Symmetry

11 pages, 255 KiB

Open AccessArticle

New Sufficient Conditions for p-Valent Functions

by Mamoru Nunokawa, Janusz Sokół and Edyta Trybucka

Symmetry 2025, 17(7), 1141; https://doi.org/10.3390/sym17071141 - 16 Jul 2025

A function that is holomorphic in a set E, in the complex plane, is called p-valent in E if, for every complex number, w, the equation f (z) = w has, at most, p roots in E. [...] Read more.

A function that is holomorphic in a set E, in the complex plane, is called p-valent in E if, for every complex number, w, the equation f (z) = w has, at most, p roots in E. In this paper, we established some sufficient conditions for holomorphic functions in the unit disk |z| < 1 to be at most p-valent in the unit disk or p-valent or p-valent starlike in the unit disk. Full article

(This article belongs to the Section Mathematics)

20 pages, 2632 KiB

Open AccessArticle

Data-Driven Attack Detection Mechanism Against False Data Injection Attacks in DC Microgrids Using CNN-LSTM-Attention

by Chunxiu Li, Xinyu Wang, Xiaotao Chen, Aiming Han and Xingye Zhang

Symmetry 2025, 17(7), 1140; https://doi.org/10.3390/sym17071140 - 16 Jul 2025

Abstract

This study presents a novel spatio-temporal detection framework for identifying False Data Injection (FDI) attacks in DC microgrid systems from the perspective of cyber–physical symmetry. While modern DC microgrids benefit from increasingly sophisticated cyber–physical symmetry network integration, this interconnected architecture simultaneously introduces significant [...] Read more.

This study presents a novel spatio-temporal detection framework for identifying False Data Injection (FDI) attacks in DC microgrid systems from the perspective of cyber–physical symmetry. While modern DC microgrids benefit from increasingly sophisticated cyber–physical symmetry network integration, this interconnected architecture simultaneously introduces significant cybersecurity vulnerabilities. Notably, FDI attacks can effectively bypass conventional Chi-square detector-based protection mechanisms through malicious manipulation of communication layer data. To address this critical security challenge, we propose a hybrid deep learning framework that synergistically combines: Convolutional Neural Networks (CNN) for robust spatial feature extraction from power system measurements; Long Short-Term Memory (LSTM) networks for capturing complex temporal dependencies; and an attention mechanism that dynamically weights the most discriminative features. The framework operates through a hierarchical feature extraction process: First-level spatial analysis identifies local measurement patterns; second-level temporal analysis detects sequential anomalies; attention-based feature refinement focuses on the most attack-relevant signatures. Comprehensive simulation studies demonstrate the superior performance of our CNN-LSTM-Attention framework compared to conventional detection approaches (CNN-SVM and MLP), with significant improvements across all key metrics. Namely, the accuracy, precision, F1-score, and recall could be improved by at least 7.17%, 6.59%, 2.72% and 6.55%. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Applied Machine Learning and Neural Networks for Hybrid Energy Ship Power System)

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21 pages, 733 KiB

Open AccessArticle

A Secure and Privacy-Preserving Approach to Healthcare Data Collaboration

by Amna Adnan, Firdous Kausar, Muhammad Shoaib, Faiza Iqbal, Ayesha Altaf and Hafiz M. Asif

Symmetry 2025, 17(7), 1139; https://doi.org/10.3390/sym17071139 - 16 Jul 2025

Abstract

Combining a large collection of patient data and advanced technology, healthcare organizations can excel in medical research and increase the quality of patient care. At the same time, health records present serious privacy and security challenges because they are confidential and can be [...] Read more.

Combining a large collection of patient data and advanced technology, healthcare organizations can excel in medical research and increase the quality of patient care. At the same time, health records present serious privacy and security challenges because they are confidential and can be breached through networks. Even traditional methods with federated learning are used to share data, patient information might still be at risk of interference while updating the model. This paper proposes the Privacy-Preserving Federated Learning with Homomorphic Encryption (PPFLHE) framework, which strongly supports secure cooperation in healthcare and at the same time providing symmetric privacy protection among participating institutions. Everyone in the collaboration used the same EfficientNet-B0 architecture and training conditions and keeping the model symmetrical throughout the network to achieve a balanced learning process and fairness. All the institutions used CKKS encryption symmetrically for their models to keep data concealed and stop any attempts at inference. Our federated learning process uses FedAvg on the server to symmetrically aggregate encrypted model updates and decrease any delays in our server communication. We attained a classification accuracy of 83.19% and 81.27% when using the APTOS 2019 Blindness Detection dataset and MosMedData CT scan dataset, respectively. Such findings confirm that the PPFLHE framework is generalizable among the broad range of medical imaging methods. In this way, patient data are kept secure while encouraging medical research and treatment to move forward, helping healthcare systems cooperate more effectively. Full article

