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Neurovascular coupling (NVC) is the tight relationship between changes in cerebral blood flow and neural activation. NVC can be evaluated non-invasively using transcranial Doppler ultrasound (TCD)-measured changes in brain activation (cerebral blood velocity [CBv]) using different cognitive tasks and stimuli. This study used a novel approach to analyzing CBv changes occurring in response to 20 tasks from the Addenbrooke's Cognitive Examination III in 40 healthy individuals. The novel approach compared various information entropy families (permutation, Tsallis, and Rényi entropy) and statistical complexity measures based on disequilibrium. Using this approach, we found the majority of the attention, visuospatial, and memory tasks from the Addenbrooke's Cognitive Examination III that showed lower statistical complexity values when compared with the resting state. On the entropy-complexity (HC) plane, a receiver operating characteristic curve was used to distinguish between baseline and cognitive tasks using the area under the curve. Best area under the curve values were 0.91 ± 0.04, p =.001, to distinguish between resting and cognitively active states. Our findings show that brain hemodynamic signals captured with TCD can be used to distinguish between resting state (baseline) and cognitive effort (stimulation paradigms) using entropy and statistical complexity as an alternative method to traditional techniques such as coherent averaging of CBv signals. Further work should directly compare these analysis methods to identify the optimal method for analyzing TCD-measured changes in NVC. [ABSTRACT FROM AUTHOR]

The aim of this article is to define a new flexible statistical model to examine the COVID‐19 data sets that cannot be modeled by the inverse exponential distribution. A novel extended distribution with one scale and three shape parameters is proposed using the generalized alpha power family of distributions to derive the generalized alpha power exponentiated inverse exponential distribution. Some important statistical properties of the new distribution such as the survival function, hazard function, quantile function, rth$$ r\mathrm{th} $$ moment, Rényi entropy, and order statistics are all derived. The method of maximum likelihood estimation is used to estimate the parameters of the new distribution. The performance of the estimators are assessed through Monte Carlo simulation, which shows that the maximum likelihood method works well in estimating the parameters. The GAPEIEx distribution was applied to COVID‐19 data sets in order to access the flexibility and adaptability of the distribution, and it happens to perform better than its submodels and other well‐known distributions. [ABSTRACT FROM AUTHOR]

We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called correlation dimension (or Rényi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent. [ABSTRACT FROM AUTHOR]

We consider the solution of one dimensional Schrödinger Dunkl equation for energies and eigenfunctions. Then we provide analytical expressions for various information theoretic measures. For a given density function, quantities such as position expectation value, entropic moment, disequilibrium, Rényi entropy, Shannon entropy, Tsallis entropy, Fisher information are presented. Next, a few relative information measures corresponding to two density functions, like relative entropy, relative Fisher, relative Rényi, relative Tsallis, along with their associated Jensen divergences such as Jensen–Shannon divergence, Jensen–Fisher divergence, Jensen–Rényi divergence, Jensen–Tsallis divergence are treated. Sample results are provided in graphical form. Dependence of these quantities on the Dunkl parameter μ shows distinct features for μ < 0 and μ > 0. [ABSTRACT FROM AUTHOR]

In this work, the holographic dark energy model is constructed by using the non‐extensive nature of the Schwarzschild black hole via the Rényi entropy. Due to the non‐extensivity, the black hole can be stable under the process of fixing the non‐extensive parameter. A change undergoing such a process would then motivate us to define the energy density of the Rényi holographic dark energy (RHDE). As a result, the RHDE with choosing the characteristic length scale as the Hubble radius provides the late‐time expansion without the issue of causality. Remarkably, the proposed dark energy model contains the non‐extensive length scale parameter additional to the standard ΛCDM$\Lambda{\rm CDM}$ model. The cosmic evolution can be characterized by comparing the size of the Universe to this length scale. Moreover, the preferable value of the non‐extensive length scale is determined by fitting the model to recent observations. The results of this work would shed light on the interplay between the thermodynamic description of the black hole with non‐extensivity and the classical gravity description of the evolution of the Universe. [ABSTRACT FROM AUTHOR]

The shortest distance between the first $ n $ iterates of a typical point can be quantified with a log rule for some dynamical systems admitting Gibbs measures. We show this in two settings. For topologically mixing Markov shifts with at most countably infinite alphabet admitting a Gibbs measure with respect to a locally Hölder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the Rényi entropy. For interval maps with a Gibbs-Markov structure, we prove a similar rule relating the correlation dimension of Gibbs measures with the shortest distance between two iterates in the orbit generated by a typical point. [ABSTRACT FROM AUTHOR]

