Entropy

44 pages, 1577 KB

Open AccessArticle

The Capacity Gains of Gaussian Channels with Unstable Versus Stable Autoregressive Noise

by Charalambos D. Charalambous, Christos Kourtellaris, Stelios Louka and Sergey Loyka

Entropy 2025, 27(12), 1264; https://doi.org/10.3390/e27121264 - 18 Dec 2025

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In this paper, we consider Cover’s and Pombra’s formulation of feedback capacity of additive Gaussian noise (AGN) channels, with jointly Gaussian nonstationary and nonergodic noise. We derive closed-form feedback capacity formulas, using Karush–Kuhn–Tucker (KKT) conditions and convergence properties of difference Riccati equations to [...] Read more.

In this paper, we consider Cover’s and Pombra’s formulation of feedback capacity of additive Gaussian noise (AGN) channels, with jointly Gaussian nonstationary and nonergodic noise. We derive closed-form feedback capacity formulas, using Karush–Kuhn–Tucker (KKT) conditions and convergence properties of difference Riccati equations to limiting algebraic Riccati equations of filtering theory, for unstable and stable autoregressive (AR) noise. Surprisingly, the capacity formulas depend on the parameters of the AR noise, its pole

c \in (- \infty, \infty)

and noise variance

K_{W} \in (0, \infty)

, and the total transmit power

κ \in [0, \infty)

, indicating substantial gains for the unstable noise region

\forall c^{2} \in (1, \infty), \forall κ > κ_{m i n} \overset{▵}{=} \frac{K_{W} (1 + \sqrt{4 c^{2} - 3})}{2 {(c^{2} - 1)}^{2}}

compared to its complement region. In particular, feedback capacity is distinguished by three regimes, as follows. Regime 1,

\forall c^{2} \in (1, \infty), \forall κ > κ_{m i n}

: the optimal channel input includes an innovations part, the capacity increases as

| c | > 1

increases, while

κ_{m i n}

and the allocated transmit power decrease. Regime 2,

\forall c^{2} \in (1, \infty), \forall κ \leq κ_{m i n}

, Regime 3,

\forall c \in [- 1, 1], \forall κ \in [0, \infty)

(complement of Regime 1): the innovations part of the optimal channel is asymptotically zero and the capacity is fundamentally different compared to Regime 1. The differences of capacity formulas for Regimes 1, 2 and 3 are directly related to their operational meaning: (i) Regime 1 is an ergodic capacity while Regimes 2 and 3 are nonergodic capacities; (ii) Regime 1 is achieved by an asymptotically stationary channel input with a non-zero innovations part, while Regimes 2 and 3 are achieved by an asymptotically zero innovations part. The gains of capacity for Regime 1 are attributed to the high correlation of noise samples compared to stable noise and the use of an informative innovations part by the optimal channel input, which make possible the prediction of future noise samples from past samples, unlike memoryless noise. Our results provide answers to certain open questions regarding the validity of capacity formulas of stable noise that appeared in the literature. Full article

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

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31 pages, 4844 KB

Open AccessArticle

GAME-YOLO: Global Attention and Multi-Scale Enhancement for Low-Visibility UAV Detection with Sub-Pixel Localization

by Ruohai Di, Hao Fan, Yuanzheng Ma, Jinqiang Wang and Ruoyu Qian

Entropy 2025, 27(12), 1263; https://doi.org/10.3390/e27121263 - 18 Dec 2025

Viewed by 358

Abstract

Detecting low-altitude, slow-speed, small (LSS) UAVs is especially challenging in low-visibility scenes (low light, haze, motion blur), where inherent uncertainties in sensor data and object appearance dominate. We propose GAME-YOLO, a novel detector that integrates a Bayesian-inspired probabilistic reasoning framework with Global Attention [...] Read more.

Detecting low-altitude, slow-speed, small (LSS) UAVs is especially challenging in low-visibility scenes (low light, haze, motion blur), where inherent uncertainties in sensor data and object appearance dominate. We propose GAME-YOLO, a novel detector that integrates a Bayesian-inspired probabilistic reasoning framework with Global Attention and Multi-Scale Enhancement to improve small-object perception and sub-pixel-level localization. Built on YOLOv11, our framework comprises: (i) a visibility restoration front-end that probabilistically infers and enhances latent image clarity; (ii) a global-attention-augmented backbone that performs context-aware feature selection; (iii) an adaptive multi-scale fusion neck that dynamically weights feature contributions; (iv) a sub-pixel-aware small-object detection head (SOH) that leverages high-resolution feature grids to model sub-pixel offsets; and (v) a novel Shape-Aware IoU loss combined with focal loss. Extensive experiments on the LSS2025-DET dataset demonstrate that GAME-YOLO achieves state-of-the-art performance, with an AP@50 of 52.0% and AP@[0.50:0.95] of 32.0%, significantly outperforming strong baselines such as LEAF-YOLO (48.3% AP@50) and YOLOv11 (36.2% AP@50). The model maintains high efficiency, operating at 48 FPS with only 7.6 M parameters and 19.6 GFLOPs. Ablation studies confirm the complementary gains from our probabilistic design choices, including a +10.5 pp improvement in AP@50 over the baseline. Cross-dataset evaluation on VisDrone-DET2021 further validates its generalization capability, achieving 39.2% AP@50. These results indicate that GAME-YOLO offers a practical and reliable solution for vision-based UAV surveillance by effectively marrying the efficiency of deterministic detectors with the robustness principles of Bayesian inference. Full article

(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)

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15 pages, 3214 KB

Open AccessArticle

Transfer Irreversibilities in the Lenoir Cycle: FTT Design Criteria with

ε - NTU

by Ricardo T. Páez-Hernández, Juan Carlos Pacheco-Paez, Juan Carlos Chimal-Eguía, Delfino Ladino-Luna and Javier Contreras-Sánchez

Entropy 2025, 27(12), 1262; https://doi.org/10.3390/e27121262 - 18 Dec 2025

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Abstract

This work extends the steady flow Lenoir cycle within finite-time thermodynamics (FTT) by incorporating heat transfer irreversibilities through the

ε - N T U

formalism and a non-isentropic expansion modeled via the expander isentropic efficiency

η_{E}

. The total conductance

U_{T}

[...] Read more.

