Entropy

12 pages, 478 KB

Open AccessArticle

The Amount of Data Required to Recognize a Writer’s Style Is Consistent Across Different Languages of the World

by Boris Ryabko, Nadezhda Savina, Yeshewas Getachew Lulu and Yunfei Han

Entropy 2025, 27(10), 1039; https://doi.org/10.3390/e27101039 - 4 Oct 2025

In this paper, we apply an information-theoretic method proposed by Ryabko and Savina (therefore called the RS-method), based on the use of data compression, to recognize the individual author’s style of a writer across four languages from different language groups and families. In [...] Read more.

In this paper, we apply an information-theoretic method proposed by Ryabko and Savina (therefore called the RS-method), based on the use of data compression, to recognize the individual author’s style of a writer across four languages from different language groups and families. In this paper, the presented method was used to study fiction texts in Russian (East Slavic group of languages of the Indo-European language family), Amharic (South Ethiosemitic group of the Semitic language family), Chinese (Sinitic group of the Sino-Tibetan language family) and English (West Germanic language group of the Indo-European language family). It was found that the amount of data necessary for recognizing an author’s style is almost the same for all four languages, i.e., the amount of data is invariant across different language groups. The results obtained are of interest to computer science, literary studies, linguistics and, in particular, computational linguistics. Full article

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

15 pages, 3708 KB

Open AccessArticle

Multiscale Permutation Time Irreversibility Analysis of MEG in Patients with Schizophrenia

by Dengxuan Bai, Muxuan Xue, Yining Wang, Zhen Zhang, Xiaoli Chen, Wenpo Yao and Jun Wang

Entropy 2025, 27(10), 1038; https://doi.org/10.3390/e27101038 - 4 Oct 2025

Abstract

The use of questionnaire survey results as a clinical diagnostic method for schizophrenia lacks a certain degree of objectivity; thus, markers of schizophrenia in different brain signals have been widely investigated. The objective of this investigation was to explore potential markers of schizophrenia [...] Read more.

The use of questionnaire survey results as a clinical diagnostic method for schizophrenia lacks a certain degree of objectivity; thus, markers of schizophrenia in different brain signals have been widely investigated. The objective of this investigation was to explore potential markers of schizophrenia by investigating nonequilibrium features in magnetoencephalography (MEG) signals. We propose a new method to quantify the nonequilibrium features of MEG signals: the multiscale permutation time irreversibility (MsPTIRR) index. The results revealed that the MsPTIRR indices of the MEG recordings of patients with schizophrenia were significantly lower than those of the healthy controls (HCs). Moreover, the MsPTIRR indices of the MEG recordings of patients with schizophrenia and HCs differed significantly in the frontal, occipital, and temporal lobe regions. Furthermore, the MsPTIRR indices of the MEG recordings differed significantly between patients with schizophrenia and HCs in the

θ

,

α

and

β

bands. Abnormal nonequilibrium features mined in MEG recordings using the MsPTIRR index may be used as potential markers for schizophrenia, assisting in the clinical diagnosis of this disorder. Full article

(This article belongs to the Section Entropy and Biology)

15 pages, 1341 KB

Open AccessArticle

The Wave–Particle Dualism of Photons as Seen from an Informational Point of View

by J. Gerhard Müller

Entropy 2025, 27(10), 1037; https://doi.org/10.3390/e27101037 - 3 Oct 2025

Abstract

This paper deals with J. A. Wheeler’s proposal that each piece of reality owes its existence to observation—an approach to physics, which implies that all physical entities at their bottom are informational in character. Focusing on the double-slit experiment with photons, which is [...] Read more.

