Advances in Mathematics and Applications : Celebrating 50 Years of the Institute of Mathematics, Statistics and Scientific Computing, University of Campinas

This book celebrates the 50th anniversary of the Institute of Mathematics, Statistics and Scientific Computing (IMECC) of the University of Campinas, Brazil, by offering reviews of selected research developed at one of the most prestigious mathematics institutes in Latin America. Written by senior professors at the IMECC, it covers topics in pure and applied mathematics and statistics ranging from differential geometry, dynamical systems, Lie groups, and partial differential equations to computational optimization, mathematical physics, stochastic process, time series, and more. A report on the challenges and opportunities of research in applied mathematics - a highly active field of research in the country - and highlights of the Institute since its foundation in 1968 completes this historical volume, which is unveiled in the same year that the International Mathematical Union (IMU) names Brazil as a member of the Group V of countries with the most relevant contributions in mathematics.-- Provided by publisher Preface 5 Contents 7 Contributors 9 ``And Now We're in 2018...'' 11 Applied Mathematics in Brazil: Challenges and Opportunities 18 1 Highlights of a Career in Applied Mathematics 18 2 Some Historical Notes 22 3 Problem-Oriented Science 24 4 Funding of Education of Human Resources 25 5 Funding for Research, Development and Innovation 26 6 New Trends and Scenarios 28 7 Challenges and Opportunities 29 8 Summary and Conclusions 31 Nomenclature/Acronyms 31 The Biomathematics in IMECC 33 33 Review of Some of the Papers Published by the Biomathematics Group of IMECC: Unicamp 37 Fuzzy Theory and Biomathematics 53 Phase Field: A Methodology to Model Complex Material Behavior 74 1 Introduction 74 2 Some Representative References 79 3 Diffusification Approach 80 4 The Energetic Variational Approach 82 4.1 Phase Field Equations (Isothermal Processes) 82 4.2 Phase Field Equation Coupled with the Equation for the Macroscopic Motion (Isothermal Processes) 85 5 The Entropy Approach 90 5.1 General Governing Equations 91 5.2 Constitutive Relations 95 5.3 Further Commentaries 103 References 105 Spherical Codes from Lattices 111 1 Introduction 111 2 Lattices 112 2.1 Quotient of Lattices and q-Ary Codes 114 3 Spherical Codes 115 4 Flat Tori 116 5 Commutative Group Codes 118 6 Constructive Spherical Codes from Lattices 121 6.1 Commutative Group Codes Obtained from Quotient of Lattices with Good Packing Density 121 6.2 Optimum Commutative Group Codes 123 6.3 A Heuristic Method for Large Number of Points and Higher Dimensions 125 7 Spherical Codes on Torus Layers 127 8 Continuous Constructions 130 8.1 Continuous Curves and Secrecy 132 References 133 Nonvariational Semilinear Elliptic Systems 136 1 Introduction 136 2 On Variational Methods 137 3 Nonvariational Elliptic Systems 138 3.1 Estimates Using Hardy-Type Inequalities 139 3.2 Estimates Using Moving Planes 142 3.3 The Blow-Up Method 143 4 Liouville Theorems 147 References 152 Perfect Simulation and Convex Mixture of Context Trees 157 1 Introduction 157 2 Stochastic Chains with Unbounded Memory 159 2.1 Transition Kernels and Compatible Chains 160 2.2 Probabilistic Context Tree 163 3 Convex Mixtures of Kernels 164 3.1 Continuous Case: Convex Mixture of Markov Kernels 165 3.2 Dropping the Continuity Assumption 166 3.2.1 The Context Tree Assumption 166 3.2.2 Convex Mixture of Probabilistic Context Trees 167 4 Perfect Simulation Based on Convex Mixture of Unbounded Probabilistic Context Trees 168 4.1 The Convex Mixture of Unbounded Probabilistic Context Trees 169 4.2 Proof of Theorem 1: Construction of a Triplet 170 4.2.1 Definition of the First Partition of [0,1[ 170 4.2.2 Definition of the Second Partition of [0,1[ 171 4.2.3 Definition of the Triplet of Parameters ({λk}k≥-1,{p-1(a)}aA,{(τk,pτk)}k≥0) 171 4.2.4 What About vτ0 Having Infinite Size? 172 4.2.5 Proof of Theorem 1 172 4.3 Coupling from the Past 