{"title":"Applied Mathematical Sciences","description":"\u003cp\u003eDelve into the world of advanced mathematical techniques with the Applied Mathematical Sciences series. Ideal for researchers and graduate students, these texts offer deep insights into diverse applications and theories.\u003c\/p\u003e","products":[{"product_id":"elliptic-functions-and-applications-book-derek-f-lawden-9780387969657","title":"Elliptic Functions and Applications","description":"The subject matter of this book formed the substance of a mathematical se am which was worked by many of the great mathematicians of the last century. The mining metaphor is here very appropriate, for the analytical tools perfected by Cauchy permitted the mathematical argument to penetra te to unprecedented depths over a restricted region of its domain and enabled mathematicians like Abel, Jacobi, and Weierstrass to uncover a treasurehouse of results whose variety, aesthetic appeal, and capacity for arousing our astonishment have not since been equaled by research in any other area. But the circumstance that this theory can be applied to solve problems arising in many departments of science and engineering graces the topic with an additional aura and provides a powerful argument for including it in university courses for students who are expected to use mathematics as a tool for technological investigations in later life. Unfortunately, since the status of university staff is almost wholly determined by their effectiveness as research workers rather than as teachers, the content of undergraduate courses tends to reflect those academic research topics which are currently popular and bears little relationship to the future needs of students who are themselves not destined to become university teachers. Thus, having been comprehensively explored in the last century and being undoubtedly difficult .","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":49617386537233,"sku":"CIN0387969659G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52478610080017,"sku":"NLS9780387969657","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":52735257379089,"sku":"NIN9780387969657","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387969659.jpg?v=1750878702"},{"product_id":"ordinary-and-delay-differential-equations-book-r-d-driver-9780387902319","title":"Ordinary and Delay Differential Equations","description":"This textbook is designed for the intermediate-level course on ordinary differential equations offered at many universities and colleges. It treats, as standard topics of such a course: existence and uniqueness theory, linear s- terns, stability theory, and introductory phase-plane analysis of autonomous second order systems. The unique feature of the book is its further inc- sion of a substantial introduction to delay differential eq- tions. Such equations are motivated by problems in control theory, physics, biology, ecology, economics, inventory c- trol, and the theory of nuclear reactors. The surge of interest in delay differential equations during the past two or three decades is evidenced by th- sands of research papers on the subject and about 20 published books devoted in whole or in part to these equations. The v * ...books include those of Myskis [1951] , El' sgol' c [1955] and [1964], Pinney [1958], Krasovskil [1959], Bellman and Cooke [1963], Norkin [1965], Halanay [1966], Oguztoreli [1966], Lakshmikantham and Leela [1969], Mitropol'skir and Martynjuk [1969], Martynjuk [1971], and Hale [1971], plus a number of symposium and seminar proceedings published in the U.S. and the U. S.S.R. These books have influenced the present textbook.","brand":"WoB","offers":[{"title":"GB \/ WELL_READ \/ INTERNAL","offer_id":49641317794065,"sku":"GOR013168333","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ GOOD \/ SBYB","offer_id":50350383530257,"sku":"CIN0387902317G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52125410722065,"sku":"NLS9780387902319","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387902317.jpg?v=1751038791"},{"product_id":"course-in-elasticity-book-b-m-fraeijs-de-veubeke-9780387904283","title":"A Course in Elasticity","description":"This book is based on lecture notes of the late Professor de Veubeke. The subject is presented at a level suitable for graduate students in engineering, physics, or mathematics. Some exposure to linear algebra, complex analysis, variational calculus, or basic continuum mechanics would be helpful. The first third of the book contains the fundamentals of the theory of elasticity. Kinematics of continuous media, the notions of stress and equilibrium, conservation of energy, 'and the elastic constitutive law are each treated first in a nonlinear context, then specialized to the linear case. The remainder of the book is given to three classic applications of the theory, each supplemented by original re­ sults based on the use of complex variables. Each one of the three topics - Saint-Venant's theory of prismatic beams, plane deformations, and the bending of plates - is first pre­ sented and analyzed in general, then rounded out with numerous specific and sometimes novel examples. The following notational conventions are generally in force, except where noted to the contrary: lower case boldface letters denote vectors or triples of Cartesian co­ ordinates, upper case boldface letters denote 3 x 3 matrices, repeated lower case Latin subscripts are summed over (1,2,3), and non-repeated lower case Latin subscripts are assumed to range over (1,2,3).","brand":"WoB","offers":[{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":49656814797073,"sku":"GOR013183107","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52593630544145,"sku":"NLS9780387904283","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/038790428X.jpg?v=1751071692"},{"product_id":"partial-differential-equations-i-book-michael-e-taylor-9783031338588","title":"Partial Differential Equations I","description":"This is the first of three volumes on partial differential equations. 