This paper deals with J. A. Wheeler’s proposal that each piece of reality owes its existence to observation—an approach to physics, which implies that all physical entities at their bottom are informational in character. Focusing on the double-slit experiment with photons, which is the key evidence for the wave–particle dualism of photons, this paper follows Wheeler’s observational approach and interprets this experiment as a question posed to nature. Considering how the enquiry regarding the wave–particle duality of photons is answered by nature, it is shown that experimental questions are being answered by nature in the form of spatiotemporal patterns of elementary observations (EOs) which are binary pieces of information, produced by the dissipation of energy. Working through this line of thought, Wheeler’s statements of “binary information gain”, “observer participance” and the “impossibility of continuum idealizations of physical laws” are elucidated and connections to the Landauer Principle are made. Full article

(This article belongs to the Special Issue The Landauer Principle in Physics, Biophysics, Engineering and Computer Science: From Foundations of Thermodynamics to Computer Engineering)

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14 pages, 471 KB

Open AccessArticle

A Dissipative Phenomenon: The Mechanical Model of the Cosmological Axion Influence

by Ferenc Márkus and Katalin Gambár

Entropy 2025, 27(10), 1036; https://doi.org/10.3390/e27101036 - 2 Oct 2025

Abstract

The appearance of a negative mass term in the classical, non-relativistic Klein–Gordon equation deduced from mechanical interactions describes a repulsive interaction. In the case of a traveling wave, this results in an increase in amplitude and a decrease in the wave propagation velocity. [...] Read more.

The appearance of a negative mass term in the classical, non-relativistic Klein–Gordon equation deduced from mechanical interactions describes a repulsive interaction. In the case of a traveling wave, this results in an increase in amplitude and a decrease in the wave propagation velocity. Since this leads to dissipation, it is a symmetry-breaking phenomenon. After the repulsive interaction is eliminated, the system evolves towards the original state. Given that the interactions within the system are conservative, it would be assumed that even the original state is restored. The analysis to be presented shows that a wave with a lower angular frequency than the original one is transformed back to a slightly larger amplitude. This description is a suitable model of the axion effect, during which an electromagnetic wave interacts with a repulsive field and becomes of a continuously lower frequency. Full article

(This article belongs to the Special Issue Dissipative Physical Dynamics)

16 pages, 959 KB

Open AccessArticle

Research on the Consensus Convergence Rate of Multi-Agent Systems Based on Hermitian Kirchhoff Index Measurement

by He Deng and Tingzeng Wu

Entropy 2025, 27(10), 1035; https://doi.org/10.3390/e27101035 - 2 Oct 2025

Abstract

Multi-agent systems (MAS) typically model interaction topologies using directed or undirected graphs when analyzing consensus convergence rates. However, as system complexity increases, purely directed or undirected networks may be insufficient to capture interaction heterogeneity. This paper adopts hybrid networks as interaction topology to [...] Read more.

Multi-agent systems (MAS) typically model interaction topologies using directed or undirected graphs when analyzing consensus convergence rates. However, as system complexity increases, purely directed or undirected networks may be insufficient to capture interaction heterogeneity. This paper adopts hybrid networks as interaction topology to investigate strategies for improving consensus convergence rates. We propose the Hermitian Kirchhoff index, a novel metric based on resistance distance, to quantify the consensus convergence rates and establish its theoretical justification. We then examine how adding or removing edges/arcs affects the Hermitian Kirchhoff index, employing first-order eigenvalue perturbation analysis to relate these changes to algebraic connectivity and its associated eigenvectors. Numerical simulations corroborate the theoretical findings and demonstrate the effectiveness of the proposed approach. Full article

(This article belongs to the Section Complexity)

16 pages, 1005 KB

Open AccessArticle

A Two-Step Machine Learning Approach Integrating GNSS-Derived PWV for Improved Precipitation Forecasting

by Laura Profetto, Andrea Antonini, Luca Fibbi, Alberto Ortolani and Giovanna Maria Dimitri

Entropy 2025, 27(10), 1034; https://doi.org/10.3390/e27101034 - 2 Oct 2025

Abstract

Global Navigation Satellite System (GNSS) meteorology has emerged as a valuable tool for atmospheric monitoring, providing high-resolution, near-real-time data that can significantly improve precipitation nowcasting. This study aims to enhance short-term precipitation forecasting by integrating GNSS-derived Precipitable Water Vapor (PWV)—a key indicator of [...] Read more.

