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ldquo9781118853986prerdquo mdash 2015421 mdash 911 mdash page vi mdash 6

ldquo9781118853986prerdquo mdash 2015421 mdash 911 mdash page i mdash 1

MATHEMATICAL AND COMPUTATIONALMODELING

ldquo9781118853986prerdquo mdash 2015421 mdash 911 mdash page ii mdash 2

PURE AND APPLIED MATHEMATICS

A Wiley Series of Texts Monographs and Tracts

Founded by RICHARD COURANTEditors Emeriti MYRON B ALLEN III PETER HILTON HARRYHOCHSTADT ERWIN KREYSZIG PETER LAX JOHN TOLAND

A complete list of the titles in this series appears at the end of this volume

ldquo9781118853986prerdquo mdash 2015421 mdash 911 mdash page iii mdash 3

MATHEMATICAL ANDCOMPUTATIONALMODELING

With Applications in Natural andSocial Sciences Engineeringand the Arts

Edited by

RODERICK MELNIKWilfrid Laurier UniversityWaterloo Ontario Canada

ldquo9781118853986prerdquo mdash 2015421 mdash 911 mdash page iv mdash 4

Copyright copy 2015 by John Wiley amp Sons Inc All rights reserved

Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada

No part of this publication may be reproduced stored in a retrieval system or transmitted in any form orby any means electronic mechanical photocopying recording scanning or otherwise except aspermitted under Section 107 or 108 of the 1976 United States Copyright Act without either the priorwritten permission of the Publisher or authorization through payment of the appropriate per-copy fee tothe Copyright Clearance Center Inc 222 Rosewood Drive Danvers MA 01923 (978) 750-8400 fax(978) 750-4470 or on the web at wwwcopyrightcom Requests to the Publisher for permission shouldbe addressed to the Permissions Department John Wiley amp Sons Inc 111 River Street HobokenNJ 07030 (201) 748-6011 fax (201) 748-6008 or online at httpwwwwileycomgopermissions

Limit of LiabilityDisclaimer of Warranty While the publisher and author have used their best efforts inpreparing this book they make no representations or warranties with respect to the accuracy orcompleteness of the contents of this book and specifically disclaim any implied warranties ofmerchantability or fitness for a particular purpose No warranty may be created or extended by salesrepresentatives or written sales materials The advice and strategies contained herein may not be suitablefor your situation You should consult with a professional where appropriate Neither the publisher norauthor shall be liable for any loss of profit or any other commercial damages including but not limited tospecial incidental consequential or other damages

For general information on our other products and services or for technical support please contact ourCustomer Care Department within the United States at (800) 762-2974 outside the United States at(317) 572-3993 or fax (317) 572-4002

Wiley also publishes its books in a variety of electronic formats Some content that appears in print maynot be available in electronic formats For more information about Wiley products visit our web site atwwwwileycom

Library of Congress Cataloging-in-Publication Data

Mathematical and computational modeling with applications in natural and social sciences engineeringand the arts edited by Roderick Melnik Wilfrid Laurier University Waterloo Ontario Canada

pages cmIncludes bibliographical references and indexISBN 978-1-118-85398-6 (cloth)

1 Mathematical models I Melnik Roderick editorQA401M3926 2015511prime8mdashdc23

2014043267

Set in 1012pts Times Lt Std by SPi Publisher Services Pondicherry India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

1 2015

ldquo9781118853986prerdquo mdash 2015421 mdash 911 mdash page v mdash 5

Teachers who educate children deserve more honor than parents who merely gavebirth for bare life is furnished by the one the other ensures a good life

(Aristotle)

To my parents who were my first and most devoted teachers


ldquo9781118853986prerdquo mdash 2015421 mdash 1132 mdash page vii mdash 7

CONTENTS

LIST OF CONTRIBUTORS xiii

PREFACE xv

SECTION 1 INTRODUCTION 1

1 Universality of Mathematical Models in Understanding NatureSociety and Man-Made World 3Roderick Melnik

11 Human Knowledge Models and Algorithms 312 Looking into the Future from a Modeling Perspective 713 What This Book Is About 1014 Concluding Remarks 15References 16

SECTION 2 ADVANCED MATHEMATICAL AND COMPUTATIONALMODELS IN PHYSICS AND CHEMISTRY 17

2 Magnetic Vortices Abrikosov Lattices and AutomorphicFunctions 19Israel Michael Sigal

21 Introduction 19

ldquo9781118853986prerdquo mdash 2015421 mdash 1132 mdash page viii mdash 8

viii CONTENTS

22 The GinzburgndashLandau Equations 20221 GinzburgndashLandau energy 21222 Symmetries of the equations 21223 Quantization of flux 22224 Homogeneous solutions 22225 Type I and Type II superconductors 23226 Self-dual case κ=1

radic2 24

227 Critical magnetic fields 24228 Time-dependent equations 25

23 Vortices 25231 n-vortex solutions 25232 Stability 26

24 Vortex Lattices 30241 Abrikosov lattices 31242 Existence of Abrikosov lattices 31243 Abrikosov lattices as gauge-equivariant states 34244 Abrikosov function 34245 Comments on the proofs of existence results 35246 Stability of Abrikosov lattices 40247 Functions γδ(τ) δ gt 0 42248 Key ideas of approach to stability 45

25 Multi-Vortex Dynamics 4826 Conclusions 51Appendix 2A Parameterization of the equivalence classes [L] 51Appendix 2B Automorphy factors 52References 54

3 Numerical Challenges in a Cholesky-Decomposed Local CorrelationQuantum Chemistry Framework 59David B Krisiloff Johannes M Dieterich Florian Libischand Emily A Carter

