Symmetry, Volume 15, Issue 8 (August 2023) – 147 articles

Graph embedding transforms high-dimensional graphs into a lower-dimensional vector space while preserving their structural information and properties. Context-sensitive graph embedding, in particular, performs well in tasks such as link prediction and ranking recommendations. However, existing context-sensitive graph embeddings have limitations: they require additional information, depend on community algorithms to capture multiple contexts, or fail to capture sufficient structural information. In this paper, we propose a novel Graph Embedding with Similarity Metric Learning (GESML). The core of GESML is to learn the optimal graph structure using an attention-based symmetric similarity metric function and establish association relationships between nodes through top-k pooling. Its primary advantage lies in not requiring additional features or multiple contexts, only using the symmetric similarity metric function and pooling operations to encode sufficient topological information for each node. Experimental results on three datasets involving link prediction and node-clustering tasks demonstrate that GESML significantly improves learning for all challenging tasks relative to a state-of-the-art (SOTA) baseline. Full article

A significant addition to fuzzy set theory for expressing uncertain data is an n,m-th power root fuzzy set. Compared to the n^th power root, Fermatean, Pythagorean, and intuitionistic fuzzy sets, n,m-th power root fuzzy sets can cover more uncertain situations due to their greater range of displayed membership grades. When discussing the symmetry between two or more objects, the innovative concept of an n,m-th power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, a Pythagorean fuzzy set, and a n^th power root fuzzy set. In this study, we demonstrate a number of additional operations on n,m-th power root fuzzy sets along with a number of their special aspects. Additionally, to deal with choice information, we create a novel weighted aggregated operator called the n,m-th power root fuzzy weighted power average (FWPA${}_m^n$) across n,m-th power root fuzzy sets and demonstrate some of its fundamental features. To rank n,m-th power root fuzzy sets, we also define the score and accuracy functions. Moreover, we use this operator to identify the countries with the best standards of living and show how we can select the best option by contrasting aggregate results using score values. Finally, we contrast the results of the FWPA${}_m^n$ operator with the square-root fuzzy weighted power average (SR-FWPA), the n^th power root fuzzy weighted power average (nPR-FWPA), the Fermatean fuzzy weighted power average (FFWPA), and the n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) operators. Full article

The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. Motivated by the several applications of the mentioned evolution equation, in this investigation, the Laplace residual power series method (LRPSM) is employed to analyze the fractional S-KdV equation in the framework of the Caputo operator. By incorporating both the Caputo operator and fractional derivatives into the mentioned evolution equation, we can account for memory effects and non-local behavior. The LRPSM is a powerful analytical technique for the solution of fractional differential equations and therefore it is adapted in our current study. In this study, we prove that the combination of the residual power series expansion with the Laplace transform yields precise and efficient solutions. Moreover, we investigate the behavior and properties of the (un)symmetric solutions to the fractional S-KdV equation using extensive numerical simulations and by considering various fractional orders and initial fractional values. The results contribute to the greater comprehension of the interplay between quantum mechanics and nonlinear dynamics in fractional systems and shed light on wave phenomena and symmetry soliton solutions in such equations. In addition, the proposed method successfully solves fractional differential equations with the Caputo operator, providing a valuable computational instrument for the analysis of complex physical systems. Moreover, the obtained results can describe many of the mysteries associated with the mechanism of nonlinear wave propagation in plasma physics. Full article

In the present article, we investigate the free convective flow of a ternary hybrid nanofluid in a two-phase inclined channel saturated with a porous medium. The flow has been propelled using the pressure gradient, thermal radiation, and buoyancy force. The flow model’s governing equations are resolved using the regular perturbation approach. The governing equations are solved with the help of the regular perturbation method. Polyethylene glycol and water (at a ratio of 50%:50%) fill up Region I, while a ternary hybrid nanofluid based on zirconium dioxide, magnesium oxide, and carbon nanotubes occupies Region II. The ternary hybrid nanofluids are defined with a mixture model in which three different shapes of nanoparticles, namely spherical, platelet, and cylindrical, are incorporated. The consequences of the most significant variables have been examined using both visual and tabular data. The main finding of this work is that utilising a ternary hybrid nanofluid at the plate y = 1 increases the rate of heat transfers by 753%, demonstrating the potential thermal efficiency. The overall heat and volume flow rates are amplified by buoyant forces and viscous dissipations and dampened by the thermal radiation parameter. The optimum enhancement of temperature is achieved by the influence of buoyancy forces. A ternary nanofluid region experiences the maximum temperature increase compared to a clear fluid region. To ensure the study’s efficiency, we validated it with prior studies. Full article

