Entropy

25 pages, 2890 KB

Open AccessArticle

Prediction Method for Fault-Induced Frequency Response Characteristics in Wind-Integrated Power Systems Using Wide-Area Measurement Data

by Yi Hu, Jinglin Luo, Tao Wang, Xiaoqin Lv, Yufei Teng, Xiaopeng Li and Jian Li

Entropy 2025, 27(11), 1134; https://doi.org/10.3390/e27111134 - 2 Nov 2025

Abstract

The decoupling properties and low-inertia characteristics of large-scale wind power have heightened concerns regarding power grid frequency stability, particularly as modern power systems impose stringent frequency regulation requirements on wind integration, leading to an increased complexity of frequency response characteristics under fault conditions. [...] Read more.

The decoupling properties and low-inertia characteristics of large-scale wind power have heightened concerns regarding power grid frequency stability, particularly as modern power systems impose stringent frequency regulation requirements on wind integration, leading to an increased complexity of frequency response characteristics under fault conditions. To address this challenge in high-wind-penetration grids, this paper proposes a post-fault frequency dynamics analysis method capable of concurrently accommodating multi-wind-speed scenarios through three key innovations: the linearization of traditional AC system components (including network equations, composite load models, and generator prime mover-governor systems) to establish nodal power increment equations; the development of wind turbine frequency regulation models under diverse wind conditions using small-signal analysis, incorporating regional operational disparities and refined by information entropy-based reliability quantification for adaptive parameter adjustment; and the derivation of the system state equation for post-fault frequency response using wide-area measurement system (WAMS) data, yielding an analytical model that captures region-specific regulation characteristic disparities for physically faithful frequency analysis. Validation via tailored IEEE 39-node simulations convincingly demonstrates the method’s effectiveness and superiority in handling fault-induced transients and wind variability. Full article

(This article belongs to the Special Issue Failure Diagnosis of Complex Systems)

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24 pages, 677 KB

Open AccessArticle

FLACON: An Information-Theoretic Approach to Flag-Aware Contextual Clustering for Large-Scale Document Organization

by Sungwook Yoon

Entropy 2025, 27(11), 1133; https://doi.org/10.3390/e27111133 - 31 Oct 2025

Abstract

Enterprise document management faces a significant challenge: traditional clustering methods focus solely on content similarity while ignoring organizational context, such as priority, workflow status, and temporal relevance. This paper introduces FLACON (Flag-Aware Context-sensitive Clustering), an information-theoretic approach that captures multi-dimensional document context through [...] Read more.

Enterprise document management faces a significant challenge: traditional clustering methods focus solely on content similarity while ignoring organizational context, such as priority, workflow status, and temporal relevance. This paper introduces FLACON (Flag-Aware Context-sensitive Clustering), an information-theoretic approach that captures multi-dimensional document context through a six-dimensional flag system encompassing Type, Domain, Priority, Status, Relationship, and Temporal dimensions. FLACON formalizes document clustering as an entropy minimization problem, where the objective is to group documents with similar contextual characteristics. The approach combines a composite distance function—integrating semantic content, contextual flags, and temporal factors—with adaptive hierarchical clustering and efficient incremental updates. This design addresses key limitations of existing solutions, including context-aware systems that lack domain-specific intelligence and LLM-based methods that require prohibitive computational resources. Evaluation across nine dataset variations demonstrates notable improvements over traditional methods, including a 7.8-fold improvement in clustering quality (Silhouette Score: 0.311 vs. 0.040) and performance comparable to GPT-4 (89% of quality) while being ~7× faster (60 s vs. 420 s for 10 K documents). FLACON achieves

O (m l o g n)

complexity for incremental updates affecting

m

documents and provides deterministic behavior, which is suitable for compliance requirements. Consistent performance across business emails, technical discussions, and financial news confirms the practical viability of this approach for large-scale enterprise document organization. Full article

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51 pages, 28544 KB

Open AccessArticle

Spatial Flows of Information Entropy as Indicators of Climate Variability and Extremes

by Bernard Twaróg

Entropy 2025, 27(11), 1132; https://doi.org/10.3390/e27111132 - 31 Oct 2025

Abstract

The objective of this study is to analyze spatial entropy flows that reveal the directional dynamics of climate change—patterns that remain obscured in traditional statistical analyses. This approach enables the identification of pathways for “climate information transport”, highlights associations with atmospheric circulation types, [...] Read more.