31 Introduction 5932 Local MRSDCI 61

321 MRSDCI 61322 Symmetric group graphical approach 62323 Local electron correlation approximation 64324 Algorithm summary 66

33 Numerical Importance of Individual Steps 6734 Cholesky Decomposition 6835 Transformation of the Cholesky Vectors 7136 Two-Electron Integral Reassembly 7237 Integral and Execution Buffer 7638 Symmetric Group Graphical Approach 7739 Summary and Outlook 87References 87

ldquo9781118853986prerdquo mdash 2015421 mdash 1132 mdash page ix mdash 9

CONTENTS ix

4 Generalized Variational Theorem in Quantum Mechanics 92Mel Levy and Antonios Gonis

41 Introduction 9242 First Proof 9343 Second Proof 9544 Conclusions 96References 97

SECTION 3 MATHEMATICAL AND STATISTICAL MODELS IN LIFEAND CLIMATE SCIENCE APPLICATIONS 99

5 A Model for the Spread of Tuberculosis with Drug-Sensitiveand Emerging Multidrug-Resistant and ExtensivelyDrug-Resistant Strains 101Julien Arino and Iman A Soliman

51 Introduction 101511 Model formulation 102512 Mathematical Analysis 107

5121 Basic properties of solutions 1075122 Nature of the disease-free equilibrium 1085123 Local asymptotic stability of the DFE 1085124 Existence of subthreshold endemic equilibria 1105125 Global stability of the DFE when the

bifurcation is ldquoforwardrdquo 1135126 Strain-specific global stability in ldquoforwardrdquo

bifurcation cases 11552 Discussion 117References 119

6 The Need for More Integrated Epidemic Modeling with Emphasison Antibiotic Resistance 121Eili Y Klein Julia Chelen Michael D Makowskyand Paul E Smaldino

61 Introduction 12162 Mathematical Modeling of Infectious Diseases 12263 Antibiotic Resistance Behavior and Mathematical

Modeling 125631 Why an integrated approach 125632 The role of symptomology 127

64 Conclusion 128References 129

ldquo9781118853986prerdquo mdash 2015421 mdash 1132 mdash page x mdash 10

x CONTENTS

SECTION 4 MATHEMATICAL MODELS AND ANALYSIS FORSCIENCE AND ENGINEERING 135

7 Data-Driven Methods for Dynamical Systems QuantifyingPredictability and Extracting Spatiotemporal Patterns 137Dimitrios Giannakis and Andrew J Majda

71 Quantifying Long-Range Predictability and Model Errorthrough Data Clustering and Information Theory 138711 Background 138712 Information theory predictability and model error 140

7121 Predictability in a perfect-model environment 1407122 Quantifying the error of imperfect models 143

713 Coarse-graining phase space to reveal long-rangepredictability 1447131 Perfect-model scenario 1447132 Quantifying the model error in long-range

forecasts 147714 K-means clustering with persistence 149715 Demonstration in a double-gyre ocean model 152

7151 Predictability bounds for coarse-grainedobservables 154

7152 The physical properties of the regimes 1577153 Markov models of regime behavior in the

15-layer ocean model 1597154 The model error in long-range predictions with

coarse-grained Markov models 16272 NLSA Algorithms for Decomposition of Spatiotemporal Data 163

721 Background 163722 Mathematical framework 165

7221 Time-lagged embedding 1667222 Overview of singular spectrum analysis 1677223 Spaces of temporal patterns 1677224 Discrete formulation 1697225 Dynamics-adapted kernels 1717226 Singular value decomposition 1737227 Setting the truncation level 1747228 Projection to data space 175

723 Analysis of infrared brightness temperature satellite datafor tropical dynamics 1757231 Dataset description 1767232 Modes recovered by NLSA 1767233 Reconstruction of the TOGA COARE MJOs 183

73 Conclusions 184References 185

ldquo9781118853986prerdquo mdash 2015421 mdash 1132 mdash page xi mdash 11

CONTENTS xi

8 On Smoothness Concepts in Regularization for Nonlinear InverseProblems in Banach Spaces 192Bernd Hofmann

81 Introduction 19282 Model Assumptions Existence and Stability 19583 Convergence of Regularized Solutions 19784 A Powerful Tool for Obtaining Convergence Rates 20085 How to Obtain Variational Inequalities 206

851 Bregman distance as error measure the benchmark case 206852 Bregman distance as error measure violating the

benchmark 210853 Norm distance as error measure 1-regularization 213

86 Summary 215References 215

9 Initial and Initial-Boundary Value Problems for First-OrderSymmetric Hyperbolic Systems with Constraints 222Nicolae Tarfulea

91 Introduction 22292 FOSH Initial Value Problems with Constraints 223

921 FOSH initial value problems 224922 Abstract formulation 225923 FOSH initial value problems with constraints 228

93 FOSH Initial-Boundary Value Problems with Constraints 230931 FOSH initial-boundary value problems 232932 FOSH initial-boundary value problems with constraints 234

94 Applications 236941 System of wave equations with constraints 237942 Applications to Einsteinrsquos equations 240

9421 EinsteinndashChristoffel formulation 2439422 AlekseenkondashArnold formulation 246

References 250

10 Information Integration Organization and NumericalHarmonic Analysis 254Ronald R Coifman Ronen Talmon Matan Gavishand Ali Haddad

