EMS Series in Industrial and Applied Mathematics Vol. 1
ISBN 978-3-03719-137-8
DOI 10.4171/137
April 2014, 466 pages, hardcover, 17 x 24 cm.
Mathematics: intellectual endeavor, production factor, key technology, key to key technologies?
Mathematics is all of these! The last three of its facets have been the focus of the research and development in the Berlin-based DFG Research Center MATHEON in the last twelve years. Through these activities MATHEON has become an international trademark for carrying out creative, application-driven research in mathematics and for cooperating with industrial partners in the solution of complex problems in key technologies.
Modern key technologies have become highly sophisticated, integrating aspects of engineering, computer, business and other sciences. Flexible mathematical models, as well as fast and accurate methods for numerical simulation and optimization open new possibilities to handle the indicated complexities, to react quickly, and to explore new options. Researchers in mathematical fields such as Optimization, Discrete Mathematics, Numerical Analysis, Scientific Computing, Applied Analysis and Stochastic Analysis have to work hand in hand with scientists and engineers to fully exploit this potential and to strengthen the transversal role of mathematics in the solution of the challenging problems in key technologies.
This book presents in seven chapters the highlights of the research work carried out in the MATHEON application areas: Life Sciences, Networks, Production, Electronic and Photonic Devices, Finance, Visualization, and Education. The chapters summarize many of the contributions, put them in the context of current mathematical research activities and outline their impact in various key technologies. To make some of the results more easily accessible to the general public, a large number of “showcases” are presented that illustrate a few success stories.
EMS Textbooks in Mathematics
ISBN 978-3-03719-138-5
DOI 10.4171/138
May 2014, 636 pages, hardcover, 16.5 x 23.5 cm.
Projective geometry is concerned with the properties of figures that are invariant by projecting and taking sections. It is considered one of the most beautiful parts of geometry and plays a central role because its specializations cover the whole of the affine, Euclidean and non-Euclidean geometries. The natural extension of projective geometry is projective algebraic geometry, a rich and active field of research. Regarding its applications, results and techniques of projective geometry are today intensively used in computer vision.
This book contains a comprehensive presentation of projective geometry, over the real and complex number fields, and its applications to affine and Euclidean geometries. It covers central topics such as linear varieties, cross ratio, duality, projective transformations, quadrics and their classifications ? projective, affine and metric ?, as well as the more advanced and less usual spaces of quadrics, rational normal curves, line complexes and the classifications of collineations, pencils of quadrics and correlations. Two appendices are devoted to the projective foundations of perspective and to the projective models of plane non-Euclidean geometries. The presentation uses modern language, is based on linear algebra and provides complete proofs. Exercises are proposed at the end of each chapter; many of them are beautiful classical results.
The material in this book is suitable for courses on projective geometry for undergraduate students, with a working knowledge of a standard first course on linear algebra. The text is a valuable guide to graduate students and researchers working in areas using or related to projective geometry, such as algebraic geometry and computer vision, and to anyone wishing to gain an advanced view on geometry as a whole.
Surveys of Modern Mathematics, Volume 7
Published: 24 April 2014
Paperback
128 pages
Since the 1950s, many new ideas and tools from algebra, and algebraic and geometric topology, have been applied to study the structure of high-dimensional differential and topological manifolds, and so today it can be difficult for beginners to delve through the literature. This volume is a helpful guide to the basic concepts and results of topology of manifolds?including the h- and s-cobordism theorems, topological invariance of rational Pontryagin classes, surgery theory, and algebraic K-theory.
This volume is part of the Surveys of Modern Mathematics book series.
Surveys of Modern Mathematics, Volume 8
Published: 24 April 2014
Paperback
160 pages
This is an interdisciplinary monograph at the cutting edges of infinite dimensional dynamical systems, partial differential equations, and mathematical physics. It discusses Y. Charles Li’s work of connecting Darboux transformations to homoclinic orbits and Melnikov integrals for integrable partial differential equations; and Artyom Yurov’s work in applying Darboux transformations to numerous areas of physics.
Of particular interest to the reader might be the brand-new methods, developed by Li in collaboration with others, of using Darboux transformations to construct homoclinic orbits, Melnikov integrals, and Melnikov vectors for integrable systems. It should be noted that integrable systems (also named soliton equations) are the infinite dimensional counterparts of finite dimensional integrable Hamiltonian systems. What the new methods reveal are the infinite dimensional phase space structures.
This work is intended for advanced undergraduates, graduate and postdoctoral students, and senior researchers in mathematics, physics, and other relevant scientific areas.
