Parameter Analysis of Multiscale Two-Dimensional Fuzzy and Dispersion Entropy Measures Using Machine Learning Classification

Two-dimensional fuzzy entropy, dispersion entropy, and their multiscale extensions (MFuzzyEn2D and MDispEn2D, respectively) have shown promising results for image classifications. However, these results rely on the selection of key parameters that may largely influence the entropy values obtained. Yet, the optimal choice for these parameters has not been studied thoroughly. We propose a study on the impact of these parameters in image classification. For this purpose, the entropy-based algorithms are applied to a variety of images from different datasets, each containing multiple image classes. Several parameter combinations are used to obtain the entropy values. These entropy values are then applied to a range of machine learning classifiers and the algorithm parameters are analyzed based on the classification results. By using specific parameters, we show that both MFuzzyEn2D and MDispEn2D approach state-of-the-art in terms of image classification for multiple image types. They lead to an average maximum accuracy of more than 95% for all the datasets tested. Moreover, MFuzzyEn2D results in a better classification performance than that extracted by MDispEn2D as a majority. Furthermore, the choice of classifier does not have a significant impact on the classification of the extracted features by both entropy algorithms. The results open new perspectives for these entropy-based measures in textural analysis.     

Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions,and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy

Let Tϵ, 0≤ϵ≤1/2, be the noise operator acting on functions on the boolean cube {0,1}n. Let f be a distribution on {0,1}n and let q>1. We prove tight Mrs. Gerber-type results for the second Rényi entropy of Tϵf which take into account the value of the qth Rényi entropy of f. For a general function f on {0,1}n we prove tight hypercontractive inequalities for the ℓ2 norm of Tϵf which take into account the ratio between ℓq and ℓ1 norms of f.     

Logical Entropy of Information Sources

In this paper, we present the concept of the logical entropy of order m, logical mutual information, and the logical entropy for information sources. We found upper and lower bounds for the logical entropy of a random variable by using convex functions. We show that the logical entropy of the joint distributions X1 and X2 is always less than the sum of the logical entropy of the variables X1 and X2. We define the logical Shannon entropy and logical metric permutation entropy to an information system and examine the properties of this kind of entropy. Finally, we examine the amount of the logical metric entropy and permutation logical entropy for maps.     

New Bounds and a Generalization for Share Conversion for 3-Server PIR

Private Information Retrieval (PIR) protocols, which allow the client to obtain data from servers without revealing its request, have many applications such as anonymous communication, media streaming, blockchain security, advertisement, etc. Multi-server PIR protocols, where the database is replicated among the non-colluding servers, provide high efficiency in the information-theoretic setting. Beimel et al. in CCC 12’ (further referred to as BIKO) put forward a paradigm for constructing multi-server PIR, capturing several previous constructions for k≥3 servers, as well as improving the best-known share complexity for 3-server PIR. A key component there is a share conversion scheme from corresponding linear three-party secret sharing schemes with respect to a certain type of “modified universal” relation. In a useful particular instantiation of the paradigm, they used a share conversion from (2,3)-CNF over Zm to three-additive sharing over Zpβ for primes p1,p2,p where p1≠p2 and m=p1·p2, and the relation is modified universal relation CSm. They reduced the question of the existence of the share conversion for a triple (p1,p2,p) to the (in)solvability of a certain linear system over Zp, and provided an efficient (in m,logp) construction of such a sharing scheme. Unfortunately, the size of the system is Θ(m2) which entails the infeasibility of a direct solution for big m’s in practice. Paskin-Cherniavsky and Schmerler in 2019 proved the existence of the conversion for the case of odd p1, p2 when p=p1, obtaining in this way infinitely many parameters for which the conversion exists, but also for infinitely many of them it remained open. In this work, using some algebraic techniques from the work of Paskin-Cherniavsky and Schmerler, we prove the existence of the conversion for even m’s in case p=2 (we computed β in this case) and the absence of the conversion for even m’s in case p>2. This does not improve the concrete efficiency of 3-server PIR; however, our result is promising in a broader context of constructing PIR through composition techniques with k≥3 servers, using the relation CSm where m has more than two prime divisors. Another our suggestion about 3-server PIR is that it’s possible to achieve a shorter server’s response using the relation CSm′ for extended Sm′⊃Sm. By computer search, in BIKO framework we found several such sets for small m’s which result in share conversion from (2,3)-CNF over Zm to 3-additive secret sharing over Zpβ′, where β′>0 is several times less than β, which implies several times shorter server’s response. We also suggest that such extended sets Sm′ can result in better PIR due to the potential existence of matching vector families with the higher Vapnik-Chervonenkis dimension.     