Scale-free brain activity, linked with learning, the integration of different time scales, and the formation of mental models, is correlated with a metastable cognitive basis. The spectral slope, a key aspect of scale-free dynamics, was proposed as a potential indicator to distinguish between different sleep stages. Studies suggest that brain networks maintain a consistent scale-free structure across wakefulness, anesthesia, and recovery. Although differences in anesthetic sensitivity between the sexes are recognized, these variations are not evident in clinical electroencephalographic recordings of the cortex. Recently, changes in the slope of the power law exponent of neural activity were found to correlate with changes in Rényi entropy, an extended concept of Shannon's information entropy. These findings establish quantifiers as a promising tool for the study of scale-free dynamics in the brain. Our study presents a novel visual representation called the Rényi entropy-complexity causality space, which encapsulates complexity, permutation entropy, and the Rényi parameter q. The main goal of this study is to define this space for classical dynamical systems within theoretical bounds. In addition, the study aims to investigate how well different time series mimicking scale-free activity can be discriminated. Finally, this tool is used to detect dynamic features in intracranial electroencephalography (iEEG) signals. To achieve these goals, the study implementse the Bandt and Pompe method for ordinal patterns. In this process, each signal is associated with a probability distribution, and the causal measures of Rényi entropy and complexity are computed based on the parameter q. This method is a valuable tool for analyzing simulated time series. It effectively distinguishes elements of correlated noise and provides a straightforwardmeans of examining differences in behaviors, characteristics, and classifications. For the iEEG experimental data, the REM state showed a greater number of significant sex-based differences, while the supramarginal gyrus region showed themost variation across differentmodes and analyzes. Exploring scale-free brain activity with this framework could provide valuable insights into cognition and neurological disorders. The results may have implications for understanding differences in brain function between the sexes and their possible relevance to neurological disorders. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new two-parameter lifetime distribution based on the type II Topp-Leone-G family and the inverse length biased exponential distribution which is called the type II Topp-Leone inverse length biased exponential (TIITL-ILBE) distribution. Various statistical properties of the proposed distribution, including the ordinary moments, moment generating function, incomplete moments, conditional moments, vitality and mean residual life functions, moments of the residual and reversed residual lifetime functions and Rényi entropy are derived. The method of maximum likelihood for estimation and some simulation results for the model parameters of the TIITL-ILBE distribution under complete samples is investigated. Also, The methods of maximum likelihood and Bayesian for estimation and some simulation results for the model parameters of the TIITL-ILBE distribution under joint censored samples is investigated. The real life data set results utilizing that the proposed distribution exhibits greater flexibility and evaluated empirically under complete and joint censored data. [ABSTRACT FROM AUTHOR]

We propose a new family of distributions called the Zografos-Balakrishnan type-I heavy-tailed-G (ZBTIHT-G) distributions. A special model of the proposed family, namely Zografos-Balakrishnan type-I heavy-tailed-Weibull (ZBTIHT-W) model is thoroughly studied. Statistical properties of the new family of distributions including, among others, the hazard rate function, quantile function, moments, distribution of order statistics and Rényi entropy are presented. The maximum likelihood method of estimation is used for estimating the model parameters and Monte Carlo simulation is conducted to examine the performance of the estimators of the model parameters. The flexibility and importance of the new family of distributions are demonstrated by means of applications to real data sets. [ABSTRACT FROM AUTHOR]

We prove that a globally hyperbolic smooth spacetime endowed with a C1-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even C1,1, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics. [ABSTRACT FROM AUTHOR]

We give systematic ways of defining monotone quantum relative entropies and (multi-variate) quantum Rényi divergences starting from a set of monotone quantum relative entropies. Interestingly, despite its central importance in information theory, only two additive and monotone quantum extensions of the classical relative entropy have been known so far, the Umegaki and the Belavkin-Staszewski relative entropies, which are the minimal and the maximal ones, respectively, with these properties. Using the Kubo-Ando weighted geometric means, we give a general procedure to construct monotone and additive quantum relative entropies from a given one with the same properties; in particular, when starting from the Umegaki relative entropy, this gives a new one-parameter family of monotone (even under positive trace-preserving (PTP) maps) and additive quantum relative entropies interpolating between the Umegaki and the Belavkin-Staszewski ones on full-rank states. In a different direction, we use a generalization of a classical variational formula to define multi-variate quantum Rényi quantities corresponding to any finite set of quantum relative entropies (D q x ) x ∈ X and real weights (P (x)) x ∈ X summing to 1, as Q P b , q ((ϱ x) x ∈ X) : = sup τ ≥ 0 ⁡ { Tr τ − ∑ x P (x) D q x (τ ‖ ϱ x) }. We analyze in detail the properties of the resulting quantity inherited from the generating set of quantum relative entropies; in particular, we show that monotone quantum relative entropies define monotone Rényi quantities whenever P is a probability measure. With the proper normalization, the negative logarithm of the above quantity gives a quantum extension of the classical Rényi α -divergence in the 2-variable case (X = { 0 , 1 } , P (0) = α). We show that if both D q 0 and D q 1 are lower semi-continuous, monotone, and additive quantum relative entropies, and at least one of them is strictly larger than the Umegaki relative entropy then the resulting barycentric Rényi divergences are strictly between the log-Euclidean and the maximal Rényi divergences, and hence they are different from any previously studied quantum Rényi divergence. [ABSTRACT FROM AUTHOR]

Dempster-Shafer theory (DST) has attracted wide attention in many fields thanks to its strong advantages over probability theory. Whereas the uncertainty measure of basic belief assignment (BBA) in DST is an open and essential problem. The main goal of this article is to propose a new belief Rényi entropy for the uncertainty measure of BBA, which is inspired by generalized Rényi entropy in DST. The proposed belief Rényi entropy satisfies some desirable properties of uncertainty measure. Furthermore, the proposed belief Rényi entropy can be degraded to Rényi entropy when BBA is transformed into a probability distribution. Finally, a new decision-making method is designed based on the proposed belief Rényi entropy. The validity of the proposed belief entropy is verified by some numerical examples and its application to decision-making. [ABSTRACT FROM AUTHOR]

