, t) =
R
J(x, y)u(y, t)dy, where J is a nonnegative function defined in , with J(x, y) =J(y, x). Moreover, h0(x) =
R
J(x, y)dy is a bounded function in L1(). One of the main character-istics of problem (1), is that it admits constant solutions in space. We give some criteria on the convexityof the function f , to prove that the instability with respect to the linearization, implies the instability ofthe equilibria of the nonlinear problem (1). We also prove that any continuous nonconstant equilibriumsolution of (1) is, if it exists, unstable when f is convex.
138
15 Sessao tematica: Otimizacao
139
Programacao das palestras
Sessao: Otimizacao
Segunda-feira Auditorio
14:00 - 14:30 T. MartiniAceleracao quasi-Newton para um problema de ponto fixo provenientede Tomografia por Emissao de Positrons
14:30 - 15:00 T. C. SilvaPrimal-dual fixed point algorithm apllied for ill conditioned RidgeRegression problems
15:00 - 15:30 B. H. CervelinDeterminacao de parametros de um modelo de aviario atravesde otimizacao contınua sem derivadas
16:00 - 16:30 P. S. FerreiraA globally convergent derivative-free filter algorithm for nonlinearconstrained problems
16:30 - 17:00 D. G. FerreiraO algoritmo PSIFA em problemas de otimizacao com ruıdo na funcaoobjetivo e restricoes lineares degeneradas
17:00 - 17:30 R. LopesAceleracao dos metodos de descenso coordenado por blocosvia funcao identificadora
Terca-feira Auditorio
10:30 - 11:00 F. N. C. SobralEnsuring poisedness in derivative-free trust-region algorithms
11:00 - 11:30 L. E. A. SimoesConvergencia Local de Metodos Amostrais
11:30 - 12:00 G. N. GrapigliaHybrid derivative-free methods for composite nonsmooth optimization
16:00 - 16:30 G. A. PericaroAn improvement to the HLRF algorithm for structural reliability
16:30 - 17:00 T. A. SenneApproximated solution of linear systems arising from topologyoptimization of structures
17:00 - 17:30 A. LaurainStructural Optimization using the Level Set Method
Quarta-feira Auditorio
10:00 - 10:30 R. LobatoEmpacotamento de elipsoides
10:30 - 11:00 D. S. GoncalvesA fixed-point method for computing the approximate projection ontothe positive semidefinite cone
11:00 - 11:30 R. BehlingAceleracao do metodo de Douglas-Rachford
11:30 - 12:00 J. L. GardenghiComplexidade em Programacao Nao Linear
15:30 - 16:00 M. C. ScarabelloMetodo baseado em q-gradiente para problemas de otimizacao de grande porte
16:00 - 16:30 L. F. BuenoUma estrategia de descida para problemas de equilıbrio
140
Structural Optimization using the Level Set Method
Antoine LaurainUniversity of Sao Paulo (Brazil)
Abstract
In this talk we present the level set method and how it can be used to perform structuraloptimization. The goal is to minimize an objective functional which depends on the geometryof a structure, whose behaviour is governed by linearized elasticity. To compute the sensitivityof the functional with respect to perturbations of the geometry, we use the concept of shapederivative. We use a novel approach which consists in using a volume expression for the shapederivative instead of the usual boundary expression. We explain how the level set method canbe adapted to use this volume expression. We show the advantages of this approach from anumerical point of view, we discuss in particular the fact that it is easier to implement thanthe usual approach in level set methods. We illustrate the eciency of this approach with acompact code written for educational purposes.
141
Determinacao de parametros de um modelo de aviario
atraves de otimizacao contınua sem derivadas
B. H. Cervelin, D. Conti, M. A. Diniz-Ehrhardt, J. M. MartınezUniversidade Estatual de Campinas (Unicamp)
D. T. DetschUniversidade Federal do Parana (UFPR)
Abstract
O Brasil e um dos maiores produtoes de carne de frango do mundo, e e o primeiro emrelacao as exportacoes. A taxa de crescimento dos frangos depende de varios fatores, algunsassociados a condicoes ambientais. Temperatura, umidade e concentracao de amonia devemser controladas para que o conforto maximo dos frangos seja obtido. Tipicamente, aviarios noestado de Sao Paulo controlam a temperatura atraves de ventilacao gerada por exaustores.
Nos ultimos anos produtores tem recorrido a matematicos, cientistas da computacao, fısicose engenheiros para a construcao de modelos matematicos com o objetivo de automatizar aproducao.
Um dos modelos, apresentado na tese de doutorado de D. Detsch, descreve uma EDP quemodela a temperatura em aviario representado unidimensionalmente, com parede esquerda edireita, no qual a temperatura e propagada por um processo de difusao, sujeito a condicoesde fronteira dadas pela temperatura externa do aviario e condicoes iniciais dadas por variossensores no aviario. Os exaustores sao modelados como fontes de calor que contribuem para areducao da temperatura. Uma fonte natural de calor e dada pelos animais, que varia durantesuas vidas.
Assumiremos que a funcao de controle depende dos exaustores, e que os controladorespossuem um numero finito de estados possıveis d0, d1, . . . , dN . Para cada possıvel estado doscontroles dj uma funcao udj (x, t) = ↵ 0.05jp3 e associada. Assim a EDP do modelo possuitres parametros que precisam ser adaptados para dados reais antes (ou durante) a aplicacao domodelo em um aviario real. Esses parametros correspondem ao coeficiente de difusao dentrodo aviario, o coeficiente de difusao nas paredes e o efeito de cada exaustor na variacao datemperatura em uma unidade de tempo.
Podemos aplicar tecnicas de otimizacao para encontrar esses coeficientes. E, apesar denao ser impossıvel de se avaliar as derivadas da funcao de merito que avalia a qualidadeda aproximacao em relacao aos parametros, a necessidade de mudar a estrutura do modelodurante a pesquisa nos levou a aplicar metodos sem derivadas, o que nos forneceu flexibilidadee velocidade de execucao satisfatoria.
Nesse trabalho descreveremos o comportamento de varios algoritmos sem-derivadas paraencontrar os parametros do modelo EDP usando dados reais correspondentes a um aviariolocalizado em Jundiaı (Sao Paulo, Brasil). Os resultados desse estudo determinaram nossadecisao final sobre o tipo de minimizacao que deve ser aplicado em situacoes praticas daquipara frente.
142
O algoritmo PSIFA em problemas de otimizacao com ruıdo
na funcao objetivo e restricoes lineares degeneradas
M. A. Diniz-Ehrhardt, D. G. Ferreira, S. A. SantosUniversidade Estadual de Campinas
Resumo
PSIFA (Pattern Search Implicit Filtering Algorithm) e um algoritmo que nao faz uso dederivadas, combinando uma abordagem de busca padrao com ideias do metodo de filtragemimplıcita. E voltado a problemas de otimizacao com ruıdo na funcao objetivo e restricoeslineares. Resultados de convergencia global foram demostrados tanto para o caso nao dege-nerado quanto para problemas degenerados, usando hipoteses enfraquecidas em relacao aosmetodos de Busca Padrao (R. Lewis e V. Torczon, 2000) e Filtragem Implıcita (C. T. Kelley,2011). Realizamos testes numericos com diversos graus de degeneracao nas restricoes linearese considerando dois tipos de ruıdos, um nao suave que satisfaz as hipoteses de convergenciado metodo, e um ruıdo “branco” que se aproxima mais das situacoes encontradas na pratica.Apresentamos uma analise comparativa com os resultados dos metodos de Busca Padrao e deFiltragem Implıcita.
143
A fixed-point method for computing the approximate
projection onto the positive semidefinite cone
J. B. Francisco, D. S. GoncalvesUniversidade Federal de Santa Catarina
juliano, douglas @mtm.ufsc.br
Abstract
The projection of a symmetric matrix onto the positive semidefinite cone is an importantproblem with application in many di↵erent areas such as economy, physics and optimization.This problem has analytical solution, but it relies on the eigendecomposition of a given sym-metric matrix which clearly becomes prohibitive for larger dimension and dense matrices. Wepresent a fixed-point iterative method for computing an approximation of such projection.Each iteration requires matrix-matrix products whose costs may be much less than O(n3)for certain structured matrices. Numerical experiments showcase the attractiveness of theproposed approach for sparse symmetric banded matrices.
144
Ensuring poisedness in derivative-free trust-region algorithms
L. G. PedrosoFederal University of Parana (Brazil)
F. N. C. SobralState University of Maringa (Brazil)
Abstract
When a set of points in Rn is well poised, important theoretical and practical propertiesarise in derivative-free trust-region algorithms. Models built with well poised sets use to begood approximations of the true objective function. Some well established techniques exist tobuild and maintain well poised sets for models based on Lagrange polynomials. However, whenmodels come from di↵erent strategies, such approach may demand unnecessary calculations.In this talk, we discuss the use of QR factorization in order to maintain well poised sets forlinear interpolation models in unconstrained derivative-free optimization. The properties oforthogonal matrices allow a simple calculation to decide which point should be removed andwhich point would improve the poisedness of the set. Some numerical experiments will bepresented.
This work is supported by CNPq, proc. 458360/2014-7
145
Hybrid derivative-free methods for composite nonsmooth
optimization
Geovani N. GrapigliaFederal University of Parana (Brazil)
Abstract
In this talk, the combination of global solvers and derivative-free trust-region algorithmsis considered for minimizing a composite function (x) = f(x) + h(c(x)), where f and c aresmooth and h is convex but may be non-smooth. Global convergence results and worst-casecomplexity bounds are discussed. The performance of the algorithms is illustrated consideringthe parameter estimation of biochemical models.
146
An improvement to the HLRF algorithm for structural
reliability
G. A. Pericaro, S. R. SantosUniversidade Estadual do Parana (Brazil)
A. A. Ribeiro, L. C. MatioliUniversidade Federal do Parana (Brazil)
Abstract
In this work we briefly discuss some concepts of structural reliability as well as the op-timization algorithm that is commonly used in this context, called HLRF. We show thatthe HLRF algorithm is a particular case of the SQP method, in which the Hessian of theLagrangian is approximated by an identity matrix. Motivated by this fact, we propose theBFGS-HLRF algorithm that considers the BFGS update formula to approximate the Hes-sian. The algorithm proposed is as simple as the HLRF algorithm, since it requires just onefunction and gradient evaluation at each iteration and the new iterate is given by a recursiveformula. Comparative numerical experiments on a set of problems selected from the literatureare presented to illustrate the performance of the algorithm and the results indicate that theHLRF-BFGS algorithm has the advantage of being more robust and ecient with respect tothe function and gradient evaluation, than HLRF.
147
Complexidade em Programacao Nao Linear
E. G. Birgin, J. L. GardenghiUniversidade de Sao Paulo (Brasil)
J. M. Martınez, S. A. SantosUniversidade Estadual de Campinas (Brasil)
Ph. L. TointUniversidade de Namur (Belgica)
Resumo
Complexidade em programacao nao linear tem por objetivo contar quantas iteracoes (ouquantas avaliacoes de funcao, ou alguma outra medida quantitativa) sao necessarias para queum metodo convirja a um ponto com determinadas propriedades para um dado problema.
No presente trabalho, estudamos e estabelecemos a complexidade de avaliacao de metodosque se utilizam de modelos de ordem superior para resolucao de problemas de minimizacaosem restricoes.
Do ponto de vista teorico, estabelecemos a complexidade de avaliacao de um algoritmoque utiliza modelos de ordem p para minimizacao de uma funcao p vezes continuamente di-ferenciavel. Vimos que sua complexidade possui relacao direta com a ordem dos modelosusados. Os resultados detalhados foram publicados em E. G. Birgin, J. L. Gardenghi, J. M.Martınez, S. A. Santos e Ph. L. Toint. “Worst-case evaluation complexity for unconstrai-ned nonlinear optimization using high-order regularized models”, Mathematical Programming,disponıvel online em http://dx.doi.org/10.1007/s10107-016-1065-8.
Do ponto de vista pratico, experimentamos numericamente uma implementacao computa-cional do algoritmo para o caso em que p = 3. Exibiremos resultados numericos que ilustramo bom desempenho deste algoritmo do ponto de vista de quantidade de avaliacoes de funcao.
148
Convergencia Local de Metodos Amostrais
Lucas E. A. Simoes, Sandra A. SantosUniversity of Campinas (Brazil)
Elias S. HelouUniversity of Sao Paulo (Brazil)
Abstract
Uma quantidade significativa de problemas dependem da minimizacao de funcoes naosuaves. Consequentemente, muitos metodos de otimizacao tem sido desenvolvidos com oobjetivo de lidar com este tipo de funcao. Recentemente, um algoritmo conhecido como Gra-dient Sampling (GS) vem ganhando atencao pelos seus bons resultados numericos e por suaforma intuitiva de funcionamento, generalizando o metodo de maxima descida para funcoesnao suaves. Desta forma, este estudo possui duas frentes: analisar a convergencia local doalgoritmo GS e apresentar uma nova proposta de metodo amostral que possui, sob algumascondicoes, uma convergencia local superlinear.
149
Uma estrategia de descida para problemas de equilıbrio
Luıs Felipe BuenoUniversidade Federal de Sao Paulo (Brasil)
Abstract
Varias aplicacoes sao modeladas como problemas de Equilıbrio de Nash. A maioria dosmetodos para abordar estes problemas sao baseados nas desigualdades variacionais associadas.Sob as hipoteses de convexidade, sao muito discutidos resultados sobre existencia, unicidade econvegencia de algoritmos desvenvolvidos para este tipo de problema. Por outro lado, muitopouco e conhecido sobre metodos que explorem a estrutura de otimizacao do problema, o que eum fator chave para problemas nao convexos. Neste trabalho apresentamos um algoritmo queusa este conceito, impondo uma condicao de descida para aceitacao do ponto. A ideia principalpara este algoritmo e que cada jogador toma a decisao de acordo com a decisao previstapara o outro. Mostraremos que o metodo proposto pode ser interpretado como o metodo deNewton para a desigualdade variacional associada ao problema de equilıbrio. Alguns exemplosilustrativos serao apresentados.
150
Metodo baseado em q-gradiente para problemas de
otimizacao de grande porte
M. C. Scarabello, A. C. Soterroni, E. J. C. Gouvea, R. L. Galski, F. M. RamosInstituto Nacional de Pesquisas Espaciais (INPE)
Resumo
Recentemente foi proposta uma generalizacao do metodo da maxima descida utilizando oconceito de q-gradiente, denominada metodo q-G. Esse metodo, que utiliza o vetor q-gradientepara determinar a direcao de busca em problemas de otimizacao global contınua mono-objetivo,se mostrou competitivo quando aplicado em funcoes teste de ate 30 dimensoes. De formaanaloga aos metodos baseados em gradiente, o metodo baseado em q-gradiente, de modogeral, necessita de pelo menos N + 1 avaliacoes da funcao objetivo por iteracao, onde N e adimensao do problema a ser otimizado. Devido a esta caracterıstica, o metodo q-G tem o seudesempenho deteriorado com o aumento da dimensionalidade dos problemas. Com o intuito desolucionar problemas de otimizacao com um grande numero de variaveis de decisao foi geradauma versao rapida do metodo q-G, denominada metodo Fq-G. Este metodo e baseado no uso deperturbacoes simultaneas para calcular uma aproximacao do vetor q-gradiente, abordagem queexige apenas duas avaliacoes da funcao objetivo por iteracao, independentemente do valor deN . Assim como no metodo q-G, no algoritmo do Fq-G o processo de busca muda gradualmentede global, no inıcio do procedimento iterativo, para local no final. Alem disso, sao utilizadasperturbacoes gaussianas para garantir a convergencia do metodo Fq-G para o mınimo globalem um sentido probabilıstico. O metodo Fq-G foi aplicado em 7 funcoes testes com ate1000 variaveis de decisao propostas na competicao para problemas de otimizacao de grandeporte do IEEE Congress on Evolutionary Computation (CEC) de 2008. Os resultados foramcomparados com 7 Algoritmos Evolutivos participantes da competicao e apontam o potencialdo metodo do Fq-G para resolver problemas de otimizacao de alta dimensionalidade.
151
A globally convergent derivative-free filter algorithm for
nonlinear constrained problems
P. S. FerreiraFederal University of Technology – Parana (Brazil)
E. W. Karas, M. Sachine,Federal University of Parana (Brazil)
F. N. C. SobralState University of Maringa (Brazil)
Abstract
We present an algorithm for solving constrained optimization problems in which the deriva-tives of the objective function are not available. The algorithm mixes an inexact restorationframework with filter techniques. Each iteration is decomposed in two independent phases: afeasibility phase which reduces an infeasibility measure without evaluations of the objectivefunction, and an optimality phase which reduces the objective function value. As the algo-rithm does not make explicit use of the objective function derivatives, the optimality stepis computed by derivative-free trust-region internal iterations. The trust-region models canbe linear or quadratic and constructed by any technique, such as polynomial interpolation orsupport vector regression, since the gradient of the model is a reasonable approximation of thegradient of the objective function at the current point. Assuming this and classical assump-tions, we prove that the full steps are ecient in the sense that near a feasible nonstationarypoint, the decrease in the objective function is relatively large, ensuring the global convergenceresults of the algorithm. To finish we present numerical experiments to show the e↵ectivenessof the proposed method.
152
Empacotamento de elipsoides
E. G. Birgin,Universidade de Sao Paulo
R. D. Lobato, J. M. MartınezUniversidade Estadual de Campinas
Resumo
O problema de empacotamento de elipsoides consiste em arranjar uma dada colecao deelipsoides dentro de um determinado conjunto, de forma que os elipsoides nao se sobrepo-nham. Esse problema tem diversas aplicacoes em fısica, engenharia de materiais e biologia,por exemplo. Apresentaremos modelos de programacao nao-linear contınuos e diferenciaveise algoritmos para o empacotamento de elipsoides no espaco n-dimensional. Tambem apresen-taremos alguns experimentos numericos.
153
Aceleracao do metodo de Douglas-Rachford
R. Behling, L. R. dos SantosUniversidade Federal de Santa Catarina
Abstract
Neste trabalho nos propomos uma tecnica de aceleracao do celebrado metodo de Douglas-Rachford para encontrar um ponto comum a um numero finito de conjuntos convexos numespaco de Hilbert. Comparamos nosso esquema com o metodo classico de projecoes alternadase variantes aceleradas encontradas na literatura.
154
Aceleracao dos metodos de descenso coordenado por blocos
via funcao identificadora
Ronaldo Lopes, Sandra A. Santos , Paulo J. S. SilvaUniversity of Campinas (Brazil)
Abstract
Apresentamos um metodo de descenso coordenado por blocos, baseado no artigo [Pe-ter Richtarik, Martin Takac, Iteration complexity of randomized block-coordinate descentmethods for minimizing a composite function, Math. Program. (2014)], que se aplica aoproblema de minimizacao
minx f(x) + (x) (1)
s.a. l x u
em que f e uma funcao suave e e uma funcao convexa, possivelmente nao suave, com umaestrutura separavel por blocos.Usando o conceito de funcao identificadora apresentado no artigo [Francisco Facchinei, AndreasFischer, Christian Kanzow, On the accurate identification of active constraints, SIAM Journalon Optimization (1998)], propomos uma funcao identificadora para (1), capaz de encontraras restricoes ativas do problema. Adicionalmente, para (x) = kxk1, > 0, essa funcaoconsegue descobrir as coordenadas nulas de um ponto estacionario de (1).Explorando a informacao dessa funcao identificadora, construımos um metodo de descensocoordenado aleatorio nao uniforme, que produziu resultados numericos promissores para ocaso (x) = kxk1, > 0.
