Sum Of Entropy Research Articles

Mass Dependence of the Entropy Product and Sum

For black holes with multiple horizons, the area product of all horizons has been proven to be mass independent in many cases. Counterexamples were also found in some occasions. In this paper, we first prove a theorem derived from the first law of black hole thermodynamics and a mathematical lemma related to the Vandermonde determinant. With these arguments, we develop some general criteria for the mass independence of the entropy product as well as the entropy sum. In particular, if a $d$-dimensional spacetime is spherically symmetric and its radial metric function $f(r)$ is a Laurent series in $r$ with the lowest power $\ensuremath{-}m$ and the highest power $n$, we find the criterion is extremely simple: The entropy product is mass independent if and only if $m\ensuremath{\ge}d\ensuremath{-}2$ and $n\ensuremath{\ge}4\ensuremath{-}d$. The entropy sum is mass independent if and only if $m\ensuremath{\ge}d\ensuremath{-}2$ and $n\ensuremath{\ge}2$. Compared to previous works, our method does not require an exact expression of the metric. Our arguments turn out to be useful even for rotating black holes. By applying our theorem and lemma to a Myers--Perry black hole with spacetime dimension $d$, we show that the entropy product/sum is mass independent for all $d>4$, while it is mass dependent only for $d=4$, i.e., the Kerr solution.

Thermodynamic Relations for the Entropy and Temperature of Multi-Horizon Black Holes

We present some entropy and temperature relations of multi-horizons, even including the "virtual" horizon. These relations are related to product, division and sum of entropy and temperature of multi-horizons. We obtain the additional thermodynamic relations of both static and rotating black holes in three and four dimensional (A)dS spacetime. Especially, a new dimensionless, charges-independence and $T_+S_+=T_-S_-$ like relation is presented. This relation does not depend on the mass, electric charge, angular momentum and cosmological constant, as it is always a constant. These relations lead us to get some interesting thermodynamic bound of entropy and temperature, including the Penrose inequality which is the first geometrical inequality of black holes. Besides, based on these new relations, one can obtain the first law of thermodynamics and Smarr relation for all horizons of black hole.

A stable approach for model order selection in nonnegative matrix factorization

In order to find the correct model order in non-negative matrix factorization (NMF), an algorithm called automatic relevance determination (ARD) is proposed in Tan and Fevotte (2013). The algorithm explores the similarities of the NMF components and removes redundant ones iteratively. However, the algorithm can yield over-parsimonious representations where ground truth patterns can be grouped into one single component to cause superposition. In this paper, mixed entropy regularized NMF (MER-NMF) is proposed to overcome the above problem. In MER-NMF, the objective function of NMF is regularized by minimizing a mixed entropy of the coefficient matrix which is a weighted sum of two parts: the entropy of all the entries and the entropy of the row sums of the coefficient matrix. With the mixed entropy regularization, the algorithm tends to yield sharper activations of the components for each sample. By combining MER-NMF and ARD-NMF, correct number of components can always be selected according to our experiments.

Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime

We propose that holographic entanglement entropy can be calculated at arbitrary orders in the bulk Planck constant using the concept of a "quantum extremal surface": a surface which extremizes the generalized entropy, i.e. the sum of area and bulk entanglement entropy. At leading order in bulk quantum corrections, our proposal agrees with the formula of Faulkner, Lewkowycz, and Maldacena, which was derived only at this order; beyond leading order corrections, the two conjectures diverge. Quantum extremal surfaces lie outside the causal domain of influence of the boundary region as well as its complement, and in some spacetimes there are barriers preventing them from entering certain regions. We comment on the implications for bulk reconstruction.