Quantum systems are vulnerable against environmental interactions which may result in depletion of potential resources. The decay in the resources may be to the extent that they cannot be retrieved even with a non-local unitary action on the whole system. Two important figure of merits in the context of quantum information are the fully entangled fraction (FEF) and conditional entropy of a composite quantum system. While FEF plays a key role in teleportation, negativity of conditional entropy assumes significance in state merging and dense coding. A state may lose such merits and may move to an absolute regime, where even a global unitary fails to reclaim those merits. In the present work, we probe the action of some quantum channels in two qubits and two qudits and find that some quantum states move to the absolute regime under the action. Since global unitary operations are unable to retrieve them back to the non-absolute regime, we provide a prescription for the retrieval using an entanglement swapping network. We also provide an explicit illustration of our prescription. Furthermore, we extend the notion of absoluteness to conditional Rényi entropies and find the required condition for a state to have absolute conditional Rényi entropy non-negative property. Exploiting the Bloch-Fano decomposition of density matrices, we characterize such states. We then extend the work to include the marginals of a tripartite system and provide for their characterization with respect to the aforementioned absolute properties. [ABSTRACT FROM AUTHOR]

Frequency-modulated (FM) signals, prevalent across various applied disciplines, exhibit time-dependent frequencies and a multicomponent nature necessitating the utilization of time-frequency methods. Accurately determining the number of components in such signals is crucial for various applications reliant on this metric. However, this poses a challenge, particularly amidst interfering components of varying amplitudes in noisy environments. While the localized Rényi entropy (LRE) method is effective for component counting, its accuracy significantly diminishes when analyzing signals with intersecting components, components that deviate from the time axis, and components with different amplitudes. This paper addresses these limitations and proposes a convolutional neural network-based (CNN) approach for determining the local number of components using a time–frequency distribution of a signal as input. A comprehensive training set comprising single and multicomponent linear and quadratic FM components with diverse time and frequency supports has been constructed, emphasizing special cases of noisy signals with intersecting components and differing amplitudes. The results demonstrate that the estimated component numbers outperform those obtained using the LRE method for considered noisy multicomponent synthetic signals. Furthermore, we validate the efficacy of the proposed CNN approach on real-world gravitational and electroencephalogram signals, underscoring its robustness and applicability across different signal types and conditions. [ABSTRACT FROM AUTHOR]

One of the most popular techniques for changing the purpose of an image or resizing a digital image with content awareness is the seam-carving method. The performance of image resizing algorithms based on seam machining shows that these algorithms are highly dependent on the extraction of importance map techniques and the detection of salient objects. So far, various algorithms have been proposed to extract the importance map. In this paper, a new method based on Rényi entropy is proposed to extract the importance map. Also, a deep learning network has been used to detect salient objects. The simulator results showed that combining Rényi's importance map with a deep network of salient object detection performed better than classical seam-carving and other extended seam-carving algorithms based on deep learning. [ABSTRACT FROM AUTHOR]

A system can be considered to be reliable if it operates successfully for a long period of time and has fewer uncertainties during its lifespan. In other words, lower the uncertainty of a random variable implies higher reliability [see Ebrahimi N. How to measure uncertaintyin the residual life time distribution. Sankhya: Indian J Stat SerA. 1996;58:48–57]. Motivated by this, the present study considers a generalized entropy function, namely the Rényi entropy in the quantile framework as a measure of uncertainty and proposes two nonparametric estimators for its computation. Asymptotic properties of the estimators are established under suitable regularity conditions. Simulation study and real data analysis are carried out to compare the performance and usefulness of the estimators. [ABSTRACT FROM AUTHOR]

The notion of weighted Renyi's entropy for truncated random variables has recently been proposed in the information-theoretic literature. In this paper, we introduce a generalized measure of it for double truncated distribution, namely weighted generalized interval entropy (WGIE), and study it in the context of reliability analysis. Several properties, including monotonicity, bounds and uniqueness of WGIE are investigated. Moreover, the proposed measure is estimated using parametric approach and a simulation study is carried out to demonstrate the performance of the estimates for a real data set. The role of WGIE in reliability modeling has been investigated. We also provide an application of the proposed concept in coding theory. [ABSTRACT FROM AUTHOR]

We reveal the continuous change of information dynamics patterns in anyonic-PT symmetric systems that originates from the continuity of anyonic-PT symmetry. We find there are three information dynamics patterns for anyonic-PT symmetric systems: damped oscillations with an overall decrease (increase) and asymptotically stable damped oscillations, which are three-fold degenerate and are distorted using the Hermitian quantum Rényi entropy or distinguishability. It is the normalization of the non-unitary evolved density matrix that causes the degeneracy and distortion. We give a justification for non-Hermitian quantum Rényi entropy being negative. By exploring the mathematics and physical meaning of the negative entropy in open quantum systems, we connect negative non-Hermitian quantum Rényi entropy and negative quantum conditional entropy, paving the way to rigorously investigate negative entropy in open quantum systems. [ABSTRACT FROM AUTHOR]