This work extends the steady flow Lenoir cycle within finite-time thermodynamics (FTT) by incorporating heat transfer irreversibilities through the

ε - N T U

formalism and a non-isentropic expansion modeled via the expander isentropic efficiency

η_{E}

. The total conductance

U_{T}

(sum for the two heat exchangers) is partitioned between hot and cold units using

u_{L} = U_{L} / U_{T}

, with

U_{T} = U_{H} + U_{L}

. For each triplet (

τ = T_{H} / T_{L}

,

U_{L}

,

U_{T}

), we closed the cycle by determining

T_{1}

, the working fluid temperature at the cooler outlet and heater inlet,

T_{2}

, the heater outlet and expander inlet, and

T_{3}

, the expander outlet and cooler inlet. Using these states, we compute the heat rates

{\dot{Q}}_{12}

,

{\dot{Q}}_{31}

and the net power

P

. In addition to the thermal efficiency

η

, the following extended objective functions are evaluated: the efficient power

E_{F}

, the ecological efficiency

ϕ

, and the second law efficiency

η_{I I}

. Parametric sweeps on

u_{L}

for

τ ϵ \{3.25, 3.75\}

and

U_{T} ϵ \{2.5,5.0, 7.5,10\} k W

show unimodal curves for P(

u_{L}

) and maxima. A robust result places the optima of

P

,

η

,

E_{F}

,

ϕ

, and

η_{I I}

in a distribution band at

u_{L} ~ 0.6

. This guideline offers clear design guidance for allocating exchange area in heat recovery and microgeneration, maximizing power, high η, and exergetic utilization with contained entropic penalty. Full article

(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)

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15 pages, 272 KB

Open AccessArticle

Comprehension as Purification in Reading

by Miho Fuyama

Entropy 2025, 27(12), 1261; https://doi.org/10.3390/e27121261 - 17 Dec 2025

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Abstract

When reading a novel or poem, readers sometimes gain comprehension or experiences that cannot be expressed in language yet are felt as holistic. Previous studies focused on the linguistically expressible aspects of text comprehension. In this study, we propose a new hypothesis, the [...] Read more.

When reading a novel or poem, readers sometimes gain comprehension or experiences that cannot be expressed in language yet are felt as holistic. Previous studies focused on the linguistically expressible aspects of text comprehension. In this study, we propose a new hypothesis, the purification comprehension hypothesis, that seeks to explain how a reader constructs indescribable and coherent comprehension using quantum probability theory. This hypothesis regards the reading process as purification, in which the reader’s initial interpretation state is mixed, and the reader incorporates external systems, such as the interpretation of other parts of the text or prior knowledge, to purify their state. Therefore, the dimensionality of the state increases and von Neumann entropy decreases through purification. We also highlight two types of reading based on this hypothesis: purification and deterministic. Our model contributes to studies on reading by bridging humanities and scientific studies, provides implications for cognition models that aim to minimize Shannon entropy, and has the potential to apply cognition related to other modalities and media, such as music and art. Full article

(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering IV)

17 pages, 301 KB

Open AccessArticle

Entropy and Variational Formulation of Relativistic Fluid Dynamics

by Asher Yahalom

Entropy 2025, 27(12), 1260; https://doi.org/10.3390/e27121260 - 16 Dec 2025

Viewed by 269

Abstract

In this work, the earlier variational analysis of classical non-barotropic flows is extended to the special relativistic non-barotropic case. Specifically, we present a new Eulerian variational formulation for relativistic non-barotropic flows, based on six functions. This allows the canonical derivation of the energy–momentum [...] Read more.

In this work, the earlier variational analysis of classical non-barotropic flows is extended to the special relativistic non-barotropic case. Specifically, we present a new Eulerian variational formulation for relativistic non-barotropic flows, based on six functions. This allows the canonical derivation of the energy–momentum tensor. Full article

(This article belongs to the Special Issue Entropy in Relativistic Fluid Dynamics and Magnetohydrodynamics and Its Implications to the Cosmological Dark Energy and Dark Matter Problems)

27 pages, 5343 KB

Open AccessArticle

A Multi-Feature Fusion-Based Two-Stage Method for Airport Crater Extraction from Remote Sensing Images

by Yalun Zhao, Derong Chen and Jiulu Gong

Entropy 2025, 27(12), 1259; https://doi.org/10.3390/e27121259 - 16 Dec 2025

Viewed by 198

Abstract

The accurate extraction of damage information around airport runways is crucial for the rapid development of subsequent damage effect assessment work and the timely formulation of the ensuing operational plan. However, the presence of dark interference areas such as trees and shadows in [...] Read more.

The accurate extraction of damage information around airport runways is crucial for the rapid development of subsequent damage effect assessment work and the timely formulation of the ensuing operational plan. However, the presence of dark interference areas such as trees and shadows in the background, as well as the increased irregularity at the edge of the crater due to the proximity to the crater, pose challenges to the accurate extraction of the crater area in high entropy images. In this paper, we present a multi-feature fusion-based two-stage method for airport crater extraction from remote sensing images. In stage I, we designed an edge arc segment grouping and matching strategy based on the shape characteristics of craters for preliminary detection. In stage II, we established a crater model based on the regional distribution characteristics of craters and used the marked point processing method for crater detection. In addition, during the step of calculating the magnitude of the edge gradient, we proposed a near-region search strategy, which enhanced the ability of the proposed method to accurately extract craters with irregular shapes. In the test images, the proposed method accurately extracts craters located around and within the runways. Among them, the average recall R and precision P of the proposed method for extracting all craters around the airport runways reached 89% and 87%, respectively, and the average recall R and precision P of the proposed method for extracting craters inside the runways reached 94% and 92%, respectively. Meanwhile, the results of comparative tests showed that our method outperformed other representative algorithms in terms of both crater extraction recall and extraction precision. 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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12 pages, 571 KB

Open AccessArticle

Partition Function Zeros of the Spin-One Ising Model on the Honeycomb Lattice in the Complex Temperature Plane

by Seung-Yeon Kim

Entropy 2025, 27(12), 1258; https://doi.org/10.3390/e27121258 - 15 Dec 2025

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Abstract

The spin-one Ising model on the honeycomb lattice has never been solved exactly in spite of its simplicity. Even its exact critical temperature is not known. The exact integer values for the density of states of the spin-one Ising model on the [...] Read more.

