Now published by Sage The new Twelfth Edition of Social Psychology by Saul Kassin, Steven Fein, and Hazel Rose Markus captures the excitement of this dynamic and responsive field in our ever-changing world. The authors highlight the most exciting and important foundational and contemporary research, while every chapter also uniquely investigates the influences of culture and social class. In this enthusiastic introduction to social psychology, students delve into their own passion drivers, from favorite sports teams to social media to their own political perspectives, dispelling misconceptions and understanding the scientific foundations that explain our daily interactions and social behaviors. This textbook shows students how social psychology— its theories, research methods, and basic findings—has never been more relevant or more important.
Stunning recent results by Host–Kra, Green–Tao, and others, highlight the timeliness of this systematic introduction to classical ergodic theory using the tools of operator theory. Assuming no prior exposure to ergodic theory, this book provides a modern foundation for introductory courses on ergodic theory, especially for students or researchers with an interest in functional analysis. While basic analytic notions and results are reviewed in several appendices, more advanced operator theoretic topics are developed in detail, even beyond their immediate connection with ergodic theory. As a consequence, the book is also suitable for advanced or special-topic courses on functional analysis with applications to ergodic theory. Topics include: • an intuitive introduction to ergodic theory • an introduction to the basic notions, constructions, and standard examples of topological dynamical systems • Koopman operators, Banach lattices, lattice and algebra homomorphisms, and the Gelfand–Naimark theorem • measure-preserving dynamical systems • von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem • strongly and weakly mixing systems • an examination of notions of isomorphism for measure-preserving systems • Markov operators, and the related concept of a factor of a measure preserving system • compact groups and semigroups, and a powerful tool in their study, the Jacobs–de Leeuw–Glicksberg decomposition • an introduction to the spectral theory of dynamical systems, the theorems of Furstenberg and Weiss on multiple recurrence, and applications of dynamical systems to combinatorics (theorems of van der Waerden, Gallai,and Hindman, Furstenberg’s Correspondence Principle, theorems of Roth and Furstenberg–Sárközy) Beyond its use in the classroom, Operator Theoretic Aspects of Ergodic Theory can serve as a valuable foundation for doing research at the intersection of ergodic theory and operator theory
This book introduces functional analysis at an elementary level without assuming any background in real analysis, for example on metric spaces or Lebesgue integration. It focuses on concepts and methods relevant in applied contexts such as variational methods on Hilbert spaces, Neumann series, eigenvalue expansions for compact self-adjoint operators, weak differentiation and Sobolev spaces on intervals, and model applications to differential and integral equations. Beyond that, the final chapters on the uniform boundedness theorem, the open mapping theorem and the Hahn-Banach theorem provide a stepping-stone to more advanced texts. The exposition is clear and rigorous, featuring full and detailed proofs. Many examples illustrate the new notions and results. Each chapter concludes with a large collection of exercises, some of which are referred to in the margin of the text, tailor-made in order to guide the student digesting the new material. Optional sections and chapters supplement the mandatory parts and allow for modular teaching spanning from basic to honors track level.
This investigation introduces a new description and classification for the set of all self-adjoint operators (not just those defined by differential boundary conditions) which are generated by a linear elliptic partial differential expression $A(\mathbf{x}, D)=\sum_{0\, \leq\, \left s\right \, \leq\,2m}a_{s} (\mathbf{x})D DEGREES{s}\;\text{for all}\;\mathbf{x}\in\Omega$ in a region $\Omega$, with compact closure $\overline{\Omega}$ and $C DEGREES{\infty }$-smooth boundary $\partial\Omega$, in Euclidean space $\mathbb{E} DEGREES{r}$ $(r\geq2).$ The order $2m\geq2$ and the spatial dimensio
Semiconductor quantum optics is on the verge of moving from the lab to real world applications. When stepping from basic research to new technologies, device engineers will need new simulation tools for the design and optimization of quantum light sources, which combine classical device physics with cavity quantum electrodynamics. This thesis aims to provide a holistic description of single-photon emitting diodes by bridging the gap between microscopic and macroscopic modeling approaches. The central result is a novel hybrid quantum-classical model system that self-consistently couples semi-classical carrier transport theory with open quantum many-body systems. This allows for a comprehensive description of quantum light emitting diodes on multiple scales: It enables the calculation of the quantum optical figures of merit together with the simulation of the spatially resolved current flow in complex, multi-dimensional semiconductor device geometries out of one box. The hybrid system is shown to be consistent with fundamental laws of (non-)equilibrium thermodynamics and is demonstrated by numerical simulations of realistic devices.
