Essential results of functional analysis

7R4. Systems of Conservation Laws 2: Geometric Structures, Oscillations and Mixed Problems. - D Serre (Ecole Normale Superieure, Lyon, France). Cambridge UP, Cambridge, UK. 2000. 269 pp. ISBN 0-521-63330-3. $74.95. Reviewed by Y Horie (Los Alamos Natl Lab, Group X-7, MS D413, Los Alamos NM 87545).Conservation laws arise in many areas of continuum physics or mechanics and are described by certain types of nonlinear partial differential equations. Solving these PDEs for non-dissipative systems with appropriate initial and boundary values is known as the Cauchy problem or the mixed problem. This two-volume set of books is concerned with the modern mathematical treatment of the mixed problem. According to the author, the coverage of the book is comparable to that of Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Springer, Berlin, 1984), but the subject matters are dealt with “with more detail, but less animation.” Though important, the phenomenon of relaxation and kinetic formulation are not included. Volume 1 dealt with the fundamentals, including such topics as hyperbolicity, Lax shock, the Glimm scheme, and viscosity solution. The second volume, consisting of Chapters 8 through 15, is concerned with questions such as: the convergence of the viscosity method, stability for the Cauchy problem, admissible boundary conditions, and the multi-dimensional mixed problem. But the author cautions that “the materials in the second volume represent a significant part of the active research in the last fifteen years in the field, and are intended more for research workers and less for students.” Whether dealing with fundamentals or the state-of-the art research, this book is definitely written for post-graduate students and active researchers in mathematics. All the basic ideas are discussed fully, but for nonspecialists guiding from well-versed instructors may be helpful to wade through the materials. The readers are expected to have a good mathematical background in contemporary functional analysis. Keywords such as hyperbolic systems, shock waves, viscosity, and initial and boundary value problems are familiar to physical scientists and engineers, but the questions treated in the book are primarily mathematical: existence, uniqueness, convergence, stability, and consistency, etc. The book, however, has something illuminating to offer even to those who may not have the appetite or requisite mathematics to follow theorems and lemmas in depth. It shows what it means to think with the equations from the mathematical point of view. Half-baked knowledge is always dangerous, but this reviewer, who is not a mathematician, was still able to gain new appreciation of and fresh insight into the kinds of questions he does not normally worry about. Good examples are the treatment of stability types and admissible boundary conditions on a finite domain in mathematical terms. His formal training ends at the level of classic books by such people as Courant and Friedrichs, Whitham, and Courant and Hilbert, but he could not help wonder at the mathematical transformation of the questions one used to think intuitively about on a physical basis. At the same time, as marked by Courant (Methods of Mathematical Physics, Volume 1), one fears an increasing gap between contemporary mathematics and the interest of many scientists and engineers. Systems of Conservation Laws 2 is primarily for researchers in mathematics who are interested in the current state of knowledge and students in advanced graduate courses in PDEs. The book is also recommended to non-mathematicians as a means of getting a broad brush view of the conservation equations, nonlinear wave propagation, and shock waves from the mathematical point of view. The book is definitely recommended for university libraries.