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Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis. The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids.  The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds.     Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC.  Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”  (Peter Lax, SIAM review, June 1998)","brand":"WoB","offers":[{"title":"GB \/ NEW \/ GARDNERS","offer_id":49745774706961,"sku":"NGR9783031336997","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/3031336992.jpg?v=1751429016"},{"product_id":"partial-differential-equations-iii-book-michael-e-taylor-9783031339271","title":"Partial Differential Equations III","description":"The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. 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The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.  The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters.  It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids.  The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds. Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC.  Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”  (Peter Lax, SIAM review, June 1998)","brand":"WoB","offers":[{"title":"GB \/ NEW \/ GARDNERS","offer_id":49745780801809,"sku":"NGR9783031339271","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/3031339274.jpg?v=1751420981"},{"product_id":"partial-differential-equations-book-fritz-john-9780387906096","title":"Partial Differential Equations","description":"From the reviews of the earlier editions: “An excellent second-reading text.  Should be accessible to any mathematician.  Highly recommended.” – The Mathematical Gazette    “An excellent text upon which to build and introduction to partial differential equations at the graduate level.  In addition…recommended for the nonspecialist.” – American Mathematical Monthly    Partial Differential Equations is a very well accepted introduction to the subject.  In it, the author identifies the significant aspects of the theory and explores them with a limited amount of machinery from mathematical analysis.  Now, in this fourth edition, the book has again been updated with an additional chapter on Lewy’s example of a linear equation with solutions.","brand":"WoB","offers":[{"title":"US \/ VERY_GOOD \/ SBYB","offer_id":49876019413265,"sku":"CIN0387906096VG","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ GOOD \/ SBYB","offer_id":50351192932625,"sku":"CIN0387906096G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":52082282692881,"sku":"GOR012358356","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52335700148497,"sku":"NLS9780387906096","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":53439032787217,"sku":"NIN9780387906096","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ LIKE_NEW \/ INTERNAL","offer_id":53489556881681,"sku":"GOR011156924","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387906096.jpg?v=1751390523"},{"product_id":"stability-and-transition-in-shear-flows-book-peter-j-schmid-9781461265641","title":"Stability and Transition in Shear Flows","description":"The field of hydrodynamic stability has a long history, going back to Rey­ nolds and Lord Rayleigh in the late 19th century. Because of its central role in many research efforts involving fluid flow, stability theory has grown into a mature discipline, firmly based on a large body of knowledge and a vast body of literature. The sheer size of this field has made it difficult for young researchers to access this exciting area of fluid dynamics. For this reason, writing a book on the subject of hydrodynamic stability theory and transition is a daunting endeavor, especially as any book on stability theory will have to follow into the footsteps of the classical treatises by Lin (1955), Betchov \u0026amp; Criminale (1967), Joseph (1971), and Drazin \u0026amp; Reid (1981). 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The main goal of this work is to present the variety of image analysis applications and the precise mathematics involved. It is intended for two audiences. The first is the mathematical community, to show the contribution of mathematics to this domain and to highlight some unresolved theoretical questions. The second is the computer vision community, to present a clear, self-contained, and global overview of the mathematics involved in image processing problems. The book is divided into five main parts. Chapter 1 is a detailed overview. Chapter 2 describes and illustrates most of the mathematical notions found throughout the work. Chapters 3 and 4 examine how PDEs and variational methods can be successfully applied in image restoration and segmentation processes. Chapter 5, which is more applied, describes some challenging computer vision problems, such as sequence analysis or classification. This book will be useful to researchers and graduate students in mathematics and computer vision.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50109253779729,"sku":"CIN0387953264G","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387953264.jpg?v=1751292578"},{"product_id":"geometry-of-minkowski-spacetime-book-gregory-l-naber-9781441978370","title":"The Geometry of Minkowski Spacetime","description":"This book offers a presentation of the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics.  It treats, in addition to the usual menu of topics one is accustomed to finding in introductions to special relativity, a wide variety of results of more contemporary origin. These include Zeeman’s characterization of the causal automorphisms of Minkowski spacetime, the Penrose theorem on the apparent shape of a relativistically moving sphere, a detailed introduction to the theory of spinors, a Petrov-type classification of electromagnetic fields in both tensor and spinor form, a topology for Minkowski spacetime whose homeomorphism group is essentially the Lorentz group, and a careful discussion of Dirac’s famous Scissors Problem and its relation to the notion of a two-valued representation of the Lorentz group.  