Pseudo Riemannian Geometry Delta Invariants And Applications

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Pseudo-riemannian Geometry, Delta-invariants And Applications

Author : Bang-yen Chen
Publisher : World Scientific
ISBN 13 : 9814462489
Total Pages : 510 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Pseudo-riemannian Geometry, Delta-invariants And Applications by : Bang-yen Chen

Download or read book Pseudo-riemannian Geometry, Delta-invariants And Applications written by Bang-yen Chen and published by World Scientific. This book was released on 2011-03-23 with total page 510 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included.The second part of this book is on δ-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as δ-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between δ-invariants and the main extrinsic invariants. Since then many new results concerning these δ-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades.

Pseudo-Riemannian Geometry, [delta]-invariants and Applications

Author : Bang-yen Chen
Publisher : World Scientific
ISBN 13 : 9814329630
Total Pages : 510 pages
Book Rating : 4.8/5 (143 download)

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Book Synopsis Pseudo-Riemannian Geometry, [delta]-invariants and Applications by : Bang-yen Chen

Download or read book Pseudo-Riemannian Geometry, [delta]-invariants and Applications written by Bang-yen Chen and published by World Scientific. This book was released on 2011 with total page 510 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included.The second part of this book is on ë-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as ë-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between ë-invariants and the main extrinsic invariants. Since then many new results concerning these ë-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades.

Geometry of CR-Submanifolds

Author : Aurel Bejancu
Publisher : Springer Science & Business Media
ISBN 13 : 9789027721945
Total Pages : 202 pages
Book Rating : 4.7/5 (219 download)

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Book Synopsis Geometry of CR-Submanifolds by : Aurel Bejancu

Download or read book Geometry of CR-Submanifolds written by Aurel Bejancu and published by Springer Science & Business Media. This book was released on 1986-07-31 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can us;; Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

The Laplacian on a Riemannian Manifold

Author : Steven Rosenberg
Publisher : Cambridge University Press
ISBN 13 : 9780521468312
Total Pages : 190 pages
Book Rating : 4.4/5 (683 download)

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Book Synopsis The Laplacian on a Riemannian Manifold by : Steven Rosenberg

Download or read book The Laplacian on a Riemannian Manifold written by Steven Rosenberg and published by Cambridge University Press. This book was released on 1997-01-09 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Differential Geometry Of Warped Product Manifolds And Submanifolds

Author : Bang-yen Chen
Publisher : World Scientific
ISBN 13 : 9813208945
Total Pages : 517 pages
Book Rating : 4.8/5 (132 download)

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Book Synopsis Differential Geometry Of Warped Product Manifolds And Submanifolds by : Bang-yen Chen

Download or read book Differential Geometry Of Warped Product Manifolds And Submanifolds written by Bang-yen Chen and published by World Scientific. This book was released on 2017-05-29 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson-Walker models, are warped product manifolds.The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson-Walker's and Schwarzschild's.The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century.The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.

Differential Geometry

Author : Loring W. Tu
Publisher : Springer
ISBN 13 : 3319550845
Total Pages : 358 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Differential Geometry by : Loring W. Tu

Download or read book Differential Geometry written by Loring W. Tu and published by Springer. This book was released on 2017-06-01 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.

Foliations and the Geometry of 3-Manifolds

Author : Danny Calegari
Publisher : Oxford University Press on Demand
ISBN 13 : 0198570082
Total Pages : 378 pages
Book Rating : 4.1/5 (985 download)

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Book Synopsis Foliations and the Geometry of 3-Manifolds by : Danny Calegari

Download or read book Foliations and the Geometry of 3-Manifolds written by Danny Calegari and published by Oxford University Press on Demand. This book was released on 2007-05-17 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique reference, aimed at research topologists, gives an exposition of the 'pseudo-Anosov' theory of foliations of 3-manifolds. This theory generalizes Thurston's theory of surface automorphisms and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Significant themes returned to throughout the text include the importance of geometry, especially the hyperbolic geometry of surfaces, the importance of monotonicity, especially in1-dimensional and co-dimensional dynamics, and combinatorial approximation, using finite combinatorical objects such as train-tracks, branched surfaces and hierarchies to carry more complicated continuous objects.

Semi-Riemannian Geometry With Applications to Relativity

Author : Barrett O'Neill
Publisher : Academic Press
ISBN 13 : 0080570577
Total Pages : 483 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Semi-Riemannian Geometry With Applications to Relativity by : Barrett O'Neill

Download or read book Semi-Riemannian Geometry With Applications to Relativity written by Barrett O'Neill and published by Academic Press. This book was released on 1983-07-29 with total page 483 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest.

An Invitation to Alexandrov Geometry

Author : Stephanie Alexander
Publisher : Springer
ISBN 13 : 3030053121
Total Pages : 95 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis An Invitation to Alexandrov Geometry by : Stephanie Alexander

Download or read book An Invitation to Alexandrov Geometry written by Stephanie Alexander and published by Springer. This book was released on 2019-05-08 with total page 95 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.

