Symmetry

16 pages, 973 KiB

Open AccessArticle

Sandwich Theorems for a New Class of Complete Homogeneous Symmetric Functions by Using Cyclic Operator

by Intissar Abdulhur Kadum, Waggas Galib Atshan and Areej Tawfeeq Hameed

Symmetry 2022, 14(10), 2223; https://doi.org/10.3390/sym14102223 - 21 Oct 2022

Cited by 2 | Viewed by 1296

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete’s theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric [...] Read more.

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete’s theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric functions class involving an ordered cyclic operator. In addition, certain sandwich theorems are found. Full article

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

10 pages, 300 KiB

Open AccessArticle

Generalizations of Topological Decomposition and Zeno Sequence in Fibered n-Spaces

by Susmit Bagchi

Symmetry 2022, 14(10), 2222; https://doi.org/10.3390/sym14102222 - 21 Oct 2022

Viewed by 1067

Abstract

The space-time geometry is rooted in the Minkowski 4-manifold. Minkowski and Euclidean topological 4-manifolds behave differently in view of compactness and local homogeneity. As a result, Zeno sequences are selectively admitted in such topological spaces. In this paper, the generalizations of topologically fibered [...] Read more.

The space-time geometry is rooted in the Minkowski 4-manifold. Minkowski and Euclidean topological 4-manifolds behave differently in view of compactness and local homogeneity. As a result, Zeno sequences are selectively admitted in such topological spaces. In this paper, the generalizations of topologically fibered n-spaces are proposed to formulate topological decomposition and the formation of projective fibered n-subspaces. The concept of quasi-compact fibering is introduced to analyze the formation of Zeno sequences in topological n-spaces (i.e., n-manifolds), where a quasi-compact fiber relaxes the Minkowski-type (algebraically) strict ordering relation under topological projections. The topological analyses of fibered Minkowski as well as Euclidean 4-spaces are presented under quasi-compact fibering and topological projections. The topological n-spaces endowed with quasi-compact fibers facilitated the detection of local as well as global compactness and the non-analytic behavior of a continuous function. It is illustrated that the 5-manifold with boundary embedding Minkowski 4-space transformed a quasi-compact fiber into a compact fiber maintaining generality. Full article

(This article belongs to the Special Issue Topological Structures and Analysis with Applications)

25 pages, 2181 KiB

Open AccessArticle

MULBER: Effective Android Malware Clustering Using Evolutionary Feature Selection and Mahalanobis Distance Metric

by Pradeepkumar Duraisamy Soundrapandian and Geetha Subbiah

Symmetry 2022, 14(10), 2221; https://doi.org/10.3390/sym14102221 - 21 Oct 2022

Cited by 1 | Viewed by 1540

Abstract

Symmetric and asymmetric patterns are fascinating phenomena that show a level of co-existence in mobile application behavior analyses. For example, static phenomena, such as information sharing through collaboration with known apps, is a good example of a symmetric model of communication, and app [...] Read more.

Symmetric and asymmetric patterns are fascinating phenomena that show a level of co-existence in mobile application behavior analyses. For example, static phenomena, such as information sharing through collaboration with known apps, is a good example of a symmetric model of communication, and app collusion, where apps collaborate dynamically with unknown malware apps, is an example of a serious threat with an asymmetric pattern. The symmetric nature of app collaboration can become vulnerable when a vulnerability called PendingIntent is exchanged during Inter-Component Communication (ICC). The PendingIntent (PI) vulnerability enables a flexible software model, where the PendingIntent creator app can temporarily share its own permissions and identity with the PendingIntent receiving app. The PendingIntent vulnerability does not require approval from the device user or Android OS to share the permissions and identity with other apps. This is called a PI leak, which can lead to malware attacks such as privilege escalation and component hijacking attacks. This vulnerability in the symmetric behavior of an application without validating an app’s privileges dynamically leads to the asymmetric phenomena that can damage the robustness of an entire system. In this paper, we propose MULBER, a lightweight machine learning method for the detection of Android malware communications that enables a cybersecurity system to analyze multiple patterns and learn from them to help prevent similar attacks and respond to changing behavior. MULBER can help cybersecurity teams to be more proactive in preventing dynamic PI-based communication threats and responding to active attacks in real time. MULBER performs a static binary analysis on the APK file and gathers approximately 10,755 features, reducing it to 42 key features by grouping the permissions under the above-mentioned four categories. Finally, MULBER learns from these multivariate features using evolutionary feature selection and the Mahalanobis distance metric and classifies them as either benign or malware apps. In an evaluation of 22,638 malware samples from recent Android APK malware databases such as Drebin and CICMalDroid-2020, MULBER outperformed others by clustering applications based on the Mahalanobis distance metric and detected 95.69% of malware with few false alarms and the explanations provided for each detection revealed the relevant properties of the detected malware. Full article

(This article belongs to the Special Issue Information Technologies and Electronics Ⅱ)

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28 pages, 45232 KiB

Open AccessArticle

Remarks on Fractal-Fractional Malkus Waterwheel Model with Computational Analysis

by Liliana Guran, Esra Karataş Akgül, Ali Akgül and Monica-Felicia Bota

Symmetry 2022, 14(10), 2220; https://doi.org/10.3390/sym14102220 - 21 Oct 2022

Cited by 4 | Viewed by 1326

Abstract

In this paper, we investigate the fractal-fractional Malkus Waterwheel model in detail. We discuss the existence and uniqueness of a solution of the fractal-fractional model using the fixed point technique. We apply a very effective method to obtain the solutions of the model. [...] Read more.

In this paper, we investigate the fractal-fractional Malkus Waterwheel model in detail. We discuss the existence and uniqueness of a solution of the fractal-fractional model using the fixed point technique. We apply a very effective method to obtain the solutions of the model. We prove with numerical simulations the accuracy of the proposed method. We put in evidence the effects of the fractional order and the fractal dimension for a symmetric Malkus Waterwheel model. Full article

(This article belongs to the Section Mathematics)

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14 pages, 398 KiB

Open AccessArticle

Hidden Euclidean Dynamical Symmetry in the U(n + 1) Vibron Model

by Yu Zhang, Zi-Tong Wang, Hong-Di Jiang and Xin Chen

Symmetry 2022, 14(10), 2219; https://doi.org/10.3390/sym14102219 - 21 Oct 2022

Cited by 4 | Viewed by 1104

Abstract

Based on the boson realization of the Euclidean algebras, it is found that the E(n) dynamical symmetry (DS) may emerge at the critical point of the U(n)-SO(

n + 1

) quantum phase transition. To justify this finding, we [...] Read more.

