Entropy doi: 10.3390/e28101144

Authors: Samira Ebrahimi Ebrahimi AslMamaghani Hossein Rangani Jahromi Rosario Lo Lo Franco

Relativistic motion introduces decoherence that limits quantum parameter estimation. Earlier treatments of bipartite Unruh–DeWitt detectors have often restricted the reduced state to the X-state form. We consider single-qubit and isolated-ancilla two-qubit probes whose reduced state is not restricted to this form, moving along a trajectory that combines uniform acceleration with a transverse velocity. The quantum Fisher information for an initially encoded phase and for a charging time across the three velocity regimes is investigated. The two parameters respond to acceleration in distinct ways, and the isolated ancilla improves both estimations over different portions of the parameter space. Phase estimation remains nearly unaffected by the transverse velocity of the probe, whereas at short drive durations, the charging-time QFI reverses with acceleration. The separable regime that follows entanglement loss is not devoid of metrological value: both parameters can still be estimated, and the discord of the reduced state remains nonzero. These results point to quantum correlations beyond entanglement as a resource for parameter estimation in accelerated probes.

Entropy doi: 10.3390/e28101143

Authors: Yousef Alnafisah M. A. Sohaly

This paper develops a stochastic framework for epidemic spreading on regular networks based on the Fokker–Planck equation and spectral graph theory. Starting from an Itô stochastic differential equation, we derive the corresponding Fokker–Planck equation and characterize the stationary distribution of epidemic fluctuations. A fundamental connection between the stationary covariance structure and the Kirchhoff index is established, showing that network topology plays a central role in determining the magnitude of stochastic variability. Using Lyapunov methods, we prove exponential stability of the disease-free equilibrium and show that the convergence rate is governed by the algebraic connectivity of the underlying network. In addition, an exact description of the mean-square dynamics is obtained, revealing how the spectral properties of regular graphs control both transient behavior and long-term fluctuations. Numerical experiments on complete, cycle, circulant, and random regular graphs confirm the theoretical predictions and illustrate the influence of network connectivity on epidemic variability and relaxation dynamics.

Entropy doi: 10.3390/e28101142

Authors: Aaron Adair

I derive a Lorentzian correlation kernel from a maximum-entropy principle on an abstract possibility space, assuming no spacetime or field theory. Maximizing Shannon entropy over positive phase rates at fixed mean gives p(ω)=Ω−1e−ω/Ω, and averaging relative phase coherence yields ρ(σ)=1/(1+Ω2σ2). Given a posited array of pairwise comparison intervals, classical multidimensional scaling returns a screen, its metric, and its dimension; the dimension is an output, and the test can fail, which makes the existence of a geometry falsifiable in principle. A postulated finite invariant speed supplies the Lorentzian cone. Under a stated locality assumption the Fisher coordinate of the ensemble is a shift-symmetric scalar that admits no potential within the Fisher sector; with Lovelock’s theorem this gives Gμν+Λgμν=(8πG/c4)Tμν, with the coupling normalized thermodynamically and G an input. A Gaussian treatment of the patches solves for the correlation scale, giving λG≃1.302ℓP. Treating Planck patches as binary variables on a square lattice instead sets λ=2ℓP, the largest scale binary patches can realize, before any entropy comparison. At that scale the maximum-entropy Ising multi-information agrees with 1−1/(4ln2)=0.6393, the Bekenstein–Hawking deficit, within 0.05%: transfer-matrix bounds give 0.6135, and 0.6389 under one assumption; a pre-registered extrapolation gives 0.6391. No anti-de Sitter asymptotics are used.

Entropy doi: 10.3390/e28101141

Authors: Rafał Rak

This study introduces the Visibility-Graph-Based Relative Strength Index (VGRSI), a causal indicator that selects and implicitly weights price increments using backward natural visibility. Independent parameters specify the aggregation window and the backward visibility range. Two aggregation variants use the magnitudes and counts of positive and negative contributions to produce values on a 0–100 scale. The VGRSI was tested in an automated walk-forward experiment based on 30-day optimisation windows followed by 7-day walk-forward windows. This study covered DJI30, EUR/USD, and XAU/USD over 503 trading days in 2024–2025. The VGRSI strategy produced positive cumulative net profits for all three instruments, whereas the tested classical-RSI strategy and the VGRSI variant without visibility selection produced cumulative losses under the same walk-forward procedure and trading conditions. Because the analysis covered only three instruments and one historical period, the results should be treated as preliminary.

Entropy doi: 10.3390/e28101140

Authors: Tuğçe Hancı Fatma Gül Akgül

Accelerated life testing is an efficient methodology that enables the acquisition of failure time data in a shorter period by subjecting test units to different stress levels, compared to tests conducted under normal operating conditions. In this study, it is assumed that the lifetime of a product under a constant stress level follows an exponentiated half-logistic distribution and the model parameters, reliability function and mean time to failure function are estimated using eleven different frequentist methods, namely, maximum likelihood, least squares, weighted least squares, Cramér–von Mises, Anderson–Darling, right-tail Anderson–Darling, left-tail Anderson–Darling, second-order left-tail Anderson–Darling, maximum product of spacings, minimum spacing absolute distance, and minimum spacing absolute-log distance. The performances of the estimators are compared through an extensive Monte Carlo simulation study. Furthermore, a real dataset is analyzed to evaluate the practical effectiveness of the proposed methods.

Entropy doi: 10.3390/e28101139

Authors: Huilong Ren

Path-dependent entropic Lagrangians formulate irreversible evolution through terminal variations of accumulated process power. This paper establishes a mathematical basis for calibrated terminal spending and weak diffusion power-port decompositions. Endpoint calibration and cocycle additivity determine a unique oriented spending increment, while local selector fields resolve its channel contributions. The diffusion power-port decomposition extends from Sobolev fields with normal-trace boundary power to finite-energy fields through a distributional balance component. These constructions are combined in a synchronized mixed thermo-diffusion functional. A Cahn–Hilliard specialization illustrates the weak channel structure and the associated energy balance.

Entropy doi: 10.3390/e28101138

Authors: Qian Yin Haoxian Li Zhishuang Wang Kaiyao Wang Jingmin Li Zhiyong Hong

The spread of epidemics in real social systems is often coupled with competitive information diffusion, higher-order group interactions, and dynamic contact behaviors. To capture these mechanisms, we develop a bilayer competitive information–epidemic coupled spreading framework. The upper layer is a weighted higher-order network incorporating both pairwise spreading and 2-simplex-driven group spreading, while the lower layer is a weighted activity-driven network describing time-varying physical contacts regulated by a policy relaxation parameter. Based on the microscopic Markov chain framework, we establish node-level evolution equations and derive a spectral condition for the local invasion threshold, which is evaluated numerically. Monte Carlo simulations show good agreement with the predicted steady-state densities. These findings indicate that enhancing the diffusion of the more protective information reduces steady-state epidemic prevalence and raises the epidemic threshold, whereas preference for the less protective information weakens the overall protective effect. Higher-order interactions exert asymmetric effects on competing information, while social tie weights and activity-driven contacts also significantly influence epidemic spreading. Moreover, the epidemic threshold is jointly affected by recovery probability, activity heterogeneity, information-layer topology, and policy regulation. These findings reveal the cross-layer coupling between higher-order information diffusion and temporal physical contacts.

Entropy doi: 10.3390/e28101137

Authors: Swetha Pandithahalli Mahadevaswamy Prasanna Bantaganahalli Thimmappa

Wireless Sensor Networks (WSNs) remain vulnerable to clone-node attacks, in which malicious replicas use the identity or credentials of legitimate sensor nodes. This study presents an intelligent learning-based approach for clone-node detection using behavioral characteristics related to communication, topology, traffic, distance, and energy consumption. Since the WSN-DS dataset does not contain a native clone-node class, a controlled synthetic clone-node scenario was developed from legitimate sensor observations by preserving node identities while introducing behavioral changes associated with replication. A group-aware evaluation strategy was used to evaluate Logistic Regression, Random Forest, 1D-CNN, and a correlation-enhanced DCNN under the same experimental conditions. The results showed different detection characteristics among the evaluated models. Random Forest achieved the strongest overall classification performance, while Logistic Regression also provided strong discrimination with a low false-positive rate. The deep-learning models achieved high clone-detection recall, with the correlation-enhanced DCNN reaching 99.46% recall. These results demonstrate that different learning approaches provide different advantages for clone-node detection and highlight the importance of considering accuracy, detection sensitivity, and false-positive behavior together. The findings provide a useful basis for developing more reliable and adaptive intelligent clone-node detection methods for resource-constrained WSN environments.

Entropy doi: 10.3390/e28101136

Authors: Ahmad Nazzal Hanada Taha-Thomure

Lexical-disambiguation tasks with supplied context do not reveal whether readers would choose to seek evidence before deciding. Therefore, we adapted the spatial T-maze in the active-inference package pymdp to show how agents seek and use contextual information under lexical uncertainty. Agents could choose between two readings or request a contextual hint. We performed two main sets of simulations. First, deterministic simulations used fully informative hints to reproduce context seeking followed by lexical commitment. A hint shifted belief over two readings from equal probabilities (0.50, 0.50) to certainty (1.00, 0.00), reducing entropy by one bit. Second, lexical decision trials were repeated with informative and uninformative hints across 32 paired simulation runs. Agents learned the relationship between the intended reading and the observed hint. Informative experience increased both the probability and simulated frequency of requesting context. This increase persisted when trials ended after one response and feedback. A hint information-gain ablation, in which policies were rescored without crediting the information expected from hints, eliminated the learning-related increase in hint-request probability. These simulations demonstrate how experience changes the expected informational value of context within a lexical task, providing a computational proof of concept for subsequent behavioral experiments.

Entropy doi: 10.3390/e28101135

Authors: Wanjing He Yufeng Zhuang Yanzhu Hu

Network growth and node-removal robustness are often compared while node arrivals and link budgets also vary. We hold these two quantities fixed using the email-Eu-core and CollegeMsg temporal communication networks and vary only the selection of older attachment targets. Uniform, degree-only, activity-only, combined degree-activity, and hybrid kernels are evaluated with 30 independent graph realizations per setting. Random failures use 50 deletion orders per graph; intentional attacks use static degree, exact static betweenness, and degree recalculated after every deletion. Uncertainty is summarized by realization-level 95% confidence intervals. Under static degree attack, the mean integrated robustness falls from 0.456 to 0.313 for email-Eu-core and from 0.388 to 0.176 for CollegeMsg when uniform attachment is replaced by the combined kernel. Both single-factor kernels are less damaging than their combination, and the ranking of the five counterfactual rules is unchanged under adaptive degree removal. The combined kernel has higher intact-network global efficiency but lower random-failure robustness than uniform attachment in both data sets; no universal reversal between random and intentional attacks is observed. Susceptibility peaks are finite-network, grid-dependent fragmentation indicators, not estimates of thermodynamic critical points. The results identify how retrospective target-selection counterfactuals redistribute structural vulnerability and separate this effect from dynamical recovery.

Entropy doi: 10.3390/e28101134

Authors: Linghao Zhang Qing Zhang Bo Pang Siyu Xiang Wenjuan Wu

Autoregressive probability models improve the rate–distortion performance of learned image compression, but their spatial conditioning introduces serial neural inference and high decoding latency. This paper presents a latency-oriented learned image compression framework that jointly addresses sequential entropy estimation, decoder computation, and dense intermediate activation responses. First, a dependency-decoupled entropy model estimates all latent distribution parameters in parallel from hyperprior statistics, eliminating the neural autoregressive loop. Second, a lightweight synthesis transform uses compact convolutional mappings, reduced channel widths, and a short progressive reconstruction path. Third, a differentiable activation-magnitude regularizer penalizes the normalized L1 magnitude of intermediate decoder activations and is introduced gradually during rate–distortion training to encourage sparse feature responses. The activation regularizer is treated as a representation-level training constraint rather than as a direct estimator of dense-convolution arithmetic. Experiments on Kodak and BSD100 show that the proposed model consistently improves on the hyperprior baseline over the evaluated bitrate range. On Kodak, it achieves BD-rate reductions of 7.9% and 9.7% relative to hyperprior in terms of PSNR and MS-SSIM, respectively. At approximately 0.30 bpp, the proposed model achieves 32.94 dB PSNR and 0.9745 MS-SSIM with 8.10 M parameters, 165.3 GMACs, and a decoding time of 35.6±1.0 ms. Compared with the serial autoregressive joint baseline at a comparable operating point, this corresponds to a 95.0% reduction in decoding latency.

Entropy doi: 10.3390/e28101133

Authors: Hai Xu Yanan Zhou Haoxiang Liu Qi Yuan

Due to the complex practical working conditions, fault diagnosis is very challenging in rotary machinery. Rotary machinery is often used in places with a high temperature, strong corrosion, or where acceleration sensors are hard to install. Thus, acoustical signal processing becomes a powerful tool for diagnosing faults in rotary machinery. This paper proposes a novel strategy called intrinsic multidimensional Gnome entropy (IMGE) analysis that can extract the quantitative features of vibro-acoustic signals. The method consists of signal analysis using variational mode decomposition (VMD), and feature extraction using the proposed mean Gnome entropy (MGE). Multidimensional MGE solves the issue of negative computation values in the original algorithm and improves the accuracy and stability of entropy in multiple dimensions. VMD is used to decompose vibro-acoustic signals into a series of band-limited intrinsic mode functions (BLIMFs) denoting the dynamical characteristics of signals. Multidimensional MGE values of BLIMFs are extracted as quantitative features and inputted into a convolutional neural network (CNN) to diagnose faults in rotary machinery. The improved performance of the proposed framework under the tested experimental conditions is verified by theoretical derivation, numerical simulations, and research on practical applications.

Entropy doi: 10.3390/e28101132

Authors: Wei Gao

The free-energy machine (FEM) relaxes discrete combinatorial optimization by minimizing expected energy plus a temperature-dependent negative-entropy penalty over product distributions. This contribution investigates the computational and mathematical role of that entropy term. At high temperature, entropy drives the optimizer toward the uniform product law; explicit relative-entropy and total-variation bounds quantify this effect, while an influence condition and a restricted Hessian characterize uniqueness, contraction, and the first possible loss of stability. As temperature decreases, the entropy coefficient vanishes. Uniform and variational convergence then connect free energy to the original discrete objective, obtaining an inverse-temperature excess-energy bound, ground-state concentration, and entropy selection among degenerate optima. Extraction bounds, path tracking, perturbation stability, and factorwise complexity further distinguish the effects of entropy, annealing, and replica selection. A batched annealed FEM algorithm is evaluated on exactly enumerable Ising, MaxCut, and Max-3SAT instances; the results reveal the predicted entropy-to-energy transition and show problem-dependent benefits of annealing and replication. For empirical-loss energies, a PAC-Bayes inequality additionally calibrates inverse temperature to sample size. The analysis characterizes entropy-dominated, instability, and energy-dominated regimes without attributing global guarantees to the nonconvex optimizer.

