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27 pages, 1260 KB

Open AccessArticle

High-Dimensional Semiparametric Analysis of Differential Network Analysis for Matrix-Variate

by Tianying Li and Lu Niu

Symmetry 2026, 18(2), 364; https://doi.org/10.3390/sym18020364 - 15 Feb 2026

Viewed by 505

Abstract

Differential network analysis provides a powerful framework for characterizing how biological networks change across different conditions or external stimuli. With recent advances in high-throughput technologies, matrix-valued data have become increasingly common in biostatistics and medical research. However, most existing differential network methods are [...] Read more.

Differential network analysis provides a powerful framework for characterizing how biological networks change across different conditions or external stimuli. With recent advances in high-throughput technologies, matrix-valued data have become increasingly common in biostatistics and medical research. However, most existing differential network methods are designed for vector-valued observations. As a result, they do not fully exploit the structural information inherent in matrix-valued data. In addition, many of these approaches rely on multivariate normality, an assumption that is often violated in practice, motivating the need for robust inference procedures. In this paper, we develop a semiparametric matrix-variate differential network model that accommodates heavy-tailed or non-Gaussian data distributions. We introduce a rank-based D-trace loss with an

ℓ_{1}

penalty to directly estimate the spatial differential partial correlation matrix. The resulting optimization problem is solved using an efficient ADMM algorithm. We establish the theoretical properties of the proposed estimator, including its consistency under high-dimensional scaling. Extensive simulation studies and real-data analyses further demonstrate that our robust procedure substantially outperforms existing non-robust alternatives. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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48 pages, 2334 KB

Open AccessArticle

Symmetry-Aware Optimized Fuzzy Deep Reinforcement Learning-GRU for Load Balancing in Smart Power Grids

by Mohammad Mahdi Mohammad, Mojdeh Sadat Najafi Zadeh, Seyedkian Rezvanjou, Nuria Serrano, Francisco Hernando-Gallego, Diego Martín and José Vicente Álvarez-Bravo

Symmetry 2026, 18(2), 343; https://doi.org/10.3390/sym18020343 - 12 Feb 2026

Viewed by 778

Abstract

The rapid growth of renewable integration and active consumer participation has made modern power grids increasingly complex and dynamic, where maintaining balanced and efficient energy distribution remains a central challenge. This paper introduces a symmetry-aware optimized fuzzy deep reinforcement learning-gated recurrent unit (OF-DRL-GRU) [...] Read more.

The rapid growth of renewable integration and active consumer participation has made modern power grids increasingly complex and dynamic, where maintaining balanced and efficient energy distribution remains a central challenge. This paper introduces a symmetry-aware optimized fuzzy deep reinforcement learning-gated recurrent unit (OF-DRL-GRU) model that exploits the natural symmetry and asymmetry in demand–generation behavior to achieve stable and adaptive load balancing. The proposed architecture consists of four core modules: a fuzzy logic layer that formulates symmetrically distributed membership functions for interpretable and balanced state transitions; a DRL agent that governs decision actions through a symmetry-preserving reward mechanism balancing exploration and exploitation; a GRU network that models temporal symmetries while performing controlled symmetry-breaking during dynamic fluctuations to enhance generalization; and an improved multi-objective biogeography-based optimization (IMOBBO) algorithm that optimizes fuzzy parameters and model hyper-parameters through adaptive migration alternating between symmetry preservation and deliberate asymmetry, ensuring efficient convergence and global diversity. The synergy among these modules forms a unified symmetry-aware optimization paradigm, reflecting how symmetric structures sustain stability while purposeful asymmetry enhances robustness and adaptivity. The proposed framework is evaluated using three benchmark datasets (UK-DALE, Pecan Street, and REDD) and compared against several advanced and competitive models. Experimental outcomes show that the proposed OF-DRL-GRU model achieves 99.23% accuracy, 99.69% recall, and 99.83% area under the curve (AUC), alongside faster runtime, lower variance, and improved convergence stability. These results demonstrate that incorporating symmetry–asymmetry principles within AI-driven optimization significantly enhances interpretability, resilience, and energy efficiency, paving the way for intelligent, self-adaptive load management in next-generation smart grids. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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24 pages, 979 KB

Open AccessArticle

Analytic Solutions and Solvability of the Polyharmonic Cauchy Problem in

R^{n}

by Iqbol Ergashevich Niyozov, Davron Aslonqulovich Juraev, Rakib Feyruz Efendiev, Davron Shokirovich Fozilov and Ebrahim E. Elsayed

Symmetry 2026, 18(1), 56; https://doi.org/10.3390/sym18010056 - 28 Dec 2025

Viewed by 612

Abstract

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in

R^{n}

, highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions of Laplace and biharmonic equations, [...] Read more.