By a metric fixed point theorem we mean an existence result for a fixed point of a mapping f under conditions which depend on a metric d, and which are not invariant when we replace d by an equivalent metric. The best known metric fixed point theorem is the Banach theorem, also called the contractive mapping principle: “every contraction from a complete metric space into itself has a (unique) fixed point”. It is clear that a contractive mapping can lose this property if d is replaced by an equivalent metric. The Banach theorem is a basic tool in functional analysis, nonlinear analysis and differential equations. If we relax the contractive condition, requiring only that the mapping be nonexpansive, that is, d(f (x), f (y)) ≤ d(x, y), then trivial examples show that the Banach theorem need no longer hold. This failure may have been the reason why no significant result about the existence of fixed points for nonexpansive mappings was obtained for many years. However, in 1965, Browder [Br1 and Br2], Göhde [Go] and Kirk [Ki1] proved the following results: “let X be a Banach space, C a closed, convex and bounded subset of X and T : C → C a nonexpansive mapping. If X is either a Hilbert space, or a uniformly convex Banach space or a reflexive Banach space with normal structure, then T has a fixed point”. This result is, in some sense, surprising because it uses convexity hypotheses (more usual in topological fixed point theory) and geometric properties of the Banach spaces (commonly used in linear functional analysis, but rarely considered in nonlinear analysis prior to this time). The above results were the starting point for a new mathematical field: the application of the geometric theory of Banach spaces to fixed point theory. The texts [GK1], [AK], [KZ] and [Z] constitute excellent surveys of this theory.KeywordsBanach SpaceFixed Point TheoremNonexpansive MappingFixed Point TheoryConvex Banach SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This work is concerned with a kind of optimal control problem for a size-structured biological population model. Well-posedness of the state system and an adjoint system are proved by means of Banach’s fixed point theorem. Existence and uniqueness of optimal control are shown by functional analytical approach. Optimality conditions describing the optimal strategy are established via tangent and normal cones technique. The results are of the first ones for this novel structure.

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics.

It is shown that there is a nonseparable Banach space on which every operator has the form S + pi where p is a scalar and S is an operator with separable range.An open problem in the theory of Banach spaces is the construction of a Banach space with the property that all operators from it to itself are the sum of a multiple of the identity and a compact operator.Since compact operators have separable range a related question is to find a (nonseparable) Banach space on which all operators are the sum of a multiple of the identity and an operator with separable range.This was first done by Shelah using an extra set-theoretic axiom in [1].It is the purpose of this paper, which is the continuation of research which began in [3], to prove the same result without appealing to axioms other than those of ZFC.The following notation will be used.The integers will be considered as ordinals so that n Gm means the same as n < m.The set of integers is denoted as w instead of N and the first uncountable ordinal is wi.The symbols [X]x and [X]<A denote, respectively, the set of A sized subsets of X and less than A sized subsets of X.The set of functions from Y to X is denoted YX.In particular, WlR, when endowed with the usual pointwise operations, is the real vector space of "wi-tuples".For X ui the characteristic function of X will be denoted by lx-Let us begin by reviewing the construction from [2] of a Banach space on which every operator is the sum of an operator with separable range and one which is diagonal with respect to a fixed basis.Let 21 [u>i]<N be such that:i. ua = wi.2. 21 is closed under subsets.It is possible to define a function || ||a on ">R by ||/||*=sup{{|/(a)|;aGA};AG2l}.Let B = {UlR; ||/||a is finite} and note that (B, || ||a) is a normed linear space.Of course if 21 = [wi]<"' then B = /i(wi) but we will assume that there is a family 53 [wi]<w such that: 3. 93 is closed under subsets.4. an8c [uji]1.5.For all families of disjoint pairs {{a(a),b(a)}; a G u>i} and integers k there is X G [ui]k such that {a(a); a G X} G 21 and {6(a); a G X} G 03.

It is shown that every compact group G is a -simplicial space where Q is any field of characteristic zero.As a consequence it follows that G satisfies a variation of the Lefschetz fixed point theorem.It has been known for some time that the Lefschetz fixed point theorem applies to a few spaces other than just ANR spaces, especially if some care is taken to use coefficients in certain fields [2].The case of all compact groups provides a broad class of spaces which may not have local connectivity of any order.It is shown that every compact group G satisfies the Lefschetz fixed point theorem when coefficients for the homology groups are taken in a field of characteristic zero.1. Theorem.Let G be a compact group.Then G is an inverse limit of Lie groups G = proj lim GM and if f : G" -> G is a homomorphism belonging to this inverse system, then there is a commutative diagram of covering compact Lie groups Presented to the Society, January 18,