Global Navigation Satellite System (GNSS) meteorology has emerged as a valuable tool for atmospheric monitoring, providing high-resolution, near-real-time data that can significantly improve precipitation nowcasting. This study aims to enhance short-term precipitation forecasting by integrating GNSS-derived Precipitable Water Vapor (PWV)—a key indicator of atmospheric moisture—with traditional meteorological observations. A novel two-step machine learning framework is proposed that combines a Random Forest (RF) model and a Long Short-Term Memory (LSTM) neural network. The RF model first estimates current precipitation based on PWV, surface weather parameters, and auxiliary atmospheric variables. Then, the LSTM network leverages temporal dependencies within the data to predict precipitation for the subsequent hour. This hybrid method capitalizes on the RF’s ability to model complex nonlinear relationships and the LSTM’s strength in handling time series data. The results demonstrate that the proposed approach improves forecasting accuracy, particularly during extreme weather events such as intense rainfall and thunderstorms, outperforming conventional models. By integrating GNSS meteorology with advanced machine learning techniques, this study offers a promising tool for meteorological services, early warning systems, and disaster risk management. The findings highlight the potential of GNSS-based nowcasting for real-time decision-making in weather-sensitive applications. Full article

(This article belongs to the Special Issue Entropy in Machine Learning Applications, 2nd Edition)

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36 pages, 714 KB

Open AccessArticle

Security, Privacy, and Linear Function Retrieval in Combinatorial Multi-Access Coded Caching with Private Caches

by Mallikharjuna Chinnapadamala and B. Sundar Rajan

Entropy 2025, 27(10), 1033; https://doi.org/10.3390/e27101033 - 1 Oct 2025

Abstract

We consider combinatorial multi-access coded caching with private caches, where users are connected to two types of caches: private caches and multi-access caches. Each user has its own private cache, while multi-access caches are connected in the same way as caches are connected [...] Read more.

We consider combinatorial multi-access coded caching with private caches, where users are connected to two types of caches: private caches and multi-access caches. Each user has its own private cache, while multi-access caches are connected in the same way as caches are connected in a combinatorial topology. A scheme is proposed that satisfies the following three requirements simultaneously: (a) Linear Function Retrieval (LFR), (b) content security against an eavesdropper, and (c) demand privacy against a colluding set of users. It is shown that the private caches included in this work enable the proposed scheme to provide privacy against colluding users. For the same rate, our scheme requires less total memory accessed by each user and less total system memory than the existing scheme for multi-access combinatorial topology (no private caches) in the literature. We derive a cut-set lower bound and prove optimality when

r \geq C - 1

. For

r < C - 1

, we show a constant gap of 5 under certain conditions. Finally, the proposed scheme is extended to a more general setup where different users are connected to different numbers of multi-access caches, and multiple users are connected to the same subset of multi-access caches. Full article

(This article belongs to the Special Issue Information and Coding Theory for Distributed Learning, Storage, Scheduling, and Security)

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15 pages, 479 KB

Open AccessArticle

Security of Quantum Key Distribution with One-Time-Pad-Protected Error Correction and Its Performance Benefits

by Roman Novak

Entropy 2025, 27(10), 1032; https://doi.org/10.3390/e27101032 - 1 Oct 2025

Abstract

In quantum key distribution (QKD), public discussion over the authenticated classical channel inevitably leaks information about the raw key to a potential adversary, which must later be mitigated by privacy amplification. To limit this leakage, a one-time pad (OTP) has been proposed to [...] Read more.