155
Primal-dual fixed point algorithm apllied for ill conditioned
Ridge Regression problems
T. C. SilvaFederal Technological University of Parana (Brazil)
A. A. RibeiroFederal University of Parana (Brazil)
G. A. PericaroState University of Parana (Brazil)
Abstract
In this work we study the primal and dual formulations of the regularized least squaresproblem, in special norm L2, named Ridge Regression. We state the Gradient method for thecorresponding primal-dual problem through a fixed point approach. For this formulation wepresent an original convergence analysis involving the spectral radius of a suitable iterationmatrix. The main contribution of this work is to accelerate the studied method by meansof a memory-like strategy. Some conections with the spectral properties of the associatediteration matrices were established in order to prove the convergence of the accelerated method.Preliminary experiments showed the good performance of the proposed algorithm when appliedto solve ill conditioned problems, providing better results than the Conjugate Gradient methodfor this class of problems.
156
Approximated solution of linear systems arising from
topology optimization of structures
T. A. SenneFederal University of Sao Paulo (UNIFESP - Brazil)
F. A. M. Gomes, S. A. SantosState University of Campinas (UNICAMP - Brazil)
Abstract
A common problem in topology optimization is the minimization of the compliance of astatic structure, subject to a volume constraint. No matter what method is used to solvethis problem, the computational cost is dominated by the solution of the system of equationsassociated with the equilibrium conditions. If the structure has an elastic material and issubjected to small displacements, one linear system need to be solved at each iteration of theoptimization algorithm, in order to obtain the nodal displacements, which are used to evaluatethe objective function. Since the sti↵ness matrix of the structure is symmetric and positivedefinite, this linear system is frequently solved using the Cholesky factorization, although thismethod can be expensive for large-scale problems. To overcome this diculty, Amir, Bendsøe& Sigmund (2009) presented a strategy based on the combined approximations approach,proposed by Kirsch (1991), which consists basically in reusing the Cholesky factorization.In this work, we combine this approach with the Sequential Piecewise Linear Programmingmethod, proposed by Gomes & Senne (2014), and discuss the reduction of the computationalcost of the solution of the linear systems when this strategy is applied to a classical problem.
157
158
16 Sessao tematica: Sistemas Dinamicos
159
Programacao das palestras
Sessao tematica: Sistemas Dinamicos
Segunda-feira
14:00 - 14:50 Iris OliveiraDynamics nearby a typical cycle of a Filippov system
14:50 - 15:30 Pedro CardinComportamento assintotico de ciclos limites em sistemas de Lienard
16:00 - 16:40 Jairo Kras MengueHolonomic probabilities and entropy
16:40 - 17:20 Durval TononExistencia de orbitas periodicas simetricas para o problema dos trescorpos colineares com carga eletrica
Terca-feira
10:30 - 11:10 Amanda de LimaO Teorema do Limite Central para Funcoes de Weierstrass Generalizadas
11:10 - 11:50 Tiago de CarvalhoAnalise qualitativa de sistemas biologicos modelados por Campos de Vetores Descontınuos.
16:00 - 16:40 Bernardo de CarvalhoHomeomorfismos n-expansivos com a propriedade de sombreamento
16:40 - 17:30 Paulo VarandasGibbs measures and entropy: convergence from a topological andprobabilistic viewpoints
160
Gibbs measures and entropy: convergence from a topological
and probabilistic viewpoints
Paulo VarandasUniversidade Federal da Bahia - UFBA (Brazil)
Abstract
The notion of Gibbs measures arises naturally in the context of thermodynamic formalismin dynamical systems. Given a continuous map f on a compact metric space X and a continu-ous potential : M ! R one of the main goals in the thermodynamic formalism in dynamicalsystems is to determine the existence (and uniqueness) of invariant measures attaining theequality on the variational principle
Ptop(f,) = supµ2M1(f)
nhµ(f) +
Z dµ
o,
referred as equilibrium states. The classical thermodynamic formalism for uniformly hyper-bolic dynamics developed by Sinai, Ruelle and Bowen in the seventies, all equilibrium statesare Gibbs measures. As one can read above, in order to describe the thermodynamic formalismit is crucial to compute the measure theoretical entropy hµ(f) of an f -invariant probabilitymeasure µ and there are natural questions concerning the computability of entropy. In thepresent talk I will discuss almost sure estimates for the error term and large deviations esti-mates for the convergence to entropy in both uniform and non-uniform expanding contexts.Other topological and geometrical aspects of Gibbs measures will also be discussed. Thisreports on a joint work with Y. Zhao (Suzhou University).
161
Anlise qualitativa de sistemas biolgicos modelados por
Campos de Vetores Descontnuos.
Tiago de CarvalhoUniversidade Estadual Paulista – UNESP (Brazil)
Abstract
Sistemas biolgicos sujeitos a mudanas repentinas na sua evoluo podem ser modelados us-ando Campos de Vetores Descontnuos (ou suaves por partes). Por exemplo: modelos predadorpresa com refgio, modelos envolvendo um predador e duas presas onde o predador tem prefer-ncia por determinada presa de acordo com a oferta das mesmas, populaes de pragas/bactriassujeitas a controles qumicos ou biolgicos, entre outros. Muitos destes problemas so modeladosutilizando-se equaes diferenciais ordinrias bastante complicadas e de difcil soluo. Desta forma,faz-se conveniente a utilizao de ferramentas geomtricas/qualitativas para se obter informaesa respeito da evoluo do modelo. A anlise qualitativa do retrato de fase de tais sistemas din-micos descontnuos (determinao de pontos de equilbrio, ciclos limite, modos deslizantes, etc)pode revelar o comportamento a longo prazo do sistema, sem a necessidade de se resolverexplicitamente as equaes envolvidas.
162
Dynamics nearby a typical cycle of a Filippov system
D.D. Novaes, M.A. Teixeira, I.O. ZeliUniversity of Campinas (Brazil)
T.M. SearaUniversitat Politecnica de Catalunya (Spain)
Keywords: Filippov systems, crossing cycle, sliding cycle.
Abstract
In this work we describe the bifurcation diagram of a typical cycle of a piecewise smoothplanar dynamical system (Filippov system). Roughly speaking this object consists in a tra-jectory connecting typical singularities known as fold-fold points.
163
Existencia de orbitas periodicas simetricas para o problema
dos tres corpos colineares com carga eletrica
D. J. Tonon,Universidade Federal de Goias (Brasil)
J. LlibreUniversidad Autonoma de Barcelona (Espanha)
Abstract
Nesta palestra estudamos a existencia de orbitas simetricas periodicas do problema dos3-corpos quando cada corpo possui massa e uma carga eletrica. A principal tecnica aplicadaneste estudo e o metodo de continuacao de Poincare.
Um dos problemas mais relevantes na area de Sistemas Dinamicos e o problema dos n-corpos e muitos trabalhos foram feitos para compreender sua dinamica. Assim, o estudo desuas orbitas periodicas e um dos principais objetivos. Neste trabalho, focalizamos a atencao nasorbitas periodicas do problema 3-corpos quando estes sao colineares e carregadas eletricamente.
Recentemente, muitas tecnicas e metodos distintos foram usados para provar a existenciade orbitas periodicas para o problema dos n-corpos, por exemplo, teoria da media, analisenumerica, funcoes de Melnikov, formas normais, metodos variacionais, entre outros. Um dosprimeiros estudos analıticos sobre a existencia de orbitas periodicas para o problema dos n-corpos foi feito por Poincare e nesse trabalho nos aplicaremos seu metodo para estudar asorbitas periodicas simetricas do problema dos 3-corpos colineares carregados eletricamente.
O metodo de continuacao de Poincare consiste em fornecer uma solucao para o sistemacom um parametro igual a zero e dar condicoes para estender esta solucao a pequenos valoresdo parametro.
164
Holonomic probabilities and entropy
Jairo K. MengueUniversidade Federal do Rio Grande do Sul (Brazil)
Abstract
We will present a possible definition of entropy for holonomic probabilities and some of itsproperties. We assume that X and Z are compact metric spaces and the iterated functionsystem x : Z ! Z, x 2 X is defined using uniform contractions, that is, there exists someconstant 0 < < 1 such that
d(x1(z1), x2(z2)) [d(x1, x2) + d(z1, z2)] 8xi, zi.
We say that a probability 2 P (X Z) is holonomic if
Zg(x(z))d(x, z) =
Zg(z)d(x, z), for any g 2 C(Z).
Given a probability ↵ 2 P (X) with supp(↵) = X we define the entropy of a holonomicprobability with respect to ↵ as
H↵() := sup
Zc d, c is Lipschitz and
Zec(x,z) d↵(x) = 1 8 z 2 Z
.
On the other hand, given a continuous function c(x, z) we define the pressure of c with respectto ↵ as
P↵(c) = sup
Zc d + H↵(), is holonomic
.
If c is Lipschitz then eP↵(c) is equal to the spectral radius of the operator Lc,↵ : C(Z) ! C(Z)given by
Lc,↵( )(z) =
Z
X
ec(x,z) (x(z)) d↵(x).
The entropy is a concave function on the convex set of holonomic probabilities, but thereare examples where it is not an ane function. As a consequence, it is possible to constructexamples where no extremal points of the convex set attain the supremum that defines P↵(c).
165
O Teorema do Limite Central para Funcoes de Weierstrass
Generalizadas
A. de Lima, D. SmaniaUniversidade de Sao Paulo (Brazil)
Abstract
Sejam f : S1 ! S1 uma transformacao expansora do cırculo de classe C2+ e v : S1 ! Ruma funcao de classe C1+. Consideremos a equacao cohomologica torcida
v(x) = ↵(f(x)) Df(x)↵(x),
a qual possui uma unica solucao ↵. Mostramos que ↵ ou e de classe C1+ ou nao possuiderivada em ponto algum. Alem disso, ↵ nao possui derivada em ponto algum se e somentese = () 6= 0, onde
(x) =
v0(x) + ↵(x)D2f(x)
Df(x)
e
2() = limn!1
Z Pnj=0 f j
pn
!2
dµ.
Aqui, µ e a unica probabilidade invariante absolutamente contınua de f . Mostramos que se ↵nao possui derivada em ponto algum entao
limh!0
µ
(x :
↵(x + h) ↵(x)
`hp
log |h| y
)=
1p2
Z y
1e
t2
2 dt.
para alguma constante ` > 0. Em particular ↵ nao e uma funcao Lipschitz em qualquersubconjunto com medida de Lebesgue positiva.
166
Homeomorfismos n-expansivos com a propriedade de
sombreamento
Bernardo CarvalhoUniversidade Federal de Vicosa (Brasil)
Abstract
A dinamica de homeomorfismos expansivos com a propriedade de sombreamento pode serbastante complicada mas esta razoavelmente bem compreendida. Esses sistemas sao chamadosde topologicamente hiperbolicos, por sua dinamica ser similar a dinamica hiperbolica. Umageneralizacao da propriedade de expansividade que tem tido bastante atencao recentemente ea propriedade de n-expansividade.
Dizemos que um homeomorfismo f , definido em um espaco metrico (X, d), e n-expansivo(n 2 N) se existe c > 0 tal que para todo x 2 X o conjunto
(x, c) := y 2 X ; d(fn(x), fn(y)) c, n 2 Z
contem no maximo n pontos diferentes de X. Em poucas palavras, os homeomorfismos n-expansivos podem admitir n orbitas diferentes movendo-se juntas mas nao admitem n + 1.Nessa palestra falarei sobre um trabalho em conjunto com Welington Cordeiro (IMPA), publi-cado recentemente no Journal of Di↵erential Equations, onde analisamos a dinamica de home-omorfismos n-expansivos com a propriedade de sombreamento, definidos em espacos metricoscompactos.
167
Comportamento assintotico de ciclos limites em sistemas de
Lienard
Pedro T. CardinUniversidade Estadual Paulista – FEIS/UNESP (Brazil)
Douglas D. NovaesUniversidade Estadual de Campinas – UNICAMP (Brazil)
Abstract
Consideramos famılias a um–parametro 2 R de equacoes de Lienard. Estudamos aexistencia e unicidade de solucoes periodicas para todos os valores positivos de e, principal-mente, estudamos o comportamento assintotico de tal solucao periodica para valores pequenose grandes de . Para provar nosso principal resultado usamos a teoria de oscilacoes de rela-xamento e a teoria da media. Mais especificamente, a teoria de oscilacoes de relaxamento eapropriada para estudar as solucoes periodicas para valores grandes de e a teoria da mediapara valores pequenos de . Em particular, nossas hipoteses nos permitem estabelecer um linkentre essas duas teorias.
168
17 Sessao tematica: Teoria da Aproximacao
169
Programacao das palestras
Sessao especial: Teoria da Aproximacao
Segunda-feira - Sala 224
14:00 - 14:45 T. JordaoK-funcionals of fractional order: characterizations by the rate ofapproximation of operators
14:45 - 15:30 J. Borrego-MorellAn asymptotic expansion for the eigenpolynomials of a class oflinear differential operators
16:00 - 16:45 J. C. FerreiraOn positive and trace class integral operators
16:45 - 17:30 A. C. ChavezBounding Sn(t) on the Riemann hypothesis
17:30 - 18:15 R. L. B. StabileRelations between Kolmogorov n-Widths and Entropy Numbersof Sets of Smooth Functions on the Torus Td
Terca-feira - Sala 124
10:30 - 11:15 M. H. CastroUpper bounds for eigenvalues of positive integral operators on the sphere
11:15 - 12:00 F. R. RafaeliUma Revisita ao Teorema de Markov e Variacoes
16:00 - 16:30 J. C. GuellaFrom Schoenberg coefficients to Schoenberg functions: a unifying framework
16:30 - 17:15 S. W. NoorA Bergman space approach to the Riemann hypothesis
17:15 - 18:00 E. CarneiroFrontiers of Fourier analysis, approximation theory and number theory:recent progress and open problems
170
Bounding Sn(t) on the Riemann hypothesis
Andres Chirre ChavezIMPA - Instituto Nacional de Matematica Pura e Aplicada (Brazil)
Abstract
Let S(t) denote the argument of the Riemann zeta-function at the point 1/2+it. Let Sn(t)be the n-th antiderivative of S(t) (adding a suitable constant cn at each step). In 1924, J.Littlewood established, under the Riemann hypothesis, that
Sn(t) log t
(log log t)n+1,
and this estimate has never been improved in its order of magnitude over the last 92 years. Thee↵orts have focused in improving the implicit constant in this estimate. In this talk we willshow how to obtain the best (up to date) form of all of these estimates. This involves the useof certain special entire functions of exponential type. This is an aplication of approximationtheory in analytic number theory. This is a joint work with Emanuel Carneiro (IMPA).
171
Frontiers of Fourier analysis, approximation theory and
number theory: recent progress and open problems
Emanuel CarneiroIMPA (Brazil)
Abstract
In this talk I will present some optimization problems in Fourier analysis and approxima-tion theory that appear in connection with other fields, such as number theory and partialdi↵erential equations. These include, for instance, approximations by bandlimited functions,sharp inequalities for the fractional laplacian, optimal sphere packings and sharp root uncer-tainty principles. I intend to present some of the recent progress in these problems and leavean extensive list of open problems that I find particularly interesting.
172
Uma Revisita ao Teorema de Markov e Variacoes
Fernando Rodrigo RafaeliUniversidade Federal de Uberlandia
Abstract
Nesta breve revisao didatica discutiremos novos e classicos resultados sobre monotonicidadee entrelaamento de zeros, no contexto de polinomios ortogonais, com especial atencao a obrade A. Markov cuja origem e de 1886.
173
An asymptotic expansion for the eigenpolynomials of a class
of linear di↵erential operators
J. Borrego–MorellUniversidade Federal do Rio de Janeiro, Polo de Xerem (Brasil)
Abstract
Consider a M–th order linear di↵erential operator
L(M) =MX
k=0
k(z)dk
dzk,
where M is a monic complex polynomial such that deg[M ] = M and kM1k=0 are complex
polynomials such that deg[k] k, 0 k M 1. We obtain an asymptotic expansion forthe eigenpolynomials of L(M) on some compact subsets of C.
174
On positive and trace class integral operators
J. C. FerreiraFederal University of Alfenas (Brazil)
Abstract
Let K : H ! H being a compact linear operator on a Hilbert space H. It follows fromspectral theorem that there are n and n, orthonormal sets of H, such that
K(f) =1X
n=1
snhf,niH n, f 2 H,
where s2n are eigenvalues of K K, with K(n) = sn n and K( n) = snn. If
kKkS1 =1X
j=1
sn(K) < 1,
then we say that K is in the trace class S1(H) and the trace
Tr(K) =X
2BhK(),iH1 < 1,
for one (and any) orthonormal basis B, is well defined (Barry Simon - Trace Ideals and TheirApplications, Mathematical Surveys and Monographs, 120, American Mathematical Society(2005)). In particular, if K is also selfadjoint then
Tr(K) =1X
n=1
n
and we can define the determinant
Det(I + K) =1Y
n=1
(1 + n),
where n are eigenvalues of K (counting multiplicities) and I is the identity operator.In this work we discuss some conditions to ensure that a positive integral operator has
a trace, by using reproducing kernel Hilbert space theory (please see J. C. Ferreira and V.A. Menegatto - Positive definiteness, reproducing kernel Hilbert spaces and beyond, Ann. ofFunct. Anal., 4, 64-88 (2013)).
In fact we analyze operators K : L2(X, ) ! L2(X, ), given by
K(f)(x) =
Z
X
K(x, y)f(y) d(x), f 2 L2(X, ), x 2 X,
for which the kernel K is positive definite, that is,
nX
i,j=1
cicjK(xi, xj) 0,
whenever n 1, x1, x2, . . . , xn are in X and c1, c2, . . . , cn are complex numbers. Under thiscontext, we like to know what conditions ensures the trace formula
Tr(K) =
Z
X
K(x, x)d(x).
This results is related to papers: Brian Je↵eries - Traceability of Positive Integral Operator.Integr. Equ. Oper. Theory, 84, 419-428 (2016) and J. C. Ferreira, at all - On the nuclearityof integral operators, Positivity, 13, 519-541 (2009)).
175
From Schoenberg coecients to Schoenberg functions: a
unifying framework
J.C. Guella, V. A. MenegattoICMC-USP - Sao Carlos
Abstract
A function K mapping a product space Z Z into C is termed a positive definite kernelon Z if
nX
µ,=1
cµcK(zµ, z) 0,
for n 1, distinct points z1, z2, . . . , zn in Z and complex scalars c1, c2, . . . , cn. The strictpositive definiteness of a positive definite kernel K demands that the inequalities above bestrict when at least one of the ci is nonzero.
