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Green's Functions with Applications (Advances in Applied Mathematics)
by Dean G. DuffySince publication of the first edition over a decade ago, Green’s Functions with Applications has provided applied scientists and engineers with a systematic approach to the various methods available for deriving a Green’s function. This fully revised Second Edition retains the same purpose, but has been meticulously updated to reflect the current state of the art. The book opens with necessary background information: a new chapter on the historical development of the Green’s function, coverage of the Fourier and Laplace transforms, a discussion of the classical special functions of Bessel functions and Legendre polynomials, and a review of the Dirac delta function. The text then presents Green’s functions for each class of differential equation (ordinary differential, wave, heat, and Helmholtz equations) according to the number of spatial dimensions and the geometry of the domain. Detailing step-by-step methods for finding and computing Green’s functions, each chapter contains a special section devoted to topics where Green’s functions particularly are useful. For example, in the case of the wave equation, Green’s functions are beneficial in describing diffraction and waves. To aid readers in developing practical skills for finding Green’s functions, worked examples, problem sets, and illustrations from acoustics, applied mechanics, antennas, and the stability of fluids and plasmas are featured throughout the text. A new chapter on numerical methods closes the book. Included solutions and hundreds of references to the literature on the construction and use of Green's functions make Green’s Functions with Applications, Second Edition a valuable sourcebook for practitioners as well as graduate students in the sciences and engineering.
Green's Kernels and Meso-Scale Approximations in Perforated Domains
by Alexander Movchan Vladimir Maz'Ya Michael NievesThere are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary. Examples include perforated domains and bodies with defects of different types. The accurate direct numerical treatment of such problems remains a challenge. Asymptotic approximations offer an alternative, efficient solution. Green's function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems. The uniformity of the asymptotic approximations is the principal point of attention. We also show substantial links between Green's functions and solutions of boundary value problems for meso-scale structures. Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusions. The main focus of the present text is on two topics: (a) asymptotics of Green's kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions. The novel feature of these asymptotic approximations is their uniformity with respect to the independent variables. This book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations.
Grenzwerte oder infinitesimale Zahlen?: Über Einstiege in die Analysis und ihren Hintergrund (essentials)
by Karl Kuhlemann Thomas BedürftigIn diesem Buch wird der Weg von den infinitesimalen Größen bei Leibniz über die Grenzwerte zu den infinitesimalen Zahlen der Nichtstandardanalysis skizziert. Die begrifflichen Probleme der Grenzwerte werden diskutiert und der Einstieg in die Analysis mit infinitesimalen und infiniten Zahlen vorgestellt. Ein Vergleich mit dem Grenzwerteinstieg zeigt die Möglichkeiten der infinitesimalen Zahlen. Der Grenzwertformalismus entfällt. Der Einstieg in die Analysis wird arithmetisch und zugleich anschaulich. Da die heutige Analysis in Grenzwerten geschrieben ist, geht es nicht um eine Entscheidung, sondern um Offenheit. Infinitesimale Zahlen erweitern das Repertoire, vertiefen und verändern das mathematische Denken und öffnen den Blick, nicht zuletzt auf die Grenzwerte.
Grey Forecasting: Mechanism, Models and Applications (Series on Grey System)
by Naiming Xie Baolei WeiThis book aims to present an overview of grey system models for time series modelling and forecasting. It is about modelling and forecasting time series with ordinary differential equations, especially when the available samples are extremely limited. Grey system models (GSM) develop sequence operators to nonparametrically identify the underlying dynamics from the limited observations. This book concerns about two important modelling themes, small sample and poor information. The former focuses on the mechanism and methodology of GSMs for small-sample real-number time series, and the latter on the uncertainty quantification of grey number together with its small-sample modelling principles. In this book, a broad entry point to applied data science for students majoring in economic, management science, and engineering is applied, covering a wide range of topics from basic introductory material up to research-level techniques.