The spin-one Ising model on the honeycomb lattice has never been solved exactly in spite of its simplicity. Even its exact critical temperature is not known. The exact integer values for the density of states of the spin-one Ising model on the

L \times 2 L

honeycomb lattice are enumerated up to

L = 14

. The partition function zeros in the complex temperature plane of the spin-one Ising model on the

L \times 2 L

honeycomb lattice are exactly obtained, using the density of states. The properties of the partition function zeros in the complex temperature plane are related to the behaviors of various thermodynamic functions, in particular, their singular behaviors. The unknown properties of the spin-one Ising model on the honeycomb lattice are investigated, based on its partition function zeros in the complex temperature plane. Full article

(This article belongs to the Special Issue Ising Model—100 Years Old and Still Attractive)

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25 pages, 992 KB

Open AccessPerspective

Towards Pragmatist Thermodynamics: An Essay on the Natural Philosophy of Entropy and Sustainability

by Carsten Herrmann-Pillath

Entropy 2025, 27(12), 1257; https://doi.org/10.3390/e27121257 - 15 Dec 2025

Viewed by 351

Abstract

Classical thermodynamics (CT) has become integrated into everyday life, especially through its applications in engineering. In contrast, out-of-equilibrium thermodynamics (OET) is often viewed as a fundamental science that seems distant from daily experiences. While “energy” is a familiar term in households, “entropy,” which [...] Read more.

Classical thermodynamics (CT) has become integrated into everyday life, especially through its applications in engineering. In contrast, out-of-equilibrium thermodynamics (OET) is often viewed as a fundamental science that seems distant from daily experiences. While “energy” is a familiar term in households, “entropy,” which refers to degraded energy, remains enigmatic. This gap in understanding has significant implications for developing effective sustainability practices. CT typically emphasizes the efficiency of individual systems that produce work, often overlooking the entropy production that occurs within larger, interconnected systems. This paper aims to establish a philosophical framework that transforms OET into what is referred to as “lived thermodynamics.” This framework is grounded in pragmatism, particularly drawing from the early synthesis of thermodynamics and evolutionary theory proposed by Charles S. Peirce. A central aspect of this approach involves shifting the focus from traditional “systems” to out-of-equilibrium assemblages. In these assemblages, the physical trends of entropy production are often interrupted and redirected by evolutionary innovations and random events. The evolving envelope of open systems within these assemblages manifests an increasing rate of entropy production. This synthesis of thermodynamics and evolutionary theory builds on Lotka’s pioneering contributions and contemporary theories, particularly Vermeij’s work on the evolution of power. The framework introduces a sustainability criterion based on entropy. By applying this criterion, OET can evolve into “lived thermodynamics,” fostering a holistic understanding of energy use in devices and technological systems while considering the broader implications of entropy production in the out-of-equilibrium assemblages in which we live. Full article

(This article belongs to the Section Multidisciplinary Applications)

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18 pages, 582 KB

Open AccessArticle

Construction of Space-Filling Asymmetrical Marginally Coupled Designs

by Weiping Zhou, Miaomiao Meng, Min Li and Xue Yang

Entropy 2025, 27(12), 1256; https://doi.org/10.3390/e27121256 - 13 Dec 2025

Viewed by 225

Abstract

Marginally coupled designs (MCDs) are very suitable for computer experiments with both qualitative and quantitative factors. An MCD consisting of two subdesigns—one for the qualitative factors and the other for the quantitative factors—is said to be symmetrical or asymmetrical when the qualitative factor [...] Read more.

Marginally coupled designs (MCDs) are very suitable for computer experiments with both qualitative and quantitative factors. An MCD consisting of two subdesigns—one for the qualitative factors and the other for the quantitative factors—is said to be symmetrical or asymmetrical when the qualitative factor subdesign is equal-level or mixed-level, respectively. Although symmetrical MCDs have been studied extensively recently, investigations of asymmetrical MCDs are still relatively scarce. In this paper, based on space-filling symmetrical MCDs or space-filling Latin hypercube designs (LHDs), we propose four approaches to constructing a series of space-filling asymmetrical MCDs. The obtained asymmetrical MCDs can inherit the low-dimensional space-filling properties of these symmetrical MCDs or these LHDs. Moreover, the resulting asymmetrical MCDs are flexible in terms of their run sizes. A numerical study is conducted to compare and evaluate the performance of the proposed designs in computer experiments. Full article

(This article belongs to the Special Issue Number Theoretic Methods in Statistics: Theory and Applications)

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11 pages, 1697 KB

Open AccessArticle

The Effect of Additive and Multiplicative Cyclic Perturbations on Noise-Induced Tipping Dynamics

by Igor A. Khovanov and Natasha A. Khovanova

Entropy 2025, 27(12), 1255; https://doi.org/10.3390/e27121255 - 13 Dec 2025

Viewed by 329

Abstract

The dynamics of systems near tipping points attract considerable attention in the context of climate change, ecological regime shifts, disease spreading, and other complex systems undergoing transitions. In particular, the duration and cause of transitions between states remain subjects of ongoing debate. We [...] Read more.

The dynamics of systems near tipping points attract considerable attention in the context of climate change, ecological regime shifts, disease spreading, and other complex systems undergoing transitions. In particular, the duration and cause of transitions between states remain subjects of ongoing debate. We address these questions by applying the large-fluctuation framework to analyse noise-induced transitions in a widely studied tipping model describing dynamics near a fold bifurcation. As complex systems are typically not in equilibrium, we include cyclic perturbations representing, for example, diurnal variations, seasonal cycles, solar activity oscillations, and Milankovitch cycles in the climate system. We investigate how the frequency and type of cyclic perturbation influence noise-induced transitions between states by examining the fluctuational force. Two types of periodic perturbations, additive and multiplicative, representing B- and R-tipping, are considered. We show, first, that depending on the type of cyclic perturbation, the fluctuations need to be synchronised with different perturbation phases to induce the transition. Secondly, we demonstrate that the transition duration depends on the perturbation frequency: when the periodic perturbation is slower than the system’s relaxation rate, the transition occurs within a single oscillatory cycle, whereas high-frequency perturbations can significantly prolong the transition time. Full article

(This article belongs to the Special Issue From Order to Disorder: Superfluidity, Stochastic Processes, and the Dynamics of Life—Dedicated to Professor Peter McClintock on the Occasion of His 85th Birthday)

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22 pages, 492 KB

Open AccessArticle

Measuring Statistical Dependence via Characteristic Function IPM

by Povilas Daniušis, Shubham Juneja, Lukas Kuzma and Virginijus Marcinkevičius

Entropy 2025, 27(12), 1254; https://doi.org/10.3390/e27121254 - 12 Dec 2025

Viewed by 539

Abstract

We study statistical dependence in the frequency domain using the integral probability metric (IPM) framework. We propose the uniform Fourier dependence measure (UFDM) defined as the uniform norm of the difference between the joint and product-marginal characteristic functions. We provide a theoretical analysis, [...] Read more.

