Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems
Oxidation Potential of 2,6-Dimethyl-1,4-dihydropyridine Derivatives Estimated by Structure Descriptors
In Silico Comparison of Quantum and Bioactivity Parameters of a Series of Natural Diphenyl Acetone Analogues, and In Vitro Caco-2 Studies on Three Main Chalcone Derivatives
A Survey of Dynamical and Gravitational Lensing Tests in Scale Invariance: The Fall of Dark Matter?
Magnetohydrodynamic Waves in Asymmetric Waveguides and Their Applications in Solar Physics—A Review

Journal Description

Symmetry

Symmetry is an international, peer-reviewed, open access journal covering research on symmetry/asymmetry phenomena wherever they occur in all aspects of natural sciences. Symmetry is published monthly online by MDPI.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within SCIE (Web of Science), Scopus, CAPlus / SciFinder, Inspec, Astrophysics Data System, and other databases.
Journal Rank: JCR - Q2 (Multidisciplinary Sciences) / CiteScore - Q1 (General Mathematics )
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 17.3 days after submission; acceptance to publication is undertaken in 3.7 days (median values for papers published in this journal in the second half of 2024).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Testimonials: See what our editors and authors say about Symmetry.

Impact Factor: 2.2 (2023); 5-Year Impact Factor: 2.3 (2023)

Imprint Information Journal Flyer Open Access ISSN: 2073-8994

Latest Articles

30 pages, 595 KiB

Open AccessArticle

Dual-Performance Multi-Subpopulation Adaptive Restart Differential Evolutionary Algorithm

by Yong Shen, Yunlu Xie and Qingyi Chen

Symmetry 2025, 17(2), 223; https://doi.org/10.3390/sym17020223 (registering DOI) - 3 Feb 2025

Abstract

To cope with common local optimum traps and balance exploration and development in complex multi-peak optimisation problems, this paper puts forth a Dual-Performance Multi-subpopulation Adaptive Restart Differential Evolutionary Algorithm (DPR-MGDE) as a potential solution. The algorithm employs a novel approach by utilising the fitness and historical update frequency as dual-performance metrics to categorise the population into three distinct sub-populations: PM (the promising individual set), MM (the medium individual set) and UM (the un-promising individual set). The multi-subpopulation division mechanism enables the algorithm to achieve a balance between global exploration, local exploitation and diversity maintenance, thereby enhancing its overall optimisation capability. Furthermore, the DPR-MGDE incorporates an adaptive cross-variation strategy, which enables the dynamic adjustment of the variation factor and crossover probability in accordance with the performance of the individuals. This enhances the flexibility of the algorithm, allowing for the prioritisation of local exploitation among the more excellent individuals and the exploration of new search space among the less excellent individuals. Furthermore, the algorithm employs a collision-based Gaussian wandering restart strategy, wherein the collision frequency serves as the criterion for triggering a restart. Upon detecting population stagnation, the updated population is subjected to optimal solution-guided Gaussian wandering, effectively preventing the descent into local optima. Through experiments on the CEC2017 benchmark functions, we verified that DPR-MGDE has higher solution accuracy compared to newer differential evolution algorithms, and proved its significant advantages in complex optimisation tasks with the Wilcoxon test. In addition to this, we also conducted experiments on real engineering problems to demonstrate the effectiveness and superiority of DPR-MGDE in dealing with real engineering problems. Full article

(This article belongs to the Special Issue Symmetry in Intelligent Algorithms)