On the Validity of Detrended Fluctuation Analysis at Short Scales

Detrended Fluctuation Analysis (DFA) has become a standard method to quantify the correlations and scaling properties of real-world complex time series. For a given scale ℓ of observation, DFA provides the function F(ℓ), which quantifies the fluctuations of the time series around the local trend, which is substracted (detrended). If the time series exhibits scaling properties, then F(ℓ)∼ℓα asymptotically, and the scaling exponent α is typically estimated as the slope of a linear fitting in the logF(ℓ) vs. log(ℓ) plot. In this way, α measures the strength of the correlations and characterizes the underlying dynamical system. However, in many cases, and especially in a physiological time series, the scaling behavior is different at short and long scales, resulting in logF(ℓ) vs. log(ℓ) plots with two different slopes, α1 at short scales and α2 at large scales of observation. These two exponents are usually associated with the existence of different mechanisms that work at distinct time scales acting on the underlying dynamical system. Here, however, and since the power-law behavior of F(ℓ) is asymptotic, we question the use of α1 to characterize the correlations at short scales. To this end, we show first that, even for artificial time series with perfect scaling, i.e., with a single exponent α valid for all scales, DFA provides an α1 value that systematically overestimates the true exponent α. In addition, second, when artificial time series with two different scaling exponents at short and large scales are considered, the α1 value provided by DFA not only can severely underestimate or overestimate the true short-scale exponent, but also depends on the value of the large scale exponent. This behavior should prevent the use of α1 to describe the scaling properties at short scales: if DFA is used in two time series with the same scaling behavior at short scales but very different scaling properties at large scales, very different values of α1 will be obtained, although the short scale properties are identical. These artifacts may lead to wrong interpretations when analyzing real-world time series: on the one hand, for time series with truly perfect scaling, the spurious value of α1 could lead to wrongly thinking that there exists some specific mechanism acting only at short time scales in the dynamical system. On the other hand, for time series with true different scaling at short and large scales, the incorrect α1 value would not characterize properly the short scale behavior of the dynamical system.     

Electron Kinetic Entropy across Quasi-Perpendicular Shocks

We use Magnetospheric Multiscale (MMS) data to study electron kinetic entropy per particle Se across Earth’s quasi-perpendicular bow shock. We have selected 22 shock crossings covering a wide range of shock conditions. Measured distribution functions are calibrated and corrected for spacecraft potential, secondary electron contamination, lack of measurements at the lowest energies and electron density measurements based on plasma frequency measurements. All crossings display an increase in electron kinetic entropy across the shock ΔSe being positive or zero within their error margin. There is a strong dependence of ΔSe on the change in electron temperature, ΔTe, and the upstream electron plasma beta, βe. Shocks with large ΔTe have large ΔSe. Shocks with smaller βe are associated with larger ΔSe. We use the values of ΔSe, ΔTe and density change Δne to determine the effective adiabatic index of electrons for each shock crossing. The average effective adiabatic index is 〈γe〉=1.64±0.07.     

Constraints on Non-Flat Starobinsky f(R) Dark Energy Model

We study the viable Starobinsky f(R) dark energy model in spatially non-flat FLRW backgrounds, where f(R)=R−λRch[1−(1+R2/Rch2)−1] with Rch and λ representing the characteristic curvature scale and model parameter, respectively. We modify CAMB and CosmoMC packages with the recent observational data to constrain Starobinsky f(R) gravity and the density parameter of curvature ΩK. In particular, we find the model and density parameters to be λ−1<0.283 at 68% C.L. and ΩK=−0.00099−0.0042+0.0044 at 95% C.L., respectively. The best χ2 fitting result shows that χf(R)2≲χΛCDM2, indicating that the viable f(R) gravity model is consistent with ΛCDM when ΩK is set as a free parameter. We also evaluate the values of AIC, BIC and DIC for the best fitting results of f(R) and ΛCDM models in the non-flat universe.     

