Entropy doi: 10.3390/e28090970

Authors: Chengyi Jin Luyuan Chen Hao Li

Distance measures in random permutation set (RPS) theory are crucial for characterizing inconsistency among permutation-based information distributions. However, existing RPS discrepancy measures do not explicitly distinguish ordering conflicts according to their positional importance under propensity semantics. To address this issue, this paper proposes a new RPS distance, termed MEOWA-KTC, by combining maximum-entropy-based ordered weighted averaging (MEOWA) weights with Kendall’s tau coefficient (KTC). Specifically, MEOWA-KTC constructs a top-weighted similarity between permutation events by using KTC to evaluate the ordinal consistency of corresponding sub-permutations and MEOWA weights controlled by an adjustable orness parameter to emphasize discrepancies at leading positions. Additionally, a spectral correction is applied to ensure that the proposed distance satisfies the metric axioms. Numerical examples and ablation results demonstrate the positional sensitivity of the proposed distance and the respective contributions of MEOWA weighting and KTC. Based on this distance, a fusion model is further developed to derive source support degrees and fusion weights from pairwise RPS distances. In the threat-assessment application, the proposed method produces stable decisions and generally larger decision margins than the benchmark methods. Monte Carlo experiments further demonstrate its robustness to mass-distribution and permutation-order noise.

Entropy doi: 10.3390/e28090969

Authors: António M. Lopes

This paper presents a coarse-grained framework for analyzing the long-term structural evolution of soccer leagues. This framework draws on ideas from non-equilibrium statistical physics. It combines latent-strength modeling, interaction rules, and large-scale observables to explore the rise of hierarchical organization in competitive leagues. Teams interact based on a Bradley–Terry-type stochastic model. This model relies on teams’ latent competitive strengths, which are derived from match outcomes employing maximum-likelihood estimation. Experiments on 38 soccer leagues around the world show persistent hierarchical organization. This is evident in varied latent strengths, uneven competitive balance, and stable macro-state organization from season to season. Even with differences in history, geography, and competition format, many leagues display surprisingly similar large-scale structural patterns. These findings support the idea of shared principles that guide the evolution of competitive leagues and highlight the value of using a statistical physics perspective to describe how macro-level organization arises from repeated team interactions.

Entropy doi: 10.3390/e28090968

Authors: Wenhao Zhou Zhiwei Zhang Siyu Lin

Alliance network embeddedness provides firms with access to external knowledge and resources, yet its innovation implications vary across firms and network contexts. This study examines how network structure and combinations of embedding characteristics are associated with corporate innovation performance. Based on 335 firm-level observations from Chinese listed biopharmaceutical manufacturing firms, the study first identifies heterogeneous alliance network environments through community detection and K-Means clustering. Four network types are identified: dyadic, ringlike, star, and complex alliances. Classification and regression trees (CART) are then employed to extract interpretable, threshold-based decision rules linking network embedding characteristics to high and non-high innovation performance. The results show that no single network characteristic is consistently associated with innovation performance across alliance types. In dyadic alliances, moderate cooperation intensity is associated with high innovation performance, whereas ringlike alliances exhibit conditional associations involving cooperation intensity and partner centrality. Star alliances are characterized by configurations involving cooperation breadth and network position, while complex alliances exhibit more multidimensional combinations of structural and relational conditions. The findings indicate that the innovation relevance of alliance network embeddedness is network-type-specific and configuration-dependent. As a complementary robustness analysis, fuzzy-set qualitative comparative analysis broadly supports several core configurational patterns identified by CART, while also revealing alternative configurations, particularly in complex alliances. The study demonstrates that understanding alliance network embeddedness requires attention to network context, empirical thresholds, and combinations of network characteristics rather than isolated network attributes.

Entropy doi: 10.3390/e28090967

Authors: Salvatore Monaco Dorothée Normand-Cyrot

While continuous-time Hamiltonian dynamics are naturally energy preserving with a symplectic flow, their discrete-time counterparts enhance either geometric or energy preservation properties, but rarely both within a unified framework. It is the object of this paper to more deeply investigate this question. In both linear and nonlinear settings, necessary and sufficient conditions characterizing discrete Hamiltonian dynamics that are conservative and symplectic are derived. The relationship with exact sampled models of continuous-time Hamiltonian dynamics are investigated, showing that such models, that preserve both energy and symplectic structures, do not generally fit into the proposed canonical form. Generalized Hamiltonian structures are, thus, introduced. On these bases, Hamiltonian integrators that preserve both the energy and the symplectic structure up to a prescribed order in the sampling period, are constructed. Some simulations on nonlinear test cases illustrate the theoretical findings.

Entropy doi: 10.3390/e28090966

Authors: Guanghai Liu Zhe Zhang Wang Fu Cui Zhang Boyu Wang Yanfeng Su Zhijian Cai

In this paper, an optical color zero-watermarking scheme based on robust bimodal biometric keys and phase-shifting digital holography is proposed. The color watermark is first encrypted into three amplitude ciphertexts through an optical encryption framework combining grating modulation, Fresnel-domain double random phase encoding (DRPE), and phase-shifting digital holography, where the phase masks are generated from biometric keys derived from the iris and three-dimensional (3D) face features of the encryption user. These high-level biometric features are extracted by a bimodal biometric high-order feature extraction network (BBHEN), including an iris high-order data extraction network and a 3D face high-order data extraction network. The extracted features of the color host image are then XORed with the corresponding ciphertexts, and the results are merged to construct a single zero-watermark image containing both host and watermark information. During extraction, biometric authentication is first performed to verify the identity of the decryption user. Only authorized users can recover the original watermark through zero-watermark reconstruction and extraction; otherwise, the process is terminated. Numerical simulations demonstrate the effectiveness, security, and robustness of the proposed scheme, particularly the strong protection capability of the bimodal biometric keys.

Entropy doi: 10.3390/e28090965

Authors: D. S. Fonte L. H. A. Monteiro

This study proposes a game-theoretic agent-based model to investigate cooperation and competition in workplace environments. The iterated prisoner’s dilemma is implemented on a two-dimensional lattice, where agents interact locally and accumulate wealth over time. Two types of agents are considered: fixed probabilistic cooperators, who adopt a constant cooperation probability, and adaptive probabilistic cooperators, whose behavior depends on their accumulated wealth and reputation. Population composition and wealth distribution are quantified using Shannon entropy and the Gini coefficient. Numerical simulations show that fixed cooperators tend to predominate and accumulate higher average wealth. The simulations also show that increasing tolerance to defections raises average wealth and reduces inequality. In contrast, a higher cooperation probability increases wealth but may also amplify inequality. These results highlight the role of reputation and performance targets in shaping cooperation in organizational environments.

Entropy doi: 10.3390/e28090964

Authors: Wenxia Xu Liang Xue Liping Zhu Jiaofen Li Guodong Li

Chaotic systems have been widely investigated for color image encryption because of their nonlinear dynamics, initial-condition sensitivity, and pseudorandom behavior. However, locating numerically robust parameter regions in high-dimensional hyperchaotic systems remains difficult, while many DNA-based schemes employ fixed or weakly varying rules. This study proposes a color image encryption scheme combining a 6D Lorenz–Sprott system optimized by particle swarm optimization (PSO), symbol-level feedback-dependent DNA transformation, and bidirectional cross-channel chained diffusion. The second-largest Lyapunov exponent is used as the optimization objective to locate parameter sets with at least two positive exponents. The selected system has the Lyapunov spectrum (0.8118, 0.2736, −0.0013, −4.8957, −8.5499, −12.6746) and retains two positive exponents under refined numerical settings and ±1% single-parameter perturbations. One hundred independently initialized sequences satisfy all 15 categories of the NIST SP 800-22 test suite. Tests on six images and three secret keys achieve exact reconstruction in all 18 cases. Across 180 randomly located one-bit plaintext perturbations, the mean NPCR and UACI are 99.6089% and 33.4554%, respectively. For the tested Baboon case, perturbing any initial-state component by approximately 10−14 prevents meaningful plaintext recovery. Ablation results further demonstrate the contributions of dynamic DNA transformation and bidirectional diffusion. The proposed scheme therefore provides reproducible hyperchaotic parameter modulation and strong empirical statistical and differential performance.

Entropy doi: 10.3390/e28090963

Authors: Mohamed Seif Mohamed Andrea J. Goldsmith

We study hierarchical spectral graph clustering under edge differential privacy (DP) through the lens of iterative eigenvector estimation on adjacency matrices. We propose a differentially private recursive spectral framework, where each binary partition is obtained via a rank-one noisy power method applied to induced adjacency sub-matrices. At each iteration, carefully calibrated Gaussian noise is injected into the matrix–vector multiplication, ensuring (ε,δ)-edge DP under cumulative privacy accounting across both power iterations and recursive hierarchy levels while preserving the essential convergence properties of the classical power method. We provide a non-asymptotic analysis of the resulting noisy iterations, characterizing the trade-off between privacy and accuracy via explicit bounds on the eigenvector estimation error. In particular, we quantify how the noise variance, number of iterations, eigengap, and hierarchy depth jointly influence the accuracy of each recursive split and the overall clustering performance. Empirical evaluations on synthetic and real-world networks validate the theoretical predictions and demonstrate that the proposed method achieves strong multi-scale clustering performance under meaningful privacy budgets.

Entropy doi: 10.3390/e28090962

Authors: Salim Bouzebda Sultana Didi

This paper develops asymptotic theory for kernel estimation of density-weighted conditional functionals and regression derivatives when responses are missing at random (MAR) and the observations form a strictly stationary ergodic process. Sequential MAR and positivity identify the complete-data conditional target through an inverse-probability-weighted pseudo-response, while the fully observed covariate density and its derivatives are estimated without unnecessary response weighting. A martingale-predictable decomposition yields uniform almost-sure rates, pointwise Gaussian limits, variance expansions, studentization, and AMISE results under explicit projective/maximal, conditional-moment, conditional-density, and variance-stabilization conditions. These quantitative assumptions are additional to stationarity and ergodicity: the results are not asserted for arbitrary stationary ergodic sequences. Exact-quotient and multi-index identities transfer the primitive-estimator theory to regression derivatives, and feasible propensity estimation contributes an explicit additional remainder. Monte Carlo experiments show that stronger dependence, weak response probabilities, higher derivative order, propensity misspecification, and smoothing bias can materially degrade finite-sample performance; undersmoothing improves centring but need not eliminate coverage distortion at moderate sample sizes.

Entropy doi: 10.3390/e28090961

Authors: Luoxin Wang Mengmeng Wang Baisuo Jin Yuehua Wu

In this paper, we study the generalized nearly isotonic optimization (GNIO) model and its dynamic programming solution (GNIO-DP). We introduce randomness into the GNIO-DP algorithm, enabling its first application to change-point detection and resulting in an O(n) complexity multiple change-point detection method. At the same time, we provide the theoretical properties of the change-point detection and prove the reliability and effectiveness of the GNIO-DP algorithm for this task. The simulation results show that our method has strong change-point detection ability. Compared with traditional methods, our method is faster in most scenarios.

Entropy doi: 10.3390/e28090960

Authors: Tomaz Nunes Morsinaldo Medeiros Marianne Silva João Carlos N. Bittencourt Daniel G. Costa Ivanovitch Silva

In some scenarios, motorsport simulators have been used to enable the controlled acquisition of dense telemetry with high similarity to real-world data, reducing cost when assessing driving performance. However, although popular, performance analyses traditionally treat human control as deterministic and overlook the stochasticity of driving behavior. In fact, existing coaching methods which improve driving performance have to deal with two distinct outcomes: a driver who restructures his race control strategy and a driver who merely repeats it faster. This article presents a Behavior-First framework for interpretable driver behavior analysis that separates them. We characterize control signals with two information-theoretic descriptors: Jensen–Shannon divergence, which quantifies distributional distance from a proficiency-matched reference and whose square root satisfies the triangle inequality, and Permutation Entropy to measure the ordinal complexity of the input sequence. A deterministic, physics-informed heuristic layer then identifies kinematic performance gaps and emits structured tokens that a Large Language Model translates into natural-language coaching narratives. We evaluated the framework in an exploratory case study. The three beginners who received generated coaching messages and the single uncoached comparison participant exhibited different lap-time and information-theoretic trajectories. Because the groups were small and non-randomized, these observations describe within-driver evolution and do not estimate a causal coaching effect.

Entropy doi: 10.3390/e28090959

Authors: Hui Xiao Liying Hou Jiantang Liu

γ-photon tomography provides a non-contact approach for reconstructing and visualizing flow-field parameters. However, the resulting grayscale images often exhibit blurred boundaries and weak texture features, causing conventional colorization methods such as DeOldify to produce cross-region color diffusion and boundary color overflow. To address this, this paper proposes a γ-photon flow-field image colorization algorithm based on the Hybrid Swin Colorization Network (HSCN). A hybrid dual-stream encoder composed of a Swin Transformer semantic stream and a central difference convolution (CDC) gradient branch is combined with cross-stage gradient injection and a spatially gated adaptive fusion mechanism to enhance the perception of high-frequency structures at flow-field boundaries and suppress color overflow. The effectiveness of the algorithm is evaluated in terms of colorization quality and flow-field temperature-parameter inversion using γ-photon flow-field images of two CFD-simulated flow patterns, a large-scale vortical wake and a horizontal wake. The proposed method achieves PSNR, SSIM, FID, and MAE values of 38.7422, 0.9372, 10.7344, and 0.0085, respectively. Compared with DeOldify, PSNR and SSIM are improved by 24.30% and 11.89%, while FID and MAE are reduced by 42.98% and 60.47%, respectively. In addition, HSCN achieved a MAPE of 12.65% across 15 boundary and temperature-transition locations in three representative samples, compared with 31.24% for DeOldify and 28.70% for DDColor.

Entropy doi: 10.3390/e28090958

Authors: Rui Huang

Quantum federated learning (QFL) enables collaborative training of variational quantum circuits across decentralized clients, but client drift under non-IID data distributions degrades standard Federated Averaging (FedAvg). Existing aggregation methods either introduce tunable hyperparameters that depend on unknown data heterogeneity or incur structural costs such as order-dependent error propagation. We propose Kernel-Weighted Aggregation (KWA), a hyperparameter-free aggregation strategy that constructs a non-parametric kernel density estimate over the received client parameter vectors at each communication round. Clients in high-density regions receive greater voting power, and isolated clients are automatically attenuated. The kernel bandwidth is the median of all pairwise squared distances. We evaluate KWA against six state-of-the-art QFL methods and five classical robust baselines on five binary MNIST digit-pair tasks under IID, Dir(0.5), and Dir(0.1) distributions with a unified four-qubit benchmark. Under IID data, KWA recovers the performance of FedAvg. Under Dir(0.5), KWA attains the highest average accuracy (0.7120), followed by FedAvg (0.7093). The margin at N=4 clients is not statistically significant. It grows to +0.023 at N=16 in the client scaling experiment. Under this distribution, the five classical robust baselines also rank below FedAvg. Under Dir(0.1), KWA exceeds FedAvg by 0.028 on average, with a pooled paired t-test p=0.068. Larger-circuit and three-class experiments show that the advantage does not yet transfer consistently beyond the four-qubit binary setting. KWA thus provides a hyperparameter-free aggregation strategy that recovers FedAvg under IID conditions, adapts to client drift, and requires no manually tuned hyperparameters.

Entropy doi: 10.3390/e28090957

Authors: Lijing Ma Shaofei Zhang Wei Zhang Jiacheng Yin Yilong Wu

Proactive lane-change risk assessment requires estimating whether an intended maneuver will lead to unsafe interactions before the maneuver is completed. This is challenging in naturalistic driving data because actual crashes are extremely rare, and binary collision labels provide little discriminative information for learning risk. We propose IntentDiff, an intent-conditioned diffusion framework for proactive lane-change risk assessment. The framework uses predicted future trajectories as the basis for risk estimation. A vectorized scene context learning module combines a VectorNet backbone with a Vector Quantized Variational Autoencoder (VQ-VAE) to map agent–map interactions into discrete intent codes. These codes organize complex traffic situations into interpretable intent prototypes and provide semantic guidance for trajectory generation. Conditioned on the learned intent code, a diffusion model generates kinematically consistent multimodal trajectories of the target vehicle. On the forecast trajectories, Monte Carlo rear-end risk is evaluated against the four bounding vehicles and fused into a Lane-Change Risk Index (LCRI). On the highD dataset, the framework attains an average displacement error of 0.42 m over a 5-s horizon. The forecast-based LCRI agrees closely with the index computed from realized future trajectories, indicating that most high-risk lane changes can be identified before the maneuver is completed. Grouping LCRI by intent code further reveals systematic variation in risk across lane-change maneuvers, suggesting that the learned codebook captures risk-relevant interaction patterns in addition to maneuver semantics.