In 1942, I. J. Schoenberg published his seminal paper characterizing the continuous andisotropic positive definite kernels on spheres: f : [1, 1] ! C is continuous and the kernel(x, y) 2 Sd Sd ! f(x · y) is positive definite, where · is the usual inner product in Rd+1 if,and only if,
f(t) =1X
k=0
adkP d
k (t), t 2 [1, 1],
where all the coecients adk are nonnegative, P d
k denotes the usual Gegenbauer polynomial ofdegree k attached to the rational number (d 1)/2 and
Pk ad
kP dk (1) < 1.
The characterization for strict positive definiteness in the case d 2 appeared in 2003 in awork of Menegatto et all: a positive definite kernel from Schoenberg’s class is strictly positivedefinite if, and only if, the index set k : ad
k > 0 contains infinitely many even and infinitelymany odd integers.
We were able to generalize this two results in the following way:Let X be a non empty set. For a kernel ((u, x), (v, y)) 2 (X Sd)2 ! f(u, v, x · y), where
each function f(u, v, ·) is continuous in [1, 1], the following assertions are equivalent:(i) The kernel is positive definite. ;(ii) Each function f(u, v, ·) has a series representation in the form
f(u, v, s) =1X
k=0
adk(u, v)P d
k (s), s 2 [1, 1],
where adk(u, v) 2 C, k 2 Z+, u, v 2 X, each kernel (u, v) 2 X X ! ad
k(u, v) is positive
definite and1X
k=0
adk(u, u)P d
k (1) < 1, u 2 X.
And on the strict positive definiteness setting:Let ((u, x), (v, y)) 2 (XSd)2 ! f(u, v, x·y) be a kernel as before. The following assertions
are equivalent:(i) The kernel is strictly positive definite;(ii) If u1, u2, . . . , up are distinct points in X and c is a nonzero vector in Cp, then
k 2 Z+ : cthad
k(uµ, u)ip
µ,=1c > 0
contains infinitely many even and infinitely many odd integers.
176
Upper bounds for eigenvalues of positive integral operators
on the sphere
M. H. Castro
Universidade Federal de Uberlandia
T. JordaoICMC-Universidade de Sao Paulo
Partially supported by FAPEMIG (APQ-00474-14) [email protected]
Abstract
We get sharp estimates for eigenvalues and singular values of positive integral operatorsgenerated by infinitely many times Laplace-Beltrami di↵erentiable kernels on the unit spherein Rm+1.
177
Relations between Kolmogorov n-Widths and Entropy
Numbers of Sets of Smooth Functions on the Torus Td
R. L. B. StabileIFSP (Brazil)
S. A. TozoniUNICAMP (Brazil)
AbstractLet A be a convex, compact and centrally symmetric subset of a Banach space X. The
Kolmogorov n-width of A in X, denoted by dn(A, X) is defined by
dn(A, X) = infXn
supx2A
infy2Xn
kx ykX ,
where Xn runs over all subspaces of X of dimension n.The nth entropy number of A in X, denoted by en(A, X), is defined as the infimum of all
positive such that there exist x1, ..., x2n1 in X satisfying
A 2n1[
k=1
(xk + BX),
where BX denote the closed unit ball of X.If Y is other Banach space and If T 2 L(X, Y ) is a compact operator, then the Kolmogorov
n-width dn(T ) and the nth entropy number en(T ) are defined as
dn(T ) = dn(T (BX), Y ) and en(T ) = en(T (BX), Y ),
respectively.In this work, we obtain estimates for Kolmogorov n-width and for entropy numbers of
multiplier operators (1) =|k|(ln |k|)
k2Zd and (2) =
ne|k|r
ok2Zd
for > 0,
0 < r 1 and 0, with |k| =k21 + · · · + k2
d
1/2.
We have that (1)Up are sets of finitely di↵erentiable functions on Td, in particular, (1)Up
are Sobolev-type classes if = 0, and (2)Up are sets of infinitely di↵erentiable (0 < r < 1) oranalytic (r = 1) functions on Td, where Up denotes the closed unit ball of Lp(Td).
It was proved in (Kushpel, A., Stabile, R. L. B., Tozoni, A., 2014) and (Stabile, R. L. B.,Tozoni, A., 2016) (submitted) that, for 2 p, q < 1, 0, > 0 and 0 < r 1,
dn((1)Up, Lq) en((1)Up, Lq) n/d(ln n), dn((2)Up, Lq) eRnr/d
andek((2)Up, Lq) eCkr/(d+r)
,
where R = d(d/2)/2d/2
r/d
and C = d/(d+r)
(d + r)d(d/2)(ln 2)
2rd/2
r/(d+r)
.
It was discovered in (Birman, M. S., Solomyak, M. Z., 1967) that on the circle S1 = T1,for Sobolev (or Sobolev type) classes W
p = W p (T1), that
dn(W p , Lq) en(W
p , Lq).
We show that, for 2 p, q < 1 and for all classes of finitely di↵erentiable functions (1)Up onthe torus Td, in particular for Sobolev type classes, the Kolmogorov n-width an the entropynumbers have the same asymptotic behavior.
However, for the classes of infinitely di↵erentiable (0 < r < 1) or analytic (r = 1) functions(2)Up on the torus, the Kolmogorov n-widths and the entropy numbers are essentially di↵erentand your relationship is given by
dn((2)Up, Lq) en((2)Up, Lq).
178
A Bergman space approach to the Riemann hypothesis
S. Waleed NoorUniversity of Campinas (Brazil)
Abstract
The Nyman-Beurling criterion is a reformulation of the Riemann Hypothesis that askswhether a particular family of functions is complete in the Hilbert space L2([0, 1]). In thistalk, the objective is to show that much of Nyman-Beurling theory and the current researchactivity around it may be approached very naturally via Bergman space theory, i.e via aHilbert space of holomorphic functions on the open unit disk in C.
179
K-funcionals of fractional order: characterizations by the rate
of approximation of operators
T. JordaoUniversity of Sao Paulo (Brazil)
Xingping SunMissouri State University (USA)
Abstract
It is designed two types of a general type of spherical mean operators, depending on a realnumber as parameter, and they are employed to approximate Lp class functions. One of themis very useful theoretically, it generalizes a class of combinations of average operators foundedin Dai & Ditzian (2004). The second one brings us a more useful configuration in the sense ofapplications, and by applications here we mean computacional calculations.
We show that optimal orders of approximation of these operators are achieved via appro-priately defined K-functionals of fractional orders. Asymptotic relations between the rate ofapproximation of the new operators and the K-functional of fractional order were stablished.When the parameter we work with is taken as a natural number the general type of sphericalmean operator, the K-funcional and also the result relating such objects turn out the same asin Dai & Ditzian (2004) which introduces a class of “multi-layered” spherical mean operators.More details can be found in Jordao & Sun (2015).
180
18 Sessao tematica: Trends in Geometry and Topology
181
Programacao das palestras
Sessao especial: Trends in Geometry and Topology.
Terca-feira Sala 224
09h - 10h Palestra plenaria 310h - 10h30 Pausa do cafe10h30 - 11h Mike Deutsch
Correspondences of holomorphic curves in the Mobius group11h - 11h30 Pedro Frejlich
Variations on the pullback construction11h30 - 12h Roberto Mossa
Symplectic capacities of Hermitian symmetric spaces12h - 14h Almoco14h - 15h30 Mesa-redonda Genero15h30 - 16h Pausa de cafe16h - 17h Jason Lotay
Laplacian flow in G2 geometry17h - 18h Nicolau Sarquis
The space of min-max hypersurfaces for analytic metrics with Ric > 0
Quarta-feira Sala 224
10h - 11h Pedram HekmatiGauge theory on foliated manifolds
11h - 12h Mario Garcia FernandezThe Strominger system of partial differential equations
12h - 14h Almoco14h - 15h Palestra plenaria 415h - 15h30 Pausa de cafe15h30 - 16h Viviana del Barco
T-duality and mirror symmetry on nilmanifolds16h - 16h30 Lazaro Rodrıguez
G2 instantons over Milnor’s quintic links16h30 - 17h Rafael Leao
Immersions and spinors17h - 17h30 Encerramento
182
Immersions and spinors
Rafael LeaoUnicamp (Brazil)
Abstract
Title: Immersions and spinorsAbstract: In this talk we will show some recent results about how spinors that satisfies certainDirac equation can be used to construct isometric immersions of manifolds on Rn. This typeof construction can be understood as a generalization of the Weierstrass map for complexsurfaces.
183
Variations on the pullback construction
Pedro FrejlichPUC-Rio (Brazil)
Abstract
In this talk, we explain how the pullback construction is the central feature in realizationand normal form problems in the realm of Dirac manifolds (and, more generally, Lie alge-broids), and how it leads to some new operations in the category of Dirac manifolds. Jointwork with I. Marcut (U. Radboud, Nijmegen)
184
Correspondences of holomorphic curves in the Mobius group
Mike DeutschUFRJ (Brazil)
Abstract
The classical Lie transform establishes a correspondence between null curves in the 3-quadric Q3 and contact curves in projective 3-space CP3, a sort of codimension 2 generaliza-tion of the usual duality for plane algebraic curves. In this talk we describe an alternative suchcorrespondence of gauge theoretic origin obtained by viewing Q3 as the conformal compacti-fication of SL(2, C) and CP3 as the contact compactification of PSL(2, C), the Mobius groupof the 2-sphere.? Although it breaks both the Sp(4, C) and Plucker-relation symmetries, itdoes preserve enough symmetry to build interesting projective invariants, the utility of whichwe exhibit by constructing surfaces and monopoles in real hyperbolic 3-manifolds.
185
G2 instantons over Milnor’s quintic links
Lazaro RodrıguezUnicamp (Brazil)
Abstract
Every quintic Milnor link K on a real 9-sphere is a nontrivial Sasakian circle fibration overa Calabi-Yau 3-fold. It admits a canonical co-closed G2 structure reflecting locally the Calabi-Yau geometry, which induces a natural Yang-Mills theory with many important features ofthe torsion-free case, such as a Chern-Simons formalism, topological energy bounds and adeformation theory of index 0. In particular, we obtain G2 instantons on bundles over K fromHermitian Yang-Mills connections over the quintic Calabi-Yau, thus relating in a concreteexample the well-known Donaldson-Thomas invariant with a conjectural G2 instanton count(joint work with O.Calvo-Andrade, H.N.Sa Earp)
186
The space of min-max hypersurfaces for analytic metrics with
Ric > 0
Nicolau SarquisUniversity of Toronto (Canada)
Abstract
We show that the space of min-max minimal hypersurfaces is non-compact when the mani-fold has an analytic metric, positive Ricci curvature and dimension 3 ≤ n ≤ 7. When combinedwith the recent work of F.C. Marques and A. Neves, we then deduce some new properties re-garding the infinitely many minimal hypersurfaces they found. When the metric is bumpyinstead of analytic we show that there are minimal hypersurfaces with arbitrarily large index+ area.
187
T-duality and mirror symmetry on nilmanifolds
Viviana del BarcoUnicamp/Rosario
Abstract
Mirror symmetry predicts that for certain pairs of manifolds M and M∨, the symplecticgeometry of M is related with the complex geometry of M∨. There are some formalizationsof this idea: homological mirror symmetry introduced by Kontsevich at categorical level; anda more geometrical description presented by Strominger, Yau and Zaslow via T -duality.
Gualtieri introduced in his thesis generalized complex structures, which has complex andsymplectic geometries as particular cases. This suggests that mirror symmetry can be under-stood in this context. In fact, in a joint work with Cavalcanti they describe how to transportgeneralized complex structures between T -dual pairs, obtaining clear examples of mirror sym-metry.
In this work we study T -duality in the context of nilmanifolds. These are compact manifoldscarrying a natural torus bundle structure. The gerbe and geometrical structures to be mappedare considered to be invariant by the nilpotent Lie group action so it allows us to work at a Liealgebra level. We present explicit examples that permits us to give a new understanding tothe subject. This is an ongoing research, joint with Leonardo Soriani Alves and Lino Gramafrom IMECC-UNICAMP.
188
Laplacian flow in G2 geometry
Jason LotayUniversity College London (UK)
Abstract
A key challenge in Riemannian geometry is to find Ricci-flat metrics on compact manifolds.All non-trivial examples of such metrics have special holonomy, and the only special holonomymetrics which can occur in odd dimensions must be in dimension 7 and have holonomy G2. Iwill describe a proposed geometric flow method for finding metrics with holonomy G2, calledthe Laplacian flow, and recent progress on this flow. This is joint work with Yong Wei (ANU).
189
The Strominger system of partial differential equations
Mario Garcia-FernandezCSIC (Spain)
Abstract
The Strominger system of partial differential equations has its origins in supergravity inphysics, and it was first considered in the mathematics literature in a seminal paper by Liand Yau. The mathematical study of this PDE has been proposed by Yau as a natural gener-alization of the Calabi problem for non-Kahlerian complex manifolds, and also in relation toReid’s fantasy on the moduli space of projective Calabi-Yau threefolds. There is a conjecturalrelation between the Strominger system and conformal field theory, which arises in a certainphysical limit in compactifications of the heterotic string theory.
In this talk I will give a brief introduction to this topic, describing the links with thetheory of balanced metrics in hermitian geometry, the Hermite-Yang-Mills equations, and itsorigins in physics. I will also cover some recent developments in the moduli problem andthe interrelation of the Strominger system with generalized geometry, via the cohomologicalnotion of string class.
190
Symplectic capacities of Hermitian symmetric spaces
Roberto MossaUniversidade Federal de Santa Caterina (Brazil)
Abstract
In the first part of the talk I will recall the definitions of Gromov width and of Hermitiansymmetric spaces and their main properties. In the second part after recalling the definition ofa positive Jordan triple system I will compute the Gromov width of the Hermitian symmetricspaces. The results and the computations are part of a joint work with A.Loi, F.Zuddas (J.Symp. Geom. 2015).
191
Gauge theory on foliated manifolds
Pedram HekmatiIMPA (Brazil)
Abstract
The Yang-Mills instantons admit several natural extensions to higher dimensions. In thistalk I will first review a few examples and then focus on contact instantons, which arise ascritical points of a super Yang-Mills theory on 5-dimensional contact manifolds. The modulispace is controlled by the geometry transverse to the Reeb foliation and in particular, thedimension is given by the index of a transversally elliptic complex.
192
193
19 Posters apresentados
1. Analice Costacurta Brandi
2. Darwin Castillo
3. Juliana Honda Lopes
4. Elaine Silva
5. Larissa Meirelles
6. Jose Vanterler
7. Jose Lucas Pereira Luz
8. Juilson Paiva
9. Juan Carlos Ortiz Chata
10. Juliana Martins
11. Graziane Teodoro
12. Giane Rampasso
13. Gilcilene Sanchez de Paulo
14. Guilherme Aguilar
15. Marcel Moro
16. Antonio Marcos Duarte
17. Llohann Speranca
18. Luverci Ferreira
19. Paula Olga Gneri
20. Clayton Silva
21. Raquel Ribeiro
22. Charles Ferreira
23. Beatriz Liara Carreira
24. Eduardo Boer
25. Rodolfo Begiato
26. Anderson Oliveira
27. Robson Araujo
28. Anderson Gonzaga
29. Altair Tosti
30. Alessandro Bagatini
194
31. Adriana Nicoli
32. Samara Costa
33. Saulo Luiz
34. Solange Santos
35. Wellington Santos
36. Evelin Manoel
37. Tiago Yuzo Miyaoka
38. Gislaine Pericaro
39. Tatiane cazarin da Silva
40. Patricia Massae Kitani
41. Atila Correia
42. Esau Alves de Sousa
43. Stefania Jarosz
44. Oscar Armando Hernandez Morales
45. Jhoel Sandoval
46. Roberto Alonso
47. Eric Ossami
48. Camila Isoton
49. Cristian Schmidt
50. Tiago Luiz Ferrazza
51. Alison Melo
52. Luiz Carlos Santos Junior
53. Aydee Lopez Santana
54. Claudemir Fideles Bezerra Junior
55. Gabriel Gomes Figueiredo
56. Rafael Sterza
57. Karen Oliveira
58. Laison Furlan
59. Leodan Torres
60. Ana Flavia Lima
61. Letıcia dos Santos Silva
62. Felipe Felix Souto
63. Jessica Ventura da Silva
195
Analise de Estabilidade do Escoamento de Poiseuille
Bidimensional
A. C. Brandi, E. S. GervazoniUniversidade Estadual Paulista “Julio de Mesquita Filho” (Brazil)
Resumo
A dinamica de fluidos computacional e uma area com relevante interesse industrial nas areasquımicas, alimentıcias e petrolıferas. Entre os diversos problemas surgidos nessa area esta otratamento de escoamentos de fluidos nao-Newtonianos, em especıfico, os fluidos viscoelasticos.Desse modo, existe um grande interesse em desenvolver metodos numericos capazes de simularescoamentos de fluidos viscoelasticos para poder prever, a custos baixos e com bons resultados,o comportamento do escoamento de um fluido durante o processo industrial. Porem, essa naoe uma tarefa facil pois nestes problemas as equacoes constitutivas que modelam um fluidoviscoelastico sao equacoes matematicas complexas e de difıcil tratamento em contornos dodomınio computacional. Ha muitas aplicacoes cientıficas e industriais em que a estabilidade doescoamento laminar e a transicao para a turbulencia sao relevantes. Sendo assim, e importanteinvestigar a fısica da estabilidade e a transicao laminar-turbulenta. O modelo matematicoadotado para o fluido nao-Newtoniano do tensor extra-tensao e o Oldroyd-B. Nesse contexto,este trabalho investiga a taxa de crescimento de propagacao das ondas de Tollmien-Schlichtingpara o escoamento de Poiseuille. Para isso, sera utilizada a Simulacao Numerica Direta paraanalisar a estabilidade de escoamentos de fluidos viscoelasticos e comparar com a de fluidosNewtonianos, a fim de contribuir para as dificuldades encontradas neste problema. As equacoesde Navier-Stokes, juntamente com a equacao constitutiva Oldroyd-B sao desenvolvidas usandoo modelo numerico de diferencas finitas. Os resultados mostram que as taxas de amplificacaoobtidas diferem do fluido Newtoniano a medida que o numero de Weissenberg aumenta. A fimde avaliar as taxas maximas de amplificacao, diferentes valores de parametros adimensionaissao testados para escoamentos de fluidos Newtonianos e nao-Newtonianos.
196
Resultados de existencia de solucoes para problemas elıpticos
assintoticamente lineares
Marcos Tadeu de Oliveira Pimenta, Anderson dos Santos GonzagaUniversidade Estadual Paulista (Brazil)
Palavras chaves: Metodos variacionais, problemas assintoticamente lineares, equacoes diferenciaisparciais elıpticas.
Resumo
Neste trabalho procuramos estabelecer a solucao nodal para uma versao estacionaria daequacao de Schrodinger nao-linear, com nao-linearidade do tipo assintoticamente linear:
−∆u+ λu = f(u) (1)
em RN , para N ≥ 3 e λ > 0 com as hipoteses:
(f1) f ∈ C(R,R);
(f2) f(−t) = −f(t), para todo t ∈ R;
(f3) limt→0
f(t)t
= 0;
(f4) existe s > 0 tal que limt→+∞
f(t)t
= 1s
e f(t)t≤ 1
s, para todo t ∈ R;
(f5) f(t)t
e uma funcao crescente para todo t > 0;
e ainda assumimos uma condicao tipo nao-quadratica.
limt→+∞
[f(t)t− 2F (t)] = +∞ e [f(t)t− 2F (t)] ≥ 0, para todo t ∈ R,
onde
F (t) =
∫ t
0
f(z)dz.
