Search Results (291)

Utilizing a partitioning method based on equal (or unequal) probabilities—without incorporating the alpha-cluster (α-cluster) model—allows for the derivation of diverse topological configurations of nuclear fragments resulting from fragmentation. Subsequently, we predict the multiplicity distribution of nuclear fragments for specific excited nuclei, such as ${}^9\mathrm{Be}^{*}$, ${}^{12}\mathrm{C}^{*}$, and ${}^{16}\mathrm{O}^{*}$, which can be formed as nuclear remnants in electron–nucleus ($eA$) collisions at high energy. Based on the $\alpha$-cluster model, an $\alpha$-cluster structure may result in deviations in the multiplicity distributions of nuclear fragments with charge $Z=2$, compared to those predicted by the partitioning methods. Furthermore, in the framework of Tsallis statistics, the nonextensive generalized temperature, entropy index, and q-entropy are obtained from the multiplicity distribution of nuclear fragments with a given charge number. Our work shows that fragmentation of nuclear remnants in electron–nucleus collisions at high energy is a nonextensive process. Full article

This paper introduces a q-deformed extension of the Lindley distribution. This extension is obtained by replacing the classical exponential with the q-exponential function from Tsallis non-extensive statistical techniques. This transformation offers more control over the tail behavior of the distribution, providing a transition between exponential and power-law decay patterns. Such flexibility is particularly useful when modeling right-skewed data with excess kurtosis, where classical models may not adequately describe heavy-tailed and highly skewed data. We derive several key properties, including the quantile function, expressed by the Lambert–Tsallis function Wq, the raw and incomplete moments, the probability-weighted moments, and the Tsallis entropy. The distribution of order statistics is also investigated. For parameter estimation, we employ several frequentist methods and conduct extensive Monte Carlo simulation studies to assess and compare their performance. Finally, applications to real-world datasets show that the q-deformed Lindley model is practically superior and more flexible than the classical Lindley distribution and other well-known models. Full article

Arctic sea ice is a critical indicator of global climate dynamics, directly influencing maritime navigation, polar biodiversity, and offshore engineering safety. The precise mapping of diverse ice types, such as frazil ice, slush, melt ponds, and open water, is essential for environmental monitoring; however, it remains a formidable challenge in satellite remote sensing. These difficulties arise from low-contrast imagery, overlapping spectral signatures, and the subtle textural nuances characteristic of polar regions. Traditional entropy-based thresholding techniques often falter when segmenting these complex scenes, as they typically rely on Gaussian distribution assumptions that do not align with the stochastic nature of Arctic data. To address these limitations, this paper presents a novel unsupervised segmentation framework based on symmetric cross-entropy (SCE). Unlike standard directional measures, SCE provides a more robust objective function for multi-level thresholding by simultaneously maximizing intra-class cohesion and minimizing inter-class ambiguity. The proposed method uses an optimized search strategy to identify intensity levels that best delineate complex Arctic features. We conducted an extensive entropy-based comparative study that benchmarked SCE against 25 state-of-the-art entropy measures, including Shannon, Kapur, Rényi, Tsallis, and Masi entropies. Our experimental results demonstrate that the SCE method: (i) achieves superior accuracy by consistently outperforming established models in segmentation precision and boundary definition; (ii) provides visual clarity by producing segments with significantly reduced noise, making them ideal for identifying small-scale melt ponds and slush zones; and (iii) demonstrates computational robustness by providing stable threshold values even in datasets with non-Gaussian class distributions and poor illumination. Ultimately, these improvements deliver high-quality ice feature data that enhance risk assessment, operational planning, and predictive modeling. This research marks a major step forward in Arctic sea studies and introduces a valuable new tool for wider image processing and computer vision communities. Full article