Based on the boson realization of the Euclidean algebras, it is found that the E(n) dynamical symmetry (DS) may emerge at the critical point of the U(n)-SO(

n + 1

) quantum phase transition. To justify this finding, we provide a detailed analysis of the transitional Hamiltonian in the U(

n + 1

) vibron model in both quantal and classical ways. It is further shown that the low-lying structure of

^{82}

Kr can serve as an excellent empirical realization of the E(5) DS, which provides a specific example of the Euclidean DS in experiments. Full article

(This article belongs to the Special Issue Symmetry in Nuclear Physics: Model Calculations, Advances and Applications)

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16 pages, 11347 KiB

Open AccessArticle

High-Order Topology-Enhanced Graph Convolutional Networks for Dynamic Graphs

by Jiawei Zhu, Bo Li, Zhenshi Zhang, Ling Zhao and Haifeng Li

Symmetry 2022, 14(10), 2218; https://doi.org/10.3390/sym14102218 - 21 Oct 2022

Cited by 6 | Viewed by 1562

Abstract

Understanding the evolutionary mechanisms of dynamic graphs is crucial since dynamic is a basic characteristic of real-world networks. The challenges of modeling dynamic graphs are as follows: (1) Real-world dynamics are frequently characterized by group effects, which essentially emerge from high-order interactions involving [...] Read more.

Understanding the evolutionary mechanisms of dynamic graphs is crucial since dynamic is a basic characteristic of real-world networks. The challenges of modeling dynamic graphs are as follows: (1) Real-world dynamics are frequently characterized by group effects, which essentially emerge from high-order interactions involving groups of entities. Therefore, the pairwise interactions revealed by the edges of graphs are insufficient to describe complex systems. (2) The graph data obtained from real systems are often noisy, and the spurious edges can interfere with the stability and efficiency of models. To address these issues, we propose a high-order topology-enhanced graph convolutional network for modeling dynamic graphs. The rationale behind it is that the symmetric substructure in a graph, called the maximal clique, can reflect group impacts from high-order interactions on the one hand, while not being readily disturbed by spurious links on the other hand. Then, we utilize two independent branches to model the distinct influence mechanisms of the two effects. Learnable parameters are used to tune the relative importance of the two effects during the process. We conduct link predictions on real-world datasets, including one social network and two citation networks. Results show that the average improvements of the high-order enhanced methods are 68%, 15%, and 280% over the corresponding backbones across datasets. The ablation study and perturbation analysis validate the effectiveness and robustness of the proposed method. Our research reveals that high-order structures provide new perspectives for studying the dynamics of graphs and highlight the necessity of employing higher-order topologies in the future. Full article

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11 pages, 5700 KiB

Open AccessArticle

Design of an Optically Transparent Microwave Absorber Based on Coding Metasurface

by Senfeng Lai, Guiyang Liu, Yanpei Guo and Yang Liu

Symmetry 2022, 14(10), 2217; https://doi.org/10.3390/sym14102217 - 21 Oct 2022

Cited by 6 | Viewed by 1465

Abstract

In this paper, a metamaterial absorber with a checkerboard patterned ITO (indium tin oxide) film as the surface is obtained by using flexible and optically transparent wave-absorbing material ITO–PET (polyethylene terephthalate), and a coding arrangement of two basic coding units based on the [...] Read more.

In this paper, a metamaterial absorber with a checkerboard patterned ITO (indium tin oxide) film as the surface is obtained by using flexible and optically transparent wave-absorbing material ITO–PET (polyethylene terephthalate), and a coding arrangement of two basic coding units based on the APS-PSO (Array Pattern Synthesis -Particle Swarm Optimization) algorithm. The surface structure of the absorber consists of ITO rectangular patch structures and ITO circular patch structures (110 Ω/sq). The ITO rectangular patch structures and ITO circular patch structures are symmetrical. The middle layer is made up of two layers of PET and one layer of PMMA, and the bottom surface is covered with a layer of low square resistance ITO film (8 Ω/sq). The experimental results, which are consistent with the simulation results, show that the absorber has superior performance: over 90% absorptance in the 5.06–9.01 GHz band, high transmittance, and a −10 dBsm RCS (radar cross-section) reduction in the 5.3–8.7 GHz band. This design also has polarization insensitivity and angular stability. Full article

(This article belongs to the Special Issue Advances in Metamaterial and Symmetry/Asymmetry)

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30 pages, 3406 KiB

Open AccessArticle

Topological Data Analysis of m-Polar Spherical Fuzzy Information with LAM and SIR Models

by Rukhsana Kausar, Shaista Tanveer, Muhammad Riaz, Dragan Pamucar and Cirovic Goran

Symmetry 2022, 14(10), 2216; https://doi.org/10.3390/sym14102216 - 20 Oct 2022

Cited by 3 | Viewed by 1228

Abstract

The concept of m-polar spherical fuzzy sets (mPSFS) is a combination of m-polar fuzzy sets (mPFS) and spherical fuzzy sets (SFS). An mPSFS is an optimal strategy for addressing multipolarity and fuzziness in terms of ordered triples of positive membership grades (PMGs), negative [...] Read more.

The concept of m-polar spherical fuzzy sets (mPSFS) is a combination of m-polar fuzzy sets (mPFS) and spherical fuzzy sets (SFS). An mPSFS is an optimal strategy for addressing multipolarity and fuzziness in terms of ordered triples of positive membership grades (PMGs), negative membership grades (NMGs), and neutral grades (NGs). In this study, the innovative concept of m-polar spherical fuzzy topology (mPSF-topology) is proposed for data analysis and information aggregation. We look into the characteristics and results of mPSF-topology with the help of several examples. Topological structures on mPSFSs help with both the development of new artificial intelligence (AI) tools for different domain strategies and the study of different kinds of uncertainty in everyday life problems. These strategies make it possible to recognise and look into a situation early on, which helps professionals to reduce certain risks. In order to address various group decision-making issues in the m-polar spherical fuzzy domain, one suggestion has been to apply an extended linear assignment model (LAM) along with the SIR method known as superiority and inferiority ranking methodology in order to analyze road accident issues and dispute resolution. In addition, we examine the symmetry of optimal decision and perform a comparative study between the research carried out using the suggested methodology and several existing methods. Full article

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12 pages, 2201 KiB

Open AccessFeature PaperArticle

Peculiarities of Electron Wave Packet Dynamics in Planar Nanostructures in the Presence of Magnetic and Electric Fields

by Darya Starodubtseva and Olga Tikhonova

Symmetry 2022, 14(10), 2215; https://doi.org/10.3390/sym14102215 - 20 Oct 2022

Viewed by 1224

Abstract

Currently, spatially localized electron densities and currents are considered to be candidates for use in the encoding of quantum information. For this reason, the control of their temporal dynamics is an important task. In this work, the spatiotemporal evolution of an electron wave [...] Read more.