(This article belongs to the Special Issue Exploring Symmetry in Wireless Communication)

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19 pages, 284 KiB

Open AccessArticle

Local and Global Solutions of the 3D-NSE in Homogeneous Lei–Lin–Gevrey Spaces

by Lotfi Jlali

Symmetry 2025, 17(7), 1138; https://doi.org/10.3390/sym17071138 - 16 Jul 2025

Abstract

This paper investigates the existence and uniqueness of local and global solutions to the incompressible three-dimensional Navier–Stokes equations within the framework of homogeneous Lei–Lin–Gevrey spaces

X_{a, γ}^{ρ} (R^{3})

, where [...] Read more.

This paper investigates the existence and uniqueness of local and global solutions to the incompressible three-dimensional Navier–Stokes equations within the framework of homogeneous Lei–Lin–Gevrey spaces

X_{a, γ}^{ρ} (R^{3})

, where

ρ \in [- 1, 0), a > 0

, and

γ \in (0, 1)

. These function spaces combine the critical scaling structure of the Lei–Lin spaces with the exponential regularity of Gevrey classes, thereby enabling a refined treatment of analytic regularity and frequency localization. The main results are obtained under the assumption of small initial data in the critical Lei–Lin space

X^{ρ} (R^{3})

, extending previous works and improving regularity thresholds. In particular, we establish that for suitable initial data, the Navier–Stokes system admits unique solutions globally in time. The influence of the Gevrey parameter

γ

on the high-frequency behavior of solutions is also discussed. This work contributes to a deeper understanding of regularity and decay properties in critical and supercritical regimes. Full article

(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)

18 pages, 2823 KiB

Open AccessArticle

Quasi-Periodic Dynamics and Wave Solutions of the Ivancevic Option Pricing Model Using Multi-Solution Techniques

by Sadia Yasin, Fehaid Salem Alshammari, Asif Khan and Beenish

Symmetry 2025, 17(7), 1137; https://doi.org/10.3390/sym17071137 - 16 Jul 2025

Abstract

In this research paper, we study symmetry groups, soliton solutions, and the dynamical behavior of the Ivancevic Option Pricing Model (IOPM). First, we find the Lie symmetries of the considered model; next, we use them to determine the corresponding symmetry groups. Then, we [...] Read more.

In this research paper, we study symmetry groups, soliton solutions, and the dynamical behavior of the Ivancevic Option Pricing Model (IOPM). First, we find the Lie symmetries of the considered model; next, we use them to determine the corresponding symmetry groups. Then, we attempt to solve IOPM by means of two methods. We provide some wave solutions and give further details of the solution using 2D and 3D graphs. These results are interpreted as important clarifications in financial mathematics and deepen our understanding of the dynamics involved during the pricing of options. Secondly, the quasi-periodic behavior of the two-dimensional dynamical system and its perturbed system are plotted using Python software (Python 3.13.5 version). Various frequencies and amplitudes are considered to confirm the quasi-periodic behavior via the Lyapunov exponent, bifurcation diagram, and multistability analysis. These findings are particularly in consonance with current research that investigates IOPM as a nonlinear wave alternate for normal models and the importance of graphical representations in the understanding of financial derivative dynamics. We, therefore, hope to fill in the gaps in the literature that currently exist about the use of multi-solution methods and their effects on financial modeling through the employment of sophisticated graphical techniques. This will be helpful in discussing matters in the field of financial mathematics and open up new directions of investigation. Full article

(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)

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20 pages, 578 KiB

Open AccessArticle

Improved Median Estimation in Stratified Surveys via Nontraditional Auxiliary Measures

by Abdulaziz S. Alghamdi and Fatimah A. Almulhim

Symmetry 2025, 17(7), 1136; https://doi.org/10.3390/sym17071136 - 15 Jul 2025

Abstract

This research focuses on estimating the population median within a stratified random sampling framework by using robust statistical measures with transformation-based methodologies. An efficient estimator aims to minimize both the bias and the variance, thereby reducing the overall mean squared error (MSE, leading [...] Read more.