Traditional mean–variance portfolio optimization proves inadequate for cryptocurrency markets, where extreme volatility, fat-tailed return distributions, and unstable correlation structures undermine the validity of variance as a comprehensive risk measure. To address these limitations, this paper proposes a unified entropy-based portfolio optimization framework grounded in the Maximum Entropy Principle (MaxEnt). Within this setting, Shannon entropy, Tsallis entropy, and Weighted Shannon Entropy (WSE) are formally derived as particular specifications of a common constrained optimization problem solved via the method of Lagrange multipliers, ensuring analytical coherence and mathematical transparency. Moreover, the proposed MaxEnt formulation provides an information-theoretic interpretation of portfolio diversification as an inference problem under uncertainty, where optimal allocations correspond to the least informative distributions consistent with prescribed moment constraints. In this perspective, entropy acts as a structural regularizer that governs the geometry of diversification rather than as a direct proxy for risk. This interpretation strengthens the conceptual link between entropy, uncertainty quantification, and decision-making in complex financial systems, offering a robust and distribution-free alternative to classical variance-based portfolio optimization. The proposed framework is empirically illustrated using a portfolio composed of major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—based on weekly return data. The results reveal systematic differences in the diversification behavior induced by each entropy measure: Shannon entropy favors near-uniform allocations, Tsallis entropy imposes stronger penalties on concentration and enhances robustness to tail risk, while WSE enables the incorporation of asset-specific informational weights reflecting heterogeneous market characteristics. From a theoretical perspective, the paper contributes a coherent MaxEnt formulation that unifies several entropy measures within a single information-theoretic optimization framework, clarifying the role of entropy as a structural regularizer of diversification. From an applied standpoint, the results indicate that entropy-based criteria yield stable and interpretable allocations across turbulent market regimes, offering a flexible alternative to classical risk-based portfolio construction. The framework naturally extends to dynamic multi-period settings and alternative entropy formulations, providing a foundation for future research on robust portfolio optimization under uncertainty. Full article

The Ujlayan–Dixit (UD) fractional calculus provides a powerful fractional extension of the Lomax distribution, offering a suitable framework for representing complex behaviors beyond classical approaches. In this paper, we adopt the UD fractional Lomax distribution and establish its statistical theory. Based on the adopted density, we derive closed-form expressions for the cumulative distribution, survival, and hazard functions, as well as the mode. Several UD fractional statistical measures of the Lomax random variable are derived, including the fractional moments, fractional information theoretic measures, including UD fractional Shannon and Tsallis entropy measures, and the probability density function of the $k^{th}$ order statistic under the UD fractional framework. Finally, a real data application concerning the time to break down an insulating fluid is used to illustrate the usefulness of the proposed distribution in modeling real data applications. The fitting performance of the suggested model is compared with several extensions of the Lomax distribution. The comparative results show that the UD fractional Lomax distribution outperforms several well-known extensions of Lomax distribution. This framework provides researchers with many robust tools for advanced reliability assessment, uncertainty quantification, and risk modeling, providing insights into phenomena not captured by the classical Lomax distribution. Moreover, when the fractional parameter $q\to 1^{-}$, the proposed approach converges to the classical Lomax results, bridging fractional and classical perspectives. Full article

The paper explored the fractal, nonlinear and chaotic dynamics between oil prices, inflation, economic growth and unemployment in Turkiye from 1960 to 2024 and examined how energy market volatility propagated through the macroeconomy via complex, regime-dependent mechanisms. It developed a chaotic regression method and employed entropy-based measures (Shannon, Rényi and Tsallis), Lyapunov exponents, Lorenz and Rössler attractors, Julia set diagnostics and the chaos Granger causality test (Hiemstra–Jones). By nesting entropy, chaos and causality within a unified framework, it contributed methodological innovations and practical insights to the energy–economy literature. The chaotic regression results revealed that oil price shocks generated asymmetric and nonlinear responses in inflation, unemployment and growth that were characterized by chaos and sensitivity to initial conditions and demonstrated that oil shocks act as catalysts for nonlinear propagation and fractal macroeconomic dynamics. Julia set results determined that unemployment can be explained by inflation fractal size. Hiemstra–Jones method determined unidirectional causality from oil to both inflation, economic growth and unemployment. According to the results, adopting nonlinear and chaos-based modeling approaches is essential to understand the macroeconomic consequences of energy shocks. For policymakers, the evidence determined that the costs of disinflation or inflation control are sensitive to energy market volatility. The paper contributed to the energy–economy-econometrics literature by integrating entropy, chaos and causality analyses into the oil price–macroeconomy nexus by offering both methodological innovations and practical insights. Full article

