CBMS-NSF Regional Conference Series in Applied Mathematics – serie

A concise survey of the current state of knowledge in 1972 about solving elliptic boundary-value eigenvalue problems with the help of a computer. This volume provides a case study in scientific computing - the art of utilizing physical intuition, mathematical theorems and algorithms, and modern computer technology to construct and explore realistic models of problems arising in the natural sciences and engineering.

The applications of the theory of Optimal Control of distributed parameters is an extremely wide field and, although a large number of questions remain open, the whole subject continues to expand very rapidly. The author does not attempt to cover the field but does discuss a number of the more interesting areas of application.

Acquaints the specialist in relativity theory with some global techniques for the treatment of space-times and will provide the pure mathematician with a way into the subject of general relativity.

An exploration of the interrelated fields of design of experiments and sequential analysis with emphasis on the nature of theoretical statistics and how this relates to the philosophy and practice of statistics.

Presents a coherent body of theory for the derivation of the sampling distributions of a wide range of test statistics. Emphasis is on the development of practical techniques. A unified treatment of the theory was attempted, e.g., the author sought to relate the derivations for tests on the circle and the two-sample problem to the basic theory for the one-sample problem on the line. The Markovian nature of the sample distribution function is stressed, as it accounts for the elegance of many of the results achieved, as well as the close relation with parts of the theory of stochastic processes.

Provides a common setting for various methods of bounding the eigenvalues of a self-adjoint linear operator and emphasizes their relationships. A mapping principle is presented to connect many of the methods. The eigenvalue problems studied are linear, and linearization is shown to give important information about nonlinear problems. Linear vector spaces and their properties are used to uniformly describe the eigenvalue problems presented that involve matrices, ordinary or partial differential operators, and integro-differential operators.

This monograph provides a relatively brief introduction to conjugate duality in both finite- and infinite-dimensional problems. Emphasizes the fundamental importance of the concepts of Lagrangian function, saddle-point, and saddle-value. General examples are drawn from nonlinear programming, approximation, stochastic programming, the calculus of variations, and optimal control.

Presents a theory of indexing capable of ranking index terms, or subject identifiers in decreasing order of importance. This leads to the choice of good document representations, and also accounts for the role of phrases and of thesaurus classes in the indexing process.This study is typical of theoretical work in automatic information organization and retrieval, in that concepts are used from mathematics, computer science, and linguistics. A complete theory of information retrieval may emerge from an appropriate combination of these three disciplines.

Mathematical theories of populations have appeared both implicitly and explicitly in many important studies of populations, human populations as well as populations of animals, cells and viruses. They provide a systematic way for studying a population's underlying structure.In this book, a basic model in population age structure is studied and then applied, extended and modified, to several population phenomena such as stable age distributions, self-limiting effects, and two-sex populations. Population genetics are studied with special attention to derivation and analysis of a model for a one-locus, two-allele trait in a large randomly mating population. The dynamics of contagious phenomena in a population are studied in the context of epidemic diseases.

Lectures on a unified theory of and practical procedures for the numerical solution of very general classes of linear and nonlinear two point boundary-value problems.

An introduction to aspects of the theory of dynamical systems based on extensions of Liapunov's direct method. The main ideas and structure for the theory are presented for difference equations and for the analogous theory for ordinary differential equations and retarded functional differential equations. The latest results on invariance properties for non-autonomous time-varying systems processes are presented for difference and differential equations.

A unified discussion of the formulation and analysis of special methods of mixed initial boundary-value problems. The focus is on the development of a new mathematical theory that explains why and how well spectral methods work. Included are interesting extensions of the classical numerical analysis.

Explores modern topics in graph theory and its applications to problems in transportation, genetics, pollution, perturbed ecosystems, urban services, and social inequalities. The author presents both traditional and relatively atypical graph-theoretical topics to best illustrate applications.

Focuses on finding the minimum number of arithmetic operations needed to perform the computation and on finding a better algorithm when improvement is possible. The author concentrates on that class of problems concerned with computing a system of bilinear forms.Results that lead to applications in the area of signal processing are emphasized, since (1) even a modest reduction in the execution time of signal processing problems could have practical significance; (2) results in this area are relatively new and are scattered in journal articles; and (3) this emphasis indicates the flavor of complexity of computation.

This monograph deals with aspects of the computer programming process that involve techniques derived from mathematical logic. The author focuses on proving that a given program produces the intended result whenever it halts, that a given program will eventually halt, that a given program is partially correct and terminates, and that a system of rewriting rules always halts. Also, the author describes the intermediate behavior of a given program, and discusses constructing a program to meet a given specification.

Draws together some mathematical ideas that are useful in population genetics, concentrating on a few aspects which are both biologically relevant and mathematically interesting.

Finite elasticity is a theory of elastic materials that are capable of undergoing large deformations. This theory is inherently nonlinear and is mathematically quite complex. This monograph presents a derivation of the basic equations of the theory, a discussion of the general boundary-value problems, and a treatment of several interesting and important special topics such as simple shear, uniqueness, the tensile deformations of a cube, and antiplane shear. The monograph is intended for engineers, physicists, and mathematicians.

A monograph on some of the ways geometry and analysis can be used in mathematical problems of physical interest. The roles of symmetry, bifurcation, and Hamiltonian systems in diverse applications are explored.

This book deals with the mathematical side of the theory of shock waves. The author presents what is known about the existence and uniqueness of generalized solutions of the initial value problem subject to the entropy conditions. The subtle dissipation introduced by the entropy condition is investigated and the slow decay in signal strength it causes is shown.