Grey Game Theory and Its Applications in Economic Decision-Making (Systems Evaluation, Prediction, and Decision-Making)
by Yi Lin Sifeng Liu Zhigeng Fang Hongxing ShiTo make the best decisions, you need the best information. However, because most issues in game theory are grey, nearly all recent research has been carried out using a simplified method that considers grey systems as white ones. This often results in a forecasting function that is far from satisfactory when applied to many real situations. Grey Ga
Grey Systems Analysis: Methods, Models and Applications (Series on Grey System)
by Sifeng LiuThis open access book is the 2nd edition involving the update on data, methods and models of Grey Systems. It covers up-to-date theoretical and applied advances in grey systems from across the world, and vividly presents the reader with the overall picture of this new theory and its frontier research. Many of the concepts, models and methods in the book are original by the author, including kernel, degree of greyness of grey number, simplified form of grey number, general grey number and the operation system; the axiomatic system of buffer operators and a series of weakening and strengthening buffer operators; a series of grey relational analysis models, including grey absolute, relative, synthetic, similarity, closeness, negative, three dimension, and grey relational analysis model for cross-sequences, etc.; grey fixed weight clustering model, grey evaluation models based on center-point and end-point mixed possibility functions; original difference grey model (ODGM), even difference grey model (EDGM), discrete grey model (DGM), fractional grey models, self-memory grey models; multi-attribute weighted intelligent grey target decision models, kernel weight vector group and the weighted comprehensive clustering coefficient vector, and spectrum analysis of sequence operators, etc. The revision includes: (1) Added new achievements made in recent years, such as the moving average denoise operator, a series of negative grey relational models, grey relational model for cross-sequences, standard uncertainty numbers and their operations, adaptive Grey Prediction Models and so on; (2) Important data related to the development of grey system theory has been updated; (3) Research reviews have been added to each chapter, and a large number of references have been added; (4)Updated application examples of commonly used models and methods. This book will be appropriate as a reference and/or textbook for courses of grey system theory for graduate students or high level undergraduate students, majoring in various fields of natural sciences, social sciences and engineering technology. It can also be utilized by researchers and technicians in research institutions, business entities, and government agencies.
Grid Computing: Infrastructure, Service, and Applications
by Lizhe Wang Wei Jie Jinjun ChenIdentifies Recent Technological Developments Worldwide The field of grid computing has made rapid progress in the past few years, evolving and developing in almost all areas, including concepts, philosophy, methodology, and usages. Grid Computing: Infrastructure, Service, and Applications reflects the recent advances in this field, covering the research aspects that involve infrastructure, middleware, architecture, services, and applications. Grid Systems Across the Globe The first section of the book focuses on infrastructure and middleware and presents several national and international grid systems. The text highlights China Research and Development environment Over Wide-area Network (CROWN), several ongoing cyberinfrastructure efforts in New York State, and Enabling Grids for E-sciencE (EGEE), which is co-funded by the European Commission and the world’s largest multidisciplinary grid infrastructure today. The second part of the book discusses recent grid service advances. The authors examine the UK National Grid Service (NGS), the concept of resource allocation in a grid environment, OMIIBPEL, and the possibility of treating scientific workflow issues using techniques from the data stream community. The book describes an SLA model, reviews portal and workflow technologies, presents an overview of PKIs and their limitations, and introduces PIndex, a peer-to-peer model for grid information services. New Projects and Initiatives The third section includes an analysis of innovative grid applications. Topics covered include the WISDOM initiative, incorporating flow-level networking models into grid simulators, system-level virtualization, grid usage in the high-energy physics environment in the LHC project, and the Service Oriented HLA RTI (SOHR) framework. With a comprehensive summary of past advances, this text is a window into the future of this nascent technology, forging a path for the next generation of cyberinfrastructure developers.