101 Introduction 254102 Empirical Intrinsic Geometry 257

1021 Manifold formulation 2591022 Mahalanobis distance 261

103 Organization and Harmonic Analysis of DatabasesMatrices 263

ldquo9781118853986prerdquo mdash 2015421 mdash 1132 mdash page xii mdash 12

xii CONTENTS

1031 Haar bases 2641032 Coupled partition trees 265


SECTION 5 MATHEMATICAL METHODS IN SOCIAL SCIENCESAND ARTS 273

11 Satisfaction Approval Voting 275Steven J Brams and D Marc Kilgour

111 Introduction 275112 Satisfaction Approval Voting for Individual Candidates 277113 The Game Theory Society Election 285114 Voting for Multiple Candidates under SAV

A Decision-Theoretic Analysis 287115 Voting for Political Parties 291

1151 Bullet voting 2911152 Formalization 2921153 Multiple-party voting 294

116 Conclusions 295117 Summary 296References 297

12 Modeling Musical Rhythm Mutations with Geometric Quantization 299Godfried T Toussaint

121 Introduction 299122 Rhythm Mutations 301

1221 Musicological rhythm mutations 3011222 Geometric rhythm mutations 302

123 Similarity-Based Rhythm Mutations 3031231 Global rhythm similarity measures 304


INDEX 309

ldquo9781118853986prerdquo mdash 2015421 mdash 914 mdash page xiii mdash 13

LIST OF CONTRIBUTORS

Julien Arino Department of Mathematics University of Manitoba WinnipegCanada

Steven J Brams Department of Politics New York University New York NY USA

Emily A Carter Department of Mechanical and Aerospace Engineering PrincetonUniversity Princeton NJ USA and Program in Applied and Computational Mathe-matics and Andlinger Center for Energy and the Environment Princeton UniversityPrinceton NJ USA

Julia Chelen Center for Advanced Modeling in the Social Behavioral and HealthSciences Department of Emergency Medicine Johns Hopkins University BaltimoreMD USA

Ronald R Coifman Mathematics Department Yale University New Haven CTUSA

Johannes M Dieterich Department of Mechanical and Aerospace EngineeringPrinceton University Princeton NJ USA

Matan Gavish Statistics Department Stanford University Palo Alto CA USA

Dimitrios Giannakis Courant Institute of Mathematical Sciences New YorkUniversity New York NY USA

Antonios Gonis Physics and Life Sciences Lawrence Livermore NationalLaboratory Livermore CA USA

Ali Haddad Mathematics Department Yale University New Haven CT USA

ldquo9781118853986prerdquo mdash 2015421 mdash 914 mdash page xiv mdash 14

xiv LIST OF CONTRIBUTORS

Bernd Hofmann Faculty of Mathematics Technische Universitaumlt ChemnitzChemnitz Germany

D Marc Kilgour Department of Mathematics Wilfrid Laurier University WaterlooOntario Canada

Eili Y Klein Center for Advanced Modeling in the Social Behavioral and HealthSciences Department of Emergency Medicine Johns Hopkins University BaltimoreMD USA and Center for Disease Dynamics Economics amp Policy Washington DCUSA

David B Krisiloff Department of Chemistry Princeton University Princeton NJUSA

Mel Levy Department of Chemistry Duke University Durham NC USA Depart-ment of Physics North Carolina AampT State University Greensboro NC USAand Department of Chemistry and Quantum Theory Group School of Science andEngineering Tulane University New Orleans LO USA

Florian Libisch Department of Mechanical and Aerospace Engineering PrincetonUniversity Princeton NJ USA

Andrew J Majda Courant Institute of Mathematical Sciences New YorkUniversity New York NY USA

Michael D Makowsky Center for Advanced Modeling in the Social Behavioraland Health Sciences Department of Emergency Medicine Johns Hopkins UniversityBaltimore MD USA

Roderick Melnik The MS2Discovery Interdisciplinary Research Institute M2NeTLaboratory and Department of Mathematics Wilfrid Laurier University WaterlooOntario Canada

Israel Michael Sigal Department of Mathematics University of Toronto OntarioCanada

Paul E Smaldino Center for Advanced Modeling in the Social Behavioral andHealth Sciences Department of Emergency Medicine Johns Hopkins UniversityBaltimore MD USA

Iman A Soliman Department of Mathematics Cairo University Giza Egypt

Ronen Talmon Mathematics Department Yale University New Haven CT USA

Nicolae Tarfulea Department of Mathematics Computer Science and StatisticsPurdue University Calumet Hammond IN USA

Godfried T Toussaint Department of Computer Science New York UniversityAbu Dhabi Abu Dhabi United Arab Emirates

ldquo9781118853986prerdquo mdash 2015421 mdash 914 mdash page xv mdash 15

PREFACE

Mathematical and computational modeling has become a major driving force in sci-entific discovery and innovation covering an increasing range of diverse applicationareas in the natural and social sciences engineering and the arts Mathematicalmodels methods and algorithms have been ubiquitous in human activities from theancient times till now The fundamental role they play in human knowledge as wellas in our well-being is indisputable and it continues to grow in its importance

Significant sources of some of the most urgent and challenging problems thehumanity faces today are coming not only from traditional areas of mathemat-ics applications in natural and engineering sciences but also from life behavioraland social sciences We are witnessing an unprecedented growth of model-basedapproaches in practically every domain of human activities This expands furtherinterdisciplinary horizons of mathematical and computational modeling providingnew and strengthening existing links between different disciplines and human activi-ties Integrative holistic approaches and systemsndashscience methodologies are requiredin an increasing number of areas of human endeavor In its turn such approaches andmethodologies require the development of new state-of-the-art mathematical modelsand methods