This research focuses on estimating the population median within a stratified random sampling framework by using robust statistical measures with transformation-based methodologies. An efficient estimator aims to minimize both the bias and the variance, thereby reducing the overall mean squared error (MSE, leading to more reliable outcomes. We introduce an improved class of proposed estimators that utilizes transformation techniques to effectively address data variability and enhance estimation accuracy. To evaluate their performance, we derive expressions for bias and mean square error (MSE) up to the first-order approximation for both existing and newly developed estimators, establishing theoretical conditions for their effectiveness. Additionally, the proposed estimators are compared with traditional methods using simulated populations generated from different probability distributions and actual datasets. The results indicate that the newly introduced estimators improve precision and efficiency in median estimation, yielding more reliable outcomes. When assessed against conventional estimators, the findings demonstrate that the new estimators outperform in terms of the percent relative efficiency criterion. Full article

(This article belongs to the Section Mathematics)

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38 pages, 4803 KiB

Open AccessReview

Charge Density Waves in Solids—From First Concepts to Modern Insights

by Danko Radić

Symmetry 2025, 17(7), 1135; https://doi.org/10.3390/sym17071135 - 15 Jul 2025

Abstract

We present a brief overview of the field of charge density waves (CDW) in condensed systems with focus set to the underlying mechanisms behind the CDW ground state. Our intention in this short review is not to count all related facts from the [...] Read more.

We present a brief overview of the field of charge density waves (CDW) in condensed systems with focus set to the underlying mechanisms behind the CDW ground state. Our intention in this short review is not to count all related facts from the vast volume of literature about this decades-old and still developing field, but rather to pinpoint the most important, mostly theoretical ones, presenting the development of the field. Starting from the “early days”, mainly based on weakly coupled, chain-like quasi-1D systems and Peierls instability, in which the Fermi surface nesting has been the predominant and practically paradigmatic mechanism of the CDW ground state stabilisation, we track the change in paradigms while entering the field of layered quasi-2D systems, with Fermi surface far away from the nesting regime, in which rather strong, essentially momentum-dependent interactions and particular reconstructions of the Fermi surface become essential. Examples of real quasi-1D materials, such as organic and inorganic conductors like Bechgaard salts or transition metal trichalcogenides and bronzes, in which experiment and theory have been extremely successful in providing detailed understanding, are contrasted to layered quasi-2D materials, such as

{high-T}_{c}

superconducting cuprates, intercalated graphite compounds or transition metal dichalcogenides, for which the theory explaining an onset of the CDWs constitutes a frontier of this fast-evolving field, strongly boosted by development of modern ab initio calculation methods. Full article

(This article belongs to the Section Physics)

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13 pages, 1294 KiB

Open AccessArticle

From Complex to Quaternions: Proof of the Riemann Hypothesis and Applications to Bose–Einstein Condensates

by Jau Tang

Symmetry 2025, 17(7), 1134; https://doi.org/10.3390/sym17071134 - 15 Jul 2025

Abstract

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the [...] Read more.

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the Riemann ξ(s) function. This formulation reveals that all nontrivial zeros of the zeta function must lie along the critical line Re(s) = 1/2, offering a constructive and algebraic resolution to this fundamental conjecture. Our method is built on convexity and symmetrical principles that generalize naturally to higher-dimensional hypercomplex spaces. We also explore the broader implications of this framework in quantum statistical physics. In particular, the λ-regularized quaternionic zeta function governs thermodynamic properties and phase transitions in Bose–Einstein condensates. This quaternionic extension of the zeta function encodes oscillatory behavior and introduces critical hypersurfaces that serve as higher-dimensional analogues of the classical critical line. By linking the spectral features of the zeta function to measurable physical phenomena, our work uncovers a profound connection between analytic number theory, hypercomplex geometry, and quantum field theory, suggesting a unified structure underlying prime distributions and quantum coherence. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)

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24 pages, 1873 KiB

Open AccessArticle

Efficient Outsourced Decryption System with Attribute-Based Encryption for Blockchain-Based Digital Asset Transactions

by Rui Jin, Yuxuan Pan, Junjie Li, Yu Liu, Daquan Yang, Mengmeng Zhou and Konglin Zhu

Symmetry 2025, 17(7), 1133; https://doi.org/10.3390/sym17071133 - 15 Jul 2025

Abstract

The rapid expansion of blockchain-based digital asset trading raises new challenges in security, privacy, and efficiency. Although traditional attribute-based encryption (ABE) provides fine-grained access control, it imposes considerable computational overhead and introduces additional vulnerabilities when decryption is outsourced. To address these limitations, we [...] Read more.