Locally compact groups play an important role in many areas of mathematics as well as in physics. The class of locally compact groups admits a strong structure theory, which allows to reduce many problems to groups constructed in various ways from the additive group of real numbers, the classical linear groups and from finite groups. The book gives a systematic and detailed introduction to the highlights of that theory. In the beginning, a review of fundamental tools from topology and the elementary theory of topological groups and transformation groups is presented. Completions, Haar integral, applications to linear representations culminating in the Peter-Weyl Theorem are treated. Pontryagin duality for locally compact Abelian groups forms a central topic of the book. Applications are given, including results about the structure of locally compact Abelian groups, and a structure theory for locally compact rings leading to the classification of locally compact fields. Topological semigroups are discussed in a separate chapter, with special attention to their relations to groups. The last chapter reviews results related to Hilbert's Fifth Problem, with the focus on structural results for non-Abelian connected locally compact groups that can be derived using approximation by Lie groups. The book is self-contained and is addressed to advanced undergraduate or graduate students in mathematics or physics. It can be used for one-semester courses on topological groups, on locally compact Abelian groups, or on topological algebra. Suggestions on course design are given in the preface. Each chapter is accompanied by a set of exercises that have been tested in classes.
In The Dual Penal State, Markus Dubber addresses the rampant use of penal power in Western liberal democracies. The interference with the autonomy of the very persons upon whose autonomy the legitimacy of state power is supposed to rest is systemically normalized, rather than continuously scrutinized. The fundamental challenge of the penal paradox-the prima facie illegitimacy of modern punishment-remains unaddressed and unresolved. Focusing on the United States and Germany, and drawing on his influential account of the patriarchal origins of police power, Dubber exposes the persistence of a two-sided criminal justice regime: the dual penal state. The dual penal state combines principled punishment of equals under the rule of law, on one side, with punitive discipline of others under the rule of police, on the other. Slavery has long played a central role in drawing the line between the two sides of the dual penal state. In Europe, the slave appears in the classic and still foundational accounts of liberal punishment (from Beccaria to Kant) as the paradigmatic other beyond the protection of law, not a legal subject but a mere object of the master's or the state's discretionary discipline. In America, the patriarchal power to police portrays the continuum from the antebellum slaveholder's whipping of his slaves in private and the racial terror perpetrated by slave patrols in public, to the apartheid regime of Jim Crow and the treatment of prisoners as "slaves of the state," and eventually to the late 20th century's systemic racial violence of the “war on crime" and the widespread killing of Black suspects by an increasingly militarized and armed police force that triggered the global Black Lives Matter movement.
The aim of this volume is to provide an extensive account of the most recent advances in statistics for discretely observed Lévy processes. These days, statistics for stochastic processes is a lively topic, driven by the needs of various fields of application, such as finance, the biosciences, and telecommunication. The three chapters of this volume are completely dedicated to the estimation of Lévy processes, and are written by experts in the field. The first chapter by Denis Belomestny and Markus Reiß treats the low frequency situation, and estimation methods are based on the empirical characteristic function. The second chapter by Fabienne Comte and Valery Genon-Catalon is dedicated to non-parametric estimation mainly covering the high-frequency data case. A distinctive feature of this part is the construction of adaptive estimators, based on deconvolution or projection or kernel methods. The last chapter by Hiroki Masuda considers the parametric situation. The chapters cover the main aspects of the estimation of discretely observed Lévy processes, when the observation scheme is regular, from an up-to-date viewpoint.
