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1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse, ?1 denoted by A , such that ?1 ?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus, ?1 ?1 (A ) = A, T ?1 ?1 T (A ) =(A ) , ? ?1 ?1 ? (A ) =(A ) , ?1 ?1 ?1 (AB) = B A , T ? where A and A , respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number ? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to ?,if Ax = ?x. ?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.

Existence of huge amounts of data on the Web has developed an undeferring need to locate right information at right time, as well as to integrating information effectively to provide a comprehensive source of relevant information. There is a need to develop efficient tools for analyzing and managing Web data, and efficiently managing Web information from the database perspective. The book proposes a data model called WHOM (Warehouse Object Model) to represent HTML and XML documents in the warehouse. It defines a set of web algebraic operators for building new web tables by extracting relevant data from the Web, as well as generating new tables from existing ones. These algebraic operators are used for change detection.

Interquanta (IQ), an interactive program on quantum mechanics allows students to do their own quantum physics experiments on the computer, and to study in 3D color graphics such quantities as complex probability amplitude, eigencalues, scattering cross sections, and more. By experiencing many such computer experiments, students gain a unique, "e;hands-on"e; experience in quantum physics which is otherwise difficult to achieve. The graphic features include two-and three-dimensional graphics in the form of static frames and motion pictures.Students do no programming, and hence need no previous detailed knowledge of this. The program has a very convenient, self-explanatory user interface based on the Java software technology. The book provides a recapitulation of the basic quantum mechanical formula, a manual to the IQ program, and a complete course with more than 300 tested problems. Fully automatic demonstration sessions are provided as introduction to interactive work.Physics topics covered include free particles, bound states and scattering in various potentials in one and three space dimensions, two-particle systems, properties of special functions of mathematical physics.

Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.

My principal aim in writing this book was to present a wide range of astrophysical topics in sufficient depth to give the reader a general quantitative understanding of the subject. The book outlines cosmic events but does not portray them in detail-it provides aseries of astrophysical sketches. I think this approach befits the present uncertainties and changing views in astrophysics. The material is based on notes I prepared for a course aimed at seniors and beginning graduate students in physics and astronomy at Cornell. This course defined the level at which the book is written. For readers who are versed in physics but are unfamiliar with astronomical terminology, Appendix A is included. It gives a brief background of astronomical concepts and should be read before starting the main text. The first few chapters outline the scope of modern astrophysics and deal with elementary problems concerning the size and mass of cosmic objects. However, it soon becomes apparent that a broad foundation in physics is needed to proceed. This base is developed in Chapters 4 to 7 by using, as ex am pIes, specific astronomi- cal situations. Chapters 8 to 10 enlarge on the topics first outlined in Chapter I and show how we can obtain quantitative insights into the structure and evolution of stars, the dynamics of co~mic gases, and the large-scale behavior of the universe.

This text attempts to change the way we teach logic to beginning students. Instead of teaching logic as a subject in isolation, we regard it as a basic tool and show how to use it. We strive to give students a skill in the propo­ sitional and predicate calculi and then to exercise that skill thoroughly in applications that arise in computer science and discrete mathematics. We are not logicians, but programming methodologists, and this text reflects that perspective. We are among the first generation of scientists who are more interested in using logic than in studying it. With this text, we hope to empower further generations of computer scientists and math­ ematicians to become serious users of logic. Logic is the glue Logic is the glue that binds together methods of reasoning, in all domains. The traditional proof methods -for example, proof by assumption, con­ tradiction, mutual implication, and induction- have their basis in formal logic. Thus, whether proofs are to be presented formally or informally, a study of logic can provide understanding.

The title gives a reasonable ?rst-order approximation to what this book is about. To explain why, let's start with the expression "e;di?erential equations."e; These are essential in science and engineering, because the laws of nature t- ically result in equations relating spatial and temporal changes in one or more variables.Todevelopanunderstandingofwhatisinvolvedin?ndingsolutions, the book begins with problems involving derivatives for only one independent variable, and these give rise to ordinary di?erential equations. Speci?cally, the ?rst chapter considers initial value problems (time derivatives), and the second concentrates on boundary value problems (space derivatives). In the succeeding four chapters problems involving both time and space derivatives, partial di?erential equations, are investigated. This brings us to the next expression in the title: "e;numerical methods."e; This is a book about how to transform di?erential equations into problems that can be solved using a computer. The fact is that computers are only able to solve discrete problems and generally do this using ?nite-precision arithmetic. What this means is that in deriving and then using a numerical algorithmthecorrectnessofthediscreteapproximationmustbeconsidered,as must the consequences of round-o? error in using ?oating-point arithmetic to calculatetheanswer.Oneoftheinterestingaspectsofthesubjectisthatwhat appears to be an obviously correct numerical method can result in complete failure. Consequently, although the book concentrates on the derivation and use of numerical methods, the theoretical underpinnings are also presented andusedinthedevelopment.