Localized quantum wave packets can be produced in a variety of physical systems and are the subject of much current research in atomic, molecular, chemical, and condensed-matter physics. They are particularly well suited for studying the classical limit of a quantum-mechanical system. The motion of a localized quantum wave packet initially follows the corresponding classical motion. However, in most cases the quantum wave packet spreads and undergoes a series of collapses and revivals. We present a generic treatment of wave-packet evolution, and we provide conditions under which various types of revivals occur in ideal form. The discussion is at a level appropriate for an advanced undergraduate or first-year graduate course in quantum mechanics. Explicit examples of different types of revival structure are provided, and physical applications are discussed.

Suitable for both senior-level and first-year graduate courses, this fully revised edition provides a unique and systematic treatment of engineering dynamics that covers Newton–Euler and Lagrangian approaches. New to this edition are: two completely revised chapters on the constraints on, and potential energies for, rigid bodies, and the dynamics of systems of particles and rigid bodies; clearer discussion on coordinate singularities and their relation to mass matrices and configuration manifolds; additional discussion of contravariant basis vectors and dual Euler basis vectors, as well as related works in robotics; improved coverage of navigation equations; inclusion of a 350-page solutions manual for instructors, available online; a fully updated reference list. Numerous structured examples, discussion of various applications, and exercises covering a wide range of topics are included throughout, and source code for exercises, and simulations of systems are available online.

Thoroughly revised for its second edition, this advanced textbook provides an introduction to the basic methods of computational physics, and an overview of progress in several areas of scientific computing by relying on free software available from CERN. The book begins by dealing with basic computational tools and routines, covering approximating functions, differential equations, spectral analysis, and matrix operations. Important concepts are illustrated by relevant examples at each stage. The author also discusses more advanced topics, such as molecular dynamics, modeling continuous systems, Monte Carlo methods, genetic algorithm and programming, and numerical renormalization. It includes many more exercises. This can be used as a textbook for either undergraduate or first-year graduate courses on computational physics or scientific computation. It will also be a useful reference for anyone involved in computational research.

10 The Calculus of Variations, Functional Analysis, and Optimal Control Problems

This textbook is an introduction to the techniques of summing and nuclear norms. The author's aim is to present a clear and simple account of these ideas and to demonstrate the power of their application to a variety of Banach space questions. The style is expository and the only prerequisite is a beginner's course on Wormed linear spaces and a minimal knowledge of functional analysis. Thus, Dr Jameson is able to concentrate on important, central results and gives concrete and largely non-technical proofs, often supplying alternative proofs which both contribute something to the understanding. Final-year undergraduates and postgraduates in functional analysis will enjoy this introduction to the subject, and there are many examples and exercises throughout the text to help the reader and to demonstrate the range of application these techniques find. A list of references indicates the way for further reading.

Elliptic operators arise naturally in several different mathematical settings, notably in the representation theory of Lie groups, the study of evolution equations, and the examination of Riemannian manifolds. This book develops the basic theory of elliptic operators on Lie groups and thereby extends the conventional theory of parabolic evolution equations to a natural non-commutative context. In order to achieve this goal, the author presents a synthesis of ideas from partial differential equations, harmonic analysis, functional analysis, and the theory of Lie groups. He begins by discussing the abstract theory of general operators with complex coefficients before concentrating on the central case of second-order operators with real coefficients. A full discussion of second-order subellilptic operators is also given. Prerequisites are a familiarity with basic semigroup theory, the elementary theory of Lie groups, and a firm grounding in functional analysis as might be gained from the first year of a graduate course.

Basic results.- The representation algebra.- Positive-definite and completely positive functions.- Discrete separative semigroups.- Subsemigroups of locally compact abelian groups and weakly almost periodic functions.- Representations in Q-algebras.- Special cases of Q-representations.- Hilbert space representations.

Hilden's simple proof of Lomonosov's invariant subspace theorem

A fixed point theorem and its application in ergodic theory