Accurate predictions are essential in many areas such as corporate decision making, weather forecasting and technology forecasting. Prediction markets help to aggregate information and gain a better understanding of the future by leveraging the wisdom of the crowds. Trading prices in prediction markets thus reflect the traders’ aggregated expectations on the outcome of uncertain future events and can be used to predict the likelihood of these events. This book demonstrates that markets are accurate predictors. Results from several empirical studies reported in this work show the importance of designing such markets properly in order to derive valuable predictions. Therefore, the findings are valuable for designing future prediction markets.
A thorough introduction to solar physics based on recent spacecraft observations. The author introduces the solar corona and sets it in the context of basic plasma physics before moving on to discuss plasma instabilities and plasma heating processes. The latest results on coronal heating and radiation are presented. Spectacular phenomena such as solar flares and coronal mass ejections are described in detail, together with their potential effects on the Earth.
Developments in Solid Earth Geophysics, 4: Mathematical Aspects of Seismology introduces studies of the more advanced parts of theoretical seismology. The manuscript first ponders on contour integration and conformal transformation, methods of stationary phase and steepest descent, and series integration. Discussions focus on Love waves in heterogeneous isotropic media, Laguerre's differential equation, Hermite's differential equation, method of steepest descent, method of stationary phase, contour integration in the complex plane, and conformal transformation. The text then examines series integration, Bessel functions, Legendre functions, and wave equations. Topics include general considerations of the wave equation, expansion of a spherical wave into plane waves, common features of special functions and special differential equations, applications of Legendre functions, Legendre polynomials, Bessel's differential equation, and properties of Bessel coefficients. The book explores the influence of gravity on wave propagation, matrix calculus, wave propagation in liquid media, integral equations, calculus of variations, and integral transforms. The text is a valuable source of data for researchers wanting to study the mathematical aspects of seismology.
Migration has been one of the most pressing societal issues throughout history. Immigrant associations play a crucial role in understanding this phenomenon. They channel migration streams, influence the assimilation of their members, and serve as representatives of the entire immigrant group in society. However, they remain an understudied subject, particularly in historical research. To address this gap, this study examines German immigrant associations in New York from the 1890s to the 1930s. Through an innovative combination of statistical and textual analyses, it explores the class composition of these associations, their intricate system of mutual aid, and their political activities. This study offers insights into how specific socio-economic motivations influenced immigrant organization and collective action, including aspects such as long-distance nationalism and cross-border ethnic identity. Ultimately, based on these findings, this study demonstrates that immigrant associations played a crucial role in helping their members adapt to a new social and economic environment. Additionally, it shows why and how immigrant associations significantly shaped the image of German immigrants in American social and political life.
This book sheds new light on the work of Jean-François Champollion by uncovering a constellation of epistemological, political, and material conditions that made his decipherment of Egyptian hieroglyphs possible. Champollion’s success in understanding hieroglyphs, first published in his Lettre à M. Dacier in 1822, is emblematic for the triumphant achievements of comparative philology during the 19th Century. In its attempt to understand humanity as part of a grand history of progress, Champollion’s conception of ancient Egypt belongs to the universalistic aspirations of European modernity. Yet precisely because of its success, his project also reveals the costs it entailed: after examining and welcoming acquisitions for the emerging Egyptian collections in Europe, Champollion travelled to the Nile Valley in 1828/29, where he was shocked by the damage that had been done to its ancient cultural sites. The letter he wrote to the Egyptian viceroy Mehmet Ali Pasha in 1829 demands that excavations in Egypt be regulated, denounces European looting, and represents perhaps the first document to make a case for the international protection of cultural goods in the name of humanity.
The role of information is central to the academic debate on finance. This book provides a detailed, current survey of theoretical research into the effect on stock prices of the distribution of information, comparing and contrasting major models. It examines theoretical models that explain bubbles, technical analysis, and herding behavior. It also provides rational explanations for stock market crashes. Analyzing the implications of asymmetries in information is crucial in this area. This book provides a useful survey for graduate students.