We study statistical dependence in the frequency domain using the integral probability metric (IPM) framework. We propose the uniform Fourier dependence measure (UFDM) defined as the uniform norm of the difference between the joint and product-marginal characteristic functions. We provide a theoretical analysis, highlighting key properties, such as invariances, monotonicity in linear dimension reduction, and a concentration bound. For the estimation of the UFDM, we propose a gradient-based algorithm with singular value decomposition (SVD) warm-up and show that this warm-up is essential for stable performance. The empirical estimator of UFDM is differentiable, and it can be integrated into modern machine learning pipelines. In experiments with synthetic and real-world data, we compare UFDM with distance correlation (DCOR), Hilbert–Schmidt independence criterion (HSIC), and matrix-based Rényi’s

α

-entropy functional (MEF) in permutation-based statistical independence testing and supervised feature extraction. Independence test experiments showed the effectiveness of UFDM at detecting some sparse geometric dependencies in a diverse set of patterns that span different linear and nonlinear interactions, including copulas and geometric structures. In feature extraction experiments across 16 OpenML datasets, we conducted 160 pairwise comparisons: UFDM statistically significantly outperformed other baselines in 20 cases and was outperformed in 13. Full article

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35 pages, 25749 KB

Open AccessArticle

Ionized Keplerian Disks Demonstrating Interplay Between Strong Gravity and Magnetism

by Zdeněk Stuchlík and Jaroslav Vrba

Entropy 2025, 27(12), 1253; https://doi.org/10.3390/e27121253 - 12 Dec 2025

Viewed by 223

Abstract

Using the dynamics of charged test particles, we study the interplay of extremely strong gravitational and magnetic fields acting on ionized Keplerian disks. We assume a Schwarzschild spacetime of mass M combined with a dipole magnetic field represented by a dimensionless parameter b [...] Read more.

Using the dynamics of charged test particles, we study the interplay of extremely strong gravitational and magnetic fields acting on ionized Keplerian disks. We assume a Schwarzschild spacetime of mass M combined with a dipole magnetic field represented by a dimensionless parameter b, characterizing the influence of fields near the gravitational radius

r_{g} = 2 G M / c^{2}

. The particle dynamics can be realized in three regimes: gravitational (

b ≪ 1

), magnetic (

b ≫ 1

), and chaotic (

b \sim 1

). We demonstrate the ionization of disks that are originally both orthogonal and inclined to the magnetic field axis and consider both magnetic attraction and magnetic repulsion acting on the ionized particles. The case of secondary ionized equatorial charged disks is also discussed. The ionization in the dipole magnetic field is compared with the case of a Schwarzschild spacetime endowed with an asymptotically uniform magnetic field. The differences in the dipole and uniform fields are significant in the magnetic and chaotic regimes, while they are suppressed in the gravitational regime. Full article

(This article belongs to the Special Issue Covariance, Multiverse, and Stochastic Effects in Classical Electrodynamics, Quantum Mechanics, and Quantum Gravity)

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29 pages, 9256 KB

Open AccessArticle

MaSS-Droid: Android Malware Detection Framework Using Multi-Layer Feature Screening and Stacking Integration

by Zihao Zhang, Qiang Han and Zhichao Shi

Entropy 2025, 27(12), 1252; https://doi.org/10.3390/e27121252 - 11 Dec 2025

Viewed by 297

Abstract

In recent years, the frequent emergence of Android malware has posed a significant threat to user security. The redundancy of features in malicious software samples and the instability of individual model performance have also introduced numerous challenges to malware detection. To address these [...] Read more.

In recent years, the frequent emergence of Android malware has posed a significant threat to user security. The redundancy of features in malicious software samples and the instability of individual model performance have also introduced numerous challenges to malware detection. To address these issues, this paper proposes a malware detection framework named Mass-Droid, based on Multi-feature and Multi-layer Screening for adaptive Stacking integration. First, three types of features are extracted from APK files: permission features, API call features, and opcode sequences. Then, a three-layer feature screening mechanism is designed to effectively eliminate feature redundancy, improve detection accuracy, and reduce the computational complexity of the model. To tackle the problem of high performance fluctuations and limited generalization ability in single models, this paper proposes an adaptive Stacking integration method (Adaptive-Stacking). By dynamically adjusting the weights of base classifiers, this method significantly enhances the stability and generalization performance of the ensemble model when dealing with complex and diverse malware samples. The experimental results demonstrate that the MaSS-Droid framework can effectively mitigate overfitting, improve the model’s generalization capability, reduce feature redundancy, and significantly enhance the overall stability and accuracy of malware detection. Full article

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9 pages, 369 KB

Open AccessArticle

On the First Quantum Correction to the Second Virial Coefficient of a Generalized Lennard-Jones Fluid

by Daniel Parejo and Andrés Santos

Entropy 2025, 27(12), 1251; https://doi.org/10.3390/e27121251 - 11 Dec 2025

Viewed by 327

Abstract

We derive an explicit analytic expression for the first quantum correction to the second virial coefficient of a d-dimensional fluid whose particles interact via the generalized Lennard-Jones

(2 n, n)

potential. By introducing an appropriate change of variable, the [...] Read more.

We derive an explicit analytic expression for the first quantum correction to the second virial coefficient of a d-dimensional fluid whose particles interact via the generalized Lennard-Jones

(2 n, n)

potential. By introducing an appropriate change of variable, the correction term is reduced to a single integral that can be evaluated in closed form in terms of parabolic cylinder or generalized Hermite functions. The resulting expression compactly incorporates both dimensionality and stiffness, providing direct access to the low- and high-temperature asymptotic regimes. In the special case of the standard Lennard-Jones fluid (

d = 3

,

n = 6

), the formula obtained is considerably more compact than previously reported representations based on hypergeometric functions. The knowledge of this correction allows us to determine the first quantum contribution to the Boyle temperature, whose dependence on dimensionality and stiffness is explicitly analyzed, and enables quantitative assessment of quantum effects in noble gases such as helium, neon, and argon. Moreover, the same methodology can be systematically extended to obtain higher-order quantum corrections. Full article

(This article belongs to the Section Statistical Physics)

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15 pages, 486 KB

Open AccessArticle

Tight Bounds for Joint Distribution Functions of Order Statistics Under k-Independence

by Andrzej Okolewski and Barbara Blazejczyk-Okolewska

Entropy 2025, 27(12), 1250; https://doi.org/10.3390/e27121250 - 11 Dec 2025

Viewed by 258

Abstract

The present study investigates the problem of determining sharp bounds for key reliability and distributional characteristics associated with order statistics. We establish pointwise sharp two-sided bounds for linear combinations of joint distribution functions and joint reliability functions of selected order statistics based on [...] Read more.

