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. ; Self-contained introduction to the theory of quantum graphs First time treatment of inverse problems in detail Numerous examples from physics included Open questions at the end of several chapters
This monograph introduces an investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson's type before ...turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations.
Vector calculus is an essential mathematical tool for performing mathematical analysis of physical and natural phenomena. It is employed in advanced applications in the field of engineering and ...computer simulations. This textbook covers the fundamental requirements of vector calculus in curricula for college students in mathematics and engineering programs. Chapters start from the basics of vector algebra, real valued functions, different forms of integrals, geometric algebra and the various theorems relevant to vector calculus and differential forms. Readers will find a concise and clear study of vector calculus, along with several examples, exercises, and a case study in each chapter. The solutions to the exercises are also included at the end of the book. This is an ideal book for students with a basic background in mathematics who wish to learn about advanced calculus as part of their college curriculum and equip themselves with the knowledge to apply theoretical concepts in practical situations.
Analysis Zhang, Weigang; Press, Tsinghua University
12/2017
eBook
The book begins by introducing signals and systems, and then discusses Time-Domain analysis and Frequency-Domain analysis for Continuous-Time systems. It also covers Z-transform, state-space analysis ...and system synthesis. The author provides abundant examples and exercises to facilitate learning, preparing students for subsequent courses on circuit analysis and communication theory.
Analysis Zhang, Weigang; Press, Tsinghua University
12/2017
eBook
The book begins by introducing signals and systems, and then discusses Time-Domain analysis and Frequency-Domain analysis for Continuous-Time systems. It also covers Z-transform, state-space analysis ...and system synthesis. The author provides abundant examples and exercises to facilitate learning, preparing students for subsequent courses on circuit analysis and communication theory.
This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic ...treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.
In this paper, we give a sequent calculus GRW+∘ for the positive contraction‐less relevant logic RW+∘ and we give a proof that it is cut‐free without the use of the truth constant t. Based on GRW+∘, ...we re‐prove the decidability of the logic RW+∘.
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called Ψ‐fractional calculus. The operational ...calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.
This book presents a largely self-contained account of the main results of convex analysis, monotone operator theory, and the theory of nonexpansive operators in the context of Hilbert spaces. Unlike ...existing literature, the novelty of this book, and indeed its central theme, is the tight interplay among the key notions of convexity, monotonicity, and nonexpansiveness. The presentation is accessible to a broad audience and attempts to reach out in particular to the applied sciences and engineering communities, where these tools have become indispensable. Graduate students and researchers in pure and applied mathematics will benefit from this book. It is also directed to researchers in engineering, decision sciences, economics, and inverse problems, and can serve as a reference book.
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