Entropy doi: 10.3390/e28090956

Authors: Karl Friston Johan Medrano Tim Verbelen

We apply the free energy principle to free markets by treating the Market as a random dynamical system with an attracting set, i.e., some characteristic states. This licenses a normal form for stochastic dynamics that inherits from the Helmholtz–Hodge decomposition. Equipped with this functional form—and a suitable parameterization—one can create a generative model of fluctuations in the value of assets and accompanying indicator variables. This affords the opportunity for prospective (ex ante) prediction, scenario modelling and forecasting that could, in principle, be applied to any complex dynamical system exhibiting stochastic chaos. Here, we illustrate the application to portfolio management—in the context of financial services—and use the (posterior) predictive densities over future paths to evaluate the expected free energy that underwrites active inference. In this application, active inference reduces to risk-sensitive control, which can be used to model the optimal decision-making of an agent or investor. In this setting, an investor is characterized by their prior preferences for a high rate of return under drawdown constraints. Using numerical studies and historical financial data, we quantify the improvement in portfolio management, relative to baseline policies.

Entropy doi: 10.3390/e28090955

Authors: Tugay Karadag

Bayesian Network (BN) learning algorithms may exhibit substantially different performance across graph structures, sample sizes, and evaluation criteria. Comparative evidence on BN learning under structurally validated conditions remains limited. This study evaluates BN learning algorithms for causal discovery through a simulation-based framework that integrates multiple graph structures, sample sizes, and structural equation modeling (SEM)-based validation within a common design. Fourteen constraint-based, score-based, and hybrid algorithms were examined across five randomly generated directed acyclic graphs (DAGs) containing latent constructs and four sample sizes (n = 200, 500, 1000, and 2500). For each DAG–sample size combination, 1000 datasets were generated and validated using SEM, yielding 20,000 accepted datasets. Performance was assessed primarily by Matthews correlation coefficient (MCC), supported by directed structural Hamming distance (SHD) and F1. The results reveal substantial variation across DAG structures, sample sizes, and evaluation metrics, with mean MCC ranging from −0.43 to 0.65. Peter–Clark Stable achieved the strongest performance in several conditions, whereas hybrid algorithms were frequently among the weaker performers. Increasing sample size did not produce uniform performance gains, and no algorithm or algorithm class consistently dominated across all settings. These findings show that BN structure learning performance is strongly structure- and condition-dependent and support evaluation across multiple controlled DAG configurations.

Entropy doi: 10.3390/e28090954

Authors: Tingqiang Xie Zhonghua Li Xiujun Yan Jun Deng Duo Xu

The filling-and-emptying valve and downstream shaft are crucial components of navigation-lock systems. Under insufficient downstream submergence, air can be drawn through the shaft and trapped in the post-valve culvert, altering the flow structure and compromising hydraulic stability. A three-dimensional Reynolds-averaged Navier–Stokes/volume-of-fluid model was developed to investigate shaft aeration and entrapped-air-pocket evolution under varying inlet velocities and downstream-submergence depths. The aeration process comprises three stages: jet establishment, air-pocket formation, and air-pocket breakup and reorganization. Downstream-submergence depth determines whether a continuous air-intake pathway forms, whereas inlet velocity primarily controls aeration intensity and air-pocket persistence once the pathway is established. With decreasing submergence depth, the flow transitions successively from a water-sealed regime to a transition regime, a stable entrapped-air-pocket regime, and a strongly unsteady hydraulic-jump-like regime. For the present geometry and fixed valve opening, the transition from transient to sustained shaft aeration is identified within the downstream-submergence interval of hw = 2–5 m. Combined analyses of the air-pocket volume per unit width, pressure response, vortex structures, and shear-layer characteristics indicate that enhanced jet-induced shear is closely associated with shaft aeration and air entrapment, while pressure fluctuations are closely coupled with air-pocket formation, persistence, breakup, and reorganization.

Entropy doi: 10.3390/e28090953

Authors: Hikaru Wakaura Taiki Tanimae

The universal operator-growth hypothesis (OGH) states that, in a closed chaotic system, the Lanczos coefficients grow linearly, bn≃αn. We ask how this structure is modified when the system is coupled to a Markovian environment, so that the generator becomes non-Hermitian. Applying the Arnoldi recursion to the vectorized Lindbladian in the infinite-temperature Wightman inner product, we organize the resulting pair of growth rates αC≡αR+iαI—defined as effective slopes of the sub-diagonal and diagonal Arnoldi coefficients over a pre-registered fit window—around two statements whose logical status we delimit precisely. First, whenever the dissipator acts as D=−2γG^ with G^, a Hermitian grading (all dephasing-type baths), the diagonal obeys the identity Rean=−2γ⟨G^⟩n: the imaginary rate measures how fast the growing operator accumulates weight in the dissipation channels. Second, we prove a conditional parity theorem: if the Hamiltonian, jump operators, and seeds can be made simultaneously real in some basis (an antiunitary condition), then bn is even, and Rean is odd in γ exactly, so αR is renormalized only at O(γ2), and αI=2κ0γ follows from closed-system data alone. We exhibit a one-qubit Lindbladian that satisfies the often-assumed generator symmetry G†(γ)=−G(−γ) yet violates parity (b1=|1−γ|), showing that the extra condition is essential; all models studied here satisfy it bit-exactly. For large-q SYK, these ingredients predict αC=J−2i(q−2)γ, whose imaginary part is fixed solely by the interaction range; the first ladder step is exact, and the multi-step increments approach q−2 with system size (1.92±0.04 at N=12, q=4). Under a common fit protocol, the closed-system rates saturate by N=10 (αR(0)→0.437, 2κ0→0.224). The imaginary rate is not an independent observable at leading order—its content is its sign, which resolves how the growing operator meets its environment (opposite for spin chains and SYK).

Entropy doi: 10.3390/e28090952

Authors: Lili Zhang Zikun Han Qiubao Wang

Mode veering is a modal-interaction phenomenon found in vibration systems. For inertially coupled structures, the combined influence of coupling delay, nonlinear restoring force, and stochastic coupling perturbation remain insufficiently understood. This work analyzes an inertially coupled two-coordinate prototype in which a discrete delay, a delayed cubic stiffness, and positive multiplicative stochastic modulation all enter through the same relative-coordinate coupling channel. We formulate the delayed linear spectrum through a quasi-polynomial characteristic equation. We also characterize the veering by the two positive-frequency characteristic-root branches descending from the mechanical modes. Coupling delay shifts the veering center, alters the minimum frequency gap, and moves the tracked rightmost roots toward the stability boundary. An analytical imaginary-axis-crossing criterion is derived to determine the delay-induced stability boundary of the deterministic linearized system, and the resulting boundary is independently validated by direct multi-start characteristic-root searches and Chebyshev-collocation approximation of the DDE generator. A fixed-reference modal-coordinate representation identifies the off-diagonal modal terms associated with branch exchange while retaining the full delayed characteristic equation. A first-harmonic treatment of the delayed cubic term can yield an amplitude-dependent nonlinear veering backbone. For the stochastic problem, frozen lognormal coupling samples and a time-dependent Ornstein–Uhlenbeck-driven multiplier are constructed from the same unit-mean positive lognormal marginal law. The former is used to quantify realization-wise spectral broadening, whereas the latter retains temporal correlation and is used to evaluate finite-time branch residence and pathwise delayed-work statistics. The pathwise energy balance reveals that the delayed relative-coordinate work rate is sign-indefinite. This provides a common energy-transfer mechanism through which delay, nonlinearity, and stochastic modulation reshape mode veering in the inertially coupled system.

Entropy doi: 10.3390/e28090951

Authors: Jiang Zhang Bing Yuan Qian Zhang

The pursuit of self-evolving AI raises a critical question: when is autonomous self-improvement sustainable rather than degenerative? Drawing an analogy to von Neumann’s complexity threshold for self-reproducing automata, we argue that sustainable recursive self-improvement in large language models (LLMs) requires a functional analogue: introspection—the system’s capacity to simulate its own operations and target modifications. Grounded in Kleene’s Second Recursion Theorem, we construct such introspective self-improvement programs and prove their key properties: completeness of self-modification, necessity of the reflective architecture, undecidability of improvement in general, and equivalence with Schmidhuber’s Gödel machine under a rewrite-equivalence notion, which transfers the global optimality guarantee. An empirical review, organized around these functional criteria, suggests that current LLMs exhibit only quasi-introspection.The available evidence does not establish complete introspection in the formal sense developed here, while pointing to several candidate structural bottlenecks, including incomplete self-access, feedforward processing, and limited computational depth. We outline architectural paths toward the threshold and discuss the safety implications of crossing it.

Entropy doi: 10.3390/e28090950

Authors: Aditya Gupta Nicholas G. Polson

Robert Shiller proposed perpetual futures in 1993 to create derivative markets for assets that are illiquid or whose price cannot be observed directly. Cryptocurrency markets later built the instrument under a different funding rule. We give a single no-arbitrage result that nests both designs: the perpetual price is the present value of a benchmark flow discounted at the funding rate, so the funding rule fixes both the benchmark and the discount. A random time change represents the price as the expected spot at the first event of a clock whose intensity is the funding rate. This yields the main structural result, that stochastic volatility moves the basis only through the carry, so a volatility risk premium, and not volatility itself, can break the peg. We then read price discovery as nonlinear filtering in which the funding rule is a feedback observer whose gain is the funding intensity and the peg the fixed point of a stochastic approximation, and we give a segmented market equilibrium under which the pre-listing premium is structural rather than behavioral. In the June 2026 SpaceX market, the last pre-listing closes were $172.84 on Hyperliquid and $170.82 on Binance, compared with the listed equity’s $185 close on 18 June and the $135 bookbuilt offer. Simulation matches the pricing results to their closed forms. Generative Bayesian computation recovers the funding intensity sharply but not the softness of the anchor.

Entropy doi: 10.3390/e28090949

Authors: Lei Zhuang Yang Liu

The rapid expansion of the digital currency market and the growing role of stablecoins as potential intermediaries have brought its interconnectedness with traditional financial markets to the forefront of global financial research. Using daily data from 4 January 2021 to 30 September 2025, this study constructs a variable system with the price indices of USDT and USDC as core digital currency proxies, alongside traditional financial asset indices for stocks, bonds, and gold derived via the entropy weight method. We employ a comprehensive set of econometric techniques, including static correlation analysis, vector autoregression (VAR), impulse response functions, and extreme-event shock tests, to systematically investigate the interdependence structure, risk spillover dynamics, time-varying co-movements, and structural changes between the two markets during extreme risk episodes. The findings reveal an overall weak and asymmetric bidirectional spillover relationship between the cryptocurrency and traditional financial markets. Volatility in the digital currency market is found to be largely endogenous, with a limited capacity to transmit shocks externally. Conversely, traditional financial markets—particularly the equity market—exert a more pronounced influence on the digital currency market. Critically, under the impact of extreme risk events, the cross-market linkages exhibit structural breaks; the direction and intensity of correlation can strengthen significantly or even reverse, demonstrating a clear state-dependency. This research provides empirical evidence for understanding the functional role of digital assets within the macro-financial system, their risk transmission pathways, and their implications for systemic financial stability. The findings offer valuable theoretical and practical insights for financial regulators in designing robust cross-market risk prevention frameworks and for investors seeking to optimize asset allocation strategies.

Entropy doi: 10.3390/e28090948

Authors: Louis N. Irwin

At every level of resolution, over many orders of magnitude in time, size, and space, every aspect of the universe is constantly evolving under the pressure of the major forces of nature to resolve gradients of disparity in mass and energy. This imperative for change is channeled by two fundamental constraints: the bias of the Second Law of Thermodynamics (SLT) toward increasing entropy, and the mandate by the Principle of Least Action (PLA) that change must occur by the most direct and efficient path possible. While the SLT would seem to predict that the world would unwind rather than complicate itself, the opposite often occurs at the local level. While the evolutionary imperative drives the universe as a whole toward an ever higher level of entropy, it promotes increased local granularity and complexity to effect change in the net direction required by the SLT over the optimal path prescribed by the PLA. This provides a unifying perspective for all the complexity that astronomical and geophysical forces have created in the physical world, and that random variation and natural selection have induced in the living world―a consequence of nature’s imperative to dissipate energy as thoroughly and efficiently as possible.

Entropy doi: 10.3390/e28090947

Authors: Sihang Qin Yang Zhou Weiqi Dai Yiming Sun Weizhong Qiang

Cross-domain data flows are commonplace in regulated inter-organizational environments, where durable audit evidence must be retained without publicly exposing sensitive flow metadata. This paper presents a privacy-preserving on-chain attestation framework for recorded cross-domain data transfers in a permissioned setting. Its core data structure, termed GBFPlus, extends the Garbled Bloom Filter (GBF) with explicit occupancy indicators, constrained payloads that encode a consistency prefix and an adjacent-domain identifier, and distinct pairing-derived positions. Each domain administrator records observed inbound and outbound transfers in directional GBFPlus instances and periodically commits signed filter attestations to an append-only ledger. An authorized regulator can reconstruct candidate transfer edges from available bilateral attestations, while light clients verify ledger inclusion through Merkle proofs. A traceable anonymous attestation signature conceals the uploader’s cryptographic identity from ordinary ledger observers while retaining regulator-assisted accountability. The security analysis establishes integrity, conditional anonymity, traceability, and metadata-privacy properties for committed attestations under the stated trust assumptions, and the prototype evaluation reports the measured costs of GBFPlus and the signature operations.

Entropy doi: 10.3390/e28090946

Authors: Masaki Kobayashi

Multilayer perceptrons (MLPs) are considered as a singular model of learning machines. Singularities cause local minima and plateaus in the learning process. I/O-equivalence, where two different MLPs are regarded as the same multivariable function, is an important concept for understanding singularities in neural networks. In this paper, I/O-equivalence is extended to T-equivalence, which is a concept where two MLPs yield the same results through a transformation of input and output. We provide constructive families and procedures for obtaining T-equivalent networks of real-, complex-, and quaternion-valued neural networks. In particular, T-equivalence of quaternion-valued neural networks is much more complicated than that of the others.

Entropy doi: 10.3390/e28090945

Authors: Dazeng Yuan Xiheng Liu Bin Liu

Multi-server private information retrieval (PIR) based on function secret sharing (FSS) has emerged as a prominent paradigm for achieving sublinear communication. However, standard FSS constructions require full server participation, making them highly vulnerable to single-node fail-stop faults. Existing fault-tolerant schemes mitigate this but inevitably inflate the response overhead to scale with the database size N (e.g., O(N)). To overcome this limitation, we propose a fault-tolerant PIR (FT-PIR) protocol based on a newly designed (t,p)-threshold distributed point function (FT-DPF). By introducing a hierarchical recursive patching mechanism, our scheme transforms rigid all-party evaluations into flexible t-out-of-p reconstructions. This architecture completely decouples the response communication from N and ensures efficient client-side reconstruction via lightweight XOR aggregations. Formal analysis proves that our stateless protocol guarantees (t−1)-computational privacy under the semi-honest model. Theoretical analysis demonstrates that the proposed FT-PIR achieves a response complexity bounded by O(Fmaxlevel(t,p)). Comprehensive experimental evaluations confirm that our implementation significantly reduces practical communication and computation overheads, outperforming the state-of-the-art scheme.

Entropy doi: 10.3390/e28090944

Authors: Lei Lei Yu Shen Dawei Wang Feng Xi Yixin He Chaochao Wang Jiandong Liu

Accurate delineation of infarct core and ischemic penumbra in acute ischemic stroke primarily relies on computed tomography perfusion (CTP), where the arterial input function (AIF) is essential for reliable perfusion quantification. However, reliable and fast AIF selection remains challenging in clinical practice due to noise, vascular heterogeneity, and inter-patient variability in bolus dynamics. In this study, we propose TriAIF-RWKV, a three-stage framework for robust and automated AIF extraction. Specifically, ACSANet is first employed for spatial vascular localization using axial and channel-aware attention mechanisms, thereby narrowing the candidate arterial region and reducing the AIF search space. Then, a Dilated-RWKV network is introduced to model temporal intensity dynamics from a global sequence perspective, allowing robust identification of AIF-consistent patterns. Finally, a physiology-informed scoring strategy is used to select the optimal AIF by evaluating baseline stability, peak enhancement, and washout characteristics. Extensive experiments on CTP datasets were conducted from multiple perspectives, including AIF waveform fidelity, perfusion parameter estimation, and lesion-level analysis. The results demonstrate that the proposed method achieved high agreement with expert-selected AIFs, with a global waveform PCC of 0.973, peak correlation of 0.942, and TTP correlation of 0.973 with a mean error of 0.923 s. Furthermore, the proposed method provides more consistent downstream perfusion quantification, achieving higher consistency of CTP-derived parameters and improved lesion-to-normal tissue discrimination compared with existing approaches. These results highlight its potential for reliable clinical perfusion assessment.