Currently, spatially localized electron densities and currents are considered to be candidates for use in the encoding of quantum information. For this reason, the control of their temporal dynamics is an important task. In this work, the spatiotemporal evolution of an electron wave packet in planar nanostructure in the presence of transverse magnetic and lateral electric fields is investigated by direct analytical solution of the non-stationary Schrödinger equation. Methods to control and manage the dynamics of the spatially localized electron density distribution are developed. The production of photon-like quantum states of electrons opens up opportunities for applications similar to quantum optical and quantum information technologies but implemented with charge carriers. Quantum control of the trajectory of the electron wave packet, accompanied by dramatic suppression of its spreading, is demonstrated. This study discovered methods to manage spatially localized electron behavior in a nanostructure that allows a controllable charge quantum transfer and gives rise to new prospects for quantum nanoelectronics technology. Full article

(This article belongs to the Special Issue Quantum Dynamics in Josephson Junctions and Symmetry)

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20 pages, 5298 KiB

Open AccessFeature PaperArticle

Molecular Symmetry of Permethylated β-Cyclodextrins upon Complexation

by Kostas Bethanis, Elias Christoforides, Athena Andreou and Elias Eliopoulos

Symmetry 2022, 14(10), 2214; https://doi.org/10.3390/sym14102214 - 20 Oct 2022

Cited by 2 | Viewed by 1536

Abstract

The C_n molecular symmetry implicated by the schemes with which cyclodextrins (CDs), the well-known cyclic oligosaccharides, are introduced in the literature, is not valid. Numerous studies have shown that CDs are rather flexible with their macrocycle adopting various conformations that enable the [...] Read more.

The C_n molecular symmetry implicated by the schemes with which cyclodextrins (CDs), the well-known cyclic oligosaccharides, are introduced in the literature, is not valid. Numerous studies have shown that CDs are rather flexible with their macrocycle adopting various conformations that enable the inclusion complexation of guest molecules of various shapes. In this work, the loss and gain of the C₇ symmetry of the heptakis (2, 3, 6-tri-O-methyl)-β-CD (TM-β-CD) is investigated by means of its conformation geometrical features in its hydrated form and upon complexation with molecules of different shapes. For this, the crystal structure of the inclusion complex of a bulky guest molecule (giberellic acid) in TM-β-CD is presented for the first time and compared with the previously determined crystal structures of monohydrated TM-β-CD and the inclusion complex of a linear monoterpenoid (geraniol) in TM-β-CD. The structural investigation was complemented by molecular dynamics simulations in an explicit solvent, based on the crystallographically determined models. The crucial role of the guest, in the symmetry gain of the host, reveals a pronounced induced-fit complexation mechanism for permethylated CDs. Full article

(This article belongs to the Special Issue Recent Advance in Biochemistry, Genetics and Molecular Biology)

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15 pages, 53988 KiB

Open AccessFeature PaperArticle

Toward a New Generation of Compact Transportable Yb⁺ Optical Clocks

by Ksenia Khabarova, Denis Kryuchkov, Alexander Borisenko, Ilia Zalivako, Ilya Semerikov, Mikhail Aksenov, Ivan Sherstov, Timur Abbasov, Anton Tausenev and Nikolay Kolachevsky

Symmetry 2022, 14(10), 2213; https://doi.org/10.3390/sym14102213 - 20 Oct 2022

Cited by 9 | Viewed by 2533

Abstract

Optical atomic clocks are currently one of the most sensitive tools making it possible to precisely test the fundamental symmetry properties of spacetime and Einstein’s theory of relativity. At the same time, the extremely high stability and accuracy of compact transportable optical clocks [...] Read more.

Optical atomic clocks are currently one of the most sensitive tools making it possible to precisely test the fundamental symmetry properties of spacetime and Einstein’s theory of relativity. At the same time, the extremely high stability and accuracy of compact transportable optical clocks open new perspectives in important fields, such as satellite navigation, relativistic geodesy, and the global time and frequency network. Our project aimed to develop a compact transportable optical clock based on a single ytterbium ion. We present the first prototype of the Yb

^{+}

clock (298 kg in 1 m

^{3}

) and present several solutions aimed to improve the clock’s robustness to approach the demands of a space-qualified system. We present spectroscopic studies of a 435.5 nm quadrupole clock transition with Fourier-limited spectra of 25 Hz. The estimated instability of the output frequency at 1 GHz, which was down-converted with an optical frequency comb (OFC), is at the level of

9 \times 10^{- 15} / \sqrt{τ}

, and the long-term instability and inaccuracy are at the level of

5 \times 10^{- 16}

. As the next steps, we present a new design for the clock laser and the OFC. Full article

(This article belongs to the Special Issue Selected Papers on Lasers, Accelerators, and Optics to Celebrate the 100th Anniversary of the Birth of N.G. Basov)

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18 pages, 7625 KiB

Open AccessArticle

Impact of Irregular Heat Sink/Source on the Wall Jet Flow and Heat Transfer in a Porous Medium Induced by a Nanofluid with Slip and Buoyancy Effects

by Umair Khan, Aurang Zaib, Anuar Ishak, Samia Elattar, Sayed M. Eldin, Zehba Raizah, Iskandar Waini and Muhammad Waqas

Symmetry 2022, 14(10), 2212; https://doi.org/10.3390/sym14102212 - 20 Oct 2022

Cited by 13 | Viewed by 1559

Abstract

In many industries, extremely high-performance cooling is a crucial requirement. However, the fundamental challenge to developing energy-efficient heat transfer fluids required for cooling is insufficient thermal conductivity. In this case, the utilization of nanofluid is effective to overcome these challenges. The current study [...] Read more.