Given this wide spectrum of applications of mathematical and computational mod-eling we have selected five representative areas grouped in this book into sectionsThese sections contain 12 selective chapters written by 25 experts in their respec-tive fields They open to the reader a broad range of methods and tools important inmany applications across different disciplines The book provides details on state-of-the-art achievements in the development of these methods and tools as well astheir applications Original results are presented on both fundamental theoretical andapplied developments with many examples emphasizing interdisciplinary nature of

ldquo9781118853986prerdquo mdash 2015421 mdash 914 mdash page xvi mdash 16

xvi PREFACE

mathematical and computational modeling and universality of models in our betterunderstanding nature society and the man-made world

Aimed at researchers in academia practitioners and graduate students the bookpromotes interdisciplinary collaborations required to meet the challenges at the inter-face of different disciplines on the one hand and mathematical and computationalmodeling on the other It can serve as a reference on theory and applications of math-ematical and computational modeling in diverse areas within the natural and socialsciences engineering and the arts

I am thankful to many of my colleagues in North America Europe Asia andAustralia whose encouragements were vital for the completion of this project Specialthanks go to the referees of this volume Their help and suggestions were invaluableFinally I am very grateful to the John Wiley amp Sons editorial team and in particularSusanne Steitz-Filler and Sari Friedman for their highly professional support

Roderick MelnikWaterloo ON CanadaAugust 2014ndash2015

ldquo9781118853986S01rdquo mdash 2015421 mdash 701 mdash page 1 mdash 1

SECTION 1

INTRODUCTION


ldquo9781118853986c01rdquo mdash 2015421 mdash 1054 mdash page 3 mdash 1

1UNIVERSALITY OF MATHEMATICALMODELS IN UNDERSTANDINGNATURE SOCIETY AND MAN-MADEWORLD

Roderick MelnikThe MS2Discovery Interdisciplinary Research Institute M2NeT Laboratory and Departmentof Mathematics Wilfrid Laurier University Waterloo Ontario Canada

11 HUMAN KNOWLEDGE MODELS AND ALGORITHMS

There are various statistical and mathematical models of the accumulation of humanknowledge Taking one of them as a starting point the Anderla model we wouldlearn that the amount of human knowledge about 40 years ago was 128 times greaterthan in the year ad 1 We also know that this has increased drastically over thelast four decades However most such models are economics-based and account fortechnological developments only while there is much more in human knowledgeto account for Human knowledge has always been linked to models Such modelscover a variety of fields of human endeavor from the arts to agriculture from thedescription of natural phenomena to the development of new technologies and tothe attempts of better understanding societal issues From the dawn of human civi-lization the development of these models in one way or another has always beenconnected with the development of mathematics These two processes the develop-ment of models representing the core of human knowledge and the development ofmathematics have always gone hand in hand with each other From our knowledge

Mathematical and Computational Modeling With Applications in Natural and Social SciencesEngineering and the Arts First Edition Roderick Melnikcopy 2015 John Wiley amp Sons Inc Published 2015 by John Wiley amp Sons Inc


4 UNIVERSALITY OF MATHEMATICAL MODELS

in particle physics and spin glasses [4 6] to life sciences and neuron stars [1 5 16]universality of mathematical models has to be seen from this perspective

Of course the history of mathematics goes back much deeper in the dawn of civ-ilizations than ad 1 as mentioned earlier We know for example that as early asin the 6thndash5th millennium bc people of the Ancient World including predynas-tic Sumerians and Egyptians reflected their geometric-design-based models on theirartifacts People at that time started obtaining insights into the phenomena observed innature by using quantitative representations schemes and figures Geometry played afundamental role in the Ancient World With civilization settlements and the develop-ment of agriculture the role of mathematics in general and quantitative approaches inparticular has substantially increased From the early times of measurements of plotsof lands and of the creation of the lunar calendar the Sumerians and Babyloniansamong others were greatly contributing to the development of mathematics Weknow that from those times onward mathematics has never been developed in iso-lation from other disciplines The cross-fertilization between mathematical sciencesand other disciplines is what produces one of the most valuable parts of human knowl-edge Indeed mathematics has a universal language that allows other disciplines tosignificantly advance their own fields of knowledge hence contributing to humanknowledge as a whole Among other disciplines the architecture and the arts havebeen playing an important role in this process from as far in our history as we cansee Recall that the summation series was the origin of harmonic design This tech-nique was known in the Ancient Egypt at least since the construction of the ChephrenPyramid of Giza in 2500 BCE (the earliest known is the Pyramid of Djoser likelyconstructed between 2630 BCE and 2611 BCE) The golden ratio and Fibonaccisequence have deep roots in the arts including music as well as in the naturalsciences Speaking of mathematics H Poincare once mentioned that ldquoit is the unex-pected bringing together of diverse parts of our science which brings progressrdquo [11]However this is largely true with respect to other sciences as well and more gener-ally to all branches of human endeavor Back to Poincarersquos time it was believed thatmathematics ldquoconfines itself at the same time to philosophy and to physics and itis for these two neighbors that we workrdquo [11] Today the quantitative analysis as anessential tool in the mathematics arsenal along with associated mathematical statis-tical and computational models advances knowledge in pretty much every domainof human endeavor The quantitative-analysis-based models are now rooted firmlyin the application areas that were only recently (by historical account) considered asnon-traditional for conventional mathematics This includes but not limited to lifesciences and medicine user-centered design and soft engineering new branches ofarts business and economics social behavioral and political sciences

Recognition of universality of mathematical models in understanding nature soci-ety and man-made world is of ancient origin too Already Pythagoras taught that inits deepest sense the reality is mathematical in nature The origin of quantification ofscience goes back at least to the time of Pythagorasrsquo teaching that numbers provide akey to the ultimate reality The Pythagorean tradition is well reflected in the Galileostatement that ldquothe Book of Nature is written in the language of mathematicsrdquo Todaywe are witnessing the areas of mathematics applications not only growing rapidly in