The objective of this study is to analyze spatial entropy flows that reveal the directional dynamics of climate change—patterns that remain obscured in traditional statistical analyses. This approach enables the identification of pathways for “climate information transport”, highlights associations with atmospheric circulation types, and allows for the localization of both sources and “informational voids”—regions where entropy is dissipated. The analytical framework is grounded in a quantitative assessment of long-term climate variability across Europe over the period 1901–2010, utilizing Shannon entropy as a measure of atmospheric system uncertainty and variability. The underlying assumption is that the variability of temperature and precipitation reflects the inherently dynamic character of climate as a nonlinear system prone to fluctuations. The study focuses on calculating entropy estimated within a 70-year moving window for each calendar month, using bivariate distributions of temperature and precipitation modeled with copula functions. Marginal distributions were selected based on the Akaike Information Criterion (AIC). To improve the accuracy of the estimation, a block bootstrap resampling technique was applied, along with numerical integration to compute the Shannon entropy values at each of the 4165 grid points with a spatial resolution of 0.5° × 0.5°. The results indicate that entropy and its derivative are complementary indicators of atmospheric system instability—entropy proving effective in long-term diagnostics, while its derivative provides insight into the short-term forecasting of abrupt changes. A lag analysis and Spearman rank correlation between entropy values and their potential supported the investigation of how circulation variability influences the occurrence of extreme precipitation events. Particularly noteworthy is the temporal derivative of entropy, which revealed strong nonlinear relationships between local dynamic conditions and climatic extremes. A spatial analysis of the information entropy field was also conducted, revealing distinct structures with varying degrees of climatic complexity on a continental scale. This field appears to be clearly structured, reflecting not only the directional patterns of change but also the potential sources of meteorological fluctuations. A field-theory-based spatial classification allows for the identification of transitional regions—areas with heightened susceptibility to shifts in local dynamics—as well as entropy source and sink regions. The study is embedded within the Fokker–Planck formalism, wherein the change in the stochastic distribution characterizes the rate of entropy production. In this context, regions of positive divergence are interpreted as active generators of variability, while sink regions function as stabilizing zones that dampen fluctuations. Full article

(This article belongs to the Special Issue 25 Years of Sample Entropy)

12 pages, 585 KB

Open AccessArticle

Effect of Running Speed on Gait Variability in Individuals with Functional Ankle Instability

by Wenhui Mao, Kanglong Zhao, Xiangguo Xu, Mengzi Sun, Kai Wang, Yilin Xu and Li Li

Entropy 2025, 27(11), 1131; https://doi.org/10.3390/e27111131 - 31 Oct 2025

Abstract

To compare lower limb joint angle variability between functional ankle instability (FAI) and healthy controls (CONs) at different running speeds using linear and nonlinear methods. Fifteen males with right-side FAI and fifteen matched CONs ran on a treadmill at self-selected, 20% faster, and [...] Read more.

To compare lower limb joint angle variability between functional ankle instability (FAI) and healthy controls (CONs) at different running speeds using linear and nonlinear methods. Fifteen males with right-side FAI and fifteen matched CONs ran on a treadmill at self-selected, 20% faster, and 20% slower speeds. From 25 gait cycles, the mean coefficient of variation (CV), Sample Entropy (SampEn), and largest Lyapunov Exponent (LyE) of hip, knee, and ankle angles were computed. A two-way (two groups × three speeds) mixed-design ANOVA was applied (α = 0.05). No significant interaction effects were observed. No significant differences were observed in the CV. SampEn showed group effects: FAI had lower values in hip horizontal, knee sagittal/coronal, and ankle coronal planes, but higher in the hip sagittal plane. Speed effects showed greater SampEn in the ankle sagittal and lower in the hip coronal plane at slow speed. LyE was reduced in FAI for hip, knee, and ankle sagittal planes. Speed effects indicated higher LyE in the knee sagittal and lower in the hip coronal plane at slow speed. FAI showed reduced variability, particularly in the sagittal plane, reflecting rigid control. Slower speeds increased ankle and knee sagittal variability but decreased hip coronal variability. Full article

(This article belongs to the Special Issue Entropy Application in Biomechanics)

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27 pages, 3359 KB

Open AccessReview

Review of the Use of Entropy to Understand the Thermodynamics of Pure-Substance PCMs

by Harald Mehling

Entropy 2025, 27(11), 1130; https://doi.org/10.3390/e27111130 - 31 Oct 2025

Abstract

Materials that allow the storage of a significant amount of heat in a narrow temperature range by a solid–liquid or a solid–solid phase change are called Phase Change Materials (PCMs). Understanding the thermodynamics of PCMs is crucial in PCM R&D for identifying candidate [...] Read more.