Grid Computing: Techniques and Applications (Chapman & Hall/CRC Computational Science)
by Barry WilkinsonDesigned for senior undergraduate and first-year graduate students, Grid Computing: Techniques and Applications shows professors how to teach this subject in a practical way. Extensively classroom-tested, it covers job submission and scheduling, Grid security, Grid computing services and software tools, graphical user interfaces, workflow editors,
Grid Optimal Integration of Electric Vehicles: Examples with Matlab Implementation
by Seddik Bacha Andrés Ovalle Ahmad HablyThis book is a compilation of recent research on distributed optimization algorithms for the integral load management of plug-in electric vehicle (PEV) fleets and their potential services to the electricity system. It also includes detailed developed Matlab scripts. These algorithms can be implemented and extended to diverse applications where energy management is required (smart buildings, railways systems, task sharing in micro-grids, etc.). The proposed methodologies optimally manage PEV fleets’ charge and discharge schedules by applying classical optimization, game theory, and evolutionary game theory techniques. Taking owner’s requirements into consideration, these approaches provide services like load shifting, load balancing among phases of the system, reactive power supply, and task sharing among PEVs. The book is intended for use in graduate optimization and energy management courses, and readers are encouraged to test and adapt the scripts to their specific applications.
Grid Resource Management: Toward Virtual and Services Compliant Grid Computing (Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series)
by Lei Yu Frederic Magoules Thi-Mai-Huong NguyenGrid technology offers the potential for providing secure access to remote services, thereby promoting scientific collaborations in an unprecedented scale. Grid Resource Management: Toward Virtual and Services Compliant Grid Computing presents a comprehensive account of the architectural issues of grid technology, such as security, data management,
Grid-based Nonlinear Estimation and Its Applications
by Ming Xin Bin JiaGrid-based Nonlinear Estimation and its Applications presents new Bayesian nonlinear estimation techniques developed in the last two decades. Grid-based estimation techniques are based on efficient and precise numerical integration rules to improve performance of the traditional Kalman filtering based estimation for nonlinear and uncertainty dynamic systems. The unscented Kalman filter, Gauss-Hermite quadrature filter, cubature Kalman filter, sparse-grid quadrature filter, and many other numerical grid-based filtering techniques have been introduced and compared in this book. Theoretical analysis and numerical simulations are provided to show the relationships and distinct features of different estimation techniques. To assist the exposition of the filtering concept, preliminary mathematical review is provided. In addition, rather than merely considering the single sensor estimation, multiple sensor estimation, including the centralized and decentralized estimation, is included. Different decentralized estimation strategies, including consensus, diffusion, and covariance intersection, are investigated. Diverse engineering applications, such as uncertainty propagation, target tracking, guidance, navigation, and control, are presented to illustrate the performance of different grid-based estimation techniques.
Grothendieck Construction of Bipermutative-Indexed Categories
by Donald YauThe Grothendieck construction provides an explicit link between indexed categories and opfibrations. It is a fundamental concept in category theory and related fields with far-reaching applications. Bipermutative categories are categorifications of rings. They play a central role in algebraic K-theory and infinite loop space theory. This monograph is a detailed study of the Grothendieck construction over a bipermutative category in the context of categorically enriched multicategories, with new and important applications to inverse K-theory and pseudo symmetric E∞-algebras. After carefully recalling preliminaries in enriched categories, bipermutative categories, and enriched multicategories, we show that the Grothendieck construction over a small tight bipermutative category is a pseudo symmetric Cat-multifunctor and generally not a Cat-multifunctor in the symmetric sense. Pseudo symmetry of Cat-multifunctors is a new concept we introduce in this work. The following features make it accessible as a graduate text or reference for experts: Complete definitions and proofs. Self-contained background. Parts of Chapters 1–3, 7, 9, and 10 contain background material from the research literature. Extensive cross-references. Connections between chapters. Each chapter has its own introduction discussing not only the topics of that chapter but also its connection with other chapters. Open questions. Appendix A contains open questions that arise from the material in the text and are suitable for graduate students. This book is suitable for graduate students and researchers with an interest in category theory, algebraic K-theory, homotopy theory, and related fields. The presentation is thorough and self-contained, with complete details and background material for non-expert readers.