In many industries, extremely high-performance cooling is a crucial requirement. However, the fundamental challenge to developing energy-efficient heat transfer fluids required for cooling is insufficient thermal conductivity. In this case, the utilization of nanofluid is effective to overcome these challenges. The current study aims to examine the two-dimensional (2D) stretching wall jet heat transfer fluid flow induced by a water-based alumina nanofluid embedded in a porous medium with buoyancy force. In addition, irregular heat sink/source and slip effects are assessed. The leading partial differential equations are changed into ordinary differential equations by incorporating similarity variables, then these equations are computationally or numerically worked out via the boundary-value problem of fourth-order (bvp4c) technique. The pertinent factors influencing the symmetry of the hydrothermal performance including friction factor, velocity, and temperature profiles, are illustrated using tables and graphs. The symmetrical outcomes reveal that the velocity declines in the presence of nanoparticles, whereas the temperature uplifts both assisting and opposing flows. Moreover, the friction factor augments due to porosity while the heat transfer rate declines. Full article

(This article belongs to the Special Issue Recent Advances in Conjugate Heat Transfer)

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25 pages, 3394 KiB

Open AccessFeature PaperArticle

An Ising Model for Supercooled Liquids and the Glass Transition

by Ralph V. Chamberlin

Symmetry 2022, 14(10), 2211; https://doi.org/10.3390/sym14102211 - 20 Oct 2022

Cited by 1 | Viewed by 1385

Abstract

We describe the behavior of an Ising model with orthogonal dynamics, where changes in energy and changes in alignment never occur during the same Monte Carlo (MC) step. This orthogonal Ising model (OIM) allows conservation of energy and conservation of (angular) momentum to [...] Read more.

We describe the behavior of an Ising model with orthogonal dynamics, where changes in energy and changes in alignment never occur during the same Monte Carlo (MC) step. This orthogonal Ising model (OIM) allows conservation of energy and conservation of (angular) momentum to proceed independently, on their own preferred time scales. The OIM also includes a third type of MC step that makes or breaks the interaction between neighboring spins, facilitating an equilibrium distribution of bond energies. MC simulations of the OIM mimic more than twenty distinctive characteristics that are commonly found above and below the glass temperature,

T_{g}

. Examples include a specific heat that has hysteresis around

T_{g}

, out-of-phase (loss) response that exhibits primary (α) and secondary (β) peaks, super-Arrhenius T dependence for the α-response time (

τ_{α}

), and fragilities that increase with increasing system size (N). Mean-field theory for energy fluctuations in the OIM yields a critical temperature (

T_{c}

) and a novel expression for the super-Arrhenius divergence as

T \to T_{c}

:

\ln (τ_{α}) ~ 1 / {(1 - T_{c} / T)}^{2}

. Because this divergence is reminiscent of the Vogel-Fulcher-Tammann (VFT) law squared, we call it the “VFT2 law”. A modified Stickel plot, which linearizes the VFT2 law, shows that at high T where mean-field theory should apply, only the VFT2 law gives qualitatively consistent agreement with measurements of

τ_{α}

(from the literature) on five glass-forming liquids. Such agreement with the OIM suggests that several basic features govern supercooled liquids. The freezing of a liquid into a glass involves an underlying 2nd-order transition that is broadened by finite-size effects. The VFT2 law for

τ_{α}

comes from energy fluctuations that enhance the pathways through an entropy bottleneck, not activation over an energy barrier. Values of

τ_{α}

vary exponentially with inverse N, consistent with the distribution of relaxation times deduced from measurements of α response. System sizes found via the T dependence of

τ_{α}

from simulations and measurements are similar to sizes of independently relaxing regions (IRR) measured by nuclear magnetic resonance (NMR) for simple-molecule glass-forming liquids. The OIM elucidates the key ingredients needed to interpret the thermal and dynamic properties of amorphous materials, while providing a broad foundation for more-detailed models of liquid-glass behavior. Full article

(This article belongs to the Special Issue Thermodynamics for Finite-Size and Time-Dependent Behavior: Small Systems, Fluctuations, Correlations, and Internal Heterogeneity)

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10 pages, 279 KiB

Open AccessArticle

On r-Regular Integers (mod n^r)

by Zhengjin Bu and Zhefeng Xu

Symmetry 2022, 14(10), 2210; https://doi.org/10.3390/sym14102210 - 20 Oct 2022

Viewed by 862

Abstract

Let

ρ_{r} (n^{r})

denote the number of positive r-regular integers (

\mod n^{r}

) that are less than or equal to

n^{r}

; in this paper, we investigate some arithmetic properties of certain functions related to [...] Read more.

Let

ρ_{r} (n^{r})

denote the number of positive r-regular integers (

\mod n^{r}

) that are less than or equal to

n^{r}

; in this paper, we investigate some arithmetic properties of certain functions related to r-regular integers

(\mod n^{r})

. Then, we study the average orders and the extremal orders of

ρ_{r} (n^{r})

in connection with the divisor function and the generalized Dedekind function. Moreover, we also introduce an analogue of Cohen–Ramanujan’s sum with respect to r-regular integers

(\mod n^{r})

and show some basic properties of this function. Full article

24 pages, 10033 KiB

Open AccessArticle

Effect of Heat Treatment on the Vibration Isolation Performance of Axially Symmetric NiTi Wire Mesh Damper

by Yichuan Shao, Mingyang Su, Yaoqiang Wei, Jinyu Wang, Yiwan Wu, Xiaochao Chen, Hongbai Bai and Xin Xue

Symmetry 2022, 14(10), 2209; https://doi.org/10.3390/sym14102209 - 20 Oct 2022

Cited by 1 | Viewed by 1305

Abstract

In this paper, superelastic (SE) NiTi wire is used to fabricate axially symmetric wire mesh dampers (WMDs) with the expectation of a higher damping capacity. However, the phase transformation damping of the NiTi WMD could be suppressed by the cold-work-induced dislocation. Therefore, the [...] Read more.