This book provides a comprehensive introduction to the study of international diplomacy, covering both theory and practice. This second edition has been revised and updated, with new material on such key contemporary issues as Syria, Ukraine, migration and the South China Sea. The text summarizes and discusses the major trends in the field of diplomacy, providing an innovative theoretical approach to understanding diplomacy not as a collection of practices or a set of historical traditions, but as a form of institutionalized communication through which authorized representatives produce, manage and distribute public goods. The book: Traces the evolution of diplomacy from its beginnings in ancient Egypt, Greece and China to our current age of global diplomacy. Examines theoretical explanations about how diplomats take decisions, make relations and shape the world. Discusses normative approaches to how diplomacy ought to adapt itself to the twenty-first century, help re-make states and assist the peaceful evolution of international order. In sum, Understanding International Diplomacy provides an up-to-date, accessible and authoritative overview of how diplomacy works and, indeed, ought to work in a globalized world. This textbook will be essential reading for students of international diplomacy, and is highly recommended for students of crisis negotiation, international organizations, foreign policy and IR in general.
The contributions to the "Thirty Years Volume" represented in this volume reflect the historical focus of the ProtoSociology project. Colleagues are represented who contributed to the focus. This is also true thematically, as contributions on language theory, the philosophy of the mental, and the sociology of contemporary societies are represented. The contributions to the "Thirty Years Volume" are definitely evidence that they address central research problems of ProtoSociology, regardless of their particular epistemological interests.
This book is open access and available on www.bloomsburycollections.com. It is funded by Knowledge Unlatched. Debates about the legitimacy and 'essence' of political rule and the search for 'ideal' forms of government have been at the very heart of political thought ever since antiquity. Caesarism in the Post-Revolutionary Age explores the complex relationship between democracy and dictatorship from the 18th century onwards. More concretely, it assesses how democracy emerged as something compatible with dictatorship, both at the level of political thought and practice. Taking Caesarism – a political alternative somewhere between democracy and dictatorship – as its key concept, the book considers: * To what extent was Caesarism seen as a new post-revolutionary form of rule? * What were the flaws and perils, strengths and promises of Caesaristic regimes? * Can 19th-century Caesarism be characterised as a 'prelude' to 20th-century totalitarianism? * What is the legacy and ongoing appeal of Caesarism in the contemporary world? This study will be of value to anyone interested in modern political history, but also contemporary politics.
Microeconomics is not applied math – frameworks in this book are regularly in use in daily managerial practice and strategic decision-making. Numerous case studies cover price discrimination, economies of scale, digital business models, game theory, dealing with uncertainty, entry barriers or sunk costs – all of which are crucial for understanding market dynamics and competitive behaviour.
The Compressed Word Problem for Groups provides a detailed exposition of known results on the compressed word problem, emphasizing efficient algorithms for the compressed word problem in various groups. The author presents the necessary background along with the most recent results on the compressed word problem to create a cohesive self-contained book accessible to computer scientists as well as mathematicians. Readers will quickly reach the frontier of current research which makes the book especially appealing for students looking for a currently active research topic at the intersection of group theory and computer science. The word problem introduced in 1910 by Max Dehn is one of the most important decision problems in group theory. For many groups, highly efficient algorithms for the word problem exist. In recent years, a new technique based on data compression for providing more efficient algorithms for word problems, has been developed, by representing long words over group generators in a compressed form using a straight-line program. Algorithmic techniques used for manipulating compressed words has shown that the compressed word problem can be solved in polynomial time for a large class of groups such as free groups, graph groups and nilpotent groups. These results have important implications for algorithmic questions related to automorphism groups.
In Visual Culture and Politics in the Baltic Sea Region, Kersti Markus examines how visual rhetoric was used by the Danish rulers as an instrument in establishing supremacy in the region during the Baltic crusades.
The book covers the content of a typical higher undergraduate course of the theory of Quantum Mechanics. The focus is on the general principles of quantum mechanics and the clarification of its terminology: What exactly is a Hilbert space? What is a hermitean operator? A tensor product? An entangled state? In what sense does a wave function constitute a vector? A separate chapter discusses the many open questions regarding the interpretation of the postulates.