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25 pages, 3161 KiB

Open AccessArticle

Estimation and Bayesian Prediction for the Generalized Exponential Distribution Under Type-II Censoring

by Wei Wang and Wenhao Gui

Symmetry 2025, 17(2), 222; https://doi.org/10.3390/sym17020222 - 2 Feb 2025

Abstract

This research focuses on the prediction and estimation problems for the generalized exponential distribution under Type-II censoring. Firstly, maximum likelihood estimations for the parameters of the generalized exponential distribution are computed using the EM algorithm. Additionally, confidence intervals derived from the Fisher information matrix are developed and analyzed alongside two bootstrap confidence intervals for comparison. Compared to classical maximum likelihood estimation, Bayesian inference proves to be highly effective in handling censored data. This study explores Bayesian inference for estimating the unknown parameters, considering both symmetrical and asymmetrical loss functions. Utilizing Gibbs sampling to produce Markov Chain Monte Carlo samples, we employ an importance sampling approach to obtain Bayesian estimates and compute the corresponding highest posterior density (HPD) intervals. Furthermore, for one-sample prediction and, separately, for the two-sample case, we provide the corresponding posterior distributions, along with methods for computing point predictions and predictive intervals. Through Monte Carlo simulations, we evaluate the performance of Bayesian estimation in contrast to maximum likelihood estimation. Finally, we conduct an analysis of a real dataset derived from deep groove ball bearings, calculating Bayesian point predictions and predictive intervals for future samples. Full article

(This article belongs to the Special Issue Bayesian Statistical Methods for Forecasting)

27 pages, 10555 KiB

Open AccessArticle

Research on the Method of Calculating Anchorage Flaw Length Based on the Propagation Characteristics of Bolt Excitation Stress Waves

by Chuan Li, Chuanming Li, Qinghua Han, Wanrong Liu, Jianguo Cao, Hu Li and Yikai Wang

Symmetry 2025, 17(2), 221; https://doi.org/10.3390/sym17020221 - 2 Feb 2025

Abstract

This research aimed to detect the defects of anchoring agents’ empty slurries in anchor support. The influence of anchoring defects on the propagation law of stress waves was comprehensively investigated using laboratory tests, theoretical calculations, and other methods. The characteristic modal components with symmetry and periodicity laws were extracted by adopting a variable modal decomposition (VMD) signal decomposition method. It was found that the bottom reflection time of stress waves had an inverse function relationship with the length of the anchorage flaw. The average propagation speed of the stress wave in the free rod was obtained as 5150 m/s, and the average consolidation wave speed was 4198 m/s. The calculation method of the bolt flaw length was finally proposed. After experimental verification, the average error rate was 2.65%, which meets the requirement of testing accuracy in the engineering field, which provides a guarantee for safe production. Full article

(This article belongs to the Section Engineering and Materials)

13 pages, 10346 KiB

Open AccessArticle

Charging a Dimerized Quantum XY Chain

by Riccardo Grazi, Fabio Cavaliere, Niccolò Traverso Ziani and Dario Ferraro

Symmetry 2025, 17(2), 220; https://doi.org/10.3390/sym17020220 - 2 Feb 2025

Abstract

Quantum batteries are quantum systems designed to store energy and release it on demand. The optimization of their performance is an intensively studied topic within the realm of quantum technologies. Such optimization forces the question: how do quantum many-body systems work as quantum batteries? To address this issue, we rely on symmetry and symmetry breaking via quantum phase transitions. Specifically, we analyze a dimerized quantum XY chain in a transverse field as a prototype of an energy storage device. This model, which is characterized by ground states with different symmetries depending on the Hamiltonian parameters, can be mapped onto a spinless fermionic chain with superconducting correlations, displaying a rich quantum phase diagram. We show that the stored energy strongly depends on the quantum phase diagram of the model when large charging times are considered. Full article

(This article belongs to the Section Physics)

23 pages, 23885 KiB

Open AccessArticle

Bézier Curves and Surfaces with the Blending (α, λ, s)-Bernstein Basis

by İlhan Karakılıç, Sedef Karakılıç, Gülter Budakçı and Faruk Özger

Symmetry 2025, 17(2), 219; https://doi.org/10.3390/sym17020219 - 1 Feb 2025

Abstract

This study presents a generalized approach to Bézier curves and surfaces by utilizing the blending

(α, λ, s)

-Bernstein basis. The

(α, λ, s)

-Bernstein basis introduces shape parameters

α

λ

, and s [...] Read more.