In quantum key distribution (QKD), public discussion over the authenticated classical channel inevitably leaks information about the raw key to a potential adversary, which must later be mitigated by privacy amplification. To limit this leakage, a one-time pad (OTP) has been proposed to protect message exchanges in various settings. Building on the security proof of Tomamichel and Leverrier, which is based on a non-asymptotic framework and considers the effects of finite resources, we extend the analysis to the OTP-protected scheme. We show that when the OTP key is drawn from the entropy pool of the same QKD session, the achievable quantum key rate is identical to that of the reference protocol with unprotected error-correction exchange. This equivalence holds for a fixed security level, defined via the diamond distance between the real and ideal protocols modeled as completely positive trace-preserving maps. At the same time, the proposed approach reduces the computational requirements: for non-interactive low-density parity-check codes, the encoding problem size is reduced by the square of the syndrome length, while privacy amplification requires less compression. The technique preserves security, avoids the use of QKD keys between sessions, and has the potential to improve performance. Full article

(This article belongs to the Section Quantum Information)

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12 pages, 290 KB

Open AccessArticle

Efficient Algorithms for Permutation Arrays from Permutation Polynomials

by Sergey Bereg, Brian Malouf, Linda Morales and Ivan Hal Sudborough

Entropy 2025, 27(10), 1031; https://doi.org/10.3390/e27101031 - 1 Oct 2025

Abstract

We develop algorithms for computing permutation polynomials (PPs) using normalization, so-called F-maps and G-maps, and the Hermite criterion. This allows for a more efficient computation of PPs for larger degrees and for larger finite fields. We use this to improve some lower bounds [...] Read more.

We develop algorithms for computing permutation polynomials (PPs) using normalization, so-called F-maps and G-maps, and the Hermite criterion. This allows for a more efficient computation of PPs for larger degrees and for larger finite fields. We use this to improve some lower bounds for

M (n, D)

, the maximum number of permutations on n symbols with a pairwise Hamming distance of D. Full article

(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)

12 pages, 844 KB

Open AccessArticle

Enhance the Performance of Expectation Propagation Detection in Spatially Correlated Massive MIMO Channels via DFT Precoding

by Huaicheng Luo, Jia Tang, Zeliang Ou, Yitong Liu and Hongwen Yang

Entropy 2025, 27(10), 1030; https://doi.org/10.3390/e27101030 - 1 Oct 2025

Abstract

Expectation Propagation (EP) has emerged as a promising detection algorithm for large-scale multiple-input multiple-output (MIMO) systems owing to its excellent performance and practical complexity. However, transmit antenna correlation significantly degrades the performance of EP detection, especially when the number of transmit and receive [...] Read more.

Expectation Propagation (EP) has emerged as a promising detection algorithm for large-scale multiple-input multiple-output (MIMO) systems owing to its excellent performance and practical complexity. However, transmit antenna correlation significantly degrades the performance of EP detection, especially when the number of transmit and receive antennas is equal and high-order modulation is adopted. Based on the fact that the eigenvector matrix of the channel transmit correlation matrix approaches asymptotically to a discrete Fourier transform (DFT) matrix, a DFT precoder is proposed to effectively eliminate transmit antenna correlation. Simulation results demonstrate that for high-order, high-dimensional massive MIMO systems with strong transmit antenna correlation, employing the proposed DFT precoding can significantly accelerate the convergence of the EP algorithm and reduce the detection error rate. Full article

(This article belongs to the Special Issue Next-Generation Multiple Access for Future Wireless Communications)

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26 pages, 4789 KB

Open AccessArticle

EMAT: Enhanced Multi-Aspect Attention Transformer for Financial Time Series Forecasting

by Yingjun Chen, Wenfeng Shen, Han Liu and Xiaolin Cao

Entropy 2025, 27(10), 1029; https://doi.org/10.3390/e27101029 - 1 Oct 2025

Abstract

Financial time series prediction remains a challenging task due to the inherent non-stationarity, noise, and complex temporal dependencies present in market data. Traditional forecasting methods often fail to capture the multifaceted nature of financial markets, where temporal proximity, trend dynamics, and volatility patterns [...] Read more.