This second edition includes a new chapter on the de Sitter universe which is intended to serve two purposes. The first is to provide a gentle prologue to the steps one must take to move beyond special relativity and adapt  to the presence of gravitational fields that cannot be considered  negligible. The second is to understand some of the basic features of  a model of the empty universe that differs markedly from Minkowski spacetime, but may be recommended by recent astronomical observations suggesting that the expansion of our own universe is accelerating rather than slowing down. The treatment presumes only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and, in two appendices, some elementary point-set topology.      The first edition of the book received the 1993 CHOICE award for Outstanding Academic Title.     Reviews of first edition:      “… a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics andphysics.” (American Mathematical Society, 1993)      “Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.” (CHOICE, 1993)   “… his talent in choosing the most significant results and ordering them within the book can’t be denied. The reading of the book is, really, a pleasure.” (Dutch Mathematical Society, 1993)","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50134934716689,"sku":"CIN1441978372G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52138577953041,"sku":"NLS9781441978370","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/1441978372.jpg?v=1751272952"},{"product_id":"hopf-bifurcation-and-its-applications-book-j-e-marsden-9780387902005","title":"The Hopf Bifurcation and Its Applications","description":"The goal of these notes is to give a reasonahly com­ plete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to spe­ cific problems, including stability calculations. Historical­ ly, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. Hopf's basic paper [1] appeared in 1942. Although the term \"Poincare­ Andronov-Hopf bifurcation\" is more accurate (sometimes Friedrichs is also included), the name \"Hopf Bifurcation\" seems more common, so we have used it. Hopf's crucial contribution was the extension from two dimensions to higher dimensions. The principal technique employed in the body of the text is that of invariant manifolds. The method of Ruelle­ Takens [1] is followed, with details, examples and proofs added. Several parts of the exposition in the main text come from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we are grateful. The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations: see for example, Hale [1,2] and Hartman [1]. Of course, other methods are also available. In an attempt to keep the picture balanced, we have included samples of alternative approaches. Specifically, we have included a translation (by L. Howard and N. Kopell) of Hopf's original (and generally unavailable) paper.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50147923853585,"sku":"CIN0387902007G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52489587622161,"sku":"NLS9780387902005","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387902007.jpg?v=1751102665"},{"product_id":"foundations-of-chemical-reaction-network-theory-book-martin-feinberg-9783030038571","title":"Foundations of Chemical Reaction Network Theory","description":"Part I: Preliminaries.- Anticipating the Big Picture: Some Clues.- Chemical and Notational Preliminaries.- Reaction Networks, Kinetics, and the Induced Differential Equations.- Open Systems: Why Study Nonconservative and Otherwise Peculiar Reaction Networks?.- A Toy-Reaction-Network Zoo: Varieties of Behavior and Some Questions.- Part II: Some Principle Theorems: A First Look.- Aspects of Reaction Network Structure.- The Deficiency Zero Theorem.- Deficiency One Theory.- Concentration Robustness and Its Importance in Biology: Some More Deficiency-Oriented Theorems.- Concordant Reaction Networks: Architectures that Promote Dull, Reliable Behavior Across Broad Kinetic Classes.- The Species-Reaction Graph.- The Big Picture Revisited.- Part III: Going Deeper.- Quasi-Thermodynamic Kinetic Systems.- Detailed Balancing.- Complex Balancing.- Deficiency Zero Theory Foundations and Some Key Propositions.- Deficiency One Theory Foundations.- Concentration Robustness Foundations.- Species-Reaction Graph Foundations.","brand":"WoB","offers":[{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":50289672356113,"sku":"GOR013914149","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":53522839929105,"sku":"NLS9783030038571","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/3030038572.jpg?v=1750837859"},{"product_id":"geometric-quantization-and-quantum-mechanics-book-jedrzej-sniatycki-9780387904696","title":"Geometric Quantization and Quantum Mechanics","description":"This book contains a revised and expanded version of the lecture notes of two seminar series given during the academic year 1976\/77 at the Department of Mathematics and Statistics of the University of Calgary, and in the summer of 1978 at the Institute of Theoretical Physics of the Technical University Clausthal. The aim of the seminars was to present geometric quantization from the point of view· of its applica­ tions to quantum mechanics, and to introduce the quantum dynamics of various physical systems as the result of the geometric quantization of the classical dynamics of these systems. The group representation aspects of geometric quantiza­ tion as well as proofs of the existence and the uniqueness of the introduced structures can be found in the expository papers of Blattner, Kostant, Sternberg and Wolf, and also in the references quoted in these papers. The books of Souriau (1970) and Simms and Woodhouse (1976) present the theory of geometric quantization and its relationship to quantum mech­ anics. The purpose of the present book is to complement the preceding ones by including new developments of the theory and emphasizing the computations leading to results in quantum mechanics.