This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress–energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy–momentum tensor is nondegenerate and degenerate. For the case of a degenerate energy–momentum tensor, we employ a direct integration technique to solve the MC equations, which leads to an infinite-dimensional Lie algebra. On the other hand, when considering the nondegenerate energy–momentum tensor, the contravariant form results in a finite-dimensional Lie algebra with dimensions of either 4 or 10. However, in the case of the mixed form of the energy–momentum tensor, the dimension of the Lie algebra is infinite. Moreover, the obtained MCs are compared with those already found for covariant stress–energy. Full article

A relativistic motor exchanging momentum and energy with an electromagnetic field is studied. We discuss the advantages and challenges of this novel mover, giving specific emphasis to the more favorable (yet challenging) nano configurations. It specifically turns out that an isolated hydrogen atom in either a ground or excited state does not produce relativistic motor momentum. Full article

The complete characterization of saturated varieties of semigroups remains an unsolved problem. The primary objective of this paper is to make significant progress in this direction. We initially demonstrate that the variety of semigroups defined by the identity $axy=ayxa$ is saturated. The next main result establishes that the variety of semigroups determined by the identity $axy=ayax$ is saturated. Finally, we show that medial semigroups satisfying the identity $xy=x{y}^n$, where $n\ge 2$, are also saturated. These results collectively lead to the conclusion that epis from these saturated varieties are onto. This paper thus offers substantial progress towards the comprehensive characterization of saturated varieties of semigroups. Full article

Analytical modeling presents symmetries and aesthetic-mathematical characteristics which are not catchable in numerical computation for science and technology; nanoscience plays a significant role in unification attempts, considering also models including holistic aspects of reality. In this paper we present new discovered results about the complete analytical quantum-relativistic form of the mean square deviation of position $R^2(t)$ related to a recently introduced Drude–Lorentz-like model (DS model), already performed at classical, quantum and relativistic level. The function $R^2(t)$ gives precise information about the distance crossed by carriers (electrons, ions, etc.) inside a nanostructure, considering both quantum effects and relativistic velocities. The model has a wide scale range of applicability; the nanoscale is considered in this paper, but it holds application from sub-pico-level to macro-level because of the existence of a gauge factor, making it applicable to every oscillating process in nature. Examples of application and suggestions supplement this paper, as well as interesting developments to be studied related to the model and to one of the basic elements of a current unified holistic approach based on vacuum energy. Full article

Artificial neural networks (ANNs) are used to solve many problems, such as modeling, identification, prediction, and classification. The success of ANN is directly related to the training process. Meta-heuristic algorithms are used extensively for ANN training. Within the scope of this study, a feed-forward artificial neural network (FFNN) is trained using the marine predators algorithm (MPA), one of the current meta-heuristic algorithms. Namely, this study is aimed to evaluate the performance of MPA in ANN training in detail. Identification/modeling of nonlinear systems is chosen as the problem. Six nonlinear systems are used in the applications. Some of them are static, and some are dynamic. Mean squared error (MSE) is utilized as the error metric. Effective training and testing results were obtained using MPA. The best mean error values obtained for six nonlinear systems are 2.3 × 10⁻⁴, 1.8 × 10⁻³, 1.0 × 10⁻⁴, 1.0 × 10⁻⁴, 1.2 × 10⁻⁵, and 2.5 × 10⁻⁴. The performance of MPA is compared with 16 meta-heuristic algorithms. The results have shown that the performance of MPA is better than other algorithms in ANN training for the identification of nonlinear systems. Full article

In this paper, we take physical and health integration as the entry point. Firstly, based on the transformation mechanism of adolescent idiopathic scoliosis we construct a time delay differential model. Moreover, using the theory of characteristic equation we discuss the stability of a positive equilibrium under the delays of $\tau =0$ and $\tau \ne 0$. Furthermore, through numerical simulation, it has been verified the delay, $\tau$, exceeds a critical value, the positive equilibrium loses its stability and Hopf bifurcation occurs. Lastly, we determine that sports have a positive effect on adolescent idiopathic scoliosis, directly reducing the number of people with adolescent idiopathic scoliosis. Full article