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in

R^{n}

, highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions of Laplace and biharmonic equations, frequently arise in elasticity, potential theory, and mathematical physics, yet their Cauchy problems are inherently ill-posed. Using hyperspherical harmonics and homogeneous harmonic polynomials, whose orthogonality reflects the underlying rotational and reflectional symmetries, the study constructs explicit, uniformly convergent series solutions. Through analytic continuation of integral representations, necessary and sufficient solvability criteria are established, ensuring convergence of all derivatives on compact domains. Furthermore, newly derived Green-type identities provide a systematic method to reconstruct boundary information and enforce stability constraints. This approach not only generalizes classical Laplace and biharmonic results to higher-order polyharmonic equations but also demonstrates how symmetry governs boundary data admissibility, convergence, and analytic structure, offering both theoretical insights and practical tools for elasticity, inverse problems, and mathematical physics. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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17 pages, 2845 KB

Open AccessArticle

Poisson Mean Homogeneity: Single-Observation Framework with Applications

by Xiaoping Shi, Augustine Wong and Kai Kaletsch

Symmetry 2025, 17(10), 1702; https://doi.org/10.3390/sym17101702 - 10 Oct 2025

Viewed by 546

Abstract

Practical problems often drive the development of new statistical methods by presenting real-world challenges. Testing the homogeneity of n independent Poisson means when only one observation per population is available is considered in this paper. This scenario is common in fields where limited [...] Read more.

Practical problems often drive the development of new statistical methods by presenting real-world challenges. Testing the homogeneity of n independent Poisson means when only one observation per population is available is considered in this paper. This scenario is common in fields where limited data from multiple sources must be analyzed to determine whether different groups share the same underlying event rate or mean. These settings often exhibit underlying structural or spatial symmetries that influence statistical behavior. Traditional methods that rely on large sample sizes are not applicable. Hence, it is crucial to develop techniques tailored to the constraints of single observations. Under the null hypothesis, with large n and a fixed common mean

λ

, the likelihood ratio test statistic (LRTS) is shown to be asymptotically normally distributed, with the mean and variance being approximated by a truncation method and a parametric bootstrap method. Moreover, with fixed n and large

λ

, under the null hypothesis, the LRTS is shown to be asymptotically distributed as a chi-square with

n - 1

degrees of freedom. The Bartlett correction method is applied to improve the accuracy of the asymptotic distribution of the LRTS. We highlight the practical relevance of the proposed method through applications to wildfire and radioactive event data, where correlated observations and sparse sampling are common. Simulation studies further demonstrate the accuracy and robustness of the test under various scenarios, making it well-suited for modern applications in environmental science and risk assessment. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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17 pages, 505 KB

Open AccessArticle

On Doubly-Generalized-Transmuted Distributions

by Barry C. Arnold, Yolanda M. Gómez, Diego I. Gallardo and Héctor W. Gómez

Symmetry 2025, 17(10), 1606; https://doi.org/10.3390/sym17101606 - 27 Sep 2025

Cited by 1 | Viewed by 583

Abstract

Many parametric models can be enriched by introducing additional parameters through transmutation, mixing, or compounding techniques. In this paper, we develop the framework of doubly generalized transmutation models (DGTMs), obtained by the repeated application of rank transmutation maps and their generalizations. We show [...] Read more.

Many parametric models can be enriched by introducing additional parameters through transmutation, mixing, or compounding techniques. In this paper, we develop the framework of doubly generalized transmutation models (DGTMs), obtained by the repeated application of rank transmutation maps and their generalizations. We show that several flexible families already available in the literature can be reinterpreted as instances of double or multiple transmutation, thus unifying apparently disparate constructions under a common perspective. A key feature of DGTMs is their ability to flexibly control symmetry through parameterization, enabling more accurate modeling of asymmetric or heavy-tailed phenomena. We also discuss the potential extension of these models to the bivariate case. In addition, we introduce the gentransmuted R package, Version 1.0, which provides routines for data generation, parameter estimation, and model comparison for generalized transmutation models. Two real data applications illustrate the practical advantages of this approach, highlighting improved model fit relative to classical alternatives. Our results underscore the value of transmutation-based methods as a systematic tool for generating flexible probability distributions and advancing their computational implementation. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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14 pages, 263 KB

Open AccessArticle

PT-Symmetric Dirac Inverse Spectral Problem with Discontinuity Conditions on the Whole Axis

by Rakib Feyruz Efendiev, Davron Aslonqulovich Juraev and Ebrahim E. Elsayed

Symmetry 2025, 17(10), 1603; https://doi.org/10.3390/sym17101603 - 26 Sep 2025

Cited by 1 | Viewed by 816

Abstract

We address the inverse spectral problem for a PT-symmetric Dirac operator with discontinuity conditions imposed along the entire real axis—a configuration that has not been explicitly solved in prior literature. Our approach constructs fundamental solutions via convergent recursive series expansions and establishes their [...] Read more.