Financial time series prediction remains a challenging task due to the inherent non-stationarity, noise, and complex temporal dependencies present in market data. Traditional forecasting methods often fail to capture the multifaceted nature of financial markets, where temporal proximity, trend dynamics, and volatility patterns simultaneously influence price movements. To address these limitations, this paper proposes the Enhanced Multi-Aspect Transformer (EMAT), a novel deep learning architecture specifically designed for stock market prediction. EMAT incorporates a Multi-Aspect Attention Mechanism that simultaneously captures temporal decay patterns, trend dynamics, and volatility regimes through specialized attention components. The model employs an encoder–decoder architecture with enhanced feed-forward networks utilizing SwiGLU activation, enabling superior modeling of complex non-linear relationships. Furthermore, we introduce a comprehensive multi-objective loss function that balances point-wise prediction accuracy with volatility consistency. Extensive experiments on multiple stock market datasets demonstrate that EMAT consistently outperforms a wide range of state-of-the-art baseline models, including various recurrent, hybrid, and Transformer architectures. Our ablation studies further validate the design, confirming that each component of the Multi-Aspect Attention Mechanism makes a critical and quantifiable contribution to the model’s predictive power. The proposed architecture’s ability to simultaneously model these distinct financial characteristics makes it a particularly effective and robust tool for financial forecasting, offering significant improvements in accuracy compared to existing approaches. Full article

(This article belongs to the Special Issue Entropy, Artificial Intelligence and the Financial Markets)

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17 pages, 3129 KB

Open AccessArticle

ADP-Based Fault-Tolerant Control with Stability Guarantee for Nonlinear Systems

by Luojia Liu, Junhong Lv, Haowei Lin, Ruidian Zhan and Liming Wu

Entropy 2025, 27(10), 1028; https://doi.org/10.3390/e27101028 - 1 Oct 2025

Abstract

This paper develops the stability-guaranteed adaptive dynamic programming (ADP)-based fault tolerant control (FTC) for nonlinear systems with an actuator fault. Firstly, a fault observer is designed to identify the unknown actuator fault. Then, a critic neural network (NN) is built to approximate the [...] Read more.

This paper develops the stability-guaranteed adaptive dynamic programming (ADP)-based fault tolerant control (FTC) for nonlinear systems with an actuator fault. Firstly, a fault observer is designed to identify the unknown actuator fault. Then, a critic neural network (NN) is built to approximate the optimal control of the nominal system. Meanwhile, a stability-aware weight update mechanism is proposed based on the Lyapunov stability theorem to relax the restriction of the initial admissible control on the system stability. By integrating the nominal optimal control and the fault estimation, the stability-guaranteed ADP-based FTC is developed to eliminate the influence of the actuator fault. Furthermore, the observer errors, critic NN weight estimation errors, and the closed-loop system are all shown to exhibit uniform ultimate boundedness by using Lyapunov’s direct method. Finally, simulation examples are given to demonstrate the validity of the proposed method. Full article

(This article belongs to the Section Complexity)

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12 pages, 1262 KB

Open AccessArticle

Ordinal Spectrum: Mapping Ordinal Patterns into Frequency Domain

by Mario Chavez and Johann H. Martínez

Entropy 2025, 27(10), 1027; https://doi.org/10.3390/e27101027 - 30 Sep 2025

Abstract

Classical spectral analysis characterizes linear systems effectively but often fails to reveal the nonlinear temporal structure of chaotic dynamics. We introduce the ordinal spectrum, a frequency-domain characterization derived from the ordinal-pattern representation of a time series. Applied to both synthetic and real-world [...] Read more.

