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In the last two decades, the understanding of complex dynamical systems underwent important conceptual shifts. The catalyst was the infusion of new ideas from the theory of critical phenomena (scaling laws, renormalization group, etc.), (multi)fractals and trees, random matrix theory, network theory, and non-Shannonian information theory. The usual Boltzmann–Gibbs statistics were proven to be grossly inadequate in this context. While successful in describing stationary systems characterized by ergodicity or metric transitivity, Boltzmann–Gibbs statistics fail to reproduce the complex statistical behavior of many real-world systems in biology, astrophysics, geology, and the economic and social sciences.The aim of this Special Issue was to extend the state of the art by original contributions that could contribute to an ongoing discussion on the statistical foundations of entropy, with a particular emphasis on non-conventional entropies that go significantly beyond Boltzmann, Gibbs, and Shannon paradigms. The accepted contributions addressed various aspects including information theoretic, thermodynamic and quantum aspects of complex systems and found several important applications of generalized entropies in various systems.
Research & information: general --- Mathematics & science --- ecological inference --- generalized cross entropy --- distributional weighted regression --- matrix adjustment --- entropy --- critical phenomena --- renormalization --- multiscale thermodynamics --- GENERIC --- non-Newtonian calculus --- non-Diophantine arithmetic --- Kolmogorov-Nagumo averages --- escort probabilities --- generalized entropies --- maximum entropy principle --- MaxEnt distribution --- calibration invariance --- Lagrange multipliers --- generalized Bilal distribution --- adaptive Type-II progressive hybrid censoring scheme --- maximum likelihood estimation --- Bayesian estimation --- Lindley's approximation --- confidence interval --- Markov chain Monte Carlo method --- Rényi entropy --- Tsallis entropy --- entropic uncertainty relations --- quantum metrology --- non-equilibrium thermodynamics --- variational entropy --- rényi entropy --- tsallis entropy --- landsberg-vedral entropy --- gaussian entropy --- sharma-mittal entropy --- α-mutual information --- α-channel capacity --- maximum entropy --- Bayesian inference --- updating probabilities --- ecological inference --- generalized cross entropy --- distributional weighted regression --- matrix adjustment --- entropy --- critical phenomena --- renormalization --- multiscale thermodynamics --- GENERIC --- non-Newtonian calculus --- non-Diophantine arithmetic --- Kolmogorov-Nagumo averages --- escort probabilities --- generalized entropies --- maximum entropy principle --- MaxEnt distribution --- calibration invariance --- Lagrange multipliers --- generalized Bilal distribution --- adaptive Type-II progressive hybrid censoring scheme --- maximum likelihood estimation --- Bayesian estimation --- Lindley's approximation --- confidence interval --- Markov chain Monte Carlo method --- Rényi entropy --- Tsallis entropy --- entropic uncertainty relations --- quantum metrology --- non-equilibrium thermodynamics --- variational entropy --- rényi entropy --- tsallis entropy --- landsberg-vedral entropy --- gaussian entropy --- sharma-mittal entropy --- α-mutual information --- α-channel capacity --- maximum entropy --- Bayesian inference --- updating probabilities
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In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed.