In this paper, a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable, with uniform distribution in the interval (0, 1). We will call this new extension the slash Fréchet distribution and one of its main characteristics is that its tails are heavier than the Fréchet distribution. The general density of this distribution and some basic properties are determined. Its moments, skewness coefficients, and kurtosis are calculated. In addition, the estimation of the model parameters is obtained by the method of moments and maximum likelihood. Finally, three applications with real data are performed by fitting the new model and comparing it with the Fréchet distribution. Full article

Sensor arrays are gradually becoming a current research hotspot due to their flexible beam control, high signal gain, robustness against extreme interference, and high spatial resolution. Among them, spherical microphone arrays with complex rotational symmetry can capture more sound field information than planar arrays and can convert the collected multiple speech signals into the spherical harmonic domain for processing through spherical modal decomposition. The subspace class direction of arrival (DOA) estimation algorithm is sensitive to noise and reverberation, and its performance can be improved by introducing relative sound pressure and frequency-smoothing techniques. The introduction of the relative sound pressure can increase the difference between the eigenvalues corresponding to the signal subspace and the noise subspace, which is helpful to estimate the number of active sound sources. The eigenbeam estimation of signal parameters via the rotational invariance technique (EB-ESPRIT) is a well-known subspace-based algorithm for a spherical microphone array. The EB-ESPRIT cannot estimate the DOA when the elevation angle approaches 90°. Huang et al. proposed a two-step ESPRIT (TS-ESPRIT) algorithm to solve this problem. The TS-ESPRIT algorithm estimates the elevation and azimuth angles of the signal independently, so there is a problem with DOA parameter pairing. In this paper, the DOA parameter pairing problem of the TS-ESPRIT algorithm is solved by introducing generalized eigenvalue decomposition without increasing the computation of the algorithm. At the same time, the estimation of the elevation angle is given by the arctan function, which increases the estimation accuracy of the elevation angle of the algorithm. The robustness of the algorithm in a noisy environment is also enhanced by introducing the relative sound pressure into the algorithm. Finally, the simulation and field-testing results show that the proposed method not only solves the problem of DOA parameter pairing, but also outperforms the traditional methods in DOA estimation accuracy. Full article

Our paper explores warped product pointwise semi-slant submanifolds with a semi-symmetric metric connection in an odd-dimensional sphere and uncovers fundamental results. We also demonstrate how our findings can be applied to the homology of these submanifolds. Notably, we prove that under a specific condition, there are no stable currents for these submanifolds. This work adds valuable insights into the stability and behavior of warped product pointwise semi-slant submanifolds and sets the foundation for further research in this field. Full article

The monkeypox virus causes a respiratory illness called monkeypox, which belongs to the Poxviridae virus family and the Orthopoxvirus genus. Although initially endemic in Africa, it has recently become a global threat with cases worldwide. Using the Antangana–Baleanu fractional order approach, this study aims to propose a new monkeypox transmission model that represents the interaction between the infected human and rodent populations. An iterative method and the fixed-point theorem are used to prove the existence and uniqueness of the symmetry model’s system of solutions. It shows that the symmetry model has equilibrium points when there are epidemics and no diseases. As well as the local asymptotic stability of the disease-free equilibrium point, conditions for the endemic equilibrium point’s existence have also been demonstrated. For this purpose, the existence of optimal control is first ensured. The aim of the proposed optimal control problem is to minimize both the treatment and prevention costs, and the number of infected individuals. Optimal conditions are acquired Pontryagin’s maximum principle is used. Then, the stability of the symmetry model is discussed at monkeypox-free and endemic equilibrium points with treatment strategies to control the spread of the disease. Numerical simulations clearly show how necessary and successful the proposed combined control strategy is in preventing the disease from becoming epidemic. Full article

Let $S_s^{*}\left(e^z\right)$ denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk $\mathbb{U}$ the condition $\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f(-z)}\prec e^z(z\in \mathbb{U})$. We obtained the sharp estimate of the second-order Hankel determinants $H_{2,3}\left(f\right)$ and improved the estimate of the third-order $H_{3,1}\left(f\right)$ for this functions class $S_s^{*}\left(e^z\right)$. Full article

Our universe consists mainly of regular matter, while the amount of antimatter seems to be negligible. The origin of this difference, known as the baryon asymmetry, remains undiscovered. Since the discovery of antimatter, many experiments have been carried out to study antiparticles and to compare matter and antimatter twins. Two of the most sensitive methods in physics, radiofrequency and optical spectroscopy, can be efficiently used to search for the difference. The successful synthesis and trapping of cold antihydrogen atoms opened the possibility of significantly increasing the sensitivity of matter/antimatter tests. This brief review focuses on a hydrogen/antihydrogen comparison using other independent spectroscopic measurements of single particles in traps and other simple atomic systems like positronium. Although no significant difference is detected in today’s level of accuracy, one can push forward the sensitivity by improving the accuracy of 1S–2S positronium spectroscopy, spectroscopy of hyperfine transition in antihydrogen, and gravitational measurements. Full article