This paper addresses the critical need for precise thermal modeling in electronics, where temperature significantly impacts system reliability. We emphasize the necessity of accurate temperature measurement and uncertainty quantification in thermal imaging, a vital tool across multiple industries. Current mathematical models and uncertainty measures, such as Rényi and Shannon entropies, are inadequate for the detailed informational content required in thermal images. Our work introduces a novel entropy that effectively captures the informational content of thermal images by combining local and global data, surpassing existing metrics. Validated by rigorous experimentation, this method enhances thermal images' reliability and information preservation. We also present two enhancement frameworks that integrate an optimized genetic algorithm and image fusion techniques, improving image quality by reducing artifacts and enhancing contrast. These advancements offer significant contributions to thermal imaging and uncertainty quantification, with broad applications in various sectors. [ABSTRACT FROM AUTHOR]

In this article, we introduce a robust generalization of the generalized Topp-Leone-G (GEN-TL-G) family of distributions via the heavy-tailed technique, namely, heavy-tailed generalized Topp-Leone-G (HT-GEN-TL-G) family of distributions. Statistical properties of the HT-GEN-TL-G family of distributions including reliability functions, quantile function, density expansion, moments, moment generating function, incomplete moments, Rényi entropy, distribution of order statistics are derived. Different estimation methods including Maximum Likelihood, Anderson-Darling, Ordinary Least Squares, Weighted Least Squares, Cramér-von Mises and Maximum Product of Spacing are utilized to estimate the unknown parameters of the new distribution, and a simulation study is used to compare the results of the estimation methods. Risk measures for this distribution were also developed and finally the effectiveness of this new family of distributions was demonstrated using applications to two real data sets. Graphical and application results in this manuscript were obtained using R programming language. [ABSTRACT FROM AUTHOR]

A new generalized class of distributions called the Topp-Leone-Gompertz-G Power Series (TL-Gom-GPS) distribution is presented. The TL-Gom-GPS distribution has the ability to capture diverse data patterns such as heavy-tailed, lighttailed and skewed distributions, as well as hazard rate functions that have monotonic and non-monotonic shapes. Some statistical properties of the new class of distributions are explored. For this new class of distributions, we derived the expansion of the density function, the hazard rate and quantile functions, moments and generating function, probability weighted moments, distribution of order statistics and Rényi entropy. Monte Carlo simulations are conducted to show the performance of the estimation methods. Finally, the usefulness and flexibility of the new class of distributions is examined by means of applications to real data sets. [ABSTRACT FROM AUTHOR]

Rotating machinery fault diagnosis is of key significance for ensuring safe and efficient operation of various industrial equipment. However, under nonstationary operating conditions, the fault–induced characteristic frequencies are often time–varying. Conventional Fourier spectrum analysis is not suitable for revealing time–varying details, and nonstationary fault feature extraction methods are still in desperate need. Order spectrum can reveal the rotational–speed–related time–varying frequency components as spectral peaks in order domain, thus facilitating fault feature extraction under time–varying speed conditions. However, the speed–unrelated frequency components are still nonstationary after angular–domain resampling, thus causing wide–band features and interferences in the order spectrum. To overcome such a drawback, this work proposes a rotating machinery fault diagnosis method based on adaptive separation of time–varying components and order feature extraction. Firstly, the rotational speed is estimated by the multi–order probabilistic approach (MOPA), thus eliminating the inconvenience of installing measurement equipment. Secondly, adaptive separation of the time–varying frequency component is achieved through time–varying filtering and surrogate test. It effectively eliminates interference from irrelevant components and noise. Finally, a high–resolution order spectrum is constructed based on the average amplitude envelope of each mono–component. It does not involve Fourier transform or angular–domain resampling, thus avoiding spectral leakage and resampling errors. By identifying the fault–related spectral peaks in the constructed order spectrum, accurate fault diagnosis can be achieved. The Rényi entropy values of the proposed order spectrum are significantly lower than those of the traditional order spectrum. This result verifies the effective energy concentration and high resolution of the proposed order spectrum. The results of both numerical simulation and lab experiments confirm the effectiveness of the proposed method in accurately presenting the time–varying frequency components for rotating machinery diagnosing faults. [ABSTRACT FROM AUTHOR]

This article consists of an introduction to the theory of nonassociative geometric classical and quantum information flows defined by star products with R‐flux deformations in string gravity. Corresponding nonassociative generalizations of the concepts of classical Shannon entropy, quantum von Neumann entropy, Rényi entropy are formulated. The fundamental geometric and quantum information objects are computed following the Grigori Perelman statistical thermodynamic approach to Ricci flows and gravity theories generalized for phase spaces modeled as (co) tangent Lorentz bundles. Nonassociative parametric deformations and nonholonomic thermo‐geometric versions of statistical generating functions, their quantum analogues as density matrices are considered for deriving the entropy, energy and fluctuation functionals. This allows us to define and compute respective classical and quantum relative and conditional entropies, mutual information and nonassociative entanglement and thermodynamic information variables. The principles of nonassociative quantum geometric and information flow theory, QGIF, and study the basic properties of such quasi‐stationary models related to modified gravity theories are formulated. Applications are considered for nonassociative deformed and entangled couples of four‐dimensional (4‐d), wormholes (defined by respective spacetime and/or momentum type coordinates) and nonassociative QGIFs of 8‐d phase space generalized wormholes configurations. Finally, phase space black holes and wormholes being transversable for nonassociative qubits, quantum channels and entanglement witness are speculated; thought and laboratory experiments are discussed; and perspectives for quantum computer modeling and tests of nonassociative geometric flow and gravity theories are considered. [ABSTRACT FROM AUTHOR]