n/a --- nonautonomous (autonomous) dynamical system --- stabilization --- multi-time scale fractional stochastic differential equations --- conditional Tsallis entropy --- wavelet transform --- hyperchaotic system --- Chua’s system --- permutation entropy --- neural network method --- Information transfer --- self-synchronous stream cipher --- colored noise --- Benettin method --- method of synchronization --- topological entropy --- geometric nonlinearity --- Kantz method --- dynamical system --- Gaussian white noise --- phase-locked loop --- wavelets --- Rosenstein method --- m-dimensional manifold --- deterministic chaos --- disturbation --- Mittag–Leffler function --- approximate entropy --- bounded chaos --- Adomian decomposition --- fractional calculus --- product MV-algebra --- Tsallis entropy --- descriptor fractional linear systems --- analytical solution --- fractional Brownian motion --- true chaos --- discrete mapping --- partition --- unbounded chaos --- fractional stochastic partial differential equation --- noise induced transitions --- random number generator --- Fourier spectrum --- hidden attractors --- (asymptotical) focal entropy point --- regular pencils --- continuous flow --- Bernoulli–Euler beam --- image encryption --- Gauss wavelets --- Lyapunov exponents --- discrete fractional calculus --- Lorenz system --- Schur factorization --- discrete chaos --- Wolf method
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In the last two decades, the understanding of complex dynamical systems underwent important conceptual shifts. The catalyst was the infusion of new ideas from the theory of critical phenomena (scaling laws, renormalization group, etc.), (multi)fractals and trees, random matrix theory, network theory, and non-Shannonian information theory. The usual Boltzmann–Gibbs statistics were proven to be grossly inadequate in this context. While successful in describing stationary systems characterized by ergodicity or metric transitivity, Boltzmann–Gibbs statistics fail to reproduce the complex statistical behavior of many real-world systems in biology, astrophysics, geology, and the economic and social sciences.The aim of this Special Issue was to extend the state of the art by original contributions that could contribute to an ongoing discussion on the statistical foundations of entropy, with a particular emphasis on non-conventional entropies that go significantly beyond Boltzmann, Gibbs, and Shannon paradigms. The accepted contributions addressed various aspects including information theoretic, thermodynamic and quantum aspects of complex systems and found several important applications of generalized entropies in various systems.
ecological inference --- generalized cross entropy --- distributional weighted regression --- matrix adjustment --- entropy --- critical phenomena --- renormalization --- multiscale thermodynamics --- GENERIC --- non-Newtonian calculus --- non-Diophantine arithmetic --- Kolmogorov–Nagumo averages --- escort probabilities --- generalized entropies --- maximum entropy principle --- MaxEnt distribution --- calibration invariance --- Lagrange multipliers --- generalized Bilal distribution --- adaptive Type-II progressive hybrid censoring scheme --- maximum likelihood estimation --- Bayesian estimation --- Lindley’s approximation --- confidence interval --- Markov chain Monte Carlo method --- Rényi entropy --- Tsallis entropy --- entropic uncertainty relations --- quantum metrology --- non-equilibrium thermodynamics --- variational entropy --- rényi entropy --- tsallis entropy --- landsberg—vedral entropy --- gaussian entropy --- sharma—mittal entropy --- α-mutual information --- α-channel capacity --- maximum entropy --- Bayesian inference --- updating probabilities --- n/a --- Kolmogorov-Nagumo averages --- Lindley's approximation --- Rényi entropy --- rényi entropy --- landsberg-vedral entropy --- sharma-mittal entropy
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The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties.
entanglement indicators --- generalized uncertainty principle --- Tsallis entropy --- linear entropy --- quantum-classical relationship --- Wigner–Yanase–Dyson skew information --- deep learning --- spinors in quantum and classical physics --- quantum mechanics --- entropy --- original Bell inequality --- Bohmian dynamics --- qudit states --- square integrable --- uncertainty relation --- quantum bound --- uncertainty relations --- foundations of quantum mechanics --- Born probability rule --- Rényi entropy --- energy quantization --- quantum foundations --- Born rule --- measure of classicality --- minimal observable length --- quantum information --- Kochen–Specker theorem --- neuromorphic computing --- bell inequalities --- successive measurements --- Gleason theorem --- continuous variables --- quantum memory --- perfect correlation/anticorrelation --- quantum trajectory --- quantum computing --- high performance computing --- Quantum Hamilton-Jacobi Formalism --- quantum uncertainty
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The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties.
entanglement indicators --- generalized uncertainty principle --- Tsallis entropy --- linear entropy --- quantum-classical relationship --- Wigner–Yanase–Dyson skew information --- deep learning --- spinors in quantum and classical physics --- quantum mechanics --- entropy --- original Bell inequality --- Bohmian dynamics --- qudit states --- square integrable --- uncertainty relation --- quantum bound --- uncertainty relations --- foundations of quantum mechanics --- Born probability rule --- Rényi entropy --- energy quantization --- quantum foundations --- Born rule --- measure of classicality --- minimal observable length --- quantum information --- Kochen–Specker theorem --- neuromorphic computing --- bell inequalities --- successive measurements --- Gleason theorem --- continuous variables --- quantum memory --- perfect correlation/anticorrelation --- quantum trajectory --- quantum computing --- high performance computing --- Quantum Hamilton-Jacobi Formalism --- quantum uncertainty
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The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties.