The rapid expansion of blockchain-based digital asset trading raises new challenges in security, privacy, and efficiency. Although traditional attribute-based encryption (ABE) provides fine-grained access control, it imposes considerable computational overhead and introduces additional vulnerabilities when decryption is outsourced. To address these limitations, we present EBODS, an efficient outsourced decryption framework that combines an optimized ABE scheme with a decentralized blockchain layer. By applying policy matrix optimization and leveraging edge decryption servers, EBODS reduces the public key size by 8% and markedly accelerates computation. Security analysis confirms the strong resistance of EBODS to collusion attacks, making it suitable for resource-constrained digital asset platforms. Full article

(This article belongs to the Special Issue Advanced Studies of Symmetry/Asymmetry in Cybersecurity)

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19 pages, 865 KiB

Open AccessArticle

Improved SBM Model Based on Asymmetric Data—Mathematical Evaluation and Analysis of Green Innovation Efficiency

by Limei Chen, Yao Yao and Can Yang

Symmetry 2025, 17(7), 1132; https://doi.org/10.3390/sym17071132 - 15 Jul 2025

Abstract

Green innovation has become a core driving force for promoting sustainable development, making the accurate evaluation of enterprises’ green innovation efficiency an important research topic. Based on the Environmental, Social, and Governance (ESG) framework, this paper improves the SBM model to overcome shortcomings [...] Read more.

Green innovation has become a core driving force for promoting sustainable development, making the accurate evaluation of enterprises’ green innovation efficiency an important research topic. Based on the Environmental, Social, and Governance (ESG) framework, this paper improves the SBM model to overcome shortcomings such as homogeneity in traditional SBM models during efficiency evaluation. By introducing an asymmetric slack measure, it breaks through the limitation of efficiency value ceilings, enabling gradient ranking of decision-making units and precisely distinguishing between efficient and inefficient enterprises, thereby better assessing the green innovation efficiency of hydrogen energy companies. The study shows that the improved SBM model significantly enhances the accuracy of enterprise efficiency evaluation. The contribution of this paper lies in constructing an improved SBM model integrated within the ESG framework, compensating for the lack of environmental dimensions in traditional evaluation methods, addressing issues of efficiency homogeneity and the static nature of the frontier, and achieving optimized ranking of frontier-efficient enterprises. Full article

(This article belongs to the Section Mathematics)

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27 pages, 14879 KiB

Open AccessArticle

Research on AI-Driven Classification Possibilities of Ball-Burnished Regular Relief Patterns Using Mixed Symmetrical 2D Image Datasets Derived from 3D-Scanned Topography and Photo Camera

by Stoyan Dimitrov Slavov, Lyubomir Si Bao Van, Marek Vozár, Peter Gogola and Diyan Minkov Dimitrov

Symmetry 2025, 17(7), 1131; https://doi.org/10.3390/sym17071131 - 15 Jul 2025

Abstract

The present research is related to the application of artificial intelligence (AI) approaches for classifying surface textures, specifically regular reliefs patterns formed by ball burnishing operations. A two-stage methodology is employed, starting with the creation of regular reliefs (RRs) on test parts by [...] Read more.

The present research is related to the application of artificial intelligence (AI) approaches for classifying surface textures, specifically regular reliefs patterns formed by ball burnishing operations. A two-stage methodology is employed, starting with the creation of regular reliefs (RRs) on test parts by ball burnishing, followed by 3D topography scanning with Alicona device and data preprocessing with Gwyddion, and Blender software, where the acquired 3D topographies are converted into a set of 2D images, using various virtual camera movements and lighting to simulate the symmetrical fluctuations around the tool-path of the real camera. Four pre-trained convolutional neural networks (DenseNet121, EfficientNetB0, MobileNetV2, and VGG16) are used as a base for transfer learning and tested for their generalization performance on different combinations of synthetic and real image datasets. The models were evaluated by using confusion matrices and four additional metrics. The results show that the pretrained VGG16 model generalizes the best regular reliefs textures (96%), in comparison with the other models, if it is subjected to transfer learning via feature extraction, using mixed dataset, which consist of 34,037 images in following proportions: non-textured synthetic (87%), textured synthetic (8%), and real captured (5%) images of such a regular relief. Full article