HUMAN KNOWLEDGE MODELS AND ALGORITHMS 5

more traditional natural and engineering sciences but also in social and behavioralsciences as well It should be noted that the term ldquouniversalityrdquo is also used in theliterature in different more specific and narrow contexts For example in statisticalmechanics universality is the observation that there are properties for a large class ofsystems that are independent of the dynamical details of the system A pure mathe-matical definition of a universal property is usually given based on representations ofcategory theory Another example is provided by computer science and computabil-ity theory where the word ldquouniversalrdquo is usually applied to a system which is Turingcomplete There is also a universality principle a system property often modeled byrandom matrices These concepts are useful for corresponding mathematical or sta-tistical models and are subject of many articles (see eg [2ndash71416] and referencestherein) For example the authors of Ref [2] discuss universality classes for com-plex networks with possible applications in social and biological dynamic systemsA universal scaling limit for a class of Ising-type mathematical models is discussed inRef [6] The concept of universality of predictions is discussed in Ref [14] within theBayesian framework Computing universality is a subject of discussions in Ref [3]while universality in physical and life sciences are discussed in Refs [7] and [5]respectively Given a brief historical account demonstrating the intrinsic presence ofmodels in human knowledge from the dawn of civilizations ldquouniversalityrdquo here isunderstood in a more general Aristotlersquos sense ldquoTo say of what is that it is notor of what is not that it is is false while to say of what is that it is and of whatis not that it is not is truerdquo The underlying reason for this universality lies withthe fact that models are inherently linked to algorithms From the ancient times tillnow human activities and practical applications have stimulated the developmentof model-based algorithms If we note that abstract areas of mathematics are alsobased on models it can be concluded that mathematical algorithms have been at theheart of the development of mathematics itself The word ldquoalgorithmrdquo was derivedfrom Al-Khwarizmi (c 780 ndash c 850) a mathematician astronomer and geographerwhose name was given to him by the place of his birth (Khwarezm or Chorasmia)The word indicated a technique with numerals Such techniques were present inhuman activities well before the ninth century while specific algorithms mainly stim-ulated by geometric considerations at that time were also known Examples includealgorithms for approximating the area of a given circle (known to Babylonians andIndians) an algorithm for calculating π by inscribing and then circumscribing a poly-gon around a circle (known to Antiphon and Bryson already in the fifth century bc)Euclidrsquos algorithm to determine the greatest common divisor of two integers andmany others Further development of the subject was closely interwoven with appli-cations and other disciplines It led to what in the second part of the twentieth centurywas called by E Wigner as ldquothe unreasonable effectiveness of mathematics in the nat-ural sciencesrdquo In addition to traditional areas of natural sciences and engineering thetwentieth century saw an ever increasing role of mathematical models in the life andenvironmental sciences too This development was based on earlier achievementsIndeed already during the 300 bc Aristotle studied the manner in which speciesevolve to fit their environment His works served as an important stepping stone in thedevelopment of modern evolutionary theories and his holistic views and teaching that



ldquothe whole is more than the sum of its partsrdquo helped the progress of systems sciencein general and systems biology in particular A strong growth of genetics and popu-lation biology in the twentieth century effectively started from the rediscovery of GMendelrsquos laws in 1900 (originally published in 1865ndash1866) and a paramount impe-tus for this growth to be linked with mathematical models was given by R A FisherrsquosFundamental Theorem of Natural Selection in 1930 This result was based on a par-tial differential equation (PDE) expressing the rate of fitness increase for any livingorganism Mathematical models in other areas of life sciences were also develop-ing and included A J Lotka and V Volterrarsquos predatorndashprey systems (1925ndash1931)A A Malinovskyrsquos models for evolutionary genetics and systems analysis (1935)R Fisher and A Kolmogorov equation for gene propagation (1937) A L Hodgkinand A F Huxleyrsquos equations for neural axon membrane potential (1952) to name justa few New theories such as self-organization and biological pattern formation haveappeared demonstrating the powerful cross-fertilization between mathematics andthe life sciences (see additional details in Ref [1]) More recently the ready availabil-ity of detailed molecular functional and genomic data has led to the unprecedenteddevelopment of new data-driven mathematical models As a result the tools of math-ematical modeling and computational experiment are becoming largely important intodayrsquos life sciences The same conclusion applies to environmental earth and cli-mate sciences as well Based on the data since 1880 by now we know that globalwarming has been mostly caused by the man-made world with its emission from theburning of fossil fuels environmental pollution and other factors In moving for-ward we will need to solve many challenging environmental problems and the roleof mathematical and computational modeling in environmental earth and climatesciences will continue to increase [13]

Mathematical models and algorithms have become essential for many profes-sionals in other areas including sociologists financial analysts political scientistspublic administration workers and the governments [12] with this list continuing togrow Our discussion would be incomplete if we do not mention here a deep connec-tion between mathematics and the arts Ancient civilizations including EgyptiansMesopotamians and Chinese studied the mathematics of sound and the AncientGreeks investigated the expression of musical scales in terms of the ratios of smallintegers They considered harmony as a branch of science known now as musicalacoustics They worked to demonstrate that the mathematical laws of harmonics andrhythms have a fundamental character not only to the understanding of the world butalso to human happiness and prosperity While a myriad of examples of the intrin-sic connection between mathematics and the arts are found in the Ancient Worldundoubtedly the Renaissance brought an enriched rebirth of classical ancient worldcultures and mathematical ideas not only for better understanding of nature but alsofor the arts Indeed painting three-dimensional scenes on a two-dimensional canvaspresents just one example where such a connection was shown to be critical Notonly philosophers but artists too were convinced that the whole universe includ-ing the arts could be explained with geometric and numerical techniques There aremany examples from the Renaissance period where painters were also mathemati-cians with Piero della Francesca (c1415ndash1492) and Leonardo da Vinci (1452ndash1519)