A condicao de interesse aqui esta em provar a existencia de uma solucao mudanca de sinalpara s ∈ (0, 1
λ) que sao radialmente simetricas. Comecamos por meio de uma colagem de duas
solucoes, uma em uma bola de raio fixo e a outra em um domınio exterior ilimitado. Nossoproblema provem do fato de minimizar o problema e provar que a energia mınima, e obtidapor algumas funcoes nesse subconjunto e, em seguida provamos que a restricao e natural, ouseja, que sao realmente solucoes da equacao, pode-se tambem obter informacoes sobre o seuındice de Morse. A fim de contornar a possıvel falta de regularidade deste, e crucial aplicar oLema de Deformacao e usar o Teorema de Miranda.
197
Poster para II Congresso Brasileiro de Jovens Pesquisadores
em Matematica Pura e Aplicada
Alison M. V.D. L. MeloUnivasf (Brasil)
Keywords: Folheacoes, Fluxos Estocasticos, Orientacao Transversal.
Thematic session (optional): Sistemas Dinamicos.
Resumo
Topologia das Folheacoes e Decomposicao de Fluxos Estocasticos.
A apresentacao sera baseada no artigo Topology of foliations and decomposition of stochas-tic flows de Melo, Morgado e Ruffino (2016). Neste trabalho consideramos uma variedadediferenciavel M munida de duas folheacoes H (horizontal) e V (vertical) transversais entre si.Isto e, em cada ponto x de M existe uma folha horizontal e uma folha vertical passando porx e que se intersectam transversalmente em x.
Considere um fluxo estocastico de difeomorfismos φt, em M , que pode ser dado, por exem-plo, pela solucao de uma equacao diferencial estocastica (e.d.e.). De fato, estamos consi-derando e.d.es de Stratonovich da seguinte forma: dxt =
∑mr=1Xr dW r
t , onde os Xr saocampos diferenciaveis sobre M e W r sao movimentos brownianos independentes. O fluxode difeomorfismos φt gerado pela solucao desta equacao da origem a um sistema dinamicoestocastico.
No trabalho Decomposition of stochastic flows with automorphism of subbundles compo-nent. Stochastics and Dynamics (2012), Catuogno, Silva e Ruffino, demonstram que quandoM e compacta e possıvel obter uma decomposicao para um fluxo estocastico φt, dado poruma e.d.e., de modo que uma das componentes preserva a folheacao horizontal, e a outracomponenente preserva a folheacao vertical. Mais precisamente φt = ξt ψt, onde, ξt fixa cadafolha de H e ψ fixa cada folha de V ate um tempo de parada τ . Decomposicoes de naturezasemelhante sao apresentadas em Stochastic flows and stochastic differential equations (1988)de H. Kunita, em Decomposition of stochastic flows and Lyapunov exponents (2000) de M.Liao, por exemplo.
A motivacao inicial para este tipo de decomposicao e o caso em que o fluxo originalmentepreserva energia e portanto as trajetorias estao contidas em nıveis de energia. Depois desofrer uma perturbacao por um campo transversal, o fluxo perde o comportamento folheado,mas uma decomposicao do tipo proposto permite estudar separadamente a componente quepreserva energia e a componente transversal.
No trabalho a ser apresentado, discutimos os aspectos topologicos deste tipo de decom-posicao de fluxos. Por exemplo, uma das condicoes nessessarias para que φt seja decomponıvele que para todo x ∈ M e para todo t, tenhamos que φt(x) ∈ A(x), onde A(x) e o conjuntodos pontos de M que podem ser atingidos, partindo-se de x e tomando-se um caminho quee a concatenacao de um caminho vertical (contido em uma folha vertical) com um caminhohorizontal (contido em uma folha horizontal). Nosso principal resultado afirma o seguinte:Teorema: Quando H e transversalmente orientavel, um fluxo e decomponıvel se e somentese preserva a orientacao transversal de H.
Tal caracterizacao expoe a profunda conexao entre a topologia das folheacoes e a decom-posicao de fluxos estocasticos. Como aplicacao deste resultado obtemos uma condicao sobresubdeterminantes do fluxo linearizado associado a φt suficiente para que este seja decom-ponıvel. A saber,Proposicao: Suponha que a folheacao horizontal e transversalmente orientavel. Entao se o
fluxo linearizado Yt satisfazn−k+1,...,n
detl1,...,lk
[I : DX : i]·l1,...,lk
detn−k+1,...,n
(Yt) = 0 para todo n−k+1 ≤
i ≤ n e todo (l1, ..., lk) exceto talvez (n−k+1, ..., n). Entao φt e decomponıvel para todo t ≥ 0.198
Grupo de Lodha-Moore e Conjectura de von Neumann-Day
A. S. O. TostiUniversidade Estadual de Campinas (Brasil)
Abstract
A Conjectura de von Neumann-Day afirma que um grupo G e nao-mediavel se, e somentese, G contem um subgrupo que seja grupo livre sobre dois geradores. Enquanto trabalhavacom o Paradoxo de Banach-Tarski, John von Neumann definiu o conceito de grupo mediavel edemonstrou que nenhum grupo mediavel contem um grupo livre de posto dois. A conjecturafoi demonstrada ser falsa por Alexander Ol’shanskii (Uspekhi Matem. Nauk (1980), 35 (4),199-200).
O grupo de Lodha-Moore foi introduzido por Yash Lodha e Justin Moore (Groups, Geome-try, and Dynamics 10.1 (2016): 177-200). Tal grupo e nao-mediavel, finitamente apresentado(tres geradores e nove relacoes) e livre de torcao, alem de ser o contraexemplo mais simplesconhecido para a conjectura. Para definı-lo, consideremos a aplicacao real a(t) := t + 1 e osseguintes homeomorfismos sobre R:
b(t) :=
t, se t ≤ 0,t
1−t, se 0 ≤ t ≤ 1
2,
3− 1t, se 1
2≤ t ≤ 1,
t+ 1, se t ≥ 1.
e c(t) :=
2t1+t
, se 0 ≤ t ≤ 1,
t, caso contrario.
O grupo G0 de Lodha-Moore e gerado por a(t), b(t) e c(t), isto e, G0 := 〈a(t), b(t), c(t)〉.Como ja mencionado, foi demonstrado seguinte resultado sobre esse grupo:
Teorema 0.1 (Lodha-Moore, 2016) O grupo G0 e nao-mediavel e finitamente apresen-tado. Alem disso, G0 nao contem subgrupos livres nao-abelianos.
199
Determinacao de grupos fuchsianos casados a constelacoes de
sinais atraves da uniformizacao de curvas algebricas planares
Anderson Jose de Oliveira, Reginaldo Palazzo JuniorUniversidade Estadual de Campinas (Unicamp)
[email protected], [email protected]
Palavras-chave: equacoes diferenciais fuchsianas, serie de Farey, transformacoes de Mobius, su-perfıcies, grupos fuchsianos.
Resumo
Uma equacao diferencial do tipo fuchsiana e caracterizada por conter pontos regularese/ou pontos singulares regulares no plano complexo. Quando todos os pontos singulares saoregulares e distintos a superfıcie associada e compacta e orientavel. O quociente entre duassolucoes linearmente independentes conduz a uma transformacao linear ou uma transformacaode Mobius. O conjunto formado por todos os quocientes de solucoes linearmente independentessatisfazendo determinadas condicoes constitui um grupo. Whittaker estabelece uma interes-sante relacao entre o coeficiente da derivada de maior ordem de uma equacao diferencial dotipo fuchsiana com uma superfıcie de Riemann cujas singularidades estao associadas a umacurva algebrica do tipo y2 = x2g+1−1, onde g denota o genero da superfıcie, a qual pretende-seuniformizar. Os demais coeficientes das derivadas de menor ordem sao polinomios racionaisque devem satisfazer determinadas condicoes para que o conjunto dos quocientes das solucoeslinearmente independentes forme um grupo de transformacoes de Mobius hiperbolicas, de ondese extrai os geradores desse grupo e, consequentemente a identificacao do grupo fuchsiano as-sociado. O objetivo deste trabalho e apresentar um procedimento para a determinacao dogrupo fuchsiano associado a constelacao de sinais a ser utilizada na transmissao da informacaobem como a sua contrucao. Isso se verificara atraves do mergulho do grafo completo biparti-cionado associado ao canal discreto sem memoria cujo mergulho possibilitara a determinacaonao so do genero da correspondente superfıcie de Riemann como da curva algebrica planarassociada a essa superfıcie atraves dos elementos da sequencia de Farey (singularidades). Auniformizacao dessa curva algebrica planar atraves da equacao diferencial fuchsiana fornecerao grupo fuchsiano associado a constelacao de sinais e com isso possibilitando a construcao dareferida constelacao de sinais.
200
A topological lower bound for the energy of a unit vector field
on a closed hypersurface of the Euclidean space. The
3-dimensional case
A. V. Nicoli, F. G. B. BritoUniversidade Federal do ABC (Brasil)
A. GomesUniversidade do Estado de So Paulo (Brasil)
Abstract
Let M be a closed oriented 3-dimensional immersed hypersurface of the Euclidean space.Consider ν : M → S3 as its normal map, and let ∇ and 〈·, ·〉 be the Levi-Civita connec-tion and the induced Riemannian metric on M , respectively. For a smooth unit vectorfield ~v on M , its energy is defined as the energy of the map ~v : M → TM . Followinghttps://arxiv.org/abs/1609.04900arXiv:1609.04900, we deduce a lower bound for the energy
functional in terms of the degree deg(ν) and S := maxx∈M||S(~v)x||, where S is the Wein-garten operator of M . It reads
E(~v) > 1
2Svol(S3)deg(ν) +
3
2vol(M).
201
Caracterizacao Cohomologica de uma famılia de Fibrados
Vetoriais sobre a Hipersuperfıcie Quadrica Q2l+1
A. L. Santana, S. Marchesi (Orientador)Instituto de Matematica, Estatıstica e Computacao Cientıfica
Universidade Estadual de Campinas (Brasil)
Palavras Chave: Fibrados vetoriais, Fibrados cohomologia, Posto baixo, Monadas, Quadricas,Caracterizacao cohomologica.
Sessao Tematica: Geometria algebrica.
Abstract
Um problema bem conhecido na geometria algebrica e fornecer exemplos de fibrados vetori-ais sobre uma variedade algebrica com posto baixo (posto menor que a dimensao da variedade).Neste trabalho construimos uma famılia de fibrados vetoriais de posto baixo sobre uma hiper-superfıcie quadrica de dimensao ımpar, utilizando o fibrado cohomologia associado a umamonada especial. Com este intuito, fixamos V um C−espaco vetorial (2l + 3)−dimensionale consideramos Q2l+1 ⊂ P2l+2 = P(V ) a hipersuperfıcie quadrica suave de dimensao ımpar.Sejam k, c inteiros positivos, focamos no estudo de fibrados vetoriais E sobre Q2l+1 que saodefinidos como cohomologia de uma monada da forma
M• := 0→ OcQ2l+1(−k)
α→ O2l+2cQ2l+1
β→ OcQ2l+1(k)→ 0.
No caso particular k = 1, o fibrado E e conhecido e chama-se fibrado linear sobre Q2l+1. Vejapor exemplo [L. Costa and R. M. Miro-Roig, Moduli spaces of linear bundles on hyperquadrics,2015]
Uma vez introduzida a famılia, estudamos as caracterizacoes de tipo cohomologico destesfibrados. Uma caracterizacao cohomologica dos fibrados lineares sobre as hipersuperfıciesquadricas foi primeiramente considerado no artigo [L. Costa and R. M. Miro-Roig, Mon-ads and instanton bundles on smooth hyperquadrics, 2009], estendemos este resultado para1 ≤ k ≤ l. Posteriormente, a caracterizacao cohomologica intrınseca de monadas sobre qual-quer variedade projetiva foi abordada em [P. Macias and H. Soares, Cohomological character-ization of monads, 2014], em particular, mostrou-se outra caracterizacao dos fibrados linearessobre hipersuperfıcies quadricas. Utilizamos Teoria de helices para estender esta ultima car-acterizacao quando 1 ≤ k ≤ l.
202
ANALISE DAS CONDICOES DE OTIMALIDADE DE UM
PROBLEMA DE CONTROLE OTIMO DISCRETO
C. Isoton, L. B. dos SantosUniversidade Federal do Parana (Brasil)
M.A. Rojas-Medar, V. Vivanco-Orellanas,Universidad de Tarapaca, Universidad Catolica de la Santıssima Concepcion (Chile)
Resumo
O estudo das condicoes de otimalidade e um tema importante em Analise Variacional, emOtimizacao e tambem em Controle Otimo. Existe uma gama de artigos na literatura que tra-tam do estudo das condicoes de otimalidade para estes problemas; veja Hilscher, R.; Zeidan, V.(Discrete optimal control: second order optimality conditions, J. Difference Equ. Appl. 2002).Nossa enfase neste trabalho e apresentar condicoes de otimalidade para o problema de controleotimo discreto (PCD) - nao linear, com restricoes de igualdade e desigualdade sobre o estadoe o controle e uma condicao de contorno - para isto usaremos alguns resultados provados emMarinkovic (Optimality conditions in discrete optimal control problems with state constraints,N Func Anal and Opt. 2007). Os resultados obtidos se baseiam na teoria desenvolvida porBoltyanskii (Optimal control of discrete systems, John Wiley-Sons. 1978) e generalizada emArutyunov (Necessary optimality conditions for discrete optimal control problems, Comput.Math. Cybernet. 2005). Isto e feito por meio de uma conveniente reinterpretacao do problema(PCD), cujas funcoes envolvidas sao todas diferenciaveis, como um problema de programacaomatematica, ao qual aplicamos o formalismo de Dubovitskii-Milyutin, veja Girsanov (Lectu-res on mathematical theory of extremum problems, Springer-Verlag. 1972). O conceito deKT-invexidade, que foi introduzido por Martin (The essence of invexity, J. Optim. Th. Appl.1985) para problemas de programacao matematica e posteriormente generalizado por Osuna-Gomez et al. (Invex functions and generalized convexity in multiobjective programming, J.Optim. Th. Appl. 1998) para problemas multiobjetivos, nao somente e interessante para aobtencao das condicoes suficientes de otimalidade, mas tambem nos proporciona uma com-pleta caracterizacao de otimalidade. Neste trabalho, mostra-se que um problema de controleotimo discreto e PM-invexo se e somente se todo processo admissıvel que satisfaz as condicoesdo Princıpio do Maximo Discreto e um processo otimo. Alem disso, apresentamos um teoremadas condicoes suficientes de otimalidade atraves das derivadas de segunda ordem.
203
Metodo de Diferencas Finitas Exponencial Aplicado na
Solucao de Problemas Nao Lineares em Regime Estacionario
B. L. Carreira, A. C. BrandiUniversidade Estadual Paulista “Julio de Mesquita Filho” (Unesp)
Keywords: Diferencas finitas exponencial, problemas estacionarios, equacoes nao lineares.
Thematic session: Analise Numerica e Dinamica de Fluidos Computacional.
Abstract
Problemas estacionarios independem da evolucao temporal e portanto sao justamente aque-les representados por equacoes diferenciais parciais elıpticas. Diversos fenomenos fısicos podemser modelados por esse tipo de equacao e grande parte dessas modelagens envolvem equacoesnao lineares. Alem disso, existe ainda a questao sobre a existencia de metodos analıticos com-patıveis com a equacao envolvida. Quando a abordagem analıtica nao e possıvel recorre-seaos metodos numericos, cujo estudo e aplicacao foram impulsionados justamente pela neces-sidade de obtencao de solucoes aproximadas. O metodo mais antigo e popular para a solucaonumerica de equacoes diferenciais parciais e o metodo de diferencas finitas.
Este trabalho consiste no estudo de um novo metodo, conhecido como metodo de diferencasfinitas exponencial de alta ordem para a solucao numerica de problemas estacionarios mode-lados por equacoes nao lineares. Considerando a equacao de Poisson bidimensional:
∇2u =∂2u
∂x2+∂2u
∂y2= f(x, y), (1)
discretizada numa regiao quadrada Ω = (x, y) : a 6 x 6 b, a 6 y 6 b com condicao decontorno do tipo Dirichlet u(x, y) = g(x, y) em ∂Ω, sendo ∂Ω a fronteira da regiao Ω.
A princıpio o metodo de diferencas finitas exponencial utiliza nove pontos e e dado pelaequacao (2):
a0(ui+1,j + ui−1,j + ui,j+1 + ui,j−1) + a1(ui+1,j+1 + ui−1,j+1 + ui+1,j−1 + ui−1,j−1) +
+a2ui,j = b0h2fi,je
φ(h), (2)
sabendo que a0, a1, a2 e b0 sao constantes, a variavel h representa o espacamento nas direcoesx e y, e φ trata-se de uma funcao diferenciavel.
O desenvolvimento do metodo e baseado em Serie de Taylor, Serie de Mac Lauren e ex-pansao exponencial. Expandindo a equacao (2) e realizando as devidas substituicoes, descobre-se o valor das variaveis presentes na expressao, alem de uma aproximacao para a funcao φ.Assim a equacao do novo metodo na sua forma final pode ser escrita como:
4(ui+1,j + ui−1,j + ui,j+1 + ui,j−1) + (ui+1,j+1 + ui−1,j+1 + ui+1,j−1 + ui−1,j−1) +
−20ui,j = 6h2fi,jeh2∇2fi,j
12fi,j . (3)
O laplaciano presente na equacao (3) e aproximado por diferencas finitas de segunda ordem,como visto a seguir:
h2∇2fi,j = fi+1,j + fi−1,j − 4fi,j + fi,j+1 + fi,j−1. (4)
A fim de verificar o metodo, simulacoes numericas foram realizadas para problemas comcondicoes de contorno definidas, tendo em vista testar e avaliar seu desempenho. As apro-ximacoes obtidas foram comparadas com resultados numericos presentes na literatura. Comisso, busca-se mostrar o quanto o metodo e eficiente e discutir sua competitividade em relacaoa outros metodos mais populares, como o metodo de diferencas finitas.
204
Sobre a expansão de uma função na esfera unitáriacomplexa em harmônicos esféricos
C. F. dos Santos, S. A. TozoniUniversidade Estadual de Campinas (Brasil)
Abstract
Um dos objetivos deste trabalho é expor parte de um estudo sobre harmônicos esféricoscomplexos, baseado em resultados retirados, principalmente, de [C. P. Oliveira, Aproximaçãona esfera unitária de Cq, q ≥ 2. Tese de Doutorado em Matemática, ICMC-USP, 2003].Os harmônicos esféricos complexos são definidos como restrições de polinômios bihomogêneosharmônicos à esfera unitária complexa Ωd de Cd, onde d ≥ 2 é um inteiro fixado. IdentificandoΩd com a esfera unitária S2d−1 de R2d, tem-se a medida de superfície σd sobre Ωd. Isto dáorigem ao produto interno
(f, g) =
∫
Ωd
fg dσd,
com respeito ao qual o espaço de Hilbert L2(Ωd) admite um conjunto ortonormal completoformado por harmônicos esféricos. Assim, cada função f ∈ L2(Ωd) se expressa unicamentecomo uma série
f =
∞∑
m=0
∞∑
n=0
Y (m,n), (1)
onde cada Y (m,n) é um harmônico esférico de bigrau (m,n) e a série acima converge incondi-cionalmente na norma de L2(Ωd).