Entropy doi: 10.3390/e28090943

Authors: Ryan A. Peterson Sarah M. Bird Logan M. Harris Patrick J. Breheny Joseph E. Cavanaugh

The concept of ranked sparsity, originally introduced in the context of penalized regression, arises in modeling applications when an expected disparity exists in the quality of information between different feature sets. Its presence can cause traditional and modern model selection methods to fail because such procedures commonly presume “covariate equipoise”—that each potential parameter is equally worthy of entering into the final model. However, this presumption does not always hold, especially in the presence of derived variables or with highly disparate feature sets (i.e., multi-modal data). For instance, when all possible interactions are considered as candidate predictors, the sheer number of them grossly inflates the number of false discoveries, resulting in unnecessarily complex and difficult-to-interpret models with many (truly spurious) interactions. In this work, we motivate a ranked sparsity extension to the Bayesian Information Criterion (RBIC) that requires a stronger level of evidence in order to allow certain variables (e.g., interactions vs main effects and genetic vs clinical covariates) into a model. We compare the performance of RBIC relative to competing methods for selecting polynomials and interactions in a simulation study and in two applications, showing that stepwise selection guided by RBIC produces better-predicting, more transparent models (with fewer false interactions) compared to existing alternatives.

Entropy doi: 10.3390/e28090942

Authors: Sadullah Çelik Muhammet Ali Köroğlu Cemile Zehra Köroğlu

The rule of law can be considered as a multidimensional institutional phenomenon, which emerges through interplay between legal, governance and administrative institutions. The paper offers an exploratory statistical modeling approach to find empirical patterns in national rule-of-law profiles according to the 2024 World Justice Project (WJP) Rule of Law Index. Eight dimensions of the index are considered to identify differences between countries and similarities of their multidimensional institutional performance. Principal Component Analysis reveals strong associations between eight dimensions, which are structured along the same performance institutional scale; the first principal component explains 85.7% of the overall variation and two principal components explain 92.5% of it. K-Means, hierarchical and DBSCAN clustering methods are then used to examine the empirical similarities between countries. While the six-cluster solution of K-Means offers distinct group descriptions, low bootstrap stability of this solution suggests that these groups cannot be regarded as fixed rule-of-law regimes. In addition, the Random Forest analysis reveals Regulatory Enforcement, Absence of Corruption, and Criminal Justice as the three dimensions, which contribute to the empirical differentiation of the described profiles the most. In general, the results imply that international variations in rule-of-law performance are viewed as heterogeneous locations in a multidimensional institution space, rather than as stable and distinct legal systems. The above-presented methodology allows for an exploratory approach to analyze international variations in rule-of-law performance that considers the limitations of cross-section data and instability of clusters.

Entropy doi: 10.3390/e28080941

Authors: Xiaoliang Feng Zhouliner Gao Teng Liu

Distributed maximum correntropy Kalman filters improve robustness to non-Gaussian noise, but existing variants generally introduce consensus through average or weighted fusion without explicitly separating the innovation residual from the state-disagreement residual in a dimensionally consistent objective. This paper proposes a Distributed Maximum Correntropy Kalman Filter with Innovation and Consensus Weighting Terms (DMCKF-IW-CWT). The innovation and consensus residuals are normalized separately and mapped by Gaussian kernels, after which the resulting information matrices are incorporated into a fixed-point local update. Posterior covariance intersection (CI) is then used to fuse neighboring estimates without requiring the unavailable cross-covariances. A sufficient contraction condition is given for the fixed-point iteration. In a five-node benchmark with 500 independent Monte Carlo runs and 1000 sampling steps, the proposed method obtains overall, transient, and steady-state MAEs of 0.172210, 0.188271, and 0.168195, respectively, corresponding to reductions of 0.254%, 0.526%, and 0.178% relative to DMCKF-W; the paired 95% confidence intervals of all three differences remain below zero. The consensus RMS is further reduced by 5.371%. Additional tests involving five noise families, packet loss and communication noise, a four-state nonlinear model, and systems with up to eight states and twenty nodes confirm the numerical convergence and extensibility of the framework.

Entropy doi: 10.3390/e28080939

Authors: Xingchi Yan Yan Yu Ning Li

Cross-condition bearing fault diagnosis requires models that remain reliable under load-induced distribution shifts while providing transparent and traceable reasoning. Conventional belief rule bases (BRBs) may repeatedly use correlated vibration evidence during inference, and their Cartesian-product rule construction can rapidly increase rule-base complexity. This study proposes a correlation-decoupled interval belief rule base (CD-IBRB) for cross-condition bearing fault diagnosis. Seven diagnostically relevant time-domain features are selected using XGBoost and transformed into a less-correlated feature space through a Kendall-rank-correlation-guided matrix estimated exclusively from the source training data. Attribute-wise referential points and intervals are then constructed from the transformed training attributes, allowing the rule base to grow additively rather than combinatorially. Initial belief distributions are obtained from interval-level class distributions. The projection covariance matrix adaptation evolution strategy (P-CMA-ES) jointly optimizes the belief degrees, rule reliabilities, and rule weights, while evidential reasoning aggregates the activated interval rules to produce the final diagnostic result. In the primary cross-load bearing experiment, CD-IBRB achieved an accuracy of 0.9702 and a macro-averaged F1 score of 0.9703. It outperformed the strongest BRB variant and data-driven baseline by 7.70 and 6.10 percentage points in accuracy, respectively. Ablation experiments confirmed that removing parameter optimization or attribute decoupling reduced accuracy to 0.9053 and 0.9303, respectively. Additional cross-load and noise-injection experiments further demonstrated the stability of CD-IBRB under load shifts and input disturbances. Across five public multiclass datasets, CD-IBRB achieved a mean accuracy of 0.9004 and consistently outperformed the compared BRB variants. These results demonstrate that CD-IBRB provides a compact, uncertainty-aware, and traceable framework for cross-condition bearing fault diagnosis.

Entropy doi: 10.3390/e28080940

Authors: Ibrahim Senturk Metin Bilge Tahsin Oner

This paper addresses the problem of formulating generalized, non-extensive information-theoretic measures on finite non-distributive algebraic structures equipped with Riečan states, with particular emphasis on product Sheffer stroke basic algebras. Our approach formalizes finite summations, admissible partitions, refinement relations, and Sheffer stroke joint refinement candidates by using the primitive Sheffer stroke operation, with partition and marginalization properties imposed under the stated product and admissibility assumptions. By leveraging the state-theoretic properties of Riečan states, we construct baseline Shannon and logical entropies alongside algorithmic procedures for their computational evaluation. As the main result, we introduce and analytically characterize a parametric Tsallis entropy functional over these basic algebras. We prove its fundamental properties, including bounding inequalities, state concavity, monotonicity under refinement, subadditivity (for α>1), conditional chain-type identities under the relevant joint refinement marginalization assumptions, and exact analytical convergence to the classical Shannon limit as the entropic index α→1. Furthermore, under a state-dependent statistical independence condition, we show that the joint Tsallis entropy satisfies a pseudo-additive relation. By defining the Tsallis mutual information and the associated pseudo-additive residual, we isolate the deviation of a joint Sheffer stroke refinement from the factorized model determined by its marginal Riečan-state distributions. This residual is intended as a state-dependent algebraic indicator of deviations from the factorized Tsallis pseudo-additive model; it is not claimed to be an operational contextuality witness, a contextuality inequality, an entanglement measure, or a physical implementation criterion.

Entropy doi: 10.3390/e28080938

Authors: Yiming An Yanshu Yu Weidong Li Orest Kochan

Binary code similarity detection (BCSD) ranks candidates but does not quantify the reliability of an already selected Top-1 match. We study this post-retrieval problem in a known-source, closed-set protocol: the target Top-1 is frozen before same-source cross-compilation views are queried, so auxiliary evidence audits cannot replace it. A frozen 34-variable map feeds a low-capacity logistic model with project-grouped cross-fitting, Platt calibration, and training-side threshold selection. On 413 families from 16 projects, cross-view evidence improved discrimination over target score/margin features. GCC-O0 was a dominant-anchor regime: Full showed no statistically resolved ROC-AUC gain over Primary-anchor, whereas Clang-O0 benefited from complementary non-primary evidence. On 240 project-identity-disjoint families from 55 projects, the design-locked structural branch accepted 75/240 GCC and 99/240 Clang candidates (31.3%/41.3% coverage) with no observed family-level errors. Correspondence mismatch reduced discrimination toward chance. Corrected TF-IDF remained supportive because correction followed label access. The contribution of this paper is a versioned candidate-preserving audit interface with explicit evidence and deployment boundaries, but not a universal retrieval improvement or distribution-free guarantee.

Entropy doi: 10.3390/e28080937

Authors: Yi Liang Xiaoduo Wang Han Liu Hao Wang

Geopolitical conflicts, resource-protection policies, and unexpected disruptions have heightened supply-security concerns across the global nickel industry chain. This study constructs a multilayer trade network based on complex network theory to characterize structural evolution across the upstream, midstream, and downstream segments and applies a cascading-failure model to simulate the propagation of supply risks. There are four main findings: (1) The global nickel trade network exhibits pronounced layer heterogeneity, with the midstream layer acting as the principal amplifier of cascading failure risks. (2) Nodes with high centrality and broad cross-layer participation largely coincide with the countries that generate the largest systemic risks. (3) A small group of countries controls most trade flows and dominates risk transmission. (4) The center of systemic risk is shifting from traditional industrial and trading economies toward resource suppliers and countries that integrate resource extraction with processing. These findings support a risk-governance strategy based on diversified supply sources, dynamic monitoring of critical nodes, improved resilience in midstream smelting and refining, strategic resource stockpiling, and the circular utilization of nickel resources.

Entropy doi: 10.3390/e28080936

Authors: Xinyi Xie Jia-Ning Zhu Bo Wan

Biorthogonal quantum geometry is often read through scalar tensor components. In non-Hermitian bands, however, the biorthogonal contraction can lose the ordering of the left-right interband matrix elements from which a scalar component is formed. We study this information loss for spectrally separated, diagonalizable two-band Bloch Hamiltonians. For a specified control parameter, the Hamiltonian variation defines a local response vertex. In the instantaneous biorthogonal eigenbasis, the interband part of this vertex is completely specified by two ordered matrix elements, whereas the corresponding equal-parameter scalar QGT component retains only their product. This separation leads to a local classification of interband vertices into no-interband, Hermitian-locked, generic complex-transverse, and complex-null cases. On a complex-null branch, the scalar component can vanish even though one ordered interband matrix element remains nonzero. We identify this as a local chiral-vertex mechanism in a vertex-resolved geometric response kernel, distinct from generic non-Hermiticity or exceptional-point proximity. Nonreciprocal SSH, a two-dimensional complex-spin–orbit lattice, and a kz-only chiral ladder stack realize the same mechanism in one, two, and three dimensions, while diagonal and gain–loss-like vertices provide nonselective comparisons.

Entropy doi: 10.3390/e28080935

Authors: Na Zhao Chao Dai Guolin Yang Ting Luo Nifei Xiong Jian Wang

Accurately identifying influential nodes is essential for analyzing network structures and optimizing information propagation. Existing methods predominantly rely on single indicators such as degree, H-index, or k-shell, inherently limiting their ability to capture a node’s true influence. Recent hybrid centrality approaches attempt to address this by combining multiple local and global attributes; however, they typically integrate features through simple weighting or superposition, failing to characterize the intrinsic synergy among structural properties. Furthermore, they often quantify neighbor contributions too coarsely, overlook the regulatory role of edge strength, and some suffer from high computational complexity, limiting scalability. To overcome these deficiencies, we propose WKDH, a novel influential node identification method based on multi-attribute neighbor contributions. WKDH fuses local structural attributes (degree and H-index) with global structural attributes (k-shell) via a multiplicative weighted synergy model, simultaneously capturing local connection “quantity,” local connection “quality,” and global core-layer position. By transforming neighbors’ comprehensive characteristics into regulated contribution degrees, WKDH mitigates excessive self-attribute interference and accurately reflects the actual propagation potential of edges. Notably, the method achieves linear computational complexity of O(m). Experimental results on nine real-world and six artificial networks demonstrate that WKDH outperforms nine established indicators in terms of node influence ranking, identification of high-influence nodes, and measuring propagation capability. Moreover, WKDH exhibits strong universality across diverse network structures, as it operates without parameter tuning.

Entropy doi: 10.3390/e28080934

Authors: Chenglong Zhu Xiaolong Chen Xinming Guo Wei Sun

Maglev trains are susceptible to various types of operational challenges, including track irregularities, load variations, and actuator faults. It is evident that these factors can compromise suspension performance and even pose a serious risk to operational safety. This paper proposes a prescribed performance event-triggered fault-tolerant control method for the electromagnetic suspension system of a maglev train subject to multiple constraints. A projection-based adaptive extended state observer is designed to estimate the unknown gain caused by actuator faults and load variations, as well as the external disturbance. In light of the disparity in upper and lower safety margins inherent to the suspension gap error, arising from track irregularities, an asymmetric prescribed performance function and an error transformation are devised to ensure that the gap tracking error perpetually complies with the asymmetric prescribed performance constraint. In addressing the issue of rapid variations in the suspension gap, the vertical velocity is also constrained through the implementation of prescribed performance, resulting in a joint constraint framework that encompasses both the gap tracking error and the vertical motion. A dynamic event-triggered mechanism has been incorporated into the backstepping design with a view to reducing unnecessary control updates under limited communication resources, while Zeno behavior has been excluded from the closed-loop system. Within this framework, a dynamic gain adjustment mechanism with an explicitly bounded rate of variation is further developed to achieve smoother gain adaptation. The uniform ultimate boundedness of all closed-loop signals is demonstrated through Lyapunov stability analysis under the prescribed multiple constraints. The efficacy of the proposed method is demonstrated through comparative simulation results.

Entropy doi: 10.3390/e28080933

Authors: Nida Oruç Ünal Muzaffer Göztaş Doğan Yıldız

Feature selection is a critical step in regression problems where a large number of continuous explanatory variables explain the same target through different dependency structures. Classical filters may remain sensitive to a single form of dependence, a single scale, or a specific discretization scheme; generalized entropy measures, on the other hand, typically require the parameters to be fixed at a single point. This study proposes a framework that evaluates the Sharma–Mittal entropy volumetrically across a two-dimensional parameter region rather than for a single parameter pair. For the continuous target and explanatory variables, the marginal, joint, and conditional densities are obtained using a Gaussian kernel density estimation; the conditional entropy and information gain surfaces are integrated across the region Ω = [0.05, 0.95]2 in the α-β plane to define three indices: PICSME, which measures the conditional uncertainty volume; PIGSME, which measures the gain volume; and NIGSME, which is the ratio of this gain to the total entropy volume of the target. The method is supported by bandwidth consistency and the renormalization of conditional densities; thus, the issue of negative gain that can occur in the continuous variables is resolved, yielding positive and interpretable scores across all six datasets. It is formally demonstrated that the fact that the three indices produce the same ranking is not an empirical observation but rather the result of a monotonicity relationship valid under a fixed target entropy volume. The method is compared with Pearson and Spearman correlations, the Shannon information gain, mutual information, and random forest variable importance across six regression datasets (Airfoil Self-Noise, AirQualityUCI, BodyFat, Meteorology, Concrete, and WineQualityWhite) that differ in their sample size, dimensions, and application domain. The evaluation is not limited to ranking consistency; the out-of-sample prediction performance is measured using least-squares models on the top-k subsets, with rankings calculated from the training partition. The findings show that NIGSME exhibits a performance comparable to that of built-in filters, outperforms them on the Concrete and Meteorology datasets, and never ranks as the weakest method on any dataset. The results demonstrate that volumetric entropy metrics defined across the entire parameter space provide a feature-ranking tool that is independent of parameter selection for continuous variables.

Entropy doi: 10.3390/e28080932

Authors: Claudio Cremaschini Ramesh Radhakrishnan Gerald Cleaver

A novel theoretical study of Boltzmann-Shannon entropy arising in information-statistic theory applied to black-hole physics is proposed. The invariant setting implemented is represented by the manifestly-covariant quantum-gravity theory expressed in canonical Hamiltonian form. In such a framework the appropriate statistical interpretation relies on the configuration-space quantum expectation value of physical observables over the 4− scalar quantum-gravity probability density function (PDF). A representation for the black-hole Boltzmann-Shannon entropy is obtained for a Gaussian PDF profile and by establishing simultaneously a relationship between the black-hole invariant energy-content and the mean value of the quantum-gravity nonlinear Bohm potential. This yields a non-trivial functional dependence of the Boltzmann-Shannon entropy on the black-hole surface area, to be interpreted as a quantum statistical entropy counting black-hole bulk quantum-gravity states. The mathematical setting is shown to preserve manifest covariance and be self-contained within quantum-gravity realm. Comparisons with literature treatments dealing with thermodynamic or kinetic-statistical entropies that lead to the Bekenstein-Hawking black-hole surface entropy linear relation or its proposed quantum modifications are discussed.