We address the inverse spectral problem for a PT-symmetric Dirac operator with discontinuity conditions imposed along the entire real axis—a configuration that has not been explicitly solved in prior literature. Our approach constructs fundamental solutions via convergent recursive series expansions and establishes their linear independence through a constant Wronskian. We derive explicit formulas for transmission and reflection coefficients, assemble them into a PT-symmetric scattering matrix, and demonstrate how both spectral and scattering data uniquely determine the underlying complex-valued, discontinuous potentials. Unlike classical treatments, which assume smoothness or limited discontinuities, our framework handles full-axis discontinuities within a non-Hermitian setting, proving uniqueness and providing a constructive recovery algorithm. This method not only generalizes existing inverse scattering theory to PT-symmetric discontinuous operators but also offers direct applicability to optical waveguides, metamaterials, and quantum field models where gain–loss mechanisms and zero-width resonances are critical. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

21 pages, 10319 KB

Open AccessArticle

A Nonconvex Fractional Regularization Model in Robust Principal Component Analysis via the Symmetric Alternating Direction Method of Multipliers

by Zhili Ge, Siyu Zhang, Xin Zhang and Yingying Xu

Symmetry 2025, 17(10), 1590; https://doi.org/10.3390/sym17101590 - 24 Sep 2025

Viewed by 777

Abstract

This paper addresses the NP-hard problem of solving the rank of a matrix in Robust Principal Component Analysis (RPCA) by proposing a nonconvex fractional regularization approximation. Compared to existing convex regularization (which often yields suboptimal solutions) and nonconvex regularization (which typically requires parameter [...] Read more.

This paper addresses the NP-hard problem of solving the rank of a matrix in Robust Principal Component Analysis (RPCA) by proposing a nonconvex fractional regularization approximation. Compared to existing convex regularization (which often yields suboptimal solutions) and nonconvex regularization (which typically requires parameter selection), the proposed model effectively avoids parameter selection while preserving scale invariance. By introducing an auxiliary variable, we transform the problem into a nonconvex optimization problem with a separable structure. We use a more flexible Symmetric Alternating Direction Method of Multipliers (SADMM) to arrive at a solution and provide a rigorous convergence proof. In numerical experiments involving synthetic data, image recovery, and foreground–background separation for surveillance video, the proposed fractional regularization model demonstrates high computational accuracy, and its performance is comparable to that of many state-of-the-art algorithms. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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18 pages, 3433 KB

Open AccessArticle

Mathematical Modelling of Electrode Geometries in Electrostatic Fog Harvesters

by Egils Ginters and Patriks Voldemars Ginters

Symmetry 2025, 17(9), 1578; https://doi.org/10.3390/sym17091578 - 21 Sep 2025

Cited by 1 | Viewed by 1535

Abstract

This paper presents a comparative mathematical analysis of electrode configurations used in active fog water harvesting systems based on electrostatic ionization. The study begins with a brief overview of fog formation and typology. It also addresses the global relevance of fog as a [...] Read more.

This paper presents a comparative mathematical analysis of electrode configurations used in active fog water harvesting systems based on electrostatic ionization. The study begins with a brief overview of fog formation and typology. It also addresses the global relevance of fog as a decentralized water resource. It also outlines the main methods and collector designs currently employed for fog water capture, both passive and active. The core of the work involves solving the Laplace equation for various electrode geometries to compute electrostatic field distributions and analyze field line density patterns as a proxy for potential water collection efficiency. The evaluated configurations include centered rod–cylinder, symmetric parallel multi-rod, and asymmetric wire–plate layouts, with emphasis on identifying spatial regions of high field line convergence. These regions are interpreted as likely trajectories of charged droplets under Coulombic force influence. The modeling approach enables preliminary assessment of design efficiency without relying on time-consuming droplet-level simulations. The results serve as a theoretical foundation prior to the construction of electrode layouts in the portable HygroCatch experimental harvester and provide insight into how field structure correlates with fog water harvesting performance. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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26 pages, 365 KB

Open AccessArticle

Exact Solutions of Maxwell Vacuum Equations in Petrov Homogeneous Non-Null Spaces

by Valery V. Obukhov

Symmetry 2025, 17(9), 1574; https://doi.org/10.3390/sym17091574 - 20 Sep 2025

Cited by 1 | Viewed by 848

Abstract

The classification of exact solutions of Maxwell vacuum equations for pseudo-Riemannian spaces with spatial symmetry (homogeneous non-null spaces in Petrov) in the presence of electromagnetic fields invariant with respect to the action of the group of space motions is summarized. A new classification [...] Read more.

