Spectral Analysis On Graph Like Spaces
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Spectral Analysis On Graph Like Spaces
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Author : Olaf Post
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-01-06
Spectral Analysis On Graph Like Spaces written by Olaf Post and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-06 with Mathematics categories.
Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.
Spectral Analysis On Graph Like Spaces
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Author : Olaf Post
language : en
Publisher: Springer
Release Date : 2012-01-05
Spectral Analysis On Graph Like Spaces written by Olaf Post and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-05 with Mathematics categories.
Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.
Spectral Analysis Of Growing Graphs
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Author : Nobuaki Obata
language : en
Publisher: Springer
Release Date : 2017-02-17
Spectral Analysis Of Growing Graphs written by Nobuaki Obata and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-02-17 with Science categories.
This book is designed as a concise introduction to the recent achievements on spectral analysis of graphs or networks from the point of view of quantum (or non-commutative) probability theory. The main topics are spectral distributions of the adjacency matrices of finite or infinite graphs and their limit distributions for growing graphs. The main vehicle is quantum probability, an algebraic extension of the traditional probability theory, which provides a new framework for the analysis of adjacency matrices revealing their non-commutative nature. For example, the method of quantum decomposition makes it possible to study spectral distributions by means of interacting Fock spaces or equivalently by orthogonal polynomials. Various concepts of independence in quantum probability and corresponding central limit theorems are used for the asymptotic study of spectral distributions for product graphs.This book is written for researchers, teachers, and students interested in graph spectra, their (asymptotic) spectral distributions, and various ideas and methods on the basis of quantum probability. It is also useful for a quick introduction to quantum probability and for an analytic basis of orthogonal polynomials.
Spectral Analysis Of Growing Graphs
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Author : Nobuaki Obata
language : en
Publisher:
Release Date : 2017
Spectral Analysis Of Growing Graphs written by Nobuaki Obata and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Distribution (Probability theory) categories.
The Spectral Analysis Of Time Series
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Author : Lambert H. Koopmans
language : en
Publisher: Elsevier
Release Date : 1995-05-18
The Spectral Analysis Of Time Series written by Lambert H. Koopmans and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-05-18 with Mathematics categories.
To tailor time series models to a particular physical problem and to follow the working of various techniques for processing and analyzing data, one must understand the basic theory of spectral (frequency domain) analysis of time series. This classic book provides an introduction to the techniques and theories of spectral analysis of time series. In a discursive style, and with minimal dependence on mathematics, the book presents the geometric structure of spectral analysis. This approach makes possible useful, intuitive interpretations of important time series parameters and provides a unified framework for an otherwise scattered collection of seemingly isolated results. The books strength lies in its applicability to the needs of readers from many disciplines with varying backgrounds in mathematics. It provides a solid foundation in spectral analysis for fields that include statistics, signal process engineering, economics, geophysics, physics, and geology. Appendices provide details and proofs for those who are advanced in math. Theories are followed by examples and applications over a wide range of topics such as meteorology, seismology, and telecommunications. Topics covered include Hilbert spaces; univariate models for spectral analysis; multivariate spectral models; sampling, aliasing, and discrete-time models; real-time filtering; digital filters; linear filters; distribution theory; sampling properties ofspectral estimates; and linear prediction. Hilbert spaces univariate models for spectral analysis multivariate spectral models sampling, aliasing, and discrete-time models real-time filtering digital filters linear filters distribution theory sampling properties of spectral estimates linear prediction
A Guide To Spectral Theory
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Author : Christophe Cheverry
language : en
Publisher: Springer Nature
Release Date : 2021-05-06
A Guide To Spectral Theory written by Christophe Cheverry and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-05-06 with Mathematics categories.