By using the Choi–Jamiołkowski isomorphism, we propose a well-defined coherence measure of quantum channels based on the generalized α -z-relative Rényi entropy. In addition, we present an alternative coherence measure of quantum channels by quantifying the commutativity between the channels and the completely dephasing channels with the generalized α -z-relative Rényi entropy. Some elegant properties of the measures are illustrated in detail. Explicit formulas of these coherence measures are derived for some detailed typical quantum channels. [ABSTRACT FROM AUTHOR]

We compute the surface defect contribution to Rényi entropy and supersymmetric Rényi entropy in six dimensions. We first compute the surface defect contribution to Rényi entropy for free fields, which verifies a previous formula about entanglement entropy with surface defect. Using conformal map to S β 1 × H d − 1 we develop a heat kernel approach to compute the defect contribution to Rényi entropy, which is applicable for p-dimensional defect in general d-dimensional free fields. Using the same geometry S β 1 × H 5 with an additional background field, one can construct the supersymmetric refinement of the ordinary Rényi entropy for six-dimensional (2, 0) theories. We find that the surface defect contribution to supersymmetric Rényi entropy has a simple scaling as polynomial of Rényi index in the large N limit. We also discuss how to connect the free field results and large N results. [ABSTRACT FROM AUTHOR]

We investigate the entanglement and Rényi entropies for the (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method. The central charge is determined from the asymptotic scaling properties of both entropies. We also examine the consistency between the entanglement entropy and the nth-order Rényi entropy with n → 1. [ABSTRACT FROM AUTHOR]

We provide the sandwiched Rényi divergence of order α ∈ (1 2 , 1) , as well as its induced quantum information quantities, with an operational interpretation in the characterization of the exact strong converse exponents of quantum tasks. Specifically, we consider (a) smoothing of the max-relative entropy, (b) quantum privacy amplification, and (c) quantum information decoupling. We solve the problem of determining the exact strong converse exponents for these three tasks, with the performance being measured by the fidelity or purified distance. The results are given in terms of the sandwiched Rényi divergence of order α ∈ (1 2 , 1) , and its induced quantum Rényi conditional entropy and quantum Rényi mutual information. This is the first time to find the precise operational meaning for the sandwiched Rényi divergence with Rényi parameter in the interval α ∈ (1 2 , 1) . [ABSTRACT FROM AUTHOR]

We consider three operations on the positive definite cones of C ∗ -algebras which are Schatten p-norm of various versions of the quantum Rényi relative entropy. The main results concern how Jordan ∗ -isomorphisms between C ∗ -algebras can be characterized by the preservation of the Schatten p-norms of those operations. [ABSTRACT FROM AUTHOR]

Knowledge of how to define and estimate the dependence among multivariate hydrological series is essential for detecting abrupt changes in the dependence. In this paper, a new method (BMCTC) is proposed to detect all possible abrupt change points in the dependence among multivariate hydrological series. The total correlation estimated by the matrix-based Renyi's alpha-order entropy functional is firstly introduced to define and measure the dependence strength among multivariate hydrological series. Then, the moving cut total correlation (MCTC) sequence is built by the moving window technique, which is used to measure changes in the dependence strength among multivariate hydrological series. Finally, the Bernaola-Galvan algorithm is used to detect all change points of the MCTC sequence. Simulations are performed to compare the effectiveness of BMCTC with Pearson correlation (BMCPC) and Spearman correlation (BMCSC), Cramer-von Mises (CvM) and copula-based likelihood-ratio (CLR). The results show that all change points are detected by BMCTC regardless of the samples size, but wrong change points or no change points are detected by other methods in most cases. BMCTC is applied to detect change points in the dependence among annual runoff, precipitation and sediment discharge series in the Xiliugou and the Kuyehe River, China. It is found that the dependence among runoff, precipitation and sediment discharge changed abruptly in 1980 and 1996 in the Kuyehe River and in 1999 in the Xiliugou River. These changes are mainly caused by human activities such as construction of water conservancy projects and coal mining. [ABSTRACT FROM AUTHOR]

Complex-valued expression models have been widely used in the application of intelligent decision systems. However, there is a lack of entropy to measure the uncertain information of the complex-valued information. Therefore, how to reasonably measure the uncertain information of the complex-valued information is a gap to be filled. In this paper, inspired by the Rényi entropy, we propose the complex-valued Rényi entropy, which measures uncertain information of the complex-valued probability under the framework of complex number, and this is also the first time to measure uncertain information in the complex space. The complex-valued Rényi entropy contains the features of the classical Rényi entropy, i.e., the complex-valued Rényi entropy corresponds to different information functions with different parameters q. Moreover, complex-valued Rényi entropy has some properties, such as non-negativity, monotonicity and etc. Some numerical examples can demonstrate the flexibility and reasonableness of the complex-valued Rényi entropy. [ABSTRACT FROM AUTHOR]