The classification of exact solutions of Maxwell vacuum equations for pseudo-Riemannian spaces with spatial symmetry (homogeneous non-null spaces in Petrov) in the presence of electromagnetic fields invariant with respect to the action of the group of space motions is summarized. A new classification method is used, common to all homogeneous zero spaces of Petrov. The method is based on the use of canonical reper vectors and on the use of a new approach to the systematization of solutions. The classification results are presented in a form more convenient for further use. Using the previously made refinement of the classification of Petrov spaces, the classification of exact solutions of Maxwell vacuum equations for spaces with the group of motions

G_{3} (V I I I)

is completed. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

13 pages, 556 KB

Open AccessFeature PaperArticle

Fractal Complexity and Symmetry in Lava Flow Emplacement

by Antonio F. Miguel

Symmetry 2025, 17(9), 1502; https://doi.org/10.3390/sym17091502 - 10 Sep 2025

Cited by 2 | Viewed by 745

Abstract

This study presents a cohesive physical model that predicts lava flow morphology by establishing a quantitative link between a lava’s yield strength and its geometric complexity, measured by a prefractal dimension. The model is founded on the principle of symmetry, where the potential [...] Read more.

This study presents a cohesive physical model that predicts lava flow morphology by establishing a quantitative link between a lava’s yield strength and its geometric complexity, measured by a prefractal dimension. The model is founded on the principle of symmetry, where the potential for fracturing and complexity peaks at an intermediate yield strength. This peak in complexity, observed with a predicted prefractal dimension (D_pf) of 1.15 for terrestrial ‘a’ā-like lava, arises from a critical state where a balance between gravitational driving forces and internal resistance allows for the formation of intricate margins. The model demonstrates that as lavas deviate from this optimal strength, becoming either too fluid (pāhoehoe, D_pf = 1.05) or too rigid (rhyolite, D_pf = 1.07), their morphology becomes progressively simpler, representing a symmetrical decline in complexity. Our approach also incorporates the overriding influence of topographic confinement and the temporal evolution of complexity as the lava cools. Validated against terrestrial lavas and successfully applied to lower-gravity environments, the model predicts a reduction in complexity for similar flows on Mars (D_pf = 1.13) and the Moon (D_pf = 1.09), providing a tool for interpreting volcanic processes grounded in the fundamental principles of symmetry and complexity. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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14 pages, 284 KB

Open AccessArticle

Symmetric Analysis of Stability Criteria for Nonlinear Systems with Multi-Delayed Periodic Impulses: Intensity Periodicity and Averaged Delay

by Yao Lu, Dehao Ruan and Quanxin Zhu

Symmetry 2025, 17(9), 1481; https://doi.org/10.3390/sym17091481 - 8 Sep 2025

Cited by 5 | Viewed by 947

Abstract

This paper investigates the pth moment exponential stability of random impulsive delayed nonlinear systems (RIDNS) with multiple periodic delayed impulses. Moreover, the continuous dynamics are described by random delay differential equations whose random disturbances are driven by second-order moment processes. Using the periodic [...] Read more.

This paper investigates the pth moment exponential stability of random impulsive delayed nonlinear systems (RIDNS) with multiple periodic delayed impulses. Moreover, the continuous dynamics are described by random delay differential equations whose random disturbances are driven by second-order moment processes. Using the periodic impulsive intensity (PII), average delay time (ADT), average impulsive delay (AID), as well as the Lyapunov method, we present some pth exponential stability criteria for impulsive random delayed nonlinear systems with multiple delayed impulses. Furthermore, the criterion is unified, which is not only applicable to stable or unstable original systems but also takes into account the coexistence of stabilizing and destabilizing impulses. The periodic structure of impulses and their intensities introduces an intrinsic temporal symmetry, which plays a critical role in determining the stability behavior of the system. This symmetry-based perspective highlights the fundamental impact of regularly recurring impulsive actions on system dynamics. Several illustrated examples are given to verify the effectiveness of our results. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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11 pages, 250 KB

Open AccessArticle

The Denseness of the Closure of Some Nyman–Beurling Linear Manifolds Implies the Absence of Zeroes of Certain Combinations of Riemann Zeta-Functions in the Critical Strip

by Sergey K. Sekatskii

Symmetry 2025, 17(9), 1391; https://doi.org/10.3390/sym17091391 - 26 Aug 2025

Viewed by 1814

Abstract

The famous Nyman–Beurling theorem states that the absence of zeroes in the Riemann zeta-function in the half-plane Res > 1/p, p > 1, is equivalent to the circumstance in which the closure of the linear manifold of the functions [...] Read more.

