Analysis in Banach Spaces: Volume III: Harmonic Analysis and Spectral Theory

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This third volume of Analysis in Banach Spaces offers a systematic treatment of Banach space-valued singular integrals, Fourier transforms, and function spaces. It further develops and ramifies the theory of functional calculus from Volume II and describes applications of these new notions and tools to the problem of maximal regularity of evolution equations. The exposition provides a unified treatment of a large body of results, much of which has previously only been available in the form of research papers. Some of the more classical topics are presented in a novel way using modern techniques amenable to a vector-valued treatment. Thanks to its accessible style with complete and detailed proofs, this book will be an invaluable reference for researchers interested in functional analysis, harmonic analysis, and the operator-theoretic approach to deterministic and stochastic evolution equations.

Metric Spaces and Related Analysis

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This book offers the comprehensive study of one of the foundational topics in Mathematics, known as Metric Spaces. The book delivers the concepts in an appropriate and concise manner, at the same time rich in illustrations and exercise problems. Special focus has been laid on important theorems like Baire\'s Category theorem, Heine–Borel theorem, Ascoli–Arzela Theorem, etc, which play a crucial role in the study of metric spaces. The additional chapter on Cofinal completeness, UC spaces and finite chainability makes the text unique of its kind. This helps the students in: taking the secondary step towards analysis on metric spaces, realizing the connection between the two most important classes of functions, continuous functions and uniformly continuous functions, understanding the gap between compact metric spaces and complete metric spaces. Readers will also find brief discussions on various subtleties of continuity like subcontinuity, upper semi-continuity, lower semi-continuity, etc. The interested readers will be motivated to explore the special classes of functions between metric spaces to further extent. Consequently, the book becomes a complete package: it makes the foundational pillars strong and develops the interest of students to pursue research in metric spaces. The book is useful for third and fourth year undergraduate students and it is also helpful for graduate students and researchers. Sample Chapter(s) Foreword Chapter 1: Fundamentals of Analysis Contents: Fundamentals of Analysis Continuity and Some Stronger Notions Complete Metric Spaces Compactness Weaker Notions of Compactness Real-Valued Functions on Metric Spaces Connectedness Readership: Undergraduate and graduate students, researchers in the areas of real analysis, analysis on metric spaces, real functions, topology, functional analysis.

Seeing Four-Dimensional Space and Beyond Using Knots!

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According to string theory, our universe exists in a 10- or 11-dimensional space. However, the idea the space beyond 3 dimensions seems hard to grasp for beginners. This book presents a way to understand four-dimensional space and beyond: with knots! Beginners can see high dimensional space although they have not seen it. With visual illustrations, we present the manipulation of figures in high dimensional space, examples of which are high dimensional knots and n-spheres embedded in the (n+2)-sphere, and generalize results on relations between local moves and knot invariants into high dimensional space. Local moves on knots, circles embedded in the 3-space, are very important to research in knot theory. It is well known that crossing changes are connected with the Alexander polynomial, the Jones polynomial, HOMFLYPT polynomial, Khovanov homology, Floer homology, Khovanov homotopy type, etc. We show several results on relations between local moves on high dimensional knots and their invariants. The following related topics are also introduced: projections of knots, knot products, slice knots and slice links, an open question: can the Jones polynomial be defined for links in all 3-manifolds? and Khovanov-Lipshitz-Sarkar stable homotopy type. Slice knots exist in the 3-space but are much related to the 4-dimensional space. The slice problem is connected with many exciting topics: Khovanov homology, Khovanv–Lipshits–Sarkar stable homotopy type, gauge theory, Floer homology, etc. Among them, the Khovanov–Lipshitz–Sarkar stable homotopy type is one of the exciting new areas; it is defined for links in the 3-sphere, but it is a high dimensional CW complex in general. Much of the book will be accessible to freshmen and sophomores with some basic knowledge of topology. Sample Chapter(s) Introduction Chapter 1: Local Moves on Knots: For Beginners Chapter 2: Four-Dimensional Space ℝ4 Contents: About the Author Acknowledgment Introduction Local Moves On Knots: For Beginners Four-Dimensional Space ℝ4 Local Moves in Higher-Dimensional Space: For Beginners Knotted-Objects in 4-Space and Beyond Local Moves on High-Dimensional Knots and Related Invariants The Alexander and Jones Polynomials of One-Dimensional Links in ℝ3 Local Moves and Knot Polynomials in Higher Dimensions Important Topics in Knot Theory References Index Readership: Freshmen and sophomores of science and people with similar mathematical background; general public interested in science, sci-fiction, and high-dimensional space; novice researchers in knot theory.