Differential Geometry in Physics

Author : Gabriel Lugo
Publisher :
ISBN 13 : 9781469669250
Total Pages : 372 pages
Book Rating : 4.6/5 (692 download)

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Book Synopsis Differential Geometry in Physics by : Gabriel Lugo

Download or read book Differential Geometry in Physics written by Gabriel Lugo and published by . This book was released on 2021-10-15 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Geometry in Physics is a treatment of the mathematical foundations of the theory of general relativity and gauge theory of quantum fields. The material is intended to help bridge the gap that often exists between theoretical physics and applied mathematics. The approach is to carve an optimal path to learning this challenging field by appealing to the much more accessible theory of curves and surfaces. The transition from classical differential geometry as developed by Gauss, Riemann and other giants, to the modern approach, is facilitated by a very intuitive approach that sacrifices some mathematical rigor for the sake of understanding the physics. The book features numerous examples of beautiful curves and surfaces often reflected in nature, plus more advanced computations of trajectory of particles in black holes. Also embedded in the later chapters is a detailed description of the famous Dirac monopole and instantons. Features of this book: * Chapters 1-4 and chapter 5 comprise the content of a one-semester course taught by the author for many years. * The material in the other chapters has served as the foundation for many master's thesis at University of North Carolina Wilmington for students seeking doctoral degrees. * An open access ebook edition is available at Open UNC (https: //openunc.org) * The book contains over 80 illustrations, including a large array of surfaces related to the theory of soliton waves that does not commonly appear in standard mathematical texts on differential geometry.

Noncommutative Geometry, Quantum Fields and Motives

Author : Alain Connes
Publisher : American Mathematical Soc.
ISBN 13 : 1470450453
Total Pages : 810 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Noncommutative Geometry, Quantum Fields and Motives by : Alain Connes

Download or read book Noncommutative Geometry, Quantum Fields and Motives written by Alain Connes and published by American Mathematical Soc.. This book was released on 2019-03-13 with total page 810 pages. Available in PDF, EPUB and Kindle. Book excerpt: The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.

Geometry of Submanifolds

Author : Bang-Yen Chen
Publisher : Courier Dover Publications
ISBN 13 : 0486832783
Total Pages : 193 pages
Book Rating : 4.4/5 (868 download)

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Book Synopsis Geometry of Submanifolds by : Bang-Yen Chen

Download or read book Geometry of Submanifolds written by Bang-Yen Chen and published by Courier Dover Publications. This book was released on 2019-06-12 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first two chapters of this frequently cited reference provide background material in Riemannian geometry and the theory of submanifolds. Subsequent chapters explore minimal submanifolds, submanifolds with parallel mean curvature vector, conformally flat manifolds, and umbilical manifolds. The final chapter discusses geometric inequalities of submanifolds, results in Morse theory and their applications, and total mean curvature of a submanifold. Suitable for graduate students and mathematicians in the area of classical and modern differential geometries, the treatment is largely self-contained. Problems sets conclude each chapter, and an extensive bibliography provides background for students wishing to conduct further research in this area. This new edition includes the author's corrections.

Eigenfunctions of the Laplacian on a Riemannian Manifold

Author : Steve Zelditch
Publisher : American Mathematical Soc.
ISBN 13 : 1470410370
Total Pages : 410 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Eigenfunctions of the Laplacian on a Riemannian Manifold by : Steve Zelditch

Download or read book Eigenfunctions of the Laplacian on a Riemannian Manifold written by Steve Zelditch and published by American Mathematical Soc.. This book was released on 2017-12-12 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.

Introduction to Smooth Manifolds

Author : John M. Lee
Publisher : Springer Science & Business Media
ISBN 13 : 0387217525
Total Pages : 646 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Introduction to Smooth Manifolds by : John M. Lee

Download or read book Introduction to Smooth Manifolds written by John M. Lee and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 646 pages. Available in PDF, EPUB and Kindle. Book excerpt: Author has written several excellent Springer books.; This book is a sequel to Introduction to Topological Manifolds; Careful and illuminating explanations, excellent diagrams and exemplary motivation; Includes short preliminary sections before each section explaining what is ahead and why

Strings and Geometry

Author : Clay Mathematics Institute. Summer School
Publisher : American Mathematical Soc.
ISBN 13 : 9780821837153
Total Pages : 396 pages
Book Rating : 4.8/5 (371 download)

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Book Synopsis Strings and Geometry by : Clay Mathematics Institute. Summer School

Download or read book Strings and Geometry written by Clay Mathematics Institute. Summer School and published by American Mathematical Soc.. This book was released on 2004 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains selection of expository and research article by lecturers at the school. Highlights current interests of researchers working at the interface between string theory and algebraic supergravity, supersymmetry, D-branes, the McKay correspondence andFourer-Mukai transform.

Grid Homology for Knots and Links

Author : Peter S. Ozsváth
Publisher : American Mathematical Soc.
ISBN 13 : 1470417375
Total Pages : 423 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Grid Homology for Knots and Links by : Peter S. Ozsváth

Download or read book Grid Homology for Knots and Links written by Peter S. Ozsváth and published by American Mathematical Soc.. This book was released on 2015-12-04 with total page 423 pages. Available in PDF, EPUB and Kindle. Book excerpt: Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.

Metric Spaces of Non-Positive Curvature

Author : Martin R. Bridson
Publisher : Springer Science & Business Media
ISBN 13 : 3662124947
Total Pages : 665 pages
Book Rating : 4.6/5 (621 download)

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Book Synopsis Metric Spaces of Non-Positive Curvature by : Martin R. Bridson

Download or read book Metric Spaces of Non-Positive Curvature written by Martin R. Bridson and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 665 pages. Available in PDF, EPUB and Kindle. Book excerpt: A description of the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries. The theory of these objects is developed in a manner accessible to anyone familiar with the rudiments of topology and group theory: non-trivial theorems are proved by concatenating elementary geometric arguments, and many examples are given. Part I provides an introduction to the geometry of geodesic spaces, while Part II develops the basic theory of spaces with upper curvature bounds. More specialized topics, such as complexes of groups, are covered in Part III.