The present study investigates the problem of determining sharp bounds for key reliability and distributional characteristics associated with order statistics. We establish pointwise sharp two-sided bounds for linear combinations of joint distribution functions and joint reliability functions of selected order statistics based on k-independent and identically distributed random variables. The proposed framework is general and also applies to arbitrarily dependent observations. The obtained results provide exact bounds for the expected values of functions of order statistics corresponding to finite-valued random variables. Furthermore, the study yields the best possible upper and lower bounds for the joint reliability function of semicoherent systems with shared exchangeable k-independent components. Full article

(This article belongs to the Special Issue Statistical Inference: Theory and Methods)

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20 pages, 1214 KB

Open AccessArticle

Three-Basis Loop-Back QKD: A Passive Architecture for Secure and Scalable Quantum Mobile Networks

by Luis Adrián Lizama-Pérez and Patricia Morales-Calvo

Entropy 2025, 27(12), 1249; https://doi.org/10.3390/e27121249 - 11 Dec 2025

Viewed by 269

Abstract

The Loop-Back Quantum Key Distribution (LB-QKD) protocol establishes a bidirectional architecture in which a single photon travels forth and back through the same optical channel. Unlike conventional one-way schemes such as BB84, Alice performs both state preparation and measurement, while Bob acts as [...] Read more.

The Loop-Back Quantum Key Distribution (LB-QKD) protocol establishes a bidirectional architecture in which a single photon travels forth and back through the same optical channel. Unlike conventional one-way schemes such as BB84, Alice performs both state preparation and measurement, while Bob acts as a passive polarization modulator and reflector. This design eliminates detectors at Bob’s side, minimizes synchronization requirements, and enables compact, low-power implementations suitable for quantum-mobile and IoT platforms. An extended three-basis configuration

{X, Y, Z}

is introduced, preserving the simplicity of the two-basis scheme while improving noise tolerance through enhanced orthogonality-based filtering. Analytical modeling shows that the effective protocol error decreases from

E_{protocol}^{(2)} = e / 2

to

E_{protocol}^{(3)} = e / 3

, achieving a 33% improvement in noise resilience. Despite its slightly lower sifting efficiency (

η = 1 / 6

), the total information gain reaches

G = 0.26

bits per pulse, maintaining post-sifting throughput comparable to BB84. The protocol doubles the tolerable QBER of conventional QKD, sustaining secure operation up to

22 %

for two bases and approximately

47.58 %

for three bases. Its passive, self-verifying architecture enhances resistance to man-in-the-middle, photon-number-splitting, and side-channel attacks, providing a scalable and energy-efficient framework for secure key distribution and authentication in next-generation mobile and distributed quantum networks. Full article

(This article belongs to the Special Issue New Advances in Quantum Communications and Quantum Computing)

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20 pages, 1577 KB

Open AccessArticle

Unraveling the Network Signatures of Oncogenicity in Virus–Human Protein–Protein Interactions

by Francesco Zambelli, Vera Pancaldi and Manlio De Domenico

Entropy 2025, 27(12), 1248; https://doi.org/10.3390/e27121248 - 11 Dec 2025

Viewed by 316

Abstract

Background: Climate change, urbanization, and global mobility increase the risk of emerging infectious diseases with pandemic potential. There is a need for rapid methods that can assess their long-term effects on human health. In silico approaches are particularly suited to study processes that [...] Read more.

Background: Climate change, urbanization, and global mobility increase the risk of emerging infectious diseases with pandemic potential. There is a need for rapid methods that can assess their long-term effects on human health. In silico approaches are particularly suited to study processes that may manifest years later, under the assumption that perturbed biomolecular interactions underlie these outcomes. Here we focus on viral oncogenicity—the ability of viruses to increase cancer risk—which accounts for about 15% of global cancer cases. Methods: We characterize viruses through multilayer representations of protein–protein interaction (PPI) networks reconstructed from the human interactome. Statistical analyses of topological features, combined with interpretable machine learning models, are used to distinguish oncogenic from non-oncogenic viruses and to identify proteins with potential central role in these processes. Results: Our analysis reveals clear statistical differences between the network properties of oncogenic and non-oncogenic viruses. Furthermore, the machine learning approach enables classification of virus–host interaction networks and identification of relevant subsets of proteins associated with oncogenesis. Functional enrichment analysis highlights mechanisms related to viral oncogenicity, including chromatin structure and other processes linked to cancer development. Conclusions: This framework enables virus classification and highlights mechanisms underlying viral oncogenicity, providing a foundation for investigating long-term health effects of emerging pathogens. Full article

(This article belongs to the Special Issue Statistical and Physics-Based Interpretation of AI and Machine Learning Methods for Biomedical Data)

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26 pages, 360 KB

Open AccessArticle

Entropic Dynamics Approach to Quantum Electrodynamics

by Ariel Caticha

Entropy 2025, 27(12), 1247; https://doi.org/10.3390/e27121247 - 11 Dec 2025

Viewed by 334

Abstract

Entropic Dynamics (ED) is a framework that allows one to derive quantum theory as a Hamilton–Killing flow on the cotangent bundle of a statistical manifold. These flows are such that they preserve the symplectic and the metric geometries; they explain the linearity of [...] Read more.

Entropic Dynamics (ED) is a framework that allows one to derive quantum theory as a Hamilton–Killing flow on the cotangent bundle of a statistical manifold. These flows are such that they preserve the symplectic and the metric geometries; they explain the linearity of quantum mechanics and the appearance of complex numbers. In this paper the ED framework is extended to deal with local gauge symmetries. More specifically, on the basis of maximum entropy methods and information geometry, for an appropriate choice of ontic variables and constraints, we derive the quantum dynamics of radiation fields interacting with charged particles. Full article

(This article belongs to the Special Issue SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday)

19 pages, 332 KB

Open AccessArticle

A Lyapunov-Based Analysis on the Almost Periodicity of Impulsive Conformable Reaction–Diffusion Neural Networks with Distributed Delays

by Ivanka Stamova, Gani Stamov and Cvetelina Spirova

Entropy 2025, 27(12), 1246; https://doi.org/10.3390/e27121246 - 11 Dec 2025

Viewed by 281

Abstract

The focus of this research is the qualitative behavior of a reaction–diffusion neural network with distributed delays and conformable derivatives under impulsive perturbations. In particular, the almost periodic behavior of the proposed model is studied using a Lyapunov-based approach. By constructing an appropriate [...] Read more.