In this paper, we examine the average Rényi entropy S α of a subsystem A when the whole composite system AB is a random pure state. We assume that the Hilbert space dimensions of A and AB are m and mn, respectively. First, we compute the average Rényi entropy analytically for m = α = 2 . We compare this analytical result with the approximate average Rényi entropy, which is shown to be very close. For general case, we compute the average of the approximate Rényi entropy S ~ α (m , n) analytically. When 1 ≪ n , S ~ α (m , n) reduces to ln m - α 2 n (m - m - 1) , which is in agreement with the asymptotic expression of the average von Neumann entropy. Based on the analytic result of S ~ α (m , n) , we plot the ln m -dependence of the Rényi information derived from S ~ α (m , n) . It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing α and eventually disappears in the limit of α → ∞ . The physical implication of the result is briefly discussed. [ABSTRACT FROM AUTHOR]

In parametric statistical modeling, it is essential to create generalizations of current statistical distributions that are more flexible when modeling actual data sets. Therefore, this study introduces a new generalized lifetime model named the odd Weibull Inverse Gompertz distribution (OWIG). The OWIG is derived by combining the odd Weibull family of distributions with the inverse Gompertz distribution. Essential statistical properties are discussed, including reliability functions, moments, Rényi entropy, and order statistics. The proposed OWIG is particularly significant as its hazard rate functions exhibit various monotonic and nonmonotonic shapes. This enables OWIG to model different hazard behaviors more commonly observed in natural phenomena. OWIG's parameters are estimated and its flexibility in predicting unique symmetric and asymmetric patterns is shown by analyzing real-world applications from psychology, environmental, and medical sciences. The results demonstrate that the proposed OWIG is an excellent candidate as it provides the most accurate fits to the data compared with some competing models. [ABSTRACT FROM AUTHOR]

We study the Rényi entanglement entropies along the massless renormalisation group flow that connects the tricritical and critical Ising field theories. Similarly to the massive integrable field theories, we derive a set of bootstrap equations, from which we can analytically calculate the twist field form factors in a recursive way. Additionally, we also obtain them as a non-trivial 'roaming limit' of the sinh-Gordon theory. Then the Rényi entanglement entropies are obtained as expansions in terms of the form factors of these branch point twist fields. We find that the form factor expansion of the entanglement entropy along the flow organises in two different kind of terms. Those that couple particles with the same chirality, and reproduce the entropy of the infrared Ising theory, and those that couple particles with different chirality, which provide the ultraviolet contributions. The massless flow under study possesses a global ℤ2 spin-flip symmetry. We further consider the composite twist fields associated to this group, which enter in the study of the symmetry resolution of the entanglement. We derive analytical expressions for their form factors both from the bootstrap equations and from the roaming limit of the sinh-Gordon theory. [ABSTRACT FROM AUTHOR]

Ball bearing is used to provide free rotation around a fixed axis. Various kinds of faults exist in the bearing such as inner race fault, outer race fault and ball race fault. One very effective method to diagnose the bearing fault is vibration signal analysis. Empirical mode decomposition (EMD) has been used for ball-bearing fault diagnosis in mechanical systems using vibration signal analysis. Classification of the ball-bearing fault has always been a challenging task. Various classification schemes such as Extreme learning machine (ELM), K-means Clustering, and Support vector machine (SVM), have been reported in the literature for ball-bearing fault classification. SVM is restricted by multiclass classification efficiency, and ELM is restricted by the longer training. In this paper, the entropy analysis of the wavelet coefficient obtained from the third level decomposition of the residue signal (obtained after subtracting the highest frequency component from the raw signal) has been done for ball-bearing fault classification. A comparative analysis of the wavelet coefficient based on entropy measurement has been presented here. High fault classification accuracy has been achieved using the proposed method for the detection of ball-bearing fault. Shannon entropy, Average Shannon entropy, and Renyi's entropy are parameters for the justification of the proposed approach. The result shows the best wavelet to be chosen among the available discrete wavelet based on various entropy measurements. [ABSTRACT FROM AUTHOR]

Analytically continuing the von Neumann entropy from Rényi entropies is a challenging task in quantum field theory. While the n-th Rényi entropy can be computed using the replica method in the path integral representation of quantum field theory, the analytic continuation can only be achieved for some simple systems on a case-by-case basis. In this work, we propose a general framework to tackle this problem using classical and quantum neural networks with supervised learning. We begin by studying several examples with known von Neumann entropy, where the input data is generated by representing Tr ρ A n with a generating function. We adopt KerasTuner to determine the optimal network architecture and hyperparameters with limited data. In addition, we frame a similar problem in terms of quantum machine learning models, where the expressivity of the quantum models for the entanglement entropy as a partial Fourier series is established. Our proposed methods can accurately predict the von Neumann and Rényi entropies numerically, highlighting the potential of deep learning techniques for solving problems in quantum information theory. [ABSTRACT FROM AUTHOR]

In this paper, a new class of distributions called the T-X θ family of distributions for bounded—(0,1)—and unbounded— (0 , ∞) —supported random variables is suggested. Some special sub-models of the proposed family are utilized. A new sub-model is selected to be studied in details. The statistical properties of the suggested family including quantile function, moments, moment generating function, order statistics and Rényi entropy are discussed. The maximum likelihood method is provided to estimate the parameters of the distribution and a Monte Carlo simulation study is used. The discretized T-X θ family provided many sub-families and sub-models. In addition, eight real data sets are utilized to demonstrate the flexibility of the proposed continuous and discrete family's multiple sub models. [ABSTRACT FROM AUTHOR]