In the last decades of the 17th century, the feast of Christmas in Lutheran Germany underwent a major transformation when theologians and local governments waged an early modern "war on Christmas," discouraging riotous pageants and carnivalesque rituals in favor of more personal and internalized expressions of piety. Christmas rituals, such as the "Heilig Christ" plays and the rocking of the child (Kindelwiegen) were abolished, and Christian devotion focused increasingly on the metaphor of a birth of Christ in the human heart. John Sebastian Bach's Christmas Oratorio, composed in 1734, both reflects this new piety and conveys the composer's experience living through this tumult during his own childhood and early career. Markus Rathey's book is the first thorough study of this popular masterpiece in English. While giving a comprehensive overview of the Christmas Oratorio as a whole, the book focuses on two themes in particular: the cultural and theological understanding of Christmas in Bach's time and the compositional process that led Bach from the earliest concepts to the completed piece. The cultural and religious context of the oratorio provides the backdrop for Rathey's detailed analysis of the composition, in which he explores Bach's compositional practices, for example, his reuse and parodies of movements that had originally been composed for secular cantatas. The book analyzes Bach's original score and sheds new light on the way Bach wrote the piece, how he shaped musical themes, and how he revised his initial ideas into the final composition.
How can inter-organizational cooperation’s in supply chains be improved? This book analyzes the interdependencies between the use of information and communication technologies and the efficient form of IT-based coordination in supply chains and/or supply chain networks. Using the results of an empirical survey done in the European automotive industry, the status quo and the future of different information and communication technologies used to support supply chain operations is shown. The results on a general level were complemented by case studies which give examples demonstrating the specific behavior of individual companies and technological concepts in the automotive industry. On the methodological basis of transaction costs theory, network effect theory and empirical findings, an economic model of “information logistics” is developed, that can be applied to determine optimal IT-configuration and information flows and thus to analyze efficiency of networked cooperation forms. At the example of vehiclespecific supply chains and industry networks it is shown, how the overall network costs necessary to exchange information between business partners can be improved.
This book contains a systematic and partly axiomatic treatment of the holomorphic functional calculus for unbounded sectorial operators. The account is generic so that it can be used to construct and interrelate holomorphic functional calculi for other types of unbounded operators. Particularly, an elegant unified approach to holomorphic semigroups is obtained. The last chapter describes applications to PDE, evolution equations and approximation theory as well as the connection with harmonic analysis.
This definitive assessment of Cormac McCarthy’s novels captures the interactions among the literary and mythic elements, the social dynamics of violence, and the natural world in The Orchard Keeper, Child of God, Outer Dark, Blood Meridian, and The Road. Elegantly written and deeply engaged with previous scholarship as well as interviews with the novelist, this study provides a comprehensive introduction to McCarthy’s work while offering an insightful new analysis. Drawing on René Girard’s mimetic theory, mythography, thermodynamics, and information science, Markus Wierschem identifies a literary apocalypse at the center of McCarthy’s work, one that unveils another buried deep within the history, religion, and myths of American and Western culture.
This book focuses on the phenomena of inertia and gravitation, one objective being to shed some new light on the basic laws of gravitational interaction and the fundamental nature and structures of spacetime. Chapter 1 is devoted to an extensive, partly new analysis of the law of inertia. The underlying mathematical and geometrical structure of Newtonian spacetime is presented from a four-dimensional point of view, and some historical difficulties and controversies - in particular the concepts of free particles and straight lines - are critically analyzed, while connections to projective geometry are also explored. The relativistic extensions of the law of gravitation and its intriguing consequences are studied in Chapter 2. This is achieved, following the works of Weyl, Ehlers, Pirani and Schild, by adopting a point of view of the combined conformal and projective structure of spacetime. Specifically, Mach’s fundamental critique of Newton’s concepts of ‘absolute space’ and ‘absolute time’ was a decisive motivation for Einstein’s development of general relativity, and his equivalence principle provided a new perspective on inertia. In Chapter 3 the very special mathematical structure of Einstein’s field equations is analyzed, and some of their remarkable physical predictions are presented. By analyzing different types of dragging phenomena, Chapter 4 reviews to what extent the equivalence principle is realized in general relativity - a question intimately connected to the ‘new force’ of gravitomagnetism, which was theoretically predicted by Einstein and Thirring but which was only recently experimentally confirmed and is thus of current interest.