LOOKING INTO THE FUTURE FROM A MODELING PERSPECTIVE 7

among them Nowadays the arts and mathematics are closely interconnected con-tinuing to enrich each other There are many architectural masterpieces and paintingsthat have been preserved based on the implementation of sophisticated mathemati-cal models Efficient computer graphics algorithms have brought a new dimension tomany branches of the modern arts while a number of composers have incorporatedmathematical ideas into their works (and the golden ratio and Fibonacci numbers areamong them) Musical applications of number theory algebra and set theory amongother areas of mathematics are well known

While algorithms and models have always been central in the developmentof mathematical sciences providing an essential links to the applications theirimportance has been drastically amplified in the computer age where the role ofmathematical modeling and computational experiment in understanding nature andour world becomes paramount

12 LOOKING INTO THE FUTURE FROM A MODELING PERSPECTIVE

Although on a historical scale electronic computers belong to a very recent inventionof humans the first computing operations were performed from ancient times by peo-ple themselves From abacus to Napierrsquos Bones from the Pascaline to the LeibnitzrsquosStepped Reckoner from the Babbagersquos Difference (and then Analytic) Engine to theHollerithrsquos Desk invention as a precursor to IBM step by step we have drasticallyimproved our ability to compute Today modern computers allow us to increase pro-ductivity in intellectual performance and information processing to a level not seen inthe human history before In its turn this process leads to a rapid development of newmathematics-based algorithms that are changing the entire landscape of human activ-ities penetrating to new and unexpected areas As a result mathematical modelingexpands its interdisciplinary horizons providing links between different disciplinesand human activities It becomes pervasive across more and more disciplines whilepractical needs of human activities and applications as well as the interface betweenthese disciplines human activities mathematics and its applications stimulate thedevelopment of state-of-the-art new methods approaches and tools A special men-tion in this context deserves such areas as social behavioral and life sciences Theever expanding range of the two-way interaction between mathematical modelingand these disciplines indicates that this interaction is virtually unlimited Indeedtaking life sciences as an example the applications of mathematical algorithmsmethods and tools in drug design and delivery genetic mapping and cell dynam-ics neuroscience and bionanotechnology have become ubiquitous In the meantimenew challenges in these disciplines such as sequencing macromolecules (includ-ing those already present in biological databases) provide an important catalyst forthe development of new mathematics new efficient algorithms and methods [1]Euclidian non-Euclidian and fractal geometries as well as an intrinsic link betweengeometry and algebra highlighted by R Descartes through his coordinate systemhave all proved to be very important in these disciplines while the discovery ofwhat is now known as the Brownian motion by Scottish botanist R Brown has



revolutionized many branches of mathematics Game theory and the developmentsin control and cybernetics were influenced by the developments in social behav-ioral and life sciences while the growth of systems science has provided one of thefundamentals for the development of systems biology where biological systems areconsidered in a holistic way [1] There is a growing understanding that the interac-tions between different components of a biological system at different scales (egfrom the molecular to the systemic level) are critical Biological systems provide anexcellent example of coupled systems and multiscale dynamics A multiscale spa-tiotemporal character of most systems in nature science and engineering is intrinsicdemonstrating complex interplay of its components well elucidated in the literature(eg [8913] and references therein) In life sciences the number of such examplesof multiscale coupled systems and associated problems is growing rapidly in manydifferent albeit often interconnected areas Some examples are as follows

bull Complex biological networks genomics cellular systems biology and systemsbiological approaches in other areas studies of various organs their systemsand functions

bull Brain dynamics neuroscience and physiology developmental biology evolutionand evolutionary dynamics of biological games

bull Immunology problems epidemiology and infectious diseases drug develop-ment delivery and resistance

bull Properties dynamics and interactions at various length and time scales inbio-macromolecules including DNA RNA proteins self-assembly and spatio-temporal pattern formation in biological systems phase transitions and so on

Many mathematical and computational modeling tools are ubiquitous They areuniversal in a sense that they can be applied in many other areas of human endeavorsLife sciences have a special place when we look into the future developments of math-ematical and computational modeling Indeed compared to other areas for examplethose where we study physical or engineering systems our knowledge of biologicalsystems is quite limited One of the reasons behind this is biological system com-plexity characterized by the fact that most biological systems require dealing withmultiscale interactions of their highly heterogeneous parts on different time scales

In these cases in particular the process of mathematical and computational mod-eling becomes frequently a driving source for the development of hierarchies ofmathematical models This helps determine the range of applicability of modelsWhat is especially important is that based on such hierarchies mathematical mod-els can assist in explaining the behavior of a system under different conditions andthe interaction of different system components Clearly different models for the samesystem can involve a range of mathematical structures and can be formalized with var-ious mathematical tools such as equation- or inequality-based models graphs andlogical and game theoretic models We know by now that the class of the modelsamenable to analytical treatments while keeping assumptions realistic is strikinglysmall when compared to the general class of mathematical models that are at the fore-front of modern science and engineering [10] As a result most modern problems are



treated numerically in which case the development of efficient algorithms becomescritical As soon as such algorithms are implemented on a computer we can run themodel multiple times under varying conditions helping us to answer outstandingquestions quicker and more efficiently providing us an option to improve the modelwhen necessary Model-algorithm-implementation is a triad which is at the heart ofmathematical modeling and computational experiment It is a pervasive powerfultheoretical and practical tool covering the entire landscape of mathematical applica-tions [10] This tool will play an increasingly fundamental role in the future as we cancarry out mathematical modeling and computational experiment even in those caseswhen natural experiments are impossible At the same time given appropriate valida-tion and verification procedures we can provide reliable information more quicklyand with less expense compared to natural experiments The two-way interactionsbetween new developments in information technology and mathematical modelingand computational experiment are continuously increasing predictive capabilities andresearch power of mathematical models