Em cada ponto w ∈ Ωd, o termo Y (m,n)(w) em (1) pode ser expresso como o produtointerno de f com o harmônico zonal Z(m,n)
w , sendo este último o único harmônico esférico debigrau (m,n), a menos de constante multiplicativa, que permanece invariante pela composiçãocom transformações unitárias em Cd que preservam w. O harmônico zonal pode ser expressocomo
Z(m,n)w (z) =
dimHm,nωd
r mod m−nei(m−n)θP(d−2,m−n)m∧n (2r2 − 1),
onde m∧n = minm,n, 〈z, w〉 = reiθ com r ≥ 0 e θ ∈ [0, 2π], Hm,n denota o espaço vetorialdos harmônicos esféricos de bigrau (m,n) e P (α,β)
k denota o k-ésimo polinômio de Jacobi deparâmetro (α, β) com a normalização P (α,β)
k (1) = 1.Em geral, a validade da expressão (1) em Lp(Ωd), 1 ≤ p < ∞, não ocorre. Surge, pois,
o problema de fazer com que tal expansão seja válida por algum método de somabilidadede séries que generalize a noção usual de convergência. O segundo objetivo deste trabalho éapresentar um teorema referente à somabilidade pelo método de Cesàro da série (1) para fem Lp(Ωd). Para obter este resultado, é feito o uso de estimativas para os as somas de Cesàrodos polinômios de Jacobi apresentadas por Bordin, Kushpel, Levesley e Tozoni em [J. Func.Anal. 202 (2003), 307-326].
205
Primeness Property for Central Polynomials of Verbally
Prime Algebras
Claudemir Fideles Bezerra JniorFederal University of Campina Grande (Brazil)
Abstract
Let F be an infinite field. The primeness property for central polynomials of Mn(F )was established by A. Regev, i.e., if the product of two polynomials in distinct variables iscentral then each factor is also central. In this work we consider the analogous property forMn(F ) and determine, within the elementary gradings with commutative neutral component,the ones that satisfy this property, namely the crossed product gradings. Next we considerMn(R), where R admits a regular grading, with a grading such that Mn(F ) is a homogeneoussubalgebra and provide sufficient conditions - satisfied by Mn(E) with the trivial grading - toprove that Mn(R) has the primeness property if Mn(F ) does. We also prove that the algebrasMa,b(E) satisfy this property for ordinary central polynomials. Hence we conclude that, overa field of characteristic zero, every verbally prime algebra has the primeness property.
206
An Introduction to Categorification of Quantum sl(2)
C. SilvaUniversity of Campinas (Brazil)
A. Moura (Advisor)University of Campinas (Brazil)
Keywords: categorification, Grothendieck ring, quantum groups, nilHecke ring, flag varieties.
Thematic session (optional): Algebra.
Abstract
The term categorification, introduced by Louis Crane and Igor Frenkel, refers to the pro-cess of replacing set-theoretic notions by the corresponding category-theoretic analogues. Setsare replaced by categories, elements by objects, relations between sets by morphisms betweenobjects, functions by functors and relations between functions by natural transformation offunctors.
We present a categorification U , constructed by Aaron Lauda, of the quantum mod-ified algebra U (a version of the quantum group Uq(sl2). U is a 2-category whose splitGrothendieck ring is isomorphic to the algebra U . Lauda also defined for each positiveinteger N a representation ΓN of U which categorifies the irreducible (N + 1)-dimensionalrepresentation of U .
207
Existencia de Solucoes para uma classe de Equacoes
Diferenciais Parciais Elıpticas Nao Lineares envolvendo o
Operador p-Laplaciano
E. S. Boer, J. D. B. de GodoiUniversidade Federal de Santa Maria (Brasil)
Abstract
No presente trabalho garantimos a existencia de ao menos uma solucao fraca para o seguinteproblema
− Mp u+ c(x)|u|p−2u = f(x, u), se x ∈ Ω|∇u|p−2 ∂u
∂η= g(x, u), se x ∈ ∂Ω
, (1)
onde c, f, g,Ω e ∂Ω satisfazem certas condicoes e p ∈ (1,∞). Para atingir tal objetivo, con-sideramos os problemas de Steklov e de Neumann, associados ao problema (1). Garantimos aexistencia de um primeiro autovalor de Steklov (µ1) e de um primeiro autovalor de Neumann(λ1) e, a partir destes, determinamos a regiao do plano λµ, onde ha solucoes para (1). Estassao obtidas atraves de tecnicas variacionais aplicadas ao funcional Ip : W 1,p(Ω)→ R, definidopor
Ip(u) =1
p
∫
Ω
|∇u|p + c(x)updx−∫
Ω
F (x, u)dx−∫
∂Ω
G(x, u)dσ,∀u ∈W 1,p(Ω),
onde F (x, u) =u∫0
f(x, s)ds e G(x, u) =u∫0
g(x, s)ds.
208
Stability of the Phase Transition of Critical-Field Ising Model
on Cayley trees under Inhomogeneous External Fields
R. BissacotUniversity of Sao Paulo (Brazil)
E. O. EndoUniversity of Sao Paulo (Brazil), University of Groningen (The Netherlands)
A.C.D. van EnterUniversity of Groningen (The Netherlands)
Abstract
We consider the ferromagnetic Ising model with spatially dependent external fields on aCayley tree, and as the closest equivalent of the question considered in [1, 2], we investigatehow fast the external fields can asymptotically approach to the homogeneous critical externalfield hc in order to still be able to see a phase transition. Our results extend earlier results byRozikov [4] and Ganikhodjaev [3].
We show that, if the external fields are of the form hn = −hc − εn, with εn positive,decreasing, decaying to zero and satisfying the following condition,
limn→∞
n∑
j=1
(n∑
i=j
εi
)2
<∞, (1)
then the model undergoes a phase transition at low temperature. On the other hand, when thecondition is violated, and the sum diverges, we will obtain uniqueness of the Gibbs measurefor the perturbed model.
References
[1] R. Bissacot, M. Cassandro, L. Cioletti and E. Presutti. Phase Transitions in FerromagneticIsing Models with Spatially Dependent Magnetic Fields. Communications in MathematicalPhysics, 336, No. 1, 41–53 (2015).
[2] R. Bissacot and L. Cioletti. Phase Transition in Ferromagnetic Ising Models with Non-uniform External Magnetic Fields. Journal of Statistical Physics, 139, Issue 5, 769–778,(2010).
[3] N. Ganikhodjaev. Phase Transition in Ferromagnetic Lattice Models on a Cayley Tree withNon-uniform External Magnetic Fields.International Conference on Operations Researchand Statistics Proceedings, 106–110 Singapore Global Science and Technology Forum (2011).
[4] U.A. Rozikov. Gibbs Measures on Cayley Trees. World Scientific, (2013).
209
On the influence of matricial representation of individuals for
continuous genetic algorithms in RBF neural networks
training
J. F. da MotaParana State University (Brazil)
P. H. Siqueira, L. V. de SouzaFederal University of Parana (Brazil)
Keywords: RBF Neural Networks, Genetic Algorithms, Classification.
Thematic session: Operations Research or Computational Intelligence.
Abstract
One of the issues of modeling a RBFNN - Radial Basis Function Neural Network con-sists of determining the weights of the output layer, usually represented by a rectangularmatrix. The inconvenient characteristic at this stage it’s the calculation of the pseudo-inverseof the activation values matrix. This operation may become computationally expensive andcause rounding errors when the amount of variables is large or the activation values form anill-conditioned matrix so that the model can misclassify the patterns. In this paper we’vecompared two GA - Genetic Algorithm - based approaches for computing the weight matrixof a RBFNN - Radial Basis Function Neural Network. The first, named GA, was based inMichalewicz’s Operators for Continuous Genetic Algorithms and the other, named modGA,was based in extending these operators to matricial individuals, consequently proposing newoperators. The main objective was verifying if the new approach could reduce the number ofiterations (generations) necessary to compute the weight matrix. Six datasets was tested andin 50% of them, that hypothesis was confirmed.
210
Slowly non-dissipative partial differential equations and
non-compact global attractors
Esau A. SousaFederal University of Paraıba (Brazil)[email protected]
Keywords: slowly non-dissipative equations, semigroups, attractors, blow-up.
Thematic session: Nonlocal equations and applications.
Abstract
In this work we study slowly non-dissipative and partial differential equations non-compactglobal attractors, we give a brief overview history on the theme, examples and we prove someresults of the theory of non-compact global attractors for slowly non-dissipative systems.
211
Estudo teorico de Maquinas de Vetores de Suporte aplicado
ao problema de analise de credito
Evelin Heringer Manoel KrulikovskiUniversidade Federal do Parana (Brasil)
Orientadores: Mael Sachine e Ademir Alves RibeiroUniversidade Federal do Parana (Brasil)
Resumo
Neste trabalho abordaremos uma tecnica de aprendizagem de maquina, as Maquinas deVetores de Suporte (SVM), para o problema de analise de credito, que e um problema de classi-ficacao binaria. Atraves desta tecnica, encontraremos uma funcao de decisao, que determinarase um cliente e um bom ou mau pagador. O objetivo geral sera realizar um estudo teoricosobre o SVM, pois queremos fundamentar matematicamente sua utilizacao na aplicacao ci-tada acima, a qual envolve um problema de otimizacao quadratico e convexo. Para facilitar aanalise, faremos uso da teoria de dualidade Lagrangiana. Pretendemos fazer uma comparacaoentre dois tipos de funcoes Kernel: Polinomial e Gaussiano, usadas para resolver o problemaquando nao for possıvel encontrar uma funcao de decisao.
212
Culm Nitrogen Estimation with Foliar Dry Matter in
Bamboo Forest
A. M. Fragoso, L. C. Santos Junior, I. A. GuerriniUniversidade Estadual Paulista (Brazil)
L. ZimbackInstituto Florestal (Brazil)
C. RobinUniversity of Northampton (United Kingdom)
Abstract
Estimation evolving indirect parameters could be necessary in forest aspects and couldit changed to other agricultural species. Nitrogen and dry matter are useful to determinedeficiency and to estimate the correct nitrogen fertilizers. The aim of this work is to show therelation between foliar nitrogen dry matter and culm nitrogen (kg.ha-1). We use data from thepaper: Biomass and nutrient distribution in a highland bamboo forest in southwest Ethiopia:implications for management, with authors K. Embaye M. Weih, S. Ledin, L. Christersson(2004), from Forest Ecology Management. The resulting equation is: from the data, X =(4 × 10−3, 71 × 10−3, 266 × 10−3), where x is Nitrogen in foliar dry matter kg.ha-1 and Y =(40×10−3, 63×10−3, 196×10−3) where y= Nitrogen in the culm, and dividing by the spacing10x10 meters. The equation is
[E(Y ) = 29.7616 × 10−3 + 0.61487 × 10−3X]
Adjusted R-squared: 0.9752, p-value: 0.07111. The results showed that although there isa strait correlation between the dependent and independent variables, we can do the measurefoliar nitrogen to estimate culm nitrogen without cut the plant. We can do an equipment tomeasure nitrogen foliar, with the equation of an specific plant a calibrating him. To makethis we need specifics patters of measures like fertilizer dosages, climate changes, anthropictechniques and its influences in the plant. We need a better data to decrease error and comparewith real situations. With a possible form factor between equation and real situations. A singleleaf estimation is needed.
213
Grafos Aleatorios de Erdos-Renyi
G.G.FigueiredoUniversidade Federal Fluminense (Brasil)
Abstract
Nesse trabalho vamos estudar as componentes conexas dos Grafos Aleatorios de Erdos-Renyi e tambem como o processo de formacao dessas componentes podem ser descritas pelosprocessos de ramificacao.
O que vamos estudar mais afundo e que quando a expectativa do grau e muito menor que1, as componentes sao pequenas e a maior componente nesse caso e da ordem de log(n). Agoraquando a expectativa do grau e maior do que 1, existe uma componente conexa gigante quecontem uma proporcao positiva de todos os vertices.
214
Critérios para uma derivada fracionária
G. S. Teodoro, E. Capelas de OliveiraDepartamento de Matemática Aplicada
IMECC - UNICAMP13083-859 Campinas - SP
[email protected]@ime.unicamp.br
Palavras-chave: Cálculo Fracionário, Derivada fracionária.
Sessão Temática: Cálculo Fracionário.
Resumo
O cálculo fracionário, nomenclatura utilizada para cálculo de ordem não inteira, tem semostrado importante e, em muitos casos, imprescindível na discussão de problemas advindosde diversas áreas da ciência, ganhando popularidade e importância considerável durante asúltimas três décadas principalmente devido a suas aplicações atraentes. Existe mais de umaformulação para a derivada fracionária e esse número de definições vem aumentando ao passardo tempo, sendo cada uma delas mais adequada a um certo contexto físico. Com isso nosdeparamos com a seguinte questão: Que critérios um operador deve satisfazer para que estepossa ser considerado uma derivada fracionária?
Em 1975 Ross [ ROSS, B., A brief history and exposition of the fundamental theory of frac-tional calculus. Fractional Calculus and its Applications, 57, 1-36, (1975)] apresentou cincocritérios para que um dado operador seja considerado uma derivada fracionária e em 2015Ortigueira e Tenreiro Machado [ORTIGUEIRA, M. D., MACHADO, J.A.T., What is a frac-tional derivative?. Journal of Computational Physics, 293, 4-13, (2015)] reformularam essescritérios tendo em vista a necessidade de uma derivada fracionária satisfazer a generalizaçãoda regra de Leibniz, a saber,
Dα(f(x)g(x)) =
∞∑
k=0
(α
k
)Dkf(t)Dα−kg(t),
sendo a notação Dα usada para representar uma derivada fracionária qualquer de ordem α.Nesse trabalho faremos um pararelo entre os critérios propostos por Ross e Ortigueira/TenreiroMachado bem como apresentaremos um operador que satisfaça a esses critérios.
215
Solucao positiva de uma equacao de Schrodinger
assintoticamente linear no infinito via variedade de Pohozaev
Juan Carlos Ortiz ChataUniversidade Estadual Paulista ”Julio De Mesquita Filho”(Brasil)
Palavras-Chave: Palavras-chave: assintoticamente linear; identidade de Pohozaev; compaci-dade e concentracao; sequencia de Cerami; funcao baricentro.
Resumo
Neste trabalho teorico em Equacoes deferenciais parciais elıpticas, iremos apresentar otrabalho das autoras L.A.Maia e R.Lehrer de tıtulo ”Positive solutions of asymptotically linearequations via Pohozaev manifold ”que trata-se de uma abordagem diferente e mais geral abusca de solucoes positivas da equacao de Schrodinger assintoticamente linear no infinito
−∆u+ λu = a(x)f(u) em RN ,
para N ≥ 3 e λ > 0.Nos assumiremos as seguintes condicao para a funcao a :
(A1) a ∈ C2(RN ,R+) com infx∈RN
a(x) > 0;
(A2) lim|x|→∞
a(x) = a∞ > λ;
(A3) ∇a(x).x ≥ 0 para todo x ∈ RN , com a desigualdade estrita para un subconjunto demedida de Lebesgue positiva de RN ;
(A4) a(x) + ∇a(x).xN
< a∞, para todo x ∈ RN ;
(A5) a(x) + x.H(x).xN
≥ 0, para todo x ∈ RN , onde H representa a matriz hessiana da funcaoa.
Depois neste trabalho assumiremos outra condicao (A6), o qual requere que o supremo de|a∞ − a(x)| nao seja grande.
Alem disso consideremos as seguintes hipoteses sobre a funcao f :
(f1) f ∈ C2(R+,R+), lims→0
f(s)s
= 0;
(f2) lims→∞
f(s)s
= 1;
(f3) se F (s) =∫ s
0f(t)dt e Q(s) = 1
2f(s)s− F (s), entao existe uma constante D ≥ 1 tal que
0 < Q(s) ≤ DQ(t) para todo 0 < s ≤ t e lims→∞
Q(s) =∞.
Nos provaremos o seguinte resultado de existencia.
Teorema 1 Assumindo (A1) − (A6) e f(1) − f(3). Entao a equacao (1.1) possui solucaopositiva u ∈ H1(RN ).
Alem disso, vamos provar o seguinte teorema de nao existencia.
Teorema 2 Assumindo (A1)−(A4) e f(1)−f(3). Entao, p = infu∈P
I(u) nao e um nıvel crıtico
para a funcional I. Em particular , o ınfimo nao e atingido.
De onde observarmos que pelo Teorema 2 o ınfimo restrito a variedade de Pohozav nao eatingido, assim nossa busca de solucoes positivas sao em nıveis maiores de energia que seramostrado no Teorema 1.
216
SEMINARIO DE ALGEBRA
SUBGRUPOS NORMAIS DE INDICE PRIMO EMGRUPOS LIMITES
JHOEL ESTEBANY SANDOVAL GUTIERREZUniversidade de Brasilia
02/12/2016
14:30 Horas
Auditorio do MAT
Abstract.Motivated by their study of pro-p limit groups, D.H. Kochloukova and P.A. Zalesskii
formulated in [Remark after Thm. 3.3][1] a question concerning the minimum number ofgenerators d(N) of a normal subgroup N of prime index p in a non-abelian limit groupG (cf. Question*). It is shown that the analogous question for the rational rank hasan affirmative answer. From this result one may conclude that the original question ofD.H. Kochloukova and P.A. Zalesskii has an affirmative answer if the abelianization Gab
of G is torsion free and d(G) = d(Gab), or if G is a special kind of one-relator group.
Question*Let G be a non-abelian limit group, and let U be a normal subgroup of G of prime
index p. Does this imply that d(U) > d(G)?
Referencias
[1] Kochloukova, D. H. and Zalesskii, P. A., Subgroups and homology of extensions ofcentralizers of pro-p groups, Mathematische Nachrichten, 288 (2015), 604–618.
217
Sobre a propriedade de Schur em espacos de Banach duais
Jose Lucas Pereira Luiz; Geraldo BotelhoUniversidade Federal de Uberlandia (Brasil)
Resumo
Neste trabalho apresentamos algumas caracterizacoes da propriedade de Schur em espacosde Banach duais e tambem uma aplicacao.
Introducao: Dizemos que uma sequencia (xn)n converge fracamente para x em um espaco de
Banach E, e denotamos este fato por xnw−→ x, se ϕ(xn) −→ ϕ(x) para todo ϕ ∈ E′, onde E′ e o
dual topologico ( ou simplesmente o dual ) de E. E facil observar que a convergencia em normaimplica na convergencia fraca, porem a recıproca nem sempre e verdadeira. Um exemplo classicodesse fato e a sequencia dos vetores unitarios canonicos no espaco c0 das sequencias de escalaresque convergem para zero.