The phase‐space relative Rényi entropy is introduced using the information theoretical and thermodynamic view of density functional theory. In the special case of constant inverse temperature the phase‐space relative Rényi entropy is a sum of the position‐space relative Rényi entropy and a term arising from the momentum space. This quantity can be considered as a measure of similarity. It includes more information than the position‐space measures, since it also incorporates momentum‐space knowledge. [ABSTRACT FROM AUTHOR]

Biodiversity plays a pivotal role in sustaining ecosystem processes, encompassing diverse biological species, genetic types and the intricacies of ecosystem composition. However, the precise definition of biodiversity at the individual level remains a challenging endeavour. Hill numbers, derived from Rényi's entropy, have emerged as a popular measure of diversity, with a recent unified framework extending their application across various levels, from genetics to ecosystems. In this study, we employ a computational approach to exploring the diversity of mitochondrial heteroplasmy using real‐world data. By adopting Hill numbers with q = 2, we demonstrate the feasibility of quantifying mitochondrial heteroplasmy diversity within and between individuals and populations. Furthermore, we investigate the alpha diversity of mitochondrial heteroplasmy among different species, revealing heterogeneity at multiple levels, including mitogenome components and protein‐coding genes (PCGs). Our analysis explores large‐scale mitochondrial heteroplasmy data in humans, examining the relationship between alpha diversity at the mitogenome components and PCGs level. Notably, we do not find a significant correlation between these two levels. Additionally, we observe significant correlations in alpha diversity between mothers and children in blood samples, exceeding the reported R2 value for allele frequency correlations. Moreover, our investigation of beta diversity and local overlay similarity demonstrates that heteroplasmy variant distributions in different tissues of children more closely resemble those of their mothers. Through systematic quantification and analysis of mitochondrial heteroplasmy diversity, this study enhances our understanding of heterogeneity at multiple levels, from individuals to populations, providing new insights into this fundamental dimension of biodiversity. [ABSTRACT FROM AUTHOR]

One of the most important applications of statistical models is in analyzing survival data. In this study, we developed the Gamma Type Two Half Logistic Topp-Leone-G model using the technique earlier proposed by Zografos and Balakrishnan. Different characteristics of the proposed distribution are obtained. In order to estimate the model parameters based on complete and censored data, the maximum likelihood estimation method is used. Through Monte Carlo simulation, the performance of the estimators is evaluated. The proposed distribution's potential significance and applicability are empirically demonstrated using actual datasets. We found that our new distribution is a very competitive model for describing both complete and censored observations in survival analysis. The work demonstrated that in certain cases, our new model performed better than other parametric models with the same number of parameters. [ABSTRACT FROM AUTHOR]

Simulating the von Neumann and second-order Rényi entropies of driven Mott insulators, we examine the entanglement dynamics of the η -pairing state in the one-dimensional half-filled Hubbard model after pulse irradiation. Corresponding data are obtained by means of the infinite time-evolving block decimation technique directly in the thermodynamic limit. For optimal pump-pulse parameters the temporal course of the entanglement entropies shows the characteristic entanglement growth due to the increase of η -pairing correlations. [ABSTRACT FROM AUTHOR]

This study introduces a novel gray-level co-occurrence matrix (GLCM) feature called modified Rényi–Deng entropy, denoted as E RD α T f . The practical application involves the utilizations of a digital elevation model raster dataset and diagnosing melanoma dataset. E RD α T f 's performance was compared with other GLCM texture features (including entropy, angular second moment, energy, dissimilarity, contrast, homogeneity, variance, and correlation) at various scale parameters ( α = 0 , α → 1 , α = 2 and α = 3 ) through Pearson correlation analysis. Visualization of all features was achieved using ArcGIS. The results demonstrate significant associations between all GLCM texture features, except for correlation, and the parametric measures of E RD α T f . Notably, entropy shows the strongest correlations with E RD α T 0 measures, while dissimilarity and homogeneity are most closely associated with E RD 0 T 1 , E RD → 1 T 1 and E RD 2 T 1 . Entropy and E RD → 1 T 0 exhibit identical distribution patterns, given that Rényi–Deng entropy converges to Deng entropy when α → 1, and Deng entropy transitions to Shannon entropy when α → 1 and f = 0 . Angular second moment and energy also display high correlations with E RD α T f measures, indicating that angular second moment and energy increase as E RD α T f measures decrease. In conclusion, the modified Rényi–Deng entropy effectively characterizes grid-based textural features, with the exception of correlation. Therefore, it can be employed as a GLCM texture feature by utilizing alfa scale parameters ranging from 0 to infinity. [ABSTRACT FROM AUTHOR]

Magic is an important part of quantum resource theory. The quantitative study of magic is a difficult but important task, which means a suitable magic measure is needed. We generalize the magic measure introduced by Leone et al. (Phys Rev Lett 128:050402, 2022) and extend the Pauli operators acting on qubits to the discrete Heisenberg–Weyl operators (i.e., the generalized Pauli operators) acting on qudits. We show that it is a suitable magic measure. We use this measure to calculate some related qutrit magic states and compare it with other measures. Subsequently, a tighter upper bound of this measure is obtained by using the group covariant symmetric informationally complete (SIC) states. Moreover, we extend this measure to mixed states and calculate the corresponding mixed magic states. Then, it is shown that there are different fundamental limits for the distillation of four qutrit magic states by using the quantum hypothesis testing at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new quantity called SVD entanglement entropy. This is a generalization of entanglement entropy in that it depends on two different states, as in pre- and post-selection processes. This SVD entanglement entropy takes non-negative real values and is bounded by the logarithm of the Hilbert space dimensions. The SVD entanglement entropy can be interpreted as the average number of Bell pairs distillable from intermediates states. We observe that the SVD entanglement entropy gets enhanced when the two states are in the different quantum phases in an explicit example of the transverse-field Ising model. Moreover, we calculate the Rényi SVD entropy in various field theories and examine holographic calculations using the AdS/CFT correspondence. [ABSTRACT FROM AUTHOR]