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50350408466705,"sku":"CIN0387904697G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52593064149265,"sku":"NLS9780387904696","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387904697.jpg?v=1751357953"},{"product_id":"box-splines-book-carl-de-boor-9780387941011","title":"Box Splines","description":"Compactly supported smooth piecewise polynomial functions provide an efficient tool for the approximation of curves and surfaces and other smooth functions of one and several arguments. Since they are locally polynomial, they are easy to evaluate. Since they are smooth, they can be used when smoothness is required, as in the numerical solution of partial differential equations (in the Finite Element method) or the modeling of smooth sur­ faces (in Computer Aided Geometric Design). Since they are compactly supported, their linear span has the needed flexibility to approximate at all, and the systems to be solved in the construction of approximations are 'banded'. The construction of compactly supported smooth piecewise polynomials becomes ever more difficult as the dimension, s, of their domain G ~ IRs, i. e. , the number of arguments, increases. In the univariate case, there is only one kind of cell in any useful partition, namely, an interval, and its boundary consists of two separated points, across which polynomial pieces would have to be matched as one constructs a smooth piecewise polynomial function. This can be done easily, with the only limitation that the num­ ber of smoothness conditions across such a breakpoint should not exceed the polynomial degree (since that would force the two joining polynomial pieces to coincide). In particular, on any partition, there are (nontrivial) compactly supported piecewise polynomials of degree ~ k and in C(k-l), of which the univariate B-spline is the most useful example.","brand":"WoB","offers":[{"title":"US \/ VERY_GOOD \/ SBYB","offer_id":50350519976209,"sku":"CIN0387941010VG","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52333651558673,"sku":"NLS9780387941011","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387941010.jpg?v=1750878686"},{"product_id":"partial-differential-equations-book-michael-eugene-taylor-9780387946511","title":"Partial Differential Equations","description":"This second in the series of three volumes builds upon the basic theory of linear PDE given in volume 1, and pursues more advanced topics. 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Early drafts were used for several single semester courses on singular perturbation meth­ ods given at Rensselaer, and a more complete version was used for a one year course at the Technische Universitat Wien. Some portions have been used for short lecture series at Universidad Central de Venezuela, West Vir­ ginia University, the University of Southern California, the University of California at Davis, East China Normal University, the University of Texas at Arlington, Universita di Padova, and the University of New Hampshire, among other places. As a result, I've obtained lots of valuable feedback from students and listeners, for which I am grateful. This writing continues a pattern. Earlier lectures at Bell Laboratories, at the University of Edin­ burgh and New York University, and at the Australian National University led to my earlier works (1968, 1974, and 1978). All seem to have been useful for the study of singular perturbations, and I hope the same will be true of this monograph. I've personally learned much from reading and analyzing the works of others, so I would especially encourage readers to treat this book as an introduction to a diverse and exciting literature. The topic coverage selected is personal and reflects my current opin­ ions. An attempt has been made to encourage a consistent method of ap­ proaching problems, largely through correcting outer limits in regions of rapid change. Formal proofs of correctness are not emphasized.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50350925447441,"sku":"CIN038797556XG","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52477625598225,"sku":"NLS9780387975566","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/038797556X.jpg?v=1751071715"},{"product_id":"geometry-of-minkowski-spacetime-book-gregory-l-naber-9780387978482","title":"The Geometry of Minkowski Spacetime","description":"This book offers a presentation of the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics.  It treats, in addition to the usual menu of topics one is accustomed to finding in introductions to special relativity, a wide variety of results of more contemporary origin. These include Zeeman's characterization of the causal automorphisms of Minkowski spacetime, the Penrose theorem on the apparent shape of a relativistically moving sphere, a detailed introduction to the theory of spinors, a Petrov-type classification of electromagnetic fields in both tensor and spinor form, a topology for Minkowski spacetime whose homeomorphism group is essentially the Lorentz group, and a careful discussion of Dirac's famous Scissors Problem and its relation to the notion of a two-valued representation of the Lorentz group.  This second edition includes a new chapter on the de Sitter universe which is intended to serve two purposes. The first is to provide a gentle prologue to the steps one must take to move beyond special relativity and adapt  to the presence of gravitational fields that cannot be considered  negligible. The second is to understand some of the basic features of  a model of the empty universe that differs markedly from Minkowski spacetime, but may be recommended by recent astronomical observations suggesting that the expansion of our own universe is accelerating rather than slowing down. The treatment presumes only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and, in two appendices, some elementary point-set topology. The first edition of the book received the 1993 CHOICE award for Outstanding Academic Title. Reviews of first edition: \"! a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics.\" (American Mathematical Society, 1993) \"Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.\" (CHOICE, 1993) \"! his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure.\" (Dutch Mathematical Society, 1993)","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50350934851857,"sku":"CIN0387978488G","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387978488.jpg?v=1750696333"},{"product_id":"dynamics-numerical-explorations-book-helena-e-nusse-9780387982649","title":"Dynamics: Numerical Explorations","description":"The Dynamics program and handbook allows the reader to explore nonlinear dynamics and chaos by the use of illustrated graphics. It is suitable for research and educational needs. This new edition allows the program to run 3 times faster on the processes that are time consuming. Other major changes include: There will be an add-your-own equation facility. This means it will be unnecessary to have a compiler. PD and Lyanpunov exponents and Newton method for finding periodic orbits can all be carried out numerically without adding specific code for partial derivatives. The program will support color postscript. New menu system in which the user is prompted by options when a command is chosen. This means that the program is much easier to learn and to remember in comparison to current version. Mouse support is added. The program will be able to use the expanded memory available on modern PC's. This means pictures will be higher resolution. There are also many minor chan ce much of the source code will be available on the web, although some of ges such as zoom facility and help facility. Due to limited spa it willr emain on the disk so that the unix users still have to purchase the book. This will allow minor upgrades for Unix users.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50350962868497,"sku":"CIN0387982647G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52341576892689,"sku":"NLS9780387982649","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":52951981130001,"sku":"NIN9780387982649","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387982647.jpg?v=1751006761"},{"product_id":"perturbation-methods-in-applied-mathematics-book-j-kevorkian-9780387905075","title":"Perturbation Methods in Applied Mathematics","description":"This book is a revised and updated version, including a substantial portion of new material, of J. D. Cole's text Perturbation Methods in Applied Mathe­ matics, Ginn-Blaisdell, 1968. We present the material at a level which assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate level course on the subject. The applied mathematician, attempting to understand or solve a physical problem, very often uses a perturbation procedure. In doing this, he usually draws on a backlog of experience gained from the solution of similar examples rather than on some general theory of perturbations. The aim of this book is to survey these perturbation methods, especially in connection with differ­ ential equations, in order to illustrate certain general features common to many examples. The basic ideas, however, are also applicable to integral equations, integrodifferential equations, and even to_difference equations. In essence, a perturbation procedure consists of constructing the solution for a problem involving a small parameter B, either in the differential equation or the boundary conditions or both, when the solution for the limiting case B = 0 is known. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of B.","brand":"WoB","offers":[{"title":"US \/ WELL_READ \/ SBYB","offer_id":50351181463825,"sku":"CIN0387905073A","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52477753196817,"sku":"NLS9780387905075","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":52735252955409,"sku":"NIN9780387905075","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387905073.jpg?v=1751390523"},{"product_id":"introduction-to-functional-differential-equations-book-jack-k-hale-9780387940762","title":"Introduction to Functional Differential Equations","description":"The present book builds upon an earlier work of J. Hale, \"Theory of Func­ tional Differential Equations\" published in 1977. We have tried to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a complete new presentation of lin­ ear systems (Chapters 6~9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global at­ tractors was completely revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (see Chapters 1, 2, 3, 9, and 10). Chapter 12 is completely new and contains a guide to active topics of re­ search. In the sections on supplementary remarks, we have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive. Jack K. Hale Sjoerd M. Verduyn Lunel Contents Preface............................................................ v Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1. Linear differential difference equations . . . . . . . . . . . . . . 11 . . . . . . 1.1 Differential and difference equations. . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . 1.2 Retarded differential difference equations. . . . . . . . . . . . . . . . 13 . . . . . . . 1.3 Exponential estimates of x( ¢,f) . . . . . . . . . . . . . . . . . . . . . 15 . . . . . . . . . . 1.4 The characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . 17 . . . . . . . . . . . . 1.5 The fundamental solution. . . . . . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . 1.6 The variation-of-constantsformula............................. 23 1. 7 Neutral differential difference equations . . . . . . . . . . . . . . . . . 25 . . . . . . . 1.8 Supplementary remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 34 . . . . . . . . . . . . . 2. Functional differential equations: Basic theory . . . . . . . . 38 . . 2.1 Definition of a retarded equation. . . . . . . . . . . . . . . . . . . . . . 38 . . . . . . . . . 2.2 Existence, uniqueness, and continuous dependence . . . . . . . . . . 39 . . . 