entanglement indicators --- generalized uncertainty principle --- Tsallis entropy --- linear entropy --- quantum-classical relationship --- Wigner–Yanase–Dyson skew information --- deep learning --- spinors in quantum and classical physics --- quantum mechanics --- entropy --- original Bell inequality --- Bohmian dynamics --- qudit states --- square integrable --- uncertainty relation --- quantum bound --- uncertainty relations --- foundations of quantum mechanics --- Born probability rule --- Rényi entropy --- energy quantization --- quantum foundations --- Born rule --- measure of classicality --- minimal observable length --- quantum information --- Kochen–Specker theorem --- neuromorphic computing --- bell inequalities --- successive measurements --- Gleason theorem --- continuous variables --- quantum memory --- perfect correlation/anticorrelation --- quantum trajectory --- quantum computing --- high performance computing --- Quantum Hamilton-Jacobi Formalism --- quantum uncertainty --- entanglement indicators --- generalized uncertainty principle --- Tsallis entropy --- linear entropy --- quantum-classical relationship --- Wigner–Yanase–Dyson skew information --- deep learning --- spinors in quantum and classical physics --- quantum mechanics --- entropy --- original Bell inequality --- Bohmian dynamics --- qudit states --- square integrable --- uncertainty relation --- quantum bound --- uncertainty relations --- foundations of quantum mechanics --- Born probability rule --- Rényi entropy --- energy quantization --- quantum foundations --- Born rule --- measure of classicality --- minimal observable length --- quantum information --- Kochen–Specker theorem --- neuromorphic computing --- bell inequalities --- successive measurements --- Gleason theorem --- continuous variables --- quantum memory --- perfect correlation/anticorrelation --- quantum trajectory --- quantum computing --- high performance computing --- Quantum Hamilton-Jacobi Formalism --- quantum uncertainty
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This book presents the current views of leading physicists on the bizarre property of quantum theory: nonlocality. Einstein viewed this theory as “spooky action at a distance” which, together with randomness, resulted in him being unable to accept quantum theory. The contributions in the book describe, in detail, the bizarre aspects of nonlocality, such as Einstein–Podolsky–Rosen steering and quantum teleportation—a phenomenon which cannot be explained in the framework of classical physics, due its foundations in quantum entanglement. The contributions describe the role of nonlocality in the rapidly developing field of quantum information. Nonlocal quantum effects in various systems, from solid-state quantum devices to organic molecules in proteins, are discussed. The most surprising papers in this book challenge the concept of the nonlocality of Nature, and look for possible modifications, extensions, and new formulations—from retrocausality to novel types of multiple-world theories. These attempts have not yet been fully successful, but they provide hope for modifying quantum theory according to Einstein’s vision.