Materials that allow the storage of a significant amount of heat in a narrow temperature range by a solid–liquid or a solid–solid phase change are called Phase Change Materials (PCMs). Understanding the thermodynamics of PCMs is crucial in PCM R&D for identifying candidate materials, developing new PCMs, and optimizing known PCMs. In this work, a review of the use of entropy to understand the thermodynamics of pure substances as PCMs is performed. Among pure substances, water, alkanes, alkanols, and fatty acids are well-known. Because they give valuable information, elements are also included. While phase change enthalpy and temperature are easy to comprehend and are directly used for application, the opposite holds for entropy. Thus, entropy usually receives little attention. However, as this review shows, entropy is of central importance, and even if it is not analyzed explicitly, then it is implicitly included in the data. If explicitly used, it can reveal crucial information. This is shown by a review of analysis tools and their results from analyzing typical PCMs. The review shows that if entropy is used systematically, a significant improvement in the understanding of the thermodynamics of PCMs is possible. Full article

(This article belongs to the Section Entropy Reviews)

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45 pages, 750 KB

Open AccessArticle

The Price Equation Reveals a Universal Force–Metric–Bias Law of Algorithmic Learning and Natural Selection

by Steven A. Frank

Entropy 2025, 27(11), 1129; https://doi.org/10.3390/e27111129 - 31 Oct 2025

Abstract

Diverse learning algorithms, optimization methods, and natural selection share a common mathematical structure despite their apparent differences. Here, I show that a simple notational partitioning of change by the Price equation reveals a universal force–metric–bias (FMB) law: [...] Read more.

Diverse learning algorithms, optimization methods, and natural selection share a common mathematical structure despite their apparent differences. Here, I show that a simple notational partitioning of change by the Price equation reveals a universal force–metric–bias (FMB) law:

Δ θ = M f + b + ξ

. The force

f

drives improvement in parameters,

Δ θ

, in proportion to the slope of performance with respect to the parameters. The metric

M

rescales movement by inverse curvature. The bias

b

adds momentum or changes in the frame of reference. The noise

ξ

enables exploration. This framework unifies natural selection, Bayesian updating, Newton’s method, stochastic gradient descent, stochastic Langevin dynamics, Adam optimization, and most other algorithms as special cases of the same underlying process. The Price equation also reveals why Fisher information, Kullback–Leibler divergence, and d’Alembert’s principle arise naturally in learning dynamics. By exposing this common structure, the FMB law provides a principled foundation for understanding, comparing, and designing learning algorithms across disciplines. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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28 pages, 1080 KB

Open AccessArticle

Interactions Among Morphology, Word Order, and Syntactic Directionality: Evidence from 55 Languages

by Wenchao Li and Haitao Liu

Entropy 2025, 27(11), 1128; https://doi.org/10.3390/e27111128 - 31 Oct 2025

Abstract

This study investigates interactions among morphology, word order, and syntactic directionality across 55 languages from 11 families. We quantify morphological richness (moving-average mean size of paradigm), word order flexibility (entropy), and syntactic directionality (dependency direction), linking linguistic structure to information-theoretic principles. Analyses show [...] Read more.

This study investigates interactions among morphology, word order, and syntactic directionality across 55 languages from 11 families. We quantify morphological richness (moving-average mean size of paradigm), word order flexibility (entropy), and syntactic directionality (dependency direction), linking linguistic structure to information-theoretic principles. Analyses show that morphological richness is only weakly related to word order entropy and does not provide a robust predictor after statistical correction. Rich morphology facilitates the predictability of syntactic functions. Languages with richer morphology consistently favor head-final structures, whereas minimally inflected languages lean toward head-initial patterns, indicating that syntactic directionality is more closely associated with morphological complexity than with surface word order. Overall, the findings indicate that languages maintain a balance between redundancy and flexibility in optimizing information transmission, providing quantitative evidence for efficiency-driven trade-offs in human language. Full article

(This article belongs to the Section Complexity)

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24 pages, 406 KB

Open AccessArticle

Kicked General Fractional Lorenz-Type Equations: Exact Solutions and Multi-Dimensional Discrete Maps

by Vasily E. Tarasov

Entropy 2025, 27(11), 1127; https://doi.org/10.3390/e27111127 - 31 Oct 2025

Abstract

Lorenz-type systems are dissipative dynamical systems that are described by three nonlinear equations with derivatives of the first order and are capable of exhibiting chaotic behavior. The generalization of Lorenz-type equations by using general fractional derivatives (GFDs) and periodical kicks is proposed. GFDs [...] Read more.