Entropy doi: 10.3390/e28080931

Authors: Wei-Xin Liu Zhan-Peng Lu Tao Wang

We theoretically investigate geometric Stückelberg interferometry in a doubly driven optical lattice clock (OLC). By tuning the relative phase between the two driving fields, we control the relative sign of the effective coupling strengths at the avoided crossings. Within the adiabatic-impulse model, we analyze the time evolution of the two-level system, where nonadiabatic transitions occur only near the crossing points and adiabatic evolution takes place between them. We show that, besides the usual dynamical phase and the Stokes phase, a gauge-invariant noncyclic geometric phase contributes to the final transition probability. This geometric contribution yields a stable π-phase shift in the Stückelberg interference fringes. Moreover, we demonstrate that, under realistic experimental conditions, this geometric Stückelberg interferometer remains insensitive to inhomogeneities in atom-light coupling arising from the finite temperature of the atomic ensemble. Our results provide a general framework for engineering and detecting geometric phases on the OLC platform.

Entropy doi: 10.3390/e28080930

Authors: Shuangshuang Yang Peilun Liu Wenlong Zhu

Aspect-based Sentiment Analysis (ABSA) is a fine-grained sentiment classification task that aims to predict the sentiment polarity associated with aspect terms in sentences. Traditional methods based on syntactic and semantic dependency trees are insufficient for capturing contextual sentence features. To address this, we propose a Knowledge-Enhanced and Multidimensional Feature Fusion (KEMFF) model for ABSA, which captures sentiment feature representations across multiple dimensions, including syntax, semantics, and knowledge. First, the pre-trained model RoBERTa is used to obtain embeddings of sentences and aspect terms. Then, a syntactic dependency parser and a graph convolutional network are utilized to learn syntactic features. Meanwhile, an Abstract Meaning Representation (AMR)-based parser is employed to construct semantic relations, and axial attention is used to aggregate incoming and outgoing semantic dependencies. Furthermore, external knowledge is embedded, and an attention mechanism is employed to obtain aspect-specific knowledge representations, thereby complementing syntactic and semantic representations with external lexical knowledge. Finally, multidimensional features are fused and passed to a softmax classifier for predicting sentiment polarities. Unlike previous models that mainly focus on either syntax–semantic fusion or knowledge-enhanced graph propagation, KEMFF explicitly models syntax, semantics, and lexical knowledge in three parallel branches and aligns them into a unified aspect-level representation. Experiments on Laptop14, Restaurant14, and Twitter datasets show that KEMFF achieves the best performance among the compared baselines on Laptop14 and Restaurant14, and it obtains competitive results on Twitter.

Entropy doi: 10.3390/e28080929

Authors: Juan Yang Jingjing Lu Jianbo Ji Tao Wang

BP decoding of BATS codes may stop when no residual batch satisfies the full-row-rank condition. To resume BP decoding, this paper proposes an Important-Packet-Guided Left-Nullspace Rank-Completion (LNRC) method. LNRC first identifies the unrecovered source packet that connects to the largest number of undecoded batches, denotes it as the Important Packet, and uses it as a guidance packet to locate the undecoded batches containing it as repair candidates. For a selected batch with rank deficit one, the destination computes a nonzero left-null vector and selects a local repair coordinate that provides the missing independent direction. The destination sends the corresponding global source-packet index and finite-field coefficient through a reliable reverse feedback-control link, and the source returns the scaled repair packet through a reliable forward repair-data link. The associated completion column increases the target residual transfer-matrix rank by one and makes the batch BP-decodable. Thus, LNRC exploits the column-space structure of the target undecoded batch to select the repair coordinate, rather than selecting the Important Packet solely by the number of connected undecoded batches. Under equal encoding redundancy, simulations show that LNRC achieves a lower packet error rate (PER) compared with conventional Important Packet feedback, with average relative PER reductions of approximately 0.40–12.17% across the evaluated settings.

Entropy doi: 10.3390/e28080928

Authors: Matteo Gori Roberto Franzosi Giulio Pettini Marco Pettini

We review a theoretical and experimental programme with the aim of understanding two intimately related fundamental phenomena in biophysics: (i) the classical analogue of Fröhlich phonon condensation in macromolecules driven out of thermal equilibrium and (ii) the consequent activation of long-range resonant electrodynamic intermolecular forces. Both phenomena are underpinned by explicit Hamiltonian models. The first is derived by applying the time-dependent variational principle (TDVP) to the quantum Wu–Austin model, producing a fully classical Hamiltonian in action-angle variables whose nonlinear rate equations exhibit a nonequilibrium phase transition: the channelling of supplied energy into the lowest-frequency collective mode. The second is grounded in a classical electrodynamic Hamiltonian for two coupled oscillating dipoles whose normal-mode structure predicts long-range (∼1/r3) resonant interactions, absent at thermal equilibrium but activated by out-of-equilibrium collective oscillations. We also discuss a complementary Hamiltonian approach that connects Fröhlich’s rate equations directly to Hamilton’s equations of motion, clarifying the role of bath-mediated nonlinear coupling and the conditions for strong condensation at room temperature. In addition, the TDVP is applied to a Davydov–Holstein–Fröhlich Hamiltonian describing electron–phonon motion along the backbone of a specific DNA sequence and its cognate restriction enzyme, EcoRI: the time-domain Fourier cross-spectrum of the resulting electron currents exhibits a sharp co-resonance peak for the canonical recognition sequence that disappears upon randomisation, providing a sequence-specific electrodynamic signature of DNA–protein recognition. Experimental evidence from THz near-field spectroscopy, fluorescence correlation spectroscopy, and direct observation of protein clustering is reviewed in relation to these theoretical predictions. The results establish a coherent physical picture suggesting that metabolic energy supply can play a role in driving macromolecules into coherently oscillating states that activate selective, distance-reaching electrodynamic forces capable of contributing to the organisation of biochemical reactions in living matter.

Entropy doi: 10.3390/e28080927

Authors: Pengcheng Cai Canjv Lu Ying Huang Yi Xie

Accurately identifying critical nodes in complex networks and applying targeted protection strategies significantly enhances network security. Traditional importance metrics rely on a single topological feature and cannot fully capture node influence. Existing multi-attribute fusion methods integrate multiple structural sources but typically use fixed weights or predefined rules, failing to adaptively adjust attribute contributions based on local and global network characteristics, which limits their generalization across diverse networks. To address this, we propose the CRITIC-based Objective Weighting and Multi-Metric Fusion Method (COWMF). COWMF first builds a Graph Attention Network with Virtual Global–Local Integration (GAT-VGL), taking four low-complexity topological metrics, degree centrality (DC), H-index, degree and neighborhood information centrality (DNC), and k-shell, as input. Through a learnable attention mechanism, GAT-VGL adaptively aggregates multi-hop neighborhood information and explicitly incorporates global structural information via a virtual node to achieve whole-graph topological awareness, generating a global influence score with good discriminative power and high computational efficiency. This score is then integrated with DC and DNC into an improved CRITIC-based objective weighting fusion scheme, enabling adaptive synergy among local connectivity, semi-local radiation, and global structure. Experiments on six real-world networks of varying types and scales show that COWMF demonstrates relatively stable and competitive performance in both simulated attack and susceptible-infected-recovered (SIR) spreading simulations, two complementary experiments, demonstrating satisfactory disruptive capability and propagation influence. Its importance scores exhibit high monotonicity across all networks, with good discriminative power.

Entropy doi: 10.3390/e28080926

Authors: Wenjun Song Hao Shen Xuncai Zhang Chengye Zou

Traditional image encryption methods based on static permutation and diffusion are vulnerable to structural cryptanalysis and often exhibit limited robustness under imperfect communication conditions. To address these issues, this paper proposes a robust topology-evolving image encryption algorithm driven by complex hyperchaotic dynamics for secure visual data transmission. First, a two-dimensional Rosenbrock–Schwefel hyperchaotic map is constructed to generate high-quality pseudorandom sequences for both permutation and diffusion. Based on this map, a bidirectional oscillatory spatial permutation mechanism governed by a dynamic linked-list topology is developed. Unlike fixed-path permutation strategies, the proposed topology continuously evolves with the system state during image traversal, thereby increasing nonlinear path complexity and improving resistance to structural attacks. Furthermore, a plaintext-dependent adaptive diffusion mechanism is designed to enhance sensitivity to plaintext variations and produce a strong global avalanche effect. Experimental results demonstrate that the proposed algorithm achieves favorable encryption performance, with an information entropy of up to 7.9994, a Number of Pixels Change Rate (NPCR) of 99.6076%, and a Unified Average Changing Intensity (UACI) of 33.4683%. In addition, the algorithm maintains good recovery performance under cropping attacks and noise interference, indicating its robustness and applicability for secure image transmission in complex communication environments.

Entropy doi: 10.3390/e28080925

Authors: Abdelmoula El-Yazizi Yaroslav Koshka

A local-valley (LV)-centered approach to assessing the quality of sampling from Restricted Boltzmann Machines (RBMs) was applied to the latest generation of the D-Wave quantum annealer. D-Wave and Gibbs samples from a classically trained RBM were obtained at conditions relevant to contrastive-divergence-based RBM learning. The samples were compared for the number of LVs to which they belonged and the energy of the corresponding local minima. No significant (desirable) increase in the number of the LVs has been achieved by decreasing the D-Wave annealing time. At any training epoch, the states sampled by the D-Wave belonged to a somewhat higher number of LVs than in the Gibbs sampling. However, many of those LVs found by the two techniques differed. For high-probability sampled states, the two techniques were (unfavorably) less complementary and more overlapping. Nevertheless, many potentially “important” local minima, i.e., those having intermediate, even if not high, probability values, were found by only one of the two sampling techniques while missed by the other. The two techniques overlapped less at later than earlier training epochs, which is precisely the stage of the training when modest improvements to the sampling quality could make meaningful differences for the RBM trainability. The results of this work may explain the failure of previous investigations to achieve substantial (or any) improvement when using D-Wave-based sampling. However, the results reveal some potential for improvement, e.g., using a combined classical–quantum approach.

Entropy doi: 10.3390/e28080924

Authors: Juan Carlos Boschero Rares Adrian Oancea Luca Mazzarella Hugo Doeleman Simon Cramer

Federated learning enables multiple organizations to collaboratively analyze data while retaining local control over their datasets, making it attractive for applications such as healthcare. Distributed quantum computing provides a natural framework for such workflows, allowing geographically separated quantum processors to execute joint computations using shared entanglement while preserving data privacy. In this work, we investigate the feasibility of satellite-enabled distributed quantum computing for federated quantum learning. As a representative application, we consider a distributed distance-based quantum classifier in which multiple parties contribute local data through quantum operations. To support this application, we develop a hybrid space-ground quantum network architecture in which satellites distribute entanglement between distant ground stations. The communication layer is combined with a noise-aware neutral-atom processor model, enabling a system-level analysis that captures both network and hardware constraints. Simulation results across varying network sizes, feature dimensions, and coherence regimes show that classifier performance is jointly determined by communication resources, processor noise, and data representation. In low-coherence regimes, decoherence destroys the classifier’s discriminative signal, whereas high-coherence regimes reveal limitations arising from feature-space conditioning and feature redundancy. These results demonstrate that satellite quantum networks could support distributed quantum learning over long distances, while highlighting the importance of coherence time, entanglement-distribution performance, and learning-aware data encoding for future large-scale deployments.

Entropy doi: 10.3390/e28080923

Authors: Rongqian Wang Le Wang Zelin Kong Xiaoping Ma Yuxin Xu Ziming Wang Jia Tan Xiang Hao Jincheng Lu

We study the thermoelectric performance and stationary quantum correlations of a coherent double quantum dot heat engine driven solely by two conventional electronic reservoirs. The role of coherence is isolated by comparing the fully coherent dynamics with those under an energy-conserving pure dephasing channel that does not alter the system energy. Reducing the dephasing strength enhances the particle current, heat current, output power, and thermodynamic efficiency over a broad voltage range. After optimizing the electrochemical load and the dot energy levels, we find that coherence primarily amplifies the attainable power and efficiency without significantly relocating the optimal operating region. Although appreciable interdot coherence already exists at moderate Coulomb interaction, stationary entanglement emerges only when Coulomb blockade sufficiently suppresses the mixed-state contribution from the empty and doubly occupied states. We further construct a transport-based lower bound on the concurrence, providing an experimentally accessible entanglement witness that avoids full state tomography. These findings establish a clear hierarchy among energy filtering, quantum coherence, and Coulomb blockade in a minimal quantum thermoelectric device.

Entropy doi: 10.3390/e28080922

Authors: Shansong Song Wei Han Bing Wan Xiangyi Liu Xichao Su Chao Li Yunyang Cao

To address ambiguous maneuver boundaries, background interference, and uneven multi-sensor quality in long-duration flight parameter recordings, this paper proposes an adaptive flight maneuver boundary localization algorithm that integrates spectral entropy-weighted multi-channel spectrogram fusion with attitude-constrained structural correction. Multi-channel Short-Time Fourier Transform (STFT) spectrograms are first constructed from flight parameter time series. Spectral entropy (SE) is introduced to quantify the uncertainty of each channel’s time–frequency energy distribution and is combined with the maneuver activation ratio (MAR) and the linear contrast ratio (LCR) to form objective credibility weights, thereby suppressing channels dominated by aerodynamic turbulence and high frequency structural vibration. Normal overload soft gating and logarithmic noise floor subtraction are then applied to obtain an enhanced fused spectrogram, from which candidate intervals are extracted by low band energy thresholding. Finally, roll and pitch angle steady-state priors refine the event structure through local boundary refinement, cross-segment expansion/chain merging, and semantic post-processing, recovering continuous maneuvers fragmented by instantaneous energy valleys. On the held-out test sorties (SE_018–SE_020; 61 annotated intervals), the proposed algorithm achieves Precision, Recall, and F1-scores of 0.967. On the full primary corpus of 20 sorties (461 intervals), used for ablation and sensitivity analyses, the corresponding figures are Precision 0.934, Recall 0.959, and F1 0.946, with start and end boundary mean absolute errors of 1.484 s and 1.471 s. Under the same IoU protocol, consistent superiority is observed against learning-based baselines, and an independent external set of 10 sorties yields F1 = 0.938. The results indicate that entropy-constrained multi-sensor time–frequency fusion mainly improves maneuver/background separability, whereas attitude-constrained structural correction restores the integrity of long continuous maneuvers.

Entropy doi: 10.3390/e28080921

Authors: Rawan Abdullah Andrei Halanay Lara Abou Orm

We study an optimal control framework for chemotherapy administration in patients with chronic lymphocytic leukemia (CLL) while accounting for immune regulation and treatment-induced allergic reactions. The analysis is based on a previously developed nonlinear delay differential equation model describing the interactions between leukemic cells, immune populations, antigen-presenting cells, and cytokine dynamics, with three distinct biological delays. The chemotherapy infusion rate is introduced as a time-dependent control variable and optimized to reduce leukemic burden, shift the helper T-cell balance toward a Th1-dominant configuration associated with lower hypersensitivity risk, and preserve immune competence. Existence of an optimal control is established for arbitrary delays and horizon, without the commensurability hypothesis required by reductions in delay systems to higher-dimensional delay-free ones; the argument uses only that the control enters the dynamics affinely and the running cost concavely. Necessary optimality conditions are derived via Pontryagin’s Maximum Principle for systems with delays, and the resulting eleven-dimensional adjoint system, which carries advanced arguments generated by the three delays, is written out explicitly. A contraction estimate for the associated sweep operator yields both uniqueness of the optimal control on a short horizon and geometric convergence of the numerical scheme. The optimality system is solved by a forward–backward sweep adapted to the delayed setting, with documented convergence and grid independence and sensitivity analysis over kinetic parameters, delays, initial conditions and objective weights. The optimized schedule is compared not only with the untreated case and a low constant dose, but also with a constant infusion delivering the same cumulative exposure, so that the reported benefit is attributable to the temporal distribution of the dose rather than to its total amount. At equal exposure, the optimal schedule reaches each therapeutic milestone earlier—Th1 dominance 0.9 days sooner and a 90% leukemic reduction 1.6 days sooner—and attains a terminal leukemic burden lower by a factor of 2.25; a constant infusion of the same total dose reaches a comparable configuration later. The benefit of adaptive scheduling in this model is therefore principally one of rate of response at fixed drug exposure. We emphasize that the absolute Th2 population is not reduced by treatment; the reduction in hypersensitivity risk arises from the resulting Th1-dominant relative balance rather than from direct Th2 suppression. To characterize the therapeutic outcome in information-theoretic terms, we describe the two competing goals as distributional balances: an allergy axis, given by the Th1/Th2/Treg distribution, and a leukemia axis, given by the immune/leukemic distribution, each measured by its Shannon entropy and its Kullback–Leibler divergence to a healthy reference profile. These quantities are used in two roles. As diagnostics, they are evaluated along the computed trajectories, and the ordering of dosing strategies is shown to be robust across twenty alternative reference profiles. As an objective, the combined divergence is then taken as the running cost of a second optimal control problem; because it depends on the leukemic population only through a normalized fraction, it prescribes a markedly gentler schedule that administers 37% of the drug and still achieves a 93% leukemic reduction, against 98% for the population-based formulation. These results suggest that adaptive, immune-aware chemotherapy scheduling may accelerate disease control at fixed drug exposure, and that information-theoretic objectives offer a scale-free alternative formulation of the therapeutic goal.

