In 1836-1837 Sturm and Liouville published a series of papers on second order linear ordinary differential operators, which started the subject now known as the Sturm-Liouville problem. In 1910 Hermann Weyl published an article which started the study of singular Sturm-Liouville problems. Since then, the Sturm-Liouville theory remains an intensely active field of research, with many applications in mathematics and mathematical physics. The purpose of the present book is (a) to provide a modern survey of some of the basic properties of Sturm-Liouville theory and (b) to bring the reader to the forefront of knowledge about some aspects of this theory. To use the book, only a basic knowledge of advanced calculus and a rudimentary knowledge of Lebesgue integration and operator theory are assumed. An extensive list of references and examples is provided and numerous open problems are given. The list of examples includes those classical equations and functions associated with the names of Bessel, Fourier, Heun, Ince, Jacobi, Jorgens, Latzko, Legendre, Littlewood-McLeod, Mathieu, Meissner, Morse, as well as examples associated with the harmonic oscillator and the hydrogen atom. Many special functions of applied mathematics and mathematical physics occur in these examples.
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.
In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular Sturm-Liouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded self-adjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained. In this monograph the authors summarize some of these directions discussing self-adjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasi-differential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, self-adjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and self-adjoint characterizations are extended to two-interval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson.
Norm inequalities relating (i) a function and two of its derivatives and (ii) a sequence and two of its differences are studied. Detailed elementary proofs of basic inequalities are given. These are accessible to anyone with a background of advanced calculus and a rudimentary knowledge of the Lp and lp spaces. The classical inequalities associated with the names of Landau, Hadamard, Hardy and Littlewood, Kolmogorov, Schoenberg and Caravetta, etc., are discussed, as well as their discrete analogues and weighted versions. Best constants and the existence and nature of extremals are studied and many open questions raised. An extensive list of references is provided, including some of the vast Soviet literature on this subject.
Norm inequalities relating (i) a function and two of its derivatives and (ii) a sequence and two of its differences are studied. Detailed elementary proofs of basic inequalities are given. These are accessible to anyone with a background of advanced calculus and a rudimentary knowledge of the Lp and lp spaces. The classical inequalities associated with the names of Landau, Hadamard, Hardy and Littlewood, Kolmogorov, Schoenberg and Caravetta, etc., are discussed, as well as their discrete analogues and weighted versions. Best constants and the existence and nature of extremals are studied and many open questions raised. An extensive list of references is provided, including some of the vast Soviet literature on this subject.
In 1836-1837 Sturm and Liouville published a series of papers on second order linear ordinary differential operators, which started the subject now known as the Sturm-Liouville problem. In 1910 Hermann Weyl published an article which started the study of singular Sturm-Liouville problems. Since then, the Sturm-Liouville theory remains an intensely active field of research, with many applications in mathematics and mathematical physics. The purpose of the present book is (a) to provide a modern survey of some of the basic properties of Sturm-Liouville theory and (b) to bring the reader to the forefront of knowledge about some aspects of this theory. To use the book, only a basic knowledge of advanced calculus and a rudimentary knowledge of Lebesgue integration and operator theory are assumed. An extensive list of references and examples is provided and numerous open problems are given. The list of examples includes those classical equations and functions associated with the names of Bessel, Fourier, Heun, Ince, Jacobi, Jorgens, Latzko, Legendre, Littlewood-McLeod, Mathieu, Meissner, Morse, as well as examples associated with the harmonic oscillator and the hydrogen atom. Many special functions of applied mathematics and mathematical physics occur in these examples.
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.
In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular Sturm-Liouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded self-adjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained. In this monograph the authors summarize some of these directions discussing self-adjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasi-differential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, self-adjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and self-adjoint characterizations are extended to two-interval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson.
In recent years there has been growing interest in the nucleon-nucleon correl ations inside nuclei. In many respects the motions of the nucleons can be very well described by an overall mean field, so that the motion of each nucleon is governed by the mean field due to all the other nucleons. This concept underlies the Fermi-gas, Hartree-Fock and shell models and has enabled a range of nuclear properties to be calculated, often to surprising accuracy. It gradually became clear, however, that these mean-field models are limited by the effects due to the very strong interactions between the nucleons that occur at short distances; these are the short-range correlations. They are responsible for instance for the high-momentum components in the nucleon momentum dis tribution, and prevent the simultaneous description of the nuclear density and momentum distributions by the same mean field. It thus becomes necessary to develop methods for including the effects of nucleon correlations in nuclei, and these are the main subject of this book. Some related problems of nuclear structure were discussed in an earlier book by the same authors: Nucleon Momentum and Density Distributions in Nuclei (Clarendon Press, Oxford, 1988). The main aim of that book was to study the effects of nucleon-nucleon correlations, both short-range and tensor, on the nucleon momentum distribution, which is particularly sensitive to these correl ations, and on the nucleon density distribution.