Entropy doi: 10.3390/e28101131

Authors: Zhongyi Tang Xiaoqiang Wang Bingchun Li

Hyperspectral unmixing aims to recover endmembers and their corresponding abundances from mixed pixels, which is a key task in quantitative hyperspectral remote sensing. Although convolutional neural networks (CNNs) are effective in modeling local spatial–spectral features, their ability to capture long-range spatial dependencies remains limited. Transformers can enhance global feature interactions but typically incur high computational complexity. To address these issues, this study proposes an abundance-guided Mamba network for hyperspectral unmixing, termed AGMNet. Built upon an interpretable autoencoder-based unmixing framework, the proposed method uses CNNs to extract local spatial–spectral features and introduces a lightweight spatial Mamba branch to incorporate whole-image contextual information. Meanwhile, an abundance estimation mechanism with physical constraints and a linear decoder consistent with the linear mixing model are employed to achieve physically interpretable endmember-abundance decomposition. Experiments on three public datasets, namely Samson, Jasper Ridge, and Urban, show that AGMNet provides accurate and stable abundance recovery, achieving the lowest average root mean square error (RMSE) across all three datasets. Its spectral angle distance (SAD) performance is more dataset-dependent: AGMNet obtains the best average SAD on Urban and competitive SAD values on Samson and Jasper Ridge, but it does not consistently achieve the lowest SAD for every endmember. These results indicate that AGMNet is particularly effective for abundance recovery while retaining physically interpretable spectral reconstruction, although further improvement of endmember spectral-shape consistency remains necessary.

Entropy doi: 10.3390/e28101130

Authors: Marzieh Hassanshahi Varposhti Mahyar Shahsavari Marcel van Gerven

Spiking Neural Networks (SNNs) promise ultra-low-power, event-driven computation; however, current efforts to integrate self-attention into spiking transformers predominantly rely on global attention mechanisms, introducing growing token complexity when scaling and diluting discriminative feature representations. We introduce Spike-Driven Neighborhood Attention (SDNA) over the full k×k neighborhood in two forms: Pairwise SDNA, which gates each query–neighbor interaction before aggregating, and Linear SDNA, which exploits the absence of a softmax in spiking attention to pool the key–value interaction over the window and gate it once, evaluating the spiking gate per position rather than per query–neighbor pair. Both operators are placed in a fully local hierarchical backbone that increases the receptive field without any global attention layer. Across static and neuromorphic benchmarks the two operators are competitive with global spike-driven transformers at comparable or lower parameter counts. Pairwise SDNA achieves the highest accuracy among the compared spike-driven transformers on CIFAR-10 at 97.24%, while Linear SDNA achieves the highest accuracy on N-Caltech101 at 89.65%. These results show that competitive performance can be achieved using a fully local hierarchical attention design, without requiring an explicit global-attention stage.

Entropy doi: 10.3390/e28101129

Authors: Majed Albaity Muhammad Rahim

Renewable power generation plays a central role in the transition toward sustainable, low-carbon energy systems. However, selecting an appropriate technology remains challenging because technical, economic, environmental, and operational criteria may conflict, while expert judgments may involve uncertainty and diversity. To address these challenges, this study proposes a fractional fuzzy multi-criteria group decision-making framework for evaluating and ranking renewable power generation technologies under uncertain expert assessments. A new family of aggregation operators is developed by incorporating fractional fuzzy information into the Einstein operational framework, providing a mechanism for representing and aggregating uncertain assessments. An entropy-based weighting mechanism is further employed to determine criterion weights from the available decision information. The proposed components are integrated into a unified decision procedure that combines expert weighting, fractional fuzzy aggregation, criterion weighting, and distance-based TOPSIS assessment within a common computational framework. The numerical case study considers technical, economic, environmental, and operational criteria for renewable power generation technology selection. A sensitivity analysis examines the influence of the fractional parameters on the resulting rankings, while a comparative analysis with existing decision-making approaches is conducted to examine the consistency of the obtained results. The findings demonstrate the application of the proposed framework under the specified assessment conditions and illustrate the role of fractional fuzzy information and Einstein-based aggregation in the decision process. The study provides an integrated approach for incorporating uncertain expert information into renewable energy technology selection.

Entropy doi: 10.3390/e28101128

Authors: Yicheng Yu Wangyu Wu Renwen Ma

Low-rank adaptation (LoRA) fine-tunes a large model with a low-rank update over frozen weights. The update travels as a separate adapter file that is easy to copy, alter, and re-upload; so, a hub may have to decide whether a disputed file derives from a registered adapter. Since an altered copy no longer matches the original byte for byte, the usual tool is a fingerprint, secret measurements of the weights that survive moderate edits. Verifying a fingerprint currently requires disclosing either the measurements, which a copier can then evade, or the possibly confidential weights. ZK-AdapterMark is, to our knowledge, the first to verify a LoRA fingerprint in zero knowledge: a hub confirms that a disputed file passes a registered fingerprint without learning the secret or the weights. The proof is practical because our fingerprint is expressed over the two low-rank factors, making the complete proof circuit up to 44 times smaller than rebuilding the update. The acceptance threshold fixes a relative-entropy bound on the probability of accepting an unrelated adapter, which gives it an information-theoretic footing. Across ten model families, the fingerprint survives moderate quantization, noise, and rank reduction, and no pair among 904 independently trained adapters is accepted.

Entropy doi: 10.3390/e28101127

Authors: Vera Pecorino Giovanni Calabrò Alessandro Pluchino Giuseppe Inturri Salvatore Cavalieri

This study applies Transfer Entropy (TE) to floating car data from the urban road network of Catania (Italy) to investigate directional dependencies in traffic flows and identify relevant patterns of information transfer. Unlike traditional correlation-based approaches, TE captures asymmetric directed information dependencies between road segments, enabling the identification of predictive relationships among congestion states across the network. We constructed TE-based influence networks using traffic speed data collected over 15 min intervals across 86 road segments over six weeks. The analysis examined information transfer using history lengths of 1, 2, and 3 observations, corresponding to 15, 30, and 45 min of past information, revealing distinct patterns between time windows. Key findings include the identification of recurring information-transfer relationships across multiple history lengths, and road segments with comparatively high outgoing or incoming information strength. Notably, the morning and evening time windows differ in network density, total strength and highest-ranked nodes, although some of these patterns depend on the discretisation and history length adopted, highlighting the dynamic nature of urban traffic systems. This approach shifts focus from traditional traffic prediction to network prioritisation, allowing the identification of candidate locations for enhanced traffic monitoring with minimal data requirements. The methodology provides practical insights for optimising sensor placement and informing further traffic-management analysis based on predictive influence patterns rather than congestion levels alone.

Entropy doi: 10.3390/e28101124

Authors: Ralph V. Chamberlin

Nanothermodynamics is the theory describing how large systems subdivide into equilibrium distributions of small subsystems. A key ingredient is Hill’s subdivision potential (E) that ensures adherence to the 1st and 2nd laws of thermodynamics in systems of any size. In this review and reassessment, it is emphasized that nanothermodynamics gives new insight into many experiments, theories, and simulations. Measurements establishing the need for nanothermodynamics show thermodynamic heterogeneity from multiple effective temperatures (Ti) inside most types of materials. One theoretical result that requires E=0 is the stable solution of Ising’s original model for finite chains of interacting spins, a solution Ising could not have found 40 years before Hill’s work. Another result is a novel solution to Gibbs’ paradox that makes the entropy of the semiclassical ideal gas exactly extensive, even for small systems. Molecular dynamics simulations reveal how a standard fluctuation relation is modified when local degrees of freedom fluctuate faster than their coupling to the heat bath, consistent with the measured thermodynamic heterogeneity. Simulations of a Creutz-like model, containing Ising spins coupled to an explicit heat bath of Einstein oscillators, are used to study the 2nd law. It is found that maximizing the total entropy (St) requires an intrinsically irreversible step, providing a counterexample to the usual claim that statistical mechanics emerges from reversible dynamics. Furthermore, fluctuations of this model are best described by Einstein’s reversal of Boltzmann’s relation and the 2nd law, not by recent fluctuation theorems.

Entropy doi: 10.3390/e28101126

Authors: Wiktor Bohdan Daszczuk Stanisław Jerzy Niepostyn

Entropy-based metrics can provide a quantitative representation of the information content of UML models, but their practical usefulness depends on a systematic and reproducible calibration of the parameters used to construct the underlying probability distribution. This paper presents FBS26, an extension of the FBS metric that introduces calibrated weights for UML element kinds while retaining the normalized Shannon entropy formulation. The proposed calibration procedure combines three criteria: avoiding false lack-of-information cases, reducing non-monotonic behavior of the metric, and achieving a pronounced decrease in the metric for balanced diagram series. The criteria are applied according to a predefined lexicographic priority. The approach is evaluated using a controlled series of UML class diagrams and 123 UML class diagrams from four industrial software projects, selected from a repository containing more than 2000 UML diagrams from 18 projects. Architectural maturity labels, assigned independently by an experienced software architect, serve as reference data for examining empirical thresholds for structural information. The results show that systematic weight calibration removes the undesirable behavior observed for the considered FBS16 configurations and provides a more stable basis for comparing UML class diagrams. The industrial analysis further indicates differences in the structural entropy ranges of commonly distinguished diagram categories. FBS26, therefore, provides a quantitative information-content measure that can support expert-based assessment of the architectural maturity of UML class diagrams.

Entropy doi: 10.3390/e28101125

Authors: Xingyan Lu Ze Liu Xinyu Li Yan Xiong Jun Luo Huabing Zhou

The effective tuning of extensive, high-dimensional parameter sets within nuclear power simulation systems presents considerable challenges for conventional, experience-based methodologies. Such approaches often exhibit limitations including slow convergence rates, susceptibility to local optima, and inadequate adaptability across diverse scenarios. This study introduces an automated parameter tuning strategy employing the Soft Actor–Critic (SAC) reinforcement learning approach. A SAC-based agent is formulated, where state representations, action spaces, and reward functions are specifically defined to tune parameters under predefined constraints. This facilitates the automated adjustment of complex parameter configurations. Comparative experimental evaluations indicate that the proposed SAC method reaches a solution quality comparable to that of the established baseline techniques while producing markedly more consistent results across independent runs, offering a reliable and repeatable methodology for parameter tuning in this domain.

Entropy doi: 10.3390/e28101123

Authors: Hengzhou Xu Dongshu Chang Jinlan Guo Mengmeng Xu Qian Wang

This paper studies a family of (3,29)-regular bipartite graphs introduced by Tanner for low-density parity-check (LDPC) codes. For a prime p≡1(mod87), the graph has 3p left vertices and 29p right vertices. In the associated LDPC code, the right vertices (there are 29p of them, each of degree 3) serve as the variable nodes, while the left vertices (3p vertices, each of degree 29) serve as the check nodes. Using the Euclidean division algorithm, we completely determine the girth characterization of these graphs. A detailed cycle analysis yields necessary and sufficient conditions for the existence of cycles of lengths 4, 6, 8 and 10, which are translated into polynomial equations over Fp involving a primitive 87th root of unity. By computing integer resultants via the Euclidean division algorithm, we identify the prime sets S6, S8 and S10 for which the girth equals 6, 8 or 10, respectively. The sets are constructed recursively so that no prime associated with a shorter cycle is included in a set corresponding to a longer cycle. For all other admissible primes, the girth attains the maximum possible value of 12. Finally, we construct explicit LDPC codes from these graphs and evaluate their error-correcting performance via extrinsic information transfer (EXIT) charts and additive white Gaussian noise (AWGN) simulations. The results indicate competitive performance in the simulated high-rate setting and suggest potential for high-rate applications.

Entropy doi: 10.3390/e28101122

Authors: Yosef Akhtman

The paper derives the fundamental physical units in a universe defined over a finite holographic substrate: a finite totality whose capacity bounds the state count of every embedded observer. The units are anchored at the substrate’s four horizons: the Planck length, momentum, time and energy. The constants c, ℏ, G, kB follow uniquely from this quartet, over realisations stated with their falsifiers: c the ratio across the two Fourier pairs, ℏ the product within each conjugate pair, kB the mixed product, G the normalisation to the totality. The quartet admits one cancellation identity and no further independent relation; on an admissible substrate the constants are exact residues read as unit-face magnitudes. Beneath the quartet, dimensional analysis is a graded modular domain algebra on the framed shell, recovering the classical calculus exactly inside the sub-capacity window. The domain of ℏ is the unit flag; its torsion-free surrogate is the classical mass dimension, temperature inherits the acceleration domain, and the flag cancels in every count-valued comparison. Two predictions are staked: a resolution floor on single coherent registration and the exact constancy of the constants’ dimensionless ratios. Every theorem of the paper is numerically verified and formalized in Lean 4.

Entropy doi: 10.3390/e28101121

Authors: Duowen Chen Haoze Lv Xinze Li Wenjie Liang Kailin Bai Xin Zhao Linlin Cao Xiaolong Zhou

Aiming at the difficulties in extracting gear fault feature information and the unsatisfactory performance of commonly used machine learning classification models in terms of feature separability and stable within-distribution recognition, this paper proposes a fault diagnosis method combining variational mode decomposition (VMD) optimized by the coati optimization algorithm (COA) and a BP neural network optimized by the sparrow search algorithm (SSA). Firstly, a composite index based on permutation entropy and mutual information entropy is adopted as the fitness function, and the COA is used to optimize the key parameters of VMD to improve the decomposition accuracy of the VMD algorithm. Secondly, the intrinsic mode function (IMF) components sensitive to fault information are selected for signal reconstruction according to the time–frequency-domain comprehensive evaluation factor algorithm, and the refined composite multiscale fuzzy entropy (RCMFE) is used to extract the features of the reconstructed signal to form a state feature set. Finally, the SSA is employed to optimize the parameters of the BP model to achieve accurate diagnosis of gear working conditions and fault types. The proposed method is verified using measured gear single and compound fault data and compared with commonly used signal decomposition methods and machine learning classification models. The results show that the proposed method can effectively extract gear fault feature information and has the advantages of high accuracy, good feature separability and stable within-distribution recognition. It can achieve ideal diagnosis accuracy for both single and compound gear faults, providing a certain reference for solving such problems.

Entropy doi: 10.3390/e28101120

Authors: Terrence Lingzhu Zhao Zhou Shangnan

Conservation laws reshape quantum information dynamics by restricting how scrambling can occur. In monitored circuits, this raises a natural dynamical question: when a symmetry constraint is gradually relaxed, how quickly do the entanglement dynamics forget the symmetric regime? We address this question in one-dimensional monitored random circuits by driving the two-qubit gate ensemble from exact U(1) conservation toward unconstrained Haar dynamics. The response is not instantaneous: the entanglement retains a finite-depth memory of the conserving dynamics and crosses over toward the unconstrained behavior over a rate-dependent time window. Operational 25%, 50%, and 75% crossover times all move to later circuit depths for slower drives. At p=0.4, the same rate ordering is also observed through L=32 in a finite-time matrix-product-state consistency check with a limited trajectory ensemble. By contrast, the weak fixed perturbations studied here, ϵ(t)=ϵ0≤0.1, remain close to the conserving reference over the same finite observation window. Within this finite-time window, the driven crossover is therefore not reproduced merely by waiting under the fixed perturbations tested. Purification dynamics from a maximally mixed state exhibit the same qualitative response. We therefore interpret the results as a finite-time dynamical crossover from symmetry-constrained dynamics toward less constrained scrambling, rather than as evidence for a universal crossover exponent or a microscopic noise law. The study provides a controlled setting for asking how long symmetry-dependent information-processing behavior persists as a conservation law is deliberately weakened.