Entropy doi: 10.3390/e28080919

Authors: Zheng Yang Guohao Li Yali Xue

Multi-agent reinforcement learning has emerged as a promising paradigm for cooperative unmanned aerial vehicle (UAV) swarm coordination. However, existing communication-aware MARL methods primarily focus on communication topology, message routing, and message aggregation, while the information content of the exchanged messages is often only implicitly controlled. In realistic UAV networks, inter-agent communication is constrained by limited bandwidth, communication range, energy consumption, and packet loss. It is therefore desirable for each UAV to transmit compact and task-relevant information rather than dense and redundant latent features. In this paper, we propose IB-CEMARL, an information-bottleneck-guided, communication-efficient multi-agent reinforcement learning framework for UAV swarms. We formulate inter-UAV communication as a minimal sufficient message-learning problem in which each UAV encodes its local observation into a stochastic bottleneck message before exchanging information with its neighbors. Cauchy–Schwarz divergence-based quadratic mutual information is adopted as a unified dependence measure to jointly regularize message compression, preserve decision-relevant information, and reduce statistical redundancy among neighboring UAV messages. Extensive experiments demonstrate that IB-CEMARL achieves superior cooperative performance, reduced message redundancy, and stronger robustness compared with representative communication-aware MARL baselines. In particular, IB-CEMARL improves the average return by 4.9% and reduces inter-message dependence by 29.0% compared with the KL-IB-MARL baseline while maintaining efficient communication under constrained bandwidth settings.

Entropy doi: 10.3390/e28080920

Authors: Xiaojie Lu Hui Lou Xiaoyang Jin Bianmei Zhang

To evaluate whether nonlinear descriptors of electroencephalogram (EEG) signals support interpretable XGBoost classification and to determine how analysis window duration affects performance. A secondary analysis of the public Bonn EEG dataset was performed. Nine nonlinear features were extracted from non-overlapping 1, 5, 10, and 20 s windows after an original-recording-level train/validation/test split, and a multiclass XGBoost model was interpreted with class-specific SHAP values. The model achieved 93.3% overall accuracy; the class-specific AUC values were 0.978 for Z/O, 0.978 for N/F, and 0.984 for S. Across the four fixed-split duration conditions, the 1 s condition had the lowest descriptive performance, whereas the 5, 10, and 20 s conditions were broadly comparable; no uniquely optimal duration was established. The nonlinear-feature/XGBoost framework provides interpretable benchmark segment classification evidence. Because EEG is modeled as a stochastic process and the dataset is small and heterogeneous, the SHAP attributions do not establish physiological causality or clinical diagnostic validity.

Entropy doi: 10.3390/e28080918

Authors: Rashid Hussain Zahid Hussain Mehboob Ali Małgorzata Przybyła-Kasperek

Picture Fuzzy Sets (PiFSs) have appeared as an effective tool to tackle ambiguity in decision-making and offer greater flexibility than traditional extensions of Fuzzy Sets (FSs). Under the framework of evidence theory (ET), the concepts of belief and plausibility significantly boost the representative capacity of PiFSs, which enables the management of uncertain and ambiguous data. We constructed both distance and similarity measures specifically for Belief and Plausible Picture Fuzzy Sets (BP-PiFSs). The constructed measures detect the differences and connections between BP-PiFSs and addressed the key shortcomings in current methodologies. They are mathematically validated and applied to real-world scenarios, such as fault detection in complex systems and antenna design optimization, where managing uncertainty is critical. A modified decision-making method, Belief and Plausible SMART (BP-SMART), extends the classical SMART approach to more effectively handle multi-criteria decision-making (MCDM) in uncertain environments. Numerical evaluations across pattern recognition, clustering, fault detection, and MCDM demonstrates the effectiveness and robustness of the suggested framework, contributing significantly to both the theoretical and practical development of fuzzy set theory.

Entropy doi: 10.3390/e28080917

Authors: Bing Liu Xun Su Binghong Yan Anqi Liu

Post-quantum cryptography has gained urgent attention as quantum computing poses fundamental threats to traditional public-key cryptosystems. Code-based cryptography stands out as a robust post-quantum candidate, but most existing schemes are built on Hamming distance, whereas Lee distance provides a more natural error model for specific communication channels like phase-modulation channels. This paper presents the Lee–Stern zero-knowledge identification protocol, which extends the classic Stern protocol from the binary Hamming metric to the Lee metric over arbitrary prime fields. We adopt the state-of-the-art LMMT-ISD attack framework to conduct rigorous security re-evaluation and derive necessary parameter bounds for standard post-quantum security levels. Extensive experiments analyze how code length and prime modulus affect the protocol’s overheads, showing that the proposed scheme achieves equivalent security with notably shorter code length and smaller public key size than the original binary Stern protocol.

Entropy doi: 10.3390/e28080916

Authors: Ruizhe Shen Zichang Hao Ching Hua Lee

In this work, we benchmark two prominent quantum algorithms: Quantum Imaginary-Time Evolution (QITE) and the Quantum Approximate Optimization Algorithm (QAOA) for obtaining the ground state of Ising-type Hamiltonians. Specifically, we apply them to the Markowitz portfolio optimization problem in quantitative finance, on both digital quantum computers and local quantum simulators with controllable two-qubit errors (noise). In noiseless settings, we find that QAOA achieves excellent convergence to the optimal results. Under noisy conditions, the QITE method exhibits greater robustness and stability, though it incurs substantially more classical numerical cost. In contrast, we demonstrate that QAOA offers better scalability and can still yield robust results if the noise can be effectively mitigated. Our findings provide valuable insights into the trade-offs between scalability and noise tolerance and demonstrate the practical potential of quantum algorithms for solving real-world optimization problems on near-term quantum devices.

Entropy doi: 10.3390/e28080915

Authors: M. Rodrigo Pinheiro Mario J. Pinheiro

We develop a tensor-based dynamical framework for monetary flows in multi-sector, multi-agent economies and quantify the information destroyed when the monetary state is reduced to a scalar aggregate. The state is a third-order tensor encoding capital flows across sectors, agent classes, and time; deviations from equilibrium obey a tensor-indexed Langevin (multivariate Ornstein–Uhlenbeck) equation with a coupling operator and channel-specific friction rates. Using standard Lyapunov theory, we assemble a stability and convergence framework for the induced vectorized system, with a bound stated so as to remain valid for the non-normal system matrices generated by asymmetric economic coupling, and characterize the stochastically forced case in the mean-square sense. Shannon entropy, Kullback–Leibler divergence, and sector–agent mutual information measure the structural information discarded by scalar aggregation. We then study a stylized, heuristically calibrated 3×3 economy subject to a shock inspired by the 2007–2009 crisis; we emphasize at the outset that the figures reported below are properties of that calibration and are not empirical estimates. In this scenario Finance absorbs an 18.9% peak capital loss while Manufacturing and Services suffer 5.8% and 3.9% secondary drops, against an aggregate contraction of only 8.6%; the Kullback–Leibler divergence of the sector–agent flow distribution recovers systematically later than the aggregate signal, a lag that is positive in 96.6% of a 1000-draw Monte Carlo ensemble, although its magnitude is calibration-dependent. Under a symmetric exit rule, a deficit-targeted stimulus restores equilibrium substantially faster than a share-weighted uniform stimulus in 100% of the ensemble while spending strictly less—its realized expenditure saturates below the uniform budget because it self-terminates as deficits close—and attains integrated disequilibrium within 18% of the exact linear-quadratic optimum at equal control effort while requiring no knowledge of the system matrix. The ordinal conclusions—aggregation masks the epicenter, structure lags the aggregate, and deficit targeting dominates uniformity—are robust across a wide neighborhood of the calibration, and identify the disaggregated state as the object that stabilization policy needs and that scalar aggregation destroys.

Entropy doi: 10.3390/e28080914

Authors: Mohammad Abedi

The standard models of stock-price dynamics and option valuation rest on stochastic processes postulated at the outset; here, we lay down an entropic-inference framework that derives these processes rather than assuming them, by making explicit the information each one encodes. A symmetry comes first: markets reward returns rather than price levels, which selects the logarithm of price as the dynamical variable. The price then evolves through two channels, a continuous one carrying the constraints of continuity and directionality, and a jump channel carrying the arrival rate and the first two moments of the jump size. Because these constraints act on disjoint parts of the microstate, the channels factorize as a theorem, and the dynamics is the Merton jump-diffusion, with Geometric Brownian Motion as its no-jump limit; the log-price density obeys a Kolmogorov–Feller equation, of which the Fokker–Planck equation is the no-jump limit. The same principle, now imposing no-arbitrage through the mean log-return, selects the Esscher transform from among the many martingale measures an incomplete market admits, here derived rather than borrowed; the premium then satisfies Merton’s partial integro-differential equation, and the risk-neutral mixture of lognormals generates the implied-volatility smile, the Black–Scholes results returning when jumps vanish. What changes from one model to the next is never the inference but the information supplied to it.

Entropy doi: 10.3390/e28080913

Authors: Guoxu Liu Dong Mu Tianyu Li Mingqian Sun Liu Chen

Counts of trade links are often used as evidence of diversification, yet they say little about how value is distributed across those links. Using bilateral flows for selected lithium, cobalt and nickel products among 72 economies from 2010 to 2024, we built directed, value-weighted networks and examined them with multiscale entropy measures, lagged formation models and disruption tests. Here, entropy is used in the information-theoretic sense to measure how evenly trade value, network weight or motif participation is distributed across routes and structural modes; for route-value entropy, exp(H) is the effective number of equally weighted routes. The number of lithium links increased from 180 to 290, but its entropy-effective route count declined from 38.68 to 12.75. Nickel displayed a similar divergence, falling from 118.84 to 21.81 effective routes as links increased, whereas cobalt moved in the opposite direction. Across layer-years, flow entropy was associated with the share of trade retained under targeted attack (ρ = 0.754; Holm-adjusted p = 0.001). Binary dependence between lithium and nickel rose over time, although their weighted divergence was still 0.907 in 2024. In the China-removal experiment at the largest capacity margin, 95.7% of nodes survived but only 33.8% of trade value remained. For these product baskets, a larger network therefore need not be a more diversified one, and preserved connectivity can coexist with substantial economic loss.

Entropy doi: 10.3390/e28080912

Authors: Bing Wu Sheng Zhang Jiangnan Zhou Mengen Xu Qiuming Wang Yirong Zeng Ka Sun Fenglian Yuan

Dynamic link prediction, which aims to infer future edges from historical network structures, is a fundamental task in dynamic network analysis. Traditional models fail to capture high-order information, while existing methods neglect the distinct temporal evolution patterns between low-order and high-order structures, thereby limiting prediction accuracy. To address these issues, we propose DC-GHCN, a dynamic link prediction model based on a dual-channel architecture that integrates Graph Convolutional Network (GCN) and Hypergraph Convolutional Network (HGCN). Firstly, we extract closed motifs from dynamic network snapshots to construct an initial hypergraph, then refine it via nested motif pruning and node weight compensation strategies. Secondly, we design a dual-channel architecture: the GCN channel learns low-order structural features, while the HGCN channel learns high-order structural features. Furthermore, two independent Gated Recurrent Units (GRUs) separately model the temporal evolution of the two channels. Finally, the model employs a gating mechanism to adaptively fuse the dual-channel node representations for link prediction. Experiments on five real-world dynamic network datasets demonstrate that DC-GHCN outperforms baseline models, validating the effectiveness of the proposed model in dynamic link prediction.

Entropy doi: 10.3390/e28080910

Authors: Antonio F. Miguel Vinicius R. Pepe Luiz A. O. Rocha

The spontaneous emergence of macroscopic order in driven, far-from-equilibrium complex fluids lacks a generalized framework capable of bridging continuous and discrete symmetry-breaking transitions. In this study, we propose a non-equilibrium phenomenological framework that synthesizes irreversible thermodynamics, coupled Landau–de Gennes potential expansions, and active hydrodynamics. The formulation employs a single tensorial order parameter, a nonlinear state-dependent jamming mobility closure, and a generalized set of dimensionless groups to map the non-equilibrium phase space. The model predicts a sequential symmetry-breaking cascade and reproduces the emergence of polar heliconical smectic and antiferroelectric phases in driven liquid crystals, as well as the transition from isotropic active gases to macroscopic fluid flocks and active Wigner crystals in purely repulsive Janus colloids. Across these systems, a dimensionless active torque number acts as the principal bifurcation parameter, suggesting that their macroscopic structural transitions are governed by a common balance between thermodynamic and kinematic effects rather than by the details of their microscopic interactions.

Entropy doi: 10.3390/e28080911

Authors: Arnaldo Spalvieri

In this work, we compare three formulations of thermodynamic entropy for a non-interacting bosonic gas: (i) the grand-canonical Bose–Einstein entropy, (ii) the finite-N canonical entropy obtained from the exact partition function (Ziff, Uhlenbeck, and Kac construction), and (iii) the entropy of a multinomial distribution with Boltzmann categorical probabilities and temperature determined from Clausius’ equation. It is well-known that the grand-canonical Bose–Einstein systematically overestimates entropy of canonical systems in regimes where particle-number fluctuations are significant. The exact canonical entropy correctly enforces the particle-number constraint, but recent experimental results suggest that it also overestimates particle-number fluctuations below the crossover temperature. The multinomial distribution is less common in thermodynamics. It addresses in a mathematically exact way the puzzle of the famous −log(N!) term introduced by Gibbs as a deus ex machina in discussions of thermodynamic entropy. One remarkable consequence is that the entropy of the multinomial distribution overcomes the issue of negative entropy at low temperature that affects the Gibbs and the Sackur–Tetrode entropies at low temperature. The analysis presented in the paper shows that the multinomial distribution, equipped with a categorical distribution calibrated in such a way that the resulting multinomial entropy fits Clausius’ equation, provides accurate approximations to the canonical entropy in the classical regime, while it is smaller than the canonical entropy below the crossover temperature. One feature of the multinomial distribution is that it predicts lower peak variance of the number of particles in the ground state than the canonical distribution. This is in agreement with recent experimental results; hence, this paper identifies the thermodynamically calibrated multinomial distribution as a candidate alternative to the canonical distribution for thermodynamic bosonic entropy in finite systems.

Entropy doi: 10.3390/e28080909

Authors: Zakiah I. Kalantan Sulafah M. S. Binhimd Asmaa M. Abd AL-Fattah Asmaa A. Ahmed Gannat R. AL-Dayian Abeer A. EL-Helbawy Mervat K. Abd Elaal

Entropy is a fundamental measure of uncertainty in reliability and lifetime analysis, and the structure of the observed data intrinsically influences its estimation. In many practical applications, inference relies on ordered or record-based samples, for which generalized order statistics and their dual form provide a comprehensive and unifying framework encompassing order statistics, reversed order statistics, and record values as special cases. Despite substantial progress in entropy estimation and lifetime modeling, little attention has been devoted to a unified entropy estimation framework for flexible lifetime models under generalized ordered sampling schemes. This paper investigates the impact of generalized order statistics and their dual form on the estimation of entropy measures for the exponentiated generalized Pham distribution. The proposed distribution extends the classical Pham model through additional shape flexibility, enabling it to accommodate diverse reliability behaviors and heterogeneous tail characteristics. Several fundamental properties are derived, and maximum likelihood estimation and corresponding confidence intervals for model parameters and entropy measures are developed under both generalized order statistics and dual generalized order statistics frameworks. The general results are further specialized to order statistics, reversed order statistics, and upper and lower record values. Applications to two real datasets demonstrate that the proposed distribution provides an excellent fit compared with competing models, as confirmed by goodness-of-fit measures. The findings underscore the pivotal structural role of generalized order statistics and their dual form in entropy-based inference, particularly for record or partially observed data, where the sampling design critically affects uncertainty quantification.