Looking into the future from a modeling perspective we should also point outthat such predictive capabilities and research power allow us to deal with com-plex systems that have intrinsically interconnected (coupled) parts interacting innontrivial dynamic manner In addition to life behavioral and social sciences men-tioned earlier such systems arise in many other areas including but not limitedto fusion and energy problems materials science and chemistry high energy andnuclear physics cosmology and astrophysics earth climate environmental andsustainability sciences

In addition to the development of new models and efficient algorithms the suc-cess of predictive mathematical modeling in applications is dependent also on furtheradvances in information sciences and the development of statistical probabilistic anduncertainty quantification methods Uncertainty comes from many different sourcesamong which we will mention parameters with uncertain values uncertainty in themodel as a representation of the underlying phenomenon process or system anduncertainty in collectingprocessingmeasurements of data for model calibration Thetask of quantifying and mitigating these uncertainties in mathematical models leadsto the development of new statisticalstochastic methods along with methods forefficient integration of data and simulation

Further to supporting theories and increasing our predictive capabilities mathe-matical and computational modeling can often suggest sharper natural experimentsand more focused observations providing in their turn a check to the model accu-racy Natural experiments and results of observations may produce large amountsof data sets that can intelligently be processed only with efficient mathematicaldata mining algorithms and powerful statistical and visualization tools [15] Theapplication of these algorithms and tools requires a close collaboration between dif-ferent disciplines As a result observations and experiments theory and modelingreinforce each other leading together to our better understanding of phenomenaprocesses and systems we study as well as to the necessity of even more close inter-actions between mathematical modeling computational analyses and experimentalapproaches



13 WHAT THIS BOOK IS ABOUT

The rest of the book consists of 4 main sections containing 11 state-of-the-art chap-ters on applications of mathematical and computational modeling in natural andsocial sciences engineering and the arts These chapters are based on selected invitedcontributions from leading specialists from all over the world Inevitably given thevast range of research areas within the field of mathematical and computational mod-eling the book such as this can present only selective topics At the same time theseselective topics open to the reader a broad spectrum of methods and tools important inthese applications and ranging from infectious disease dynamics and epidemic mod-eling to superconductivity and quantum mechanical challenges from the models forvoting systems to the modeling of musical rhythms The book provides both theoreti-cal advances in these areas of applications as well as some representative examples ofmodern problems from these applications Following this introductory section eachremaining section with its chapters stands alone as an in-depth research or a surveywithin a specific area of application of mathematical and computational modeling Wehighlight the main features of each such chapter within four main remaining sectionsof this book

bull Advanced Mathematical and Computational Models in Physics andChemistry This section consists of three chapters

ndash This section is opened by a chapter written by I M Sigal who addresses themacroscopic theory of superconductivity Superconducting vortex states pro-vide a rich area of research In the 1950s A Abrikosov solved the GinzburgndashLandau (GL) equations in an applied magnetic field for certain values of GLparameter (later A Abrikosov received a Nobel Prize for this work) This ledto what is now known as the famous vortex solution characterized by the factthat the superconducting order parameter contains a periodic lattice of zeros Inits turn this led to studies of a new mixed Abrikosov vortex phase between theMeissner state and the normal state The area keeps generating new interestingresults in both theory and application For example unconventional vortex pat-tern formations (eg vortex clustering) were recently discovered in multibandsuperconductors (eg [17] and references therein) Such phenomena whichare of both fundamental and practical significance present a subject of manyexperimental and theoretical works Recently it was shown that at low temper-atures the vortices form an ordered Abrikosov lattice both in low and in highfields The vortices demonstrate distinctive modulated structures at interme-diate fields depending on the effective intervortex attraction These and otherdiscoveries generate an increasing interest to magnetic vortices and Abrikosovlattices Chapter by I M Sigal reminds us that the celebrated GL equationsform an integral part namely the Abelian-Higgs component of the standardmodel of particle physics having fundamental consequences for many areasof physics including those beyond the original designation area of the modelNot only this chapter reviews earlier works on key solutions of the GL model


WHAT THIS BOOK IS ABOUT 11

but it presents some interesting recent results Vortex lattices their existencestability and dynamics are discussed demonstrating also that automorphicfunctions appear naturally in this context and play an important role

ndash A prominent role in physics and chemistry is played by the Hartree-Fockmethod which is based on an approximation allowing to determine the wavefunction and the energy of a quantum many-body system in a stationary stateMore precisely the Hartree-Fock theoretical framework is based on the varia-tional molecular orbital theory allowing to solve Schroumldingerrsquos equation insuch a way that each electron spatial distribution is described by a singleone-electron wave function known as molecular orbital While in the clas-sical Hartree-Fock theory the motion of electrons is uncorrelated correlatedwavefunction methods remedy this drawback The second chapter in this sec-tion is devoted to a multireference local correlation framework in quantumchemistry focusing on numerical challenges in the Cholesky decompositioncontext The starting point of the discussion presented by D K Krisiloff J MDieterich F Libisch and E A Carter is based on the fact that local correlationmethods developed for solving Schroumldingerrsquos equation for molecules have areduced computational cost compared to their canonical counterparts Hencethe authors point out that these methods can be used to model notably largerchemical systems compared to the canonical algorithms The authors analyzein detail local algorithmic blocks of these methods