While quantum multifractality has been widely studied in the physics literature and is by now well understood from the point of view of physics, there is little work on this subject in the mathematical literature. I will report on the proof of multifractal scaling laws for arithmetic Seba billiards. I will explain the mathematical approach to defining the Rényi entropy associated with a sequence of eigenfunctions and sketch how arithmetic methods permit us to obtain a precise asymptotic in the semiclassical regime and how this allows us to compute the fractal exponents explicitly. Moreover, I will discuss how the symmetry relation for the fractal exponent is related to the functional equation of certain zeta functions. [ABSTRACT FROM AUTHOR]

The results of information analysis of the time series of satellite monitoring of the state of the interplanetary magnetic field provided by the database of the Goddard Space Flight Center. In the arsenal of analysis methods, the main attention is paid to the control of nonextensive properties when scaling time samples created on the basis of the 2001–2022 time series. The reconstruction of the modality of the probability-distribution function for the components of the interplanetary magnetic field in the considered ensemble of realizations is demonstrated. The results of the reconstruction of the kinetics of the Shannon–Boltzmann–Gibbs, Tsallis, and Rényi entropy measures are presented. In a nonextensive approximation, the influence of q-deformation of the phase space on the stochastic regimes of the system under study. In this work, for the first time, the kinetics of information discrepancy and the kinetics of entropy measures of the ensemble of realizations of the values of the interplanetary magnetic field are studied together. The need for approval is indicated by the q-deformation parameters of the phase space, the reference and controlled subsystems, including when using asymptotic approximations in forecasting methods using neural-network algorithms and deep-learning algorithms. [ABSTRACT FROM AUTHOR]

The phase-space approach based on the Wigner distribution function is used to study the quantum dynamics of the three families of the Schrödinger cat states identified as the even, odd, and Yurke–Stoler states. The considered states are formed by the superposition of two Gaussian wave packets localized on opposite sides of a smooth barrier in a dispersive medium and moving towards each other. The process generated by this dynamics is analyzed regarding the influence of the barrier parameters on the nonclassical properties of these states in the phase space below and above the barrier regime. The performed analysis employs entropic measure resulting from the Wigner–Rényi entropy for the fixed Rényi index. The universal relation of this entropy for the Rényi index equal one half with the nonclassicality parameter understood as a measure of the negative part of the Wigner distribution function is proved. This relation is confirmed in the series of numerical simulations for the considered states. Furthermore, the obtained results allowed the determination of the lower bound of the Wigner–Rényi entropy for the Rényi index greater than or equal to one half. [ABSTRACT FROM AUTHOR]

We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian motions, and compare the entropies of one-dimensional distributions of these processes. [ABSTRACT FROM AUTHOR]

We study the entanglement transition in monitored Brownian SYK chains in the large-N limit. Without measurement the steady state n-th Rényi entropy is obtained by summing over a class of solutions, and is found to saturate to the Page value in the n → 1 limit. In the presence of measurements, the analytical continuation n → 1 is performed using the cyclic symmetric solution. The result shows that as the monitoring rate increases, a continuous von Neumann entanglement entropy transition from volume-law to area-law occurs at the point of replica symmetry unbreaking. [ABSTRACT FROM AUTHOR]

In this paper, the Additive Perks Distribution (ADP) is presented; this distribution describes the behavior of the bathtub curve, which is one of the most deeply rooted concepts in reliability analysis. The proposed distribution is based on the sum of the positive and negative hazard rate functions of the Perks distribution. ADP's statistical properties include Measures of Central Tendency, moments, moment generating function, statistical order, residual lifetime, reversed residual lifetime, Rényi entropy, time-varying, and life-stress relationship modeling presented. For estimating the ADP parameters, the Maximum Likelihood Method is used. On the other hand, the APD is tested in three case studies and compared with other distributions that can describe the failure rates in a bathtub curve. The results from the case studies showed that the APD is a good choice for describing the failure times in the shape of a bathtub curve. [ABSTRACT FROM AUTHOR]

The utilization of automated techniques for processing and analyzing medical images can offer significant support to doctors in their diagnostic and therapeutic practices. This study focuses on the complex task of detecting edges in medical images. To this end, we propose novel multilevel approaches based on quantum image representations of the one-dimensional and two-dimensional histograms of the gray-level distributions. The quantum Rényi entropy is employed to quantify the quantum information present in the two histogram types. The particle swarm optimization algorithm is utilized to identify the optimal threshold values. The performance of the proposed methods is evaluated by comparing them to benchmark methods using a set of medical images. The numerical results substantiate the efficacy of the introduced approaches. [ABSTRACT FROM AUTHOR]