(This article belongs to the Special Issue Computer Vision, Pattern Recognition, Machine Learning, and Symmetry, 2nd Edition)

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27 pages, 481 KiB

Open AccessArticle

Hurwicz-Type Optimal Control Problem for Uncertain Singular Non-Causal Systems

by Yuefen Chen and Xin Chen

Symmetry 2025, 17(7), 1130; https://doi.org/10.3390/sym17071130 - 15 Jul 2025

Abstract

Uncertain singular non-causal systems represent a class of singular systems distinguished by the presence of regularity constraints and the involvement of uncertain variables. This paper considers optimal control problems for such systems under the Hurwicz criterion. By integrating dynamic programming with uncertainty theory, [...] Read more.

Uncertain singular non-causal systems represent a class of singular systems distinguished by the presence of regularity constraints and the involvement of uncertain variables. This paper considers optimal control problems for such systems under the Hurwicz criterion. By integrating dynamic programming with uncertainty theory, a recurrence equation is developed to address these problems. This equation has been shown to be effective in handling the optimal control problems of both linear and nonlinear uncertain singular non-causal systems, thereby enabling the derivation of analytical expressions for their corresponding optimal solutions. Moreover, the transformation of the original system into forward and backward subsystems reveals a fundamental temporal and structural symmetry, which significantly contributes to problem simplification. A detailed example is presented to illustrate the proposed results. Full article

(This article belongs to the Special Issue Symmetry in Optimal Control and Applications)

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17 pages, 1331 KiB

Open AccessArticle

A Neural Network Training Method Based on Distributed PID Control

by Kun Jiang

Symmetry 2025, 17(7), 1129; https://doi.org/10.3390/sym17071129 - 14 Jul 2025

Abstract

In the previous article, we introduced a neural network framework based on symmetric differential equations. This novel framework exhibits complete symmetry, endowing it with perfect mathematical properties. While we have examined some of the system’s mathematical characteristics, a detailed discussion of the network [...] Read more.

In the previous article, we introduced a neural network framework based on symmetric differential equations. This novel framework exhibits complete symmetry, endowing it with perfect mathematical properties. While we have examined some of the system’s mathematical characteristics, a detailed discussion of the network training methodology has not yet been presented. Drawing on the principles of the traditional backpropagation algorithm, this study proposes an alternative training approach that utilizes differential equation signal propagation instead of chain rule derivation. This approach not only preserves the effectiveness of training but also offers enhanced biological interpretability. The foundation of this methodology lies in the system’s reversibility, which stems from its inherent symmetry—a key aspect of our research. However, this method alone is insufficient for effective neural network training. To address this, we further introduce a distributed Proportional–Integral–Derivative (PID) control approach, emphasizing its implementation within a closed system. By incorporating this method, we achieved both faster training speeds and improved accuracy. This approach not only offers novel insights into neural network training but also extends the scope of research into control methodologies. To validate its effectiveness, we apply this method to the MNIST (Modified National Institute of Standards and Technology database) and Fashion-MNIST, demonstrating its practical utility. Full article

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11 pages, 263 KiB

Open AccessArticle

On the Conditions Under Which an Algebra over a Field Has a Multiplicative Identity Element

by Małgorzata Jastrzębska

Symmetry 2025, 17(7), 1128; https://doi.org/10.3390/sym17071128 - 14 Jul 2025

Abstract

The set of all annihilators in a ring forms a lattice whose structure reflects the internal properties of the ring. In contrast to the lattices of one-sided ideals, lattices of one-sided annihilators possess a particularly desirable property: symmetry. More precisely, the lattice of [...] Read more.

The set of all annihilators in a ring forms a lattice whose structure reflects the internal properties of the ring. In contrast to the lattices of one-sided ideals, lattices of one-sided annihilators possess a particularly desirable property: symmetry. More precisely, the lattice of left annihilators in a given ring is anti-isomorphic to the lattice of right annihilators. In the literature, attention is primarily focused on rings with identity. In this work, we study lattices of one-sided annihilators in rings without a multiplicative identity element. The properties of rings with an identity element usually differ significantly from those of rings without an identity. We restrict our consideration to rings that are algebras over fields. We indicate certain connections between algebras A without identity and their extensions to algebras

A^{1}

with identity (we consider the Dorroh extension). We highlight similarities and differences in the structure of the lattices of one-sided annihilators in A and those in

A^{1} .