173 4.4 The Algorithm 175 5 Complete Description of a Simple Example on A={1,2} 177 6 Recent Bibliography and Some Open Problems 179 References 181 Inference in (M)GARCH Models in the Presence of Additive Outliers: Specification, Estimation, and Prediction 183 1 Introduction 183 2 GARCH Model 184 2.1 Uncontaminated GARCH Models 185 2.2 Contaminated GARCH Model 185 2.3 Parameter and Volatility Estimation 185 2.4 Forecast Densities 186 3 MGARCH Models 187 3.1 Uncontaminated cDCC Model 187 3.2 Contaminated cDCC Model 187 3.3 Parameters and Model Estimation 188 3.4 Forecast Densities 188 4 Effects of Outliers 189 4.1 Effects on Specification 189 4.2 Effects on Estimation 189 4.3 Effects on Volatility Estimation and Prediction 190 5 Detection of Outliers 191 5.1 Lagrange Multiplier and Likelihood Ratio Tests 191 5.2 Test Based on ARMA Representation 191 5.3 Test Based on Wavelets 192 5.4 Test for MGARCH Models 192 5.5 Influential Observation 192 6 Robustness 193 6.1 Parameter Estimation 193 6.1.1 QMLt Estimator 193 6.1.2 BM Estimator 194 6.1.3 BQMLt Estimator 195 6.1.4 MT Estimator 195 6.1.5 BVT Estimator 196 6.2 Volatility Estimation 197 6.3 Correlation Estimation 197 6.4 Forecasting 199 6.4.1 Mancini and Trojani Algorithm 199 6.4.2 Trucíos, Hotta, and Ruiz Algorithm 200 6.4.3 Trucíos, Hotta, and Ruiz Algorithm: Multivariate Version 201 7 Conclusion and Final Remarks 202 References 203 Notes on Newton's Method After 1960 207 1 Introduction 207 2 Quasi-Newton Age 209 3 Linear Programming 211 4 Convergence and Complexity in Unconstrained Optimization 214 5 Newton in Constrained Optimization 219 References 221 Minimal Surfaces and Their Gauss Maps 223 1 Introduction 223 2 Stability 227 3 The Weierstrass Representation Formula 237 4 On the Image of the Gauss Map: The General Case 241 5 On the Image of the Gauss Map: The Finite Total Curvature Case 242 6 Work in Progress and Some Problems 247 References 248 Galois Theories: A Survey 250 1 A Brief Introduction 250 2 A Bit of the Starting History 251 3 A Comment 251 4 Definition Theorem 252 4.1 On Finite Field Extensions 252 4.2 On Commutative Ring Extensions 253 On Separability 253 On Strong Distinctness 254 On SG-Modules 254 4.3 On Ring Extensions 256 4.3.1 Still on Group Actions 256 4.3.2 On Hopf Actions 257 5 Correspondence Theorem 261 5.1 On Group Actions 261 5.1.1 In the Classical Galois Theory for Field Extensions 261 5.1.2 In the CHR Galois Theory for Commutative Ring Extensions 262 5.1.3 In the Grothendieck's Approach for Group Actions on Sets 263 5.1.4 In the VZ Galois Theory for Commutative Ring Extensions 264 5.1.5 In the Noncommutative Ring Context 265 5.2 On Hopf Actions 266 6 Partial Actions 267 6.1 Partial Group Actions 267 6.2 Partial Hopf Actions 269 7 Final Comments 272 References 272 On the Geometry and Topology of the Commutatorof Unit Quaternions 277 1 Introduction and Our Motivation 277 2 Duran's Idea 280 3 Linear Algebra 284 4 Infinitesimal Triality and G2"0362G2 SO"0362SO(7) 289 5 Generators of Some Homotopy Groups 290 6 Hopf Maps 292 7 The Geometry of the Commutator and Exotic Phenomena 293 8 The G-M Sphere 294 9 Exotic Involutions 296 10 Non-cancellation Phenomena 297 11 An Infinite Family of Gromoll–Meyer Spheres 298 12 Homotopy Revisited 299 References 300 Life in the Rindler Reference Frame: Does a Uniformly Accelerated Charge Radiate? Is There a Bell `Paradox'? Is Unruh Effect Real? 302 1 Introduction 302 2 Rindler Reference Frame 304 2.1 Rindler Coordinates 306 2.2 Decomposition of DR 307 2.3 Constant Proper Distance Between σ and σ 308 3 Bell `Paradox' 310 4 Does a Charge in Hyperbolic Motion Radiate? 312 4.1 The Answer Given by the Liénard-Wiechert Potential 312 4.2 Pauli's Answer 315 4.2.1 Calculation of Components of the Potentials in the R Frame 317 4.3 Is Pauli Argument Correct? 318 4.4 The Rindler (Pseudo) Energy 319 4.5 The Turakulov Solution 320 4.5.1 Does the Turakulov Solution Imply that a Charge in Hyperbolic Motion Does Not Radiate? 