The famous Nyman–Beurling theorem states that the absence of zeroes in the Riemann zeta-function in the half-plane Res > 1/p, p > 1, is equivalent to the circumstance in which the closure of the linear manifold of the functions

f (x) = \sum_{k = 1}^{n} α_{k} \{\frac{ϑ_{k}}{x}\}

, where

0 < ϑ_{k} \leq 1

, with the condition

\sum_{k = 1}^{n} a_{k} ϑ_{k} = 0

, is dense in L_p(0,1). Here, we show that if the closure of linear manifolds of the same functions but with the conditions

\sum_{k = 1}^{n} a_{k} ϑ_{k}^{l} = 0

with l = 2, 3, 4 is dense in L_p(0,1), then certain combinations of Riemann zeta-functions are free from zeroes in the half-plane Res > 1/p, p > 1—like, e.g., the function

g_{2} (s) = \frac{2}{s - 1} ζ (s - 1) + ζ (s)

for l = 2. Similar results for larger integer l can be established. The connections between the Riemann zeta-function, including the question concerning the location of its zeroes, with different symmetry aspects of numerous physical systems are well established, and recently they were highlighted also for supersymmetric quantum mechanics. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

24 pages, 333 KB

Open AccessArticle

Is Gravity Truly Balanced? A Historical–Critical Journey Through the Equivalence Principle and the Genesis of Spacetime Geometry

by Jaume de Haro and Emilio Elizalde

Symmetry 2025, 17(8), 1340; https://doi.org/10.3390/sym17081340 - 16 Aug 2025

Cited by 3 | Viewed by 2195

Abstract

We present a novel derivation of the spacetime metric generated by matter, without invoking Einstein’s field equations. For static sources, the metric arises from a relativistic formulation of D’Alembert’s principle, where the inertial force is treated as a real dynamical entity that exactly [...] Read more.

We present a novel derivation of the spacetime metric generated by matter, without invoking Einstein’s field equations. For static sources, the metric arises from a relativistic formulation of D’Alembert’s principle, where the inertial force is treated as a real dynamical entity that exactly compensates gravity. This leads to a conformastatic metric whose geodesic equation—parametrized by proper time—reproduces the relativistic version of Newton’s second law for free fall. To extend the description to moving matter—uniformly or otherwise—we apply a Lorentz transformation to the static metric. The resulting non-static metric accounts for the motion of the sources and, remarkably, matches the weak-field limit of general relativity as obtained from the linearized Einstein equations in the de Donder (or Lorenz) gauge. This approach—at least at Solar System scales, where gravitational fields are weak—is grounded in a new dynamical interpretation of the Equivalence Principle. It demonstrates how gravity can emerge from the relativistic structure of inertia, without postulating or solving Einstein’s equations. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

24 pages, 1966 KB

Open AccessArticle

A Hybrid Bayesian Machine Learning Framework for Simultaneous Job Title Classification and Salary Estimation

by Wail Zita, Sami Abou El Faouz, Mohanad Alayedi and Ebrahim E. Elsayed

Symmetry 2025, 17(8), 1261; https://doi.org/10.3390/sym17081261 - 7 Aug 2025

Cited by 5 | Viewed by 2372

Abstract

In today’s fast-paced and evolving job market, salary continues to play a critical role in career decision-making. The ability to accurately classify job titles and predict corresponding salary ranges is increasingly vital for organizations seeking to attract and retain top talent. This paper [...] Read more.

In today’s fast-paced and evolving job market, salary continues to play a critical role in career decision-making. The ability to accurately classify job titles and predict corresponding salary ranges is increasingly vital for organizations seeking to attract and retain top talent. This paper proposes a novel approach, the Hybrid Bayesian Model (HBM), which combines Bayesian classification with advanced regression techniques to jointly address job title identification and salary prediction. HBM is designed to capture the inherent complexity and variability of real-world job market data. The model was evaluated against established machine learning (ML) algorithms, including Random Forests (RF), Support Vector Machines (SVM), Decision Trees (DT), and multinomial naïve Bayes classifiers. Experimental results show that HBM outperforms these benchmarks, achieving 99.80% accuracy, 99.85% precision, 100% recall, and an F1 score of 98.8%. These findings highlight the potential of hybrid ML frameworks to improve labor market analytics and support data-driven decision-making in global recruitment strategies. Consequently, the suggested HBM algorithm provides high accuracy and handles the dual tasks of job title classification and salary estimation in a symmetric way. It does this by learning from class structures and mirrored decision limits in feature space. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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16 pages, 278 KB

Open AccessArticle

Maximal Norms of Orthogonal Projections and Closed-Range Operators

by Salma Aljawi, Cristian Conde, Kais Feki and Shigeru Furuichi

Symmetry 2025, 17(7), 1157; https://doi.org/10.3390/sym17071157 - 19 Jul 2025

Viewed by 1929

Abstract

Using the Dixmier angle between two closed subspaces of a complex Hilbert space

H

, we establish the necessary and sufficient conditions for the operator norm of the sum of two orthogonal projections,

P_{W_{1}}

and

P_{W_{2}}

, onto closed [...] Read more.