This study presents a generalized approach to Bézier curves and surfaces by utilizing the blending

(α, λ, s)

-Bernstein basis. The

(α, λ, s)

-Bernstein basis introduces shape parameters

α

λ

, and s, which allow for more flexibility and control over the curve’s shape compared to the classical Bernstein basis. The paper explores the properties of these generalized curves and surfaces, demonstrating their ability to maintain essential geometric characteristics, such as convex hull containment and endpoint interpolation, while providing enhanced control over the shape. This work aims to contribute to the fields of computer-aided geometric design and related applications by offering a robust tool for curve and surface modeling. Full article

(This article belongs to the Special Issue Computer-Aided Geometric Design and Matrices)

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Figure 1

31 pages, 457 KiB

Open AccessArticle

Safety of Uncertain Session Types with Interval Probability

by Bogdan Aman and Gabriel Ciobanu

Symmetry 2025, 17(2), 218; https://doi.org/10.3390/sym17020218 - 1 Feb 2025

Abstract

In this article, we introduce and study a new kind of multiparty session type for a probabilistic process calculus that incorporates two forms of choice: probabilistic and nondeterministic. The main novelty of our approach is the use of interval probabilities in a type system in order to deal with uncertainty in a probabilistic process calculus. We present a decidable proof system that ensures deadlock freedom, type preservation and type safety, even when several types can be assigned to a process. The new typing system represents a conservative extension of the standard typing system based on multiparty session types (removing the probabilities from processes does not affect their well-typedness). We also define a probabilistic bisimulation between processes that are typed by using the same sorting and typing. Full article

(This article belongs to the Section Mathematics)

42 pages, 2052 KiB

Open AccessArticle

BCA: Besiege and Conquer Algorithm

by Jianhua Jiang, Xianqiu Meng, Jiaqi Wu, Jun Tian, Gaochao Xu and Weihua Li

Symmetry 2025, 17(2), 217; https://doi.org/10.3390/sym17020217 - 1 Feb 2025

Abstract

This paper introduces a bio-inspired meta-heuristic algorithm, the Besiege and Conquer Algorithm (BCA), developed to tackle complex and high-dimensional optimization problems. Drawing inspiration from the concept of symmetry and guerrilla warfare strategies, the BCA incorporates four core components: besiege, conquer, balance, and feedback. The besiege strategy strengthens exploration, while the conquer strategy enhances exploitation. Balance and feedback mechanisms maintain a dynamic equilibrium between these capabilities, ensuring robust optimization performance. The algorithm’s effectiveness is validated through benchmark test functions, demonstrating superior results in comparison with existing methods, supported by Friedman rankings and Wilcoxon signed-rank tests. Beyond theoretical and experimental validation, the BCA showcases its real-world relevance through applications in engineering design and classification problems, addressing practical challenges. These results underline the algorithm’s strong exploration, exploitation, and convergence capabilities and its potential to contribute meaningfully to diverse real-world domains. Full article

(This article belongs to the Section Mathematics)

11 pages, 382 KiB

Open AccessCommunication

Thioureas Derived from (S)-1-(2-pyridyl)ethylamine Enantiomer: Synthesis and Selected Applications as an Organocatalyst

by Jacek Chrzanowski, Luca Sancineto, Malgorzata Deska, Michal Rachwalski and Jozef Drabowicz

Symmetry 2025, 17(2), 216; https://doi.org/10.3390/sym17020216 - 31 Jan 2025

Abstract

In order to expand the group of chiral thiourea structures, several optically active thioureas derived from the (S)-1-(2-pyridyl)ethylamine enantiomer were prepared via its reaction with achiral or optically active isothiocyanates. To show their synthetic potential as chiral auxiliaries the isolated thioureas were tested as an optically active organocatalyst in the asymmetric version of the selected aldol condensation and addition of diethylzinc to benzaldehyde. The observation of asymmetric induction in these model reactions encourages further research on the use of this group of thioureas in asymmetric versions of multicomponent reactions and cycloadditions. The mechanistic aspects of the reactions under study are also briefly discussed. Full article