Entropy doi: 10.3390/e28080908

Authors: Libin Wang Dmitry Golubev Yuri M. Galperin Jukka P. Pekola

The performance of low-temperature detectors utilizing thermal effects is determined by their energy relaxation properties. Usually, heat transport experiments in mesoscopic structures are carried out in the steady state, where temperature gradients do not change in time. Here, we present an experimental study of dynamic thermal relaxation in a mesoscopic system—thin metallic film. We find that thermal relaxation of hot electrons in copper and silver films is characterized by several time constants, and that the annealing of the films changes them. In most cases, two time constants are observed, and we can model the system by introducing an additional thermal reservoir coupled to the film electrons. We determine the specific heat of this reservoir and its coupling to the electrons. We suspect that multiscale thermal relaxation arises from the complicated morphology of the films, in which the electron–phonon coupling strength in grains with different orientations varies.

Entropy doi: 10.3390/e28080907

Authors: Yuhang Li Xifan Chen Jiale Shi Guocheng Lv Ye Jin

Motivated by the increasing security requirements of next-generation satellite-terrestrial communication systems and the emergence of Open Radio Access Network (O-RAN) architectures, this paper presents a secrecy analysis of a novel reconfigurable intelligent surface (RIS)-assisted mixed free-space optical (FSO) and radio frequency (RF) satellite downlink transmission system within an O-RAN-enabled non-terrestrial network (NTN) framework. The inherent broadcast nature of RF transmissions presents significant eavesdropping risks, which serves as the primary impetus for this study. We analyze the combined effects of imperfect channel state information (CSI) and random link blockage within such integrated networks. The impact of discrete phase shift constraints at the RIS is also investigated. Closed-form expressions are derived for three key performance metrics: connection outage probability (COP), secrecy outage probability (SOP), and the probability of positive secrecy capacity (PPSC). Through high signal-to-noise ratio (SNR) asymptotic analysis, corresponding asymptotic expressions are obtained, and all analytical results are validated via extensive Monte Carlo simulations. Our findings demonstrate that: (i) Link blockage probability and channel estimation accuracy jointly govern the secrecy performance floor. (ii) Increasing the number of RIS elements enhances physical-layer security by driving both the COP and SOP toward their theoretical lower bounds. (iii) Improving channel estimation accuracy diminishes the eavesdropper’s channel advantage and improves the overall system security. These results offer valuable insights for designing secure mixed FSO/RF satellite-terrestrial systems within O-RAN-enabled NTN architectures that effectively balance connectivity and confidentiality.

Entropy doi: 10.3390/e28080906

Authors: Sandor M. Molnar Joseph R. Godfrey

We develop a KvN/Weyl/Wigner/Moyal phase-space formulation for shock-containing compressible, irrotational, barotropic Euler flow. Smooth branches are represented by branchwise Wigner distributions, while piecewise-smooth entropy-admissible shocks generate an interface-supported defect in the weak phase-space balance. This defect is concentrated on the moving shock surface and is weighted by the normal relative transport flux between the one-sided branches. An exact planar constant-state three-dimensional example shows how the same mass flux is transferred between distinct velocity-space supports and how its moments recover the classical jump structure. We also introduce a shock solution of the one-dimensional Burgers equation with a triangular initial profile as an exactly solvable reduced benchmark. In this example, the shock trajectory, transported branch weights, branchwise Wigner transforms, and a two-component localized phase-space defect are obtained in closed form. The construction is a restricted branchwise representation of Euler shocks already selected by the Rankine–Hugoniot and entropy conditions; it is not a new admissibility criterion or a complete global Wigner theory across discontinuities. The formulation separates smooth phase-space evolution from singular interface contributions within a unified construction and provides a compact diagnostic description of shock-supported phase-space structure.

Entropy doi: 10.3390/e28080905

Authors: Yiting Hao Dongqing Cao Wenhao Gui

Aiming at the multi-constraint three-dimensional bin packing problem for mixed vehicle types in urban logistics, where traditional exact algorithms are limited by NP-hard computational complexity and practical engineering constraints, this study proposes a progressive optimization framework for vehicle loading and fleet allocation optimization. First, a heuristic loading algorithm based on the extreme point method and greedy strategy is developed to maximize single-vehicle loading efficiency by balancing space and weight utilization. Second, an NSGA-II based evolutionary framework with sequential encoding is constructed to minimize fleet size while improving loading balance for single-vehicle-type optimization. Third, a three-stage hybrid algorithm integrating greedy packing, enumerative search, and tail vehicle replacement is designed to optimize mixed-vehicle fleet composition and minimize total transportation cost. Experimental results demonstrate that the proposed heuristic achieves high composite loading performance across vehicle types, and the evolutionary framework significantly reduces fleet size compared with theoretical lower bounds. Under mixed-fleet optimization, the model identifies cost-effective vehicle configurations that outperform single-type dispatching strategies. Sensitivity analysis reveals that cargo composition, particularly the number of fragile items, is the most critical factor affecting system performance, while validation on 16 vehicle types confirms the robustness and practical generalizability of the method. This study verifies the effectiveness and stability of heuristic-evolutionary hybrid optimization methods, providing a reliable decision-making reference for logistics enterprises in vehicle selection, cargo allocation, and transportation planning.

Entropy doi: 10.3390/e28080904

Authors: Gianmarco Baldini

The knowledge of the Line of Sight (LOS) or Not Line of Sight (NLOS) propagation condition is useful information in wireless communication system. Such knowledge can be inferred by the analysis of the signal, by using specific signal structures (e.g., preambles) or by the application of machine learning (ML) algorithms. In recent times, deep learning (DL) has been applied with success to the classification of LOS/NLOS conditions but with a significant computational time, which can be a practical issue in computing constrained devices. On the other hand, ML relies on the identification of key discriminating features, which can enhance the classification performance. This paper explores the application of entropy metrics to this classification problem. Beyond Shannon entropy, researchers have developed various entropy metrics in recent years in various domains (e.g., healthcare), but they have been scarcely applied to UWB LOS/NLOS classification to the best of the author’s knowledge. This paper addresses this gap by applying entropy metrics in combination with ML classifiers to the public eWINE dataset, characterised by seven different propagation environments where UWB signals were transmitted and recorded in LOS and NLOS conditions. The results presented in this paper show that entropy metrics can significantly enhance the LOS/NLOS classification accuracy and can produce an overall competitive performance. In addition, this paper presents a novel instance selection approach based on the use of entropy metrics, which is demonstrated to significantly outperform even the direct application of some DL algorithms on the basis of the results presented in the literature on the same eWine data set. To summarise the novelty aspects of this study, for the first time in the literature, this study presents an extensive analysis of the discriminative advantage (discrimination index) of entropy measures introduced in the research literature in other domains (e.g., mechanical problems, analysis of physiological signals) in UWB multipath environments for UWB LOS/NLOS classification. In addition, this study presents for the first time the application of an ensemble instance selection algorithm based on entropy measures to the problem of UWB LOS/NLOS classification to handle “noise” or “boundary” samples in the data set, thereby improving model generalisation.

Entropy doi: 10.3390/e28080903

Authors: Zakaryae Ezzouine Adil Salbi Mohamed Abouzahir Ilham Elmourabit Adil Brouri Sébastien Roy

Reliable navigation in GPS-denied environments remains a critical challenge for autonomous vehicles (AVs), particularly in complex indoor and urban settings. GPS-based localization systems often fail under these conditions, highlighting the need for resilient multimodal solutions. In this article, we present a radar-assisted tracking system that integrates LiDAR and inertial measurements within a sensor-fusion architecture to achieve robust navigation. The principal methodological contribution is a unified tracking and prediction framework that combines Bayesian state estimation with learning-based temporal prediction, enabling accurate tracking while continuously forecasting the slave robot’s short-term future state from mapping observations generated by the master robot, with a typical end-to-end perception-to-action latency of 20–60 ms. The communication and prediction forecasting module operates with an update interval below 35 ms, enabling real-time cooperative robotic operation. Sensor data are fused through a pipeline incorporating Gaussian Mixture Models (GMMs) for post-processing, which helps mitigate the limitations associated with individual sensors during edge processing. Moreover, Kalman filtering is employed to mitigate sensor noise and drift, thereby improving state estimation accuracy through trajectory smoothing. The fused spatiotemporal information is subsequently exploited by a Convolutional Recurrent Neural Network (CRNN) coupled with a Nonlinear Autoregressive model with eXogenous Inputs (NARX) to model the robot’s motion dynamics and provide short-horizon state prediction. Through simulations and real-world indoor experiments conducted in GPS-denied environments, we validate the system’s ability to provide accurate and continuous pose estimation with low localization errors. Experimental results show that the proposed framework achieves root-mean-square errors of 0.12 m, 0.15 m, and 0.28 m along the X, Y, and Z axes, respectively, while maintaining sub-meter maximum position deviations throughout the evaluated trajectories. These results confirm that the proposed framework provides reliable localization and predictive state estimation for cooperative robotic navigation in indoor GPS-denied environments. Future work will investigate outdoor validation and extend the framework to additional data-driven decision-making models for future robotic services.

Entropy doi: 10.3390/e28080902

Authors: Chunhui Liu Bilin Shao Dawen Nie Ning Tian Hongbin Dai Huibin Zeng Wei Zhao Xue Zhao Xinyu Liu Caiyun Qin

Wind power integration into mountainous power grids amplifies probabilistic forecasting challenges arising from strong non-stationarity, multi-source meteorological redundancy and frequent curtailment events. To address the limitations of existing approaches, this paper proposes a multi-horizon probabilistic forecasting framework integrating multi-perspective entropy-weighted fusion, permutation-entropy-guided decomposition, and residual-anchored probability modelling. First, an Entropy-Weighted Multi-criteria Permutation Feature Importance (EW-MPFI) module fuses KSG mutual information, Tree-SHAP, and elastic-net permutation importance through entropy-based weighted aggregation, distilling 23-dimensional meteorological inputs into eight informative features while suppressing single-criterion bias. Then, a three-stage decomposition strategy applies ICEEMDAN primary decomposition, permutation-entropy and sample-entropy guided band reconstruction, and SSA secondary refinement on high-frequency components, achieving complexity-aligned multi-scale separation. Finally, a decomposition-aware patch-based Transformer backbone (DPC-Former) generates three-quantile point forecasts, upon which an NGBoost residual layer models the conditional distribution via natural-gradient optimization in the information-geometric parameter space. Case studies on a 130 MW mountainous wind farm in Sichuan, China, covering 8736 15-min samples with 566 curtailment samples (6.48% of the dataset), show that, under the partition-wise offline batch-evaluation protocol, the proposed framework achieves an NMAE of 5.21%, an NCRPS of 3.74%, and a PICP80 of 0.84 across forecasting horizons from 15 min to 4 h. Ablation analysis attributes NMAE improvements of 25.36% and 24.57% to the decomposition and feature-selection modules, respectively, while 50-seed ensembling further reduces NCRPS, NMAE, and NRMSE by 7.40%, 7.00%, and 13.70% relative to single-seed training. A fixed-checkpoint test-block diagnostic further shows limited sensitivity at approximately weekly and three-day decomposition cadences, but a material degradation at a one-day cadence. The reported metrics should therefore be interpreted as offline best-case results rather than as performance under strictly causal real-time deployment.

Entropy doi: 10.3390/e28080901

Authors: Ziyang Wang Junliang Du

Large pretrained language models can generate fluent free-text explanations for natural language reasoning tasks, but these explanations may contain redundant, irrelevant, or unsupported information. In this paper, we propose a faithful explanation regeneration framework based on a Cauchy–Schwarz mixture information bottleneck. The proposed method compresses noisy explanations into a structured bottleneck representation while preserving task-relevant and decision-supporting information. Instead of using a unimodal Gaussian prior, we introduce a Gaussian mixture prior and employ the Cauchy–Schwarz divergence as a tractable compression regularizer. Furthermore, a faithfulness-aware objective is introduced to encourage the learned representation to remain aligned with the task decision. Experiments on free-text explanation benchmarks demonstrate that the proposed method improves explanation quality, conciseness, and faithfulness while providing an information-theoretic compression–preservation framework.

Entropy doi: 10.3390/e28080898

Authors: Neri Merhav

A variable-to-variable (V2V) length code parses a source sequence into phrases of variable length and maps each phrase to a binary codeword of, generally, a different random length. After encoding n phrases, the realized compression ratio Rn=Λn/Σn—total codeword length over total source-symbol count—is the finite-sample counterpart of the code’s asymptotic rate ρ, to which it converges only as n→∞. This paper first derives exact formulas for all integer moments of Rn for a given discrete memoryless source (DMS). Specifically, we obtain a closed-form formula for every moment E{Rnk} as a one-dimensional integral involving only single-phrase moment generating functions of the pair (L,l)—the phrase length, in source symbols, and codeword length, in bits. From these moments we derive an Edgeworth approximation to the cumulative distribution function (CDF) of Rn that is substantially more accurate than the central limit theorem (CLT) approximation. Using the Laplace method of integration, we also derive explicit closed-form formulas for the bias constant C=limn→∞n(E{Rn}−ρ) and for the variance constant limn→∞n·Var{Rn}. The analysis extends to Markov sources via state-indexed matrices with a redundancy formula obtained in closed form. On the coding-theoretic side, we cast V2V length codes as finite-state encoders and apply a generalized Kraft inequality for a compression-rate lower bound, and give a structural decomposition of the bias coefficient that separates cleanly across variable-to-fixed (V2F) length codes, fixed-to-variable (F2V) length codes, and V2V length codes. Applied to the Khodak code of Bugeaud, Drmota, and Szpankowski, this decomposition shows that its improved performance is reflected in its smaller bias constant.

Entropy doi: 10.3390/e28080900

Authors: Zhengyi Wang Feng Gao Yunqing Xu Xiaohui Wang Jingyang Fang

Single-path transmission optimization is a core task for resource scheduling and operation of complex networks, which requires coordinated optimization of transmission cost and flow. Classical algorithms bear heavy computational loads in high-dimensional decision spaces as networks grow. This paper constructs a hybrid quantum model integrating quantum approximate optimization algorithm (QAOA) and cubic spline interpolation. Paths, discrete flows, and trade-off coefficients are unified within a quadratic unconstrained binary optimization (QUBO) model. Least-squares fitting converts native parameters into QUBO coefficients, whose fitting errors are measured to verify robustness and penalty sensitivity, and auxiliary variables eliminate high-order terms to exponentially cut qubit consumption. QAOA narrows the feasible range via global coarse search, and cubic spline interpolation further yields precise continuous flow values. Powered by quantum superposition for parallel full-space exploration, the framework avoids repeated modeling for separate bias coefficients. Mixed integer programming (MIP) and genetic algorithm (GA) are adopted as comparative benchmarks. For the small-scale network instance, the relative error between the proposed method and the global optimum solved by MIP is less than 1%. For the large-scale case, the overall error of our approach remains within an acceptable range even when discrepancies exist between results yielded by classical algorithms.

Entropy doi: 10.3390/e28080899

Authors: Yuanzheng Zhu Qiao Hu

Approximate transport maps can facilitate exploration of a Bayesian target distribution, but the resulting samples generally do not follow that distribution. To address this problem, we develop residual-slice correction, a sampling framework that combines slice sampling with an approximate transport map held fixed during sampling. Each iteration uses a slice variable to represent the residual left by the map and updates the state while preserving the conditional distribution on the resulting feasible set. To assess sampling efficiency, we derive a lower bound on the corrected chain’s Dirichlet-form gap using a reference Markov kernel. The bound separates movement within each feasible set, the transport–reference comparison, and reference mixing, while projected diagnostics examine the first two factors. Numerical experiments show that residual-slice correction recovers summaries and shape diagnostics distorted by approximate transport; they also show that the choice of Markov update within each feasible set substantially affects mixing efficiency, and that the corrected chains have lower serial dependence after normalizing-flow training. Overall, the framework retains the geometric benefits of approximate transport while preserving the target distribution.

Entropy doi: 10.3390/e28080897

Authors: Ming Liu Fengxiao Huang Siqi Zhang Feng Yang Wei Zhao Junling Liu Hong Li

We propose a physics-informed neural network (PINN) scheme for designing shortcuts to adiabaticity in a three-level non-Hermitian quantum system. The PINN is used to solve an inverse control problem in which the state amplitudes and the auxiliary driving field are learned simultaneously from the Schrödinger residual, the initial and target population constraints and a probability conservation constraint on the control pulse. The learned compensation field counteracts the loss of the intermediate state and enables high-fidelity population inversion in an open-system setting. Importantly, the imposed probability conservation is treated as an auxiliary constraint along the learned trajectory rather than as an intrinsic property of the non-Hermitian Hamiltonian. Under this constrained evolution, the Hamiltonian expectation value evaluated on the obtained state remains real within numerical accuracy. Numerical simulations and independent propagation with the fitted control field verify the population inversion and exhibit strong generalization capability over a range of coupling strengths and dissipation rates, as verified by retraining the network independently for each parameter set. The results demonstrate that PINNs provide a flexible inverse design tool for non-Hermitian shortcut to adiabaticity protocols when the governing dynamics and physical constraints are explicitly incorporated into the loss function.