Published: 12 May 2014
Paperback
151 pages
(incl. 10 color pages)
Lively and engaging articles from the lecturers and the participants of the 20th Gokova Geometry-Topology Conference, held on the shores of Gokova Bay, Turkey, in May of 2013.
This volume is part of the Gokova Geometry-Topology Conferences book series.
ISBN: 978-1-118-72140-7
224 pages
September 2014
Symmetry analysis has played a crucial role in the construction of exact solutions to differential equations. In fact, all the standard techniques for solving first order ordinary differential equations (ODEs) can be explained by symmetry analysis. The usefulness of symmetry analysis is readily evident, and the beauty of the method is that it is both fairly easy to learn and is very algorithmic, making it amenable to a variety of computer algebra packages such as Maple? or Mathematica.Symmetry Analysis of Differential Equations: An Introductionfills a gap in the literature by providing introductory coverage of both ODEs and partial differential equations (PDEs), explores select advanced topics, and contains platiful problems with complete solutions. Chapter coverages includes: ordinary differential equations with Lie's invariance condition, standard integration techniques, infinitesimal operator and higher order equations, second order questions, and ODE systems; partial differential equations with first and second order equations, systems of PDEs, higher dimensional PDEs, and nonclassical symmetries; and compatibility with nonclassical symmetry analysis and first order compatibility.
Preface i
Acknowledgements iii
Dedication iv
1 An Introduction 1
1.1 What is a symmetry? 1
1.2 Lie Groups 4
1.3 Invariance of Differential Equations 6
1.4 Some Ordinary Differential Equations 8
1.5 Exercises 11
2 Ordinary Differential Equations 13
2.1 Infinitesimal Transformations 16
2.2 Lie’s Invariance Condition 19
2.2.1 Exercises 22
2.3 Standard Integration Techniques 23
2.3.1 Linear Equations 24
2.3.2 Bernoulli Equation 25
2.3.3 Homogeneous Equations 26
2.3.4 Exact Equations 27
2.3.5 Riccati Equations 30
2.3.6 Exercises 31
2.4 Infinitesimal Operator and Higher Order Equations 32
2.4.1 The Infinitesimal Operator 32
2.4.2 The Extended Operator 32
2.4.3 Extension to Higher Orders 33
2.4.4 First Order Infinitesimals (revisited) 33
2.4.5 Second Order Infinitesimals 34
2.4.6 The Invariance of Second Order Equations 35
2.4.7 Equations of arbitrary order 36
2.5 Second Order Equations 36
2.5.1 Exercises 46
2.6 Higher Order Equations 47
2.6.1 Exercises 51
2.7 ODE Systems 52
2.7.1 First Order Systems 52
2.7.2 Higher Order Systems 56
2.7.3 Exercises 60
3 Partial Differential Equations 62
3.1 First Order Equations 62
3.1.1 What do we do with the symmetries of PDEs? 65
3.1.2 Direct Reductions 68
3.1.3 The Invariant Surface Condition 70
3.1.4 Exercises 71
3.2 Second Order PDEs 71
3.2.1 Heat Equation 71
3.2.2 Laplace’s Equation 76
3.2.3 Burgers’ Equation and a Relative 80
3.2.4 Heat equation with a source 85
3.2.5 Exercises 91
3.3 Higher Order PDEs 93
3.3.1 Exercises 98
3.4 Systems of PDEs 99
3.4.1 First order systems 99
3.4.2 Second order systems 103
3.4.3 Exercises 106
3.5 Higher Dimensional PDEs 107
3.5.1 Exercises 113
4 Nonclassical Symmetries and Compatibility 114
4.1 Nonclassical Symmetries 114
4.1.1 Invariance of the Invariant Surface Condition 116
4.1.2 The nonclassical method 117
4.2 Nonclassical Symmetry Analysis and Compatibility 125
4.3 Beyond Symmetries Analysis ? General compatibility 126
4.3.1 Compatibility with First Order PDEs - Charpit’s Method 127
4.3.2 Compatibility of systems 134
4.3.3 Compatibility of the nonlinear heat equation 136
4.4 Exercises 137
4.5 Concluding Remarks 138
Solutions 139
References 145
See More
Daniel J. Arrigo, PhD, is Professor in the Department of Mathematics at the University of Central Arkansas. The author of over thirty journal articles, his research interests include the construction of exact solutions of PDEs; symmetry analysis of nonlinear PDEs; and solutions to physically important equations, such as the nonlinear heat equations and the governing equations modeling of granular materials and nonlinear elasticity. Dr. Arrigo received the Oklahoma-Arkansas Section of the Mathematical Association of America’s “Award for Distinguished Teaching of College or University Mathematics” in 2008.