In this paper, superelastic (SE) NiTi wire is used to fabricate axially symmetric wire mesh dampers (WMDs) with the expectation of a higher damping capacity. However, the phase transformation damping of the NiTi WMD could be suppressed by the cold-work-induced dislocation. Therefore, the NiTi WMDs were heat-treated and then tested by a hydraulic universal testing machine. The NiTi WMD is found to achieve higher damping capacity when heat-treated at 200 °C. However, the WMD heat-treated at 250 °C suffers from a sharp decline in the loss factor in exchange for an improvement in the stiffness. The sine sweep test was then conducted to examine the dependency of the WMD’s vibration isolation performance upon the heat treatment temperature and the excitation acceleration. The NiTi WMD outperforms the 304 stainless steel (SS 304) WMD in damping capacity only when the excitation acceleration magnitude is less than 1.5 g. The stiffness of NiTi WMD can be improved without significantly compromising its damping capacity by heat treatment at 200 °C for 30 min. The present work carries out comprehensive measurements of the NiTi WMD’s response to dynamic mechanical test and sine sweep test and addresses how heat treatment influences the stiffness and damping capacity of the SE NiTi WMD. Full article

(This article belongs to the Section Engineering and Materials)

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13 pages, 495 KiB

Open AccessArticle

Application of the Learning Automaton Model for Ensuring Cyber Resiliency

by Maxim Kalinin, Tigran Ovasapyan and Maria Poltavtseva

Symmetry 2022, 14(10), 2208; https://doi.org/10.3390/sym14102208 - 20 Oct 2022

Cited by 2 | Viewed by 1621

Abstract

This work addresses the functional approach to ensuring cyber resiliency as a kind of adaptive security management. For this purpose, we propose a learning automaton model capable of self-learning and adapting to changes while interacting with the external environment. Each node in the [...] Read more.

This work addresses the functional approach to ensuring cyber resiliency as a kind of adaptive security management. For this purpose, we propose a learning automaton model capable of self-learning and adapting to changes while interacting with the external environment. Each node in the under-controlled system has a set of probable actions with respect to neighboring nodes. The same actions are represented in the graph of the learning automaton, but the probabilities of actions in the graph model are permanently updated based on the received reinforcement signals. Due to the adaptive reconfiguration of the nodes, the system is able to counteract the cyberattacks, preserving resiliency. The experimental study results for the emulated wireless sensor network (WSN) are presented and discussed. The packets loss rate stays below 20% when the number of malicious nodes is 20% of the total number of nodes, while the common system loses more than 70% of packets. The network uptime with the proposed solution is 30% longer; the legitimate nodes detect malicious nodes and rebuild their interaction with them, thereby saving their energy. The proposed mechanism allows ensuring the security and functional sustainability of the protected system regardless of its complexity and mission. Full article

(This article belongs to the Special Issue Complex Systems Modeling Using Graphs and Symmetry/Asymmetry)

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20 pages, 414 KiB

Open AccessArticle

Synchrotron Radiation Taking External Influences into Account

by Vladimir Cheslavovich Zhukovsky

Symmetry 2022, 14(10), 2207; https://doi.org/10.3390/sym14102207 - 20 Oct 2022

Viewed by 1110

Abstract

In this paper, we demonstrate how various external forces influence the effect of the radiation of a charged particle. As a particular example, we obtained a solution to the Dirac equation for an electron in a constant homogeneous magnetic field and by taking [...] Read more.

In this paper, we demonstrate how various external forces influence the effect of the radiation of a charged particle. As a particular example, we obtained a solution to the Dirac equation for an electron in a constant homogeneous magnetic field and by taking into account the anomalous magnetic moment and influence of possible Lorentz invariance violation in minimal CPT-odd form. Based on the solution found, we calculated the synchrotron radiation (SR) characteristics and predicted possible observable effects attributable to the Lorentz invariance violation. As another example, we calculated the stimulated synchrotron radiation in the presence of the field of an electromagnetic wave and taking into account the inhomogeneity of an external magnetic field. Moreover, the superposition of two electromagnetic waves was also considered taking into account the properties of radiated electromagnetic waves. We also point out a way to use a corresponding semiclassical solution to the Dirac equation to obtain synchrotron radiation without approximating the radiative amplitudes themselves. This last way of calculating might be of use for studying SR in real circumstances of radiation in an astrophysical magnetic field and in electron accelerators, where electron trajectories are far from being circular. Full article

(This article belongs to the Special Issue Advances in Synchrotron and Undulator Radiation Studies Ⅱ)

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14 pages, 777 KiB

Open AccessArticle

Perturbed Newton Methods for Solving Nonlinear Equations with Applications

by Ioannis K. Argyros, Samundra Regmi, Stepan Shakhno and Halyna Yarmola

Symmetry 2022, 14(10), 2206; https://doi.org/10.3390/sym14102206 - 20 Oct 2022

Cited by 4 | Viewed by 1137

Abstract

Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed [...] Read more.

Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed forms of them can be found only in special cases. Several iterative methods were developed whose convergence was established under very general conditions. Numerous applications are also provided to solve systems of nonlinear equations and differential equations appearing in the aforementioned areas. The ball convergence analysis was developed for the King-like and Jarratt-like families of methods to solve equations under the same set of conditions. Earlier studies have used conditions up to the fifth derivative, but they failed to show the fourth convergence order. Moreover, no error distances or results on the uniqueness of the solution were given either. However, we provide such results involving the derivative only appearing on these methods. Hence, we have expanded the usage of these methods. In the case of the Jarratt-like family of methods, our results also hold for Banach space-valued equations. Moreover, we compare the convergence ball and the dynamical features both theoretically and in numerical experiments. Full article

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

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12 pages, 265 KiB

Open AccessArticle

A Study on Special Kinds of Derivations in Ordered Hyperrings

by Yongsheng Rao, Saeed Kosari, Aysha Khan and Nategh Abbasizadeh

Symmetry 2022, 14(10), 2205; https://doi.org/10.3390/sym14102205 - 19 Oct 2022

Cited by 2 | Viewed by 931

Abstract

In this study, we concentrate on an important class of ordered hyperstructures with symmetrical hyperoperations, which are called ordered Krasner hyperrings, and discuss strong derivations and homo-derivations. Additionally, we apply our results on nonzero proper hyperideals to the study of derivations of prime [...] Read more.

