Society for Industrial and Applied Mathematics: Applied Geometry

Architectural Geometry

Helmut Pottmann

Architectural Geometry.

Linear and Nonlinear Functional Analysis with Applications

Philippe G. Ciarlet

Linear and Nonlinear Functional Analysis with Applications.

Back Matter

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Other Titles in Applied Mathematics, Page 777-833.
The back matter includes the Bibliographical Notes, Bibliograpy, Main Notations and Index

CHAPTER 1: REAL ANALYSIS AND THEORY OF FUNCTIONS: A QUICK REVIEW

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Other Titles in Applied Mathematics, Page 1-41.
Introduction This first chapter constitutes a quick review of real analysis, which traditionally comprises: set theory, the axiom of choice, and the construction of the sets ℝ and ℝn; the basic properties of topological and metric spaces, such as those related to the notions of continuity, compactness, completeness, connectedness, and simple-connectedness; the Tietze-Urysohn extension theorem (a crucial use of which will be made at several places in Chapter 9); and the construction and the main properties of the Lebesgue measure and Lebesgue integral in ℝn: the Radon-Nikodym theorem; Fatou’s lemma; the Beppo Levi monotone convergence theorem; the Lebesgue dominated convergence theorem; Tonelli’s and Fubini’s theorems; volumes, areas, and lengths in ℝn; and the change of variable formula in multiple integrals.

CHAPTER 2: NORMED VECTOR SPACES

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Other Titles in Applied Mathematics, Page 43-122.
Introduction Linear functional analysis constitutes the subject of Chapters 2-5. More specifically, the aim of the present chapter is to establish basic properties that hold in any normed vector space, complete or not. Then Chapter 3 will be devoted to complete normed vector spaces and Chapter 4 to normed vector spaces, complete or not, whose norm is derived from an inner product. Finally, Chapter 5 will address more elaborate properties of these spaces, assembled under the appellation “great theorems.”

CHAPTER 3: BANACH SPACES

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Other Titles in Applied Mathematics, Page 123-172.
Introduction Banach spaces, i.e., complete normed vector spaces, play a central role in linear and nonlinear functional analysis. The aim of this chapter is to establish their most immediate basic properties.

CHAPTER 4: INNER-PRODUCT SPACES AND HILBERT SPACES

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Other Titles in Applied Mathematics, Page 173-229.
Introduction Among infinite-dimensional normed vector spaces, inner-product spaces, and especially Hilbert spaces, i.e., complete inner-product spaces, such as their archetypes, the spaces ℓ2 and L2(Q) (Section 4.2), are by far “the best.”

CHAPTER 5: THE “GREAT THEOREMS” OF LINEAR FUNCTIONAL ANALYSIS

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Other Titles in Applied Mathematics, Page 231-303.
Introduction This chapter is devoted to the proofs of most of the “great theorems” of linear functional analysis. Their common characteristic is that they hinge on one, or on both, of two fundamental results: Baire’s theorem (Theorem 5.1-2) and the Hahn-Banach theorem in a normed vector space (Theorem 5.9-1).

CHAPTER 6: LINEAR PARTIAL DIFFERENTIAL EQUATIONS

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Other Titles in Applied Mathematics, Page 305-449.
Introduction In this chapter, we only consider partial differential equations where all the variables are “space variables,” i.e., coordinates of points in an open subset of RN; we do not consider “time-dependent” problems.

CHAPTER 7: DIFFERENTIAL CALCULUS IN NORMED VECTOR SPACES

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Other Titles in Applied Mathematics, Page 451-573.
Introduction Nonlinear functional analysis begins in earnest with this chapter, which is centered on the notion of derivability of mappings between arbitrary normed vector spaces.

CHAPTER 8: DIFFERENTIAL GEOMETRY IN ℝn

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Other Titles in Applied Mathematics, Page 575-656.
Introduction Why such a chapter? Simply because, even though differential geometry in ℝn may be correctly viewed as only a brief introduction to differential geometry in general, its modest scope already provides beautiful existence and uniqueness theorems for two highly nonlinear systems of partial differential equations, a topic at the core of nonlinear functional analysis. Besides, its frequent usage of notions from differential calculus makes it a natural sequel to the previous chapter.