Entropy doi: 10.3390/e28080896

Authors: Hongyin Li Yang Yue Jiahao Li Zhongning Guo Ming Wu

Motion image deblurring remains challenging because many existing models rely on complex architectures, leading to high computational cost and parameter redundancy, particularly under non-uniform blur in real-world scenes. To mitigate these limitations, we propose a Lightweight Asymmetric Convolutional Residual Network (LACR) for efficient motion image deblurring. LACR introduces an asymmetric convolutional residual module that combines local spatial embedding with horizontal and vertical asymmetric refinement, enabling direction-sensitive blur modeling with reduced spatial redundancy. A shallow deep feature fusion mechanism is further designed to integrate low-level convolutional cues with deep restoration representations, thereby complementing low-frequency structural information with high-frequency texture details. Experiments on four benchmark datasets show that LACR improves the reconstruction of edges, textures, and structural details while maintaining a lightweight design. Compared with representative lightweight deblurring methods under consistent evaluation settings, LACR achieves an average PSNR gain of 0.38 dB and reduces computational cost by more than 20%. Quantitative and qualitative results demonstrate that LACR achieves a favorable balance between restoration quality and computational efficiency.

Entropy doi: 10.3390/e28080895

Authors: Enrique Raúl Olguín-Rodríguez Carlos Gershenson Enrique Vázquez-Semadeni

The Initial Mass Function (IMF) describes the distribution of stellar masses formed in a stellar cluster and constitutes a fundamental constraint for theories of star formation. In this work, we investigate the emergence of the IMF slope using a minimal agent-based model inspired by preferential attachment mechanisms. The model represents stars as accretion centers embedded in a reservoir of infalling material, where mass growth occurs through competitive accretion at a rate proportional to a tunable power of the stellar mass, and where fragmentation of accretion centers is allowed. We systematically explore the effects of accretion and fragmentation probabilities on the resulting mass distribution. The simulations show that competitive accretion alone leads to the dominance of a single massive object, whereas the inclusion of fragmentation regulates mass growth and produces a stable mass spectrum. For a specific range of accretion exponents, the resulting stellar mass function exhibits a logarithmic slope close to the Salpeter value. These results indicate that the interplay between fragmentation and mass-dependent accretion can produce a power-law slope comparable to the observed IMF under the simplified assumptions of the proposed agent-based model.

Entropy doi: 10.3390/e28080894

Authors: Daniel Markthaler Kai Peter Birke

The Ising model provides a fundamental setting for investigating the emergence of phase transitions from simple interacting degrees of freedom. The current characterization study serves to investigate central requirements for phase transitions in terms of connectivity, i.e., the degree of coupled interactions between interaction sites. The impact of increasing connectivity between 1D-Ising chains mapped onto 2D-lattices with free boundary conditions were studied systematically, using exact free energy calculations. Starting from a reference system of non-interacting 1D-chains, interaction bonds between chains are introduced successively until the fully connected N×N-lattice is obtained. Two distinct construction schemes are analyzed, which differ in the connectivity of the intermediate partially coupled systems. The resulting free energies of the graphs along these paths are evaluated and compared with respect to their convergence behavior as a function of system size. We find that, despite topological differences between the schemes, strikingly, they converge to the same limiting straight line for increasing N when analyzed in terms of residual free energy differences. These findings provide insight into the relationship between interaction structure and thermodynamic behavior and suggest that appropriately chosen construction paths may serve as a basis for efficient extrapolation strategies toward the thermodynamic limit.

Entropy doi: 10.3390/e28080893

Authors: Atsushi Masumori Hiroki Sato Takashi Ikegami

Growing Neural Cellular Automata (GNCA) are capable of robust self-maintenance and self-repair, yet the internal dynamical mechanisms that support these capabilities remain poorly understood. Here, we investigate the role of internal fluctuations—temporal micro-variability of hidden channel states—in a trained GNCA model, hypothesizing that they constitute a functional component of the dynamics rather than merely residual stochastic noise. We analyzed the trained model through dynamical-systems analysis (low-dimensional embedding and recurrence analysis of collective state trajectories) and information-theoretic analysis (transfer entropy and partial information decomposition), including its response to localized damage and to suppression of small-magnitude updates. These analyses show that internal fluctuations are spatially structured, dynamically coupled to an attracting collective state, and associated with distributed small-magnitude updates that contribute to damage recovery. Damage induces a global deviation in latent state space followed by gradual re-convergence, and suppressing distributed small-magnitude updates associated with baseline fluctuation dynamics outside a permissive radius that encompasses the majority of the cells significantly impairs recovery. Transfer entropy analysis characterizes a spatially differentiated repair response: corrective inward flow near the damage site coexists with outward perturbation propagation at greater distances. Partial information decomposition further suggests a regime shift from synergy-dominant resting computation to redundancy-increased coordination during recovery. These findings indicate that GNCA self-maintenance and self-repair emerge from high-dimensional nonlinear collective dynamics in which internal fluctuations serve as a functional component supporting information flow, coordination, and return toward an attracting recurrent state.

Entropy doi: 10.3390/e28080892

Authors: Francine da Silva Borges André Andrade Longaray

Advances in the decision-making field and the growing adoption of hybrid mathematical methods, combined with practitioners’ and managers’ interest in more efficient systems, have intensified the search for improved combinations of mathematical modeling capable of supporting robust, effective decision processes. In this context, it becomes possible to incorporate different methods for assigning substitution rates in multicriteria methods, thereby reducing uncertainty in the application of such models. The purpose of this study was to examine research that applies the MACBETH method (measuring attractiveness by a categorical-based evaluation technique) in combination with other weighting techniques, particularly the information entropy method. The literature mapping evaluated the evolution of studies on this topic, the temporal progression of research, the most productive authors, and the journals most aligned with the theme while also identifying the techniques and sectors in which they are applied. The review further highlighted benefits, limitations, and future challenges and developments in the field. The research was conducted through a systematic literature review, with analysis conducted using a bibliometric approach complemented by a meta-synthesis. The findings reveal a growing number of publications in recent years, along with the main techniques combined for assigning substitution rates. The results also highlight the sectors of application, leading journals, and emerging trends in the domain of hybrid multicriteria methods. Given the limited research on this topic, this is considered a valuable contribution.

Entropy doi: 10.3390/e28080891

Authors: Dawlah Alsulami

This paper introduces the Complementary Asymmetric-Odds Weibull (CAO–W) distribution, a new three-parameter distribution that improves the accuracy of modeling lifetime data with complex behavior. The proposed distribution is based on an asymmetric transformation that combines the baseline distribution with its complement, providing more flexibility when modeling a variety of hazard patterns, such as increasing, decreasing, and bathtub-shaped. Some statistical properties of the CAO–W distribution were studied including a mathematical and graphical analysis of the hazard rate function (HRF). The model parameters were estimated using four widely used estimation approaches: maximum likelihood (ML), least squares (LS), maximum product of spacings (MPS), and the Cramér-von-Mises (CVM). The efficiency of these approaches in estimating the model parameters was investigated through simulation studies under different scenarios. Moreover, the proposed distribution was applied to four real datasets, and compared to some flexible distributions to demonstrate its ability to provide a good fit for lifetime data in survival and reliability analysis applications. Finally, a change point analysis, based on the Minimum Information Criterion (MIC), is also conducted to highlight the flexibility of the proposed model.

Entropy doi: 10.3390/e28080889

Authors: Manik Awale Shrirang Pund Hassan S. Bakouch Aishwarya Ghodake Amira F. Daghestani Souha K. Badr

In this article, we propose a first-order integer-valued autoregressive (INAR(1)) model based on binomial thinning, in which the innovation sequence follows a zero–one–two-modified Poisson (ZOTMPo) distribution. The proposed model accommodates overdispersion and allows for inflation or deflation at low-count values, particularly at zero, one, and two, which are commonly observed in public health count time series. We derive the main probabilistic properties of the model and estimate the unknown parameters using the conditional maximum likelihood (CML) method. A Monte Carlo simulation study is conducted to evaluate the finite-sample performance of the estimators. Furthermore, we establish the information-theoretic properties of the model, specifically deriving the Shannon entropy and conditional entropy bounds to quantify the dynamical complexity and predictability of the stochastic process. The practical utility of the model is illustrated using two real-world datasets on dengue fever incidence and Escherichia coli (E. coli) enteritis. Model performance is assessed using standard information criteria and forecast accuracy measures, as well as the Euclidean distance between observed and fitted probabilities for zero, one, and two. Diagnostic checks, including analysis of residual autocorrelation, cumulative periodograms, and jump process behavior, provide further confirmation of the fitted model’s adequacy. The results indicate that the proposed ZOTMPo–INAR(1) model provides an effective framework for modeling overdispersed count time series with a modified low-count structure.

Entropy doi: 10.3390/e28080890

Authors: Yifan Gao Yanhang Chen Shuyu Dai Bo Cui

We investigate the dissipative dynamics of a one-dimensional sawtooth-shaped Bose–Hubbard model subjected to an external magnetic flux and staggered single-particle dissipation. By combining the Lindblad master equation with a mean-field decoupling and further reducing the dynamics to an effective three-site model, we derive the nonlinear evolution equations that govern the system. Our results reveal that the magnetic flux, acting through the next-nearest-neighbor hopping, determines the preferential direction of particle flow, while the imbalance in dissipation forces the steady-state population to accumulate at lattice sites with weaker loss. Furthermore, we find that two-particle dissipation accelerates the relaxation process when it becomes negative (i.e., gain), whereas positive two-particle loss suppresses localization. These findings demonstrate that directional localization and relaxation dynamics can be controlled by the sign of the next-nearest-neighbor hopping t′ and the magnetic phase, providing a tunable scheme for engineering dissipative quantum states in optical lattices.

Entropy doi: 10.3390/e28080888

Authors: Pinru Chen Fengyun Li Jun Zheng Weiqing Xu

Blending hydrogen into existing natural-gas pipeline networks provides a practicable route toward future low-carbon applications. A Venturi-type mixer is a classical high-efficiency static gas-mixing device, and clarifying the effects of its structural parameters is important for efficient transport and downstream combustion stability. In this study, numerical simulations were performed in ANSYS Fluent 2024 R1. The contraction angle, throat length, and diffuser angle were selected as representative structural variables. First, the independent effects of these variables on the mixing process were examined through single-factor simulations. Then, three key levels of the three structural parameters were selected to establish a Box–Behnken experimental matrix for response-surface modeling. Based on the numerical results, entropy weighting and a genetic algorithm were used for multi-objective optimization, and the final solution was verified using the TOPSIS method. The results show that the optimized Venturi-type mixing device with optimized parameters of a contraction angle of 20.7°, a throat length of 60 mm, and a diffuser angle of 5° can reduce flow energy loss while maintaining high mixing uniformity. The diffuser angle is the dominant geometric parameter affecting both energy loss and mixing behavior. Compared with the reference central-point structure design, the overall TOPSIS score of the optimized structure increased from 0.41 to 0.82; the pressure loss decreased from 258.94 Pa to 206 Pa, corresponding to a reduction of approximately 20%; and the final-section mixing uniformity decreased only slightly, from 97.85% to 97.43%.

Entropy doi: 10.3390/e28080887

Authors: Lei An Jinping Dai

Quantifying the direction, strength and temporal clustering of cross-border financial risk contagion calls for methods that go beyond linear correlation. We suggest a two-layer framework that brings together transfer entropy and a multivariate Hawkes self-exciting point process on a time-varying, directed, weighted network. In the first layer, one-to-one transfer entropies of sovereign credit default swap spreads are estimated with a bias-corrected k nearest neighbour estimator, and this step detects nonlinear and directional information transfer between spreads. The second layer is a multivariate Hawkes process that models how extreme loss events arrive and mutually excite one another across countries, and it gives an excitation intensity matrix, encoding the way a tail event in one country raises the likelihood of an instantaneous hazard occurring in another. By merging these two layers, we obtain a composite, directed, weighted adjacency matrix in which the weights of the edges reflect both information flow and event clustering. We introduce a network-level contagion intensity index and split it into direct, indirect and feedback terms using the graph Laplacian spectrum. Von Neumann graph entropy together with the spectral gap ratio serve as entropy-based measures of the complexity and fragility of the evolving network. We validate the choice of Shannon-type entropy through a Tsallis q-sensitivity analysis, and we verify the nonlinear dependence structure of the data using BDS tests and maximal Lyapunov exponent estimates. Three empirical findings emerge from analysing 20 sovereign CDS markets from January 2015 to December 2025: (i) directional risk spillover signals derived based on transfer entropy are more timely than those derived from variance decomposition; (ii) the Hawkes excitation component amplifies measured contagion intensity by 35 to 58 percent during the COVID-19 shock and the 2022 European energy crisis relative to a transfer-entropy-only baseline; (iii) von Neumann graph entropy reaches historically extreme values 7 to 12 trading days before the peak drawdown in a Global Sovereign Bond Index. These results hold across rolling window lengths, significance thresholds, alternative entropy functionals and alternative Hawkes kernels.

Entropy doi: 10.3390/e28080886

Authors: Yuxuan Zhang Shuchang Wang Wei Yang

Search-based software engineering (SBSE) tackles critical optimization problems in software engineering, including the next release problem (NRP) and feature selection problem (FSP). Traditional heuristic approaches and integer linear programming (ILP) methods work well for small- to medium-scale problems but face growing computational cost as instances scale up. We investigate quantum annealing (QA) as an optimization subroutine for multi-objective SBSE problems. We propose two QA-based algorithms tailored to different problem scales. For small-scale problems, we reformulate multi-objective optimization (MOO) as single-objective optimization (SOO) using penalty-based mappings for quantum processing. For large-scale problems that exceed current hardware capacity, we employ a decomposition strategy guided by maximum energy impact (MEI) that partitions the problem into smaller sub-QUBOs, integrating QA with a steepest-descent method for local search. Applied to NRP and FSP, our approaches are benchmarked against the heuristic NSGA-II, IBEA, and MOEA/D, as well as the ILP-based ϵ-constraint method. The experimental results reveal that while our methods produce fewer non-dominated solutions than ϵ-constraint, they achieve substantial reductions in execution time. Compared to the evolutionary baselines, our methods achieve competitive solution quality with lower runtime on the instances that they can encode. The penalty-based QUBO formulation fails to reach feasible regions on constraint-dense FSP instances, limiting the current applicability of the approach. QA is a promising but still hardware-limited component for multi-objective SBSE workflows, rather than a wholesale replacement for classical solvers.

Entropy doi: 10.3390/e28080885

Authors: Hui Xu

Over the past two decades, high-entropy alloys (HEAs) have revolutionized the traditional alloy design paradigm dominated by a single primary base element [...]

Entropy doi: 10.3390/e28080884

Authors: Yiyang Wu Hongqiu Zhu

Dynamic vehicular integrated sensing and communication (ISAC) requires representations that remain compact, decision-relevant, and structurally stable as mobility rewires interference and sensing relations. This paper presents IC-GMRO, a conditional information-bottleneck graph-clustering framework for structured representation learning in multi-agent resource optimization. At each control epoch, vehicles, roadside units, targets, and typed interactions form a temporal heterogeneous graph. A context-conditioned variational bottleneck suppresses nuisance variation while retaining action-relevant information; balanced soft graph clusters then convert the latent space into reusable coordination codes. Feasibility-masked policies jointly select association, beam, resource block, transmit power, and sensing-time ratio. The analysis distinguishes representation-level information guarantees from the idealized potential and projected-dual arguments used only to motivate the practical neural updates. Controlled simulations and component ablations show improved utility, sensing success, latency robustness, and cross-density robustness relative to greedy, flat, and graph-only baselines.