323 5 The Equivalence Principle 325 6 Some Comments on the Unruh Effect 327 6.1 Minkowski and Fulling-Unruh Quantization of the Klein-Gordon Field 327 6.2 ``Deduction'' of the Unruh Effect 332 7 Conclusions 338 Appendix 1: Some Notations and Definitions 339 Appendix 2: C Algebras and the Unruh ``Effect'' 342 References 347 Flag Type of Semigroups: A Survey 350 1 Control Sets and Flag Type 350 2 Topological Properties 355 3 Maximal Semigroups 356 4 Integration on Semigroups 357 4.1 Poisson Spaces 357 4.2 Characteristic Function 358 4.3 Moment Lyapunov Exponents 360 5 Controllability and Transitive Actions 361 6 Dynamical Systems 364 References 368 Generic Singularities of 3D Piecewise Smooth Dynamical Systems 372 1 Introduction 372 2 Setting the Problem 374 2.1 Filippov Systems 374 2.2 Σ-Equivalence 375 2.3 Reversible Mappings 376 3 Generic Singularities 376 4 Statement of the Main Results 378 5 Fold–Fold Singularity 379 5.1 A Normal Form 379 5.2 Sliding Dynamics 381 6 Proofs of Theorems 3 and 4 382 7 Proofs of Theorems 4, 5 and Corollary 4.1 397 7.1 Hyperbolic Fold–Fold 397 7.2 Parabolic Fold–Fold 398 7.3 Proof of Theorem 4 402 7.4 Proof of Theorem 5 402 7.5 Proof of Corollary 4.1 402 References 402 Appendix A Non-smooth Dynamical Systems (NSDS): Reflections and Guidelines 404 A.1 Introduction 404 A.1.1 Some Words from Mauricio Peixoto 405 A.2 Some of Non-smooth Dynamical Systems 405 A.3 Miscellaneous in Geometric and Qualitative Theory in Non-smooth Dynamical Systems 406 A.4 Conclusion 407 References 408 Front Matter ....Pages i-x “And Now We’re in 2018...” (João Frederico da Costa Azevedo Meyer)....Pages 1-7 Applied Mathematics in Brazil: Challenges and Opportunities (Martin Tygel)....Pages 9-23 The Biomathematics in IMECC (Rodney Carlos Bassanezi)....Pages 25-65 Phase Field: A Methodology to Model Complex Material Behavior (José Luiz Boldrini)....Pages 67-103 Spherical Codes from Lattices (Sueli I. R. Costa, João E. Strapasson, Cristiano Torezzan)....Pages 105-129 Nonvariational Semilinear Elliptic Systems (Djairo G. de Figueiredo)....Pages 131-151 Perfect Simulation and Convex Mixture of Context Trees (Nancy L. Garcia, Sandro Gallo)....Pages 153-178 Inference in (M)GARCH Models in the Presence of Additive Outliers: Specification, Estimation, and Prediction (Luiz Koodi Hotta, Carlos Trucíos)....Pages 179-202 Notes on Newton’s Method After 1960 (José Mario Martínez)....Pages 203-218 Minimal Surfaces and Their Gauss Maps (Francesco Mercuri, Luquesio P. M. Jorge)....Pages 219-245 Galois Theories: A Survey (Antonio Paques)....Pages 247-273 On the Geometry and Topology of the Commutator of Unit Quaternions (Alcibiades Rigas, Dan A. Agüero Cerna)....Pages 275-299 Life in the Rindler Reference Frame: Does a Uniformly Accelerated Charge Radiate? Is There a Bell ‘Paradox’? Is Unruh Effect Real? (Waldyr A. Rodrigues Jr., Jayme Vaz Jr.)....Pages 301-348 Flag Type of Semigroups: A Survey (Luiz A. B. San Martin)....Pages 351-372 Generic Singularities of 3D Piecewise Smooth Dynamical Systems (Marco Antonio Teixeira, Otávio M. L. Gomide)....Pages 373-404 Back Matter ....Pages 403-407 This book celebrates the 50th anniversary of the Institute of Mathematics, Statistics and Scientific Computing (IMECC) of the University of Campinas, Brazil, by offering reviews of selected research developed at one of the most prestigious mathematics institutes in Latin America. Written by senior professors at the IMECC, it covers topics in pure and applied mathematics and statistics ranging from differential geometry, dynamical systems, Lie groups, and partial differential equations to computational optimization, mathematical physics, stochastic process, time series, and more. A report on the challenges and opportunities of research in applied mathematics - a highly active field of research in the country - and highlights of the Institute since its foundation in 1968 completes this historical volume, which is unveiled in the same year that the International Mathematical Union (IMU) names Brazil as a member of the Group V of countries with the most relevant contributions in mathematics.-- Back cover