Using the Dixmier angle between two closed subspaces of a complex Hilbert space

H

, we establish the necessary and sufficient conditions for the operator norm of the sum of two orthogonal projections,

P_{W_{1}}

and

P_{W_{2}}

, onto closed subspaces

W_{1}

and

W_{2}

, to attain its maximum, namely

∥ P_{W_{1}} + P_{W_{2}} ∥ = 2 .

These conditions are expressed in terms of the geometric relationship and symmetry between the ranges of the projections. We apply these results to orthogonal projections associated with a closed-range operator via its Moore–Penrose inverse. Additionally, for any bounded operator T with closed range in

H

, we derive sufficient conditions ensuring

∥ T T^{†} + T^{†} T ∥ = 2,

where

T^{†}

denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges and their algebraic structure governs norm extremality and extends a recent finite-dimensional result to the general Hilbert space setting. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

20 pages, 922 KB

Open AccessArticle

Distributed Time Delay Models: An Alternative to Fractional Calculus-Based Models for Fractional Behavior Modeling

by Jocelyn Sabatier

Symmetry 2025, 17(7), 1101; https://doi.org/10.3390/sym17071101 - 9 Jul 2025

Viewed by 1227

Abstract

This paper illustrates that distributed time delay models can exhibit fractional behaviors, addressing the limitations of fractional calculus-based models outlined in the introduction. Given the extensive results generated by these models, they present a compelling alternative to fractional models. The demonstration is done [...] Read more.

This paper illustrates that distributed time delay models can exhibit fractional behaviors, addressing the limitations of fractional calculus-based models outlined in the introduction. Given the extensive results generated by these models, they present a compelling alternative to fractional models. The demonstration is done both in discrete time and in continuous time. The two cases yield fractional behavior within a defined time/frequency range. To conclude and using two examples, the article highlights that modeling fractional behaviors using distributed delay systems allows for coherent physical interpretations, which a fractional model representation struggles to achieve. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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25 pages, 2709 KB

Open AccessArticle

Dynamics of a Modified Lotka–Volterra Commensalism System Incorporating Allee Effect and Symmetric Non-Selective Harvest

by Kan Fang, Yiqin Wang, Fengde Chen and Xiaoying Chen

Symmetry 2025, 17(6), 852; https://doi.org/10.3390/sym17060852 - 30 May 2025

Viewed by 1802

Abstract

This study investigates a modified Lotka–Volterra commensalism system that incorporates the weak Allee effect in prey and symmetric (equal harvesting effort for both species) non-selective harvesting, addressing a critical gap in ecological modeling. Unlike previous work, we rigorously examine how the interaction between [...] Read more.

This study investigates a modified Lotka–Volterra commensalism system that incorporates the weak Allee effect in prey and symmetric (equal harvesting effort for both species) non-selective harvesting, addressing a critical gap in ecological modeling. Unlike previous work, we rigorously examine how the interaction between the Allee effect and harvesting disrupts system stability, giving rise to novel bifurcation phenomena and population collapse thresholds. Using eigenvalue analysis and the Dulac–Bendixson criterion, we derive sufficient conditions for the existence and stability of equilibria. We find that harvesting destabilizes the system by inducing two saddle-node bifurcations. Notably, prey abundance can increase with greater Allee intensity under controlled harvesting—a rare and counterintuitive ecological outcome. Moreover, exceeding a critical harvesting threshold drives both species to extinction, while controlled harvesting allows sustainable coexistence. Numerical simulations support these analytical findings and identify critical parameter ranges for species coexistence. These results contribute to theoretical ecology and offer insights for designing sustainable harvesting strategies that balance exploitation with conservation. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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15 pages, 277 KB

Open AccessArticle

Harmonic Functions with Montel’s Normalization

by Jacek Dziok

Symmetry 2025, 17(5), 720; https://doi.org/10.3390/sym17050720 - 8 May 2025

Cited by 1 | Viewed by 598

Abstract

In the Geometric Theory of Analytic Functions, classes of functions with several normalizations are considered. We consider the symmetric idea for harmonic functions. Classes of harmonic functions f with normalization

f (0) = f_{\bar{z}}^{'} (0) = 0,

[...] Read more.