Classical spectral analysis characterizes linear systems effectively but often fails to reveal the nonlinear temporal structure of chaotic dynamics. We introduce the ordinal spectrum, a frequency-domain characterization derived from the ordinal-pattern representation of a time series. Applied to both synthetic and real-world datasets—including periodic, stochastic, and chaotic signals from physical, biological, and astronomical sources—the ordinal spectrum identifies the temporal scales implied in a possible chaotic behavior. By providing an interpretable, data-driven view of symbolic dynamics in the frequency domain, this approach complements state–space reconstructions and enhances the detection of nonlinear temporal organization that classical spectra may obscure. Its ability to distinguish between qualitatively different dynamics make it a useful tool for exploring complex time series across diverse scientific domains. Full article

(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)

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20 pages, 1372 KB

Open AccessArticle

A Novel Multi-Scale Entropy Approach for EEG-Based Lie Detection with Channel Selection

by Jiawen Li, Guanyuan Feng, Chen Ling, Ximing Ren, Shuang Zhang, Xin Liu, Leijun Wang, Mang I. Vai, Jujian Lv and Rongjun Chen

Entropy 2025, 27(10), 1026; https://doi.org/10.3390/e27101026 - 29 Sep 2025

Abstract

Entropy-based analyses have emerged as a powerful tool for quantifying the complexity, regularity, and information content of complex biological signals, such as electroencephalography (EEG). In this regard, EEG-based lie detection offers the advantage of directly providing more objective and less susceptible-to-manipulation results compared [...] Read more.

Entropy-based analyses have emerged as a powerful tool for quantifying the complexity, regularity, and information content of complex biological signals, such as electroencephalography (EEG). In this regard, EEG-based lie detection offers the advantage of directly providing more objective and less susceptible-to-manipulation results compared to traditional polygraph methods. To this end, this study proposes a novel multi-scale entropy approach by fusing fuzzy entropy (FE), time-shifted multi-scale fuzzy entropy (TSMFE), and hierarchical multi-band fuzzy entropy (HMFE), which enables the multidimensional characterization of EEG signals. Subsequently, using machine learning classifiers, the fused feature vector is applied to lie detection, with a focus on channel selection to investigate distinguished neural signatures across brain regions. Experiments utilize a publicly benchmarked LieWaves dataset, and two parts are performed. One is a subject-dependent experiment to identify representative channels for lie detection. Another is a cross-subject experiment to assess the generalizability of the proposed approach. In the subject-dependent experiment, linear discriminant analysis (LDA) achieves impressive accuracies of 82.74% under leave-one-out cross-validation (LOOCV) and 82.00% under 10-fold cross-validation. The cross-subject experiment yields an accuracy of 64.07% using a radial basis function (RBF) kernel support vector machine (SVM) under leave-one-subject-out cross-validation (LOSOCV). Furthermore, regarding the channel selection results, PZ (parietal midline) and T7 (left temporal) are considered the representative channels for lie detection, as they exhibit the most prominent occurrences among subjects. These findings demonstrate that the PZ and T7 play vital roles in the cognitive processes associated with lying, offering a solution for designing portable EEG-based lie detection devices with fewer channels, which also provides insights into neural dynamics by analyzing variations in multi-scale entropy. Full article

(This article belongs to the Special Issue Entropy Analysis of Electrophysiological Signals)

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12 pages, 1627 KB

Open AccessArticle

RC-LDPC-Polar Codes for Information Reconciliation in Continuous-Variable Quantum Key Distribution

by Fei Hua, Kun Chen, Wei Deng, Jing Cheng, Banghong Guo and Huanwen Xie

Entropy 2025, 27(10), 1025; https://doi.org/10.3390/e27101025 - 29 Sep 2025

Abstract

Continuous-variable quantum key distribution faces significant challenges, including quantum channel instability, particularly fluctuations in the signal-to-noise ratio (SNR) and extremely low SNR scenarios. Furthermore, non-ideal polar codes, characterized by insufficient polarization in finite-length regimes, can lead to some sub-channels being neither completely noise-free [...] Read more.

