The landlady, landlord, cat, trap, and cheese all take credit for catching the long-tailed rat who is really the only one who knows the truth of the matter.

The authors develop various applications, in particular to the study of Banach algebras where the numerical range provides an important link between the algebraic and metric structures.

The theories of quadratic forms and their applications appear in many parts of mathematics and the sciences. All students of mathematics have the opportunity to encounter such concepts and applications in their first course in linear algebra. This subject and its extensions to infinite dimen sions comprise the theory of the numerical range W(T). There are two competing names for W(T), namely, the numerical range of T and the field of values for T. The former has been favored historically by the func tional analysis community, the latter by the matrix analysis community. It is a toss-up to decide which is preferable, and we have finally chosen the former because it is our habit, it is a more efficient expression, and because in recent conferences dedicated to W(T), even the linear algebra commu nity has adopted it. Also, one universally refers to the numerical radius, and not to the field of values radius. Originally, Toeplitz and Hausdorff called it the Wertvorrat of a bilinear form, so other good names would be value field or form values. The Russian community has referred to it as the Hausdorff domain. Murnaghan in his early paper first called it the region of the complex plane covered by those values for an n x n matrix T, then the range of values of a Hermitian matrix, then the field of values when he analyzed what he called the sought-for region.

Starting with elementary operator theory and matrix analysis, this book introduces the basic properties of the numerical range and gradually builds up the whole numerical range theory. Over 400 assorted problems, ranging from routine exercises to published research results, give you the chance to put the theory into practice and test your understanding. Interspersed throughout the text are numerous comments and references, allowing you to discover related developments and to pursue areas of interest in the literature. Also included is an appendix on basic convexity properties on the Euclidean space. Targeted at graduate students as well as researchers interested in functional analysis, this book provides a comprehensive coverage of classic and recent works on the numerical range theory. It serves as an accessible entry point into this lively and exciting research area.

Mathematicians delight in finding surprising connections between seemingly disparate areas of mathematics. Finding Ellipses is a delight-filled romp across a three-way unexpected connection between complex analysis, linear algebra, and projective geometry.

This book describes recent developments as well as some classical results regarding holomorphic mappings. The book starts with a brief survey of the theory of semigroups of linear operators including the Hille-Yosida and the Lumer-Phillips theorems. The numerical range and the spectrum of closed densely defined linear operators are then discussed in more detail and an overview of ergodic theory is presented. The analytic extension of semigroups of linear operators is also discussed. The recent study of the numerical range of composition operators on the unit disk is mentioned. Then, the basic notions and facts in infinite dimensional holomorphy and hyperbolic geometry in Banach and Hilbert spaces are presented, L. A. Harris' theory of the numerical range of holomorphic mappings is generalized, and the main properties of the so-called quasi-dissipative mappings and their growth estimates are studied. In addition, geometric and quantitative analytic aspects of fixed point theory are discussed. A special chapter is devoted to applications of the numerical range to diverse geometric and analytic problems.

Development and Application of Classical Capillary Curve Theory shows readers how to improve chemical flooding recovery procedures in enhanced oil recovery (EOR) using sophisticated numerical modeling techniques for oil displacement that build on, and enhance, the accepted capillary curve in a novel way. The title presents research and technical data that amends the classical capillary curve, giving new impetus to chemical flooding research. By analyzing the capillary number experiment QL, the book models oil and water distribution in the micro-space of the reservoir core. Finally, the book discusses chemical flooding software, reservoir geological modeling, field tests, and a new technique of digital oil displacement testing being developed. - Describes how to improve chemical flooding recovery procedures using sophisticated numerical modeling techniques for oil displacement that build on, and enhance, accepted capillary curve theory - Presents the role of the capillary number experiment curve QL on oil-water distribution in the microscopic core of a reservoir - Details novel digital techniques for oil displacement testing and modeling with global applicability in enhanced oil recovery (EOR) - Gives information on chemical flooding software, reservoir geological modeling and field testing - Provides advanced digital modeling techniques alongside insights gained through many years of practical experience in Chinese oilfields