Ground States of Quantum Field Models: Perturbation of Embedded Eigenvalues (SpringerBriefs in Mathematical Physics #35)
by Fumio HiroshimaThis book provides self-contained proofs of the existence of ground states of several interaction models in quantum field theory. Interaction models discussed here include the spin-boson model, the Nelson model with and without an ultraviolet cutoff, and the Pauli–Fierz model with and without dipole approximation in non-relativistic quantum electrodynamics. These models describe interactions between bose fields and quantum mechanical matters.A ground state is defined as the eigenvector associated with the bottom of the spectrum of a self-adjoint operator describing the Hamiltonian of a model. The bottom of the spectrum is however embedded in the continuum and then it is non-trivial to show the existence of ground states in non-perturbative ways. We show the existence of the ground state of the Pauli–Fierz mode, the Nelson model, and the spin-boson model, and several kinds of proofs of the existence of ground states are explicitly provided. Key ingredients are compact sets and compact operators in Hilbert spaces. For the Nelson model with an ultraviolet cutoff and the Pauli–Fierz model with dipole approximation we show not only the existence of ground states but also enhanced binding. The enhanced binding means that a system for zero-coupling has no ground state but it has a ground state after turning on an interaction.The book will be of interest to graduate students of mathematics as well as to students of the natural sciences who want to learn quantum field theory from a mathematical point of view. It begins with abstract compactness arguments in Hilbert spaces and definitions of fundamental facts of quantum field theory: boson Fock spaces, creation operators, annihilation operators, and second quantization. This book quickly takes the reader to a level where a wider-than-usual range of quantum field theory can be appreciated, and self-contained proofs of the existence of ground states and enhanced binding are presented.
Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers
by Robert J. ZimmerRobert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology. In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.
Group Cohomology and Algebraic Cycles
by Burt TotaroGroup cohomology reveals a deep relationship between algebra and topology, and its recent applications have provided important insights into the Hodge conjecture and algebraic geometry more broadly. This book presents a coherent suite of computational tools for the study of group cohomology and algebraic cycles. Early chapters synthesize background material from topology, algebraic geometry, and commutative algebra so readers do not have to form connections between the literatures on their own. Later chapters demonstrate Peter Symonds's influential proof of David Benson's regularity conjecture, offering several new variants and improvements. Complete with concrete examples and computations throughout, and a list of open problems for further study, this book will be valuable to graduate students and researchers in algebraic geometry and related fields.
Group Decision and Negotiation in an Uncertain World: 18th International Conference, GDN 2018, Nanjing, China, June 9-13, 2018, Proceedings (Lecture Notes in Business Information Processing #315)
by Rudolf Vetschera Haiyan Xu Ye Chen Gregory KerstenThis book constitutes the refereed proceedings of the 18th International Conference on Group Decision and Negotiation, GDN 2018, held in Nanjing, China, in June 2018. The field of Group Decision and Negotiation focuses on decision processes with at least two participants and a common goal but conflicting individual goals. Research areas of Group Decision and Negotiation include electronic negotiations, experiments, the role of emotions in group decision and negotiations, preference elicitation and decision support for group decisions and negotiations, and conflict resolution principles.The 15 full papers presented in this volume were carefully reviewed and selected from 143 submissions. They were organized in topical sections named: theoretical concepts of group decision and negotiation; decision support and behavior in group decision and negotiation; and applications of group decision and negotiations.
Group Decision and Negotiation in the Era of Multimodal Interactions: 23rd International Conference on Group Decision and Negotiation, GDN 2023, Tokyo, Japan, June 11–15, 2023, Proceedings (Lecture Notes in Business Information Processing #478)
by Pascale Zaraté Liping Fang Masahide Horita Yu MaemuraThis book constitutes the refereed proceedings of the 23rd International Conference on Group Decision and Negotiation, GDN 2023, which took place in Tokyo, Japan during June 11–15, 2023.The field of Group Decision and Negotiation focuses on decision processes with at least two participants and a common goal but conflicting individual goals. Research areas of Group Decision and Negotiation include electronic negotiations, experiments, the role of emotions in group decision and negotiations, preference elicitation and decision support for group decisions and negotiations, and conflict resolution principles. This year’s conference focusses on multimodal interactions. The 11 full papers presented in this volume were carefully reviewed and selected from 102 submissions. They were organized in the following topical sections: Taking a step back: Critically re-examining technology interactions with group decision and negotiation; preference modeling and multi-criteria decision-making; and conflict modeling and distributive mechanisms.