2.3 Continuation of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 44 . . . . . . . . . . . .","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50351231828241,"sku":"CIN0387940766G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52139192582417,"sku":"NLS9780387940762","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387940766.jpg?v=1750813475"},{"product_id":"mathematical-theory-of-dilute-gases-book-carlo-cercignani-9780387942940","title":"The Mathematical Theory of Dilute Gases","description":"The idea for this book was conceived by the authors some time in 1988, and a first outline of the manuscript was drawn up during a summer school on mathematical physics held in Ravello in September 1988, where all three of us were present as lecturers or organizers.","brand":"WoB","offers":[{"title":"US \/ VERY_GOOD \/ SBYB","offer_id":50351265743121,"sku":"CIN0387942947VG","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52139268473105,"sku":"NLS9780387942940","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":53131141153041,"sku":"GOR011874890","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387942947.jpg?v=1751196518"},{"product_id":"applied-functional-analysis-book-eberhard-zeidler-9780387944425","title":"Applied Functional Analysis","description":"A theory is the more impressive, the simpler are its premises, the more distinct are the things it connects, and the broader is its range of applicability. Albert Einstein There are two different ways of teaching mathematics, namely, (i) the systematic way, and (ii) the application-oriented way. More precisely, by (i), I mean a systematic presentation of the material governed by the desire for mathematical perfection and completeness of the results. In contrast to (i), approach (ii) starts out from the question \"What are the most important applications?\" and then tries to answer this question as quickly as possible. Here, one walks directly on the main road and does not wander into all the nice and interesting side roads. The present book is based on the second approach. It is addressed to undergraduate and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems that are related to our real world and that have played an important role in the history of mathematics. The reader should sense that the theory is being developed, not simply for its own sake, but for the effective solution of concrete problems. viii Preface This introduction to functional analysis is divided into the following two parts: Part I: Applications to mathematical physics (the present AMS Vol. 108); Part II: Main principles and their applications (AMS Vol. 109).","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50351288746257,"sku":"CIN0387944427G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ NEW \/ INGRAM","offer_id":51001947816209,"sku":"NIN9780387944425","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52135656292625,"sku":"NLS9780387944425","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387944427.jpg?v=1751102673"},{"product_id":"stability-and-transition-in-shear-flows-book-peter-j-schmid-9780387989853","title":"Stability and Transition in Shear Flows","description":"The field of hydrodynamic stability has a long history, going back to Rey­ nolds and Lord Rayleigh in the late 19th century. Because of its central role in many research efforts involving fluid flow, stability theory has grown into a mature discipline, firmly based on a large body of knowledge and a vast body of literature. The sheer size of this field has made it difficult for young researchers to access this exciting area of fluid dynamics. For this reason, writing a book on the subject of hydrodynamic stability theory and transition is a daunting endeavor, especially as any book on stability theory will have to follow into the footsteps of the classical treatises by Lin (1955), Betchov \u0026amp; Criminale (1967), Joseph (1971), and Drazin \u0026amp; Reid (1981). Each of these books has marked an important development in stability theory and has laid the foundation for many researchers to advance our understanding of stability and transition in shear flows.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50351754772753,"sku":"CIN0387989854G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ NEW \/ INGRAM","offer_id":51001983795473,"sku":"NIN9780387989853","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ GARDNERS","offer_id":51157033189649,"sku":"NGR9780387989853","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ INGRAM","offer_id":52125716611345,"sku":"NLS9780387989853","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387989854.jpg?v=1751229035"},{"product_id":"theory-and-practice-of-finite-elements-book-alexandre-ern-9780387205748","title":"Theory and Practice of Finite Elements","description":"The origins of the finite element method can be traced back to the 1950s when engineers started to solve numerically structural mechanics problems in aeronautics. Since then, the field of applications has widened steadily and nowadays encompasses nonlinear solid mechanics, fluid\/structure interactions, flows in industrial or geophysical settings, multicomponent reactive turbulent flows, mass transfer in porous media, viscoelastic flows in medical sciences, electromagnetism, wave scattering problems, and option pricing (to cite a few examples). Numerous commercial and academic codes based on the finite element method have been developed over the years. The method has been so successful to solve Partial Differential Equations (PDEs) that the term \"Finite Element Method\" nowadays refers not only to the mere interpolation technique it is, but also to a fuzzy set of PDEs and approximation techniques. The efficiency of the finite element method relies on two distinct ingredi­ ents: the interpolation capability of finite elements (referred to as the approx­ imability property in this book) and the ability of the user to approximate his model (mostly a set of PDEs) in a proper mathematical setting (thus guar­ anteeing continuity, stability, and consistency properties). Experience shows that failure to produce an approximate solution with an acceptable accuracy is almost invariably linked to departure from the mathematical foundations. Typical examples include non-physical oscillations, spurious modes, and lock­ ing effects. In most cases, a remedy can be designed if the mathematical framework is properly set up.