(This article belongs to the Special Issue Asymmetric Centers in Optically Active Molecules Containing a Heteroatom)

13 pages, 1216 KiB

Open AccessArticle

Symmetry Study on Damage Inversion of Wharf Pile Foundation in Three Gorges Reservoir Area Under Ship Impact

by Liangdong Zuo, Quanbao Wang, Jia Liu and Jie Li

Symmetry 2025, 17(2), 215; https://doi.org/10.3390/sym17020215 - 31 Jan 2025

Abstract

Periodic change in reservoir water level will have a significant impact on berthing position, and the impact caused by irregular operation during berthing will cause damage to wharf pile foundations. However, most of the existing monitoring methods adopt irregular methods, so it is difficult to accurately identify and analyze the damage causes. Taking a high-piled wharf in the Three Gorges Reservoir area as an example, the uncertainty of reservoir water level change is quantitatively analyzed. By establishing a simplified parametric wharf calculation model, the data set of an inversion model of pile of a high-piled wharf under ship impact is obtained, and the inversion analysis of pile damage of a high-piled wharf under ship pile is carried out based on the artificial neural network model. The results show that the inversion model can accurately and efficiently identify the intensity of ship impact, and a low water level is better than a high water level in the identification of impact position. In this paper, the behavior of wharf structure before and after damage is analyzed symmetrically under the action of damage inducement. In summary, the inversion analysis method can basically meet the requirements of inversion identification of pile foundation damage of a high-pile wharf in a backwater fluctuation area under ship impact. Full article

(This article belongs to the Special Issue Advanced Materials, Structures, Symmetrical Design and Mechanism in Mechanical Engineering)

17 pages, 300 KiB

Open AccessArticle

Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion

by Chen Fei, Luzhen Yang and Weiyin Fei

Symmetry 2025, 17(2), 214; https://doi.org/10.3390/sym17020214 - 31 Jan 2025

Abstract

This paper focuses on the quasi-sure exponential stability of the stochastic differential delay equation driven by G-Brownian motion (SDDE-GBM): [...] Read more.

This paper focuses on the quasi-sure exponential stability of the stochastic differential delay equation driven by G-Brownian motion (SDDE-GBM):

d ξ (t) = f (t, ξ (t - κ_{1} (t))) d t + g (t, ξ (t - κ_{2} (t))) d Z (t),

where

κ_{1} (\cdot), κ_{2} (\cdot) : R_{+} \to [0, τ]

denote variable delays, and

Z (t)

denotes scalar G-Brownian motion, which has a symmetry distribution. It is shown that the SDDE-GBM is quasi-surely exponentially stable for each

τ > 0

bounded by

τ^{*}

, where the corresponding (non-delay) stochastic differential equation driven by G-Bronwian motion (SDE-GBM),

d η (t) = f (t, η (t)) d t + g (t, η (t)) d Z (t)

, is quasi-surely exponentially stable. Moreover, by solving the non-linear equation on

τ

, we can obtain the implicit lower bound

τ^{*}

. Finally, illustrating examples are provided. Full article

(This article belongs to the Special Issue Symmetric or Asymmetric Distributions and Its Applications)

18 pages, 1632 KiB

Open AccessArticle

Diffusion Mechanisms for Both Living and Dying Trees Across 37 Years in a Forest Stand in Lithuania’s Kazlų Rūda Region

by Edmundas Petrauskas and Petras Rupšys

Symmetry 2025, 17(2), 213; https://doi.org/10.3390/sym17020213 - 31 Jan 2025

Abstract

This study aimed to examine changes in the number of live and dying trees in central Lithuanian forests over time. Results were obtained using stochastic differential equations combined with the normal copula function. The examination of each tree’s individual size variables (height and diameter) showed that the mean values of dead or dying trees’ size variables had significantly lower trajectories that were particularly pronounced in mature stands. According to the data set under examination, the tree mortality rate gradually declined with age, reaching approximately 7% after 10 years. Birch trees 60–70 years old were the first species to reach the 1% mortality rate, followed by spruce trees 70–80 years old and pine trees 80–90 years old. The Maple symbolic algebra system was used to implement all results. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry of Differential Equations in Biomathematics)