Entropy doi: 10.3390/e28101119

Authors: Marcelo Bourguignon Diego I. Gallardo Roberto Vila

A two-parameter weighted beta distribution is introduced for modeling bounded data, which has many similarities to the beta distribution. We propose a class of regression models where the response is weighted beta distributed and the two shape parameters that index the weighted beta distribution are related to covariates and regression parameters. The proposed regression model is a natural strong competitor of the beta regression model. We study the mathematical and statistical properties of the distribution, and we provide a useful interpretation of the parameters. The maximum likelihood method is used for estimating the model parameters. We conduct simulation studies to investigate the performance of the maximum likelihood estimators and the asymptotic confidence intervals of the parameters. We present an application of the proposed regression model to real bounded data.

Entropy doi: 10.3390/e28101118

Authors: Xinxia Song Zhigang Chen Liqun Chen

Encrypted recurrent inference must preserve both ciphertext depth and the numerical behavior of a feedback model. We develop OneBoot GRU, which couples the validation-locked selection of degree-three Sigmoid and degree-seven Tanh domains with a CKKS GRU cell that refreshes one hidden-state ciphertext after each step. On TwoPatterns, the locked polynomial model completed finite official-test trajectories with 92.431% macro accuracy, versus 98.114% for exact activations, and 0.830% mean domain violations. On SyntheticControl, polynomial and exact macro accuracies were 89.067% and 90.133%. Direct state-error comparisons show that improved domain coverage does not always reduce recurrent error; in a separate exploratory domain extension, screening reduced qualification tail error by 74.1% and 94.6% on ECG200 and ItalyPowerDemand. A parameter profile audited against pinned 128-bit sparse-secret references completed four encrypted recurrent steps, prefix pooling, and the trained head with maximum logit error 2.91 × 10−6 relative to the plaintext polynomial circuit. In three matched four-step runs, GRU median circuit latency was 226.343 s versus 336.981 s for an implemented polynomial LSTM. Twelve additional distinct 128-step inputs were completed under an intentionally insecure structural profile, with 12/12 prediction agreement against the plaintext polynomial reference. These experiments provide measured calibration, arithmetic-fidelity, and system-cost operating points within their stated parameter and sampling scopes.

Entropy doi: 10.3390/e28101117

Authors: Chuanyan Wu Bentao Lin Qingju Zhang Jialin Zhang Rui Gao Zhi-Ping Liu

Identifying essential proteins lays a foundation for dissecting disease mechanisms and mining therapeutic targets. Existing algorithms mostly adopt unimodal inputs and struggle to capture multi-level biological interactions from genomics, subcellular localization and proteomics. We present FusionEP, which is an adaptive cross-modal deep fusion network built upon contrastive learning to predict essential proteins. FusionEP integrates gene expression profiles, subcellular localization features, and protein interaction map structures. For gene expression information, we design a six-layer Transformer with 12 attention heads and learnable positional encodings for feature extraction. For subcellular features, a multi-scale convolutional neural network (CNN) with four kernel sizes and a channel spatial attention mechanism is designed for feature extraction. A four-layer Graph Attention Network (GAT) with dynamic attention, residual connections, and target node readout functionality is designed to extract feature from protein–protein interaction (PPI) networks. FusionEP employs an adaptive fusion gate for dynamic feature aggregation and supervised contrastive learning for cross-modal alignment. Furthermore, prototype-based classification with focal loss optimization is used to address class imbalance. We evaluated the model’s performance on the yeast biological dataset. The results showed that FusionEP achieved an AUC-ROC of 0.9523 ± 0.003 in five independent experiments. FusionEP provides an end-to-end solution for multimodal protein analysis. This work sets a new method for integrating heterogeneous biological data in precision medicine applications.

Entropy doi: 10.3390/e28101116

Authors: Othmane Benyoucef Latifa Dekhici

Stacking task adapters on a frozen domain adapter is a common approach for modular specialization of large language models, usually justified by knowledge injection; whether the surface-form graders standard in this setting can measure such specialization validly has remained unexamined. Two challenges follow: isolating the domain layer’s marginal contribution under identical task training, and determining whether measured differences reflect competence or output form. In this work, both are addressed with a matched-arm DoRA ablation on a frozen Qwen 3.5 9B hybrid: a 242 M-parameter domain expert is trained on a licensed 27.8 M-token corpus, and byte-identical task adapters are trained flat, stacked (DFE-stacking), and jointly across three tasks and five conditions, with the key extraction pair replicated at a second training seed; every grader is audited by per-item failure classification and a dual-rater blind human re-grade with consensus adjudication. The principal finding is that fine-tuning shifts output form before competence: stacking yields no seen-task benefit; an apparent generalization collapse on an unseen schema reduces to a single, highly seed-stable formatting coercion that the domain layer dampens and task-mixture training eliminates, and most automated keyword-grading failures prove to be correct paraphrases, leaving no genuine capability loss detected on any instrument. Because such style drift is induced by the training condition itself, surface-sensitive grader error becomes bias aligned with experimental arms rather than noise. Three additional experiments extend the findings: the coercion replicates on Llama 3.1 8B (60% vs. 19% untuned); a pre-registered 240-item formatting-perturbation battery across four further unseen schemas and eight conventions reveals fine-tuned conditions ranking lowest under strict grading yet highest under format-normalized grading; and a matched-compute wrong-domain control shows the domain layer’s coercion dampening to be generic substrate regularization, while value integrity under the induced drift tracked the substrate’s domain. The principal contribution lies in the identification and quantification of this confound: surface-form evaluation of adapter specialization is structurally confounded, and per-item audits of the type released herein are a necessary component of specialization studies.

Entropy doi: 10.3390/e28101115

Authors: Jie Zhang Xiaojuan Xie Shungen Xiao Ziqi Wang

Urban bus networks are exposed to cascading disruptions whose propagation depends not only on network topology but also on temporally varying traffic states and passenger-flow distributions. Existing vulnerability analyses often rely on static topology or hypothetical flows, limiting their ability to characterize traffic-dependent failure dynamics. This study develops a data-driven vulnerability framework for urban bus networks by integrating a coupled road–bus network, user-equilibrium assignment and a weighted Coupled Map Lattice (CML) model. Smart-card data are used to estimate passenger origin–destination demand, while road traffic states at four representative periods are mapped onto bus links to characterize travel-speed-dependent capacity reductions. The resulting weighted networks are evaluated using degree, strength, conventional betweenness, and traffic-flow betweenness. The improved CML model jointly incorporates topological connectivity, passenger-flow characteristics, external perturbations, and flow redistribution to simulate cascading failures under targeted and random attacks. The Harbin trunk bus network is used as an empirical case study. Results show that cascading-failure patterns are strongly traffic-state dependent: under more congested conditions, failures reach their peak propagation rate earlier. Across the tested parameter settings, perturbation intensity and coupling strength influence the timing and extent of cascade propagation, with responses varying by attack strategy and traffic condition. These findings demonstrate that incorporating time-varying traffic states and actual passenger flows can reveal vulnerability patterns that static topology-based analyses may overlook, providing a basis for time-sensitive disruption management and resilience-oriented bus operations.

Entropy doi: 10.3390/e28101114

Authors: Vitaliy Kapytin Alexey Andreyev Beibit Zhumabayev Anastasiya Yanina Yekaterina Chsherbulova Stanislav Utebayev

Solar activity exhibits a dominant decadal timescale embedded in nonlinear multiscale dynamics. We present a minimal autonomous three-mode hybrid model to investigate whether such a timescale can arise from internal nonlinear energy exchange rather than external periodic forcing or direct inheritance of a prescribed modal period. The model couples a slow internal reservoir, a tachocline-associated transfer mode, and a nonlinear magnetic subsystem described by Duffing and Van der Pol–Duffing dynamics. Magnetic-energy release is implemented through a threshold-triggered reset, while a D-index-based structural energy-exchange index characterizes relative magnetic-energy changes. The system develops a persistent bounded hybrid regime with a dominant period of approximately 10 yr. Initial-condition ensembles and variations in the intrinsic modal periods demonstrate that this timescale does not simply track the prescribed 11 yr tachocline period. Parameter studies identify the release depth and magnetic recovery rate as stronger controls, supporting an accumulation–release mechanism of timescale selection. Release events exhibit organized recurrence and significant tachocline-phase preference, with localized event skipping separating spectral and release timescales. Saltation-based Lyapunov analysis reveals no robust positive largest exponent for the baseline regime over the investigated finite-time horizons. These results demonstrate that nonlinear energy exchange combined with threshold-mediated accumulation and release can autonomously generate a robust decadal collective timescale in a reduced nonlinear system.

Entropy doi: 10.3390/e28101113

Authors: Bilal Abdallah Benedek Nagy

Splicing systems (H-systems) and insertion-deletion systems are well-studied DNA computing paradigms inspired by biological operations observed in nature. They are modelled in a formal, information-theoretic manner. In these systems, splicing, insertion and deletion operations are used. In this paper, we study these operations on circular words, which naturally model cyclic DNA molecules, focusing on the relationship between algebraic operations and biologically inspired string transformations. In particular, we formally compare concatenation with circular splicing and insertion. While the splicing operation has already been defined and used for circular words in the literature, for insertion–deletion systems, e.g., circular insertion was defined to insert a linearized variant of a circular string (cyclic molecule) into a linear (ordinary) string (or linear molecule that can be represented by a string). In this paper, we provide a formal definition of insertion operators acting on circular words. We prove that if we use circular splicing and insertion in a somewhat context-independent way (using the empty word in the formal rules), these computational operations coincide with the one-wheel concatenation operation. Our results support the theoretical understanding of these operations and their biological interpretations, with implications for DNA computing and synthetic biology.

Entropy doi: 10.3390/e28101112

Authors: Zhaokun Liu Jinying Duan

A three-dimensional discrete map with state-dependent memristive feedback is constructed to examine complex dynamics persistence under finite parameter variations and perturbations. Fixed-point analysis, bifurcation diagrams, Lyapunov exponents (LEs), the 0–1 test, spectral entropy, and C0 complexity characterize the dynamics. At a=0.8, b=0.6, and c=0.4, positive finite-time maximum Lyapunov exponents support chaotic behavior while simulated trajectories remain finite. The original 1000-point scan yields 233 samples with λmax>10−3. A finer scan with step 0.0002 resolves four non-positive-LE samples within the previously reported positive-exponent region, and long-run validation retains their signs; thus, the finite grid does not establish a continuous chaotic interval. Feedback ablation distinguishes the combined model from isolated cosine and exponential components under the tested conditions. Initial-state diagnostics separate periodic initial-state equivalence and central inversion symmetry from translated attractor self-reproduction. Under selected parameter perturbations and Gaussian noise up to σ=10−3, the principal trajectory structure remains identifiable. Comparisons with the logistic and Hénon maps provide context. In National Institute of Standards and Technology (NIST) SP 800-22 testing, 146 of 148 non-overlapping template matching subtests meet the nominal 96/100 threshold, while two yield 95/100. These findings do not prove global boundedness, strict robust chaos, or cryptographic security.

Entropy doi: 10.3390/e28101111

Authors: Piotr Nowak Dariusz Gatarek

The Dupire formula is commonly regarded as a convenient analytical tool for the valuation of financial derivatives. In this paper, we continue the analytical treatment developed in our previous papers, deriving the aforementioned formula for the European call option in the space of distributions. We consider asset prices driven by Lévy-type stochastic integrals in a framework that allows flexible jump structures and contains a class of stochastic short-rate models, including the Vasicek model. We prove existence and uniqueness for the underlying SDE and establish the Dupire formula in this general setting. More explicit versions of the Dupire formulas are obtained for two models: Merton’s model with the Vasicek short rate, and a model with time-dependent jump scaling.

Entropy doi: 10.3390/e28101110

Authors: Yijiang Zou Qinghua Chen Jihui Han Longfeng Zhao

This paper investigates the non-linear characteristics and heterogeneous evolution of cross-market risk transmission in China by constructing a multi-layered analysis framework that integrates linear spillovers, non-linear transmission, and tail risk diffusion. By employing the Diebold–Yilmaz spillover index, effective transfer entropy (ETE), and quantile vector autoregression (QVAR), we systematically measure the dynamic connectedness structures and transmission pathways across key financial markets and strategic real-economy sectors. Our empirical results yield several critical insights. First, the Chinese cross-market risk network exhibits significant state-dependence and asymmetric features, where negative shocks generate stronger spillover intensities than positive ones, and systemic connectedness intensifies substantially under extreme quantile conditions, demonstrating pronounced non-linear amplification effects. Second, the high-tech sector acts as a persistent net source of non-linear information flow, which is consistent with the influence of industrial technology cycles on financial markets rather than constituting direct causal evidence. Third, the real estate market displays high network centrality, operating as a predominant net risk receiver and the ultimate heavy-bearing terminal of systemic stress. Finally, time-varying network analysis reveals that major external shocks markedly elevate network density and overall coupling, triggering synchronized risk resonance. These findings unveil the structural mechanics of risk diffusion under emerging market conditions, offering novel empirical evidence and structural insights for macroprudential regulation and cross-market risk governance.

Entropy doi: 10.3390/e28101109

Authors: Prasenjit Deb Almut Beige Lewis A. Clark

Bell-CHSH-like inequalities have been very successful in benchmarking spatial quantum correlations. However, as this paper illustrates, they are not necessarily sufficient for benchmarking temporal quantum correlations. To illustrate this, we parametrise classical and quantum stochastic finite state generators based on a single bit and a single qubit, respectively, and compare the temporal correlations of their output sequences using a Bell–CHSH-like inequality. We find that for sequential measurements by two observers, classical machines can exceed the Tsirelson bound of 22. To present at least one example in which quantum machines outperform their classical counterparts and maintain correlations for longer, we consider general scrambling operations between consecutive measurements. Our analysis shines new light on the fundamental differences between quantum and classical finite state generators and can potentially be used to identify novel resources for quantum technology applications.

Entropy doi: 10.3390/e28101108

Authors: Shixiao Xiao Jingwei Xu Yanqiu Zeng Xinyu Wu

Lithium-ion battery accelerated aging tests are costly and use small samples, while conventional censoring ignores electrochemical degradation dynamics and biases lifetime estimates. The power Rayleigh distribution has a monotonically increasing hazard rate consistent with aging from solid electrolyte interphase thickening and lithium plating. We combine it with adaptive progressive Type-II censoring that stops early sample withdrawals once failure capacity crosses the end-of-life threshold, preserving right tail degradation information. Parameters are estimated by maximum likelihood with multiple starts and limited memory BFGS, and by hybrid Metropolis–Hastings Markov chain Monte Carlo Bayesian inference, validated by Monte Carlo simulation and NASA 4 °C cycling data. Mild censoring retains roughly 10% more right tail information than heavy censoring. For point estimation, maximum likelihood has slightly lower bias for the moderate sample (n = 66), whereas Bayesian estimation is more robust under small sample censoring (n = 40); the highest posterior density credible interval is on average 12% narrower than the asymptotic interval while keeping nominal coverage. Although this distribution reparametrizes the two-parameter Weibull family with identical fit indices, its Shannon entropy and quantile functions have elementary closed forms, avoiding nested numerical integration and delivering about 740 times faster Bayesian sampling per chain with better numerical stability for onboard computation. Sensitivity analysis shows removing one extreme cell weakens the fit, so the framework suits homogeneous cell subpopulations under uniform conditions. Validation remains limited to low-temperature cycling; extending it to combined temperature and charge rate stresses is our immediate priority.