In this study, we concentrate on an important class of ordered hyperstructures with symmetrical hyperoperations, which are called ordered Krasner hyperrings, and discuss strong derivations and homo-derivations. Additionally, we apply our results on nonzero proper hyperideals to the study of derivations of prime ordered hyperrings. This work is a pioneer in studies on structures such as hyperideals and homomorphisms of an ordered hyperring with the help of derivation notation. Finally, we prove some results on 2-torsion-free prime ordered hyperrings by using derivations. We show that if d is a derivation of 2-torsion-free prime hyperring R and the commutator set

[l, d (q)]

is equal to zero for all q in R, then

l \in Z (R)

. Moreover, we prove that if the commutator set

(d (l), q)

is equal to zero for all l in R, then

(d (R), q) = 0

. Full article

(This article belongs to the Section Mathematics)

25 pages, 397 KiB

Open AccessArticle

Fractional Jensen-Mercer Type Inequalities Involving Generalized Raina’s Function and Applications

by Kamsing Nonlaopon, Muhammad Uzair Awan, Usama Asif, Muhammad Zakria Javed, Ibrahim Slimane and Artion Kashuri

Symmetry 2022, 14(10), 2204; https://doi.org/10.3390/sym14102204 - 19 Oct 2022

Cited by 1 | Viewed by 1125

Abstract

The aim of this paper is to derive some new generalized fractional analogues of Mercer type inequalities, essentially using the convexity property of the functions and Raina’s function. We also discuss several new special cases which show that our results are, to an [...] Read more.

The aim of this paper is to derive some new generalized fractional analogues of Mercer type inequalities, essentially using the convexity property of the functions and Raina’s function. We also discuss several new special cases which show that our results are, to an extent, unifying. In order to illustrate the significance of our results, we offer some interesting applications of our results to special means, error bounds, and q-digamma functions. Full article

(This article belongs to the Special Issue Advanced Computational Methods for Fractional Calculus)

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20 pages, 901 KiB

Open AccessArticle

by Zhili Huang, Qiang He, Qinglan Chen and Hongge Yue

Symmetry 2022, 14(10), 2203; https://doi.org/10.3390/sym14102203 - 19 Oct 2022

Viewed by 1416

Abstract

With regard to the interval number-based uncertain multi-attribute decision making problem, in which the attribute weights are unknown and there is no preference on decision-making alternative objects, this paper presents a new decision-making approach. In this method, Hamming distance firstly is used to [...] Read more.

With regard to the interval number-based uncertain multi-attribute decision making problem, in which the attribute weights are unknown and there is no preference on decision-making alternative objects, this paper presents a new decision-making approach. In this method, Hamming distance firstly is used to define the Hamming similarity degree of normative interval numbers, and the Hamming similarity degree of decision-making alternative objects, and then the Hamming similarity superiority relation theory to the comparison of interval numbers is proposed and some relevant results are obtained. Thus, by drawing on the idea of deviations maximization, an interval number-based decision-making object Hamming similarity programming model (IN-DMOHSPM) is established to calculate and solve the weight vector of attributes. Next, all of the selected alternative objects set is screened and sorted by using the overall Hamming similarity degree of each decision-making object compared with the ideal optimal object, and a new algorithm of Hamming similarity programming model for interval number-based multiple attribute decision-making objects is presented. Finally, the feasibility and utility of this model used in this paper are demonstrated by a numerical example. Full article

(This article belongs to the Topic Multi-Criteria Decision Making)

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19 pages, 326 KiB

Open AccessArticle

Noncommutative Bispectral Algebras and Their Presentations

by Brian D. Vasquez Campos and Jorge P. Zubelli

Symmetry 2022, 14(10), 2202; https://doi.org/10.3390/sym14102202 - 19 Oct 2022

Viewed by 1044

Abstract

We prove a general result on presentations of finitely generated algebras and apply it to obtain nice presentations for some noncommutative algebras arising in the matrix bispectral problem. By “nice presentation”, we mean a presentation that has as few as possible defining relations. [...] Read more.

We prove a general result on presentations of finitely generated algebras and apply it to obtain nice presentations for some noncommutative algebras arising in the matrix bispectral problem. By “nice presentation”, we mean a presentation that has as few as possible defining relations. This, in turn, has potential applications in computer algebra implementations and examples. Our results can be divided into three parts. In the first two, we consider bispectral algebras with the eigenvalue in the physical equation to be scalar-valued for

2 \times 2

and

3 \times 3

matrix-valued eigenfunctions. In the third part, we assume the eigenvalue in the physical equation to be matrix-valued and draw an important connection with Spin Calogero–Moser systems. In all cases, we show that these algebras are finitely presented. As a byproduct, we answer positively a conjecture of F. A. Grünbaum about these algebras. Full article

18 pages, 3225 KiB

Open AccessArticle

Instruction-Fetching Attack and Practice in Collision Fault Attack on AES

by Huilong Jiang, Xiang Zhu and Jianwei Han

Symmetry 2022, 14(10), 2201; https://doi.org/10.3390/sym14102201 - 19 Oct 2022

Cited by 1 | Viewed by 1330

Abstract

A Fault Attack (FA) is performed mainly under the data corruption model and poses a threat to security chips. Instruction corruption can enact the same purpose at the behavioral level, which is produced by interfering with the instruction system. Laser Fault Injection (LFI) [...] Read more.

A Fault Attack (FA) is performed mainly under the data corruption model and poses a threat to security chips. Instruction corruption can enact the same purpose at the behavioral level, which is produced by interfering with the instruction system. Laser Fault Injection (LFI) on program memory during the instruction-fetching process, which we refer to as an instruction-fetching attack, is studied in this paper. This process bears the ability to produce a controllable instruction-fetching fault. Our work shows the implementation of the attack and its specific application case on an 8-bit microcontroller. The main contributions of this paper include: (1) We have mapped the sensitive areas precisely to the faulted instructions via laser injection and implemented controllable instruction tampering. (2) A Collision Fault Attack (CFA) scheme based on instruction-fetching fault is proposed. (3) The impacts of the faulted instructions are fully explored, including the influence on subsequent operations and key recovery. (4) The fault mechanism of the on-chip Flash is further investigated. Instruction-fetching fault means that the controller fetches a tampered instruction from the program memory under external interference, which likely gives rise to an invalid or incorrect operation. The experiment confirms that this specific fault can induce particular types of faults that are different to realize, e.g., the byte-fault model in CFA. The realization, application and mechanism of instruction-fetching fault are discussed in detail. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Cryptography)

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15 pages, 317 KiB

Open AccessArticle

Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function

by Yusra Taj, Saira Zainab, Qin Xin, Ferdous M. O. Tawfiq, Mohsan Raza and Sarfraz Nawaz Malik

Symmetry 2022, 14(10), 2200; https://doi.org/10.3390/sym14102200 - 19 Oct 2022

Cited by 7 | Viewed by 1351

Abstract

This study introduces a subclass

S_{q s}^{*}

of starlike functions associated with the q-analogue of the sine function defined in symmetric unit disk. This article comprises the investigation of sharp coefficient bounds, and the upper bound of the third-order Hankel [...] Read more.