Lorenz-type systems are dissipative dynamical systems that are described by three nonlinear equations with derivatives of the first order and are capable of exhibiting chaotic behavior. The generalization of Lorenz-type equations by using general fractional derivatives (GFDs) and periodical kicks is proposed. GFDs allow us to use the general form of memory functions as operator kernels to describe nonlinear dynamics with memory. The exact analytical solutions of Lorenz-type equations with GFDs are derived in the general case for the wide class of nonlinearity and memory functions. Using the exact solutions, we obtain discrete maps with memory (DMMs) that describe kicked GF Lorenz-type systems with general forms of memory and nonlinearity. The proposed maps describe the exact solution of nonlinear equations with GFDs at discrete time points as the function of all past discrete moments of time. The proposed multi-dimensional DMMs are derived from kicked GF Lorenz-type equations with GFDs without any approximations. The proposed results and the method to derive multi-dimensional DMMs are derived for arbitrary dimensions. The importance and unusualness of the proposed results lies in the fact that obtained solutions for equations of the Lorenz-type system are exact analytical solutions. Full article

(This article belongs to the Special Issue Dynamics Beyond the Hamiltonian: Dissipation in Classical Metriplectic Systems and Quantum Non-Unitary Systems)

20 pages, 934 KB

Open AccessArticle

Non-Uniform Entropy-Constrained L_∞ Quantization for Sparse and Irregular Sources

by Alin-Adrian Alecu, Mohammad Ali Tahouri, Adrian Munteanu and Bujor Păvăloiu

Entropy 2025, 27(11), 1126; https://doi.org/10.3390/e27111126 - 31 Oct 2025

Abstract

Near-lossless coding schemes traditionally rely on uniform quantization to control the maximum absolute error (

L_{\infty}

norm) of residual signals, often assuming a parametric model for the source distribution. This paper introduces a novel design framework for non-uniform, entropy-aware

L_{\infty}

-oriented [...] Read more.

Near-lossless coding schemes traditionally rely on uniform quantization to control the maximum absolute error (

L_{\infty}

norm) of residual signals, often assuming a parametric model for the source distribution. This paper introduces a novel design framework for non-uniform, entropy-aware

L_{\infty}

-oriented scalar quantizers that leverages a tight and differentiable approximation of the

L_{\infty}

distortion metric and does not require any parametric density function formulations. The framework is evaluated on both synthetic parametric sources and real-world medical depth map video datasets. For smoothly decaying distributions, such as the continuous Laplacian or discrete two-sided geometric distributions, the proposed method naturally converges to near-uniform quantizers, consistent with theoretical expectations. In contrast, for sparse or irregular sources, the algorithm produces highly non-uniform bin allocations that adapt to the local distribution structure and improve rate-distortion efficiency. When embedded in a residual-based near-lossless compression scheme, the resulting codec consistently outperforms versions equipped with uniform or piecewise-uniform quantizers, as well as state-of-the-art near-lossless schemes such as JPEG-LS and CALIC. Full article

(This article belongs to the Special Issue Information Theory and Data Compression)

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17 pages, 1390 KB

Open AccessArticle

Information Entropy of Biometric Data in a Recurrent Neural Network with Low Connectivity

by David Dominguez-Carreta, Mario González-Rodríguez, Francisco B. Rodriguez, Angel Sánchez and Rubem Erichsen, Jr.

Entropy 2025, 27(11), 1125; https://doi.org/10.3390/e27111125 - 31 Oct 2025

Abstract

In this paper, we explore the storage capacity and maximal information content of a random recurrent neural network characterized by a very low connectivity. A specific set of patterns is embedded into the network according to the Hebb prescription, a fundamental principle in [...] Read more.

