Symmetry

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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 398

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

Viewed by 370

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

Viewed by 554

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 359

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 385

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 584

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

Viewed by 232

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

Viewed by 419

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 247

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

Jump to: Research

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

Viewed by 268

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 AccessReview

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 5 | Viewed by 497

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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