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This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems.

quantum graphs --- ground states --- open sets converging to metric graphs --- norm convergence of operators --- NLD --- scaling limit --- standing waves --- bound states --- networks --- localized nonlinearity --- nonlinear Schrödinger equation --- metric graphs --- convergence of spectra --- sine-Gordon equation --- NLS --- star graph --- point interactions --- Laplacians --- nonrelativistic limit --- nonlinear wave equations --- quantum graph --- soliton --- nonlinear shallow water equations --- Kre?n formula --- breather --- non-linear Schrödinger equation --- Schrödinger equation --- nodal structure

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The thematic range of this book is wide and can loosely be described as polydispersive. Figuratively, it resembles a polynuclear path of yielding (poly)crystals. Such path can be taken when looking at it from the first side. However, a closer inspection of the book’s contents gives rise to a much more monodispersive/single-crystal and compacted (than crudely expected) picture of the book’s contents presented to a potential reader. Namely, all contributions collected can be united under the common denominator of maximum-entropy and entropy production principles experienced by both classical and quantum systems in (non)equilibrium conditions. The proposed order of presenting the material commences with properly subordinated classical systems (seven contributions) and ends up with three remaining quantum systems, presented by the chapters’ authors. The overarching editorial makes the presentation of the wide-range material self-contained and compact, irrespective of whether comprehending it from classical or quantum physical viewpoints.

Research & information: general --- Physics --- multistability --- ergodicity --- Brownian motion --- tilted periodic potential --- Lévy noise --- nonequilibrium thermodynamics --- active particles --- entropy production --- dissipative structures --- quantum entanglement --- linear entropy --- coherence --- purity of states --- concurrence --- three-qubit systems --- quantum graphs --- microwave networks --- Euler characteristic --- Neumann and Dirichlet boundary conditions --- II law of thermodynamics --- Carnot principle --- Kelvin principle --- Ostwald principle --- perpetuum mobile type III --- Clausius I and II principles --- formal implication --- model theory --- spherulites --- (poly)crystal formation --- complex growing phenomenon --- soft condensed matter --- physical kinetics --- anticoherence --- entanglement --- nonlinear systems --- human serum albumin --- hyaluronan --- conformational entropy --- dihedral angles --- frequency distribution --- epidemy --- compartmental models --- computer simulation --- SARS-CoV-2-like disease spreading --- chemical computing --- network --- oscillators --- top-down design --- Oregonator model --- Japanese flag problem --- n/a --- Lévy noise

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The inverse dynamics problem was developed in order to provide researchers with the state of the art in inverse problems for dynamic and vibrational systems. Contrasted with a forward problem, which solves for the system output in a straightforward manner, an inverse problem searches for the system input through a procedure contaminated with errors and uncertainties. An inverse problem, with a focus on structural dynamics, determines the changes made to the system and estimates the inputs, including forces and moments, to the system, utilizing measurements of structural vibration responses only. With its complex mathematical structure and need for more reliable input estimations, the inverse problem is still a fundamental subject of research among mathematicians and engineering scientists. This book contains 11 articles that touch upon various aspects of inverse dynamic problems.

regenerative shock absorbers --- energy harvesting --- active control of automobile suspension systems --- railroad tracks --- track modulus --- computer simulation --- artificial neural networks --- Fiber-reinforced Foamed Urethane (FFU) --- free vibration --- impact hammer excitation technique --- high-rate dynamics --- structural health monitoring --- time-frequency analysis --- synchrosqueezing transform (SST) --- jerk --- acceleration onset --- higher-order derivatives of acceleration --- jounce --- acceleration-dot --- sports surfacing --- sand surface --- dynamic behaviour --- impact tests --- accelerometry --- greyhound racing --- equine racing --- shake table control --- vibration testing --- system identification --- inverse dynamics --- feedback linearization --- servohydraulics --- inverse problems --- quantum graphs --- delta-prime vertex conditions --- Bayesian inference --- uncertainty quantification --- dynamical systems --- inverse problem --- machine learning --- Gaussian process --- polynomial chaos --- impact force identification --- tower structure --- impact localization --- force history --- inverse algorithm --- rotor dynamic --- bearing --- centrifugal pump --- impeller diameter --- Lagrangian equations --- n/a