ndash Variational methods are in the center of the last chapter of this section writtenby M Levy and A Gonis The basic premises here lie with the Rayleigh-Ritzvariational principle which in the context of quantum mechanical appli-cations reduces the problem of determining the ground-state energy of aHamiltonian system consisting of N interacting electrons to the minimiza-tion of the energy functional The authors then move to the main part of theirresults closely connected to a fundamental element of quantum mechanicsIn particular they provide two alternative proofs of the generalization of thevariational theorem for Hamiltonians of N-electron systems to wavefunctionsof dimensions higher than N They also discuss possible applications of theirmain result

bull Mathematical and Statistical Models in Life Science Applications Thissection consists of two chapters

ndash The first chapter deals with mathematical modeling of infectious diseasedynamics control and treatment focusing on a model for the spread of tuber-culosis (TB) TB is considered to be the second highest cause of infectiousdisease-induced mortality after HIVAIDS Written by J Arino and I ASoliman this chapter provides a detailed account of a model that incorporatesthree strains namely (1) drug sensitive (2) emerging multidrug resistant and(3) extensively drug-resistant The authors provide an excellent introduction tothe subject area followed by the model analysis In studying the dynamics of



the model they characterize parameter regions where backward bifurcationmay occur They demonstrate the global stability of the disease-free equi-librium in regions with no backward bifurcation In conclusion the authorsdiscuss possible options for their model improvement and how mathematicalepidemiology contributes to our better understanding of disease transmissionprocesses and their control

ndash Epidemiological modeling requires the development and application of anintegrated approach The second chapter of this section focuses on these issueswith emphasis on antibiotic resistance The chapter is written by E Y KleinJ Chelen M D Makowsky and P E Smaldino They stress the importance ofintegrating human behavior social networks and space into infectious diseasemodeling The field of antibiotic resistance is a prime example where this isparticularly critical The authors point out that the annual economic cost to theUS health care system of antibiotic-resistant infections is estimated to be $21ndash$34 billion and given human health and economics reasons they set a task ofbetter understanding how resistant bacterial pathogens evolve and persist inhuman populations They provide a selection of historical achievements andlimitations in mathematical modeling of infectious diseases This is followedby a discussion of the integrated approach the authors advocate for in address-ing the multifaceted problem of designing innovative public health strategiesagainst bacterial pathogens The interaction of epidemiological evolutionaryand behavioral factors along with cross-disciplinary collaboration in devel-oping new models and strategies is becoming crucial for our success in thisimportant field

bull Mathematical Models and Analysis for Science and Engineering Thissection consists of four chapters

ndash The first chapter is devoted to mathematical models in climate modelingwith a major focus given to examples from climate atmosphere-ocean science(CAOS) However it covers potentially a much larger area of applications inscience and engineering Indeed as pointed out by the authors of this chapterD Giannakis and A J Majda large-scale data sets generated by dynamicalsystems arise in a vast range of disciplines in science and engineering forexample fluid dynamics materials science astrophysics earth sciences toname just a few Therefore the main emphasis of this chapter is on data-driven methods for dynamical systems aiming at quantifying predictabilityand extracting spatiotemporal patterns In the context of CAOS we are deal-ing with a system of time-dependent coupled nonlinear PDEs The dynamicsof this system takes place in an infinite-dimensional phase space where thecorresponding equations of fluid flow and thermodynamics are defined In thiscase the observed data usually correspond to well-defined physical functionsof that phase space for example temperature pressure or circulation mea-sured over a set of spatial points Data-driven methods appear to be critical in












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21 Introduction 19


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References 250






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2014043267



10 9 8 7 6 5 4 3 2 1

1 2015



(Aristotle)




CONTENTS


PREFACE xv






21 Introduction 19


viii CONTENTS


radic2 24










CONTENTS ix














x CONTENTS
















CONTENTS xi












References 250






xii CONTENTS














INDEX 309































PREFACE





xvi PREFACE






SECTION 1

INTRODUCTION






























































(Aristotle)




CONTENTS


PREFACE xv






21 Introduction 19


viii CONTENTS


radic2 24










CONTENTS ix














x CONTENTS
















CONTENTS xi












References 250






xii CONTENTS














INDEX 309































PREFACE





xvi PREFACE






SECTION 1

INTRODUCTION






























































CONTENTS


PREFACE xv






21 Introduction 19


viii CONTENTS


radic2 24










CONTENTS ix














x CONTENTS
















CONTENTS xi












References 250






xii CONTENTS














INDEX 309































PREFACE





xvi PREFACE






SECTION 1

INTRODUCTION





























































viii CONTENTS


radic2 24










CONTENTS ix














x CONTENTS
















CONTENTS xi












References 250






xii CONTENTS














INDEX 309































PREFACE





xvi PREFACE






SECTION 1

INTRODUCTION





























































CONTENTS ix














x CONTENTS
















CONTENTS xi












References 250






xii CONTENTS














INDEX 309































PREFACE





xvi PREFACE






SECTION 1

INTRODUCTION





























































x CONTENTS
















CONTENTS xi












References 250






xii CONTENTS














INDEX 309































PREFACE





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SECTION 1

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CONTENTS xi












References 250






xii CONTENTS














INDEX 309































PREFACE





xvi PREFACE






SECTION 1

INTRODUCTION





























































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