Entropy doi: 10.3390/e28101107

Authors: A. Almutairi M. A. Sohaly

Antimicrobial resistance is a major challenge in the effective treatment of bacterial infections because antibiotic exposure can selectively suppress sensitive bacteria while allowing resistant populations to persist and expand. In this study, we develop a finite-state stochastic population model for the joint dynamics of antibiotic-sensitive and antibiotic-resistant bacteria under capacity-limited treatment. The model is formulated as a bivariate continuous-time Markov chain with density-dependent logistic growth, sensitive-to-resistant mutation, differential antibiotic susceptibility, finite carrying capacity, and bounded treatment-removal capacity. Because the extinction state is absorbing, the ordinary stationary distribution of the full finite-state process is concentrated at extinction, and long-lived non-extinct behavior is therefore analyzed through transient and quasi-stationary perspectives. Exact first-moment equations are derived, showing that mutation transfers individuals from the sensitive population to the resistant population without creating additional bacterial individuals. Local stability near extinction is characterized by the growth rates rs=λs−μ−ν and rr=λr−ϵμ. The positive equilibria of the deterministic fluid approximation are then derived under the different treatment-capacity regimes, while capacity-induced critical transitions are characterized through the equilibrium equations and the singularity condition detJ=0, where J is the Jacobian of the deterministic drift. Near a nondegenerate fold, the model exhibits square-root equilibrium scaling and critical slowing down. Gillespie Monte Carlo simulations complement the analytical results by quantifying finite-horizon survival probability, extinction time, conditional bacterial burden, resistant fraction, and cumulative bacterial exposure. The resulting framework provides a mathematically consistent description of how bacterial growth, mutation, differential susceptibility, and limited treatment capacity jointly influence extinction and long-lived persistence.

Entropy doi: 10.3390/e28101102

Authors: Dongming Peng Yigang He Qianqian Liu

Distributed multichannel active noise control (ANC) is scalable, but filtered-x least-mean-square updates are highly sensitive to impulsive disturbances. This paper proposes a robust augmented-diffusion filtered-x maximum correntropy criterion algorithm (R-ADFxMCC). It keeps the fixed admissible combination of augmented diffusion and replaces the local squared-error update with a normalized correntropy-gated recursion, whose kernel width follows a median-absolute-deviation estimate of the local residual scale. The resulting influence function is bounded and redescending. Under the stated independence and linearization approximations, the analysis gives a combination-weight-independent mean stability condition and a local steady-state excess mean-square error approximation; a symmetry argument proves zero steady-state mean bias without requiring independence between the gate and the regressors. In 30-realization simulations with one validation protocol for all step sizes, the median steady-state gains over augmented-diffusion FxLMS are 25.79, 21.84, 13.80 and 6.46 dB for single-tone, multitone, narrowband and broadband references under localized Bernoulli–Gaussian impulses. The augmented-diffusion M-estimate algorithm reaches the same steady state within 0.45 dB at the validation-selected step sizes, but on single-tone and narrowband references, R-ADFxMCC remains stable at step sizes at which the M-estimate algorithm diverges. Tests with α-stable noise, reverberation, secondary-path errors and measured per-sample cost are also reported.

Entropy doi: 10.3390/e28101106

Authors: Jin Uk Park Seo-Eun Park Dongsup Jin Jin-Ho Chung

Variable-weight (also called multi-weight) optical orthogonal codes (VW-OOCs) with maximum correlation one admit a cyclic difference-packing formulation. We characterize the exact difference spectrum of an existing Chinese-remainder graph construction: each mixed-coordinate difference determines a unique candidate base pair and graph index. A pairwise balanced design covers the mixed region exactly, enabling independent completion on the two coordinate axes. For weights {3,4}, punctured finite geometries give families that are optimal for their prescribed profiles. For every prime p≡1(mod12), the length-12p family has (n3,n4)=((p+1)/2,3(p−1)/4), profile (2(p+1),3(p−1))/(5p−1), and deficiency 5. For every prime p≡1(mod30), the length-15p family has (n3,n4)=((p−1)/2,p), profile (p−1,2p)/(3p−1), and deficiency 2. Both deficiencies are the minima permitted by congruence. This study’s contribution is the exact-spectrum analysis and its uniform completion mechanism; the lengths and weight set are known, and the second completed parameter tuple follows by filling a previously known regular packing. Explicit examples are verified by exact enumeration.

Entropy doi: 10.3390/e28101105

Authors: Xiaoqian Zhang Chuancun Yin

This article proposes a new index for quantifying the degree of dependence between random vectors with the same dimension. The index takes values in [0,1] and equals zero if and only if the random vectors are sub-independent. Unlike mere uncorrelatedness, sub-independence implies a stronger form of dependence while remaining strictly weaker than full independence. The proposed index is constructed via characteristic functions and admits a simplified representation in terms of moments. We establish its theoretical properties and derive a computationally efficient formula for the corresponding empirical measure. Furthermore, we investigate the asymptotic behavior and finite-sample performance of the estimator and demonstrate its practical utility through applications in machine learning, actuarial science, and renewal theory. A numerical comparison with Pearson correlation and squared distance correlation is also given under several dependence structures. We also give a real-data application to international stock and foreign exchange returns.

Entropy doi: 10.3390/e28101104

Authors: Yuting Bai Xixian Zhang Jianlei Kong Xuebo Jin Tingli Su

Multivariate time-series forecasting models often achieve improved prediction accuracy at the cost of increased model complexity and inference latency, limiting their deployment on resource-constrained computing platforms. In addition, lower parameter counts or theoretical computational costs do not necessarily lead to faster execution on actual hardware. To address this issue, we propose a Latency-Constrained Distillation and Dynamic Pruning (LCDP) framework for efficient multivariate time-series forecasting. LCDP introduces a latency-constrained adaptive optimization strategy that jointly considers forecasting loss, knowledge-distillation loss, local feature-alignment loss, and hardware inference latency during student-model training. Local feature-guided knowledge transfer is used to enhance the student model’s ability to capture fine-grained temporal patterns, while deployment-oriented dynamic structural pruning progressively adjusts the group-wise sparsity pattern of the student model according to latency feedback from the target hardware while retaining the original dense layer dimensions. Experiments on the ETTh1, ETTm1, and ETTm2 datasets using two CPU platforms show that LCDP achieves competitive forecasting accuracy with a lightweight student inference backbone containing 7336 active parameters and requiring 50,848 dense multiply-accumulate operations. The saved student checkpoint additionally contains 2080 parameters in a training-time feature-alignment projection that is not invoked during inference. LCDP provides an effective balance between forecasting accuracy and practical inference efficiency, demonstrating its potential for low-latency multivariate time-series forecasting on CPU-based resource-constrained platforms.

Entropy doi: 10.3390/e28101103

Authors: Avel Guénin--Carlut

The Free Energy Principle (FEP) offers a physically grounded account of intentional properties in mind and life, framed in the context of Dynamical Systems Theory. Because of this generality, it was proposed as a formal basis to formalize the study of biological autonomy, autopoiesis, and enactive cognition. Yet, those approaches are centered by definition on describing how biological and cognitive agents are “laying down a path in walking”, or (re)constructing their own biological/cognitive structure through engagement with the world. Drawing from Rosen’s relational biology, we argue that such processes are by construction outside a “Newtonian” description in terms of well-defined observables and their dynamics. Rather, we draw from the thermodynamics of observation and biological computation to demonstrate the emergence of contextuality: the participatory production of observables against which the agent regulates their activity. We suggest a complementarity between Friston’s classical FEP and Fields’ relational FEP, describing, respectively, the dynamics within a biological/cognitive space and the construction of that space. Finally, we propose a structural realist understanding of this complementarity where both principles act on a physically embedded system of biological constraints, rather than an abstract mathematical space assumed to fully represent biological/cognitive organization.

Entropy doi: 10.3390/e28101101

Authors: Ellis E. Marshall Chad C. Nelmes Tony J. G. Apollaro Timothy P. Spiller Irene D’Amico

We identify different regimes of quantum state transfer in long-range coupled spin-12 systems, where naturally occurring power–law interactions enable rapid, high-fidelity transfer with minimal engineering. Across a broad range of interaction profiles, from effectively nearest-neighbour coupling to Coulomb interactions, we show how long-range connectivity fundamentally reshapes the mechanisms underlying information propagation within such systems. For effectively short-range interactions, transfer follows familiar ballistic transfer dynamics: an initially localised excitation spreads across many eigenmodes concentrated within the approximately linear region of the spectrum, enabling robust wavepacket motion. In contrast, increasing long-distance interactions via lowering the power–law exponent α (α=1−2) drives a striking transformation, where the initial state becomes confined to progressively fewer eigenmodes, ultimately reducing the dynamics to the coherent participation of only a few states corresponding to the highest eigenenergies. This spectral localisation gives rise to emergent long-range oscillations between distant sites, revealing a distinct—and faster—transfer mechanism arising from the intrinsic structure of long-range quantum interactions rather than from full-system engineering pathways.

Entropy doi: 10.3390/e28101100

Authors: Nafisa Ablat Shuzhen Yu Haijun Jiang

Considering the inevitable impact of environmental interference on rumor propagation, this paper constructs and studies a stochastic rumor propagation model in which the diffusion rate is described by the Log-normal Ornstein–Uhlenbeck (OU) process. Firstly, we rigorously prove the existence and uniqueness of global positive solutions for the model, which serves as a fundamental prerequisite for further dynamical analysis. Subsequently, by constructing an appropriate Lyapunov function and combining it with the ergodic property of the OU process, sufficient criteria for the existence of a stationary distribution of the system are given. Finally, we demonstrate that when R0e≤1, rumors will tend to become extinct at an exponential rate.

Entropy doi: 10.3390/e28101099

Authors: Mohammad Maadani Eric A. Butcher

Fractional-order PID control extends conventional PID control by allowing the integral and derivative orders to take non-integer values, thereby introducing α and β as additional tuning parameters beyond the conventional PID gains. Two strategies for fractional-order PIβDα cooperative pose control of a multi-agent rigid body system are considered assuming both undirected and directed communication networks, together with Caputo fractional derivative and integral operators. The rigid-body pose configurations are described using exponential coordinates associated with the Lie group SE(3), while the state space is TSE(3)N. Analytical stability bounds are obtained using Matignon’s theorem in terms of the proportional, derivative, and integral gains for the attitude and translational controllers, as well as the fractional orders, that guarantee global Mittag–Leffler stability. The fractional control strategy provides additional flexibility in tuning the closed-loop response beyond that of conventional integer-order PID control. Since the fractional derivative states are not directly measurable, a Luenberger observer is employed to estimate these states from the measurable pose exponential coordinates, and the resulting estimates are used in the implementation of the feedback protocols. Simulation examples of formation establishment further incorporate cohesion and collision-avoidance strategies based on artificial potential functions, illustrating the effectiveness and practical implementation of the proposed fractional PIβDα control protocols.

Entropy doi: 10.3390/e28101098

Authors: Qi Wei Shiqi Qiang Hangjun Li Shan He Junqiang Liu Longxiang Hou Yongchuan Tang

Open-world classification must account for samples whose classes are absent from the current frame of discernment. Although generalized evidence theory can represent such possibilities, the conventional base belief function assigns reference support only to nonempty propositions and therefore cannot directly support source-level treatment of unknown information. This work proposes a generalized base belief function that incorporates the open-world proposition into its reference domain and uses it to modify generalized basic probability assignments before fusion. The modified evidence is combined through rules compatible with generalized evidence theory and followed by a known/unknown decision, forming a complete classification procedure for incomplete frames. Experiments on Iris and Seeds show that source-level modification and generalized fusion jointly support known/unknown classification. On Waveform, classification remains stable as controlled perturbation increases in a larger, higher-dimensional, noisy environment. On the Wall-Following Robot dataset, the procedure gives a selective known/unknown decision across different class compositions, preserving known samples while retaining overall discrimination close to GCR. These results support the proposed GBBF-based method for open-world classification under incomplete frames.

Entropy doi: 10.3390/e28101097

Authors: Yungpil Yoo Seok-Ho Seo Si-Doek Oh Ho-Young Kwak

Various exergy-based exergoeconomic methods have been developed to evaluate the production cost of electricity in power systems. One of the unresolved issues in the field of exergoeconomics is that different exergoeconomic analysis methods can yield different unit costs of power production. In this study, the unit cost of electricity generation is comprehensively analyzed using three representative approaches: the Moran method, the specific exergy costing (SPECO) method, and the Modified Productive Structure Analysis (MOPSA) method. The objective of this study is to identify a condition under which different exergoeconomic methods yield the same production cost for a power plant. The results show that power production costs are obtained using all three exergoeconomic methods when the costs associated with exergy destruction are appropriately incorporated in the cost-balance equation. In other words, under this condition, the cost-balance equations of SPECO and MOPSA for the entire power plant reduce to the Moran equation. These findings indicate that the critical issue in exergoeconomic analysis for determining the unit cost of products is whether the cost of exergy destruction is explicitly or implicitly accounted for. Furthermore, the numerical results demonstrate that all input cost rates, including equipment investment, operation and maintenance, and fuel costs, are fully recovered through the cost flow rates of the products. It is also shown that the cost flow associated with exergy destruction ultimately appears as the cost flow of waste streams.

Entropy doi: 10.3390/e28101096

Authors: Jian-Rong Li Guang-Yan Zhong Jiang-Cheng Li

Bitcoin bubbles exhibit locally explosive dynamics, but identifying past episodes does not establish their predictability. We combine the generalized supremum augmented Dickey–Fuller test with nine probabilistic learners and three probability-output ensembles to forecast bubble states over one-, three-, and seven-day horizons. Using daily data from January 2016 to April 2026, we identify 30 maximal uninterrupted episodes comprising 455 bubble days. Development-period validation selects the lookback window and oversampling ratios before final-test evaluation; ensemble weights are estimated from development-period rolling forecasts. No model family dominates across horizons. The constrained least-squares ensemble has the lowest final-test mean squared error at one day, whereas logistic regression has the lowest error at three and seven days. The inverse-error-weighted ensemble has the highest average precision at seven days. Moving-block bootstrap intervals overlap substantially, and the 90% Model Confidence Set retains several models at every horizon. A cross-source robustness check yields 99.58% agreement in daily bubble-state classifications. These findings support horizon-dependent probabilistic risk monitoring, while the limited number of independent episodes and low recall at longer horizons constrain the practical interpretation of the forecasts.

Entropy doi: 10.3390/e28101095

Authors: Andrea Addazi

We show that a discrete quantization law for cosmological entropy emerges naturally within a meta-geometric gauge theory of gravity, when the standard de Sitter entropy relation is combined with the topological structure of the theory. Our framework is based on a pre-metric action containing a MacDowell–Mansouri term with a spontaneously broken (A)dS gauge symmetry reduced to the Lorentz subgroup. We derive an inverse proportionality between the topological θ-angle and the cosmological constant, which enforces a topological quantization condition. As a consequence, the de Sitter entropy becomes quantized and scales linearly with a topological quantum number. This quantization provides a natural entropic stabilizer and a quantum topological protection for a large cosmological constant, thereby offering a microscopic gauge-theoretic realization of the holographic naturalness conjecture. We also discuss possible emergence of the N-portrait in cosmology and a dynamical relaxation of the vacuum energy.