In the Geometric Theory of Analytic Functions, classes of functions with several normalizations are considered. We consider the symmetric idea for harmonic functions. Classes of harmonic functions f with normalization

f (0) = f_{\bar{z}}^{'} (0) = 0,

f_{z}^{'} (0) = 1

are usually considered in the geometric theory of harmonic functions. The normalization is called the classical normalization. We can obtain some interesting results by using Montel’s normalization

f (0) = f_{\bar{z}}^{'} (0) = 0,

f_{z}^{'} (ρ) - f_{\bar{z}}^{'} (ρ) = 1

, where

ρ \in [0, 1) .

In the paper, we consider the class of harmonic functions with Montel’s normalization associated with the generalized hypergeometric function. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

17 pages, 13197 KB

Open AccessArticle

Dual Graph Laplacian RPCA Method for Face Recognition Based on Anchor Points

by Shu-Ting Zhuang, Qing-Wen Wang and Jiang-Feng Chen

Symmetry 2025, 17(5), 691; https://doi.org/10.3390/sym17050691 - 30 Apr 2025

Cited by 1 | Viewed by 1139

Abstract

High-dimensional data often contain noise and undancy, which can significantly undermine the performance of machine learning. To address this challenge, we propose an advanced robust principal component analysis (RPCA) model that integrates bidirectional graph Laplacian constraints along with the anchor point technique. This [...] Read more.

High-dimensional data often contain noise and undancy, which can significantly undermine the performance of machine learning. To address this challenge, we propose an advanced robust principal component analysis (RPCA) model that integrates bidirectional graph Laplacian constraints along with the anchor point technique. This approach constructs two graphs from both the sample and feature perspectives for a more comprehensive capture of the underlying data structure. Moreover, the anchor point technique serves to substantially reduce computational complexity, making the model more efficient and scalable. Comprehensive evaluations on both GTdatabase and VGG Face2 dataset confirm that anchor-based methods maintain competitive accuracy with standard graph Laplacian approaches (within 0.5–2.0% difference) while achieving significant computational speedups of 5.7–27.1% and 12.9–14.6% respectively. The consistent performance across datasets, from controlled laboratory conditions to challenging real-world scenarios, demonstrates the robustness and scalability of the proposed anchor technique. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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15 pages, 335 KB

Open AccessArticle

On the Secant Non-Defectivity of Integral Hypersurfaces of Projective Spaces of at Most Five Dimensions

by Edoardo Ballico

Symmetry 2025, 17(3), 454; https://doi.org/10.3390/sym17030454 - 18 Mar 2025

Viewed by 569

Abstract

Let

X \subset P^{n}

, where

3 \leq n \leq 5

, be an irreducible hypersurface of degree

d \geq 2

. Fix an integer

t \geq 3

. If

n = 5

, assume

t \geq 4

and [...] Read more.

Let

X \subset P^{n}

, where

3 \leq n \leq 5

, be an irreducible hypersurface of degree

d \geq 2

. Fix an integer

t \geq 3

. If

n = 5

, assume

t \geq 4

and

(t, d) \neq (4, 2)

. Using the Differential Horace Lemma, we prove that

O_{X} (t)

is not secant defective. For a fixed X, our proof uses induction on the integer t. The key points of our method are that for a fixed X, we only need the case of general linear hyperplane sections of X in lower-dimension projective spaces and that we do not use induction on d, allowing an interested reader to tackle a specific X for

n > 5

. We discuss (as open questions) possible extensions of some weaker forms of the theorem to the case where

n > 5

. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

Review

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37 pages, 862 KB

Open AccessReview

Mathematical Modeling Techniques in Virtual Reality Technologies: An Integrated Review of Physical Simulation, Spatial Analysis, and Interface Implementation

by Junhyeok Lee, Yong-Hyuk Kim and Kang Hoon Lee

Symmetry 2026, 18(2), 255; https://doi.org/10.3390/sym18020255 - 30 Jan 2026

Viewed by 1023

Abstract

Virtual reality (VR) has emerged as a complex technological domain that demands high levels of realism and interactivity. At the core of this immersive experience lies a broad spectrum of mathematical modeling techniques. This survey explores how mathematical foundations support and enhance key [...] Read more.