The impact of convection and radiation on the thermal distribution of the wavy porous fin is examined in the present study. A hybrid model that combines the differential evolution (DE) algorithm with an artificial neural network (ANN) is proposed for predicting the heat transfer of the wavy porous fin. The equation representing the thermal variation in the wavy porous fin is reduced to its dimensionless arrangement and is numerically solved using Rung, e-Kutta-Fehlberg’s fourth-fifth order method (RKF-45). The study demonstrates the effectiveness of this hybrid model, and the results indicate that the proposed approach outperforms the ANN model with parameters obtained through grid search (GS), showcasing the superiority of the hybrid DE-ANN model in terms of accuracy and performance. This research highlights the potential of utilizing DE with ANN for improved predictive modeling in the heat transfer sector. The originality of this study is that it addresses the heat transfer problem by optimizing the selection of parameters for the ANN model using the DE algorithm. Full article

In recent years, integration of production scheduling and machine maintenance has gained increasing attention in order to improve the stability and efficiency of flowshop manufacturing systems. This paper proposes a Q-learning-based aquila optimizer (QL-AO) for solving the integrated optimization problem of blocking flowshop scheduling and preventive maintenance since blocking in the jobs processing requires to be considered in the practice manufacturing environments. In the proposed algorithmic framework, a Q-learning algorithm is designed to adaptively adjust the selection probabilities of four key population update strategies in the classic aquila optimizer. In addition, five local search methods are employed to refine the quality of the individuals according to their fitness level. A series of numerical experiments are carried out according to two groups of flowshop scheduling benchmark. Experimental results show that QL-AO significantly outperforms six peer algorithms and two state-of-the-art hybrid algorithms based on Q-Learning on the investigated integrated scheduling problem. Additionally, the proposed Q-learning and local search strategies are effective in improving its performance. Full article

We quantized the full Einstein equations in a globally hyperbolic spacetime $N=N^{n+1}$, $n\ge 3$, and found solutions of the resulting hyperbolic equation in a fiber bundle E which can be expressed as a product of spatial eigenfunctions (eigendistributions) and temporal eigenfunctions. The spatial eigenfunctions form a basis in an appropriate Hilbert space while the temporal eigenfunctions are solutions to a second-order ordinary differential equation in ${\mathbb{R}}_{+}^{}$. In case $n\ge 17$ and provided the cosmological constant $\Lambda$ is negative, the temporal eigenfunctions are eigenfunctions of a self-adjoint operator ${\widehat{H}}_0$ such that the eigenvalues are countable and the eigenfunctions form an orthonormal basis of a Hilbert space. Full article

Symmetry in designing a microstrip coupler is crucial because it ensures balanced power division and minimizes unwanted coupling between the coupled lines. In this paper, a filtering branch-line coupler (BLC) with a simple symmetrical microstrip structure was designed, analyzed and fabricated. Based on a mathematical design procedure, the operating frequency was set at 5.2 GHz for WLAN and 5G applications. Moreover, an optimization method was used to improve the performance of the proposed design. It occupied an area of 83.2 mm². Its harmonics were suppressed up to 15.5 GHz with a maximum level of −15 dB. Meanwhile, the isolation was better than −28 dB. Another advantage of this design was its high phase balance, where the phase difference between its output ports was 270° ± 0.1°. To verify the design method and simulation results, the proposed coupler was fabricated and measured. The results show that all the simulation, design methods, and experimental results are in good agreement. Therefore, the proposed design can be easily used in designing high-performance microstrip-based communication systems. Full article