The Journey Ahead At the heart of transcendental number theory lies an intriguing paradox: While essen- tially all numbers are transcendental, establishing the transcendence of a particular number is a monumental task. Thus transcendental numbers are an enigmatic species of number: We know they are all around us and yet it requires enormous effort to catch one. More often than not, they slip through our fingers and dissappear back into the dense jungle of numbers. Here we will venture to tame a few of these incredible creatures. In the pages ahead we offer an approach to transcendence that not only includes the intricate analysis but also the beautiful ideas behind the technical details. The phrase "e;classical transcendental number theory"e; in the title of this book refers to the most widely known results that were obtained in the nineteenth and early twentieth centuries. The reason for this focus is threefold. Firstly, this body of work requires only the mathematical techniques and tools familiar to advanced undergraduate mathematics students, and thus this area can be appreciated by a wide range of readers. Secondly, the ideas behind modem transcendence results are almost always an elaboration of the classical arguments we will explore here. And finally, and perhaps more importantly, this early work yields the transcendence of such admired and well-known numbers as e, rr, and even 2v'2.

Geometry of Biological Time deals with dynamics of processes that repeat themselves regularly. Such rhythmic return through a cycle of change is an ubiquitous principle of organization in living systems. In this revised and updated edition the author plans to extend the thread from 1980 to the present concentrating on areas which he personally feels have been interesting and where he feels there will be much activity in the future. This involves going through spatial biochemical, electrophysiological, and organismic dynamical systems and patterns that were discovered by pursuing the theme of phase singularities that the original book introduced. In particular the work on excitability in cell membranes has been thoroughly updated as have the references throughout the book.

This extensive volume began as a short course primarily geared toward toxicologists who want to expand their understanding of toxicologic pathology in order to be better study directors while also proving to be of great interest to other drug development scientists and regulatory reviewers. The overall goal is to help non-pathologists understand, contextualize, and communicate the pathology data and interpretations from the study pathologist in a practical and usable format. Within the book, readers will find an overview of general pathology concepts that include fundamental vocabulary and the basics of pathophysiological processes, along with numerous chapters devoted to pathology in specific organ systems as well as topics such as biomarkers, correlation of clinical pathology endpoints (chemistry and hematology) with microscopic changes, and well-known pathology findings for classes of toxic substances. Authoritative, practical, and comprehensive, Toxicologic Pathology for Non-Pathologists aims to help non-pathologists understand, converse in, and apply a basic understanding of pathology in their day-to-day careers.

The oceans are shrinking. They're not literally shrinking; warming in the last century has actually expanded the sea enough to threaten low-lying coastal lands that are vul- nerable to storm surge. During the same interval, however, events on land have increasingly affected the sea. Since in most ways the Earth is a closed system-a zero-sum planet in today's parlance-as terrestrial influence on the sea expands, the sea's influence on its own processes shrinks. Control of many crucial marine processes no longer resides within the sea. The evidence for this is abundant and, to anyone who is looking, unmis- takable. In recent decades scientists have witnessed unprecedented pertur- bations and increases in previously uncommon events that demonstrate growing terrestrial influences on the sea. Numerous marine species, from sea urchins to monk seals, have experienced devastating epidemics. The number of harmful algal blooms and jellyfishpopulation explosions is rising An hypoxic "e;dead zone"e; in the Gulf of Mexico off the mouth of the Mississippi Rivernow appears each year and grows to encompass an area as large as NewJersey. Live coral cover in shallow reefs in Florida,Jamaica, the Maldives and many other locations has severely declined. Deepwater reef- building corals, once widely distributed, have disappeared throughout much of their ranges. Researchers have discovered high concentrations of persis- tent organic pollutants in declining populations of beluga whales and polar bears, both high trophic level predators in marine food webs.

This volume is being published for two reasons. The first is to present a collection of previously published articles on the subject of programming methodology that have helped define the field and give it direction. It is hoped that the scientist in the field will find the volume useful as a reference, while the scientist in neighboring fields will find it useful in seriously acquainting himself with important ideas in programming methodology. The advanced student can also study it-either in a course or by himself -in order to learn significant material that may not appear in texts for some time. The second reason for this volume is to make public the nature and work on programming methodology of IFIP Working Group 2.3, hereafter called WG2.3. (IFIP stands for International Federation for Information Processing.) WG2.3 is one of many IFIP Working Groups that have been established to provide international forums for discussion of ideas in various areas. Generally, these groups publish proceedings of some of their meetings and occasionally they sponsor a larger conference that persons outside a group can attend. WG2.3 has been something of a maverick in this respect. From the beginning the group has shunned paperwork, reports, meetings, and the like. This has meant less pUblicity for IFIP and WG2.3, but on the other hand it has meant that meetings could be devoted almost wholly to scientific discussions.