In this paper, we explore the storage capacity and maximal information content of a random recurrent neural network characterized by a very low connectivity. A specific set of patterns is embedded into the network according to the Hebb prescription, a fundamental principle in neural learning. We thoroughly examine how various properties of the network, such as its connectivity and the level of synaptic noise, influence its performance and information retention capabilities, which is evaluated through an entropy measure. Our theoretical analyses are complemented by extensive simulations, and the results are validated through comparisons with the retrieval of real biometric patterns, including retinal vessel maps and fingerprints. This comprehensive approach provides deeper insights into the functionality and limitations of finite-connectivity neural networks and their applicability to the retrieval of complex, structured patterns. Full article

(This article belongs to the Special Issue Entropy Based Machine Learning Models)

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17 pages, 2838 KB

Open AccessArticle

Mediating Similarity: An Information-Theoretic Principle of Reference Behavior

by Qun Zhao, Menghui Yang, Guojian Xian, Jieying Bi and Tan Sun

Entropy 2025, 27(11), 1124; https://doi.org/10.3390/e27111124 - 31 Oct 2025

Abstract

While information theory is widely used to quantify knowledge combinations, the fundamental principles guiding reference selection in science remain largely unexplored. This study analyzes a large-scale journal citation network to introduce and empirically validate a principle we term “Mediating Similarity”. We posit that [...] Read more.

While information theory is widely used to quantify knowledge combinations, the fundamental principles guiding reference selection in science remain largely unexplored. This study analyzes a large-scale journal citation network to introduce and empirically validate a principle we term “Mediating Similarity”. We posit that a journal’s reference list acts as a strategic cognitive bridge, creating a more efficient informational path from its specific research identity to the broader scientific landscape. Using information-theoretic measures and computational experiments, we tested this principle and its underlying mechanisms. Our findings provide robust, multi-level evidence. First, we confirm the universality of the principle, showing that the mediated path through references is consistently more efficient than the direct path for thousands of journals. Second, perturbation experiments reveal a dual mechanism guiding reference selection: real-world reference portfolios are not merely collections of relevant works, but are synergistically optimal combinations that vastly outperform randomly assembled alternatives. This global optimization, however, operates as a robust “satisficing” strategy, balancing the search for an ideal cognitive path with the practical constraints of scientific discovery. Collectively, these findings reframe reference behavior as a strategic process of navigating a cognitive energy landscape, where journals selectively curate references to enhance their integrative capacity and innovative potential. Full article

(This article belongs to the Special Issue Aspects of Social Dynamics: Models and Concepts)

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29 pages, 7616 KB

Open AccessArticle

Dynamic Modeling and Analysis of Rotary Joints with Coupled Bearing Tilt-Misalignment Faults

by Jun Lu, Zixiang Zhu, Jie Ji, Yichao Yang, Xueyang Miao, Xiaoan Yan and Qinghua Liu

Entropy 2025, 27(11), 1123; https://doi.org/10.3390/e27111123 - 31 Oct 2025

Abstract

This study systematically analyzes the dynamic behavior of bearing tilt-misalignment coupling faults in rotary joints and establishes a high-fidelity nonlinear dynamic model for a dual-support bearing–rotor system. By integrating Hertzian contact theory, the nonlinear contact forces induced by the tilt of the inner/outer [...] Read more.

This study systematically analyzes the dynamic behavior of bearing tilt-misalignment coupling faults in rotary joints and establishes a high-fidelity nonlinear dynamic model for a dual-support bearing–rotor system. By integrating Hertzian contact theory, the nonlinear contact forces induced by the tilt of the inner/outer rings and axial misalignment are considered, and expressions for bearing forces incorporating time-varying stiffness and radial clearance are derived. The system’s vibration response is solved using the Newmark-β numerical integration method. This study reveals the influence of tilt angle and misalignment magnitude on contact forces, vibration patterns, and fault characteristic frequencies, demonstrating that the system exhibits multi-frequency harmonic characteristics under misalignment conditions, with vibration amplitudes increasing nonlinearly with the degree of misalignment. Furthermore, dynamic models for single-point faults (inner/outer ring) and composite faults are constructed, and Gaussian filtering technology is employed to simulate defect surface roughness, analyzing the modulation effects of faults on spectral characteristics. Experimental validation confirms that the theoretical model effectively captures actual vibration features, providing a theoretical foundation for health monitoring and intelligent diagnosis of rotary joints. Full article