This textbook provides a graduate-level introduction to the spectral theory of linear operators on Banach and Hilbert spaces, guiding readers through key components of spectral theory and its applications in quantum physics. Based on their extensive teaching experience, the authors present topics in a progressive manner so that each chapter builds on the ones preceding. Researchers and students alike will also appreciate the exploration of more advanced applications and research perspectives presented near the end of the book. Beginning with a brief introduction to the relationship between spectral theory and quantum physics, the authors go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators. Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm operators are introduced. The authors then develop the Grushin method to execute the spectral analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and Hilbert-Schmidt operators. The final chapter opens the discussion to several selected applications. Throughout this textbook, detailed proofs are given, and the statements are illustrated by a number of well-chosen examples. At the end, an appendix about foundational functional analysis theorems is provided to help the uninitiated reader. A Guide to Spectral Theory: Applications and Exercises is intended for graduate students taking an introductory course in spectral theory or operator theory. A background in linear functional analysis and partial differential equations is assumed; basic knowledge of bounded linear operators is useful but not required. PhD students and researchers will also find this volume to be of interest, particularly the research directions provided in later chapters.
Mathematical Technology Of Networks
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Author : Delio Mugnolo
language : en
Publisher: Springer
Release Date : 2015-07-06
Mathematical Technology Of Networks written by Delio Mugnolo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-07-06 with Mathematics categories.
Dynamical models on graphs or random graphs are increasingly used in applied sciences as mathematical tools to study complex systems whose exact structure is too complicated to be known in detail. Besides its importance in applied sciences, the field is increasingly attracting the interest of mathematicians and theoretical physicists also because of the fundamental phenomena (synchronization, phase transitions etc.) that can be studied in the relatively simple framework of dynamical models of random graphs. This volume was developed from the Mathematical Technology of Networks conference held in Bielefeld, Germany in December 2013. The conference was designed to bring together functional analysts, mathematical physicists, and experts in dynamical systems. The contributors to this volume explore the interplay between theoretical and applied aspects of discrete and continuous graphs. Their work helps to close the gap between different avenues of research on graphs, including metric graphs and ramified structures.
Spectral Geometry Of Graphs
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Author : Pavel Kurasov
language : en
Publisher: Springer Nature
Release Date : 2023-12-09
Spectral Geometry Of Graphs written by Pavel Kurasov and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-12-09 with Science categories.
This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph. The book has two central themes: the trace formula and inverse problems. The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book. To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions. The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.
Discrete And Continuous Models In The Theory Of Networks
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Author : Fatihcan M. Atay
language : en
Publisher: Springer Nature
Release Date : 2020-09-03
Discrete And Continuous Models In The Theory Of Networks written by Fatihcan M. Atay and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-09-03 with Mathematics categories.
This book contains contributions from the participants of the research group hosted by the ZiF - Center for Interdisciplinary Research at the University of Bielefeld during the period 2013-2017 as well as from the conclusive conference organized at Bielefeld in December 2017. The contributions consist of original research papers: they mirror the scientific developments fostered by this research program or the state-of-the-art results presented during the conclusive conference. The volume covers current research in the areas of operator theory and dynamical systems on networks and their applications, indicating possible future directions. The book will be interesting to researchers focusing on the mathematical theory of networks; it is unique as, for the first time, continuous network models - a subject that has been blooming in the last twenty years - are studied alongside more classical and discrete ones. Thus, instead of two different worlds often growing independently without much intercommunication, a new path is set, breaking with the tradition. The fruitful and beneficial exchange of ideas and results of both communities is reflected in this book.
Introduction To Quantum Graphs
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Author : Gregory Berkolaiko
language : en
Publisher: American Mathematical Soc.
Release Date : 2013
Introduction To Quantum Graphs written by Gregory Berkolaiko and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Mathematics categories.
A ``quantum graph'' is a graph considered as a one-dimensional complex and equipped with a differential operator (``Hamiltonian''). Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., ``meso-'' or ``nano-scale'') system that looks like a thin neighborhood of a graph. Works that currently would be classified as discussing quantum graphs have been appearing since at least the 1930s, and since then, quantum graphs techniques have been applied successfully in various areas of mathematical physics, mathematics in general and its applications. One can mention, for instance, dynamical systems theory, control theory, quantum chaos, Anderson localization, microelectronics, photonic crystals, physical chemistry, nano-sciences, superconductivity theory, etc. Quantum graphs present many non-trivial mathematical challenges, which makes them dear to a mathematician's heart. Work on quantum graphs has brought together tools and intuition coming from graph theory, combinatorics, mathematical physics, PDEs, and spectral theory. This book provides a comprehensive introduction to the topic, collecting the main notions and techniques. It also contains a survey of the current state of the quantum graph research and applications.