Group Decision and Negotiation. A Socio-Technical Perspective: 17th International Conference, GDN 2017, Stuttgart, Germany, August 14-18, 2017, Proceedings (Lecture Notes in Business Information Processing #293)
by D. Marc Kilgour Mareike SchoopThis book constitutes the refereed proceedings of the 17th International Conference on Group Decision and Negotiation, GDN 2017, held in Stuttgart, Germany, in August 2017. The field of Group Decision and Negotiation focuses on decision processes with at least two participants and a common goal but conflicting individual goals. Research areas of Group Decision and Negotiation include electronic negotiations, experiments, the role of emotions in group decision and negotiations, preference elicitation and decision support for group decisions and negotiations, and conflict resolution principles.The 14 full papers presented in this volume were carefully reviewed and selected from 87 submissions. They were organized in topical sections named: general topics in group decision and negotiation; conflict resolution; emotions in group decision and negotiation; negotiation support systems and studies; and preference modeling for group decision and negotiation. The book also contains two invited talks in full paper length.
Group Decision and Negotiation: 19th International Conference, GDN 2019, Loughborough, UK, June 11–15, 2019, Proceedings (Lecture Notes in Business Information Processing #351)
by Adiel Teixeira de Almeida Rudolf Vetschera Danielle Costa Morais Ashley CarrerasThis book constitutes the refereed proceedings of the 19th International Conference on Group Decision and Negotiation, GDN 2019, held in Loughborough, UK, in June 2019. The field of Group Decision and Negotiation focuses on decision processes with at least two participants and a common goal but conflicting individual goals. Research areas of Group Decision and Negotiation include electronic negotiations, experiments, the role of emotions in group decision and negotiations, preference elicitation and decision support for group decisions and negotiations, and conflict resolution principles. The 17 full papers presented in this volume were carefully reviewed and selected from 98 submissions. They were organized in topical sections named: preference modeling for group decision and negotiations; collaborative decision making processes; conflict resolution; behavioral OR, and negotiation support systems and studies.
Group Decision and Negotiation: 20th International Conference on Group Decision and Negotiation, GDN 2020, Toronto, ON, Canada, June 7–11, 2020, Proceedings (Lecture Notes in Business Information Processing #388)
by Liping Fang Danielle Costa Morais Masahide HoritaThis book constitutes the refereed proceedings of the 20th International Conference on Group Decision and Negotiation, GDN 2020, which was planned to be held in Toronto, ON, Canada, during June 7–11, 2020. The conference was cancelled due to the Coronavirus pandemic. Nevertheless, it was decided to publish the proceedings, because the review process had already been completed at the time the cancellation was decided. The field of Group Decision and Negotiation focuses on decision processes with at least two participants and a common goal but conflicting individual goals. Research areas of Group Decision and Negotiation include electronic negotiations, experiments, the role of emotions in group decision and negotiations, preference elicitation and decision support for group decisions and negotiations, and conflict resolution principles. The 14 full papers presented in this volume were carefully reviewed and selected from 75 submissions. They were organized in topical sections named: Conflict Resolution, Preference Modeling for Group Decision and Negotiation, Intelligent Group Decision Making and Consensus Process, Collaborative Decision Making Processes.
Group Decision and Negotiation: 22nd International Conference on Group Decision and Negotiation, GDN 2022, Virtual Event, June 12–16, 2022, Proceedings (Lecture Notes in Business Information Processing #454)
by Liping Fang Danielle Costa MoraisThis book constitutes the refereed proceedings of the 22nd International Conference on Group Decision and Negotiation, GDN 2022, which was held virtually during June 12–16, 2022. The field of Group Decision and Negotiation focuses on decision processes with at least two participants and a common goal but conflicting individual goals. Research areas of Group Decision and Negotiation include electronic negotiations, experiments, the role of emotions in group decision and negotiations, preference elicitation and decision support for group decisions and negotiations, and conflict resolution principles. This year’s conference focusses on methodological and practical issues. The 9 full papers presented in this volume were carefully reviewed and selected from 68 submissions. They were organized in the following topical sections: Preference modeling for group decision and negotiation; conflict resolution; collaborative decision making processes.