Background: Dynamic visual acuity (DVA) is functionally distinct from static visual acuity (SVA), though SVA is often used clinically as a reference. Methods: To identify EEG biomarkers for DVA, we presented participants with a high-contrast checkerboard moving horizontally at speeds ranging from 4°/s to 30°/s, engaging motion-sensitive pathways while preserving spatial detail. Six EEG features—ERPs (N200 and P300), TRCA, Hjorth activity, mean curve length, and Tsallis entropy—were extracted from eight occipito-parietal channels and evaluated for speed sensitivity. Results: Hjorth activity and Tsallis entropy showed consistent monotonic trends with respect to speed. Hjorth activity exhibited the strongest univariate correlation (r = 0.88, p < 0.05). In a Lasso regression model using all speed-sensitive features, the predicted speed correlated with actual speed at r = 0.588, with TRCA-weighted features retained for their multivariate contribution. Notably, Hjorth activity peaked at PO7/PO8 (3.558 and 1.478 µV² at 30°/s), aligning with V5/MT+ activation. Conclusion: Given its high sensitivity, neuroanatomical plausibility, and simplicity, Hjorth activity is recommended as a primary candidate for EEG-based DVA biomarker development. This study provides a foundation for objective neurophysiological evaluation of dynamic vision. Full article

Nanoscopic quantum dots exhibit discrete energy spectra and size- and shape-dependent thermal properties that cannot always be adequately described within the conventional Boltzmann–Gibbs statistical framework. In systems with strong confinement, finite size, and reduced symmetry, deviations from extensivity may emerge, affecting the occupation of energy levels and the resulting thermodynamic response. In this context, this work elucidates how GaAs quantum dot geometry, external electric fields, and nonextensive statistical effects jointly influence the thermal response of quantum dots with different geometries—cubic, cylindrical, ellipsoidal, and pyramidal. These energy levels are calculated by solving the Schrödinger equation under the effective mass approximation, employing the finite element method for numerical computation. These energy levels are then incorporated into an iterative numerical procedure to calculate the specific heat for different values of the nonextensivity parameter, thereby enabling exploration of both extensive (Boltzmann–Gibbs) and nonextensive regimes. The results demonstrate that the shape of the quantum dots strongly influences the energy spectrum and, consequently, the thermal properties, producing distinctive features such as Schottky-type anomalies and geometry-dependent shifts under an external electric field. In subextensive regimes, a discrete behavior in the specific heat emerges due to natural cutoffs in the accessible energy states. In contrast, in superextensive regimes, a smooth, saturation-like behavior is observed. These findings highlight the importance of geometry, external-field effects, and nonextensive statistics as complementary tools for tailoring the energy distribution and thermal response in nanoscopic quantum systems. Full article

Strongly convex functions form a central subclass of convex functions and have gained considerable attention due to their structural advantages and broad applicability, particularly in optimization and information theory. In this paper, we investigate the class of strongly F-convex functions, which generalizes the classical notion of strong convexity by introducing an auxiliary convex control function F. We establish several fundamental structural characterizations of this class and provide a variety of nontrivial examples such as power, logarithmic, and exponential functions. In addition, we derive refined Jensen-type and Hermite–Hadamard-type inequalities adapted to the strongly F-convex concept, thereby extending and sharpening their classical forms. As applications, we obtain new analytical inequalities and improved error bounds for entropy-related quantities, including Shannon, Tsallis, and Rényi entropies, demonstrating that the concept of strong F-convexity naturally yields strengthened divergence and uncertainty estimates. Full article

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

We develop goodness-of-fit (GOF) procedures rooted in Tsallis entropy, with a particular emphasis on multivariate exponential-power (generalized Gaussian) and q-Gaussian models. The GOF statistic compares a closed-form Tsallis entropy under the null with a non-parametric k-nearest-neighbor (k-NN) estimator. We establish consistency and mean-square convergence of the estimator under mild regularity and tail assumptions, discuss an asymptotic normality regime as $q\to 1$, and calibrate critical values by parametric bootstrap/permutation. Extensive Monte Carlo experiments report empirical size, power, and runtime. These are reported across dimensions, k, and q. An applied example illustrates practical calibration and sensitivity, which are essential for accurate measurement. Full article

Entropy-based image thresholding is one of the most widely used segmentation techniques in image processing. The Tsallis and Masi entropies are information measures that can capture long-range interactions in various physical systems. On the other hand, Shannon entropy is more appropriate for short-range correlations. In this paper, we have improved a thresholding technique based on Tsallis and Shannon formulas by using Masi entropy. Specifically, we replace the Tsallis information measure with Masi’s one, obtaining better results than the original methodology. As the proposed method depends on an entropic parameter, we designed a thresholding algorithm that incorporates a simulated annealing procedure for parameter optimization. Then, we compared our results with thresholding methods that use just Masi (or Tsallis), or a combination of them, Shannon, Sine, and Hill entropies. The comparison is enriched with a kernel version of a support vector machine, as well as a discussion of our proposal in relation to deep learning approaches. Quantitative measures of segmentation accuracy demonstrated the superior performance of our method in infrared, nondestructive testing (NDT), as well as RGB images from the BSDS500 dataset. Full article