This book constitutes the refereed proceedings of the 14th Australian Joint Conference on Artificial Intelligence, AI 2001, held in Adelaide, Australia, in December 2001. The 55 revised full papers presented together with one invited contribution were carefully reviewed and selected from a total of 100 submissions. The papers cover the whole range of artificial intelligence from theoretical and foundational issues to advanced applications in a variety of fields.
The basic idea of this introduction to the finite element method is based on the concept of explaining the complex method using only one-dimensional elements. Thus, the mathematical description remains largely simple and straightforward. The emphasis in each chapter is on explaining the method and understanding it itself. The reader learns to understand the assumptions and derivations in various physical problems in structural mechanics and to critically assess the possibilities and limitations of the finite element method. The restriction to one-dimensional elements thus enables the methodical understanding of important topics (e.g. plasticity or composite materials), which a prospective computational engineer encounters in professional practice, but which are rarely treated in this form at universities. Thus, an easy entry - also into more advanced application areas - is ensured by the concept of (a) introduction to the basics (b) exact derivation with restriction to one-dimensional elements (and in many cases also to one-dimensional problems) (c) extensive examples and advanced tasks (with short solution in the appendix). For illustration purposes, each chapter is deepened with extensively calculated and commented examples as well as with further tasks including short solutions
The third volume of Recent Advances in Otolaryngology brings clinicians and trainees fully up to date with the latest developments in the rapidly changing field of otolaryngology. Nineteen chapters present new concepts, surgical advances and imaging modalities in each of the subspecialties of otolaryngology. Each chapter begins with a brief summary of the topic, followed by all the recent advances and comprehensive references. A complete chapter is dedicated to geriatric otolaryngology, an emerging subspecialty. Written by an internationally-recognised author and editor team from the US and Europe, this comprehensive manual includes more than 270 images, illustrations and tables. The annual publication cycle of this series ensures the content is current, topical and highly relevant to clinicians and trainees. Key points New volume bringing clinicians up to date with recent advances in otolaryngology Covers all subspecialties of otolaryngology Experienced author and editor team from the US and Europe Includes more than 270 images, illustrations and tables Annual publication cycle ensures current and topical content
This exciting new book describes the use of DNA fingerprinting and its application in a wide area of plant and fungal research. It presents a thorough theoretical introduction to DNA fingerprinting, the practical aspects of extraction of DNA from the plant or fungus under study, and the statistical analysis of the data. An overview presents all species to which DNA fingerprinting is currently being applied and highlights many future technical developments.
The biography of Hans Rose, Germany's most successful WWI convoy period U-boat Ace and WWII anti-fascist naval officer. 'He was a brave man who would take chances that most of his compatriots would avoid. He played his desperate game with a certain decency.
Since Greek antiquity, the ‘barbarian’ captivates the Western imaginary and operates as the antipode against which self-proclaimed civilized groups define themselves. Therefore, the study of the cultural history of barbarism is a simultaneous exploration of the shifting contours of European identity. This two-volume co-authored study explores the history of the concept ‘barbarism’ from the 18th century to the present and illuminates its foundational role in modern European and Western identity. It constitutes an original comparative, interdisciplinary exploration of the concept’s modern European and Western history, with emphasis on the role of literature in the concept’s shifting functions. Critically responding to the contemporary popularity of the term ‘barbarian' in political rhetoric and the media, and its violent, exclusionary workings, the study contributes to a historically grounded understanding of this figure’s past and contemporary uses. It combines overviews with detailed analyses of representative works of literature, art, film, philosophy, political and cultural theory, in which “barbarism” figures prominently.
