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In order to measure and quantify the complex behavior of realworld 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 nonuniform 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 engineeringoriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and antisynchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed.
multitime scale fractional stochastic differential equations  fractional Brownian motion  fractional stochastic partial differential equation  analytical solution  nonautonomous (autonomous) dynamical system  topological entropy  (asymptotical) focal entropy point  disturbation  mdimensional manifold  geometric nonlinearity  Bernoulli–Euler beam  colored noise  noise induced transitions  true chaos  Lyapunov exponents  wavelets  Lyapunov exponents  Wolf method  Rosenstein method  Kantz method  neural network method  method of synchronization  Benettin method  Fourier spectrum  Gauss wavelets  fractional calculus  Adomian decomposition  Mittag–Leffler function  descriptor fractional linear systems  regular pencils  Schur factorization  hyperchaotic system  selfsynchronous stream cipher  permutation entropy  image encryption  wavelet transform  product MValgebra  partition  Tsallis entropy  conditional Tsallis entropy  dynamical system  discrete chaos  discrete fractional calculus  hidden attractors  approximate entropy  stabilization  Information transfer  continuous flow  discrete mapping  Lorenz system  Chua’s system  deterministic chaos  random number generator  unbounded chaos  bounded chaos  phaselocked loop  Gaussian white noise  n/a
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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, subRiemannian manifolds, and Lie groups. In parallel, in geometric mechanics, JeanMarie Souriau interpreted the temperature vector of Planck as a spacetime vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties.
uncertainty relation  Wigner–Yanase–Dyson skew information  quantum memory  Born probability rule  quantumclassical relationship  spinors in quantum and classical physics  square integrable  energy quantization  Quantum HamiltonJacobi Formalism  quantum trajectory  generalized uncertainty principle  successive measurements  minimal observable length  Rényi entropy  Tsallis entropy  deep learning  quantum computing  neuromorphic computing  high performance computing  quantum mechanics  Gleason theorem  Kochen–Specker theorem  Born rule  quantum uncertainty  quantum foundations  quantum information  continuous variables  Bohmian dynamics  entanglement indicators  linear entropy  original Bell inequality  perfect correlation/anticorrelation  qudit states  quantum bound  measure of classicality  foundations of quantum mechanics  uncertainty relations  bell inequalities  entropy  quantum computing
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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 solidstate 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 multipleworld theories. These attempts have not yet been fully successful, but they provide hope for modifying quantum theory according to Einstein’s vision.
quantum nonlocality  quantum mechanics  Stern–Gerlach experiment  quantum measurement  pre and postselected systems  retrocausal channel  channel capacity  channel entropy  axioms for quantum theory  PR box  nonlocal correlations  classical limit  retrocausality  quantum correlations  quantum bounds  nonlocality  tsallis entropy  ion channels  selectivity filter  quantum mechanics  nonlinear Schrödinger model  biological quantum decoherence  nonlocality  parity measurements  entanglement  pigeonhole principle  controlledNOT  semiconductor nanodevices  quantum transport  densitymatrix formalism  Wignerfunction simulations  nonlocal dissipation models  steering  entropic uncertainty relation  general entropies  Bell’s theorem  Einstein–Podolsky–Rosen argument  local hidden variables  local realism  nosignalling  parallel lives  local polytope  quantum nonlocality  communication complexity  optimization  KS Box  PR Box  Noncontextuality inequality  discretevariable states  continuousvariable states  quantum teleportation of unknown qubit  hybrid entanglement  collapse of the quantum state  quantum nonlocality  communication complexity  quantum nonlocality  Bell test  deviceindependent  pvalue  hypothesis testing  nonsignaling  EPR steering  quantum correlation  nonlocality  entanglement  uncertainty relations  nonlocality  entanglement  quantum
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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 nonperturbative theory, such as statistical description. Furthermore, multiscale interactions are responsible for inevitably complex dynamics in strongly nonequilibrium systems, a proper understanding of which remains a main challenge in classical physics. As a remarkable consequence of multiscale interaction, a quasiequilibrium state (the socalled selforganisation) 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, selforganisation, 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 highspeed train, solar wind and industrial application.
pipe flow boundary layer  turbulent transition  large eddy simulation  channel flow  kinetic theory  fluid dynamics  turbulence  selforganisation  shear flows  coherent structures  turbulence  stochastic processes  Langevin equation  FokkerPlanck equation  information length  trailingedge flap  control strategy  floating wind turbine  turbulence  free vortex wake  nonlocal theory  Lévy noise  Tsallis entropy  fractional Fokker–Plank equation  anomalous diffusion  hybrid (U)RANSLES  IDDES methodology  attached and separated flows  complex dynamics  microcavity laser  spatiotemporal chaos  turbulent boundary layer  low speed streaks  magnetic confinement fusion  turbulence  heat transport  Tjunction  denoise  coherent structure  continuous wavelet transform  solar wind  scaling properties  fractals  chaos  turbulence  intermittency  multifractal  thermodynamics  drop breakage  drop coalescence  local intermittency  turbulent flow  population balance equation  high efficiency impeller  Rushton turbine  energy cascade  bifurcations  Lyapunov theory  turbulence  statistical mechanics  intermittency  coherent structure  multiscale problem  selforganisation  bifurcation  nonlocality  scaling  multifractal
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Entropy theory has wide applications to a range of problems in the fields of environmental and water engineering, including river hydraulic geometry, fluvial hydraulics, water monitoring network design, river flow forecasting, floods and droughts, river network analysis, infiltration, soil moisture, sediment transport, surface water and groundwater quality modeling, ecosystems modeling, water distribution networks, environmental and water resources management, and parameter estimation. Such applications have used several different entropy formulations, such as Shannon, Tsallis, Reacutenyi Burg, Kolmogorov, Kapur, configurational, and relative entropies, which can be derived in time, space, or frequency domains. More recently, entropybased concepts have been coupled with other theories, including copula and wavelets, to study various issues associated with environmental and water resources systems. Recent studies indicate the enormous scope and potential of entropy theory in advancing research in the fields of environmental and water engineering, including establishing and explaining physical connections between theory and reality. The objective of this Special Issue is to provide a platform for compiling important recent and current research on the applications of entropy theory in environmental and water engineering. The contributions to this Special Issue have addressed many aspects associated with entropy theory applications and have shown the enormous scope and potential of entropy theory in advancing research in the fields of environmental and water engineering.