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50352829497617,"sku":"CIN0387205748G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ NEW \/ INGRAM","offer_id":51001999065361,"sku":"NIN9780387205748","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52340434370833,"sku":"NLS9780387205748","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387205748.jpg?v=1751071666"},{"product_id":"elements-of-applied-bifurcation-theory-book-yuri-kuznetsov-9780387219066","title":"Elements of Applied Bifurcation Theory","description":"The years that have passed since the publication of the first edition of this book proved that the basic principles used to select and present the material made sense. The idea was to write a simple text that could serve as a seri­ ous introduction to the subject. Of course, the meaning of \"simplicity\" varies from person to person and from country to country. The word \"introduction\" contains even more ambiguity. To start reading this book, only a moder­ ate knowledge of linear algebra and calculus is required. Other preliminaries, qualified as \"elementary\" in modern mathematics, are explicitly formulated in the book. These include the Fredholm Alternative for linear systems and the multidimensional Implicit Function Theorem. Using these very limited tools, a framewo:k of notions, results, and methods is gradually built that allows one to read (and possibly write) scientific papers on bifurcations of nonlinear dynamical systems. Among other things, progress in the sciences means that mathematical results and methods that once were new become standard and routinely used by the research and development community. Hopefully, this edition of the book will contribute to this process. The book's structure has been kept intact. Most of the changes introduced reflect recent theoretical and software developments in which the author was involved. Important changes in the third edition can be summarized as follows. A new section devoted to the fold-flip bifurcation for maps has appeared in Chapter 9.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50352847061265,"sku":"CIN0387219064G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ VERY_GOOD \/ SBYB","offer_id":51727876718865,"sku":"CIN0387219064VG","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52135590625553,"sku":"NLS9780387219066","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387219064.jpg?v=1753685988"},{"product_id":"qualitative-approach-to-inverse-scattering-theory-book-fioralba-cakoni-9781461488262","title":"A Qualitative Approach to Inverse Scattering Theory","description":"Inverse scattering theory is an important area of applied mathematics due to its central role in such areas as medical imaging , nondestructive testing and geophysical exploration. Until recently all existing algorithms for solving inverse scattering problems were based on using either a weak scattering assumption or on the use of nonlinear optimization techniques. The limitations of these methods have led in recent years to an alternative approach to the inverse scattering problem which avoids the incorrect model assumptions inherent in the use of weak scattering approximations as well as the strong a priori information needed in order to implement nonlinear optimization techniques. These new methods come under the general title of qualitative methods in inverse scattering theory and seek to determine an approximation to the shape of the scattering object as well as estimates on its material properties without making any weak scattering assumption and using essentially no a priori information on the nature of the scattering object. This book is designed to be an introduction to this new approach in inverse scattering theory focusing on the use of sampling methods and transmission eigenvalues. In order to aid the reader coming from a discipline outside of mathematics we have included background material on functional analysis, Sobolev spaces, the theory of ill posed problems and certain topics in in the theory of entire functions of a complex variable. This book is an updated and expanded version of an earlier book by the authors published by Springer titled Qualitative Methods in Inverse Scattering Theory   Review of Qualitative Methods in Inverse Scattering Theory All in all, the authors do exceptionally well in combining such a wide variety of mathematical material and in presenting it in a well-organized and easy-to-follow fashion. This text certainly complements the growing body of work in inverse scattering and should well suit both new researchers to the field as well as those who could benefit from such a nice codified collection of profitable results combined in one bound volume. 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It is geared towards graduate students and researchers in applied mathematics and can serve as a text for graduate courses.","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":50651256652049,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ GARDNERS","offer_id":50651257209105,"sku":"NGR9780387309316","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52462602748177,"sku":"NLS9780387309316","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/B005YVLHM0.jpg?v=1750845008"},{"product_id":"inverse-acoustic-and-electromagnetic-scattering-theory-book-david-colton-9783030303532","title":"Inverse Acoustic and Electromagnetic Scattering Theory","description":"The inverse scattering problem is central to many areas of science and technology such as radar, sonar, medical imaging, geophysical exploration and nondestructive testing. This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves.   