A Review of the System-Intrinsic Nonequilibrium Thermodynamics in Extended Space (MNEQT) with Applications

The review deals with a novel approach (MNEQT) to nonequilibrium thermodynamics (NEQT) that is based on the concept of internal equilibrium (IEQ) in an enlarged state space SZ involving internal variables as additional state variables. The IEQ macrostates are unique in SZ and have no memory just as EQ macrostates are in the EQ state space SX⊂SZ. The approach provides a clear strategy to identify the internal variables for any model through several examples. The MNEQT deals directly with system-intrinsic quantities, which are very useful as they fully describe irreversibility. Because of this, MNEQT solves a long-standing problem in NEQT of identifying a unique global temperature T of a system, thus fulfilling Planck’s dream of a global temperature for any system, even if it is not uniform such as when it is driven between two heat baths; T has the conventional interpretation of satisfying the Clausius statement that the exchange macroheatdeQflows from hot to cold, and other sensible criteria expected of a temperature. The concept of the generalized macroheat dQ=deQ+diQ converts the Clausius inequality dS≥deQ/T0 for a system in a medium at temperature T0 into the Clausius equalitydS≡dQ/T, which also covers macrostates with memory, and follows from the extensivity property. The equality also holds for a NEQ isolated system. The novel approach is extremely useful as it also works when no internal state variables are used to study nonunique macrostates in the EQ state space SX at the expense of explicit time dependence in the entropy that gives rise to memory effects. To show the usefulness of the novel approach, we give several examples such as irreversible Carnot cycle, friction and Brownian motion, the free expansion, etc.     

Four-Objective Optimization of an Irreversible Stirling Heat Engine with Linear Phenomenological Heat-Transfer Law

This paper combines the mechanical efficiency theory and finite time thermodynamic theory to perform optimization on an irreversible Stirling heat-engine cycle, in which heat transfer between working fluid and heat reservoir obeys linear phenomenological heat-transfer law. There are mechanical losses, as well as heat leakage, thermal resistance, and regeneration loss. We treated temperature ratio x of working fluid and volume compression ratio λ as optimization variables, and used the NSGA-II algorithm to carry out multi-objective optimization on four optimization objectives, namely, dimensionless shaft power output P¯s, braking thermal efficiency ηs, dimensionless efficient power E¯p and dimensionless power density P¯d. The optimal solutions of four-, three-, two-, and single-objective optimizations are reached by selecting the minimum deviation indexes D with the three decision-making strategies, namely, TOPSIS, LINMAP, and Shannon Entropy. The optimization results show that the D reached by TOPSIS and LINMAP strategies are both 0.1683 and better than the Shannon Entropy strategy for four-objective optimization, while the Ds reached for single-objective optimizations at maximum P¯s, ηs, E¯p, and P¯d conditions are 0.1978, 0.8624, 0.3319, and 0.3032, which are all bigger than 0.1683. This indicates that multi-objective optimization results are better when choosing appropriate decision-making strategies.     

Multiple Reflections for Classical Particles Moving under the Influence of a Time-Dependent Potential Well

We study the dynamics of classical particles confined in a time-dependent potential well. The dynamics of each particle is described by a two-dimensional nonlinear discrete mapping for the variables energy en and phase ϕn of the periodic moving well. We obtain the phase space and show that it contains periodic islands, chaotic sea, and invariant spanning curves. We find the elliptic and hyperbolic fixed points and discuss a numerical method to obtain them. We study the dispersion of the initial conditions after a single iteration. This study allows finding regions where multiple reflections occur. Multiple reflections happen when a particle does not have enough energy to exit the potential well and is trapped inside it, suffering several reflections until it has enough energy to exit. We also show deformations in regions with multiple reflection, but the area remains constant when we change the control parameter NC. Finally, we show some structures that appear in the e0e1 plane by using density plots.     

Information Shift Dynamics Described by Tsallis q = 3 Entropy on a Compact Phase Space

Recent mathematical investigations have shown that under very general conditions, exponential mixing implies the Bernoulli property. As a concrete example of statistical mechanics that are exponentially mixing we consider the Bernoulli shift dynamics by Chebyshev maps of arbitrary order N≥2, which maximizes Tsallis q=3 entropy rather than the ordinary q=1 Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd N. We show that the value of the fine structure constant αel=1/137 is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for N=3.     