The focus of this research is the qualitative behavior of a reaction–diffusion neural network with distributed delays and conformable derivatives under impulsive perturbations. In particular, the almost periodic behavior of the proposed model is studied using a Lyapunov-based approach. By constructing an appropriate Lyapunov-type function, criteria that guarantee the existence and uniqueness of an almost periodic state are provided. The established criteria extend a few existing results on the almost periodicity of conformable models and contribute to the development of the field. In addition, the notion of global conformable exponential stability is introduced and analyzed for the developed model. A suitable example is discussed. Full article

20 pages, 550 KB

Open AccessFeature PaperArticle

On Best Erasure Wiretap Codes: Equivocation Matrices and Design Principles

by Willie K. Harrison, Truman Welling, Andrew Swain and Morteza Shoushtari

Entropy 2025, 27(12), 1245; https://doi.org/10.3390/e27121245 - 9 Dec 2025

Viewed by 315

Abstract

Physical-layer security can aid in establishing secure telecommunication networks including cellular, Internet of Things, and telemetry networks, among others. Channel sounding techniques and/or telemetry systems for reporting channel conditions, coupled with superior wiretap code design are necessary to implement such secure systems. In [...] Read more.

Physical-layer security can aid in establishing secure telecommunication networks including cellular, Internet of Things, and telemetry networks, among others. Channel sounding techniques and/or telemetry systems for reporting channel conditions, coupled with superior wiretap code design are necessary to implement such secure systems. In this paper, we present recent results in best wiretap coset code design for the binary erasure wiretap channel. We define equivocation matrices, and showcase their properties and utility in constructing good, and even the best, wiretap codes. We outline the notion of equivalence for wiretap coset codes, and use it to reduce the search space in exhaustive searches for best small codes. Through example, we show that the best codes do not exist for some code sizes. We also prove that simplex codes are better than codes repeating one column multiple times in their generator matrix. Full article

(This article belongs to the Special Issue Coding for Aeronautical Telemetry)

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1 pages, 125 KB

Open AccessCorrection

Correction: Gao, H.; Yu, Q. Research on Computation Offloading and Resource Allocation Strategy Based on MADDPG for Integrated Space–Air–Marine Network. Entropy 2025, 27, 803

by Haixiang Gao and Qianlian Yu

Entropy 2025, 27(12), 1244; https://doi.org/10.3390/e27121244 - 9 Dec 2025

Viewed by 155

Abstract

Addition of an Author [...] Full article

(This article belongs to the Special Issue Space-Air-Ground-Sea Integrated Communication Networks)

25 pages, 17533 KB

Open AccessFeature PaperArticle

Mirror Descent and Exponentiated Gradient Algorithms Using Trace-Form Entropies

by Andrzej Cichocki, Toshihisa Tanaka, Frank Nielsen and Sergio Cruces

Entropy 2025, 27(12), 1243; https://doi.org/10.3390/e27121243 - 8 Dec 2025

Viewed by 664

Abstract

This paper introduces a broad class of Mirror Descent (MD) and Generalized Exponentiated Gradient (GEG) algorithms derived from trace-form entropies defined via deformed logarithms. Leveraging these generalized entropies yields MD and GEG algorithms with improved convergence behavior, robustness against vanishing and exploding gradients, [...] Read more.

This paper introduces a broad class of Mirror Descent (MD) and Generalized Exponentiated Gradient (GEG) algorithms derived from trace-form entropies defined via deformed logarithms. Leveraging these generalized entropies yields MD and GEG algorithms with improved convergence behavior, robustness against vanishing and exploding gradients, and inherent adaptability to non-Euclidean geometries through mirror maps. We establish deep connections between these methods and Amari’s natural gradient, revealing a unified geometric foundation for additive, multiplicative, and natural gradient updates. Focusing on the Tsallis, Kaniadakis, Sharma–Taneja–Mittal, and Kaniadakis–Lissia–Scarfone entropy families, we show that each entropy induces a distinct Riemannian metric on the parameter space, leading to GEG algorithms that preserve the natural statistical geometry. The tunable parameters of deformed logarithms enable adaptive geometric selection, providing enhanced robustness and convergence over classical Euclidean optimization. Overall, our framework unifies key first-order MD optimization methods under a single information-geometric perspective based on generalized Bregman divergences, where the choice of entropy determines the underlying metric and dual geometric structure. Full article

(This article belongs to the Special Issue SUURI of Information Geometry: Dedicated to SUURI Engineer Professor Shun’ichi Amari on the Occasion of His 90th Birthday)

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28 pages, 570 KB

Open AccessArticle

On the Security and Efficiency of TLS 1.3 Handshake with Hybrid Key Exchange from CPA-Secure KEMs

by Jinrong Chen, Wei Peng, Yi Wang and Yutong Bian

Entropy 2025, 27(12), 1242; https://doi.org/10.3390/e27121242 - 8 Dec 2025

Viewed by 611

Abstract

TLS 1.3 is a crucial protocol for securing modern internet communications. To facilitate a smooth transition to post-quantum security, hybrid key exchange, which combines classical key exchange algorithms with post-quantum key encapsulation mechanisms (KEMs), is proposed to enhance the security of the current [...] Read more.

TLS 1.3 is a crucial protocol for securing modern internet communications. To facilitate a smooth transition to post-quantum security, hybrid key exchange, which combines classical key exchange algorithms with post-quantum key encapsulation mechanisms (KEMs), is proposed to enhance the security of the current TLS 1.3 handshake. However, existing drafts and implementations of hybrid key exchange for TLS 1.3 primarily rely on CCA-secure KEMs (i.e., secure against chosen-ciphertext attacks) based on the Fujisaki-Okamoto (FO) transform. The re-encryption step in their decapsulation algorithms not only introduces additional performance overhead but also raises the risk of side-channel attacks. Although Huguenin-Dumittan and Vaudenay (Eurocrypt 2022) and Zhou et al. (Asiacrypt 2024) demonstrated that the weaker CPA-secure KEMs (i.e., secure against chosen-plaintext attacks) suffice for constructing a secure TLS 1.3 handshake, their analyses were limited to single-KEM settings and did not consider the hybrid key exchange scenario. This work challenges the necessity of CCA security by proving that CPA-secure KEMs are sufficient for the TLS 1.3 handshake even in the hybrid key exchange setting. We provide the first formal security proofs for this claim, covering both the classical random oracle model (ROM) and the quantum random oracle model (QROM), thereby ensuring security against quantum adversaries. To validate the practical benefits, we conduct an extensive performance evaluation based on the latest OpenSSL implementation. Our results show that using CPA-secure KEMs yields up to 44.8% performance improvement at the key exchange layer and up to approximately 9% acceleration for the full TLS 1.3 handshake. Beyond performance gains, this approach reduces the codebase’s attack surface by eliminating the re-encryption step, thereby mitigating a class of side-channel vulnerabilities. Our work positions CPA-secure KEMs as a secure, efficient, and practical alternative for standardizing and deploying post-quantum TLS 1.3 even with hybrid key exchange. Full article