This paper presents a new model that surpasses traditional distributions, specifically the three-parameter distribution of the Inverse Power Entropy Chen (IPEC) model. In comparison to the existing distributions, the latest one presents an exceptionally diverse array of probability functions. The density and hazard rate functions have characteristics indicating that the model is adaptable to many types of data. The study explores the mathematical features of the IPEC distribution, including moments with some related measures, quantile function, Rényi entropy, Tsallis entropy, and order statistics. To estimate the parameters of the IPEC model, we utilized seven classical estimation strategies, including maximum likelihood estimators, Anderson–Darling estimators, right-tail Anderson–Darling estimators, Cramér–von Mises estimators, percentile estimators, least-squares estimators, and weighted least-squares estimators. To evaluate the efficacy of these estimating approaches across varying sample sizes, Monte Carlo simulations are performed. The efficacy of each estimator is evaluated through comparisons of average relative bias and mean squared error, highlighting their suitability for the used samples. Three applications utilize real-world datasets related to medical and physical fields, demonstrating the usefulness of the new model in relation to several established competitive models. This empirical investigation further supports the utility and adaptability of the inverse power entropy Chen model in capturing the intricacies of distinct datasets, hence delivering useful insights for practitioners in numerous domains. Full article

We introduce and study the shortfall of tail-based entropy (STE), a tail-sensitive risk functional that combines expected shortfall (ES) and tail-based entropy (TE). Beyond the tail mean, STE imposes a rank-dependent penalty on tail variability, thereby capturing both the magnitude and variability of tail risk under extremes. The framework encompasses several shortfall-type measures as special cases, such as Gini shortfall, extended Gini shortfall, shortfall of cumulative residual entropy, shortfall of right-tail deviation, and shortfall of cumulative residual Tsallis entropy. We provide equivalent characterizations of STE, derive sufficient conditions for coherence, and establish monotonicity with respect to tail-variability order. As an application, we investigate STE-based capital allocation, deriving closed-form allocation formulas under elliptical and extended skew-normal distributions, along with several illustrative special cases. Finally, an empirical analysis with insurance company data illustrates the implementation and evaluates the performance of the allocation rule. Full article

It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to solve mathematical models. This paper introduces a novel methodological framework based on the SO(3) Lie method to estimate time-dependent correlation matrices (correlation flows) among three variables that have chaotic, entropy, and fractal characteristics, from 11 April 2011 to 31 December 2024 for daily data; from 10 April 2011 to 29 December 2024 for weekly data; and from April 2011 to December 2024 for monthly data. So, it develops the stochastic SO(2) Lie method into the SO(3) Lie method that aims to obtain the correlation flow for three variables with chaotic, entropy, and fractal structure. The results were obtained at three stages. Firstly, we applied entropy (Shannon, Rényi, Tsallis, Higuchi) measures, Kolmogorov–Sinai complexity, Hurst exponents, rescaled range tests, and Lyapunov exponent methods. The results of the Lyapunov exponents (Wolf, Rosenstein’s Method, Kantz’s Method) and entropy methods, and KSC found evidence of chaos, entropy, and complexity. Secondly, the stochastic differential equations which depend on S² (SO(3) Lie group) and Lie algebra to obtain the correlation flows are explained. The resulting equation was numerically solved. The correlation flows were obtained by using the defined covariance flow transformation. Finally, we ran the robustness check. Accordingly, our robustness check results showed the SO(3) Lie method produced more effective results than the standard and Spearman correlation and covariance matrix. And, this method found lower RMSE and MAPE values, greater stability, and better forecast accuracy. For daily data, the Lie method found RMSE = 0.63, MAE = 0.43, and MAPE = 5.04, RMSE = 0.78, MAE = 0.56, and MAPE = 70.28 for weekly data, and RMSE = 0.081, MAE = 0.06, and MAPE = 7.39 for monthly data. These findings indicate that the SO(3) framework provides greater robustness, lower errors, and improved forecasting performance, as well as higher sensitivity to nonlinear transitions compared to standard correlation measures. By embedding time-dependent correlation matrix into a Lie group framework inspired by physics, this paper highlights the deep structural parallels between financial markets and complex physical systems. Full article