complexity  streamflow  water level  composite multiscale sample entropy  trend  Poyang Lake basin  fourparameter exponential gamma distribution  principle of maximum entropy  precipitation frequency analysis  methods of moments  maximum likelihood estimation  flood frequency analysis  generalized gamma (GG) distribution  principle of maximum entropy (POME)  entropy theory  principle of maximum entropy (POME)  GB2 distribution  flood frequency analysis  nonpoint source pollution  ANN  entropy weighting method  datascarce  multievents  spatiotemporal variability  soil water content  entropy  arid region  joint entropy  NDVI  temperature  precipitation  groundwater depth  Hei River basin  turbulent flow  canopy flow  randomness  coherent structures  Shannon entropy  Kolmogorov complexity  entropy  information transfer  optimization  radar  rainfall network  water resource carrying capacity  forewarning model  entropy of information  fuzzy analytic hierarchy process  projection pursuit  accelerating genetic algorithm  entropy production  conditional entropy production  stochastic processes  scaling  climacogram  turbulence  water resources vulnerability  connection entropy  changing environment  set pair analysis  Anhui Province  crossentropy minimization  land suitability evaluation  spatial optimization  monthly streamflow forecasting  Burg entropy  configurational entropy  entropy spectral analysis time series analysis  entropy  water monitoring  network design  hydrometric network  information theory  entropy applications  hydrological risk analysis  maximum entropycopula method  uncertainty  Loess Plateau  entropy  water engineering  Tsallis entropy  principle of maximum entropy  Lagrangian function  probability distribution function  flux concentration relation  uncertainty  information  informational entropy  variation of information  continuous probability distribution functions  confidence intervals  precipitation  variability  marginal entropy  crop yield  Hexi corridor  flow duration curve  Shannon entropy  entropy parameter  modeling  spatial and dynamics characteristic  hydrology  tropical rainfall  statistical scaling  Tsallis entropy  multiplicative cascades  BetaLognormal model  rainfall forecast  cross entropy  ant colony fuzzy clustering  combined forecast  information entropy  mutual information  kernel density estimation  ENSO  nonlinear relation  scaling laws  power laws  water distribution networks  robustness  flow entropy  entropy theory  frequency analysis  hydrometeorological extremes  Bayesian technique  rainfall  entropy ensemble filter  ensemble model simulation criterion  EEF method  bootstrap aggregating  bagging  bootstrap neural networks  El Niño  ENSO  neural network forecast  sea surface temperature  tropical Pacific  entropy  cross elasticity  mean annual runoff  water resources  resilience  quaternary catchment  complement  substitute  entropy theory  complex systems  hydraulics  hydrology  water engineering  environmental engineering
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Entropies and entropylike quantities play an increasing role in modern nonlinear 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, entropybased coupling, and similarity analysis, along with the utilization of entropy measures as features in automatic learning and classification. Various real data applications are given.
experiment of design  empirical mode decomposition  signal analysis  similarity indices  synchronization analysis  auditory attention  entropy measure  linear discriminant analysis (LDA)  support vector machine (SVM)  auditory attention classifier  electroencephalography (EEG)  vague entropy  distance induced vague entropy  distance  complex fuzzy set  complex vague soft set  entropy, entropy visualization  entropy balance equation  Shannontype relations  multivariate analysis  machine learning evaluation  data transformation  sample entropy  treadmill walking  center of pressure displacement  dualtasking  analog circuit  fault diagnosis  cross wavelet transform  Tsallis entropy  parametric tdistributed stochastic neighbor embedding  support vector machine  information transfer  Chinese stock sectors  effective transfer entropy  market crash  system coupling  crossvisibility graphs  image entropy  geodesic distance  DempsterShafer evidence theory  uncertainty of basic probability assignment  belief entropy  plausibility transformation  weighted Hartley entropy  Shannon entropy  learning  information  novelty detection  nonprobabilistic entropy  learning systems  permutation entropy  embedded dimension  short time records  signal classification  relevance analysis  global optimization  metaheuristic  firefly algorithm  crossentropy method  coevolution  symbolic analysis  ordinal patterns  Permutation entropy  conditional entropy of ordinal patterns  KolmogorovSinai entropy  algorithmic complexity  information entropy  particle size distribution  selfsimilar measure  simulation  data analysis  entropy  entropy measures  automatic learning
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