Em 1921, o matematico J. Schur demonstrou, em seu artigo Uber lineare Transformationen inder Theorie der unendlichen Reihen, que no espaco `1 a convergencia fraca implica na convergenciaem norma, assim `1 foi o primeiro espaco em que tal implicacao foi observada. O fato e que estaimplicacao tambem vale em outros espacos de Banach, e por isso passou-se a dizer que um espaco deBanach E possui a propriedade de Schur, ou que E e um espaco de Schur, se em E a convergenciafraca de sequencias implica na convergencia em norma. O objetivo deste trabalho e demonstraralgumas caracterizacoes da propriedade de Schur em espacos de Banach duais e dar uma aplicacaode uma dessas caracterizacoes.
Resultados Principais: Os espacos de Banach E e F sao considerados sobre o corpo K, ondeK = C ou K = R. Um operador linear entre espacos de Banach T :E −→ F e dito fracamentecompacto (compacto) se T (BE) e fracamente compacto (compacto) em F , onde BE denota abola unitaria fechada de E. Um espaco de Banach E possui a propriedade de Dunford-Pettis seϕn(xn) −→ 0 sempre que (xn)n e (ϕn)n forem sequencias fracamente nulas em E e E′, respecti-vamente. Um espaco de Banach E possui uma copia de `1 se existe um isomorfismo entre `1 e umsubespaco de E.
Teorema. As seguintes afirmacoes sao equivalentes para um espaco de Banach E:(a) E′ e um espaco de Schur.(b) E tem a propriedade de Dunford-Pettis e E nao contem copia de `1.(c) Todo operador fracamente compacto T :E −→ F e compacto, qualquer que seja o espaco deBanach F .(d) Todo operador fracamente compacto T :E −→ c0 e compacto.
Alem das demonstracoes das equivalencias acima, apresentaremos tambem a seguinte aplicacao.
Proposicao: O dual do espaco `∞ nao possui a propriedade de Schur.
Demonstracao. Se (`∞)′ tivesse a propriedade de Schur, pela caracterizacao (b), `∞ nao teriacopia de `1. Mas isso nao e verdade pois `∞ contem copia de todo espaco de Banach separavel.
Note que a proposicao acima mostra, em particular, que nao e verdade que o bidual de umespaco de Schur tambem e de Schur em geral. De fato, `1 e Schur (Teorema de Schur), mas(`1)′′ = (`∞)′ nao e Schur pela proposicao acima.
218
The generalization of Herglotz variational calculus and the
contruction of a gravitational theory with a cosmological
four-vector constant
J.Paiva, M.LazoUniversidade Federal de Rio Grande
Keywords: Calculus of Variation, Generalized Euler-Lagrange Equation, Gravitation.
Abstract
Since the introduction of the Action Principle in its mature formulation by Euler, Hamiltonand Lagrange, it becomes a fundamental principle for the construction of all physical theories.In order to obtain the dynamics equations of any theory, the Lagrangian defining the Actionis constructed from the invariants of the theory. In this case, the Action itself is an invariant.Consequently, we might ask what would happen if the Lagrangian itself would be a functionof the action? As we shall show, a Lagragian depending linearly on the Action will result ina non trivial generalized gravity theory including a cosmological four-vector constant.
The Herglotz variational calculus consist in the problem of determine the path x(t) thatextremize (minimize or maximize) S(b), where S(t) is a solution of
S(t) = L(t, x(t), x(t), S(t)), t ∈ [a, b] (1)
S(a) = sa, x(a) = xa, x(b) = xb, sa, xa, xb ∈ R.
It is important to notice that (1) represents a family of differential equations since for eachfunction x(t) a different differential equation arises. Therefore, S(t) depends on x(t). Theproblem reduces to the classical fundamental problem of the calculus of variations if theLagrangian function L does not depend on S(t). Herglotz proved that a necessary conditionfor a path x(t) to be an extremizer of the variational problem (1) is given by the generalizedEuler-Lagrange equation:
∂L
∂x− d
dt
∂L
∂x+∂L
∂S
∂L
∂x= 0. (2)
In the case of the classical problem of calculus of variation one has ∂L∂S
= 0, and the differentialequation (2) reduces to the classical Euler-Lagrange equation.
In the present work we generalize the Herglotz calculus of variation for Lagrangian den-sities depending on several variables. With this generalization, we formulated a non trivialgeneralized gravity theory including a cosmological four-vector constant. We show that inthis theory the space-time manifold is viscous. Furthermore, we investigate the cosmologicalconsequences of the cosmological four-vector constant in a Bianchi I universe filed with perfectfluid.
219
A non-isothermal phase-field model with singular potential
for two incompressible fluids
J. H. Lopes, G. PlanasUNICAMP (Brazil)
Abstract
In this work we study a non-isothermal phase-field model with singular potential describingthe mixture of two viscous incompressible fluids in Ω × [0, T ], where Ω is a smooth boundeddomain in Rn, n = 2, 3, and T is a finite fixed time. We consider that the fluids have matcheddensities and the same constant viscosity and thermal conductivity. The model consists of amodified Navier-Stokes equation coupled with a phase-field equation given by an Allen-Cahntype equation, and a equation for the temperature.
Namely, we prove the existence of solutions for the following nonlinear system: for (x, t) ∈Ω× (0, T )
ut + u · ∇u− ν∆u+∇P = (−∆φ+ F ′(φ))∇φ−∆θ∇θdiv u = 0
φt + u · ∇φ = γ(∆φ− F ′(φ))θt + u · ∇θ − k∆θ = 0
(1)
where F (φ) can be a singular potential, that has the following form
F (s) = c1 ((1 + s) log(1 + s) + (1− s) log(1− s)) + c2(1− s2), s ∈ (−1, 1),
for some c1, c2 > 0. Here, u denotes the fluids velocity, P is the pressure, φ represents thephase-field variable, which describes the location of each fluid, and θ is the temperature. Theconstants ν, γ and k are the fluid viscosity, the elastic relaxation and the thermal conductivity,respectively.
We complement the system with the following initial and boundary conditions
u(0, x) = u0(x), φ(0, x) = φ0(x), θ(0, x) = θ0(x), x ∈ Ω,
u = 0,∂φ
∂η= 0
∂θ
∂η= 0 (x, t) ∈ ∂Ω× (0, T )
(2)
where η is the normal direction vector.
The main idea is to show that for a suitable initial data for phase-field variable we canseparate the solution from the singularities of the potential. Then, we can solve the problem(1)-(2) as the case when we have a non-singular potential, like the double-well potential F (s) =14(s2 − 1)2.
220
Modelo de metabolitos revela o estresse em processos de
dessecacao-reidratacao
J. M. Berbert, K. A. OliveiraCentro de Matematica, Computacao e Cognicao - CMCC
Universidade Federal do ABC - UFABC (Brasil)
Abstract
A disponibilidade de agua nos ambientes influencia diretamente a sobrevivencia dos seresvivos. A irregularidade no acesso de agua, que chamamos de processo de dessecacao-reidratacao,pode levar os indivıduos a desenvolverem adaptacoes para sobreviverem. Especies que possuemresistencia a essa oscilacao no oferecimento de agua sao chamadas de tolerantes a dessecacao.
Pesquisas recentes de Centeno et al (2016) e Suguiyama et al (2014) demonstram a in-fluencia do teor de agua nas transicoes de classes metabolicas. Assim, considerando a de-pendencia da oferta de agua na metabolomica e as seguintes classes metabolicas: acucares (S),metabolitos relacionados ao metabolismo ativo (A) e metabolitos relacionados ao acumulo dereserva (R), propomos o modelo apresentado nas Eqs. (1):
dS
dτ= αS + βf(τ)SR− ρAf(τ)S − ρRf(τ)S
dA
dτ= ρAf(τ)S − δAA
dR
dτ= ρRf(τ)S − βf(τ)SR− δRR
, (1)
ou seja, os metabolitos S entram no sistema a uma taxa α e podem ser convertidos em A, comtaxa ρA, ou em R, com taxa ρR. R pode ser reconvertido em S com taxa β. As taxas ρA, ρRe β sao taxas que sao influenciadas pela disponibilidade de agua. Ambos os metabolitos A eR apresentam taxa de decaimento δA e δR, respectivamente, devido ao gasto metabolico oua degradacao natural. Alem disso, τ representa o tempo e f(τ) e uma funcao relacionada aoprocesso de dessecacao-reidratacao.
Analises de estabilidade do modelo apresentado nas Eqs. (1) adimensionalizado garantemque o ponto de equilıbrio estavel do modelo e o ponto trivial. O segundo ponto de equilıbrionao possui sentido biologico pois admite valores negativos, ou seja, nao pode haver quantidadesde metabolitos negativas. Ou seja, indivıduos tolerantes a dessecacao se mantem em equilıbrioquando seus metabolitos vao a zero.
Analises numericas mostram que o estresse decorrente do processo de dessecacao-reidratacaofaz com que nao haja recuperacao total do indivıduo a cada ciclo, nao retornando ao estadoinicial. Considerando ainda um persistente processo de dessecacao-reidratacao, o modelo preveque a quantidade de metabolitos do sistema vai a zero. Experimentos laboratoriais corroboramas previsoes do modelo proposto.
221
Modelo de metabolitos revela o estresse em processos de
dessecacao-reidratacao
J. M. Berbert, K. A. OliveiraCentro de Matematica, Computacao e Cognicao - CMCC
Universidade Federal do ABC - UFABC (Brasil)
Abstract
A sobrevivencia dos indivıduos esta relacionada com o teor de agua disponıvel para seuconsumo. A oscilacao nessa oferta de agua e chamada de processo de dessecacao-reidratacao.Indivıduos que possuem adaptacoes evolutivas para sobreviverem em ambientes com ofertavariante de agua sao chamados de tolerantes a dessecacao. Os trabalhos de Centeno et al(2014) comprovam a influencia do teor de agua na metabolomica dos seres vivos. A partirdisso, propomos um modelo que analisa a conversao dependente de uma funcao de oferta deagua pelo tempo, f(τ), entre tres classes metabolicas: acucares (S), metabolitos relacionadosao metabolismo ativo (A) e metabolitos relacionados ao acumulo de reserva (R), onde A eR ainda possuem degradacao. Analise de estabilidade desse modelo indica que o ponto deequilıbrio estavel e o ponto trivial; ou seja, que o indivıduo em equilıbrio tem seus metabolitoszerados. As analises numericas indicam que a recuperacao pos-dessecacao nao e total o quecomprovam a existencia de estresse em processos de dessecacao-reidratacao. Experimentoslaboratoriais foram realizados e corroboram o modelo proposto.
Keywordstolerancia a desidratacao, dessecacao, modelos matematicos, dinamica de metabolitos.
222
Ideais de Operadores
Leodan Acuna TorresUniversidade Estadual de Campinas (Brasil)
Resumo
Em 1992 E. Saksman e Tylli, G. Racher, tem vindo estudando compacidade fraca paraoperadores de composicao do tipo
SRB :L(F1;E) −→ L(E1;F );SRB = R T B,
Onde E, E1, F , F1 sao espacos de Banach, R ∈ L(E;F ) e B ∈ L(E1;F1) sao operadores naonulos. Em 1999 M. Lindstrom aborda este problema do ponto de vista dos ideais de operado-res, reformulando o problema da seguinte maneira:
Dados E, E1, F , F1 espacos de Banach, R ∈ L(E;F ) e B ∈ L(E1;F1) operadores nao nulos,A um ideal de operadores e (B, ‖.‖B) um ideal Banach. Sobre quais condicoes o operador linearcontinuo
SRB :B(F1;E) −→ B(E1;F );SRB(T ) = R T B,pertence ao ideal A?.
223
On Riemannian foliations over positively curved manifolds
L. D. SperancaFederal University of Parana (Brazil)
Abstract
A basic problem in Riemannian geometry is to construct examples of manifolds with posi-tive sectional curvature. Such examples are sparse in literature. On the other hand, the set ofmanifolds with non-negative sectional curvature enjoys large classes: compact Lie groups withbi-invariant metrics, compact symmetric spaces, normal homogeneous spaces, their Cartesianproducts and quotients by free isometric group actions.
Such discrepancy suggests that only a very restrictive class of manifolds carries a metric ofpositive sectional curvature. This question of existence still makes sense in the set of manifoldswith symmetry. This paper is devoted to present a hard obstruction for existence of positivesectional curvature in a special, although common, class of metrics.
Given a compact Riemannian manifold (M, g), a foliation F in (M, g) is said to be Rieman-nian if its leaves are locally equidistant. We only consider non-singular non-trivial foliations,i.e., all leaves have the same positive dimension. We prove:
Theorem 1 Let F be an odd-codimensional Riemannian foliation with bounded holonomy ona compact manifold M . Then M has a plane with nonpositive vertizontal sectional curvature.
We say that a foliation has bounded holonomy if there is a constant that uniformly boundsthe norms of all holonomy fields with unit initial condition. This class contains principaland associated bundles with compact structure group together with all regular homogeneousfoliations defined by proper group actions.
Theorem 1 relies between two contexts. The first is a generalization of a classical resultof Berger: every Killing field in a compact even dimension manifold with positive sectionalcurvature has a zero. Given such a manifold, by considering the foliation defined by a Killingvector field, Theorem 1 recovers Berger’s result.
Wilhelm’s conjecture settles the second context:
Wilhelm’s Conjecture 1 Let π : Mn+k → Bn be a Riemannian submersion from a compactmanifold M with positive sectional curvature. Then k < n.
Theorem 1 completely rules out the possibility of a Riemannian submersion with compactholonomy group over an odd dimensional bases, implying Wilhelm’s Conjecture under thesehypothesis.
Theorem 1 also plays a role in a connection between Berger’s result to Wilhelm’s conjec-ture: if one considers only principal submersions on Wilhelm’s conjecture, then Berger’s resultguarantees that the principal group has rank 1. A regular Riemannian foliation F on M iscalled principal if the leaves of F coincides with the orbits of an action by a (not-necessarilycompact) Lie group G acting by isometries on M , with at most discrete isotropy groups.Berger’s theorem can not be directly applied in this situation since the quotient of M by anone parameter subgroup may be an orbifold, or even non-Hausdorff.
Theorem 1 provides a generalization of Berger’s result that handles this case, by consid-ering the Riemannian foliations defined by exp(h′) and exp(h), where h is a maximal abeliansubalgebra of g and h′ is a subalgebra of h of co-dimension 1. The co-dimension of one of thesefoliations must be odd.
Corollary 0.1 Let M be a compact Riemannian manifold with positive sectional curvature.If F is a principal Riemannian foliation given by the group G, then G has rank 1.
224
Determination of Synaptic Connections from Neural Spike
Trains by Two Different Approaches Using Directed
Information
Juliana M. de Assis, Francisco M. de AssisFederal University of Campina Grande (Brazil)
Keywords: Directed information, causality, spike trains, chemical synapses.
Thematic session: Matematicas da Teoria da Informacao.
Resumo
Each human neuron forms approximately 1000 synaptic connections and receives evenmore, maybe in the order of 10 000 connections. The determination of synaptic connections isuseful to understand neural circuitry and how it controls animal behaviour. The present workinvestigates the directed information rate between neural spike trains to determine chemicalsynaptic connections. Directed information rate is a relatively new concept of informationtheory that is proved to be efficient as a causality measure, in the Wiener’s sense of causality.The estimators of directed information rate compared and used here are obtained from thework of Jiao et al., 2013, and Quinn et al., 2011, where the first uses the celebrated context treeweighting algorithm of Willems et al., 1995, while the second assumes a parametric generalizedlinear model. Both estimation methods assume that the random processes involved - which inthe present work are the neural spike trains - are stationary ergodic finite-alphabet processes.The spike trains are simulated according to a model proposed by Izhikevich, 2003, with acertain known topology. Since the topology of how neurons are connected is already establishedfrom the simulations, it was possible to evaluate the ability of both approaches to indicatetrue neuronal connections, considering neural spike trains only.
225
Simulacoes computacionais em modelos epidemiologicos: um
estudo de caso do vırus Zika
T. Y. Miyaoka. J. M. R. Souza, J. F. C. A. MeyerUniversidade Estadual de Campinas
Keywords: Epidemiologia, Equacoes Diferenciais Parciais, Metodos Numericos.
Thematic session (optional): Biomatematica.
Abstract
O vırus Zika teve seus primeiros casos confirmados no Brasil no ano de 2015, nos estadosdo Rio Grande do Norte e da Bahia. Desde entao tem preocupado muito a Saude Publicabrasileira e internacional, devido a sua relacao com a sındrome de Guillain-Barre e muitoscasos de microcefalia em bebes nascidos de maes infectadas pelos vırus. A doenca tem seespalhado para outros estados do paıs, e ate mesmo para outros paıses das Americas. Alemdisso, o principal mosquito vetor de sua transmissao, Aedes aegypti, e tambem vetor de outrasduas importantes doencas, chikungunya e dengue, esta ultima podendo levar a obito. Outropossıvel vetor, que tambem ocorre no Brasil, mas em menor frequencia, e o Aedes albopictus.Casos confirmados de transmissao do vırus Zika incluem a picada do vetor, transmissao verticale por relacoes sexuais (informacoes obtidas em www.who.int, acessado em 22/09/16).
Este trabalho tem como objetivo realizar analises numericas em um modelo epidemiologicoinicial para a transmissao do vırus Zika baseado em Equacoes Diferenciais Parciais Nao Lin-eares, dado por:
dS
dt−∇ · (αS∇S) = −βSI + r (S + I)
(1− (S + I)
κ
)+ σI,
dI
dt−∇ · (αI∇I) = βSI − σI − µI,
(1)
com condicoes iniciais e de contorno apropriadas. Aqui, S denota a populacao de humanossuscetıveis a doenca, I a populacao de humanos infectados, αS e αI seus respectivos coeficientesde difusao, β a taxa de infeccao da doenca, r a taxa de reproducao intrınseca da populacao,κ sua capacidade de suporte, σ a taxa de recuperacao da doenca e µ a taxa de mortalidadeassociada.
Restringindo este modelo somente a variavel temporal, obtemos um sistema de EquacoesDiferenciais Ordinarias (EDOs), cuja analise de estabilidade (por meio dos autovalores damatriz jacobiana) pode nos dar informacoes sobre os estados estacionarios relativos a endemiae extincao da doenca. Realizamos um estudo destes estados por meio de um metodo deMonte Carlo para a geracao dos parametros β, r, κ, σ e µ. Introduzindo estes conjuntos deparametros no modelo difusivo (1), obtemos analises de suas solucoes numericas, obtidas pormeio dos metodos de Elementos Finitos e Crank–Nicolson. Realizamos tambem estudos deMonte Carlo para os parametros αS e αI , analisando o impacto de seus valores nas solucoesnumericas.
Com o estudo do sistema de EDOs, obtivemos uma estimativa de aproximadamente 6%para o numero de possıveis casos de endemia da doenca, e uma correlacao de −0.9 entre ataxa de reproducao intrınseca r e os autovalores que levam ao equilıbrio endemico, significandoque a diminuicao de reproducao levaria a um fim da epidemia. A introducao dos parametrosgerados aleatoriamente no modelo (1) nos permitiu analisar o comportamento de sua solucaonumerica, assim como a influencia dos parametros nela. Estas analises nos deram informacoessobre a capacidade dos modelos estudados em gerar resultados que possam ser utilizados paraum melhor entendimento de uma epidemia do vırus Zika.