Nonlinear analysis of random walks: a tool to analyze nucleic acid sequences

We have performed a comparison of computer-generated random sequences with tRNAs nucleotide sequences present in Bacteria represented as random walks. Nucleotide sequence data of tRNA genes were obtained from the Institute for Genomic Research (TIGR) and the GeneBank library. Random sequence data (white noise) were obtained from the algorithm by Press and Teukolsky. Random walks of nucleotide sequences were obtained by letting the orbit walk a unit step in one of four directions (down, left, right, and up), depending upon the next base (A,C,G, and T) in the sequence, and the distances from the origin calculated. The Visual Basic routines here applied to perform the analysis are presented. Relative Lempel-Ziv complexity. Entropy (sum of the positive Lyapunov indexes) and Hurst indexes of nucleotide sequences and of computer-generated random data were evaluated over the distances of their random walk. Our data show that the values of nonlinear parameters obtained from the bacteria are lower than the values of randomly generated sequences (p<0.01, p<0.05, p<0.01), meaning that the tRNA sequence is more ordered than a pure destructured random data and it owns a “memory”. The observed deviation from pure randomness should be arisen from some constraints like the secondary structure of this biologic macromolecule and/or from the peculiar origin of this macromolecule by repeated subunits. These data indicate that evolution earlier chose nonrandom “alphabets”: order together randomness were present at the dawn of life. Our method, here presented and described, provides an efficient tool to assess the amount of order/disorder in the primary structure of nucleic acid sequences.

Tri-level Thresholding using Invasive Weed Optimization based on Nonextensive Fuzzy Entropy

This study presents a tri-level thresholding method for image segmentation with invasive weed optimization (IWO) algorithm. The objective of the proposed approach is to handle the nonextensivity and vagueness of image in segmentation, in the meanwhile to reduce the computation time. In this study, the histogram of image is converted to fuzzy domain by membership function firstly. Then the thresholding method is constructed through maximizing the sum of nonextensive entropy of subsets of the each part of fuzzy histogram. The IWO algorithm is used to search the optimal thresholds to reduce the computation time in the new method. Experiments on synthetic and real-world images are given to demonstrate the effectiveness of the proposed approach compared with the other methods.

The entropy sum of (A)dS black holes in four and higher dimensions

We present the "entropy sum" relation of (A)dS charged black holes in higher-dimensional Einstein–Maxwell gravity, f(R) gravity, Gauss–Bonnet gravity and gauged supergravity. For their "entropy sum" with the necessary effect of the unphysical "virtual" horizon included, we conclude the general results that the cosmological constant dependence and Gauss–Bonnet coupling constant dependence do hold in both the four and six dimensions, while the "entropy sum" is always vanishing in odd dimensions. Furthermore, the "entropy sum" of all horizons is related to the geometry of the horizons in four and six dimensions. In these explicitly four cases, one also finds that the conserved charges M (the mass), Q (the charge from Maxwell field or supergravity) and the parameter a (the angular momentum) play no role in the "entropy sum" relations.

General entropy-like uncertainty relations in finite dimensions

We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicrú generalized -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet with cA (respectively cB) being the overlap between the elements of the POVM A (respectively B) and the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.

Moving Object Detection Based on Temporal Information

This letter presents an automatic moving object detection method in image sequences captured from videos. While we focus on extracting moving objects throughout a frame sequence, our approach does not require any prior knowledge such as the background modeling nor the interaction by users such as empirical thresholds tuning. Based on the continuous symmetric difference of the adjacent frames, we get the full resolution saliency map of the current frame, which highlights the moving objects with higher saliency values and meanwhile inhibits the saliency of the background. Then, the maximum entropy sum method is utilized to adaptively calculate the threshold to determine the candidate areas and get the reasonable attention seeds. After that, the ground truth is obtained based on the modified fuzzy growing of the attention seeds. The proposed algorithm is effective, robust and the experimental results demonstrate that it is promising for moving object detection.

The universal property of the entropy sum of black holes in all dimensions

It is proposed by Cvetic et al. [1] that the product of all horizon areas for general rotating multi-change black holes has universal expressions independent of the mass. When we consider the product of all horizon entropies, however, the mass will be present in some cases, while another new universal property [2] is preserved, which is more general and says that the sum of all horizon entropies depends only on the coupling constants of the theory and the topology of the black hole. The property has been studied in limited dimensions and the generalization in arbitrary dimensions is not straight-forward. In this Letter, we prove a useful formula, which makes it possible to investigate this conjectured universality in arbitrary dimensions for the maximally symmetric black holes in general Lovelock gravity and f(R) gravity. We also propose an approach to compute the entropy sum of general Kerr–(anti-)de-Sitter black holes in arbitrary dimensions. In all these cases, we prove that the entropy sum depends only on the coupling constants and the topology of the black hole.