Like tariffs and other border measures, national regulatory barriers impede international trade. Unlike tariffs, however, such barriers usually indicate an important domestic policy choice. This 'conflict of interest' has emerged as a crucial issue in international law, particularly with regard to services, such as telecommunications and health services. This study is the first to analyze the potential impact of incompatibilities between national regulatory regimes and the rules and obligations imposed by the General Agreement on Trade in Services (GATS). In the process of arriving at his challenging concluding theses, the author investigates such relevant concepts as the following: the political and ideological dynamics of GATS negotiations services trade liberalization in regional integration systems, particularly in EC law policies common to diverse national regulatory systems the notions of 'deregulation' and 'privatization' the human rights implications of international trade law the GATS obligations of market access, national treatment, and most-favoured-nation treatment the role of the WTO's dispute settlement organs GATS transparency obligations Professor Krajewski's study is of enormous significance to specialists in regulatory policies and instruments at all national and sectoral levels, especially in the context of ongoing GATS negotiations. As the author warns: Unless GATS negotiators and national regulators have a thorough understanding of the relationship between GATS obligations and regulatory policies and instruments, they cannot effectively use the flexible elements of GATS and could reach an agreement which they may later regret.
Remoting offers developers many ways to customize the communications process, for efficiency, security, performance and power, and allows seamless integration of components running on several computers into a single application. This book exposes the full power of remoting to developers working in mixed platform environments in a way that will ensure they have a deep understanding of what remoting is capable of, and how they can make it work the way they want.
A multi-interval quasi-differential system $\{I_{r},M_{r},w_{r}:r\in\Omega\}$ consists of a collection of real intervals, $\{I_{r}\}$, as indexed by a finite, or possibly infinite index set $\Omega$ (where $\mathrm{card} (\Omega)\geq\aleph_{0}$ is permissible), on which are assigned ordinary or quasi-differential expressions $M_{r}$ generating unbounded operators in the Hilbert function spaces $L_{r}^{2}\equiv L^{2}(I_{r};w_{r})$, where $w_{r}$ are given, non-negative weight functions. For each fixed $r\in\Omega$ assume that $M_{r}$ is Lagrange symmetric (formally self-adjoint) on $I_{r}$ and hence specifies minimal and maximal closed operators $T_{0,r}$ and $T_{1,r}$, respectively, in $L_{r}^{2}$. However the theory does not require that the corresponding deficiency indices $d_{r}^{-}$ and $d_{r}^{+}$ of $T_{0,r}$ are equal (e. g. the symplectic excess $Ex_{r}=d_{r}^{+}-d_{r}^{-}\neq 0$), in which case there will not exist any self-adjoint extensions of $T_{0,r}$ in $L_{r}^{2}$. In this paper a system Hilbert space $\mathbf{H}:=\sum_{r\,\in\,\Omega}\oplus L_{r}^{2}$ is defined (even for non-countable $\Omega$) with corresponding minimal and maximal system operators $\mathbf{T}_{0}$ and $\mathbf{T}_{1}$ in $\mathbf{H}$. Then the system deficiency indices $\mathbf{d}^{\pm} =\sum_{r\,\in\,\Omega}d_{r}^{\pm}$ are equal (system symplectic excess $Ex=0$), if and only if there exist self-adjoint extensions $\mathbf{T}$ of $\mathbf{T}_{0}$ in $\mathbf{H}$. The existence is shown of a natural bijective correspondence between the set of all such self-adjoint extensions $\mathbf{T}$ of $\mathbf{T}_{0}$, and the set of all complete Lagrangian subspaces $\mathsf{L}$ of the system boundary complex symplectic space $\mathsf{S}=\mathbf{D(T}_{1})/\mathbf{D(T}_{0})$. This result generalizes the earlier symplectic version of the celebrated GKN-Theorem for single interval systems to multi-interval systems. Examples of such complete Lagrangians, for both finite and infinite dimensional complex symplectic $\mathsf{S}$, illuminate new phenoma for the boundary value problems of multi-interval systems. These concepts have applications to many-particle systems of quantum mechanics, and to other physical problems.
An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V:H→K such that C=V∗TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor ϑ(d), expressed as a ratio of Γ functions for d even, of all d×d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.
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