226
Derived categories of functors and Fourier–Mukai transform
for quiver sheaves
P. O. GneriUTFPR
M.B.JardimUNICAMP
Keywords: Derived categories functors, Fourier–Mukai transform, Quiver sheaves.
Thematic session (optional): Algebraic Geometry.
Abstract
Let C be small category and A an arbitrary category. Consider the category C(A) whoseobjects are functors from C to A and whose morphisms are natural transformations. Let Bbe another category, and again, consider the category C(B). Now, given a functor F : A → Bwe construct the induced functor FC : C(A) → C(B). Assuming A and B to be abeliancategories, it follows that the categories C(A) and C(B) are also abelian. We have two maingoals: first, to find a relationship between the derived category D(C(A)) and the categoryC(D(A)); second relate the functors R(FC) and (RF )C : C(D(A)) → C(D(B)). We use thegeneral results obtained to prove that certain integral functors between categories of quiversheaves are derived equivalences.
227
A global hybrid derivative-free method for nonlinear
equations
R. G. BegiatoUniversidade Tecnologica Federal do Parana
A. L. CustodioUniversidade Nova de Lisboa
M. A. Gomes-RuggieroUniversidade Estadual de Campinas
Abstract
This work concerns the numerical solution of large-scale nonlinear systems of equations,when derivatives are not available for use, even if all functions involved are assumed to be con-tinuously differentiable. The large-scale systems are solved by considering an hybrid approach,based on a derivative-free iterative method, organized in three phases. The first phase is de-fined by derivative-free versions of a fixed-point method that employs spectral parameters todefine the steplength along the residual direction. The second phase consists of a matrix-freeinexact Newton method that employs the GMRES to solve the linear system that computesthe search direction. This second phase will only take place if the first one fails to find a betterpoint after a predefined number of reductions in the step size. A third phase, based on direc-tional direct search, should act whenever too many line searches have excessively decreasedthe steplenght along the inexact-Newton direction. In all stages, the criterion to accept a newpoint uses a nonmonotone decrease condition upon a merit function. Convergence results areestablished for two scenarios: for the proposed three phases method and when the last phase isdisabled. The computational performance is assessed through numerical experiments in a setof problems collected from the literature. Both the theoretical and the experimental analysiscorroborate the feasibility of the proposed strategy.
228
Transformada sumudu e o modelo fracionario de dinamica
populacional
Edmundo Capelas de Oliveira, Luverci do Nascimento FerreiraUniversidade Estadual de Campinas (Brazil)
Resumo
Apresentam-se aplicacoes da transfomada sumudu em metodos numericos, Metodo de Per-tubacao Homotopica (MPH) e o Metodo da Decomposicao de Adomian (MDA) em um modelode dinamica populacional descrito por uma equacao diferencial parcial fracionaria nao-linear
dαρ
dtα=∂2ρ2
∂x2+∂2ρ2
∂y2+ hρa(1− rρb), t ≥ 0, x, y ∈ R,
onde 0 < α ≤ 1 para t ≥ 0 e a, b, h, r, x, y ∈ R e com condicao inicial dada por ρ(x, y, 0).As tecnicas propostas pelos metodo de pertubacao homotopica usando a transformada
sumudu (MPHS) e metodo da decomposicao de Adomian usando a transformada sumudu(MADS) se mostraram eficazes produzindo os mesmos resultados e de acordo com as solucoesexatas para modelos conhecidos. O esforco computacional em ambos os metodos, dentro domodelo analisado, foi otimizado mantendo a precisao dos resultados.
229
Meta-heurıstica BRKGA aplicada ao Problema da
Diversidade Maxima
M. A. MoroUniversidade Estadual de Campinas (Brasil)
Resumo
O Problema da Diversidade Maxima (PDM) consiste em: dado um conjunto N com nelementos, a cada par de elementos (i, j) com i, j ∈ N esta associado um valor dij , denominadodiversidade entre os elementos i e j. O objetivo do problema e selecionar um subconjuntocom m elementos (m < n), tal que a soma das diversidades entre os itens selecionados sejamaxima. Neste problema temos um total de
(nm
)combinacoes de subconjuntos possıveis, e
por essa razao e considerado um problema de otimizacao combinatoria. Logo, a medida quen aumenta, torna-se inviavel analisar todas as solucoes. Este problema e classificado comoNP-completo, pois seu tempo de resolucao e exponencial no pior caso. Tal fato sugere aabordagem de heurısticas para resolve-lo quando o numero de elementos e muito grande. Emnosso estudo aplicaremos ao PDM a meta-heurıstica BRKGA (Biased Random-Key GeneticAlgorithms), que e baseada na Teoria de Evolucao das Especies.
Essencialmente, em cada iteracao de um Algoritmo Genetico, e gerado um conjunto desolucoes denominado de “populacao”. Aplica-se nesta populacao os procedimentos de crosso-ver e mutacao para gerar novas populacoes. O crossover e um processo de recombinacao desolucoes, no qual busca-se combinar duas solucoes afim de se obter uma solucao melhor. Jana mutacao sao realizadas modificacoes aleatorias nas solucoes da populacao com o intuito deaumentar a diversificacao e ampliar o espaco de busca da solucao otima. Uma funcao fitnesse associada a cada solucao para efeitos de comparacao, tal funcao geralmente e baseada nafuncao objetivo do problema. O objetivo do algoritmo e eliminar as piores solucoes no decor-rer das iteracoes, de modo que no final restem somente as melhores solucoes e a populacaoconvirga para a solucao otima.
O BRKGA diferencia-se dos outros Algoritmos Geneticos, pois utiliza um processo decodificacao de solucoes que nao permite a producao de solucoes infactıveis durante o processode crossover. A meta-heurıstica utiliza um algoritmo determinıstico responsavel por associar osvetores codificados, denominados de chaves aleatorias, a uma solucao do espaco de solucoes doproblema. Toso e Resende desenvolveram um API (Application Programming Interface), quee um conjunto de rotinas e padroes de programacao, para executar o BRKGA em C++. Comele, o BRKGA pode ser aplicado a qualquer tipo de problema a princıpio, bastando apenaso usuario implementar o decodificador e criar a funcao fitness para seu problema. Propomosneste trabalho uma decodificacao baseada em ordenacao, aliada a uma funcao fitness igual afuncao objetivo para aplica-la ao Problema da Diversidade Maxima.
230
EXPOENTES DE PI-ALGEBRAS
A. M. D. Franca1†
Universidade de Brasılia – UnBCampus Darcy Ribeiro
Brasılia - DF, 70910-900 (Brazil)[email protected] , [email protected]
27 de Setembro de 2016
Keywords: Algebras associativas, PI-algebra, PI-expoente, Crescimento, Codimensoes, Diagra-mas de Young.
Thematic session: Algebra.
Abstract
Neste trabalho, estudamos o PI-expoente de uma algebra associativa finitamente geradasobre um corpo de caracterıstica zero, como fizeram Giambruno e Zaicev (1998). O principalponto deste trabalho foi determinar uma limitacao para cn(A) sob a forma
C1nr1dn ≤ cn(A) ≤ C2n
r2dn,
para algumas constantes C1, C2, r1, r2, d > 0, onde A e uma algebra associativa finitamentegerada. Com esse resultado em maos, e possıvel responder a pergunta: “como caracterizar umaalgebra central simples em termos do comportamento das codimensoes?”. O ponto chave parasolucionar tais questionamentos esta na teoria classica das representacoes do grupo simetricoSn, mais especificamente, atraves da teoria das tabelas de Young. Tal teoria desempenhaum papel fundamental no estudo de PI-Teoria. Por fim, outro ponto que foi abordado nestetrabalho refere-se as caracterizacoes de algebras que possuem crescimento polinomial das codi-mensoes. Os resultados nesse sentido aqui apresentados foram demonstrados por Giambrunoe Zaicev (2001). Um dos objetivos foi analisar em quais situaes a sequencia de codimensoesde uma certa classe de algebras e polinomialmente limitada. Mostrou-se que, sendo A umaalgebra de dimensao finita cuja sequencia de codimensoes e polinomialmente limitada, tem-se
χn(A) =∑
λ`nmλχλ,
com mλ = 0 se |λ|−λ1 ≥ q, onde J e o radical de Jacobson de A, Jq = 0, e λ = (λ1, λ2, . . .).Vale a recıproca.
1Doutorando em Matematica, Departamento de Matematica – Universidade de Brasılia, sob orientacao da Profa.Dra. Irina Sviridova.
†Trabalho orientado pelo Prof. Dr. Antonio P. Brandao Junior, Departamento de Matematica – UniversidadeFederal de Campina Grande, sendo elaborada uma dissertacao de mestrado, com tıtulo “Expoentes de PI-AlgebrasAssociativas”.
Suporte financeiro: CAPES, CNPq. 231
Gradient estimates for solutions of singular parabolic
equations
G. C. RampassoUniversidade Estadual de Campinas, IMECC (Brazil)
Abstract
The study of parabolic equations plays a fundamental role in the development of severalfields in pure and applied mathematics, such as differential geometry, functional and har-monic analysis, infinite dimensional dynamical systems, probability, as well as in mechanics,thermodynamics, electromagnetism, among others.
Montenegro, Queiroz and Teixeira established existence and sharp regularity results forsolutions to singular elliptic equations of the order u−γ , 0 < γ < 1, with gradient dependenceand involving a forcing term. Assuming merely regularity and a growth condition of the data,Choe and Weiss proved optimal regularity and non-degeneracy estimates for the solution ofthe heat equation with strong absorption, ∂tu−∆u = −uγχu>0 in Ω× (0,∞), γ ∈ [0, 1).
The purpose this work is to obtain gradient estimates that are, in some sense, uniformwith respect to analytical properties of the right-hand side of
∂tu− Lu = −γ|∇u|muγ−1χu>0, in Ω× (0,∞)u = 1, on (Ω× 0) ∪ (∂Ω× (0,∞)),
(1)
where Ω is a smooth bounded domain in Rn, L is a uniform elliptic operator, 1 < γ ≤ 2 and0 < m < 2− γ.
This is part of the author’s Thesis Dissertation under the supervision of Olivaine S. deQueiroz.
232
Rigidez de Grafos e Geometria de Distancias
Ana Flavia da Cunha LimaUniversidade Estadual de Campinas (Brasil)
AbstractConsideramos aqui um grafo G = (V,E), aonde V = v1, ..., vr e E e um subconjunto de
pares de V (como definido anteriormente), junto com um ponto p = (p1, ..., pr) ∈ RK x ... xRK = RKr. Os pontos pi serao tratados como os vertices de G no RK , e os segmentos que osligam, como as arestas do grafo.
Definicao 1 Um sistema F = (V,E, p) e formado por um grafo G = (V,E) mapeado nospontos p1, ..., pr ∈ RK , com |V | = r.
Definicao 2 Um movimento desse sistema e uma famılia de funcoes Pi : [0, 1] → Rn, i =1, .., r, tal que:
• Pi(0) = pi, ∀i;• Pi(t) e diferenciavel em t ∈ [0, 1], ∀i;• |Pi(t), Pj(t)| = |pi, pj |, ∀t ∈ [0, 1] e vi, vj ∈ E;
Definicao 3 A funcao Pi(t) e chamada de trajetoria do ponto pi
onde |, | denota a distancia euclidiana entre dois pontos.
Definicao 4 Um movimento e rıgido se as distancias entre os vertices sao preservadas:
|Pi(t), Pj(t)| = |pi, pj |, ∀vi, vj ∈ V. (1)
e esse mesmo movimento e uma deformacao caso as distancias nao sejam todas conservadas.
Definicao 5 O sistema o qual mantem-se rıgido apos a remocao de qualquer aresta e chamadode redundantemente rıgido.
De acordo com Graver, no livro Counting on Frameworks: Mathematics to Aid the Designof Rigid Structures, a condicao para que um sistema seja rıgido e que as distancias entre ospontos permanecem sempre as mesmas.
Rigidez conecta-se com problemas relacionados a determinar posicoes de nos. No paperRigidity, Computation, and Randomization in Network Localization, de T. Eren, D.K. Golden-berg, W. Whiteley, Y. R. Yang, A. S. Morse, B. D. O. Anderson e P. N. Belhumeur e explicadocomo esses dois assuntos se relacionam, junto com algumas das definicoes apresentadas nessetrabalho.
A definicao abaixo foi retirada do paper Theory of semidefinite programming for SensorNetwork Localization, de Anthony Man-Cho So e Yinyu Ye.
Definicao 6 No Problema da Realizacao de Grafos e dado um grafo G = (V,E) e umconjunto de pesos nao-negativos dij: i, j ∈ E em suas arestas, e seu objetivo e computar arealizacao de G no espaco euclidiano RK .
Definicao 7 Um grafo G e K-conexo se nao existe um subconjunto de vertices de G de car-dinalidade K − 1 que, quando removidos, torna o grafo correspondente desconexo.
Definicao 8 Um grafo G e 2-rıgido em RK se a remocao de qualquer aresta resulta em umgrao rıgido em RK
O teorema a seguir encontra-se enunciado em Connected rigidity matroids and uniquerealizations of graphs, de Bill Jackson and Tibor Jordan, e foi provado por Hendrickson emConditions for unique graph realizations:
Teorema 1 Se um sistema generico (G, p) e uma realizacao unica de G em RK , entao
• G e o grafo completo com no maximo K + 1 vertices
• Ou entao G e K + 1-conexo e 2-rıgido.
Hendrickson conjecturou que as condicoes citadas acima sao suficientes para a realizacaounica de um grafo G no RK . De fato, elas sao suficientes no plano, contudo, Connelly em seupaper On generic global rigidity provou que esse nao e o caso para K ≤ 3.
233
Partitions into two colors and Mock Theta Functions
A. Bagatini, M. L. MatteUniversidade Federal do Rio Grande do Sul (Brazil)
A. WagnerUniversidade Estadual de Campinas (Brazil)
Keywords: Mock Theta function, integer partition, combinatorial interpretation, partition enu-meration.
Thematic session: Discrete Mathematics.
Abstract
A new combinatorial representation for partitions as two-line matrices was presented bySANTOS et al. (2011) [J. P. O. Santos, P. Mondek, A. C. Ribeiro, New two-line arraysrepresenting partitions, Annals of Combinatorics 15 (2) (2011) 341−354], allowing the authorsto obtain new results in theory of partitions. By using the same ideas, in BRIETZKE et al.(2013) [E. H. Brietzke, J. P. O. Santos, R. da Silva, Combinatorial interpretations as two-line array for the mock theta functions, Bulletin of the Brazilian Mathematical Society, NewSeries 44 (2) (2013) 233 − 253] the authors presented two-line matrix representations forthe coefficients of some Mock Theta functions, whose general terms, overlooking signs whennecessary, can be interpreted as generating functions for partitions subjected to some rules.
We consider these matrix representations and focus on Mock Theta functions ρ(q), σ(q),and λ(q), which are related to partitions into two colors. By classifying these matrices accord-ing to the sum of their second lines and organizing these numbers in a table, we are able tofind great information and discover properties till then unknown that are clearly suggested bythe table’s data.
By observing patterns on coefficients from the table, it also implies closed formulas thatrelate different kinds of partitions. Although it may not be evident, we can associate somepartitions generated by the Mock Theta function ν(q), which considers only one color, to aspecific type of partition into two colors brought forth by λ(q). As a consequence, we provean identity for ν(q) in a combinatorial way.
234
Sequential Linear Programming for Multi-materialTopology Optimization
Darwin Castillo Huamaní*, Francisco A.M. Gomes*Universidade Estadual de Campinas (Brazil)
Resumo
A otimização topológica multi-material (OTM) tem o objetivo de projetar a topologiaótima de estruturas, segundo determinado conjunto de restrições e critérios de custo, maximi-zando ou minimizando certa especificação da estrutura.
Neste trabalho, o problema de estudo é a distribuição ótima de materiais num domínio,que é representado por Ω, dado sob certas condições de fronteira e a intervenção de forçasexternas, em que nosso interesse é minimizar a flexibilidade. Para isto, reformulamos o pro-blema físico de minimizar a flexibilidade por um problema numérico de otimização baseadona teoria de elementos finitos retangulares. Assim, para o caso de um material, cada ponto dodomínio é representado por uma função discreta X : Ω→ 0, 1 onde o valor da função é 1 se oelemento pertence à estrutura ou 0 se o elemento é vazio. Devido ao alto custo computacionalde resolver problema de otimização de variáveis inteira, Bendose (1989) introduziu o métodode densidade, ou Solid Isotropic Material with Penalization (SIMP). O método SIMP apre-senta a função de interpolação continua D : [0, 1] −→ [0, 1], D(ρ) = ρp em que ρ representauma pseudo densidade em cada ponto de Ω. Para valores adequados de p > 1 , Rietz (2001),conseguimos diminuir os valores intermediários de (0, 1), que não tem um significado físico datopologia ótima.
No caso multi material, utiliza-se o método SIMP generalizado proposto por Hvejsel (2011)
para interpolar as densidades dos materiais, D(ρe) =
m∑
i=1
ρpei onde ρei representa a pseudo
densidade de um pixel de Ω com material i. A quantidade de variáveis das pseudo densidadesdos materiais fica muito grande e métodos de otimização com segunda derivada devem serevitados. Para superar isto, aplicamos o método de programação linear sequencial, versãoglobalmente convergente, Gomes e Senne (2011), que resolve uma sequencia de aproximaçãolinear de problema de otimização.
235
On lacunary power series with rational coefficients and a
Mahler’s question
Elaine C. S. Silva,University of Brasilia (Brazil)
Abstract
In 1844, Liouville showed the first examples of transcendental numbers, now known asLiouville numbers. In 1906, Maillet proved that the image of a Liouville number by a nonconstant rational function (with rational coefficients) is a Liouville number. Rational functionsare examples of algebraic functions. In 1984, Mahler asked about the existence of transcen-dental functions with this property. We present a sufficient condition given by Marques andMoreira to answer this question and also we prove a result saying that the condition givenby them is not satisfied by lacunary power series with rational coefficients. This result wasproved jointly with Marques and Ramirez.
236
The preliminary numerical studies and analytical comments
relating to FENE-CR Algebraic Extra-Stress Model
G.S. PauloFaculdade de Ciencias e Tecnologia (FCT), UNESP - Univ Estadual Paulista
Abstract
Two-dimensional flows described by differential models employ three partial differentialequations for calculating the components of the extra-stress tensor whereas an algebraic modelrequires only one partial differential equation to be solved. Therefore, algebraic models areattractive because their solutions demand less CPU time and the algebraic equations makesthe computational algorithms easier to implement.
The first AESM was developed by Mompean et al. who used the ideas for modeling turbu-lent flows of Newtonian fluids and derived an AESM model to obtain approximate solutionsof flows described by the Oldroyd-B differential model (J. Non-Newt. Fluid Mech. (1998)79:261–281). In that work, numerical results were performed in order to demonstrate theabilities of the formulation. In a subsequent paper, Mompean extended the idea the Oldroyd-B AESM to approximate the differential PTT model (Comput. Fluids (2002) 31:935–956).