A Fast and Efficient Ensemble Transfer Entropy and Applications in Neural Signals

The ensemble transfer entropy (TEensemble) refers to the transfer entropy estimated from an ensemble of realizations. Due to its time-resolved analysis, it is adapted to analyze the dynamic interaction between brain regions. However, in the traditional TEensemble, multiple sets of surrogate data should be used to construct the null hypothesis distribution, which dramatically increases the computational complexity. To reduce the computational cost, a fast, efficient TEensemble with a simple statistical test method is proposed here, in which just one set of surrogate data is involved. To validate the improved efficiency, the simulated neural signals are used to compare the characteristics of the novel TEensemble with those of the traditional TEensemble. The results show that the time consumption is reduced by two or three magnitudes in the novel TEensemble. Importantly, the proposed TEensemble could accurately track the dynamic interaction process and detect the strength and the direction of interaction robustly even in the presence of moderate noises. The novel TEensemble reaches its steady state with the increased samples, which is slower than the traditional method. Furthermore, the effectiveness of the novel TEensemble was verified in the actual neural signals. Accordingly, the TEensemble proposed in this work may provide a suitable way to investigate the dynamic interactions between brain regions.     

Pentapartite Entanglement Measures of GHZ and W-Class State in the Noninertial Frame

We study both pentapartite GHZ and W-class states in the noninertial frame and explore their entanglement properties by carrying out the negativities including 1-4, 2-3, and 1-1 tangles, the whole entanglement measures such as algebraic and geometric averages π5 and Π5, and von Neumann entropy. We illustrate graphically the difference between the pentapartite GHZ and W-class states. We find that all 1-4, 2-3 tangles and the whole entanglements, which are observer dependent, degrade more quickly as the number of accelerated qubits increases. The entanglements of these quantities still exist even at the infinite acceleration limit. We also notice that all 1-1 tangles of pentapartite GHZ state Nαβ=NαIβ=NαIβI=0 where α,β∈(A,B,C,D,E), whereas all 1-1 tangles of the W-class state Nαβ,NαIβ and NαIβI are unequal to zero, e.g., Nαβ=0.12111 but NαIβ and NαIβI disappear at r>0.61548 and r>0.38671, respectively. We notice that the entanglement of the pentapartite GHZ and W-class quantum systems decays faster as the number of accelerated particles increases. Moreover, we also illustrate the difference of von Neumann entropy between them and find that the entropy in the pentapartite W-class state is greater than that of GHZ state. The von Neumann entropy in the pentapartite case is more unstable than those of tripartite and tetrapartite subsystems in the noninertial frame.     

On the Thermodynamics of the q-Particles

Since the grand partition function Zq for the so-called q-particles (i.e., quons), q∈(−1,1), cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for q=0, and its q-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to q∈[−1,1]. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., q=0) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor 1/n! in the n term of the power series expansion with respect to the fugacity z. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons q∈(0,1).     

Analysis on Optimal Error Exponents of Binary Classification for Source with Multiple Subclasses

We consider a binary classification problem for a test sequence to determine from which source the sequence is generated. The system classifies the test sequence based on empirically observed (training) sequences obtained from unknown sources P1 and P2. We analyze the asymptotic fundamental limits of statistical classification for sources with multiple subclasses. We investigate the first- and second-order maximum error exponents under the constraint that the type-I error probability for all pairs of distributions decays exponentially fast and the type-II error probability is upper bounded by a small constant. In this paper, we first give a classifier which achieves the asymptotically maximum error exponent in the class of deterministic classifiers for sources with multiple subclasses, and then provide a characterization of the first-order error exponent. We next provide a characterization of the second-order error exponent in the case where only P2 has multiple subclasses but P1 does not. We generalize our results to classification in the case that P1 and P2 are a stationary and memoryless source and a mixed memoryless source with general mixture, respectively.     

