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
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.
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.
We consider the Hermite‐Hadamard inequality and related results on integral inequalities, in the context of fractional integrals and derivatives defined using Mittag‐Leffler kernels, specifically the ...Atangana‐Baleanu and Prabhakar models of fractional calculus.
Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the ...literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann–Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.
Abstract
We introduce a path-based cyclic proof system for first-order $\mu $-calculus, the extension of first-order logic by second-order quantifiers for least and greatest fixed points of definable ...monotone functions. We prove soundness of the system and demonstrate it to be as expressive as the known trace-based cyclic systems of Dam and Sprenger. Furthermore, we establish cut-free completeness of our system for the fragment corresponding to the modal $\mu $-calculus.
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