(This article belongs to the Special Issue Entropy-Based Fault Diagnosis: From Theory to Applications)

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18 pages, 1486 KB

Open AccessArticle

A Deep Learning-Based Ensemble System for Brent and WTI Crude Oil Price Analysis and Prediction

by Yiwen Zhang and Salim Lahmiri

Entropy 2025, 27(11), 1122; https://doi.org/10.3390/e27111122 - 31 Oct 2025

Abstract

Crude oil price forecasting is an important task in energy management and storage. In this regard, deep learning has been applied in the literature to generate accurate forecasts. The main purpose of this study is to design an ensemble prediction system based on [...] Read more.

Crude oil price forecasting is an important task in energy management and storage. In this regard, deep learning has been applied in the literature to generate accurate forecasts. The main purpose of this study is to design an ensemble prediction system based on various deep learning systems. Specifically, in the first stage of our proposed ensemble system, convolutional neural networks (CNNs), long short-term memory networks (LSTMs), bidirectional LSTM (BiLSTM), gated recurrent units (GRUs), bidirectional GRU (BiGRU), and deep feedforward neural networks (DFFNNs) are used as individual predictive systems to predict crude oil prices. Their respective parameters are fine-tuned by Bayesian optimization (BO). In the second stage, forecasts from the previous stage are all weighted by using the sequential least squares programming (SLSQP) algorithm. The standard tree-based ensemble models, namely, extreme gradient boosting (XGBoost) and random forest (RT), are implemented as baseline models. The main findings can be summarized as follows. First, the proposed ensemble system outperforms the individual CNN, LSTM, BiLSTM, GRU, BiGRU, and DFFNN. Second, it outperforms the standard XGBoost and RT models. Governments and policymakers can use these models to design more effective energy policies and better manage supply in fluctuating markets. For investors, improved predictions of price trends present opportunities for strategic investments, reducing risk while maximizing returns in the energy market. Full article

(This article belongs to the Section Multidisciplinary Applications)

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20 pages, 918 KB

Open AccessArticle

MVIB-Lip: Multi-View Information Bottleneck for Visual Speech Recognition via Time Series Modeling

by Yuzhe Li, Haocheng Sun, Jiayi Cai and Jin Wu

Entropy 2025, 27(11), 1121; https://doi.org/10.3390/e27111121 - 31 Oct 2025

Abstract

Lipreading, or visual speech recognition, is the task of interpreting utterances solely from visual cues of lip movements. While early approaches relied on Hidden Markov Models (HMMs) and handcrafted spatiotemporal descriptors, recent advances in deep learning have enabled end-to-end recognition using large-scale datasets. [...] Read more.

Lipreading, or visual speech recognition, is the task of interpreting utterances solely from visual cues of lip movements. While early approaches relied on Hidden Markov Models (HMMs) and handcrafted spatiotemporal descriptors, recent advances in deep learning have enabled end-to-end recognition using large-scale datasets. However, such methods often require millions of labeled or pretraining samples and struggle to generalize under low-resource or speaker-independent conditions. In this work, we revisit lipreading from a multi-view learning perspective. We introduce MVIB-Lip, a framework that integrates two complementary representations of lip movements: (i) raw landmark trajectories modeled as multivariate time series, and (ii) recurrence plot (RP) images that encode structural dynamics in a texture form. A Transformer encoder processes the temporal sequences, while a ResNet-18 extracts features from RPs; the two views are fused via a product-of-experts posterior regularized by the multi-view information bottleneck. Experiments on the OuluVS and a self-collected dataset demonstrate that MVIB-Lip consistently outperforms handcrafted baselines and improves generalization to speaker-independent recognition. Our results suggest that recurrence plots, when coupled with deep multi-view learning, offer a principled and data-efficient path forward for robust visual speech recognition. Full article

(This article belongs to the Special Issue The Information Bottleneck Method: Theory and Applications)

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42 pages, 7728 KB

Open AccessArticle

Low-Carbon Economic Operation of Natural Gas Demand Side Integrating Dynamic Pricing Signals and User Behavior Modeling

by Ning Tian, Bilin Shao, Huibin Zeng, Xue Zhao and Wei Zhao

Entropy 2025, 27(11), 1120; https://doi.org/10.3390/e27111120 - 30 Oct 2025

Abstract

Natural gas plays a key role in the low-carbon energy transition due to its clean and efficient characteristics, yet challenges remain in balancing economic efficiency, user behavior, and carbon emission constraints in demand-side scheduling. This study proposes a low-carbon economic operation model for [...] Read more.