Readers will learn in which ways light can be "confined" within a subwavelength region smaller than half a wavelength. Strictly within the focal spot, all degrees of freedom of light interact and manifest themselves in a dramatic way. The size and shape of the focal spot and the magnitude of side-lobes depend on the polarization state alongside phase and amplitude distributions of a light beam. Readers will learn techniques in which inhomogeneously (i.e., azimuthally and radially) polarized optical beams can be focused. In sharp focus, exotic phenomena can occur, including the negative propagation of light and a toroidal optical flow. Throughout the book, the numerical simulation is performed using the rigorous solution of Maxwell’s equations based on a Finite-Difference Time-Domain (FDTD) approach, which makes the results of modeling highly reliable. The photonic components, including optical metasurfaces, discussed in the book have been implemented using state-of-the-art techniques of electron beam writing and reactive ion-beam etching of microrelief. Two chapters are concerned with photonics hot spots, which deal with the control of light by means of optical metasurfaces and the generation of an energy backflow in the region of sharp focus of a laser beam. Another hot topic is diffractive polarization converters implemented as subwavelength diffraction gratings to convert polarization of light. By way of illustration, such converters are shown to perform linear-to-radial or linear-to-azimuthal polarization conversion. The book describes advanced photonic components fabricated by the authors to perform sharp focusing of light, including binary zone plates, binary axicons, a planar photonic crystal lens, diffraction polarization converters, and metalenses. This book is a must-have for individuals and institutions studying cutting edge optics.
Beyond the agricultural and industrial revolutions of the past, a globaltechnology revolution is currently changing the world. This book discussesthe broad, multidisciplinary, and synergistic trends in this revolution,including genomics, cloning, biomedical engineering, smart materials, agilemanufacturing, nanofabricated computation devices, and integratedmicrosystems. The revolution's effects on human health may be the most startling as breakthroughs improve both the quality and length of human life.Biotechnology will also enable us to identify, understand, manipulate,improve, and control living organisms (including ourselves). Informationtechnology is already revolutionizing our lives, especially in the developedworld, and is a major enabler of other trends. Materials technology willproduce products, components, and systems that are smaller, smarter,multi-functional, environmentally compatible, more survivable, andcustomizable. In addition, smart materials, agile manufacturing, andnanotechnology will change the way we produce devices and improve theircapabilities. The technology revolution will not be uniform in its effectacross the globe but will play out differently depending on its acceptance,investment, and a variety of issues such as bioethics, privacy, economicdisparity, cultural invasion, and social reactions. There will be no turningback, however, since some societies will avail themselves of the revolution,and globalization will thus change the environment in which each societylives.
This book is devoted to researchers and teachers, as well as graduate students, undergraduates and bachelors in engineering mechanics, nano-mechanics, nanomaterials, nanostructures and applied mathematics. It presents a collection of the latest developments in the field of nonlinear (chaotic) dynamics of mass distributed-parameter nanomechanical structures, providing a rigorous and comprehensive study of modeling nonlinear phenomena. It is written in a unique pedagogical style particularly suitable for independent study and self-education. In addition, the book achieves a good balance between Western and Eastern extensive studies of the mathematical problems of nonlinear vibrations of structural members.
Encyclopedic, definitive, and state-of-the-art in the field of vascular disease and its medical, surgical, and interventional management, Rutherford's Vascular Surgery and Endovascular Therapy offers authoritative guidance from the most respected and innovative global thought leaders and clinical and basic science experts of our time. The thoroughly revised 10th Edition, published in association with the Society for Vascular Surgery and authored by multidisciplinary and international contributors, is an outstanding reference for vascular surgeons, vascular medicine specialists, interventional radiologists and cardiologists, and their trainees who depend upon Rutherford’s in their practice. Under the expert editorial guidance of Drs. Anton N. Sidawy and Bruce A. Perler, it is quite simply the most complete and most reliable resource available on the art and science of circulatory diseases. Incorporates fundamental vascular biology, diagnostic techniques, and decision making as well as medical, endovascular, and surgical treatment of vascular disease. Features numerous concise and comprehensive diagnostic and therapeutic algorithms vital to patient evaluation and management. Covers all vascular imaging techniques, offering a non-invasive evaluation of both the morphology and hemodynamics of the vascular system. Employs a full-color layout, images and online videos, so readers can view clinical and physical findings and operative techniques more vividly. Contains fully updated and more concise chapters with a focused format and summary for each that provides a quick access to key information—ideal for consultation as well as daily practice. Includes expanded coverage of the business of vascular surgery, including a new section on the use of technology platforms and social media, and new chapters on telemedicine, the development and operation of outpatient dialysis centers and multispecialty cardiovascular centers, vascular information on the internet, and much more. Provides new content on key topics such as endovascular treatment of complex aortic disease, acute vascular occlusion in the pediatric population, outpatient vascular care, and anatomic surgical exposures for open surgical reconstructions.
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