Continuous-variable quantum key distribution faces significant challenges, including quantum channel instability, particularly fluctuations in the signal-to-noise ratio (SNR) and extremely low SNR scenarios. Furthermore, non-ideal polar codes, characterized by insufficient polarization in finite-length regimes, can lead to some sub-channels being neither completely noise-free nor fully noise-dominated. This phenomenon limits the error correction capability when such codes are applied to information reconciliation. To address these challenges, we propose a novel RC-LDPC-Polar code for the CV-QKD reconciliation algorithm. We combine the error resilience of LDPC codes with the efficiency advantages of polar coding. This scheme supports adaptive rate adjustment across varying SNR conditions. Our simulation experiments demonstrate that the RC-LDPC-Polar concatenated coding scheme achieves a lower error rate under varying SNR conditions. Meanwhile, the proposed scheme achieves a higher final key rate and a longer transmission distance. Full article

(This article belongs to the Section Quantum Information)

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14 pages, 3201 KB

Open AccessArticle

Corticomuscular Coupling Analysis in Archery Based on Transfer Entropy

by Yunrui Zhang, Yue Leng, Xiaozhi Li, Wenjing Zhang and Hairong Yu

Entropy 2025, 27(10), 1024; https://doi.org/10.3390/e27101024 - 28 Sep 2025

Abstract

Studying the information transfer between the brain and muscles during archery can help us to understand the underlying mechanisms of corticomuscular coupling during motor learning. In this study, we recruited 26 novice archers as participants and calculated the transfer entropy (TE) between their [...] Read more.

Studying the information transfer between the brain and muscles during archery can help us to understand the underlying mechanisms of corticomuscular coupling during motor learning. In this study, we recruited 26 novice archers as participants and calculated the transfer entropy (TE) between their EEG and EMG signals during the archery process. This was performed to assess the characteristics of corticomuscular coupling during archery and the impact of a period of archery training on this coupling. The results indicate that information transfer from EEG to EMG in the

α

and

β

frequency bands predominates during archery, which may be related to the roles of

α

and

β

frequency bands in inhibitory control and the sustained contraction of muscle stability. Additionally, the optimization of brain resource allocation resulting from a period of archery training is primarily reflected in the prefrontal cortex and motor cortex, where the information transfer from EEG to EMG decreases while activation related to inhibitory control increases. The intensity of corticomuscular coupling weakens with an increase in the number of arrows shot, but archery training reduces the impact of fatigue-induced changes on corticomuscular coupling. Full article

(This article belongs to the Section Entropy and Biology)

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25 pages, 9804 KB

Open AccessArticle

GLFNet: Attention Mechanism-Based Global–Local Feature Fusion Network for Micro-Expression Recognition

by Meng Zhang, Long Yao, Wenzhong Yang and Yabo Yin

Entropy 2025, 27(10), 1023; https://doi.org/10.3390/e27101023 - 28 Sep 2025

Abstract

Micro-expressions are extremely subtle and short-lived facial muscle movements that often reveal an individual’s genuine emotions. However, micro-expression recognition (MER) remains highly challenging due to its short duration, low motion intensity, and the imbalanced distribution of training samples. To address these issues, this [...] Read more.

Micro-expressions are extremely subtle and short-lived facial muscle movements that often reveal an individual’s genuine emotions. However, micro-expression recognition (MER) remains highly challenging due to its short duration, low motion intensity, and the imbalanced distribution of training samples. To address these issues, this paper proposes a Global–Local Feature Fusion Network (GLFNet) to effectively extract discriminative features for MER. Specifically, GLFNet consists of three core modules: the Global Attention (LA) module, which captures subtle variations across the entire facial region; the Local Block (GB) module, which partitions the feature map into four non-overlapping regions to emphasize salient local movements while suppressing irrelevant information; and the Adaptive Feature Fusion (AFF) module, which employs an attention mechanism to dynamically adjust channel-wise weights for efficient global–local feature integration. In addition, a class-balanced loss function is introduced to replace the conventional cross-entropy loss, mitigating the common issue of class imbalance in micro-expression datasets. Extensive experiments are conducted on three benchmark databases, SMIC, CASME II, and SAMM, under two evaluation protocols. The experimental results demonstrate that under the Composite Database Evaluation protocol, GLFNet consistently outperforms existing state-of-the-art methods in overall performance. Specifically, the unweighted F1-scores on the Combined, SAMM, CASME II, and SMIC datasets are improved by 2.49%, 2.02%, 0.49%, and 4.67%, respectively, compared to the current best methods. These results strongly validate the effectiveness and superiority of the proposed global–local feature fusion strategy in micro-expression recognition tasks. Full article