23 pages, 4620 KiB

Open AccessArticle

Chaos Regulation via Complex Nonlinear Feedback and Its Implementation Based on FPAA

by Jitong Xu, Chunbiao Li, Xiaoliang Cen, Xin Zhang and Lin Chai

Symmetry 2025, 17(2), 212; https://doi.org/10.3390/sym17020212 - 31 Jan 2025

Abstract

Complex nonlinear feedback is a key factor in the generation of chaos. In many cases, complex nonlinear functions have a higher probability for chaos producing, and correspondingly new bifurcations may be triggered in the dynamical system. Due to the difficulty in circuit implementation of complex nonlinear feedback, researchers often introduce simple nonlinear constraints to study the occurrence and evolution of chaos. In fact, the impact of complex nonlinear feedback on chaotic dynamics deserves further investigation. In this work, complex nonlinear feedback is introduced into an offset-boostable chaotic system as an example to observe and analyze its regulatory effect on the dynamics. Complex nonlinear feedback may destroy the property of symmetry of a system; therefore, we examine the evolution of chaotic attractors under the corresponding feedback and the functional transformation between bifurcation and non-bifurcation parameters as well. By fully utilizing the flexible configuration advantages of Field Programmable Analog Array (FPAA), arbitrary complex nonlinear functions are implemented to verify the chaotic dynamics. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Chaos Theory and Application)

15 pages, 301 KiB

Open AccessArticle

Inclusive Subclasses of Bi-Univalent Functions Defined by Error Functions Subordinate to Horadam Polynomials

by Tariq Al-Hawary, Basem Frasin, Daniel Breaz and Luminita-Ioana Cotîrlă

Symmetry 2025, 17(2), 211; https://doi.org/10.3390/sym17020211 - 30 Jan 2025

Abstract

In this paper, by utilizing error functions subordinate to Horadam polynomials, we introduce the inclusive subclasses

A (a, ς, r, u, η, ρ, σ),

[...] Read more.

In this paper, by utilizing error functions subordinate to Horadam polynomials, we introduce the inclusive subclasses

A (a, ς, r, u, η, ρ, σ),

B (a, ς, r, u, τ, θ)

and

C (a, ς, r, u, τ, θ)

of bi-univalent functions in the symmetric unit disk U. For functions in these subclasses, we derive estimations for the Maclaurin coefficients

| k_{2} |

and

| k_{3} |,

as well as the Fekete–Szegö functional. Additionally, some related results are also obtained. Full article

(This article belongs to the Special Issue Geometric Function Theory and Special Functions II)

27 pages, 860 KiB

Open AccessArticle

Equal Division Contribution Values of Trapezoidal Fuzzy Numbers and Their Application to Profit Allocation in Cold Chain Logistics for Agricultural Products

by Jungan Zhan, Rong Fan, Minghao Liu, Jiacai Liu and Wenjian Zhao

Symmetry 2025, 17(2), 210; https://doi.org/10.3390/sym17020210 - 29 Jan 2025

Abstract

With the acceleration of fresh food e-commerce development, cold chain logistics for agricultural products has increasingly become a research hotspot. However, limited by the number of orders accepted by enterprises, many cold chain transportation vehicles for agricultural products struggle to reach a full load. This undoubtedly increases transportation costs for agricultural product cold chain logistics enterprises. In order to reduce the cost of transportation and increase the profit of enterprises, this paper will adopt the strategy of building enterprise coalition based on cooperative game theory. By increasing the loading rate of transportation vehicles, it will increase the profit of enterprises. First, utilizing the minimization of overall dissatisfaction among players in profit allocation after coalition participation as the objective function, a model of the equal division contribution values of the trapezoidal fuzzy number will be constructed, which will be used as the profit allocation model for the players. Then, the solution of the model will be provided, and the properties including symmetry are analyzed. Second, by improving the loading rate of cold chain transport vehicles as the key and combining various parameters in the transportation stage of agricultural product cold chain logistics, the coalition profit will be calculated. Finally, using the solution of the equal division contribution value of the trapezoidal fuzzy number in the cooperative game as the allocation strategy, the obtained profit will be distributed to each enterprise participating in the coalition. The results show that when dealing with cooperative profit allocation problems in similar scenarios, the solutions of the equal division contribution value of the trapezoidal fuzzy number are highly reliable and adaptable. The method presented in this paper can not only increase the profit of enterprises but also minimize the overall dissatisfaction of all enterprises with the allocation result. Full article