We also identify conditions on annihilators in semiprimary algebras that ensure the existence of an identity element. Furthermore, we show that the existence of an identity in an algebra A is not guaranteed even if A satisfies the condition

A^{2} = A

and possesses a one-sided identity. Full article

(This article belongs to the Section Mathematics)

20 pages, 1565 KiB

Open AccessArticle

Stratified Median Estimation Using Auxiliary Transformations: A Robust and Efficient Approach in Asymmetric Populations

by Abdulaziz S. Alghamdi and Fatimah A. Almulhim

Symmetry 2025, 17(7), 1127; https://doi.org/10.3390/sym17071127 - 14 Jul 2025

Abstract

This study estimates the population median through stratified random sampling, which enhances accuracy by ensuring the proper representation of key population groups. The proposed class of estimators based on transformations effectively handles data variability and enhances estimation efficiency. We examine bias and mean [...] Read more.

This study estimates the population median through stratified random sampling, which enhances accuracy by ensuring the proper representation of key population groups. The proposed class of estimators based on transformations effectively handles data variability and enhances estimation efficiency. We examine bias and mean square error expressions up to the first-order approximation for both existing and newly introduced estimators, establishing theoretical conditions for their applicability. Moreover, to assess the effectiveness of the suggested estimators, five simulated datasets derived from distinct asymmetric distributions (gamma, log-normal, Cauchy, uniform, and exponential), along with actual datasets, are used for numerical analysis. These estimators are designed to significantly enhance the precision and effectiveness of median estimation, resulting in more reliable and consistent outcomes. Comparative analysis using percent relative efficiency (PRE) reveals that the proposed estimators perform better than conventional approaches. Full article

(This article belongs to the Section Mathematics)

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11 pages, 245 KiB

Open AccessArticle

Formulae for Generalization of Touchard Polynomials with Their Generating Functions

by Ayse Yilmaz Ceylan and Yilmaz Simsek

Symmetry 2025, 17(7), 1126; https://doi.org/10.3390/sym17071126 - 14 Jul 2025

Abstract

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator [...] Read more.

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator and functional equations of these functions. Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given. Full article

(This article belongs to the Section Mathematics)

28 pages, 1051 KiB

Open AccessArticle

Probabilistic Load-Shedding Strategy for Frequency Regulation in Microgrids Under Uncertainties

by Wesley Peres, Raphael Paulo Braga Poubel and Rafael Alipio

Symmetry 2025, 17(7), 1125; https://doi.org/10.3390/sym17071125 - 14 Jul 2025

Abstract

This paper proposes a novel integer-mixed probabilistic optimal power flow (IM-POPF) strategy for frequency regulation in islanded microgrids under uncertain operating conditions. Existing load-shedding approaches face critical limitations: continuous frameworks fail to reflect the discrete nature of actual load disconnections, while deterministic models [...] Read more.

This paper proposes a novel integer-mixed probabilistic optimal power flow (IM-POPF) strategy for frequency regulation in islanded microgrids under uncertain operating conditions. Existing load-shedding approaches face critical limitations: continuous frameworks fail to reflect the discrete nature of actual load disconnections, while deterministic models inadequately capture the stochastic behavior of renewable generation and load variations. The proposed approach formulates load shedding as an integer optimization problem where variables are categorized as integer (load disconnection decisions at specific nodes) and continuous (voltages, power generation, and steady-state frequency), better reflecting practical power system operations. The key innovation combines integer load-shedding optimization with efficient uncertainty propagation through Unscented Transformation, eliminating the computational burden of Monte Carlo simulations while maintaining accuracy. Load and renewable uncertainties are modeled as normally distributed variables, and probabilistic constraints ensure operational limits compliance with predefined confidence levels. The methodology integrates Differential Evolution metaheuristics with Unscented Transformation for uncertainty propagation, requiring only 137 deterministic evaluations compared to 5000 for Monte Carlo methods. Validation on an IEEE 33-bus radial distribution system configured as an islanded microgrid demonstrates significant advantages over conventional approaches. Results show 36.5-fold computational efficiency improvement while achieving 95.28% confidence level compliance for frequency limits, compared to only 50% for deterministic methods. The integer formulation requires minimal additional load shedding (21.265%) compared to continuous approaches (20.682%), while better aligning with the discrete nature of real-world operational decisions. The proposed IM-POPF framework successfully minimizes total load shedding while maintaining frequency stability under uncertain conditions, providing a computationally efficient solution for real-time microgrid operation. Full article