Group Explicit Methods for the Numerical Solution of Partial Differential Equations (Topics in Computer Mathematics)
by David J. EvansA new class of methods, termed "group explicit methods," is introduced in this text. Their applications to solve parabolic, hyperbolic and elliptic equations are outlined, and the advantages for their implementation on parallel computers clearly portrayed. Also included are the introductory and fundamental concepts from which the new methods are derived, and on which they are dependent. With the increasing advent of parallel computing into all aspects of computational mathematics, there is no doubt that the new methods will be widely used.
Group Inverses of M-Matrices and Their Applications
by Stephen J. Kirkland Michael NeumannGroup inverses for singular M-matrices are useful tools not only in matrix analysis, but also in the analysis of stochastic processes, graph theory, electrical networks, and demographic models. Group Inverses of M-Matrices and Their Applications highlights the importance and utility of the group inverses of M-matrices in several application areas.
Group Matrices, Group Determinants and Representation Theory: The Mathematical Legacy of Frobenius (Lecture Notes in Mathematics #2233)
by Kenneth W. JohnsonThis book sets out an account of the tools which Frobenius used to discover representation theory for nonabelian groups and describes its modern applications. It provides a new viewpoint from which one can examine various aspects of representation theory and areas of application, such as probability theory and harmonic analysis. For example, the focal objects of this book, group matrices, can be thought of as a generalization of the circulant matrices which are behind many important algorithms in information science. The book is designed to appeal to several audiences, primarily mathematicians working either in group representation theory or in areas of mathematics where representation theory is involved. Parts of it may be used to introduce undergraduates to representation theory by studying the appealing pattern structure of group matrices. It is also intended to attract readers who are curious about ideas close to the heart of group representation theory, which do not usually appear in modern accounts, but which offer new perspectives.
Group Representation for Quantum Theory
by Masahito HayashiThis book explains the group representation theory for quantum theory in the language of quantum theory. As is well known, group representation theory is very strong tool for quantum theory, in particular, angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, quark model, quantum optics, and quantum information processing including quantum error correction. To describe a big picture of application of representation theory to quantum theory, the book needs to contain the following six topics, permutation group, SU(2) and SU(d), Heisenberg representation, squeezing operation, Discrete Heisenberg representation, and the relation with Fourier transform from a unified viewpoint by including projective representation. Unfortunately, although there are so many good mathematical books for a part of six topics, no book contains all of these topics because they are too segmentalized. Further, some of them are written in an abstract way in mathematical style and, often, the materials are too segmented. At least, the notation is not familiar to people working with quantum theory. Others are good elementary books, but do not deal with topics related to quantum theory. In particular, such elementary books do not cover projective representation, which is more important in quantum theory. On the other hand, there are several books for physicists. However, these books are too simple and lack the detailed discussion. Hence, they are not useful for advanced study even in physics. To resolve this issue, this book starts with the basic mathematics for quantum theory. Then, it introduces the basics of group representation and discusses the case of the finite groups, the symmetric group, e. g. Next, this book discusses Lie group and Lie algebra. This part starts with the basics knowledge, and proceeds to the special groups, e. g. , SU(2), SU(1,1), and SU(d). After the special groups, it explains concrete applications to physical systems, e. g. , angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, and quark model. Then, it proceeds to the general theory for Lie group and Lie algebra. Using this knowledge, this book explains the Bosonic system, which has the symmetries of Heisenberg group and the squeezing symmetry by SL(2,R) and Sp(2n,R). Finally, as the discrete version, this book treats the discrete Heisenberg representation which is related to quantum error correction. To enhance readers' undersnding, this book contains 54 figures, 23 tables, and 111 exercises with solutions.