This study introduces a subclass

S_{q s}^{*}

of starlike functions associated with the q-analogue of the sine function defined in symmetric unit disk. This article comprises the investigation of sharp coefficient bounds, and the upper bound of the third-order Hankel determinant for this class. It also includes the findings of Zalcman and generalized Zalcman conjectures for functions of this class. Full article

(This article belongs to the Special Issue Applications of Symmetric Functions Theory to Certain Fields)

21 pages, 8499 KiB

Open AccessArticle

FPGA Hardware Realization of Membrane Calculation Optimization Algorithm with Great Parallelism

by Qi Song, Yourui Huang, Wenhao Lai, Jiachang Xu, Shanyong Xu, Tao Han and Xue Rong

Symmetry 2022, 14(10), 2199; https://doi.org/10.3390/sym14102199 - 19 Oct 2022

Cited by 1 | Viewed by 1291

Abstract

Aiming to investigate the disadvantage of the optimization algorithm of membrane computing (a P system) in which it is difficult to take advantage of parallelism in MATLAB, leading to a slow optimization speed, a digital-specific hardware solution (field-programmable gate array, FPGA) is proposed [...] Read more.

Aiming to investigate the disadvantage of the optimization algorithm of membrane computing (a P system) in which it is difficult to take advantage of parallelism in MATLAB, leading to a slow optimization speed, a digital-specific hardware solution (field-programmable gate array, FPGA) is proposed to design and implement the single-cell-membrane algorithm (SCA). Because the SCA achieves extensive global searches by the symmetric processing of the solution set, with independent and symmetrically distributed submembrane structures, the FPGA-hardware-based design of the SCA system includes a control module, an HSP module, an initial value module, a fitness module, a random number module, and multiple submembrane modules with symmetrical structures. This research utilizes the inherent parallel characteristics of the FPGA to achieve parallel computations of multiple submembrane modules with a symmetric structure inside the SCA, and it achieves a high degree of parallelism of rules inside the modules by using a non-blocking allocation. This study uses the benchmark Sphere function to verify the performance of the FPGA-designed SCA system. The experimental results show that, when the FPGA platform and the MATLAB platform obtain a similar calculation accuracy, the average time-consuming of the FPGA is 0.00041 s, and the average time-consuming of MATLAB is 0.0122 s, and the calculation speed is improved by nearly 40 times. This study uses the FPGA design to implement the SCA, and it verifies the advantages of the membrane-computing maximum-parallelism theory and distributed structures in computing speed. The realization platform of membrane computing is expanded, which provides a theoretical basis for further development of the distributed computing model of population cells. Full article

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27 pages, 9517 KiB

Open AccessFeature PaperArticle

Simplified Procedure for Capacity Check of Historic Monolithic Glass Windows under Soft-Body Collision/Bird-Strike

by Chiara Bedon and Maria Vittoria Santi

Symmetry 2022, 14(10), 2198; https://doi.org/10.3390/sym14102198 - 19 Oct 2022

Cited by 2 | Viewed by 1535

Abstract

Differing from present structural design procedures, most of the existing glass windows and even historic components in traditional/old buildings are not specifically designed to resist possible accidental loads. Rather thin monolithic ordinary annealed glass panels can be found in vertical non-structural envelopes, where [...] Read more.

Differing from present structural design procedures, most of the existing glass windows and even historic components in traditional/old buildings are not specifically designed to resist possible accidental loads. Rather thin monolithic ordinary annealed glass panels can be found in vertical non-structural envelopes, where they are often arranged to cover large surfaces. As such, an accidental glass fracture could originate even from rather common and moderate impact events and result in severe risk for people, due to propagation of dangerous shards from these vulnerable and fragile building components. To assess potential risks and support possible mitigation strategies, the present study is focused on the bird-strike analysis of existing/historic linearly restrained non-structural glass windows, based on a parametric Smoothed-Particle Hydrodynamics (SPH)–Finite Element (FE) model. Starting from a 1 m–wide and 1.5 m–high configuration, the attention is first given to various influencing parameters, such as impactor features (mass, 0.35–1.81 kg; impact speed, 0–40 m/s; and, thus, impact energy) and the target window (glass thickness, 4–6 mm; impact point; and, thus, glass stiffness). Local and global effects due to parametric localized bird-strikes are discussed based on non-linear dynamic numerical analyses and in terms of expected deflections, tensile stress peaks, and damage extension/severity (i.e., D1 to D3 damage levels). Scale effects are also examined for a case-study historic envelope (≈7 m in total size, 5 mm in thickness), and one of its 2.58 m × 3.3 m large glass components. Furthermore, a simplified empirical approach based on analytical formulations and normalized charts is proposed for a preliminary vulnerability assessment of historic monolithic glass envelopes, including parameters to account for impactor features and glass panel size/thickness, based on vibration-frequency considerations. Full article

(This article belongs to the Section Computer)

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20 pages, 3627 KiB

Open AccessArticle

Artificial Intelligence-Based Diabetes Diagnosis with Belief Functions Theory

by Ameni Ellouze, Omar Kahouli, Mohamed Ksantini, Haitham Alsaif, Ali Aloui and Bassem Kahouli

Symmetry 2022, 14(10), 2197; https://doi.org/10.3390/sym14102197 - 19 Oct 2022

Cited by 4 | Viewed by 1573

Abstract

We compared various machine learning (ML) methods, such as the K-nearest neighbor (KNN), support vector machine (SVM), and decision tree and deep learning (DL) methods, like the recurrent neural network, convolutional neural network, long short-term memory (LSTM), and gated recurrent unit (GRU), to [...] Read more.