CHAPTER 9: THE “GREAT THEOREMS” OF NONLINEAR FUNCTIONAL ANALYSIS

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Other Titles in Applied Mathematics, Page 657-775.
Introduction The title of this chapter is slightly misleading, for two reasons. First, such basic results as the Banach fixed point theorem, Sard’s lemma, the Newton-Kantorovich theorem, or the implicit function theorem also count among the “great theorems” of nonlinear functional analysis; yet they do not appear here (since they were treated in Chapters 3 and 7, respectively).

Front Matter

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Other Titles in Applied Mathematics, Page i-xiv.
The front matter includes the Table of Contents and Preface

Metabolic Networks, Elementary Flux Modes, and Polyhedral Cones

Isaac Klapper

Metabolic Networks, Elementary Flux Modes, and Polyhedral Cones.
In recent years, the world of molecular biology has witnessed a technology explosion, the so-called omics revolution, which has opened windows into cellular metabolic potential and activity that have already revolutionized numerous subfields of biology and medicine. At the same time, this technology (genomics, transcriptomics, etc.), intrinsically through its very nature and extrinsically through its widespread use, has generated a lot of data that begs, and has received, much statistical and mathematical attention. Such data analysis is arguably as important, if not more so, than the measurements themselves; without the appropriate parsing, much of the information potential is inaccessible.

Appendix A: A Brief Review of Relevant Linear Algebra

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Other Titles in Applied Mathematics, Page 105-111.
The following is intended as a review of material that appears in the text; for an in-depth exposition, see any linear algebra textbook, such as [16, 26].

Appendix B: Glossary of Metabolic Network Terms

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Other Titles in Applied Mathematics, Page 113-114.
Active (or inactive) reactions: referring to a flux vector v, reaction i is active (inactive) if component νi of v is nonzero (zero).

Appendix C: Notation

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Other Titles in Applied Mathematics, Page 115-115.
General: v = a vector (boldfaced), νj = the jth component of vector v (not in boldface), v1, v2, …, vk = a list of k vectors, 0 = a vector with all components equal to 0, ej = a vector with all components equal to 0 except the jth component, which is equal to 1, vT = the transpose of vector v, ‖v‖ = the standard Euclidean norm of v, A = a matrix (not in boldface), the matrix 0 is the matrix consisting of all zero entries, AT = the matrix transpose of A.

Back Matter

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Other Titles in Applied Mathematics, Page 117-122.
The back matter includes bibliography, index, and back cover.

Chapter 1: Introduction

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Other Titles in Applied Mathematics, Page 1-3.
Metabolism, the chemical reactions occurring inside cells that extract energy from nutrients (catabolism) and that build cell materials (anabolism), is central to all living cells. Characterizing its workings is a fundamental goal of the science of cell biology. Biologists conveniently organize these reactions into metabolic processes (or pathways) consisting of sequences of enzyme catalyzed reactions of chemicals (or metabolites) which operate somewhat in the manner of an assembly line, or in other cases in the manner of a disassembly line. These processing lines interconnect in complicated ways, as different pathways can share various reactions. The entire complex forms what is sometimes called a metabolic network (or map); see, for example, Figure 1.1, whose “cities” are metabolites and whose “roads” are reactions.

Chapter 2: Metabolic Networks and Stoichiometric Matrices

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Other Titles in Applied Mathematics, Page 5-14.
2.1 ▪ Basics The map in Figure 1.1 presents one way of abstracting a metabolic network, convenient for viewing but less so for computing. Alternatively, we can encode the same information into a matrix: a metabolic network S of m metabolites and n reactions is representable by an m × n stoichiometric matrix N, whose rows correspond to the m metabolites and whose columns to the n reactions. The matrix N is a mathematical construct which conveniently and concisely codifies all of the reaction balance information in the network in which element Nij is the signed stoichiometric coefficient of metabolite i in reaction j with the convention: negative for reactants, positive for products.