Entropy doi: 10.3390/e28080883

Authors: Salim Bouzebda Sultana Didi

We consider the estimation of partial derivatives of multivariate regression-type functionals from incomplete observations generated by a discrete-time strictly stationary ergodic process. The response variable is subject to a missing-at-random (MAR) mechanism, whereas the covariates are fully observed. Building upon the complete-data wavelet methodology developed in Didi and Bouzebda (2025), we construct inverse-probability-weighted empirical wavelet estimators that compensate for the selection bias induced by missing responses. When the propensity score is unknown, a feasible estimator is obtained by replacing the oracle weights with a nonparametric Nadaraya–Watson estimator. The analysis is carried out under stationary ergodicity without imposing mixing assumptions. The estimation error is decomposed into three analytically distinct components: the deterministic multiresolution approximation error, the stochastic fluctuation of the oracle inverse-probability-weighted estimator, and the additional error arising from propensity score estimation. This decomposition makes it possible to isolate the respective effects of approximation, dependence, and missingness within a unified asymptotic framework. Under explicit assumptions on the multiresolution approximation, missingness mechanism, conditional density stabilization, moment conditions, and accuracy of the propensity estimator, we establish non-asymptotic integrated mean squared error bounds together with their asymptotic rates. We further prove almost-sure uniform consistency over compact subsets of the interior of the support and derive a pointwise central limit theorem for both the oracle and feasible estimators. The limiting variance explicitly reflects the information loss induced by inverse probability weighting, and for general orthogonal projection kernels is formulated under the corresponding dyadic-phase condition. The general methodology is specialized to the estimation of first- and second-order derivatives of ordinary regression functions. A finite-sample simulation study investigates the empirical behavior of the proposed estimators under stationary ergodic dependence and MAR missingness, examines the influence of both the wavelet resolution level and the propensity-score bandwidth, evaluates the finite-sample performance of the asymptotic confidence intervals, and compares the proposed procedure with oracle, complete-case, and competing nonparametric estimators. The numerical results are consistent with the theoretical analysis and illustrate the respective contributions of wavelet approximation, inverse probability weighting, and propensity score estimation to the overall estimation error. When the propensity score is identically equal to one, the proposed methodology reduces to the corresponding complete-data wavelet estimator.

Entropy doi: 10.3390/e28080882

Authors: Alexander J. Gates

Collision statistics provide a finite-resolution view of information by measuring how often independent samples fall on the same state and form the basis of integer-order Rényi entropies. Here, we use low-order Rényi entropies to characterize finite-resolution approximations to Shannon entropy and mutual information. Specifically, we determine what population information is captured by finite collision moments, we quantify how the resulting targets differ from their Shannon counterparts, and we analyze how accurately they can be estimated from finite samples. We use the interpolation remainder to identify structural approximation error induced by extrapolating from integer-order Rényi entropies to the Shannon point. We separate this deterministic error from finite-sample estimation error: increasing sample size improves estimation of a finite-resolution target but does not eliminate its deterministic difference from Shannon entropy or mutual information. Finally, we show that finite collision moments do not generally identify Shannon entropy, and that increasing collision order shifts sensitivity toward high-probability events. Our numerical experiments illustrate the approximation–estimation trade-off and evaluate collision-based approximations alongside plug-in and Miller–Madow estimators. Together, these results provide a principled way to use low-order coincidence structure as finite-resolution information, while making explicit what finite collision moments can and cannot reveal about Shannon entropy and mutual information.

Entropy doi: 10.3390/e28080881

Authors: Shengming Wang Yunquan Song Juan Yu

This paper develops and evaluates HQER-SFA, a likelihood-based stochastic frontier specification that embeds a hybrid quantile-expectile error layer into the bilateral noise component while preserving the standard one-sided inefficiency structure. The model nests Quantile-SFA when γ=0 and extends it by normalizing the hybrid loss into a proper bilateral error density, so that likelihood inference, residual decomposition, and technical-efficiency recovery remain in a unified stochastic-frontier framework. We compare HQER-SFA with Traditional-SFA and Quantile-SFA using three processed production modules, Monte Carlo parameter-inversion experiments, an expanded R=300 robustness design, γ sensitivity and ablation checks, convergence diagnostics, and a source-assisted small-sample extension. The results demonstrate clear gains for technical-efficiency recovery: in the R=300 design, HQER-SFA attains win rates of 0.5646 for technical-efficiency RMSE and 0.5578 for technical-efficiency rank correlation, and the agricultural module shows the strongest real-data improvement under the flexible bilateral error layer. The source-assisted analysis further shows that same-domain initialization improves small-sample validation RMSE by about 2.43%, while excessive source-centered penalties should be controlled. Overall, HQER-SFA provides an interpretable and computationally feasible extension for technical-efficiency recovery, especially when bilateral noise is asymmetric, tail-sensitive, or heterogeneous across production modules.

Entropy doi: 10.3390/e28080880

Authors: Fu Tan Xiaolong Chen Ruijie Wang Chi Huang Shimin Cai

In complex network analysis, the identification of influential nodes is a fundamental issue, which is closely related to the structural robustness of the network and the dynamics of propagation processes. Current research primarily focuses on mesoscale features based on the smallest cycles or local features derived from star-shaped structures. However, the role of neighboring nodes that are connected to a given node but do not participate in its smallest cycles remains underexplored in network analysis. To address this issue, this paper proposes a hybrid centrality measure that integrates information from both smallest-cycle structures and non-smallest-cycle structures associated with each target node. The smallest-cycle structures considered in this method are identified only within the imposed local search range and do not necessarily correspond to the true smallest cycles in the full graph. Specifically, the extent of a node’s involvement in mesoscale structures is characterized by the number of the smallest cycles it participates in, while its local structural heterogeneity is represented by the number of neighboring nodes connected to it that do not belong to any smallest cycles. These two aspects are then unified into a single node importance metric through a weighted integration strategy. This paper evaluates node importance from multiple perspectives, including propagation capability analysis based on the SI model, network robustness testing through node attack simulations, and ranking accuracy assessment using Kendall correlation coefficient. The experimental results demonstrate that the proposed method achieves competitive or superior performance compared with the selected baseline methods under the experimental settings considered in this work. The findings indicate that integrating smallest-cycle and non-smallest-cycle features provides a more comprehensive characterization of a node’s role in complex networks. This study offers a novel perspective on the integration of multi-scale structural information in complex networks and presents an effective new approach for the identification of important nodes.

Entropy doi: 10.3390/e28080879

Authors: Lijesh Koottaparambil Roger A. Miller Michael M. Khonsari

Mayonnaise is a high-viscosity food emulsion whose consistency evolves during shearing due to structural rearrangement and possible emulsion destabilization. This study presents a laboratory-scale proof-of-concept for adapting an established motor current-derived accumulated entropy generation (AEG) framework as a thermodynamic descriptor for monitoring mayonnaise structure changes. First, eight reference fluids were tested using a rotating-bob viscometer at shear rates of 600, 800, and 1000 s−1 to establish the relationship between viscosity and motor current. The corrected current response showed a strong linear correlation with viscosity. The approach was then extended to commercially available mayonnaise samples. Due to the higher viscosity and structured nature of mayonnaise, testing was performed at 1000 s−1, where stable shearing could be achieved. A modified impeller-based viscometer setup was used to continuously shear the mayonnaise and monitor the motor current in situ, while rheometer measurements were performed independently to validate the corresponding viscosity changes during shearing. The motor current decreased with shearing time, consistent with the reduction in measured viscosity. The calculated AEG increased continuously and distinguished the shear stability of different mayonnaise formulations. The viscosity degradation rates of two different mayonnaises are characterized using the degradation coefficient B introduced in the degradation–entropy generation (DEG) theorem. A higher B value indicates greater structural breakdown. These results suggest that current-derived entropic parameters (B coefficient and AEG) may serve as practical, sensor-accessible descriptors for monitoring mayonnaise consistency evolution when direct torque measurement or in-line rheology is unavailable.

Entropy doi: 10.3390/e28080876

Authors: Ricardo J. Silva

This work introduces a framework in which organization is defined as the degree to which identity-defining relationships among system states are preserved under transformation or perturbation. Existing descriptors such as energy, entropy, mutual information, and divergence measures characterize magnitude, statistical dispersion, dependency, and deviation, yet do not explicitly address the persistence of recognizable structure. To address this limitation, the concepts of organizational classes, reference organizational models, organizational deviation, and recognition boundaries are introduced within a generalized state-space representation. The proposed framework treats recognizable structures as members of organizational classes whose identities are determined by defining constraints and relationships rather than by specific physical realizations. A probabilistic implementation is developed in which organizational classes are represented by reference models and organizational preservation is estimated through measures of organizational deviation. Recognition is incorporated through observer-dependent recognition boundaries that determine whether a realization remains identifiable as a member of a given class. The framework is illustrated through geometric, perceptual, and communication-based examples, including structural degradation in a maximum-entropy background, observer-dependent recognition, channel-limited observability, and a quantitative Gaussian organizational model. These examples demonstrate that entropy and organization are complementary descriptors that may evolve independently: organizational identity may degrade while occupancy statistics remain largely unchanged. The results suggest that communication and sensing systems may be interpreted not only as processes that transport energy or information, but also as systems that preserve, transform, or degrade organizational structure. By providing a descriptor for structural identity alongside entropy and information, the proposed framework offers a foundation for studying how organized structures emerge, persist, transform, and degrade across a wide range of physical, informational, and complex systems.

Entropy doi: 10.3390/e28080877

Authors: José J. Gil

Finite-dimensional density matrices contain two representation-intrinsic sectors after the real part is diagonalized, namely ordered intrinsic populations and antisymmetric imaginary coherences. This article develops exact complementarity identities showing how these sectors determine purity, spectral concentration, and entropy. Populations are described by indices of population asymmetry, while coherences are described by the Youla spectrum of the dimensionless metaspin tensor and by correlation-asymmetry indices. In the aligned class, where Youla two-planes coincide with pairs of intrinsic axes, normalized purity splits into a population hierarchy and pairwise coherence terms weighted by products of intrinsic populations. For arbitrary orientations, the coherence term is expressed as a positive semi-definite bilinear form in population-weighted Plücker coordinates. For fixed populations and pairing, increasing any Youla value sharpens the spectrum by majorization and decreases all Rényi entropies, including the von Neumann limit. For fixed ordered populations, maximum aligned cohesion is obtained by saturating adjacent population pairs. The dimensional transition of the discriminating-component cohesion bound is then interpreted as the change from one to two simultaneously saturating metaspin pairs.

Entropy doi: 10.3390/e28080878

Authors: L. López-Hernández D. A. Lara Bustillos Carlos E. Vargas V. Velázquez

In complex systems theory, there are different ways to describe a system in terms of information, such as emergence (Shannon entropy), self-organization, and complexity. These measures provide information about the dynamic behavior of a complex system. We study the differences in entropy and complexity for many-body systems undergoing a transition from a regular to a chaotic regime. To do this, we analyze the eigenvectors of the 48Ca nucleus for different quadrupole-type two-body interactions. We obtain the eigenvectors by diagonalizing the two-body Hamiltonian for 48Ca using the ANTOINE code. We then calculate the entropy and complexity for the different quadrupole-type interactions. The differences found in information entropy and complexity are clear when comparing a regular system with a chaotic one. We find that the complexity of the regular and chaotic states of 48Ca shows differences associated with its internal interactions.

Entropy doi: 10.3390/e28080875

Authors: Marcin Makowski Edward W. Piotrowski Jan L. Cieśliński

We present an analogue of the classical Gittins index for a one-armed decision problem with a continuous spectrum of payoffs. The model assumes that the decision-maker observes independent realizations of a random variable and, at each step, decides whether to accept the current opportunity or continue observing. We show that the optimal strategy takes the form of a threshold rule, while the corresponding reservation index is determined by a one-dimensional fixed-point equation with a direct decision-theoretic interpretation. The model is illustrated with an example of bookmaker betting related to horse racing and the Kelly criterion. This perspective allows the proposed index to be viewed as a threshold of informational advantage. This, in turn, points to potential applications in optimal stopping problems and decision-making under uncertainty.

Entropy doi: 10.3390/e28080874

Authors: Buriboev Abror Shavkatovich Akmal Abduvaitov Jumanov Isroil Karshiev Husan Shavkat Buriboyev Abbos Abduvaytov Aziza Akhmedova Rustam Rakhimov Obid Mavlonov Heung Seok Jeon

Ensuring information integrity requires not only reducing decoding errors but also reducing the probability that corrupted data are accepted as valid. This research presents a hybrid integrity protection system that incorporates seeded universal hash verification, algebraic manipulation detection (AMD), and a binary Reed–Muller outer code. Transmission over the binary symmetric channel BSC(p), outer encoding using RM(r, m), bounded-distance decoding, an εAMD-secure AMD layer, and a seeded 2-universal hash family with l-bit output define the model used in the analysis. Under explicitly stated freshness and conditional-independence assumptions, the system-level undetected error probability is upper-bounded by the residual decoder-miscorrection probability multiplied by the AMD acceptance bound and the seeded universal hash collision bound. A conservative alternative is also provided for settings in which the required conditional independence cannot be guaranteed. In this context, an explicit upper bound for the undetected error probability is derived. The outcome makes clear the different functions of outer coding and post-decoding verification and results in a direct dependency on the parameters r, m, p, and l. Finite-length Monte Carlo validation for a concrete instantiation based on RM(2, 5) complements the theoretical study and verifies that the hybrid construction offers a lower empirical undetected error probability compared to the comparable outer-only, AMD-only, and hash-only variations. The study does not propose new coding or verification primitives. Its contribution is a finite-length layered acceptance model and a Reed–Muller-specific undetected error analysis that incorporates the code weight distribution and bounded-distance decoding regions. The resulting spectrum-based bound distinguishes decoder miscorrection from the broader event of exceeding the guaranteed correction radius and is evaluated together with post-decoding verification and redundancy overhead. The model’s formal manipulation detection and collision guarantees are provided by AMD and universal hash layers, while Reed–Muller code parameters and their standard distance formulas are conventional.

Entropy doi: 10.3390/e28080873

Authors: Yuhui Lin Chaopeng Li Zhiwei Shen Jianfeng Liu Miao Zeng

Low-Rank Adaptation (LoRA) is a representative parameter-efficient fine-tuning method that reduces computational and memory costs without modifying the model architecture. Standard LoRA initializes matrix A from a symmetric distribution, such as Gaussian or Kaiming initialization, and matrix B to zero. Although this provides a statistically neutral starting point, it ignores the influence of task-specific input features on initialization. We propose Gamma-Moment Equalization Initialization (GME-Init), a data-aware asymmetric LoRA initialization method based on output-moment calibration. Using a small calibration set, GME-Init estimates the variance and skewness of target-layer outputs and adjusts the layer-wise initialization scale and asymmetry of LoRA weights, improving their statistical alignment with task-specific skewed representations. GME-Init operates only during initialization and does not change the LoRA architecture, trainable parameter count, training budget, or inference cost. We evaluate it on a GLUE subset with RoBERTa-base, integrate it with AdaLoRA and DoRA, and test it on VRSBench-VQA, VRSBench-Caption, and UCM-Caption using Qwen2.5-VL-3B-Instruct. Results show that GME-Init serves as a simple plug-in PEFT initialization module with no additional inference cost and that it consistently improves standard LoRA and selected LoRA-style methods across the evaluated text understanding and multimodal tasks.

Entropy doi: 10.3390/e28080872

Authors: Zhenjie Hou Xuchu Dai

Causal emergence (CE) is a phenomenon in which macrodynamics provide better effective information (EI) than microdynamics. The CE is widely used as the objective function in Markov aggregation. The existing works focus on deterministic aggregation, which may not offer a good solution since the search space of each step is finite. To solve this problem, we propose a sequential Markov probabilistic aggregation (SMPA) algorithm. We first express the aggregation problem as an optimization problem, then find that the EI is maximized when the transition probability matrix is a permutation matrix, and prove that the optimization problem is a nonconvex function of the probabilistic aggregation matrix. In the SMPA algorithm, the optimization problem is split into multiple univariate optimizations. Compared with the deterministic aggregation algorithm, SMPA can achieve better greedy solutions. The experimental results indicate that probabilistic aggregation generally performs better than deterministic aggregation.

Entropy doi: 10.3390/e28080871

Authors: Mohamed A. T. El-Shahat Reman Abo Hashem Doaa Basalamah Tmader Alballa Wael S. Abu El Azm

Estimating entropy measures under progressive censoring poses a significant challenge in reliability and lifetime analysis. Despite the widespread use of the inverse Gaussian distribution in modeling skewed lifetime data, a comprehensive inferential framework for its Shannon and Rényi entropies under progressive Type-II censoring remains absent. This paper develops a unified Bayesian framework integrating maximum likelihood and Bayesian inference under squared error, general entropy, and LINEX loss functions, employing Lindley’s approximation, importance sampling, and Markov chain Monte Carlo methods. Two parameter configurations were examined to assess robustness under varying likelihood surface complexity, along with prior sensitivity analysis. Results demonstrate that Markov chain Monte Carlo and importance sampling are the only consistently reliable methods across all scenarios, whereas maximum likelihood suffered severe bias and collapse of Wald interval coverage under challenging settings, and Lindley’s approximation exhibited numerical instability at small samples. Shannon entropy proved substantially more sensitive to parameter variation than Rényi entropy, with the LINEX and general entropy loss functions showing superior performance. The study offers clear practical guidance, validated on real lifetime data.