(This article belongs to the Special Issue Application of Information Theory to Computer Vision and Image Processing, 3rd Edition)

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34 pages, 5443 KB

Open AccessArticle

Quantum and Topological Dynamics of GKSL Equation in Camel-like Framework

by Sergio Manzetti and Andrei Khrennikov

Entropy 2025, 27(10), 1022; https://doi.org/10.3390/e27101022 - 28 Sep 2025

Abstract

We study the dynamics of von Neumann entropy driven by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation, focusing on its camel-like behavior—a hump-like entropy evolution reflecting the system’s adaptation to its environment. Within this framework, we analyze quantum correlations under decoherence and environmental interaction for three [...] Read more.

We study the dynamics of von Neumann entropy driven by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation, focusing on its camel-like behavior—a hump-like entropy evolution reflecting the system’s adaptation to its environment. Within this framework, we analyze quantum correlations under decoherence and environmental interaction for three sets of quantum states. Our results show that the sign of the entanglement entropy’s derivative serves as an indicator of the system’s drift toward either classical or quantum information exchange—an insight relevant to quantum error correction and dissipation in quantum thermal machines. We parameterize quantum states using both single-parameter and Bloch-sphere representations, where the angle

θ

on the Bloch sphere corresponds to the state’s position. On this sphere, we construct gradient and basin maps that partition the dynamics of quantum states into stable and unstable regions under decoherence. Notably, we identify a Braiding ring of decoherence-unstable states located at

θ = \frac{3 π}{4}

; these states act as attractors under a constructed Lyapunov function, illustrating the topological and dynamical complexity of quantum evolution. Finally, we propose a testable experimental setup based on camel-like entropy and discuss its connection to the theoretical framework of this entropy behavior. Full article

(This article belongs to the Special Issue Entanglement Entropy in Quantum Field Theory)

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34 pages, 9259 KB

Open AccessArticle

Dynamic Evolution and Convergence of the Coupled and Coordinated Development of Urban–Rural Basic Education in China

by Fangyu Ju, Qijin Li and Zhiyong Chen

Entropy 2025, 27(10), 1021; https://doi.org/10.3390/e27101021 - 28 Sep 2025

Abstract

Understanding the coupled and coordinated development of China’s urban and rural basic education systems is crucial for fostering their interaction and synergistic growth. Using China’s provincial panel data from 2011 to 2023, this study measures the coupled and coordinated development level of urban–rural [...] Read more.

Understanding the coupled and coordinated development of China’s urban and rural basic education systems is crucial for fostering their interaction and synergistic growth. Using China’s provincial panel data from 2011 to 2023, this study measures the coupled and coordinated development level of urban–rural basic education (CCD-URBE) via the entropy weight method, G1-method and coupling coordination degree model. On this basis, the Dagum Gini coefficient decomposition method, traditional and spatial Markov chain models, as well as convergence test models are employed for empirical research. The results show that: (1) During the study period, the CCD-URBE across the nation and the four major regions improves significantly. Both intra-regional and inter-regional disparities show a consistent downward trend. Inter-regional disparities are the main source of the overall disparities, and the contribution rate of transvariation density to the overall disparities exhibits the most significant increase. (2) The CCD-URBE demonstrates strong stability, as most regions tend to maintain their original CCD-URBE grades. Meanwhile, neighborhood grades moderate the local transition probability significantly. Neighborhoods with high CCD-URBE promote the upward improvement of the local CCD-URBE, while those with low CCD-URBE inhibit it. (3) The CCD-URBE across the nation and the four major regions shows obvious trends of σ-convergence, absolute β-convergence, and conditional β-convergence. The central region, which has lower CCD-URBE, exhibits higher convergence speed. Based on these findings, targeted policy implications are derived. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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