Stern–Gerlach experiment --- channel entropy --- non-locality --- nonsignaling --- retro-causal channel --- communication complexity --- controlled-NOT --- Bell test --- quantum measurement --- quantum mechanics --- quantum transport --- semiconductor nanodevices --- optimization --- quantum correlation --- PR Box --- non-linear Schrödinger model --- retrocausality --- entanglement --- device-independent --- Einstein–Podolsky–Rosen argument --- quantum nonlocality --- parallel lives --- PR box --- nonlocal correlations --- hypothesis testing --- quantum bounds --- channel capacity --- Wigner-function simulations --- quantum correlations --- quantum --- pre- and post-selected systems --- local hidden variables --- density-matrix formalism --- collapse of the quantum state --- local polytope --- quantum teleportation of unknown qubit --- parity measurements --- uncertainty relations --- nonlocality --- hybrid entanglement --- selectivity filter --- p-value --- steering --- axioms for quantum theory --- no-signalling --- ion channels --- KS Box --- EPR steering --- local realism --- Non-contextuality inequality --- entropic uncertainty relation --- continuous-variable states --- nonlocal dissipation models --- Bell’s theorem --- tsallis entropy --- classical limit --- general entropies --- pigeonhole principle --- biological quantum decoherence --- discrete-variable states
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There is overwhelming evidence, from laboratory experiments, observations, and computational studies, that coherent structures can cause intermittent transport, dramatically enhancing transport. A proper description of this intermittent phenomenon, however, is extremely difficult, requiring a new non-perturbative theory, such as statistical description. Furthermore, multi-scale interactions are responsible for inevitably complex dynamics in strongly non-equilibrium systems, a proper understanding of which remains a main challenge in classical physics. As a remarkable consequence of multi-scale interaction, a quasi-equilibrium state (the so-called self-organisation) can however be maintained. This special issue aims to present different theories of statistical mechanics to understand this challenging multiscale problem in turbulence. The 14 contributions to this Special issue focus on the various aspects of intermittency, coherent structures, self-organisation, bifurcation and nonlocality. Given the ubiquity of turbulence, the contributions cover a broad range of systems covering laboratory fluids (channel flow, the Von Kármán flow), plasmas (magnetic fusion), laser cavity, wind turbine, air flow around a high-speed train, solar wind and industrial application.
non-locality --- hybrid (U)RANS-LES --- channel flow --- thermodynamics --- Lévy noise --- non-local theory --- low speed streaks --- drop breakage --- pipe flow boundary layer --- bifurcation --- Langevin equation --- attached and separated flows --- anomalous diffusion --- kinetic theory --- stochastic processes --- self-organisation --- spatiotemporal chaos --- chaos --- bifurcations --- turbulent flow --- Lyapunov theory --- Rushton turbine --- turbulence --- intermittency --- information length --- denoise --- microcavity laser --- free vortex wake --- IDDES methodology --- local intermittency --- control strategy --- population balance equation --- Tsallis entropy --- coherent structures --- Fokker-Planck equation --- energy cascade --- fluid dynamics --- high efficiency impeller --- fractals --- large eddy simulation --- shear flows --- heat transport --- multifractal --- drop coalescence --- continuous wavelet transform --- T-junction --- scaling properties --- floating wind turbine --- scaling --- fractional Fokker–Plank equation --- magnetic confinement fusion --- multi-scale problem --- coherent structure --- solar wind --- trailing-edge flap --- turbulent transition --- turbulent boundary layer --- complex dynamics --- statistical mechanics
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This Special Issue brings together original research papers, in all areas of mathematics, that are concerned with inequalities or the role of inequalities. The research results presented in this Special Issue are related to improvements in classical inequalities, highlighting their applications and promoting an exchange of ideas between mathematicians from many parts of the world dedicated to the theory of inequalities. This volume will be of interest to mathematicians specializing in inequality theory and beyond. Many of the studies presented here can be very useful in demonstrating new results. It is our great pleasure to publish this book. All contents were peer-reviewed by multiple referees and published as papers in our Special Issue in the journal Symmetry. These studies give new and interesting results in mathematical inequalities enabling readers to obtain the latest developments in the fields of mathematical inequalities. Finally, we would like to thank all the authors who have published their valuable work in this Special Issue. We would also like to thank the editors of the journal Symmetry for their help in making this volume, especially Mrs. Teresa Yu.