Entropy doi: 10.3390/e28101094

Authors: Peihua Li

Wide-gap frequency-hopping sequences (WGFHSs) require a sufficiently large frequency gap between adjacent frequency points, which grants them inherent resistance against tracking jamming, barrage jamming, and multipath fading. Such sequences constitute a critical building block for enhancing the anti-jamming performance of frequency-hopping communication systems operating in complex electromagnetic environments. In practical systems, the aperiodic Hamming correlation of sequences directly affects receiver synchronization and sequence acquisition performance. Hence it bears greater practical significance than periodic Hamming correlation. However, owing to the complexity of application scenarios, related research is challenging, and existing results remain limited. In this paper, we extend the traditional Peng-Fan aperiodic bound to WGFHSs, and derive two lower bounds on the maximum aperiodic Hamming correlation of WGFHS sets. Numerical examples demonstrate that the proposed bounds are tighter than the traditional Peng–Fan aperiodic bound. This provides a more rigorous theoretical foundation for the design of WGFHS sets.

Entropy doi: 10.3390/e28101093

Authors: Jianming Wen

Time-reversed Young (TRY) interferometry holds a detector channel fixed while a source coordinate is scanned or programmed. We describe this measurement through a single accepted source–detector probability law obtained from scalar Fresnel propagation, the programmed source prior, and detector acceptance. The conventional fixed-source Young distribution, the source and detector marginals, and the fixed-detector TRY distribution are then different reductions of that common record. Standard information measures can therefore be assigned without ambiguity to the probability law actually measured. For a symmetric double-slit response with matched source and detector windows, the fixed-source and fixed-detector conditional distributions have the same entropy; the physically useful information from detector conditioning is instead expressed by grid-stable entropy deficits and Kullback–Leibler information gains. The canonical example gives a modest prior-to-posterior information gain and establishes a transparent baseline for more selective source programming, detector modes, or operating points. Finite slit width, finite-source element width, finite detector pixels, and finite-count entropy estimation are included to connect the ideal model with experiments. A phase scan then shows that conditional entropy and Fisher information characterize different properties of the same likelihood, while a two-mode coherence model separates record entropy from optical coherence entropy and distinguishes coherent superposition from incoherent mixture. The framework gives a unified statistical interpretation of fixed-detector, source-programmed Young interferometry while keeping optical reciprocity, Bayesian conditioning, local parameter sensitivity, and quantum state disturbance conceptually distinct.

Entropy doi: 10.3390/e28101092

Authors: Jorge I. Lopez Perez Cristina D. Duarte Claudio A. Delrieux

Automatic characterization of adventitious respiratory sounds is essential for developing computer-aided systems that support respiratory disease diagnosis. Commonly used acoustic descriptors, such as Mel frequency cepstral coefficients (MFCCs), primarily characterize spectral information and may not fully capture the nonlinear temporal dynamics of respiratory signals. In this work, we investigate the discriminative capability of Generalized Weighted Ordinal Pattern (GWOP) descriptors for representing adventitious respiratory sounds. Using respiratory recordings from the publicly available SPRSound pediatric database, GWOP descriptors were computed over multiple embedding dimensions, weighting factors, and time delays. An XGBoost classifier was employed exclusively as a supervised validation model to assess the discriminative information contained in the proposed feature representation, while the XGBoost Gain criterion was used to analyze the contribution of individual descriptor components. The proposed representation achieved stable performance across cross-validation folds, outperformed a conventional MFCC representation under the same evaluation protocol, and demonstrated that a reduced subset of Gain selected features preserved, and slightly improved, the classification performance while substantially reducing the dimensionality of the original feature space. These findings indicate that GWOP provides a lighter yet more informative and interpretable representation of adventitious respiratory sounds, capturing nonlinear ordinal dynamics that complement conventional spectral descriptors and offering a promising alternative for respiratory sound characterization.

Entropy doi: 10.3390/e28101091

Authors: Yuejiao Ji Xiaoping Chen

High-throughput phenotyping technologies enable plant growth to be measured repeatedly at high temporal resolution, providing rich information for studying the dynamic genetic basis of longitudinal traits. However, growth trajectories from repeated experiments contain both reproducible variation among lines and experiment-specific variation. In this study, we analyzed projected shoot area data from 356 rice lines measured in three independent experiments using an integrated genomic region association framework based on shared functional representation. We extracted the major longitudinal variation shared across experiments and characterized the covariance eigenspectra using spectral entropy and entropy-based effective rank. The first shared functional principal component explained 96.069% of shared variation, while the shared component had a normalized spectral entropy of 0.101 and an effective rank of 1.198. Regional association analysis using 200 kb genomic windows, with adjustment for population structure and leave-one-chromosome-out genetic relatedness, identified a region at 14.2–14.4 Mb on chromosome 10 that remained significant after false discovery rate correction (q=0.004). Further analyses showed time-varying regional effects and differences in association evidence across experiments, while the chromosome 10 signal was attenuated in an endpoint benchmark. Our analysis provides a method for linking longitudinal phenotypic variation shared across repeated experiments with localized genomic variation.

Entropy doi: 10.3390/e28101090

Authors: Giulio Ruffini Castaldo

Scientific discovery seeks regularities that support explanation and prediction. Compression makes their reuse explicit: shared structure is described once, while parameters specify individual cases. A model can, therefore, pay for itself through repeated use even when its description is not minimal, provided it captures reusable structure in the data. Because regularities are often easier to identify in simpler systems, a reductionist approach is often employed: discover the laws of the parts and treat them as fundamental. Yet knowing those laws does not by itself provide useful coarse-grained models of the larger systems they compose. Here we formulate Anderson’s distinction between reduction and construction for finite algorithmic observers and prove three barriers to such construction. First, an observer’s coarse-grained record may retain so much information about initial or boundary conditions that no substantially shorter description exists, even when the underlying laws are simple and known. Second, when a shorter description does exist, open-ended search can eventually find one, but there is no computable bound on how long this may take, and no algorithm that always halts with a valid description can succeed in every case. Third, every such algorithm has blind spots at all sufficiently large lengths: records it leaves unshortened even though they admit descriptions of only logarithmic length. Yet observers do discover useful models. We call the event in which an observer acquires a representation relevant to its objective and a reusable model that reveals previously unavailable regularity algorithmic emergence. We give a sufficient certificate in explicit code lengths: the pair passes when, with its own description cost included, it compresses the retained data relative to an agreed baseline and yields further savings on later observations. Even for a stream that repeats one block, no computable procedure that always halts with valid codes can guarantee finding a passing pair with the observations available whenever one exists, nor remain within a fixed number of bits of the best qualifying complete code. Favorable structure, including symmetry and restricted model classes, can nevertheless make discovery feasible.

Entropy doi: 10.3390/e28101089

Authors: Hamil Stanly Mercy Shalinie Selvaraj Riji Paul

The growing vulnerability of computer vision models to adversarial attacks has raised great concerns about the reliability of generative models in security-sensitive applications. Quantum generative models show strong potential for learning complex data distributions. However, the robustness of quantum generative models to adversarial perturbations and the effectiveness of adversarial training using quantum-classical generative architectures remain less explored. In this work, we evaluate quantum- and classical generative network-inspired adversarial training to analyse whether quantum state-space representations can enhance adversarial robustness compared with their classical counterparts. We propose HQMSGAN, a novel hybrid quantum-classical generative adversarial network with multiscaling discriminators that denoise the adversarial attacks through adversarial training. During adversarial training, the proposed model performs quantum projections to learn rich nonlinear feature representations using quantum state encoding and performs feature coupling using quantum entanglement in the quantum state space. Thus, it learns the probability distribution of both the clean and adversarially perturbed data and tries to generate pure samples by reducing the loss over iterations in the quantum realm. The experiment is carried out by inducing four different adversarial attacks and defending them by adversarial training using the resources provided by AWS Braket’s quantum simulators in the Noisy Intermediate Scale Quantum (NISQ) era. The results provide favorable performance on metrics such as Structural Similarity Index Measure (SSIM), Peak Signal-to-Noise Ratio (PSNR),

Entropy doi: 10.3390/e28101088

Authors: Jia Ouyang Jinze Liu

We study estimation of globally sparse relation probability matrices from capped binary relation sets under personalized user-level local differential privacy and Byzantine response replacement. A cardinality-calibrated joint personalized relation sketch (JPRS) releases one bit per user. The decoder combines public channel-amplitude weighting with exact sparse feasible projection. We distinguish trimmed honest information, adversarial leverage, and a response-contamination indistinguishability radius. An explicit capped binary block construction yields a sampling lower bound with an explicit low-information truncation. A separate contamination construction gives a finite-sample error floor. For precommitted corrupted identities, the global decoder admits a uniform bound against adaptive valid-output attacks. In the strong-privacy, sufficiently informative regime, sampling and contamination bounds match up to logarithms on bounded-ratio budget classes, subject to the stated sample regularity and corruption-count conditions. The theory uses global sparsity rather than low-rank structure. Across six synthetic families, the global decoder has lower mean loss than the three tested private global baselines, but improves on zero only in the Zipf setting. Three public-data benchmarks further show that low aggregate error can be driven by a merged tail category. These results delimit practical utility; finite-budget attack searches do not certify worst-case performance.

Entropy doi: 10.3390/e28101087

Authors: Gou Hosoya

Bit-interleaved coded modulation (BICM) with Gray labeling is widely used, yet its mismatched decoding metric keeps achievable rates below the coded modulation (CM) capacity, which multilevel coding attains only with multiple component codes. Delayed BICM (DBICM) bridges this gap with a single encoder and decoder by delaying coded bit subsequences so that already-decoded bits act as side information; for staircase delay vectors, its middle-region rate provably equals the CM capacity for any labeling. Under continuous transmission, however, initial and final codewords share symbols with pilot bits, and these boundary rates depend strongly on the pilot values. For pulse amplitude modulation, we design delay vectors and boundary pilots that keep every boundary rate at or above the middle-region capacity. For set-partitioning (SP) labeling, a closed-form minimum-shift pilot rule provably guarantees this condition for every staircase vector and SNR, whereas Gray labeling admits capacity-preserving pilots found by numerical search up to 32-PAM. Density evolution predicts gains approaching 1 dB over rate-matched BICM baselines for both labelings. Long continuous simulations, however, reveal severe inter-codeword error propagation under SP labeling, establishing Gray labeling as the practical choice. With per-level irregular LDPC codes, DBICM retains consistent threshold and waterfall gains at identical effective rates.

Entropy doi: 10.3390/e28101086

Authors: Massimo Blasone Fabrizio Illuminati Luciano Petruzziello Kyrylo Simonov Luca Smaldone

Macrorealism is a fundamental concept in quantum foundations, capturing basic features of our everyday experience. Among the main tools introduced to test violations of macrorealism are the Leggett–Garg inequalities. These inequalities provide necessary and sufficient conditions for a certain data set including the correlators measured in a number of different experimental runs (weak version of macrorealism). Leggett–Garg inequalities have proved useful for detecting violations of macrorealism in neutrino oscillation experiments. However, it is interesting to investigate a set of necessary and sufficient conditions for strong macrorealism—no-signaling-in-time and arrow-of-time conditions—in order to obtain a more complete characterization of macrorealism in the context of neutrino oscillations. In particular, here we extend a previous study of the no-signaling-in-time condition to the case of three-flavor neutrino oscillations, including the CP-violating phase of the Pontecorvo–Maki–Nakagawa–Sakata matrix.

Entropy doi: 10.3390/e28101085

Authors: Iddo Eliazar

Lorenz curves are a socioeconomic method of coding the statistics of positive-valued random variables. Standing at the confluence of fractals, wealth distributions, and non-Euclidean geometry, this paper presents quadrant Lorenz curves (QLCs): Lorenz curves whose shapes are that of the symmetric quarters of the p-norm unit circle. On the one hand, QLCs have a single parameter, and a clear geometric meaning. On the other hand, a comprehensive exploration establishes that QLCs display an assortment of socioeconomic and statistical behaviors. (1) They span a continuum of socioeconomic states ranging from perfect equality to perfect inequality, with three phase transitions along the continuum. (2) They display local rich fractality, local poor fractality, and double-Pareto tails. (3) They have inherent socioeconomic and statistical symmetries. The exploration includes a detailed socioeconomic-inequality analysis that illuminates QLCs from multiple inequality-indices `angles’. In a nutshell: QLCs manifest a novel class of `fractal’ statistical distributions, and the QLC features are unmatched by those of any other such class.

Entropy doi: 10.3390/e28101084

Authors: Boumediene Hamzi Marcus Hutter Houman Owhadi

Compression-based dissimilarities such as the normalized compression distance are widely used, but direct exponentiation need not produce a positive semidefinite kernel. We develop a systematic interface between prefix algorithmic information theory and three kernel discrepancy methods on finite or countable spaces: maximum mean discrepancy (MMD), the Hilbert–Schmidt independence criterion (HSIC), and a Markov-operator kernel Stein discrepancy (KSD). Explicit feature realizations guarantee positive semidefiniteness for distance-to-kernel embeddings and for an ideal Solomonoff feature-mixture kernel satisfying kSol,U(x,y)≍U2−KU(⟨x,y⟩). Zero-mass signed measures give the exact characteristicity criterion, and whether the ideal Solomonoff kernel meets that criterion turns out to depend on the universal machine: two universal prefix machines are exhibited, one whose kernel is not characteristic on a two-point set and one whose kernel separates all finite-total-variation signed measures on the countable string space. In the resulting RKHS geometry, the MMD compares compressibility profiles, the HSIC is product-kernel MMD from the joint law to the product of its marginals, and KSDTP,k(Q)=MMDk(QTP,Q) for a target-stationary Markov kernel TP. Expected algorithmic mutual information is connected to the HSIC only through explicit source-description, mutual-information, support, and kernel conditions. Ideal AIT objects, computable feature surrogates, and finite-sample errors are separated explicitly. Reproducible synthetic experiments examine the identities, landmark approximation, and selected testing alternatives at the computable and empirical layers. They provide no evidence about ideal AIT quantities or general superiority over baselines. An algebraic example also shows that a characteristic finite-support kernel can give a constant permutation statistic: identification alone does not guarantee test power.

Entropy doi: 10.3390/e28101083

Authors: Xiangyu Kang Shuiyuan Yu

Dependency distance measures the linear separation between syntactically related words but does not describe how information is distributed across their interval. We analyzed summed XGLM-2.9B word surprisal in 1,346,122 dependency spans from 22 Universal Dependencies languages at distances 4–10. Clustering 154 position-aligned profiles yielded a broadly descending type and a type with a terminal rise; 20 languages retained their assignment across distances. Leave-one-language-out comparison did not support a common three-segment structure: the one-standard-error rule selected two segments for the descending type and three for the rising type. The descending type’s initial decline disappeared after separate adjustments for sentence position and preceding context. Standardizing word-length and model subword-count distributions reduced the rising type’s mean final-word increase from +1.996 to +0.408 bits (95% interval [−0.024, +0.898]). In held-out sentences, real dependencies in eight languages independently classified by a training-set terminal decrease by 0.928 bits more than nearby nonedge intervals sharing the final part-of-speech tag (95% interval [−1.498, −0.217]). A shared part-of-speech component was negatively associated with terminal surprisal in the original descending type. Dependency-span alignment reveals recurring information profiles and locates a conditional terminal contrast for further tests of syntactic prediction.

