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Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010).The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of ½, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex ‘entropy-probability-dynamics’ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted.

special functions --- fractional calculus --- entropic functional --- mathematical physics --- applied analysis --- statistical distributions --- geometrical probabilities --- multivariate analysis

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This Special Issue is aimed at highlighting the potentialities of membrane and membrane reactor operations in various sectors of chemical engineering, based on application of the process intensification strategy. In all of the contributions, the principles of process intensification were pursued during the adoption of membrane technology, demonstrating how it may lead to the development of redesigned processes that are more compact and efficient while also being more environmental friendly, energy saving, and amenable to integration with other green processes. This Special Issue comprises a number of experimental and theoretical studies dealing with the application of membrane and membrane reactor technology in various scientific fields of chemical engineering, such as membrane distillation for wastewater treatment, hydrogen production from reforming reactions via inorganic membrane and membrane photoassisted reactors, membrane desalination, gas/liquid phase membrane separation of CO2, and membrane filtration for the recovery of antioxidants from agricultural byproducts, contributing to valorization of the potentialities of membrane operations.

membrane engineering --- hydrogen production --- CO2 conversion --- gas/liquid separation --- micro direct methanol fuel cell (µDMFC) --- porous membranes --- micro channel --- two-phase flow --- micro contactor --- separator --- water splitting --- Z-scheme --- photocatalysis --- photocatalytic membrane reactor --- hydrogen --- on-board --- steam reforming --- ethanol --- methane --- membrane reactor --- palladium --- modeling --- membrane distillation --- wastewater treatment --- membrane configuration --- fouling renewable heat sources --- membrane reactor --- Pd-based membrane --- hydrogen --- steam reforming --- solar energy --- microfiltration (MF) --- ultrafiltration (UF) --- orange press liquor --- clarification --- multivariate analysis --- advanced separations --- desalination --- hydrogel composite membranes --- ionic liquids membranes --- membrane distillation

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The Special Issue “Extractable and Non-Extractable Antioxidants” gives an updated view on antioxidants—both in their extractable and non-extractable form—in the different food groups, their products thereof, and food preparations as well as byproducts and biomass waste. The potential beneficial properties of these compounds and nutraceutical formulations are described in the various studies covered in this Special Issue.

anthocyanins --- polyphenolic compounds --- classic extraction --- ultrasound assisted extraction --- antioxidant capacity --- daidzein --- cyclodextrin --- inclusion complex --- antioxidant activity --- trans-cinnamaldehyde --- ?-cyclodextrin --- self-inclusion --- anti-inflammation --- antioxidant --- Pleurotus ostreatus --- antioxidant activity --- polyphenols --- digestibility --- fermentation --- cereals --- legumes --- Chinese mistletoes --- phenolics --- phenolic contents --- antioxidant activity --- forest residues --- phenolic compounds --- natural antioxidants --- quercitrin --- value-added by-products --- Camellia sinensis --- black teas --- catechins --- antioxidant activity --- LC-ESI-Q-TOF-MS --- principal component analysis --- extractable --- non-extractable --- antioxidant activity --- agglomerative hierarchical clustering --- principal component analysis --- multivariate analysis --- nuclear magnetic spectroscopy --- nitric oxide scavengers --- pre-column HPLC method --- Aloysia triphylla --- phenolics --- antioxidants --- dietary supplements --- botanicals --- bioactive compounds --- antioxidants --- study approach --- integrated food research --- dedicated databases --- dietary assessment. --- biorefinery --- circular economy --- grape seed --- bio-based --- chemometrics --- Cagnulari marc --- Naviglio Extractor® --- green extraction --- endothelial cell --- oxidative stress --- polyphenols --- berries jam --- phenolic acids --- flavonols glycosides --- degradation --- HPLC-ESI/MS --- antioxidant properties --- digestive enzyme --- nutrient --- polymerization --- proanthocyanidins --- eggplant --- flour --- phenolics --- antioxidant activity --- functional ingredient --- sour cherry --- anthocyanins --- extractable polyphenols --- non-extractable polyphenols --- Italian popular recipes --- food composition database --- antioxidant properties --- extractable compounds --- non-extractable compounds --- ferric reducing antioxidant power (FRAP) --- total polyphenol content (TPC) --- n/a

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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.

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 --- Shannon-type relations --- multivariate analysis --- machine learning evaluation --- data transformation --- sample entropy --- treadmill walking --- center of pressure displacement --- dual-tasking --- analog circuit --- fault diagnosis --- cross wavelet transform --- Tsallis entropy --- parametric t-distributed stochastic neighbor embedding --- support vector machine --- information transfer --- Chinese stock sectors --- effective transfer entropy --- market crash --- system coupling --- cross-visibility graphs --- image entropy --- geodesic distance --- Dempster-Shafer evidence theory --- uncertainty of basic probability assignment --- belief entropy --- plausibility transformation --- weighted Hartley entropy --- Shannon entropy --- learning --- information --- novelty detection --- non-probabilistic entropy --- learning systems --- permutation entropy --- embedded dimension --- short time records --- signal classification --- relevance analysis --- global optimization --- meta-heuristic --- firefly algorithm --- cross-entropy method --- co-evolution --- symbolic analysis --- ordinal patterns --- Permutation entropy --- conditional entropy of ordinal patterns --- Kolmogorov-Sinai entropy --- algorithmic complexity --- information entropy --- particle size distribution --- selfsimilar measure --- simulation --- data analysis --- entropy --- entropy measures --- automatic learning