Search for Sub-Solar Mass Binaries with Einstein Telescope and Cosmic Explorer

A possible detection of sub-solar mass ultra-compact objects would lead to new perspectives on the existence of black holes that are not of astrophysical origin and/or pertain to formation scenarios of exotic ultra-compact objects. Both possibilities open new perspectives for better understanding of our universe. In this work, we investigate the significance of detection of sub-solar mass binaries with components mass in the range: 10−2M⊙ up to 1M⊙, within the expected sensitivity of the ground-based gravitational waves detectors of third generation, viz., the Einstein Telescope (ET) and the Cosmic Explorer (CE). Assuming a minimum of amplitude signal-to-noise ratio for detection, viz., ρ=8, we find that the maximum horizon distances for an ultra-compact binary system with components mass 10−2M⊙ and 1M⊙ are 40 Mpc and 1.89 Gpc, respectively, for ET, and 125 Mpc and 5.8 Gpc, respectively, for CE. Other cases are also presented in the text. We derive the merger rate and discuss consequences on the abundances of primordial black hole (PBH), fPBH. Considering the entire mass range [10−2–1]M⊙, we find fPBH<0.70 (<0.06) for ET (CE), respectively.     

A Realization Approach to Lossy Network Compression of a Tuple of Correlated Multivariate Gaussian RVs

Examined in this paper is the Gray and Wyner source coding for a simple network of correlated multivariate Gaussian random variables, Y1:Ω→Rp1 and Y2:Ω→Rp2. The network consists of an encoder that produces two private rates R1 and R2, and a common rate R0, and two decoders, where decoder 1 receives rates (R1,R0) and reproduces Y1 by Y^1, and decoder 2 receives rates (R2,R0) and reproduces Y2 by Y^2, with mean-square error distortions E||Yi−Y^i||Rpi2≤Δi∈[0,∞],i=1,2. Use is made of the weak stochastic realization and the geometric approach of such random variables to derive test channel distributions, which characterize the rates that lie on the Gray and Wyner rate region. Specific new results include: (1) A proof that, among all continuous or finite-valued random variables, W:Ω→W, Wyner’s common information, C(Y1,Y2)=infPY1,Y2,W:PY1,Y2|W=PY1|WPY2|WI(Y1,Y2;W), is achieved by a Gaussian random variable, W:Ω→Rn of minimum dimension n, which makes the two components of the tuple (Y1,Y2) conditionally independent according to the weak stochastic realization of (Y1,Y2), and a the formula where di∈(0,1),i=1,…,n are the canonical correlation coefficients of the correlated parts of Y1 and Y2, and a realization of (Y1,Y2,W) which achieves this. (2) The parameterization of rates that lie on the Gray and Wyner rate region, and several of its subsets. The discussion is largely self-contained and proceeds from first principles, while connections to prior literature is discussed.     

Analyses of pp,Cu–Cu,Au–Au and Pb–Pb Collisions by Tsallis-Pareto Type Function at RHIC and LHC Energies

The parameters revealing the collective behavior of hadronic matter extracted from the transverse momentum spectra of π+, π−, K+, K−, p, p¯, Ks0, Λ, Λ¯, Ξ or Ξ−, Ξ¯+ and Ω or Ω¯+ or Ω+Ω¯ produced in the most central and most peripheral gold–gold (Au–Au), copper–copper (Cu–Cu) and lead–lead (Pb–Pb) collisions at 62.4 GeV, 200 GeV and 2760 GeV, respectively, are reported. In addition to studying the nucleus–nucleus (AA) collisions, we analyzed the particles mentioned above produced in pp collisions at the same center of mass energies (62.4 GeV, 200 GeV and 2760 GeV) to compare with the most peripheral AA collisions. We used the Tsallis–Pareto type function to extract the effective temperature from the transverse momentum spectra of the particles. The effective temperature is slightly larger in a central collision than in a peripheral collision and is mass-dependent. The mean transverse momentum and the multiplicity parameter (N0) are extracted and have the same result as the effective temperature. All three extracted parameters in pp collisions are closer to the peripheral AA collisions at the same center of mass energy, revealing that the extracted parameters have the same thermodynamic nature. Furthermore, we report that the mean transverse momentum in the Pb–Pb collision is larger than that of the Au–Au and Cu–Cu collisions. At the same time, the latter two are nearly equal, which shows their comparatively strong dependence on energy and weak dependence on the size of the system. The multiplicity parameter, N0 in central AA, depends on the interacting system’s size and is larger for the bigger system.