(This article belongs to the Section Computer)

22 pages, 312 KiB

Open AccessArticle

Four Classes of Symmetric Sums over Cyclically Binomial Products

by Marta Na Chen and Wenchang Chu

Symmetry 2025, 17(2), 209; https://doi.org/10.3390/sym17020209 - 29 Jan 2025

Abstract

Four classes of multiple symmetric sums over cyclic products of binomial coefficients are examined. By incorporating the generating function approach and recursive construction method, they are expressed analytically as coefficients of rational functions. Several recurrence relations and generating functions are explicitly determined when [...] Read more.

(This article belongs to the Special Issue Symmetry in Mathematical Theory and Simulation Methods for Backward Problems)

23 pages, 1349 KiB

Open AccessArticle

A Path-Planning Method Recommended for Multi-UAV Police Patrols Based on the Wolf Pack Optimization Algorithm Using CDRS and DRSS

by Dongxing Wang, Meijing Zhang and Zhiyang Huang

Symmetry 2025, 17(2), 208; https://doi.org/10.3390/sym17020208 - 29 Jan 2025

Abstract

Multi-UAV path planning for police patrols plays an important role in public security work, and while many path-planning algorithms have been applied in this area, all of them possess various degrees of shortcomings. To further improve the accuracy and efficiency of multi-UAV path planning for police patrols, this paper proposes a multi-UAV police patrol path-planning method based on an improved wolf pack optimization algorithm using the strategies of Composite Directional Raid and Dynamic Random Search (PMU-3PM-IWPA). Firstly, a multi-UAV police patrol path-planning model was constructed to reflect the planning problem for multi-UAV police patrol paths (PMU-3PM). Moreover, to enhance the performance of the existing wolf pack optimization algorithm, this paper proposes an improved wolf pack optimization algorithm (for short CDR-DRS-WPOA), including the Composite-Directional Raid Strategy (CDRS), aimed at enhancing global exploration capability, as well as the Dynamic Random Search Strategy (DRSS), with a view to speeding up the convergence for simple problems and heightening the optimization accuracy for difficult problems. Finally, the improved wolf pack optimization algorithm was adopted to solve the issue of multi-UAV path planning for police patrols, and numerical experiments were carried out on 20 public classical datasets as well as PMU-3PM compared with GA, PSO, WDX-WPOA, and DAF-BRS-CWOA. The results indicate that CDR-DRS-WPOA took 20~80% less time and possessed more optimization accuracy, and that PMU-3PM-IWPA based on CDR-DRS-WPOA offers excellent performance. Full article

(This article belongs to the Section Computer)

17 pages, 416 KiB

Open AccessArticle

A Simplex Model of Long Pathways in the Brain Related to the Minimalist Program in Linguistics

by Atsuhide Mori

Symmetry 2025, 17(2), 207; https://doi.org/10.3390/sym17020207 - 29 Jan 2025

Abstract

Marcolli, Chomsky, and Berwick described the minimalist program, proposed by Chomsky in generative linguistics, as an algebra of binary trees in an analogy of quantum physics on Feynman diagrams. In this paper, we proposed another model of the minimalist program based on simplicial Hodge theory by taking the relevant brain neural network into account. We focused on a long directed pathway connecting distant areas in the brain, and took the (abstract) simplex spanning the locations on the terminal area, which the signals going through the pathway can reach. The identity of each signal is represented by the symmetry of the corresponding face, consisting of locations receiving the signal simultaneously. Then, we showed that this model fits the minimalist program. Further, we calculated the spectrum and eigenspaces of the Hodge Laplacian in important cases and found their surprising rationality. According to this rationality, we could draw pictures of syntactic relations based only on the calculation without using linguistic knowledge. In addition, though word order depends on what language is used, and thus has nothing to do with the minimalist program, planar word arrangements are still possible and within the scope of our model. Full article