Virtual reality (VR) has emerged as a complex technological domain that demands high levels of realism and interactivity. At the core of this immersive experience lies a broad spectrum of mathematical modeling techniques. This survey explores how mathematical foundations support and enhance key VR components, including physical simulations, 3D spatial analysis, rendering pipelines, and user interactions. We review differential equations and numerical integration methods (e.g., Euler, Verlet, Runge–Kutta (RK4)) used to simulate dynamic environments, as well as geometric transformations and coordinate systems that enable seamless motion and viewpoint control. The paper also examines the mathematical underpinnings of real-time rendering processes and interaction models involving collision detection and feedback prediction. In addition, recent developments such as physics-informed neural networks, differentiable rendering, and neural scene representations are presented as emerging trends bridging classical mathematics and data-driven approaches. By organizing these elements into a coherent mathematical framework, this work aims to provide researchers and developers with a comprehensive reference for applying mathematical techniques in VR systems. The paper concludes by outlining the open challenges in balancing accuracy and performance and proposes future directions for integrating advanced mathematics into next-generation VR experiences. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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102 pages, 1631 KB

Open AccessReview

A Comprehensive Review on the Generalized Sylvester Equation AX − YB = C

by Qing-Wen Wang and Jiale Gao

Symmetry 2025, 17(10), 1686; https://doi.org/10.3390/sym17101686 - 8 Oct 2025

Cited by 4 | Viewed by 1933

Abstract

Since Roth’s work on the generalized Sylvester equation (GSE)

A X - Y B = C

in 1952, related research has consistently attracted significant attention. Building on this, this review systematically summarizes relevant research on GSE from five perspectives: research methods, constrained solutions, [...] Read more.

Since Roth’s work on the generalized Sylvester equation (GSE)

A X - Y B = C

in 1952, related research has consistently attracted significant attention. Building on this, this review systematically summarizes relevant research on GSE from five perspectives: research methods, constrained solutions, various generalizations, iterative algorithms, and applications. Furthermore, we provide comments on current research, put forward several intriguing questions, and offer prospects for future research trends. We hope this work can fill the gap in the review literature on GSE and offer some inspiration for subsequent studies in the field. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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85 pages, 939 KB

Open AccessReview

An Overview of Methods for Solving the System of Equations A₁XB₁ = C₁ and A₂XB₂ = C₂

by Qing-Wen Wang, Zi-Han Gao and Yu-Fei Li

Symmetry 2025, 17(8), 1307; https://doi.org/10.3390/sym17081307 - 12 Aug 2025

Cited by 7 | Viewed by 850

Abstract

This paper primarily investigates the solutions to the system of equations

A_{1} X B_{1} = C_{1}

and

A_{2} X B_{2} = C_{2}

. This system generalizes the classical equation

A X B = C

, as well [...] Read more.

This paper primarily investigates the solutions to the system of equations

A_{1} X B_{1} = C_{1}

and

A_{2} X B_{2} = C_{2}

. This system generalizes the classical equation

A X B = C

, as well as the system of equations

A X = B

and

X C = D

, and finds broad applications in control theory, signal processing, networking, optimization, and other related fields. Various methods for solving this system are introduced, including the generalized inverse method, the

vec

-operator method, matrix decomposition techniques, Cramer’s rule, and iterative algorithms. Based on these approaches, the paper discusses general solutions, symmetric solutions, Hermitian solutions, and other special types of solutions over different algebraic structures, such as number fields, the real field, the complex field, the quaternion division ring, principal ideal domains, regular rings, strongly *-reducible rings, and operators on Banach spaces. In addition, matrix systems related to the system

A_{1} X B_{1} = C_{1}

and

A_{2} X B_{2} = C_{2}

are also explored. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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81 pages, 2075 KB

Open AccessEditor’s ChoiceReview

A Comprehensive Review on Solving the System of Equations AX = C and XB = D

by Qing-Wen Wang, Zi-Han Gao and Jia-Le Gao

Symmetry 2025, 17(4), 625; https://doi.org/10.3390/sym17040625 - 21 Apr 2025

Cited by 12 | Viewed by 1916

Abstract

This survey provides a review of the theoretical research on the classic system of matrix equations

A X = C

and

X B = D

, which has wide-ranging applications across fields such as control theory, optimization, image processing, and robotics. The paper [...] Read more.

This survey provides a review of the theoretical research on the classic system of matrix equations

A X = C

and

X B = D

, which has wide-ranging applications across fields such as control theory, optimization, image processing, and robotics. The paper discusses various solution methods for the system, focusing on specialized approaches, including generalized inverse methods, matrix decomposition techniques, and solutions in the forms of Hermitian, extreme rank, reflexive, and conjugate solutions. Additionally, specialized solving methods for specific algebraic structures, such as Hilbert spaces, Hilbert

C^{*}

-modules, and quaternions, are presented. The paper explores the existence conditions and explicit expressions for these solutions, along with examples of their application in color images. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2025)

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