Entropy doi: 10.3390/e28101082

Authors: Xiao Hu Hao Qi Jing Yu Chengwu Lu Lingli Zhang

Remaining Useful Life (RUL) prediction for aircraft engines is a critical task in prognostics and health management, aiming to extract degradation information from multi-sensor operational data to support maintenance scheduling and operational risk management. Existing hybrid temporal models have explored various combinations of convolutional networks, temporal convolutions, recurrent networks, and attention mechanisms; however, there remains room for further coordination regarding the positional roles of different modules within fixed windows, bidirectional context organization, and training objectives. To address this, we propose a Multi-scale Bidirectional Query-Guided Gated Network, MBQG-Net. MBQG-Net employs one-dimensional convolution and parallel bidirectional dilated convolutions to extract local and multi-range temporal features, and utilizes multi-head Query–Value temporal attention to generate window-level context-enhanced representations prior to recurrent state aggregation. The sequence enhanced by Query-based temporal weighting and Value aggregation is then fed into a stacked bidirectional GRU for subsequent bidirectional gated state aggregation. During training, an asymmetric weighted mean squared error is adopted, assigning higher weights to RUL overestimation errors. MBQG-Net was evaluated on all four official test sets (FD001–FD004) of the NASA C-MAPSS dataset under a unified experimental protocol. It achieved the strongest overall regression performance on FD001 and FD003 and maintained competitive overall conventional regression performance on the multi-condition FD002 and FD004 subsets. The model obtained the lowest NASA Score among the compared methods on all four subsets, although the difference on FD002 was marginal. On FD004, MBQG-Net achieved a NASA Score of 930.9088, representing a 28.27% reduction relative to the second-best result. Experimental results demonstrate that MBQG-Net maintains a favorable balance between conventional regression accuracy and direction-sensitive error control across different operating-condition and fault-mode settings, offering a structurally clear hybrid temporal modeling solution for multi-sensor aircraft engine RUL prediction.

Entropy doi: 10.3390/e28101081

Authors: Gilberto Aparecido Prataviera Marcos César de Oliveira

Finite-bandwidth reservoir engineering can suppress unwanted transitions in a quantum thermal machine, but a physical filter also introduces a finite response time. We study this competition in a continuous three-level heat engine whose hot environment couples parasitically to the cold transition. The corresponding three-state rate network is solved exactly, showing that the parasitic transition produces a hot-to-cold thermodynamic short circuit and yielding a closed threshold for the loss of positive-power operation. We then retain a damped auxiliary mode explicitly as a physical spectral filter. Its Lorentzian response suppresses the detuned parasitic transition, whereas excessive narrowing limits the useful energy throughput. Independent local-GKSL and nonsecular Bloch–Redfield calculations both recover the no-leak Markovian engine in the weak-coupling limit and predict a finite maximum-power bandwidth, although its precise location is model dependent. A frequency-resolved Lorentzian-rate reduction, by contrast, has no interior optimum. The resulting design principle is that spectral selectivity can protect the useful thermodynamic cycle, but a real filter cannot be narrowed without a dynamical throughput cost.

Entropy doi: 10.3390/e28101080

Authors: Jiao Wu Xiuwen Feng Lihui Zhu Yingpeng Liu Muhua Zheng

Spreading dynamics have been extensively studied in social, communication, and epidemiological networks, leading to substantial advances in understanding spreading thresholds, outbreak patterns, temporal dynamics, and phase transitions. However, their extension to other complex systems, particularly agricultural systems, remains less systematically explored. In this mini-review, we summarize representative spreading models and theoretical approaches developed for complex networks. We first review spreading dynamics on single-layer networks and then discuss their extensions to multilayer, temporal, and higher-order networks, highlighting how increasingly complex interaction structures affect spreading thresholds, outbreak patterns, and phase transitions. We subsequently examine how these models and theoretical approaches can be transferred from social networks to complex agricultural systems. Representative applications include crop diseases and pests, livestock diseases, agricultural information and technology diffusion, and coupled disease–information processes. Finally, we discuss emerging directions for extending spreading theories to increasingly complex agricultural systems. This review may help clarify the generality and transferability of spreading theories across different networked systems and stimulate their further development in complex agricultural contexts.

Entropy doi: 10.3390/e28101078

Authors: Xuejin Zhu Shenyang Li

Social Internet of Things (SIoT) relationships enable service discovery and create pathways for malware propagation. We develop a heterogeneous susceptible–exposed–infected–recovered model coupling resource-constrained patching with reversible, trust-aware relationship suppression. A controlled next-generation operator yields a reproduction threshold, local stability, spectral monotonicity, and a uniform eradication condition. A finite-horizon security–service objective and its necessary conditions motivate joint adaptive topology and patching (JATP), a feasible sampled-data feedback heuristic combining prevalence, structural influence, trust evidence, and service importance. Under reference capacities, JATP reduces peak infection and cumulative burden by 49.43% and 38.97% relative to risk-only patching while retaining 0.8184 of weighted relationship intensity. Under a common expenditure ceiling, its objective is 59.56% above the best feasible offline nonlinear-programming value found. Explicit-network tests support aggregation under moderate heterogeneity and identify larger prediction errors under strong degree variation and susceptible-entry churn. Trust-related gains depend on alignment with model-defined propagation risk; field calibration remains open. These results characterize the operating scope of a feedback policy that coordinates two defense channels under hard resource limits. The framework links epidemic thresholds, intervention allocation, and service preservation to support the design of resource-limited SIoT containment.

Entropy doi: 10.3390/e28101079

Authors: Alexander Stahl Michael Kewming John Goold Janine Hilder Ulrich Georg Poschinger Ferdinand Schmidt-Kaler

Within the framework of microscopic thermodynamics, correlations can play a crucial role in energy extraction. We experimentally investigated the link between information encoded in tunable correlations and its utility for feedback-assisted energy transfer. We realized an agent-demon protocol involving two trapped-ion qubits and show that an appropriate feedback policy increases the average energy stored in the demon qubit by capitalizing on correlations between the constituents. For the chosen family of nearly pure resource states, we quantify both the concurrence and the feedback-assisted energy gain and observe the predicted relation between them. We also show theoretically that the same fixed protocol yields the same gain after complete dephasing in the local energy basis. The experiment establishes a concurrence–gain relation within the investigated pure-state family.

Entropy doi: 10.3390/e28101077

Authors: Xiaohu Chen Xiaoyang Liu Jiaojiao Jiang Chao Liu Xin Wang

The existing key node identification methods in complex networks rely on local topological information or global structural features, which are difficult to fully integrate multi-order local structural information and global propagation characteristics, resulting in the limitation of the accuracy and discrimination ability of node influence evaluation. In this paper, a key node ranking method combining semi-local and global influence, named SGIR, is proposed. Firstly, in the semi-local influence calculation stage, the second-order neighborhood, the third-order neighborhood, and the triangular closed structure of the node are used to construct the structural correlation matrix, and the structural energy migration mechanism is combined to quantify the local structural influence of the node, so as to enhance the expression ability of the semi-local structural features of the node. Secondly, in the global influence calculation stage, based on K-shell decomposition, a distance attenuation mechanism is introduced to weighted aggregate the influence of core nodes within the limited propagation range, and an adaptive propagation radius is designed by combining the average shortest path length of the network to effectively describe the global propagation ability of nodes. The semi-local influence and the global influence are fused to obtain the comprehensive influence ranking of nodes. Finally, comparative experiments are carried out on nine real complex networks from different fields such as communication, PowerGrid, and biological and social networks. The experimental results show that the proposed SGIR method achieves the best Kendall’s τb on most data sets, and produces a highly discriminative ranking, which verifies the rationality, effectiveness, and stability of the proposed method and provides an effective solution for the identification of critical nodes in complex networks that takes into account both semi-local structural information and global propagation ability.

Entropy doi: 10.3390/e28101076

Authors: Leopoldo Trieste Giuseppe Turchetti

The Condorcet paradox is traditionally viewed as a macro-level phenomenon of collective decision-making, showing that individually rational agents with coherent and transitive preferences may nevertheless generate socially intransitive majority outcomes. This interpretation assumes that cyclicity emerges exclusively through aggregation. In this study, we distinguish between micro-Condorcet dynamics, arising from contextual interactions within individual decision processes, and macro-Condorcet dynamics, emerging through majority aggregation. To investigate the relationship between these two levels, we introduce a minimal contextual model in which pairwise decisions are generated sequentially and linked through a contextual coupling mechanism. Although individual agents remain locally coherent and entirely classical at the microscopic level, each decision modifies the context of subsequent comparisons. Simulation results show that increasing contextuality progressively increases the probability of both micro- and macro-level Condorcet cycles. The transition between the two levels is strongly influenced by heterogeneity in decision thresholds, particularly those governing intermediate and closing comparisons. Rather than being driven primarily by initial preference relations, collective cycles emerge from heterogeneous evaluations of intermediate alternatives, which amplify contextual effects and facilitate cycle closure. A critical contextuality threshold is observed beyond which classical representability breaks down and no probability distribution over transitive preference rankings can reproduce the observed pairwise preference probabilities. These findings suggest that quantum-like probabilistic structures may emerge naturally as higher-level representations of contextual collective decision systems.

Entropy doi: 10.3390/e28101075

Authors: Arnaud Amblard Aurélien Drezet

Relativistic extensions of de Broglie–Bohm theory postulate a preferred foliation of spacetime, an additional structure essential for defining simultaneous configurations on Minkowski spacetime but conventionally believed to be empirically undetectable at quantum equilibrium. In this paper, we outline an experimental protocol for empirically detecting the preferred foliation, which is assumed to be flat for simplicity. Building on the arrival-time distributions for spin-1/2 particles predicted by Das and Dürr, we show that, in an Einstein–Podolsky–Rosen–Bohm-type experiment with spacelike-separated spin and arrival-time measurements, the observed arrival-time statistics will depend crucially on the temporal order of these measurements relative to the preferred foliation of spacetime. This dependence offers a potential experimental signature of the preferred foliation postulated by relativistic Bohmian models. Moreover, it implies the possibility of superluminal signaling.

Entropy doi: 10.3390/e28101074

Authors: Ahmed Farag Ali Nader Inan Ruben Laukkonen Pavlo Mikheenko

We develop a theoretical proposal linking vacuum stability and brain dynamics through superconductivity-inspired coherence, symmetry reduction, and the thermodynamic stabilization of low-entropy regimes. We take an unbroken SU(3) structure as a candidate stable residue of the low-temperature vacuum. At the neural level, we formulate a coarse-grained analog in which a two-fluid model with dissipative and coherence-supporting components describes brain dynamics. Specifically, the coherence-supporting component is proposed as a possible basis for the efficient binding and integration required to sustain a stable, unified conscious state. The proposal offers a common geometric language for relating physics and neuroscience with falsifiable signatures in coherence and state-dependent transitions. The main technical contribution is a computational algebraic model of conscious-state dynamics, where neural data are mapped to reconstructed state trajectories. Effective generators are inferred from those trajectories, and the two-fluid split is tested as a Cartan–root decomposition of su(3), with a rank-two commuting sector for coherence-preserving balance and six root directions for state transitions. This structure can be tested on neural data and contrasted with alternative dynamical models.

Entropy doi: 10.3390/e28101073

Authors: Guillermo Sosa-Gómez

Entropy estimation from short samples recurs in symmetric cryptography, where the reference distribution is uniform by design and no single estimator in the considered classical comparison set minimizes mean squared error (MSE) across the full range of ratios n/k. We introduce the adaptive sample-conditional entropy diagnostic (ASED), in which a compact network trained offline maps a frequency-of-frequencies descriptor of the sample to convex mixture weights over six classical estimators, at a cost of O(n+k). We prove an oracle inequality bounding the excess risk of such a mixture by the L1 error of its weights and fixed-alphabet consistency for every simplex-valued weighting rule, independently of the distribution used to train the weights. Under the uniform-null protocol, ASED attains an integrated MSE of 1.7×10−3 for bytes, compared with 3.8×10−2 for James–Stein shrinkage, a reduction that depends materially on the aggregation scheme: 95.4% for summed or averaged MSE across sample sizes versus 24% for the mean of per-size ratios. Both figures are reported together throughout, and neither is presented as the headline; the entire advantage is confined to the undersampled regime n<k, the two estimators being indistinguishable for n≥k. A locked train/validate/test evaluation with an independently written implementation reproduces the reduction at 95.1%. Because a constant output achieves zero error by construction here, we add further checks: in a post hoc analysis, ASED is non-inferior to SHR in detection power at a 0.02 margin, both pointwise and simultaneously, whereas a constant control has none, and the advantage persists on held-out configurations. The 0.02 margin was selected after inspecting the power estimates, and at n=8 and n=16, it is smaller than the variation induced by tie handling at the empirical critical value, so the non-inferiority conclusion is confirmatory only for n≥32. At n≤16, the two statistics have identical null distributions up to monotone relabeling, and no test of size 0.05 exists, so no power difference is identifiable in either direction, and the comparison is withdrawn there. For strongly non-uniform sources, the ordering reverses, and NSB dominates, delimiting ASED as a uniformity diagnostic rather than a general-purpose entropy estimator. The design choices behind ASED, the feature map, architecture, and loss weighting were made while observing results on this same evaluation protocol, so no independent model-selection split separates development from the assessment reported here. That pipeline is accordingly designated exploratory, and the locked evaluation is confirmatory. The contribution is stated as sample-conditional convex aggregation of classical estimators rather than as the particular network realizing it: a five-feature variant and a gradient-boosted surrogate perform comparably, with no paired interval separating the three. Claims are restricted to short i.i.d. samples and marginal Shannon-entropy diagnostics under a uniform null; min-entropy, unpredictability, dependence, and randomness certification are out of scope.

Entropy doi: 10.3390/e28101072

Authors: Laurent de Forges de Parny Luca Paccard Mathieu Bertrand Luca Lazzarini Valentin Leloup Raphael Aymeric Agathe Blaise Stéphanie Molin Pierre Besancenot Cyrille Laborde Mathias van den Bossche

Text Correction [...]

Entropy doi: 10.3390/e28101069

Authors: Adrian-Mihail Stoica Valentin Pană Irina Beatrice Ştefănescu

This paper presents a design methodology for the automatic control system of an airship, accounting for atmospheric disturbances and parametric modeling uncertainties. The derived design model is a linear stochastic system with multiplicative noise incorporating both the airship’s uncertain dynamics and the wind model. First, an H∞ state feedback control law is derived for the stochastic system to ensure robust stability and trajectory tracking performance. Subsequently, a robust Kalman filter is designed to estimate the turbulence model states based on available measurements. It is demonstrated that the optimal gain of this robust filter depends on the solution to a coupled system of specific Riccati and Lyapunov equations. Numerical results indicate improved heading-tracking performance for the considered turbulent simulation when the estimated wind-gust states are incorporated into the feedback law. Under parametric uncertainty, the robust Kalman-type filter also exhibits lower sensitivity of the heading-estimation error than the classical Kalman filter.

Entropy doi: 10.3390/e28101071

Authors: Alexandru Dinu Corina Nitu

We propose and analyse a hybrid chaotic pseudorandom generator that combines the logistic, tent, and Hénon maps under a deterministic entropy-greedy selector. At each step every map proposes a candidate; the selector keeps the one that maximises the plug-in Shannon entropy of a sliding output window, emits it, and advances only the winning map’s state. We benchmark all four generators plus a Mersenne Twister (MT19937) reference on uniformity (Kolmogorov–Smirnov statistic, KS), lag-correlation, entropy, Lyapunov exponent, and sensitivity. The hybrid reduces the logistic KS deviation by 26.9% and the maximum short-lag Pearson correlation by 6.5%. An explicit predictability analysis shows that the logistic and tent states are exposed, giving an adversary 92% one-step accuracy; only the hidden Hénon coordinate blocks free-running cloning, which diverges within about eight steps. Rank-uniformisation drops KS to 2.5×10−5 and passes a NIST SP 800-22 subset, while the lag-1 rank dependence is unchanged. Structural modifications—state masking for all component maps and continuous-time replacements with hidden coordinates—are discussed as routes to reducing predictability.