In this fourth edition, a number of significant additions have been made including a new chapter on transmission eigenvalues and a new section on the impedance boundary condition where particular attention has been made to the generalized impedance boundary condition and to nonlocal impedance boundary conditions. Brief discussions on the generalized linear sampling method, the method of recursive linearization, anisotropic media and the use of target signatures in inverse scattering theory have also been added.","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":50823589298449,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ GARDNERS","offer_id":50823589331217,"sku":"NGR9783030303532","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52410360561937,"sku":"NLS9783030303532","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/3030303535.jpg?v=1751415135"},{"product_id":"semigroups-of-linear-operators-and-applications-to-partial-differential-equation-book-amnon-pazy-9780387908458","title":"Semigroups of Linear Operators and Applications to Partial Differential Equations","description":"From the reviews: \"Since E. Hille and K. Yoshida established the characterization of generators of C0 semigroups in the 1940s, semigroups of linear operators and its neighboring areas have developed into a beautiful abstract theory. Moreover, the fact that mathematically this abstract theory has many direct and important applications in partial differential equations enhances its importance as a necessary discipline in both functional analysis and differential equations. In my opinion Pazy has done an outstanding job in presenting both the abstract theory and basic applications in a clear and interesting manner. The choice and order of the material, the clarity of the proofs, and the overall presentation make this an excellent place for both researchers and students to learn about C0 semigroups.\" #Bulletin Applied Mathematical Sciences 4\/85#1 \"In spite of the other monographs on the subject, the reviewer can recommend that of Pazy as being particularly written, with a bias noticeably different from that of the other volumes. Pazy's decision to give a connected account of the applications to partial differential equations in the last two chapters was a particularly happy one, since it enables one to see what the theory can achieve much better than would the insertion of occasional examples. The chapters achieve a very nice balance between being so easy as to appear disappointing, and so sophisticated that they are incomprehensible except to the expert.\" #Bulletin of the London Mathematical Society#2","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":50828702187793,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ GOOD \/ SBYB","offer_id":50828702384401,"sku":"CIN0387908455G","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ INGRAM","offer_id":52139644518673,"sku":"NLS9780387908458","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":53222548898065,"sku":"NIN9780387908458","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0387908455.jpg?v=1751102669"},{"product_id":"fractional-differential-equations-book-bangti-jin-9783030760458","title":"Fractional Differential Equations","description":"This graduate textbook provides a self-contained introduction to modern mathematical theory on fractional differential equations. 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At the time of their inception, ?nite e- ment methods were viewed primarily as a tool for solving problems in structural analysis. However, it did not take long to discover that ?nite element methods could be applied with equal success to problems in other engineering and scienti?c ?elds. Today, ?nite element methods are also in common use, and indeed are often the method of choice, for incompressible ?uid ?ow, heat transfer, electromagnetics, and advection-diffusion-reaction problems, just to name a few. Given the early conn- tion between ?nite element methods and problems engendered by energy minimi- tion principles, it is not surprising that the ?rst mathematical analyses of ?nite e- ment methods were given in the environment of the classical Rayleigh-Ritz setting. Yet again, using the fertile soil provided by functional analysis in Hilbert spaces, it did not take long for the rigorous analysis of ?nite element methods to be extended to many other settings. 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We have tried to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a complete new presentation of lin­ ear systems (Chapters 6~9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global at­ tractors was completely revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (see Chapters 1, 2, 3, 9, and 10). Chapter 12 is completely new and contains a guide to active topics of re­ search. In the sections on supplementary remarks, we have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive. Jack K. Hale Sjoerd M. Verduyn Lunel Contents Preface............................................................ v Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1. Linear differential difference equations . . . . . . . . . . . . . . 11 . . . . . . 1.1 Differential and difference equations. . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . 1.2 Retarded differential difference equations. . . . . . . . . . . . . . . . 13 . . . . . . . 1.3 Exponential estimates of x( ¢,f) . . . . . . . . . . . . . . . . . . . . . 15 . . . . . . . . . . 1.4 The characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . 17 . . . . . . . . . . . . 1.5 The fundamental solution. . . . . . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . 1.6 The variation-of-constantsformula............................. 23 1. 7 Neutral differential difference equations . . . . . . . . . . . . . . . . . 25 . . . . . . . 1.8 Supplementary remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 34 . . . . . . . . . . . . . 2. 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