(This article belongs to the Section Quantum Information)

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34 pages, 2015 KB

Open AccessArticle

Exploring the Digital Economy Innovation in the Yangtze River Delta: A Perspective of Complex Networks

by Luyun Sun, Pan Zhao and Benda Zhou

Entropy 2025, 27(12), 1241; https://doi.org/10.3390/e27121241 - 8 Dec 2025

Viewed by 388

Abstract

As a major economic engine of China, the Yangtze River Delta (YRD) region is pivotal in driving innovation across the scientific, technological, and digital economies. This study constructs the spatial associative network for digital economy innovation by treating 41 cities as nodes and [...] Read more.

As a major economic engine of China, the Yangtze River Delta (YRD) region is pivotal in driving innovation across the scientific, technological, and digital economies. This study constructs the spatial associative network for digital economy innovation by treating 41 cities as nodes and applying a gravity model adjusted for institutional distance. Subsequently, the structural characteristics of the spatial associative network and their effects were empirically explored by using complex network analysis and regression models. The findings indicate that: (1) The linkages in digital economy innovation among cities are becoming increasingly closer, and the network structure exhibits an annual increasing trend in density, connectivity, and efficiency, while hierarchy decreases; (2) The examination of network node characteristics discloses that different cities possess diverse capabilities in terms of resource aggregation, regulation, and communication. The block model analysis further categorizes the cities into four functional groups. Among them, Block I (including cities like Shanghai, Nanjing, and Hangzhou) holds the “primary” status and acts as the “core city” for digital economy innovation; (3) The attributes of the spatial associative network have a remarkable effect on both the degree of digital economy innovation and the variations among cities. Full article

(This article belongs to the Special Issue Advances in Complex Networks and Their Applications, from COMPLEX NETWORKS 2025)

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18 pages, 3445 KB

Open AccessArticle

Narrative Co-Evolution in Hybrid Social Networks: A Longitudinal Computational Analysis of Confucius Institutes

by Ming Huang, Jun-Ling Wang and Zi-Ke Zhang

Entropy 2025, 27(12), 1240; https://doi.org/10.3390/e27121240 - 8 Dec 2025

Viewed by 400

Abstract

This study investigates the complex dynamics of public discourse surrounding Confucius Institutes (CIs) across the hybrid social networks of mainstream news and social platforms from 2010 to 2023. Employing a longitudinal, multi-platform design, we analyzed news articles and tweets using a computational framework [...] Read more.

This study investigates the complex dynamics of public discourse surrounding Confucius Institutes (CIs) across the hybrid social networks of mainstream news and social platforms from 2010 to 2023. Employing a longitudinal, multi-platform design, we analyzed news articles and tweets using a computational framework combining topic modeling and sentiment analysis. Our results reveal a shared cross-platform narrative evolution from a “culture-first” to a “politics-central” orientation. However, the trajectory differed significantly: mainstream media underwent a gradual, policy-oriented shift, while social media exhibited an abrupt, nonlinear transition. Crucially, we identify an asymmetric interdependence: Twitter sentiment reliably Granger-causes mainstream media sentiment, establishing its role as a leading indicator, and systematic asymmetries in thematic framing reflect the divergent logics of each platform. The study demonstrates that public discourse on contested, state-linked institutions operates as a complex adaptive system, where bottom-up affective reactions and top-down editorial processes continuously interact in a dynamic equilibrium, ultimately co-constructing a fragmented public understanding. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

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18 pages, 2764 KB

Open AccessArticle

Optimal Transfer of Entanglement in Oscillator Chains in Non-Markovian Open Systems

by Da-Wei Luo, Edward Yu and Ting Yu

Entropy 2025, 27(12), 1239; https://doi.org/10.3390/e27121239 - 8 Dec 2025

Viewed by 299

Abstract

We consider the transfer of continuous-variable entangled states in coupled oscillator chains embedded in a generic environment. We demonstrate high-fidelity transfer via optimal control in two configurations—a linear chain and an X-shaped chain. More specifically, we use the Krotov optimization algorithm to design [...] Read more.

We consider the transfer of continuous-variable entangled states in coupled oscillator chains embedded in a generic environment. We demonstrate high-fidelity transfer via optimal control in two configurations—a linear chain and an X-shaped chain. More specifically, we use the Krotov optimization algorithm to design control fields that achieve the desired state transfer. Under environmental memory effects, the Krotov algorithm needs to be modified, since the dissipative terms in non-Markovian dynamics are generally governed by the time-dependent system Hamiltonian. Remarkably, we can achieve high-fidelity transfer by simply tuning the frequencies of the oscillators while keeping the coupling strength constant, even in the presence of open-system effects. For the system under consideration, we find that quantum memory effects can aid in the transfer of entanglement and show improvement over the memoryless case. In addition, it is possible to target a range of entangled states, making it unnecessary to know the parameters of the initial state beforehand. Full article

(This article belongs to the Section Quantum Information)

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34 pages, 4998 KB

Open AccessArticle

Resisting Memorization-Based APT Attacks Under Incomplete Information in DDHR Architecture: An Entropy-Heterogeneity-Aware RL-Based Scheduling Approach

by Xinghua Wu, Mingzhe Wang, Xiaolin Chang, Chao Li, Yixiang Wang, Bo Liang and Shengjiang Deng

Entropy 2025, 27(12), 1238; https://doi.org/10.3390/e27121238 - 7 Dec 2025

Viewed by 269

Abstract

The rapid advancement of artificial technology is giving rise to new forms of cyber threats like memorization-based APT attacks, which not only pose significant risks to critical infrastructure but also present serious challenges to conventional security architectures. As a crucial service information system [...] Read more.