Entropy doi: 10.3390/e28101070

Authors: Parmida Geranmayeh Onur Günlü

In next-generation wireless networks, communication systems are expected to go beyond simple data transmission and simultaneously provide high data rates, efficiency, and security. This requirement has motivated the extensive adoption of machine learning methods to develop intelligent and real-time network management frameworks, enabling the system to continuously monitor and react to channel variations and user behavior while maintaining efficient information delivery. In this context, the integration of machine learning with beamforming enables adaptive and data-driven beam direction selection, improving both the efficiency and security of wireless links. In this work, a 3GPP-based system model is first implemented under a no-attacker scenario, and an exhaustive search is employed as a reference to identify the best beamforming configurations. The proposed framework is then evaluated in the presence of an attacker and under different network scalability conditions. We demonstrate that the reinforcement learning-based approaches, namely Q-learning and State-Action-Reward-State-Action (SARSA), consistently outperform random selection in terms of total channel capacity, attacker-detection accuracy, and performance stability. Among the evaluated reinforcement learning methods, Q-learning achieves the best overall trade-off between detection accuracy and computational efficiency. Our results indicate that the proposed framework provides a stable, scalable, and effective solution for joint beamforming and security-aware decision-making in dynamic and adversarial wireless environments.

Entropy doi: 10.3390/e28101068

Authors: Bhadran Deepthi Bellie Sivakumar

This study explores the application of complex network theory to examine the temporal dynamics of groundwater levels in a region. It presents the first ever application of a coupled chaos theory–complex network framework to investigate the temporal dynamics and connectivity in groundwater levels in a region. For implementation, daily groundwater-level data observed over the period 2010–2014 from each of 71 wells across the United States are considered. Each well is considered as a network. In order to construct the temporal network from groundwater time series, basic concepts of chaos theory are coupled with those of complex network theory. First, the phase-space reconstruction technique is used to reconstruct the single-variable groundwater time series in a multi-dimensional phase space. Next, the optimum dimension required for the phase-space reconstruction is determined using the false nearest neighbour (FNN) algorithm. Each reconstructed vector in the phase space is considered as a node in the network, and the connections between the nodes (i.e., links) are identified based on the distance threshold between the reconstructed vectors. Four complex networks-based measures—degree centrality, betweenness centrality, closeness centrality, and clustering coefficient—are used to examine the properties of the groundwater-level network. The optimal embedding dimensions from the FNN method for the 71 groundwater level time series are found to range from 4 to 18, suggesting a wide range of complexity in the 71 groundwater level time series. However, a large majority (i.e., 62) of the time series have dimensions in the range of 4–10, suggesting low-to-medium-level complexity of the groundwater level dynamics. The results for the four network measures are degree centrality in the range 0.12–0.55, betweenness centrality in the range 604–4750, closeness centrality in the range 0.00009–0.00034, and clustering coefficient in the range 0.709–0.863. No region-specific patterns are observed in the four network measures, although, depending upon the measure, networks of wells in some states have higher/lower values and networks in some other states have a good mix of high and low values. Further, Spearman rank correlation analysis shows that the four network measures do not have significant relationships with key statistical characteristics (mean, standard deviation, and coefficient of variation) of the groundwater levels. This indicates that the differences in the network measures cannot be explained only by the basic statistical properties of the groundwater level records.

Entropy doi: 10.3390/e28101067

Authors: Ruihao Xin Sudan Bai Jiankang Fan Cong Gao Xin Feng

This paper proposes a hierarchical recurrent neural network framework that leverages a context-gated mechanism and the elastic sorting of time series indicators, addressing the strong dynamic coupling characteristics of aeroengine operational time series data. This framework addresses the coupling of multi-physical field signals and the interleaved cross-scale degradation characteristics of engine time series data. The framework achieves high-precision predictions of residual service life under complex operating conditions by deeply integrating multi-modal feature engineering with hierarchical time series modeling. An elastic ranking algorithm for time series indexes is designed to adaptively fuse multiple sensors’ time domains, frequency domains, and nonlinear characteristics, thereby constructing a dynamic feature space sensitive to load conditions. This approach addresses the limitations of traditional fixed feature combinations in characterizing variable load conditions. Additionally, a multi-scale stack dilation convolution module is introduced to exponentially enlarge the receptive field, allowing for the capture of long-range dependencies in weakly degraded signals. A bidirectional time–frequency converter decomposes the signal into low-frequency trend components and high-frequency anomaly components, enabling coupling characterization through cross-scale feature interactive gating. Furthermore, context-gated cyclic units based on hierarchical cyclic networks are constructed to hierarchically integrate multi-modal features while dynamically adjusting time-dependent weights, thereby enhancing the model’s causal reasoning ability for long sequence degradation patterns. In four typical experiments conducted on the C-MAPSS dataset, the ERCG-HRNN demonstrates significant improvements in RMSE and Score compared to existing methods.

Entropy doi: 10.3390/e28101066

Authors: Edward Bormashenko Shraga Shoval Mark Frenkel Michael Nosonovsky

We investigate the relationship between the graphical symmetry of symbols and their frequency of occurrence in culturally evolved symbolic systems and compare the observed tendencies with those found in physical structures. Capital letters of the English and Russian alphabets were classified according to their invariance under vertical reflection, horizontal reflection, and rotation through 180°. The graphical symmetry of numerical figures was analyzed across four numeral systems. Symmetry was examined for 1000 Japanese Kanji arranged into frequency-ranked groups. Spearman’s and Kendall’s rank-correlation tests revealed positive symmetry–frequency tendencies in both alphabets, although the associations did not reach conventional statistical significance. A strong and statistically significant positive association was obtained for the average symmetry of the numerical figures and their leading-digit frequencies predicted by Benford’s law. The Japanese Kanji produced the strongest statistical result. Despite their different cultural origins and graphical organization, the investigated alphabets and numeral systems exhibited surprisingly similar symmetry-class entropies, suggesting a possible common balance between graphical regularity and symbol distinguishability. For comparison, seven-bond and eight-bond colloidal clusters demonstrated the opposite tendency, with asymmetric configurations occurring more frequently and dominating their symmetry-class distributions.

Entropy doi: 10.3390/e28101065

Authors: Paolo Renati

In previous works, it has been highlighted how the confinement of electromagnetic (EM) coherent modes—arising from their coupling with the water connectome of living matter—acts as a self-emergent ‘mold’ that spatially organizes molecules, membranes, vesicles and larger cellular structures through ponderomotive and dielectrophoretic forces: the standing waves of the self-trapped fields pattern matter into nodes and antinodes—as in cymatics—driving the template-free self-assembly of compartments from the nano- to the macroscale. In this perspective report, we extend that picture and propose specific physical mechanisms by which the same confined EM modes may act as the agent of one of the deepest levels of morphogenesis: the handedness/chirality of biomolecules. We first recall that the chiral content of light is not captured by polarization (spin angular momentum, SAM) alone: the true measure of the chirality of light is the helicity (ℋ), whereas the orbital angular momentum (OAM), carried by helical-wavefront (Laguerre–Gaussian-like) modes, governs global rotational transformations. We then show that (i) the geometric (Berry) phase implied by the chiral light–matter coupling is the quantum manifestation of optical rotation, both being governed by the same molecular pseudoscalar—the Rosenfeld rotational strength—that builds the mixed electric–magnetic polarizability responsible for chiral optical forces; (ii) the decohered one-dimensional (1D) lanes generated by the Anderson–Higgs–Kibble mechanism for the self-focusing of the fields traveling through the coherent aqueous medium are themselves optical vortices carrying a quantized flux, and, therefore, natural cavities for OAM-carrying modes whose topological charge selects, through an optical OAM torque, the handedness of supramolecular helices (like α-helices, as well as more complex geometries such as several collagen types); (iii) the helicity gradient developed at the boundaries of coherence domains acts as a ‘chiral mold’ that breaks the energy degeneracy of the ‘enantiomer + field mode’ pair and exerts an enantioselective SAM torque on point-like species (amino acids and sugars), lowering the activation barrier of the preferred enantiomer. Although small, this chiral bias operates within the coherent, low-entropy medium of biological water, being thus fueled and protected against thermal noise. We conclude that the homochirality of life is a direct and necessary consequence of the QED coherence of the aqueous matrix.

Entropy doi: 10.3390/e28101064

Authors: Gregg Jaeger

Niels Bohr asserted both that (i) the properties of quantum systems are specifiable, in general, only in relation to mutually exclusive experimental arrangements which may be freely chosen by experimenters, and that (ii) observed physical behavior is objective. In a move that reduces the apparent tension between those two assertions, Bohr put forward an analogy between quantum complementarity (in effect, the dependence of property values on measurement apparatus configuration in quantum mechanics) and the principle of relativity (in effect, the dependence of property values on the inertial reference frame in special relativity). Here, Bohr’s analogy is considered in a theoretical context. The elements of quantum mechanics and special relativity involved are set side by side, the analogous elements are identified with their roles in the analogy spelled out, and a number of dissimilarities that impact the force of the analogy are indicated—for example, quantum value determination, in general, requires an uncontrollable interaction during measurement, whereas non-quantum relativistic value determination does not. These dissimilarities point to weaknesses in the analogy, which he did not highly develop, and suggest that any stronger such analogy involving the two theories must take these differences more fully into account.

Entropy doi: 10.3390/e28101063

Authors: Zhongxin Yang Chun Bao Yuanwei Bin Xiang I. A. Yang Yipeng Shi Shiyi Chen

Natural language is a complex system that exhibits robust statistical regularities. Here, we represent text as a trajectory in a high-dimensional embedding space generated by transformer-based language models, and quantify scale-dependent fluctuations along the token sequence using an embedding-step signal. Across multiple languages and corpora, the resulting power spectrum exhibits a robust power law with an exponent close to 5/3 over an extended frequency range. This scaling is observed consistently in contextual embeddings from both human-written and AI-generated text, but is absent in static word embeddings and is disrupted by randomization of token representation order. These results show that the observed scaling reflects multiscale, context-dependent organization rather than lexical statistics alone. These results identify an empirical spectral regularity in contextual language representations and provide a quantitative, model-agnostic benchmark for studying their multiscale structure.

Entropy doi: 10.3390/e28101062

Authors: Mengyao Wang Xiuyang Meng Chunling Wang

Multimodal sentiment analysis integrates text, audio, and visual signals to infer affective states. However, sentiment information shared across modalities is often entangled with modality-specific variation, and existing representation learning methods do not fully exploit the relations encoded by continuous sentiment labels. Effective cross-view alignment therefore requires structured representations and a target that captures affective relations among samples. We propose a label-aware distributional contrastive alignment framework to address these requirements. Shared-private representation learning separates common sentiment information from modality-specific cues. The shared representations form audio–text and visual–text interaction views guided by a common textual reference, while the private representations retain complementary cues for prediction. Label-Aware GCA-UOT aligns the two sets of interaction views through batch-level distributional matching. Entropic unbalanced optimal transport provides a smooth, flexible coupling, and a label-aware target guides the allocation of matching mass according to paired-view correspondence, sentiment-intensity proximity, and polarity consistency. Experiments on CMU-MOSI, CMU-MOSEI, and CH-SIMS show consistent gains in fine-grained sentiment classification. Our method improves Acc-5 by 0.5–2.9 percentage points over the strongest baseline on each dataset and achieves the best or joint-best Acc-7 among the compared methods on CMU-MOSI and CMU-MOSEI.

Entropy doi: 10.3390/e28101061

Authors: Dongdong Wang Xiongfeng Luo Haiwei Geng Xia Yu Yinbing Yang Yilin Tang

The virtual synchronous generator serves as a critical interface between renewable generation systems and the grid, providing inertia to support both active and reactive power. Based on the duality principle, a virtual permanent-capacitor synchronous generator is derived to enhance the inertia of the three-phase current-source inverter, exhibiting dynamic characteristics similar to those of a virtual permanent-magnet synchronous generator. Additionally, this work presents a unified virtual synchronous generator model applicable to both voltage-source and current-source topologies. By constructing a suitable Lyapunov function candidate, the stability criteria of the unified model under standalone operation are established. The Lyapunov function candidate is constructed as a Bregman divergence, whose monotonic decrease characterizes the convergence to synchronous equilibrium. In grid-connected mode, a frequency-locking condition is derived to ensure synchronization of the virtual synchronous generators with the grid. Eigenvalue analysis is employed to assess the stability of the unified model under frequency-locking conditions. The correctness of the derived stability conditions in standalone and grid-connected modes is confirmed through time-domain simulations.

Entropy doi: 10.3390/e28101060

Authors: Mansur Ziiatdinov Fedor Levkovich-Maslyuk Emmanuel M. Pothos

Finding the global minimum of a function with many local extrema is well known to be a challenging computational task. We propose a hybrid quantum–classical algorithm for this optimisation problem, based on a combination of Grover quantum search and the classical Newton–Raphson method. The approach is tailored to situations with a large number of false minima. The main idea is to use quantum search to efficiently locate the vicinity of the true optimum, after which a classical algorithm can solve the problem in only a few iterations. We specifically address the nontrivial question of choosing the discretisation of the function domain, which is critical for quantum approaches to optimisation of continuous functions. While we do not expect that this method will always be superior, we identify a paradigmatic case of a function for which we demonstrate that, remarkably, our hybrid method quadratically outperforms a purely classical approach (measuring performance by the number of function/oracle calls). We evaluate key performance metrics of the algorithm in numerical experiments. We furthermore discuss its possible extensions as well as the potential relevance of hybrid approaches in the context of quantum cognition.

Entropy doi: 10.3390/e28101059

Authors: Jinhong Wang Rui Wang Peng Han Weiwei Wu Lingyuan Meng Changsheng Jiang

As global energy systems pursue lower-carbon transition pathways, the rapid expansion of shale-gas development has increased subsurface fluid injection, intensifying concerns about induced seismicity and associated seismic risk. Although regional studies have documented specific induced-seismicity sequences, global patterns, regional variability, forecasting approaches, and governance practices have rarely been examined together within an integrated framework. This review synthesizes shale-gas-related induced seismicity across major production regions, including North America, Europe, and China, with particular emphasis on common patterns and contrasting seismic responses to hydraulic fracturing and wastewater disposal. Across diverse geological and operational settings, the reviewed evidence indicates that wastewater disposal is generally associated with larger-magnitude and more spatially widespread seismicity, whereas hydraulic fracturing typically produces localized seismicity but can also be associated with significant events where critically stressed faults are hydraulically connected to the stimulated zone. Building on this regional synthesis, we compare statistical, physics-based, hybrid, and machine-learning forecasting methods in terms of their operational applicability, predictive capabilities, data requirements, and limitations. However, translating these methodological advances into effective risk reduction requires their integration with robust monitoring and adaptive governance practices. Current governance measures can reduce risk, but their effectiveness depends on sufficiently sensitive monitoring, timely data exchange, regionally calibrated and adaptive operational thresholds, and continued post-injection surveillance. Future risk management should therefore move toward dynamic, physics-constrained geospatial frameworks that integrate seismic observations, injection histories, fault architecture, and regional stress conditions to support adaptive operational decision-making.