(This article belongs to the Special Issue Symmetry and Distributed Power System)

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25 pages, 1429 KiB

Open AccessArticle

A Contrastive Semantic Watermarking Framework for Large Language Models

by Jianxin Wang, Xiangze Chang, Chaoen Xiao and Lei Zhang

Symmetry 2025, 17(7), 1124; https://doi.org/10.3390/sym17071124 - 14 Jul 2025

Abstract

The widespread deployment of large language models (LLMs) has raised urgent demands for verifiable content attribution and misuse mitigation. Existing text watermarking techniques often struggle in black-box or sampling-based scenarios due to limitations in robustness, imperceptibility, and detection generality. These challenges are particularly [...] Read more.

The widespread deployment of large language models (LLMs) has raised urgent demands for verifiable content attribution and misuse mitigation. Existing text watermarking techniques often struggle in black-box or sampling-based scenarios due to limitations in robustness, imperceptibility, and detection generality. These challenges are particularly critical in open-access settings, where model internals and generation logits are unavailable for attribution. To address these limitations, we propose CWS (Contrastive Watermarking with Semantic Modeling)—a novel keyless watermarking framework that integrates contrastive semantic token selection and shared embedding space alignment. CWS enables context-aware, fluent watermark embedding while supporting robust detection via a dual-branch mechanism: a lightweight z-score statistical test for public verification and a GRU-based semantic decoder for black-box adversarial robustness. Experiments on GPT-2, OPT-1.3B, and LLaMA-7B over C4 and DBpedia datasets demonstrate that CWS achieves F1 scores up to 99.9% and maintains F1 ≥ 93% under semantic rewriting, token substitution, and lossy compression (ε ≤ 0.25, δ ≤ 0.2). The GRU-based detector offers a superior speed–accuracy trade-off (0.42 s/sample) over LSTM and Transformer baselines. These results highlight CWS as a lightweight, black-box-compatible, and semantically robust watermarking method suitable for practical content attribution across LLM architectures and decoding strategies. Furthermore, CWS maintains a symmetrical architecture between embedding and detection stages via shared semantic representations, ensuring structural consistency and robustness. This semantic symmetry helps preserve detection reliability across diverse decoding strategies and adversarial conditions. Full article

(This article belongs to the Section Computer)

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16 pages, 2159 KiB

Open AccessArticle

A General Model Construction and Operating State Determination Method for Harmonic Source Loads

by Zonghua Zheng, Yanyi Kang and Yi Zhang

Symmetry 2025, 17(7), 1123; https://doi.org/10.3390/sym17071123 - 14 Jul 2025

Abstract

The widespread integration of power electronic devices and renewable energy sources into power systems has significantly exacerbated voltage and current waveform distortion issues, where asymmetric loads—including single-phase nonlinear equipment and unbalanced three-phase power electronic installations—serve as critical harmonic sources whose inherent nonlinear and [...] Read more.

The widespread integration of power electronic devices and renewable energy sources into power systems has significantly exacerbated voltage and current waveform distortion issues, where asymmetric loads—including single-phase nonlinear equipment and unbalanced three-phase power electronic installations—serve as critical harmonic sources whose inherent nonlinear and asymmetric characteristics increasingly compromise power quality. To enhance power quality management, this paper proposes a universal harmonic source modeling and operational state identification methodology integrating physical mechanisms with data-driven algorithms. The approach establishes an RL-series equivalent impedance model as its physical foundation, employing singular value decomposition and Z-score criteria to accurately characterize asymmetric load dynamics; subsequently applies Variational Mode Decomposition (VMD) to extract time-frequency features from equivalent impedance parameters while utilizing Density-Based Spatial Clustering (DBSCAN) for the high-precision identification of operational states in asymmetric loads; and ultimately constructs state-specific harmonic source models by partitioning historical datasets into subsets, substantially improving model generalizability. Simulation and experimental validations demonstrate that the synergistic integration of physical impedance modeling and machine learning methods precisely captures dynamic harmonic characteristics of asymmetric loads, significantly enhancing modeling accuracy, dynamic robustness, and engineering practicality to provide an effective assessment framework for power quality issues caused by harmonic source integration in distribution networks. Full article

(This article belongs to the Special Issue Power Supply-Demand Asymmetry and Power Quality Issues in High-Renewable Energy Power Systems)

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