Research & information: general --- Geography --- Ostrowski inequality --- Hölder's inequality --- power mean integral inequality --- n-polynomial exponentially s-convex function --- weight coefficient --- Euler-Maclaurin summation formula --- Abel's partial summation formula --- half-discrete Hilbert-type inequality --- upper limit function --- Hermite-Hadamard inequality --- (p, q)-calculus --- convex functions --- trapezoid-type inequality --- fractional integrals --- functions of bounded variations --- (p,q)-integral --- post quantum calculus --- convex function --- a priori bounds --- 2D primitive equations --- continuous dependence --- heat source --- Jensen functional --- A-G-H inequalities --- global bounds --- power means --- Simpson-type inequalities --- thermoelastic plate --- Phragmén-Lindelöf alternative --- Saint-Venant principle --- biharmonic equation --- symmetric function --- Schur-convexity --- inequality --- special means --- Shannon entropy --- Tsallis entropy --- Fermi-Dirac entropy --- Bose-Einstein entropy --- arithmetic mean --- geometric mean --- Young's inequality --- Simpson's inequalities --- post-quantum calculus --- spatial decay estimates --- Brinkman equations --- midpoint and trapezoidal inequality --- Simpson's inequality --- harmonically convex functions --- Simpson inequality --- (n,m)-generalized convexity --- Ostrowski inequality --- Hölder's inequality --- power mean integral inequality --- n-polynomial exponentially s-convex function --- weight coefficient --- Euler-Maclaurin summation formula --- Abel's partial summation formula --- half-discrete Hilbert-type inequality --- upper limit function --- Hermite-Hadamard inequality --- (p, q)-calculus --- convex functions --- trapezoid-type inequality --- fractional integrals --- functions of bounded variations --- (p,q)-integral --- post quantum calculus --- convex function --- a priori bounds --- 2D primitive equations --- continuous dependence --- heat source --- Jensen functional --- A-G-H inequalities --- global bounds --- power means --- Simpson-type inequalities --- thermoelastic plate --- Phragmén-Lindelöf alternative --- Saint-Venant principle --- biharmonic equation --- symmetric function --- Schur-convexity --- inequality --- special means --- Shannon entropy --- Tsallis entropy --- Fermi-Dirac entropy --- Bose-Einstein entropy --- arithmetic mean --- geometric mean --- Young's inequality --- Simpson's inequalities --- post-quantum calculus --- spatial decay estimates --- Brinkman equations --- midpoint and trapezoidal inequality --- Simpson's inequality --- harmonically convex functions --- Simpson inequality --- (n,m)-generalized convexity
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Entropies and entropy-like quantities play an increasing role in modern non-linear data analysis. Fields that benefit from this application range from biosignal analysis to econophysics and engineering. This issue is a collection of papers touching on different aspects of entropy measures in data analysis, as well as theoretical and computational analyses. The relevant topics include the difficulty to achieve adequate application of entropy measures and the acceptable parameter choices for those entropy measures, entropy-based coupling, and similarity analysis, along with the utilization of entropy measures as features in automatic learning and classification. Various real data applications are given.
fault diagnosis --- empirical mode decomposition --- auditory attention --- Dempster-Shafer evidence theory --- simulation --- uncertainty of basic probability assignment --- center of pressure displacement --- particle size distribution --- multivariate analysis --- symbolic analysis --- permutation entropy --- short time records --- co-evolution --- plausibility transformation --- experiment of design --- cross-entropy method --- weighted Hartley entropy --- firefly algorithm --- embedded dimension --- entropy measure --- effective transfer entropy --- treadmill walking --- ordinal patterns --- complex fuzzy set --- entropy visualization --- belief entropy --- signal classification --- machine learning evaluation --- novelty detection --- selfsimilar measure --- Permutation entropy --- automatic learning --- cross wavelet transform --- cross-visibility graphs --- Kolmogorov-Sinai entropy --- distance --- Shannon-type relations --- Tsallis entropy --- market crash --- support vector machine (SVM) --- conditional entropy of ordinal patterns --- sample entropy --- learning --- electroencephalography (EEG) --- meta-heuristic --- entropy --- data transformation --- information entropy --- signal analysis --- synchronization analysis --- similarity indices --- data analysis --- geodesic distance --- auditory attention classifier --- entropy measures --- distance induced vague entropy --- analog circuit --- vague entropy --- complex vague soft set --- entropy balance equation --- parametric t-distributed stochastic neighbor embedding --- global optimization --- learning systems --- image entropy --- algorithmic complexity --- support vector machine --- system coupling --- relevance analysis --- Chinese stock sectors --- Shannon entropy --- linear discriminant analysis (LDA) --- information --- information transfer --- dual-tasking --- non-probabilistic entropy
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