(This article belongs to the Section Life Sciences)

14 pages, 274 KiB

Open AccessArticle

The Space of Continuous Linear Functionals on ℓ₁ Approximated by Weakly Symmetric Continuous Linear Functionals

by Taras Vasylyshyn and Andriy Zagorodnyuk

Symmetry 2025, 17(2), 206; https://doi.org/10.3390/sym17020206 - 29 Jan 2025

Abstract

We study the space of continuous linear functionals approximated by weakly symmetric continuous linear functionals on the complex Banach space

ℓ_{1}

of all absolutely summing complex sequences. We construct the sequence of groups of symmetries on

ℓ_{1},

obtain the structure [...] Read more.

We study the space of continuous linear functionals approximated by weakly symmetric continuous linear functionals on the complex Banach space

ℓ_{1}

of all absolutely summing complex sequences. We construct the sequence of groups of symmetries on

ℓ_{1},

obtain the structure of corresponding weakly symmetric continuous linear functionals, find the completion of the space, and construct for it a Schauder basis. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry: Second Edition)

9 pages, 401 KiB

Open AccessArticle

A Versatile Electronic Dimer Exhibiting PT and Anti-PT Symmetry

by Ruiqi Li and Jiayang Gu

Symmetry 2025, 17(2), 205; https://doi.org/10.3390/sym17020205 - 28 Jan 2025

Abstract

We propose a versatile electronic dimer cooperatively coupled by means of mutual induction, capacitance, and resistance. In a lot of related works, the electronic dimer is inductively coupled, with one resonator characterized by positive resistance (dissipation) and the other by negative resistance (amplification). We go beyond this picture by considering capacitive and resistive coupling, and by exploring cases where both resistances are positive, as well as a case where the resonant frequencies of the individual resonators are different. Based on analytical derivation and numerical calculations, we obtain and observe the properties of parity-time (PT), quasi-PT (QPT) and quasi-anti-PT (QAPT) symmetry by adjusting the constitutive parameters of the system. This study provides a versatile and feasible platform for observing PT/anti-PT (APT) symmetry-based phenomena and provides a foundation for further studies on finding PT/APT symmetry in more sophisticated circuits. Full article

(This article belongs to the Section Physics)

34 pages, 908 KiB

Open AccessArticle

Prioritization of Human-Centric and Sustainable City Criteria by Proportional Spherical Fuzzy Analytic Hierarchy Process

by Nursah Alkan and Cengiz Kahraman

Symmetry 2025, 17(2), 204; https://doi.org/10.3390/sym17020204 - 28 Jan 2025

Abstract

Human-centric and sustainable cities have become one of the most popular research areas today. They are based on a holistic approach that aims to both improve people’s quality of life and ensure environmental sustainability. The weighting of numerous tangible and intangible criteria used in the performance evaluation of these cities has emerged as an important problem. The symmetrical representation of vague and imprecise data using fuzzy set theory is an absolute necessity for a successful weighting process. In this study, symmetrical representation of especially intangible criteria is carried out with proportional spherical fuzzy sets. Proportional spherical fuzzy sets provide significant convenience to the expert in determining the membership, non-membership and hesitancy degrees and ensure that the assigned values are more accurate and consistent. In the study, five main human centric and sustainable city (HCSC) criteria and 26 HCSC sub-criteria determined from the literature were weighted. In addition, criteria weights were obtained by classical spherical fuzzy analytic hierarchy process (AHP) for comparison purpose. The first four most important HCSC criteria were determined as water management and conservation, employment/unemployment rate, carbon emission reduction strategies and economic sustainability, respectively. Full article

(This article belongs to the Section Mathematics)

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