Quantum uncertainties of the confined Harmonic Oscillator in position, momentum and phase‐space

Quantum uncertainties in position, momentum and phase‐space are studied in the confined Harmonic Oscillator. Standard deviations and Shannon entropies are used to quantify these uncertainties and their behaviors are compared and contrasted. We observe a minimum in the momentum space Shannon entropy as the box length is increased, a feature that is not present in the momentum space standard deviation. The behaviors of the standard deviation product and the Shannon entropy sum, which form the basis of uncertainty relationships, are also analyzed. Maxima are observed in the product as the box length increases in sharp contrast to the entropy sum. The relationship between these behaviors and that of the Shannon entropy of the phase‐space Wigner function is analyzed and discussed. An analysis of the energetic components is also performed. The results reinforce the idea that the confined Harmonic Oscillator can be considered as an intermediate model which interpolates between the Particle in a Box Model and the Harmonic Oscillator and thus contains characteristics of both models. image

A note on entropy relations of black hole horizons

Black holes with multi-horizons may provide new ways to understand the intrinsic thermodynamics. In this work, we focus on the entropy relations of black holes in three, four and higher dimensions. These entropy relations include entropy product, "part" entropy product and entropy sum. We also discuss their differences and similarities, in order to make a further study on understanding the origin of black hole entropy at the microscopic level.

Causal conditioning and instantaneous coupling in causality graphs

This study aims at providing the definitive links between Massey and Kramer’s directed information theory and Granger causality graphs, recently formalized by Eichler. This naturally leads to consider two concepts of causality that can occur in physical systems: dynamical causality and instantaneous coupling. Although it is well accepted that the former is assessed by conditional transfer entropy, the latter is often overlooked, even if it was clearly introduced and understood in seminal studies. In the bivariate case, we show that directed information is decomposed into the sum of transfer entropy and instantaneous information exchange. In the multivariate case, encountered for conditional graph modeling, such a decomposition does not hold anymore. We provide a new insight into the problem of instantaneous coupling and show how it affects the estimated structure of a graphical model that should provide a sparse representation of a complex system. This result is discussed and analyzed for the inference of causality graphs. Two synthetic examples are developed to illustrate the theoretical concepts. Practical issues are also briefly discussed on these examples and an extension of Leonenko’s k-nearest neighbors based entropy estimators is used to derive a nonparametric estimator of directed information.

The “universal property” of horizon entropy sum of black holes in four dimensional asymptotical (anti-)de-Sitter spacetime background

We present a new “universal property” of entropy, that is the “entropy sum” relation of black holes in four dimensional (anti-)de-Sitter asymptotical background. They depend only on the cosmological constant with the necessary effect of the un-physical “virtual” horizon included in the spacetime where only the cosmological constant, mass of black hole, rotation parameter and Maxwell field exist. When there is more extra matter field in the spacetime, one will find the “entropy sum” is also dependent of the strength of these extra matter field. For both cases, we conclude that the “entropy sum” does not depend on the conserved charges M , Q and J , while it does depend on the property of background spacetime. We will mainly test the “entropy sum” relation in static, stationary black hole and some black hole with extra matter source (scalar hair and higher curvature) in the asymptotical (anti-)de-sitter spacetime background. Besides, we point out a newly found counter example of the mass independence of the ”entropy product” relation in the spacetime with extra scalar hair case, while the “entropy sum” relation still holds. These result are indeed suggestive to some underlying microscopic mechanism. Moreover, the cosmological constant and extra matter field dependence of the “entropy sum” of all horizon seems to reveal that “entropy sum” is more general as it is only related to the background field. For the case of asymptotical flat spacetime without any matter source, we give a note for the Kerr black hole case in appendix. One will find only mass dependence of “entropy sum” appears. It makes us believe that, considering the dependence of “entropy sum”, the mass background field may be regarded as the next order of cosmological constant background field and extra matter field. However, fully explaining the relationship between the “entropy sum” relation and background properties still requires further exploration.