By using those ideas of Mompean, Paulo et al. derived an AESM to approximate theFENE-CR differential model (Proceedings of PANACM 2015 p. 767-774.). Moreover, analyticsolutions for the FENE-CR algebraic extra-stress model in fully developed channel flow werederived. It was found that the analytic solutions for velocity and pressure were found to be thesame solutions given in J. Non-Newt. Fluid Mech. (2014) 204:50–61, demonstrating that thisnovel algebraic model is capable of simulating shear flows defined by the classic differentialFENE-CR model. In addition, a numerical methodology to solve the flow of a FENE-CRAESM fluid was implemented and applied to solve the flow in a two-dimensional channel.The analytic solutions were employed to verify the correctness and accuracy of the proposednumerical method that presented a convergence order of 2.
In order to test the abilities of this new model for simulating viscoelastic flows in complexgeometries, preliminary numerical results in a 1:4 expansion will be presented and the bifur-cation phenomena will be discussed in comparison to those described by Rocha et al. inJ. Non-Newt. Fluid Mech. (2007) 141:1–17.
237
Consideracoes sobre as raızes de algumas equacoes trinomiais
relacionadas a problemas da Matematica Financeira
Jessica Ventura da Silva∗ , Vanessa Avansini Botta PiraniUniversidade Estadual Paulista “Julio de Mesquita Filho- FCT/UNESP (Brasil)
ventura [email protected]
Resumo
O objetivo deste trabalho e apresentar o comportamento das raızes de equacoes trinomiaisda forma
βzn − z + (1− β) = 0, (1)
onde β ∈ R, 0 < β < 1.Atraves de resultados classicos da literatura determinaremos a quantidade de raızes reais
e nao reais da equacao (1), mostrando que tais raızes sao simples.Alem disso, provaremos que as raızes da equacao (1) encontram-se no anel
1 ≤ |z| ≤ δ,onde δ e a unica raiz positiva da equacao
|β|zn − z − |1− β| = 0.
Uma motivacao para o estudo das raızes da equacao (1) tem relacao com a determinacaoda taxa de juros I de uma serie de pagamentos postecipada, representada por
PMT =FV I
(1 + I)n − 1, (2)
onde PMT e a prestacao, FV e o valor futuro e n e o numero de perıodos da serie.Atraves da mudanca de variavel z = 1 + I, podemos reescrever a equacao (2) da seguinte
forma
PMT
FVzn − z +
(1− PMT
FV
)= 0,
onde, considerando β = PMTFV
, voltamos a equacao (1).Portanto, o estudo das raızes da equacao (1) sera uma importante ferramenta para a
obtencao da taxa de juros I de uma serie de pagamentos postecipada, algo abordado naliteratura somente atraves de metodos numericos.
∗Bolsista de Mestrado da FAPESP Processo 2015/23752-3238
Estudo de uma equacao elıptica assintoticamente linear
Letıcia dos Santos Silva, Marcos Tadeu de Oliveira PimentaUniversidade Estadual Paulista Julio de Mesquita Filho - FCT/UNESP
Resumo
Neste trabalho sera mostrado a existencia de uma solucao positiva para o problema
−4u = g(x, u), u ∈ Ω (1)
com condicoes de fronteira de Dirichlet, onde Ω ⊂ Rn e um domınio limitado de classe C1,α.Definindo λ+
1 (h) := inf∫
Ω|∇u|2 : u ∈ H1
0 (Ω),∫
Ωhu2 = 1, para h ∈ L∞(Ω), tem-se as se-
guintes hipoteses:
(F1) g : Ω× [0,∞) −→ R e uma funcao de Caratheodory
(F2) g(x, 0) = 0 q. t. x ∈ Ω e existe C ∈ R tal que | g(x,s)s|≤ C q.t. x ∈ Ω, ∀s ∈ (0,∞)
(F3) λ+1 (α) > 1, onde G(x, s) :=
∫ s0g(x, ξ)dξ e α(x) := 2 lim
s→0+sup
G(x, s)
s2
(F4) β(x) := lims→∞
g(x, s)
sexiste q. t. x ∈ Ω e λ+
1 (β) < 1.
Assim, procura-se pontos crıticos para o funcional I(u) = 12
∫Ω|∇u|2 −
∫ΩG(x, u), para
u ∈ H10 (Ω). Primeiro, a partir das hipoteses (F1), (F2), (F3) e (F4), mostra-se que I satisfaz
a geometria do Passo da Montanha. Depois, supondo a existencia de sequencias Palais-Smalenao limitadas, e usando propriedades de autovalores principais de um operador linear com umafuncao peso, obtem-se que λ+
1 (β) = 1, contradizendo a hipotese (F4). Logo, toda sequenciaPalais-Smale e limitada. Por fim, com estes resultados, aplicando o Teorema do Passo daMontanha conclui-se que o problema (1) admite solucao, e que esta solucao e positiva.
239
Compatibilidade entre duas t-estruturas.
Oscar Armando Hernandez MoralesUniversity of So Paulo (Brazil)
Eduardo do Nascimento Marcos (Orientador)University of So Paulo (Brazil)
Keywords: t-estruturas, aisles, categoria triangulada.
Thematic session (optional): Algebra.
Abstract
Neste trabalho apresentaremos alguns resultados de Keller e Vossieck introduzidos em (B.Keller and D. Vossieck. Dualite de grothendieck-roos et basculement. C. R. Acad. Sci. Paris,(307):543–546, 1988). Ao investigar as relacoes entre os coracoes de duas t-estruturas sobreuma categoria triangulada Keller e Vossieck, apresentaram condicoes necessarias e suficientespara que duas t-estruturas sejam compatıveis. Este conceito permite obter uma generalizacaoda teoria da dualidade desenvolvida por Grothendieck e Roos para aneis comutativos regularese a teoria de inclinacao utilizada na investigacao de algebras de dimensao finita. Mas, antesde isso tudo precisamos definir “Ailes” (conceito que resulta equivalente ao de t-estrutura) eestabelecer algumas das suas propriedades.
240
A new algorithm for clustering based on kernel density
estimation
S. R. SantosState University of Parana (Brazil)
L. C. MatioliFederal University of Parana (Brazil)
M. KleinaFederal University of Parana (Brazil)
Abstract
In this work we present an algorithm for clustering based on univariate kernel densityestimation, named MSKL. It consists of an iterative procedure that in each step a new clusteris obtained by minimizing a smooth kernel function. Although in our applications we haveused the univariate Gaussian kernel, any smooth kernel function can be used. The proposedalgorithm has the advantage of doesn’t require a priori the number of cluster. Furthermore,the MSKL algorithm is very simple, easy to implement, well-defined and stops in a finitenumber of steps, namely, it always converges independently of the initial point. Preliminaryexperiments showed the good performance of the proposed algorithm, especially when appliedto solve practical applications with a large data sets. The results indicate that the MSKLalgorithm is competitive and fast when compared with the well-known Clusterdata and K-means algorithms, used by Matlab to clustering data.
241
Fractional Schrodinger equation with Riesz-Feller
fractional derivative for delta potentials
Stefania Jarosz, Jayme Vaz Jr.UNICAMP (Brazil)
Abstract
In this work, we study the effects of spatial fractional derivatives in one-dimensional time-independent Schrodinger equation, in the particular case of Dirac delta potentials. Initiallywe present the ordinary Schrodinger equation and its solutions for delta potentials. Next, weconsider a generalized version of the Schrodinger equation, called fractional Schrodinger equa-tion, where the Riesz fractional derivative is used, and study its solutions for delta potentials,comparing its results to those well-known from quantum mechanics. Finally, we further gener-alize the fractional Schrodinger equation by considering the Riesz-Feller fractional derivativesand also study its solutions for single and double delta potentials, again comparing the resultswith those obtained through the previous approaches.
242
Teoria de Sturm-Liouville e Aplicacoes
Felipe Felix Souto ∗
Universidade Estadual“Julio de Mesquita Filho”,Orientadora: Marta Cilene Gadotti
felipe felix [email protected]
Resumo
No estudo de certas equacoes diferenciais parciais que modelam fenomenos fısicos comoo movimento da corda vibrante, apos o uso do metodo de separacao da variaveis percebe-sea importancia de encontrar os autovalores a as autofuncoes correspondentes, o que leva na-turalmente ao estudo do problema de Sturm-Liouville. Para apresentar uma solucao desteproblema e necessario conhecer a Teoria Espectral para Operadores Hermitianos Compactosem espacos de dimensao infinita.
Baseados em Honig, C. S. - Analise Funcional e Aplicacoes, vol. 1 e 2, veremos que ometodo de separacao de variaveis nao e suficiente para resolver a equacao da corda vibrante
∂
∂x(p(x)
∂U
∂x)− q(x)U = ρ(x)
∂2U
∂t2,
com as condicoes iniciais
U(x, 0) = f(x) e∂U
∂t(x, 0) = g(x).
Neste caso, e necessario aplicar, por exemplo, essa teoria para obter uma solucao do tipo:
U(x, t) =∑
n
φn(x)[cn cos(√λnt) + dnsen(
√λnt)].
∗Bolsa: FAPESP, processo: 2015/00534-0 243
Marshall Olkin Extended Exponentiated Gamma
G. A. S. Aguilar, F. A. Moala, T. C. S. ReisFCT-UNESP
Abstract
Due to the emergence of different methods of obtaining new distributions, in the lastfew years studies on new probability distributions has become commom. The distributionExponentiated Gamma, where the model is obtained by the method
F ∗ (x) = [F (x)]θ,
where F (x) is a distribution base and θ (shape parameter) is a real and positive number. Thisdistribution has the flexibility to model monotonous and not monotone failure rates. Thus,using the method proposed by Marshall Olkin and the Exponentiated Gamma as distributionbase, we have the model developed and studied in the research, Marshall Olkin ExtendedExponentiated Gamma (MOEEG). Often, obtaining a new distribution may be advantageousdue to the different shapes of its hazard curve. With the help of the software Maple and R, itwas possible to obtain some probabilistic results of the new distribution such as n− th time,r − th moment of residual life, r − th moment of residual life inverted, stochastic ordering,entropy, mean deviation, Bonferroni and Lorenz curve, skewness, kurtosis, order statistics, andstress-strength parameter. We also tested two different methods of estimators and comparedthem.
244
On complete hypersurfaces with constant mean and scalar
curvatures in Euclidean spaces
Roberto A. R. NunezUniversidade Federal Rio Grande do Sul (Brazil)
F. X. FonteneleUniversidade Federal Fluminense (Brazil)
Abstract
Generalizing a theorem of Huang, Cheng and Wan classified the complete hypersurfacesof R4 with non-zero constant mean curvature and constant scalar curvature. In our work, weobtain results of this nature in higher dimensions. In particular, we prove that if a completehypersurface of R5 has constant mean curvature H 6= 0 and constant scalar curvature R ≥23H2, then R = H2, R = 8
9H2 or R = 2
3H2. Moreover, we characterize the hypersurface in the
cases R = H2 and R = 89H2, and provide an example in the case R = 2
3H2. The proofs are
based on the principal curvature theorem of Smyth-Xavier and a well known formula for theLaplacian of the squared norm of the second fundamental form of a hypersurface in a spaceform.
245
Construcao de reticulados Dk-rotacionados via o
homomorfismo torcido, com k ımpar
Robson R. de Araujo, Sueli I. R. da CostaState University of Campinas (Brazil)
Abstract
Reticulados sao subgrupos discretos de Rn frequentemente usados para construir bonscodigos para canais gaussianos. Um reticulado equivalente por rotacao a outro reticulado edito reticulado rotacionado. Alguns reticulados podem ser construıdos via o homomorfismotorcido com imagem em Rn e domınio em um Z-modulo de posto completo M contido no anelde inteiros de um corpo de numeros K de grau n, usando um elemento totalmente positivoβ ∈ K∩R como escala. Tal construcao e util para canais do tipo Rayleigh com desvanecimento,ja que e mais facil calcular a distancia produto mınima de um reticulado construıdo via ohomomorfismo torcido do que no caso geral.
Na literatura, ha construcoes de reticulados rotacionados Zn via o homomorfismo torcido,para n ∈ N, e de reticulados rotacionados Dk obtidos da mesma forma para k = 2m e k =(p− 1)/2, onde p e primo e m e um numero natural nao nulo. Neste trabalho, apresentamosa construcao de um reticulado rotacionado Dk para qualquer k ımpar e conjecturamos a suadistancia produto mınima.
246
Optimization problems in timeout power policies
Saulo O. D. LuizEmbedded Systems and Pervasive Computing Laboratory
Center of Electrical Engineering and InformaticsFederal University of Campina Grande58.429-900, Campina Grande, Brazil
Abstract
The energy efficiency of computer systems such as mobile phones, tablets, notebooks anddesktops may be improved by means of a power manager, which measures the system workloadand issues commands to change the system power state. The criterion for selecting the powerstate is named a power policy. For example, a power policy may reduce the computer systempower consumption at runtime by turning off an idle network interface, reducing the screenbrightness, or reducing the operation frequency or voltage of a processor.
However, energy savings may also cause performance penalties, as low power states gener-ally cause reduced performance. For example, shortly after a network interface is shut down,a notebook user may desire to browse the Internet; or a person may have difficulty using amobile phone when the screen brightness is reduced; or a desktop user may be bothered whenthe applications are executed slowly because the processor frequency was reduced. Thereforethere is a tradeoff between energy savings and performance penalties.
Timeout power policies are applied in computer systems due to its simplicity. For example,such policies are implemented in mobile phones as the time interval to turn off an idle screen.However, the timeout is usually selected based on heuristic approaches. Formally, the selectionof an optimum timeout should consider the system workload, because a wrong timeout mayincrease both the power consumption and the performance penalty.
Luiz et al. have presented a formal answer whether it is feasible or not to adopt a timeoutpower policy. Those authors have formulated an optimization problem for a timeout policyimposing constraints for both the power consumption and the performance penalty. Theworkload model was considered a probabilistic two state Markov model. Luiz et al. have alsopresented an optimization procedure for time invariant workloads with general probabilitydistributions. Later, Vilar et al. have presented an optimization procedure for time varyingworkloads with general probability distributions.
Vilar et al. have also applied the optimization procedure of Luiz et al. to four powerpolicies: the timeout policy, the predictive policy, the predictive policy without automaticturn on, and the L curve policy. Those authors have simulated and implemented the powerpolicies in an experimental platform. Considering power savings and performance penalties,the timeout policy was better than the predictive policy. For most timeout intervals selectedby Vilar et al., the timeout policy was better than the L curve policy. For only one timeoutinterval, the timeout policy was equivalent to the L curve policy and the predictive policywithout automatic turn on.
In this work, the optimization of timeout power management policies is presented, consid-ering either time invariant or time varying workloads. The selection of the timeout interval ispresented as an optimization problem, with constraints for both the power consumption andthe performance penalty.
247
Esquema de Diferenças Finitas Compactas para ProblemasEstacionários
R. L. Sterza, A. C. BrandiUniversidade Estadual Paulista “Júlio de Mesquita Filho” (UNESP)
[email protected], [email protected]
Palavras Chave: Método de diferenças finitas compactas, Equação de Poisson, Métodos numé-ricos, Matlab.Sessão temática: Análise Numérica e Dinâmica de Fluidos Computacional.
Resumo
As equações elípticas são equações diferenciais parciais e estão relacionadas com problemasde equilíbrio que não dependem, em geral, do tempo. As mais conhecidas dessas equações sãoa equação de Poisson e a equação de Laplace, cujas aplicações são as mais variadas. A primeirarepresenta o movimento de um fluido viscoso incompressível a baixa velocidade e a segunda éempregada para descrever potencial eletromagnético, por exemplo. A necessidade de obtençãode soluções aproximadas para problemas desse tipo impulsionou o estudo e a aplicação demétodos numéricos. Dessa forma, este trabalho consiste no estudo e na implementação dométodo de diferenças finitas compactas para a solução numérica de problemas modelados pelaequação de Poisson, representada por
∂2u
∂x2+∂2u
∂y2= f(x, y), (1)
e discretizada em uma região quadrada Ω = (x, y) : a ≤ x ≤ b, a ≤ y ≤ b sob condições decontorno do tipo Dirichlet, ou seja, u(x, y) = g(x, y) na fronteira ∂Ω.
A resolução do problema foi realizada através da aplicação do método de diferenças finitascompactas de ordem superior, que é desenvolvida a partir do método de série de Taylor,métodos de aproximação Padé ou métodos de interpolação Birkhoff. Os métodos compactosde alta ordem são resolvidos de forma implícita, gerando um sistema linear do tipo Ax = b.Neste trabalho foi utilizado o método compacto de 4a ordem que pode ser representado pelaseguinte equação
1
6(ui+1,j+1 + ui+1,j−1 + ui−1,j+1 + ui−1,j−1) +
2
3(ui+1,j + ui−1,j + ui,j+1 + ui,j−1) +
− 10
3ui,j =
h2
12(fi+1,j + fi−1,j + fi,j+1 + fi,j−1 + 8fi,j). (2)
Com o objetivo de avaliar a eficiência do método numérico, foram realizados testes nu-méricos para diferentes problemas encontrados na literatura com solução analítica conhecida.Os resultados numéricos obtidos com o método de diferenças finitas compactas de 4a ordemforam comparados com a solução analítica do problema e com o método de diferenças finitasnão-compacto. E ainda, para comprovar o quão melhor é o método compacto, em relação aométodo não-compacto, foi feita uma análise de convergência do método, mostrando quantasiterações foram necessárias para os métodos convergirem. Além disso, foram testados e com-parados dois métodos de resolução de sistema linear, o método direto de decomposição LU e ométodo iterativo de Gauss-Seidel, a fim de avaliar a eficiência computacional desses métodos.
248
Codigos MDS na Metrica de
Niederreiter-Rosenbloom-Tsfasman
Welington SantosUniversidade Federal do Parana
Marcelo Muniz Silva AlvesUniversidade Federal do Parana
Keywords: ρ-metrica, Distribuicoes otimas, Codigos MDS.
Thematic session (optional): Matematicas da Teoria da Informacao
Abstract
Neste trabalho apresentamos o conceito de ρ-metrica no espaco das matrizes sobre umalfabeto finito qualquer A, assim como os conceitos de codigo MDS na ρ-metrica e CodigoUniformemente Distribuıdo. Mostramos que estas duas ultimas definicoes sao equivalentes.Apresentamos o conceito de Distribuicao Otima no cubo unitario e obtemos uma relacaobiunıvoca entre Codigos MDS na ρ-metrica e Distribuicoes Otimas no cubo unitario (quandoconsideramos que nosso alfabeto A possui a estrutura de corpo). Tambem no caso em queA possui a estrutura de corpo, mostramos uma expressao para o enumerador de peso deuma Distribuicao Otima e, consequentemente, para um Codigo MDS na ρ-metrica. Por fim,exibimos uma construcao explıcita de uma famılia de codigos MDS na ρ-metrica. Os resultadosdeste trabalho encontram-se em artigos de Dougherty, S. T., and Skriganov, M. M. (2002),Rosenbloom, M, Y. T. M. A. (1997), Skriganov, M. M. (2001).
249