One of the most exciting developments in modern physics has been the discovery of the new class of oxide materials with high superconducting transition temperature. Systems with Tc well above liquid nitrogen temperature are already a reality and higher Tc's are anticipated. Indeed, the idea of a room-temperature superconductor, which just a short time ago was considered science fiction, appears to be a distinctly possible outcome of materials research. To address the need to train students and scientists for research in this exciting field, Jeffrey W. Lynn and colleagues at the University of Maryland, College Park, as well as other superconductivity experts from around the U.S., taught a graduate-level course in the fall of 1987, from which the chapters in this book were drawn. Subjects included are: Survey of superconductivity (J. Lynn).- The theory of type-II superconductivity (D. Belitz).- The Josephson effect (P. Ferrell).- Crystallography (A. Santoro).- Electronic structure (C.P. Wang).- Magnetic properties and interactions (J. Lynn).- Synthesis and diamagnetic properties (R. Shelton).- Electron pairing (P. Allen).- Superconducting devices (F. Bedard).- Superconducting properties (J. Crow, N.-P. Ong).

There is a growing social interest in developing vision-based vehicle guidance systems for improving traffic safety and efficiency and the environment. Ex- amples of vision-based vehicle guidance systems include collision warning systems, steering control systems for tracking painted lane marks, and speed control systems for preventing rear-end collisions. Like other guidance systems for aircraft and trains, these systems are ex- pected to increase traffic safety significantly. For example, safety improve- ments of aircraft landing processes after the introduction of automatic guidance systems have been reported to be 100 times better than prior to installment. Although the safety of human lives is beyond price, the cost for automatic guidance could be compensated by decreased insurance costs. It is becoming more important to increase traffic safety by decreasing the human driver's load in our society, especially with an increasing population of senior people who continue to drive. The second potential social benefit is the improvement of traffic efficiency by decreasing the spacing between vehicles without sacrificing safety. It is reported, for example, that four times the efficiency is expected if the spacing between cars is controlled automatically at 90 cm with a speed of 100 kmjh compared to today's typical manual driving. Although there are a lot of tech- nical, psychological, and social issues to be solved before realizing the high- density jhigh-speed traffic systems described here, highly efficient highways are becoming more important because of increasing traffic congestion.

Mast cells and basophils are responsible for inflammatory and allergic reactions. As such, the signals that generate these responses and how their pathways of action work are the focus of much present day research into allergy and inflammation. This book focuses primarily on the molecular mechanisms that govern mast cell and basophil cell biology and function, as well as providing a comprehensive summary of the field of signal transduction and also giving insights into areas that have therapeutic potential. The book provides detailed insights into mast cell and basophil growth and development, their activation by allergens, including details of receptor activation and downstream events, and the regulators of morphology and degranulation. The metabolic pathways involved in prostaglandin and leukotriene production are discussed as is the role of transcription factors in mast cell growth and cytokine production. Written by leaders in the field, this volume will provide the reader with an up-to-date account of a topic where the rapid progress makes conventional information gathering difficult.

This IMA Volume in Mathematics and its Applications SINGULARITIES AND OSCILLATIONS is based on the proceedings of a very successful one-week workshop with the same title, which was an integral part of the 1994-1995 IMA program on "e;Waves and Scattering. "e; We would like to thank Joseph Keller, Jeffrey Rauch, and Michael Taylor for their excellent work as organizers of the meeting. We would like to express our further gratitude to Rauch and Taylor, who served as editors of the proceedings. We also take this opportunity to thank the National Science Foun- dation (NSF), the Army Research Office (ARO) and the Office of Naval Research (ONR), whose financial support made the workshop possible. Avner Friedman Robert Gulliver v PREFACE Thestudyofsingularitiesand oscillationsofwaves has progressed along several fronts. A key common feature is the presence of a small scale in the solutions. Recent emphasis has been on nonlinear waves. Nonlinear problems are generally less amenable than linear problems to broad unified approaches. As a result there is a justifiable tendency to concentrate on problems of particular geometric or physical interest. This volume con- tains a multiplicity of approaches brought to bear on problems varying from the formation ofcaustics and the propagation ofwaves at a boundary to the examination ofviscous boundary layers. There is an examination of the foundations of the theory of high-frequency electromagnetic waves in a dielectric or semiconducting medium. Unifying themes are not entirely absent from nonlinear analysis.