Entropy doi: 10.3390/e28101058

Authors: Jile Chaoge Qinwei Yang Jingyi Li Peng Wu

Offline policy learning aims to optimize individualized decisions using historical data. However, conventional methods primarily focus on maximizing expected rewards while neglecting individual-level counterfactual harm—cases where the assigned treatment leads to worse outcomes than the control. This can result in overly aggressive policies that harm some individuals. In this article, we propose a counterfactual harm-aware policy learning framework that effectively balances expected reward with individual-level harm. To quantify harm, we introduce two measures: treatment harm rate (THR), capturing the proportion of individuals harmed, and treatment harm quantity (THQ), capturing the severity of harm, particularly for continuous outcomes. A key challenge is that both THR and THQ depend on the joint distribution of potential outcomes, which is generally unidentifiable, even in randomized controlled trials. To address this, we introduce an additive latent variable model under which we establish the identifiability of THR and THQ and develop the corresponding estimators. These estimators are then used to learn policies under explicit harm constraints. Extensive experiments demonstrate the effectiveness of the proposed approach.

Entropy doi: 10.3390/e28101057

Authors: Jianzhong Li Ping Zhong Zeying Li Qinghua Deng Chuying Yu Xinan Chen Lin Xu

An RGB-D image encryption scheme is presented by combining a newly constructed 4D-ICTM hyperchaotic system with a reformulated quaternion gyrator transform (QGT). The proposed 4D-ICTM dynamics arise from the nonlinear combination of the ICMIC and Chebyshev maps with the tanh function and modulo operation. Their dynamical and statistical properties are examined using Lyapunov exponents, bifurcation diagrams, histograms, autocorrelation analysis, and correlation-dimension computation. To handle the heterogeneous RGB-D data, a dedicated preprocessing procedure converts the RGB channels and 16-bit depth data into four 8-bit components, which are represented jointly in the quaternion domain. The reformulated QGT is evaluated in terms of numerical consistency and computational cost using numerical error, SSIM, and execution-time measurements. The resulting encryption approach is further assessed through statistical analysis, adjacent-pixel correlation, floating-frequency analysis, graphic autocorrelation, and decryption-error evaluation. The experiments show resistance to the tested attack scenarios together with competitive computational performance. An encryption throughput of 43.32 Mbps is obtained at 406 cycles/byte, indicating the potential of the proposed scheme for RGB-D image encryption applications.

Entropy doi: 10.3390/e28101056

Authors: Di Wang Chongchong Song Perry Gu Li Yao

We present an accelerated, stabilizer-based normalized gradient flow (ASNGF) framework with a nonlocal inertial (or momentum) mechanism for computing stationary states of Bose–Einstein condensates. Two schemes, ASNGF-I and ASNGF-II, are developed, and their discrete Lyapunov energy dissipation properties are established under the stated conditions. Linearized stability analyses and numerical parameter studies provide practical guidance for selecting the tunable parameters and demonstrate the important role of the nonlocal Laplacian term in the inertial dynamics. For suitable parameter scalings, the observed iteration counts are essentially insensitive to the time step over the tested range, including τ=10±60. Numerical experiments on several Gross–Pitaevskii models show that ASNGF can substantially reduce the iteration count compared with conventional normalized gradient-flow schemes for the tested problems and parameter settings. The results also reveal that its convergence behavior is problem- and parameter-dependent. The inertial mechanism can enhance exploration of non-convex energy landscapes but does not guarantee convergence to a global minimizer. These results demonstrate the potential of ASNGF as an efficient framework for stationary-state computations in BECs.

Entropy doi: 10.3390/e28101055

Authors: Francisco J. Tapiador

Jaynes’ maximum entropy (MaxEnt) principle selects, from all probability distributions consistent with a set of constraints, the one that maximizes the Shannon entropy. The principle is well-defined for classical, real-valued probabilities, but extending it to complex-valued probability theories requires care. Youssef’s complex probability framework, in which quantum mechanics is reformulated as a Bayesian theory with complex amplitudes and the observable probability is the squared modulus (Born rule), does not admit a natural MaxEnt extension: the L2 normalization of that framework is incompatible with the L1 normalization assumed by Shannon entropy, and the resulting complex-valued entropy functional cannot be maximized without additional structure absent from Youssef’s axioms. The present paper proposes a different extension of Kolmogorov’s axioms to the complex domain, replacing non-negativity with the condition that the real part of every probability be non-negative. This framework, which retains L1 normalization and recovers classical probability when the imaginary part vanishes, admits a natural MaxEnt principle: the real part of the complex Shannon entropy is a well-posed real functional, its Euler–Lagrange equations produce an explicit complex Gibbs distribution with complex Lagrange multipliers, and the classical Gibbs distribution is recovered as the special case of real multipliers. The die problem (Jaynes’ Brandeis dice example) is worked out in detail for both the fair and the loaded cases, illustrating how imaginary constraints redistribute the observable probability mass in a way that has no counterpart in classical MaxEnt.

Entropy doi: 10.3390/e28101053

Authors: Jun Li Yonghong Hao Qi Wang

We formulate the problem of estimating hydraulic conductivity from measured hydraulic head data in groundwater flow within the unified framework of parameter estimation using the extended Kalman filter (EKF) and least-squares optimization. We first formulate the EKF and its variants as problems of least-squares optimization. We then show that the successive linear estimator can be interpreted as a variant of the EKF, with its initial estimate obtained through co-Kriging. Furthermore, by showing that the EKF is equivalent to a one-step Newton iteration for solving a least-squares problem, we reformulate the hydraulic conductivity estimation problem as an associated least-squares optimization problem. Finally, we present an algorithm for estimating hydraulic conductivity from unsaturated soil data.

Entropy doi: 10.3390/e28101054

Authors: Amirarsalan Moatazedian Yauhen Yakimenka Rémi A. Chou Jörg Kliewer

We study a privacy-preserving data-sharing setting where a privatizer transforms private data into a sanitized release observed by an authorized reconstructor and two unauthorized adversaries, each with access to side information correlated with the private data. An adversary is considered stronger when it estimates the private data more accurately, thereby achieving a lower estimation loss. The privatizer maximizes the estimation loss of the stronger adversary while keeping the reconstruction distortion within a distortion budget. The difficulty is that the identity of the stronger adversary can vary with the distortion budget. Thus, optimizing the privatizer against a single adversary selected in advance can overlook the other adversary when that adversary estimates the private data more accurately. This motivates a constrained minimax formulation that accounts for both adversaries at every distortion budget. Penalized alternating updates train the privatizer and estimators. To evaluate the algorithm, we study finite-alphabet, scalar-Gaussian, ten-component Gaussian-mixture, and MNIST settings. For these settings, we compute theoretical reference curves under their respective assumptions to assess the learned sanitization mechanisms. We prove local conditional convergence bounds for a single-adversary recursion with fixed stochastic-gradient steps.

Entropy doi: 10.3390/e28101052

Authors: Hongyi Zhang Liquan Chen Huiyu Fang Jianchang Lai Xinyuan Sun

Mobile crowdsensing (MCS) leverages the sensing capabilities of mobile devices carried by a vast number of participants to accomplish large-scale urban sensing tasks. In MCS systems, participants continuously generate trajectory streams while uploading sensing data, which raises severe risks of location privacy leakage. Local differential privacy (LDP) provides strong guarantees for user-side privacy protection. However, existing approaches for real-time trajectory publishing in MCS remain inadequate; they fail to fully leverage the continuously evolving spatial density and temporal structures within trajectory streams, which leads to suboptimal utility and distorted spatiotemporal representations under strict privacy constraints. To address this issue, we propose AdaGrid-Syn, a two-layer adaptive grid framework for real-time trajectory publishing under LDP. AdaGrid-Syn dynamically adjusts spatial granularity based on changes in density distributions and transition patterns. In addition, we develop a new adaptive privacy budget allocation strategy that distributes privacy budgets according to the temporal variation of Markov transition matrices, enabling better capture of time-sensitive behavioral characteristics. Experimental results demonstrate that our framework significantly outperforms existing approaches.

Entropy doi: 10.3390/e28101051

Authors: Karol Karpiński Natalia Kruszewska J. Miguel Rubi Adam Gadomski

This Communication addresses a superlubric state in graphene-type layers in nanoscale conditions. We focus on the superlubric state based on the twofold generation and propagation of (thermal) phonons. In the first instance, phonon propagation has a local mode in the graphene-type lattice, whereas in the second instance, it can spread globally. In the former, we see that the kinematic harmonic-mean-type speed condition underlies the mechanism of phonon propagation. In the latter, an arithmetic-mean fingerprint, pointing to a constant acceleration case, can be applied. We disclose a linear phonon-propagation mechanism with an entropic force behind it. Under a passage between classical and quantum domains, this entropic force is equivalent to a Stokes-type force (linear in the sliding speed) with a resultant involvement of a quantum fluctuation–dissipation relation, including a “double quantum” of thermal energy and the apparent (quantum viz. nanoscale) viscosity. The corresponding entropy-production conditions are derived and critically discussed within this framework, establishing a direct thermodynamic link between microscopic (physical) information-processing events and mesoscopic phonon transport. The resulting unified classical–quantum description of entropy-driven phonon-mediated superlubricity constitutes the principal novelty of the present work and may be relevant for future developments in nanotechnology and advanced materials engineering.

Entropy doi: 10.3390/e28101050

Authors: Arieh Ben-Naim

The concept of solvation was originally studied within thermodynamics. This may be referred to as the macroscopic approach to solvation. It involved the transfer of solute from the gaseous phase into the solvent under well-specified thermodynamic conditions. In 1978, a new definition of the solvation process was introduced. This new definition may be referred to as the molecular or the microscopic approach to the study of solvation. Once the new definition was introduced and shown to be advantageous, all the traditional definitions were rendered obsolete. With this new definition, a host of difficult problems in chemistry and biochemistry could be studied, including the mystery associated with the entropy of solvation of inert gases in water, the problems associated with protein folding, and protein self-assembly.

Entropy doi: 10.3390/e28101049

Authors: Wei Shen Jiaqing Mo Yuhua Lin

Wireless sensor networks (WSNs) are pervasively employed in critical domains such as smart healthcare, industrial automation, and environmental surveillance, where secure access control is essential. User authentication protocols are vital for thwarting unauthorized intrusions, yet the vast majority of existing schemes rely on conventional public-key cryptography (such as ECC, RSA), which is provably insecure in the post-quantum era due to Shor’s algorithm. This motivates the design of post-quantum authentication that is also lightweight for resource-limited sensors. To address this challenge, we propose an efficient authentication and key agreement protocol based on Number Theory Research Unit (NTRU), a lattice-based cryptosystem offering proven resistance to quantum attacks. We have proven the correctness and security of our scheme through BAN logic and random oracle model and demonstrated through informal analysis that our scheme can resist various attack methods and possesses high security features. Moreover, we thoroughly evaluate our protocol against the most recent state-of-the-art schemes. The results indicate that our scheme has the lowest computational cost, and its communication cost is superior to that of most other schemes. Overall, our proposed protocol provides a robust, efficient, and future-proof authentication solution for WSNs, well-suited for deploying WSNs environments.

Entropy doi: 10.3390/e28101048

Authors: Yongjia Gao Jieyu Liu Shuanzhu Li Can Li Qiang Shen Zhaoqiang Wang

Laser strapdown inertial measurement units (LIMUs) are critical components of autonomous navigation and attitude determination systems used in mission-critical platforms such as spacecraft. Their performance directly affects the reliability and safety of space missions. To address the challenges posed by high-dimensional indicator sets, limited data, and stringent interpretability requirements, this paper proposes an interpretability-aware hierarchical Bayesian belief rule base model, termed IH-BRB, for LIMU performance evaluation. First, a five-level hierarchical evaluation structure is developed based on the physical architecture and error-propagation relationships of a LIMU, thereby reducing rule base complexity and improving inference traceability. Second, an interlayer transmission mechanism based on the expected grade scores and entropy-derived reliability is introduced to preserve essential uncertainty information and dynamically adjust the attribute weights of upper-level BRB nodes. Third, semantic consistency constraints are imposed on the conditional probability tables and rule-consequent belief distributions to prevent semantic drift during data-driven optimization. The proposed model is validated using an engineering LIMU dataset with expert-defined target grades to assess its consistency with the established grading criterion. The results show that IH-BRB maintains a competitive evaluation performance while preserving physically meaningful rule semantics and layer-wise traceability.

Entropy doi: 10.3390/e28101047

Authors: Shengna Liu Kheder Suleiman

This study investigates the transport properties of inertial Active Ornstein–Uhlenbeck particles (AOUPs) in a one-dimensional rocking washboard potential externally imposed fractional Gaussian noise (fGn). Through numerical simulations, we explore how fGn in the potential influences particle mobility. Our findings reveal anomalous transport, including absolute negative mobility (ANM) at specific active intensity levels. The Hurst exponent significantly modulates ANM, with higher values of the Hurst exponent widening the ANM window by reducing the noise’s anti-persistence (all Hurst exponents studied here satisfy H≤1/2). These results provide new insights into the mechanisms governing anomalous transport in media with power-law correlated noise. Our findings have potential applications in microfluidics and active matter systems, enabling precise manipulation and separation of active particles.

Entropy doi: 10.3390/e28101046

Authors: Ayrton Sarango Juan Aguarón Rodrigo Aznar Daniel Bordonaba Cristina Moreno-Loscertales José María Moreno-Jiménez Jorge Navarro

Decision-making in healthcare involves both clinical evidence and emotional and cognitive processes that are often difficult to formalize. While sentiment analysis has been widely used to capture emotional information in clinical interactions, it does not fully account for how emotional categories are distributed across interactions. This study proposes an entropy-based approach to complement sentiment analysis by quantifying the diversity and distribution of emotional states expressed during doctor–patient consultations. A dataset of 30 primary care consultations involving patients with hypertension (HTA) and dyslipidemia (DLM) was analyzed using natural language processing techniques. Emotional information was extracted using the NRC Emotion Lexicon, and normalized Shannon entropy was computed at the intervention level. Intervention-based, role-based, and phase-based analyses were conducted. The results show that emotional entropy varies throughout the consultation and across consultation phases. Consultation-level analyses showed higher overall entropy in HTA than in DLM consultations, while no significant overall effect of the speaker role was found. Entropy also varied significantly in consultation phases, although the disease-by-phase interaction was not significant. Robustness analyses supported the stability of these patterns, and the negligible association with sentiment valence indicates complementary information. These findings highlight entropy as a complementary measure for analyzing emotional dynamics in clinical interactions.

Entropy doi: 10.3390/e28101045

Authors: Shuang Zhao Dong Yang Xiujiao Zhang

Accurately modeling the distribution of the crude oil market is crucial in forecasting downside risk. In this paper, the flexible skewed generalized t (SGT) distribution is used to model the distribution of WTI and Brent crude oil market returns. The estimates of the unconditional SGT indicate that the empirical distributions of both WTI and Brent crude oil returns can be well fitted by the skewed t (ST) distribution. Furthermore, we relax the conventional GARCH-type models and allow the high-order conditional moments, such as the third and fourth moments, which are related to skewness and kurtosis, to be time-varying. The results show that all the conditional moments in the ST density function are statistically significant. Lastly, the efficiency of ST-GARCH-type models with time-varying high-order moments is evaluated based on the results from both in-sample and out-of-sample conditional value at risk (VaR) forecasting. We find that the proposed model generates superior estimates compared with the real VaR thresholds. These results have both theoretical and practical implications for financial institutions in formulating crude oil market risk management strategies and avoiding potential losses due to extreme risks in the crude oil market.