Invariance and Noises of Shannon Entropy for Information on Oxidative Activity of DNA in All Living Cells for Medical Diagnostics

We analyze oxidative activity of DNA due to fluorescence of chromosomes inside cells, using flow cytometry method with nanometer spatial resolution. Statistics of fluorescence is presented in histogram as frequency distributions of flashes in the dependence on their intensity and in distributions of Shannon entropy, which was defined on the base of normalized distribution of information in original histogram for frequency of flashes. We show that overall sum of entropy, i.e. total entropy E , for any histogram is invariant and has identical trends of changes all values of E(r) = lnr at reduction of histogram’ rank r. This invariance reflects informational homeostasis of chromosomes activity in multi-scale networks of entropy inside all cells in various samples of blood for DNA inside neutrophils, lymphocytes, inside all leukocytes of human and inside chicken erythrocytes for various dyes, colors and various excitations of fluorescence. Informational homeostasis of oxidative activity of 3D DNA in the full set of chromosomes inside living cells exists for any Shannon-Weaver index of biodiversity of cells, at any state of health different beings. Regulation perturbations in information activity DNA provides informational adaptability and vitality of cells at homeostasis support. Noises of entropy, during regulation of informational homeostasis, depend on the states of health in real time. The main structural reconstructions of chromosomal correlations, corresponding to self-regulation of homeostasis, occur in the most large-scale networks of entropy, for rank r<32. We show that stability of homeostasis is supported by activity of all 46 chromosomes inside cells. Patterns, hidden switching and branching in sequences of averages of H?lder and central moments for noises in regulation of homeostasis define new opportunities in diagnostics of health and immunity. All people and all aerobic beings have one overall homeostatic level for countdown of information activity of DNA inside cells. We noted very bad and dangerous properties of artificial cells with other levels of informational homeostasis for all aerobic beings in foods, medical treatment and in biotechnologies.

Optimal features selected by NSGA-II for partial discharge pulses separation based on time-frequency representation and matrix decomposition

This paper presents a feature extraction algorithm for partial discharge (PD) pulses separation using S transform (ST)-based time-frequency representation. Firstly, the algorithm acquires a series of base vectors in the frequency domain and location vectors in the time domain obtained by applying a non-negative matrix factorization (NMF)-based matrix decomposition technique to compress ST amplitude (STA) matrices of PD pulses. Then, a new group of features including sharpness, sum of derivatives, sparsity, entropy, mean value and standard deviation is extracted from the base and location vectors, which is further separated by a fuzzy C-means (FCM) clustering algorithm. Finally, non-dominated sorting genetic algorithm II (NSGA-II) is introduced as a feature selection tool to improve the FCM clustering performance and acquire the corresponding selected feature subsets. The 600 PD pulses sampled from four typical defect models are adopted for testing. It is shown that a minimum clustering error of 7.67% with 4 dimensional optimal feature subset selected by NSGA-II is achieved when NMF parameter r = 1. In addition, NSGA-II can not only reduce the feature dimension but also dramatically improve the FCM clustering performance compared with the original extracted features. The selected four features are also examined by the data of two PD sources simultaneous active. The results demonstrate that it is feasible to apply the proposed algorithm to PD pulses separation.

Dense Fractal Networks, Trends, Noises and Switches in Homeostasis Regulation of Shannon Entropy for Chromosomes’ Activity in Living Cells for Medical Diagnostics