Natural gas plays a key role in the low-carbon energy transition due to its clean and efficient characteristics, yet challenges remain in balancing economic efficiency, user behavior, and carbon emission constraints in demand-side scheduling. This study proposes a low-carbon economic operation model for terminal natural gas systems, integrating price elasticity and differentiated user behavior with carbon emission management strategies. To capture diverse demand patterns, dynamic time warping k-medoids clustering is employed, while scheduling optimization is achieved through a multi-objective framework combining NSGA-III, the entropy weight (EW) method, and the VIKOR decision-making approach. Using real-world data from a gas station in Xi’an, simulation results show that the model reduces gas supply costs by 3.45% for residential users and 6.82% for non-residential users, increases user welfare by 4.64% and 88.87%, and decreases carbon emissions by 115.18 kg and 2156.8 kg, respectively. Moreover, non-residential users achieve an additional reduction in carbon trading costs of 183.85 CNY. The findings demonstrate the effectiveness of integrating dynamic price signals, user behavior modeling, and carbon constraints into a unified optimization framework, offering decision support for sustainable and flexible natural gas scheduling. Full article

(This article belongs to the Section Multidisciplinary Applications)

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22 pages, 2682 KB

Open AccessReview

Unitary Entities Are the True “Atoms”

by Chris Jeynes and Michael Charles Parker

Entropy 2025, 27(11), 1119; https://doi.org/10.3390/e27111119 - 30 Oct 2025

Abstract

Quantitative Geometrical Thermodynamics (QGT) exploits the entropic Lagrangian–Hamiltonian canonical equations of state as applied to entities obeying the holographic principle and exhibiting Shannon information, the creation of which measures the (validly defined) “entropic purpose” of the system. QGT provides a physical description for [...] Read more.

Quantitative Geometrical Thermodynamics (QGT) exploits the entropic Lagrangian–Hamiltonian canonical equations of state as applied to entities obeying the holographic principle and exhibiting Shannon information, the creation of which measures the (validly defined) “entropic purpose” of the system. QGT provides a physical description for what we might consider the true “atoms” of physical science and has also recently enabled a number of significant advances: accounting ab initio for the chirality of DNA and the stability of Buckminsterfullerene; the size of the alpha particle (and other nuclear entities) and the lifetime of the free neutron; and the shape, structure, and stability of the Milky Way galaxy. All these entities, ranging in size over more than 38 orders of magnitude, can each be considered to be an “atom”; in particular, the size of the alpha is calculated from QGT by assuming that the alpha is a “unitary entity” (that is, than which exists no simpler). The surprising conclusion is that clearly compound entities may also be physically treated as unitary (“uncuttable”) according to a principle of scale relativity, where a characteristic size for such an entity must be specified. Since QGT is entropic, and is therefore described using a logarithmic metric (involving hyperbolic space), it is not surprising that the length scale must be specified in order to account for unitary properties and for an entity to be appropriately considered an “atom”. The contribution to physics made by QGT is reviewed in the context of the related work of others. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics, 4th Edition)

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27 pages, 430 KB

Open AccessArticle

A Monte Carlo-Based Framework for Two-Stage Stochastic Programming: Application to Bond Portfolio Optimization

by Hissah Albaqami, Mehdi Mrad, Anis Gharbi and Munevver Mine Subasi

Entropy 2025, 27(11), 1118; https://doi.org/10.3390/e27111118 - 30 Oct 2025

Abstract

This paper presents a Monte Carlo simulation-based approach for solving stochastic two-stage bond portfolio optimization problems. The main objective is to optimize the cost of the bond portfolio while making decisions on bond purchases, holdings, and sales under random market conditions such as [...] Read more.