We compared various machine learning (ML) methods, such as the K-nearest neighbor (KNN), support vector machine (SVM), and decision tree and deep learning (DL) methods, like the recurrent neural network, convolutional neural network, long short-term memory (LSTM), and gated recurrent unit (GRU), to determine the ones with the highest precision. These algorithms learn from data and are subject to different imprecisions and uncertainties. The uncertainty arises from the bad reading of data and/or inaccurate sensor acquisition. We studied how these methods may be combined in a fusion classifier to improve their performance. The Dempster–Shafer method, which uses the formalism of belief functions characterized by asymmetry to model nonprecise and uncertain data, is used for classifier fusion. Diagnosis in the medical field is an important step for the early detection of diseases. In this study, the fusion classifiers were used to diagnose diabetes with the required accuracy. The results demonstrated that the fusion classifiers outperformed the individual classifiers as well as those obtained in the literature. The combined LSTM and GRU fusion classifiers achieved the highest accuracy rate of 98%. Full article

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15 pages, 1310 KiB

Open AccessFeature PaperArticle

Attractive and Repulsive Fluctuation-Induced Pressure in Peptide Films Deposited on Semiconductor Substrates

by Galina L. Klimchitskaya, Vladimir M. Mostepanenko and Oleg Yu. Tsybin

Symmetry 2022, 14(10), 2196; https://doi.org/10.3390/sym14102196 - 19 Oct 2022

Cited by 2 | Viewed by 1092

Abstract

We consider the fluctuation-induced (Casimir) pressure in peptide films deposited on GaAs, Ge, and ZnS substrates which are either in a dielectric or metallic state. The calculations of the Casimir pressure are performed in the framework of the fundamental Lifshitz theory employing the [...] Read more.

We consider the fluctuation-induced (Casimir) pressure in peptide films deposited on GaAs, Ge, and ZnS substrates which are either in a dielectric or metallic state. The calculations of the Casimir pressure are performed in the framework of the fundamental Lifshitz theory employing the frequency-dependent dielectric permittivities of all involved materials. The electric conductivity of semiconductor substrates is taken into account within the experimentally and thermodynamically consistent approach. According to our results, the Casimir pressure in peptide films deposited on dielectric-type semiconductor substrates vanishes for some definite film thickness and is repulsive for thinner and attractive for thicker films. The dependence of this effect on the fraction of water in the film and on the static dielectric permittivity of the semiconductor substrate is determined. For the metallic-type semiconductor substrates, the Casimir pressure in peptide coatings is shown to be always repulsive. The possible applications of these results to the problem of stability of thin coatings in microdevices are discussed. Full article

(This article belongs to the Special Issue Physics and Symmetry Section: Feature Papers 2022)

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12 pages, 728 KiB

Open AccessArticle

Is Asymmetry Different Depending on How It Is Calculated?

by Mario Iglesias-Caamaño, Tania Álvarez-Yates, Javier Carballo-López, Alba Cuba-Dorado and Óscar García-García

Symmetry 2022, 14(10), 2195; https://doi.org/10.3390/sym14102195 - 19 Oct 2022

Cited by 2 | Viewed by 1531

Abstract

This study aimed to (1) determine the magnitude and direction of asymmetry in volleyball players, (2) establish asymmetry thresholds, and (3) explore differences depending on the test used and the players’ category. Twenty-nine junior and senior male volleyball players were assessed through a [...] Read more.

This study aimed to (1) determine the magnitude and direction of asymmetry in volleyball players, (2) establish asymmetry thresholds, and (3) explore differences depending on the test used and the players’ category. Twenty-nine junior and senior male volleyball players were assessed through a muscle asymmetry battery test: active knee extension test (AKE), single-leg countermovement jump (SL-CMJ), single-leg squat jump (SL-SJ), triple hop test for distance (THTD), modified 20-yard shuttle run, Y-balance test, single-leg one-repetition maximum in leg press test (1RM-SL), and lateral symmetry in radial muscle belly displacement through Tensiomyography in the biceps femoris and rectus femoris. A two-way ANOVA alongside an individual analysis of asymmetry thresholds was used to analyze the test and categorize the influence on the magnitude and the direction of asymmetry. The 1RM-SL, SL-SJ, and the lateral symmetry in radial muscle belly displacement showed a clear asymmetry towards the non-dominant side, while the AKE, SL-CMJ, and THTD showed an asymmetry towards the dominant side. The magnitude of the asymmetry was highly variable between tests (1.46–30.26%). The individualized asymmetry thresholds revealed that the percentage of asymmetrical players varied depending on the type of test used. In conclusion, the type of test used determines the magnitude and direction of asymmetry in well-trained volleyball players. Full article

(This article belongs to the Special Issue Symmetry Application in Motor Control in Sports and Rehabilitation)

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13 pages, 709 KiB

Open AccessArticle

Small Area Estimation Using a Semiparametric Spatial Model with Application in Insurance

by Seyede Elahe Hosseini, Davood Shahsavani, Mohammad Reza Rabiei, Mohammad Arashi and Hossein Baghishani

Symmetry 2022, 14(10), 2194; https://doi.org/10.3390/sym14102194 - 18 Oct 2022

Cited by 1 | Viewed by 4197

Abstract

Additional information and borrowing strength from the related sites and other sources will improve estimation in small areas. Generalized linear mixed-effects models (GLMMs) have been frequently used in small area estimation; however, the relationship between the response variable and some covariates may not [...] Read more.

Additional information and borrowing strength from the related sites and other sources will improve estimation in small areas. Generalized linear mixed-effects models (GLMMs) have been frequently used in small area estimation; however, the relationship between the response variable and some covariates may not be linear in many cases. In such cases, using semiparametric modeling, incorporating some nonlinear symmetric/asymmetric functions to the predictor seems more appropriate due to their flexibility. In addition, spatial dependence is observed between areas in many cases. Thus, using the semiparametric spatial models for small areas is of interest. This paper presents semiparametric spatial GLMMs and approximates the nonlinear component using splines to estimate the linear part. We apply our proposal for analyzing insurance data obtained from an Iranian insurance company. Our numerical illustrations will support the use of our proposal in situations where the spatial GLMMs may not be appropriate. Full article

(This article belongs to the Special Issue Symmetry in Multivariate Analysis)

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