The jackknife and the bootstrap are nonparametric methods for assessing the errors in a statistical estimation problem. They provide several advantages over the traditional parametric approach: the methods are easy to describe and they apply to arbitrarily complicated situations; distribution assumptions, such as normality, are never made.This monograph connects the jackknife, the bootstrap, and many other related ideas such as cross-validation, random subsampling, and balanced repeated replications into a unified exposition. The theoretical development is at an easy mathematical level and is supplemented by a large number of numerical examples.The methods described in this monograph form a useful set of tools for the applied statistician. They are particularly useful in problem areas where complicated data structures are common, for example, in censoring, missing data, and highly multivariate situations.

An introduction to far reaching developments in theoretical combustion, with special emphasis on flame stability, a topic that has, to date, benefited most from the application of modern asymptotic methods. The authors provide a modern view of flame theory, and a complete description of the longstanding ignition and explosion problems, including the solutions that were made available independently by Kapila and Kassoy through activation energy asymptotics, the main theme of this monograph.

There has been an explosive growth in the field of combinatorial algorithms. These algorithms depend not only on results in combinatorics and especially in graph theory, but also on the development of new data structures and new techniques for analyzing algorithms.Four classical problems in network optimization are covered in detail, including a development of the data structures they use and an analysis of their running time.Data Structures and Network Algorithms attempts to provide the reader with both a practical understanding of the algorithms, described to facilitate their easy implementation, and an appreciation of the depth and beauty of the field of graph algorithms.

Many situations exist in which solutions to problems are represented as function space integrals. Such representations can be used to study the qualitative properties of the solutions and to evaluate them numerically using Monte Carlo methods. The emphasis in this book is on the behavior of solutions in special situations when certain parameters get large or small.

A systematic, self-contained treatment of the theory of stochastic differential equations in infinite dimensional spaces. Included is a discussion of Schwartz spaces of distributions in relation to probability theory and infinite dimensional stochastic analysis, as well as the random variables and stochastic processes that take values in infinite dimensional spaces.

The soliton is a dramatic concept in nonlinear science. What makes this book unique in the treatment of this subject is its focus on the properties that make the soliton physically ubiquitous and the soliton equation mathematically miraculous. Here, on the classical level, is the entity field theorists have been postulating for years: a local traveling wave pulse; a lump-like coherent structure; the solution of a field equation with remarkable stability and particle-like properties. It is a fundamental mode of propagation in gravity-driven surface and internal waves; in atmospheric waves; in ion acoustic and Langmuir waves in plasmas; in some laser waves in nonlinear media; and in many biologic contexts, such as alpha-helix proteins.This is not an encyclopedia of information on solitons in which every sentence is interrupted by either a caveat or a reference. Rather, Newell has tried to tell the story of the soliton as he would have liked to have heard it as a graduate student, with some historical development, lots of motivation, and frequent attempts to relate the topic at hand to the big picture.The book begins with a history of the soliton from its first sighting to the discovery of the inverse scattering method and recent ideas on the algebraic structure of soliton equations. Chapter 2 focuses on the universal nature of these equations and how and why they arise in physical and engineering contexts as asymptotic solvability conditions. The third chapter deals with the inverse scattering method and perturbation theories. Chapter 4 introduces the t-function and discusses the relations between the various methods for constructing solutions to the soliton equations and their various properties. Finally, an algebraic structure for the equations is provided in Chapter 5.

A study of how complexity questions in computing interact with classical mathematics in the numerical analysis of issues in algorithm design. Algorithmic designers concerned with linear and nonlinear combinatorial optimization will find this volume especially useful.Two algorithms are studied in detail: the ellipsoid method and the simultaneous diophantine approximation method. Although both were developed to study, on a theoretical level, the feasibility of computing some specialized problems in polynomial time, they appear to have practical applications. The book first describes use of the simultaneous diophantine method to develop sophisticated rounding procedures. Then a model is described to compute upper and lower bounds on various measures of convex bodies. Use of the two algorithms is brought together by the author in a study of polyhedra with rational vertices.The book closes with some applications of the results to combinatorial optimization.

The approximation of functions of several variables continues to be a difficult problem in scientific computing because many of the algorithms required for such problems have yet to be written. This monograph is written for a broad audience of computational mathematicians and statisticians concerned with the development of algorithms or the derivation of approximations from linear projections, of which the interpolating operators are an important example. As an aid to both researchers and students, a bibliography of more than 200 titles is included.

Interfacial phenomena are commonplace in physics, chemistry, biology, and in various disciplines bridging these fields. They occur whenever a continuum is present which can exist in at least two different chemical or physical ""states"", and there is some mechanism which generates or enforces a spatial separation between these states. The separation boundary is then called an interface. In the examples studied here, the separation boundary, and its internal structure, result from the balance between two opposing tendencies: a diffusive effect which attempts to mix and smooth the properties of the material, and a physical or chemical mechanism which works to drive it to one or the other pure state.This volume is unique in that the treatment of flames, as well as internal layer dynamics ""including curvature effects"", is more detailed and systematic than in most other publications. Mathematicians and natural scientists interested in interfacial phenomena, especially flame theory, the mathematics of excitable media, electrophoresis, and phase change problems, will find Dynamics of Internal Layers and Diffusive Interfaces exceptionally useful.

The subject of multivariate splines has become a rapidly growing field of mathematical research. The author presents the subject from an elementary point of view that parallels the theory and development of univariate spline analysis. To compensate for the missing proofs and details, an extensive bibliography has been included. There is a presentation of open problems with an emphasis on the theory and applications to computer aided design, data analysis, and surface fitting. Applied mathematicians and engineers working in the areas of curve fitting, finite element methods, computer aided geometric design, signal processing, mathematical modelling, computer aided design, computer aided manufacturing, and circuits and systems will find this monograph essential to their research.