The rapid advancement of artificial technology is giving rise to new forms of cyber threats like memorization-based APT attacks, which not only pose significant risks to critical infrastructure but also present serious challenges to conventional security architectures. As a crucial service information system in railway passenger stations, the Railway Passenger Service System (RPSS) is particularly vulnerable due to its widespread terminal distribution and large attack surface. This paper focuses on two key challenges within the RPSS Cloud Center’s Double-Layer Dynamic Heterogeneous Redundancy (DDHR) architecture under such attacks: (i) the inability to accurately estimate redundant executor scheduling time, and (ii) the absence of an intelligent defense scheduling method capable of countering memorization-based attacks within a unified and quantifiable environment. To address these issues, we first establish the problem formulation of optimizing defender’s payoff under incomplete information, which applies information entropy of DDHR redundant executors to reflect attacking and defending behaviors. Then a method of estimating attacking time is proposed in order to overcome the difficulty in determining scheduling time due to incomplete information. Finally, we introduce the PPO_HE approach—a Proximal Policy Optimization (PPO) algorithm enhanced with quantifiable information Entropy and Heterogeneity of DDHR redundant executors. Extensive experiments were conducted for evaluation in terms of the two entropy-related metrics: information entropy decay amount and information entropy decay rate. Results demonstrate that the PPO_EH approach achieves the highest efficiency per scheduling operation in countering attacks and provides the longest resistance time against memorization-based attacks under identical initial information entropy conditions. Full article

(This article belongs to the Section Multidisciplinary Applications)

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23 pages, 1226 KB

Open AccessArticle

IE-MAS: Internal–External Multi-Agent Steering for Controllable Image Captioning

by Tiecheng Cai, Chao Chen, Shanshan Lin, Sibo Ju and Xiangwen Liao

Entropy 2025, 27(12), 1237; https://doi.org/10.3390/e27121237 - 7 Dec 2025

Viewed by 418

Abstract

Controllable Image Captioning (CIC) aims to generate coherent and semantically faithful textual descriptions of images while adhering to user-specified constraints. Existing methods have achieved promising results under individual constraints such as sentimental style or sentence length. However, they typically fail to handle and [...] Read more.

Controllable Image Captioning (CIC) aims to generate coherent and semantically faithful textual descriptions of images while adhering to user-specified constraints. Existing methods have achieved promising results under individual constraints such as sentimental style or sentence length. However, they typically fail to handle and satisfy multiple constraints simultaneously, as the controls often interact and interfere with one another. To overcome these challenges, we propose Internal–External Multi-Agent Steering (IE-MAS) for CIC. IE-MAS introduces an internal multimodal steering (IMS) strategy to control affective coherence within the caption, and an external multi-agent collaboration system (EMCS) to guide visual grounding and contextual alignment. From an information-theoretic view, IMS reduces uncertainty in the generation process, while EMCS strengthens the dependency between captions and visual inputs, converting the length and sentiment constraints into information gains. Together, they produce a stable balance among semantic consistency, affective expression, and length control through an adaptive steering process that dynamically balances internal linguistic control and external perceptual grounding. Experimental results demonstrate that IE-MAS effectively coordinates multiple constraints, producing captions that satisfy the length constraint and are sentimental expressive and visually faithful. Full article

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

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19 pages, 2361 KB

Open AccessArticle

Detrended Cross-Correlations and Their Random Matrix Limit: An Example from the Cryptocurrency Market

by Stanisław Drożdż, Paweł Jarosz, Jarosław Kwapień, Maria Skupień and Marcin Wątorek

Entropy 2025, 27(12), 1236; https://doi.org/10.3390/e27121236 - 6 Dec 2025

Viewed by 494

Abstract

Correlations in complex systems are often obscured by nonstationarity, long-range memory, and heavy-tailed fluctuations, which limit the usefulness of traditional covariance-based analyses. To address these challenges, we construct scale- and fluctuation-dependent correlation matrices using the multifractal detrended cross-correlation coefficient

ρ_{r}

that selectively [...] Read more.

Correlations in complex systems are often obscured by nonstationarity, long-range memory, and heavy-tailed fluctuations, which limit the usefulness of traditional covariance-based analyses. To address these challenges, we construct scale- and fluctuation-dependent correlation matrices using the multifractal detrended cross-correlation coefficient

ρ_{r}

that selectively emphasizes fluctuations of different amplitudes. We examine the spectral properties of these detrended correlation matrices and compare them to the spectral properties of the matrices calculated in the same way from synthetic Gaussian and q-Gaussian signals. Our results show that detrending, heavy tails, and the fluctuation-order parameter r jointly produce spectra, which substantially depart from the random case even under the absence of cross-correlations in time series. Applying this framework to one-minute returns of 140 major cryptocurrencies from 2021 to 2024 reveals robust collective modes, including a dominant market factor and several sectoral components whose strength depends on the analyzed scale and fluctuation order. After filtering out the market mode, the empirical eigenvalue bulk aligns closely with the limit of random detrended cross-correlations, enabling clear identification of structurally significant outliers. Overall, the study provides a refined spectral baseline for detrended cross-correlations and offers a promising tool for distinguishing genuine interdependencies from noise in complex, nonstationary, heavy-tailed systems. Full article

(This article belongs to the Special Issue Entropy, Econophysics, and Complexity)

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22 pages, 1405 KB

Open AccessArticle

Entropy-Based Evidence Functions for Testing Dilation Order via Cumulative Entropies

by Mashael A. Alshehri

Entropy 2025, 27(12), 1235; https://doi.org/10.3390/e27121235 - 5 Dec 2025

Viewed by 221

Abstract

This paper introduces novel non-parametric entropy-based evidence functions and associated test statistics for assessing the dilation order of probability distributions constructed from cumulative residual entropy and cumulative entropy. The proposed evidence functions are explicitly tuned to questions about distributional variability and stochastic ordering, [...] Read more.

This paper introduces novel non-parametric entropy-based evidence functions and associated test statistics for assessing the dilation order of probability distributions constructed from cumulative residual entropy and cumulative entropy. The proposed evidence functions are explicitly tuned to questions about distributional variability and stochastic ordering, rather than global model fit, and are developed within a rigorous evidential framework. Their asymptotic distributions are established, providing a solid foundation for large-sample inference. Beyond their theoretical appeal, these procedures act as effective entropy-driven tools for quantifying statistical evidence, offering a compelling non-parametric alternative to traditional approaches, such as Kullback–Leibler discrepancies. Comprehensive Monte Carlo simulations highlight their robustness and consistently high power across a wide range of distributional scenarios, including heavy-tailed models, where conventional methods often perform poorly. A real-data example further illustrates their practical utility, showing how cumulative entropies can provide sharper statistical evidence and clarify stochastic comparisons in applied settings. Altogether, these results advance the theoretical foundation of evidential statistics and open avenues for applying cumulative entropies to broader classes of stochastic inference problems. Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications, 2nd Edition)

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Entropy, Volume 27, Issue 12 (December 2025) – 85 articles

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