This paper presents a Monte Carlo simulation-based approach for solving stochastic two-stage bond portfolio optimization problems. The main objective is to optimize the cost of the bond portfolio while making decisions on bond purchases, holdings, and sales under random market conditions such as interest rate fluctuations and liabilities. The proposed algorithm identifies the number of randomly generated scenarios required to convert the stochastic problem into a deterministic one, subsequently solving it as a Mixed-Integer Linear Program. The practical relevance of this research is shown through an application of the proposed method to a real-world bond market. The results indicate that the proposed approach successfully minimizes costs and meets liabilities, providing a robust solution for bond portfolio optimization. Full article

(This article belongs to the Section Multidisciplinary Applications)

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37 pages, 25662 KB

Open AccessArticle

A Hyperspectral Remote Sensing Image Encryption Algorithm Based on a Novel Two-Dimensional Hyperchaotic Map

by Zongyue Bai, Qingzhan Zhao, Wenzhong Tian, Xuewen Wang, Jingyang Li and Yuzhen Wu

Entropy 2025, 27(11), 1117; https://doi.org/10.3390/e27111117 - 30 Oct 2025

Abstract

With the rapid advancement of hyperspectral remote sensing technology, the security of hyperspectral images (HSIs) has become a critical concern. However, traditional image encryption methods—designed primarily for grayscale or RGB images—fail to address the high dimensionality, large data volume, and spectral-domain characteristics inherent [...] Read more.

With the rapid advancement of hyperspectral remote sensing technology, the security of hyperspectral images (HSIs) has become a critical concern. However, traditional image encryption methods—designed primarily for grayscale or RGB images—fail to address the high dimensionality, large data volume, and spectral-domain characteristics inherent to HSIs. Existing chaotic encryption schemes often suffer from limited chaotic performance, narrow parameter ranges, and inadequate spectral protection, leaving HSIs vulnerable to spectral feature extraction and statistical attacks. To overcome these limitations, this paper proposes a novel hyperspectral image encryption algorithm based on a newly designed two-dimensional cross-coupled hyperchaotic map (2D-CSCM), which synergistically integrates Cubic, Sinusoidal, and Chebyshev maps. The 2D-CSCM exhibits superior hyperchaotic behavior, including a wider hyperchaotic parameter range, enhanced randomness, and higher complexity, as validated by Lyapunov exponents, sample entropy, and NIST tests. Building on this, a layered encryption framework is introduced: spectral-band scrambling to conceal spectral curves while preserving spatial structure, spatial pixel permutation to disrupt correlation, and a bit-level diffusion mechanism based on dynamic DNA encoding, specifically designed to secure high bit-depth digital number (DN) values (typically >8 bits). Experimental results on multiple HSI datasets demonstrate that the proposed algorithm achieves near-ideal information entropy (up to 15.8107 for 16-bit data), negligible adjacent-pixel correlation (below 0.01), and strong resistance to statistical, cropping, and differential attacks (NPCR ≈ 99.998%, UACI ≈ 33.30%). The algorithm not only ensures comprehensive encryption of both spectral and spatial information but also supports lossless decryption, offering a robust and practical solution for secure storage and transmission of hyperspectral remote sensing imagery. Full article

(This article belongs to the Section Signal and Data Analysis)

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18 pages, 2390 KB

Open AccessFeature PaperArticle

Thermal BCS-BEC Crossovers in Finite Systems

by Angelo Plastino, Flavia Pennini and Victor Apel

Entropy 2025, 27(11), 1116; https://doi.org/10.3390/e27111116 - 30 Oct 2025

Abstract

We investigate the thermal evolution of fermionic pairings in a finite-size SU(2) × SU(2) complex model, drawing an analogy to the BCS-BEC crossover in interacting quantum gases. Unlike the conventional crossover, which is driven by tuning the interaction strength, our study [...] Read more.

We investigate the thermal evolution of fermionic pairings in a finite-size SU(2) × SU(2) complex model, drawing an analogy to the BCS-BEC crossover in interacting quantum gases. Unlike the conventional crossover, which is driven by tuning the interaction strength, our study suggests that temperature alone can induce a smooth transition from weakly bound Cooper pairs (BCS-like state) to tightly bound dimers (BEC-like state). Using an exactly solvable model with a finite number of fermions, we analyze the structure of eigenstates, pairing correlations, and thermodynamic response functions. We demonstrate that different multiplet structures, characterized by distinct quasi-spin quantum numbers, become thermally accessible, effectively mimicking the crossover behavior seen in ultracold Fermi gases. Our results provide new insights into the role of thermal fluctuations in quantum pairing phenomena and suggest alternative routes for exploring crossover physics in mesoscopic and strongly correlated systems. Full article

(This article belongs to the Section Statistical Physics)

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