We analyze correlations and patterns of oxidative activity of 3D DNA at DNA fluorescence in complete sets of chromosomes in neutrophils of peripheral blood. Fluorescence of DNA is registered by method of flow cytometry with nanometer spatial resolution. Experimental data present fluorescence of many ten thousands of cells, from different parts of body in each population, in various blood samples. Data is presented in histograms as frequency distributions of flashes in the dependence on their intensity. Normalized frequency distribution of information in these histograms is used as probabilistic measure for definition of Shannon entropy. Data analysis shows that for this measure of Shannon entropy common sum of entropy, i.e. total entropy E, for any histogram is invariant and has identical trends of changes all values of E (r) = lnr at reduction of rank r of histogram. This invariance reflects informational homeostasis of chromosomes activity inside cells in multi-scale networks of entropy, for varied ranks r. Shannon entropy in multi-scale DNA networks has much more dense packing of correlations than in “small world” networks. As the rule, networks of entropy differ by the mix of normal D 2 and abnormal D > 2 fractal dimensions for varied ranks r, the new types of fractal patterns and hinges for various topology (fractal dimension) at different states of health. We show that all distributions of information entropy are divided on three classes, which associated in diagnostics with a good health or dominants of autoimmune or inflammatory diseases. This classification based on switching of stability at transcritical bifurcation in homeostasis regulation. We defined many ways for homeostasis regulation, coincidences and switching patterns in branching sequences, the averages of Hölder for deviations of entropy from homeostasis at different states of health, with various saturation levels the noises of entropy at activity of all chromosomes in support regulation of homeostasis.

Quantum statistical entropy of Schwarzchild-de Sitter spacetime

Using the quantum statistical method, we calculate quantum statistical entropy between the black hole horizon and the cosmological horizon in Schwarzchild spacetime and derive the expression of quantum statistical entropy in de Sitter spacetime. Under the Unruh-Verlinde temperature of Schwarzchild-de Sitter spacetime in the entropic force views, we obtain the expression of quantum statistical entropy in de Sitter spacetime. It is shown that in de Sitter spacetime quantum statistical entropy is the sum of thermodynamic entropy corresponding black hole horizon and the one corresponding cosmological horizon. And the correction term of de Sitter spacetime entropy is obtained. Therefore, it is confirmed that the black hole entropy is the entropy of quantum field outside the black hole horizon. The entropy of de Sitter spacetime is the entropy of quantum field between the black hole horizon and the cosmological horizon.

Mathematical study of non-ideal electrostatic correlations in equilibrium electrolytes

We undertake the mathematical analysis of a model describing equilibrium binary electrolytes surrounded by charged solid walls. The problem is formulated in terms of the electrostatic potential and the ionic concentrations which have prescribed spatial mean values. The free energy of the system is decomposed as the difference of the internal energy and entropy functionals. The entropy functional is the sum of an ideal entropy and an excess entropy, the latter taking into account non-ideality due to electrostatic correlations at low ionic concentrations and steric exclusion effects at high ionic concentrations. We derive sufficient conditions to achieve convexity of the entropy functional, yielding a convex–concave free energy functional. Our main result is the existence and uniqueness of the saddle point of the free energy functional and its characterization as a solution of the original model problem. The proof hinges on positive uniform lower bounds for the ionic concentrations and uniform upper bounds for the ionic concentrations and the electrostatic potential. Some numerical experiments are presented in the case where the excess entropy is evaluated using the mean spherical approximation.

Scaling properties of net information measures for bound states of spherical model potentials confined with finite barrier #

Using dimensional analyses, the scaling properties of the Heisenberg uncertainty relationship as well as the various information theoretical uncertainty-like relationships are derived for the bound states corresponding to the superposition of the power potential of the form \(V(r)=Zr^n+\sum_i Z_i r^{n_i}\), where Z, Z i , n, n i are parameters, in the free state as well as in the additional presence of a spherical penetrable boundary wall located at radius R The uncertainty product and all other net information measures are shown here to depend only on the parameters [s i ] defined by the ratios \(Z_i/Z^{(n_i+2)/(n+2)}\). Introduction of a finite potential, V c at the radial distance r ≥ R results in a complete set of scaling parameters given by [s i , t 1, t 2], where t 1 is given by RZ 1/(n + 2) and \(t_2= V_c /(Z)^{2/(n+2)}\). The scaling of the sum of Shannon information entropy, S T , with the nuclear charge, Z, for the hydrogen isoelectronic series confined inside an impenetrable cavity of radius R . The optimum value of S T remains fixed for all Z, while the location of R opt scales as 1/Z.