To enhance our understanding of the thermal characteristics of lithium-ion batteries and gain valuable insights into the thermal impacts of battery thermal management systems (BTMSs), it is crucial to develop precise thermal models for lithium-ion batteries that enable numerical simulations. The primary objective of creating a battery thermal model is to define equations related to heat generation, energy conservation, and boundary conditions. However, a standalone thermal model often lacks the necessary accuracy to effectively anticipate thermal behavior. Consequently, the thermal model is commonly integrated with an electrochemical model or an equivalent circuit model. This article provides a comprehensive review of the thermal behavior and modeling of lithium-ion batteries. It highlights the critical role of temperature in affecting battery performance, safety, and lifespan. The study explores the challenges posed by temperature variations, both too low and too high, and their impact on the battery’s electrical and thermal balance. Various thermal analysis approaches, including experimental measurements and simulation-based modeling, are described to comprehend the thermal characteristics of lithium-ion batteries under different operating conditions. The accurate modeling of batteries involves explaining the electrochemical model and the thermal model as well as methods for coupling electrochemical, electrical, and thermal aspects, along with an equivalent circuit model. Additionally, this review comprehensively outlines the advancements made in understanding the thermal behavior of lithium-ion batteries. In summary, there is a strong desire for a battery model that is efficient, highly accurate, and accompanied by an effective thermal management system. Furthermore, it is crucial to prioritize the enhancement of current thermal models to improve the overall performance and safety of lithium-ion batteries. Full article

In the conventional (so-called Schrödinger-picture) formulation of quantum theory the operators of observables are chosen self-adjoint and time-independent. In the recent innovation of the theory, the operators can be not only non-Hermitian but also time-dependent. The formalism (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the time-dependent states $\psi \left(t\right)$ (using Schrödinger-type equations) as well as of the time-dependent observables ${\Lambda }_j\left(t\right)$, $j=1,2,\dots ,K$ (using Heisenberg-type equations). In the unitary-evolution dynamical regime of our interest, both of the respective generators of the evolution (viz., in our notation, the Schrödingerian generator $G\left(t\right)$ and the Heisenbergian generator $\Sigma \left(t\right)$) have, in general, complex spectra. Only the spectrum of their superposition remains real. Thus, only the observable superposition $H\left(t\right)=G\left(t\right)+\Sigma \left(t\right)$ (representing the instantaneous energies) should be called Hamiltonian. In applications, nevertheless, the mathematically consistent models can be based not only on the initial knowledge of the energy operator $H\left(t\right)$ (forming a “dynamical” model-building strategy) but also, alternatively, on the knowledge of the Coriolis force $\Sigma \left(t\right)$ (forming a “kinematical” model-building strategy), or on the initial knowledge of the Schrödingerian generator $G\left(t\right)$ (forming, for some reason, one of the most popular strategies in the literature). In our present paper, every such choice (marked as “one”, “two” or “three”, respectively) is shown to lead to a construction recipe with a specific range of applicability. Full article

The development of modern technologies has made it much easier to regain the ability to walk after losing a lower limb. The variety of prosthetic feet available on the market allows for optimal choice and appropriate adjustment of the foot prosthesis to the trans-tibial amputee patient’s needs. Unfortunately, the best solutions are often not available to everyone due to their high prices. This study compares the gait patterns of patients using the new concept of an affordable dynamic foot with those of other commonly available but much more expensive foot prostheses. The kinematic and spatio-temporal parameters of gait obtained using the motion capture system were analyzed. For a clear picture of changes in bilateral deficits during gait for the pelvis, hip, knee, and ankle joints, the symmetry function was used. The results indicate that the new and cheaper concept of foot prostheses offers a very similar level of gait quality to that provided by more expensive and popular solutions. The authors suggest that the use of symmetry function thresholds of 10% does not work for amputees. Full article

In the era of rapid growth in the short video industry, it is very important to find more accurate suitable advertising promoters, namely Key Opinion Leaders, to promote the development of short video commerce. A mathematical method is needed to grade and evaluate KOL’s abilities. Only in this way can advertisers better determine the value of KOL and determine whether it is suitable for promoting its products. Moreover, in the hierarchical evaluation of KOL, there is not only structured and quantifiable information, but also a large amount of unstructured and linguistic non-quantifiable information. Therefore, this article regards unquantifiable information as an uncertain variable and uses a comprehensive evaluation method based on uncertainty theory to handle subjective uncertainty in the evaluation process. Among them, all uncertain variables are symmetric. The main contribution of this article is the provision of a new evaluation method for KOL grading. Firstly, a two-level evaluation index system for KOL was established. Secondly, the importance and annotation of the Index set are set as uncertain variables, and the KOL evaluation model is constructed. Finally, two KOLs on TikTok were selected for comparative analysis to determine the importance ranking and KOL scores of each level of indicator, verifying the effectiveness and practicality of this method. Full article

We analyze a ballistic electron transport through a corrugated (rippled) graphene system with a curvature-induced spin–orbit interaction. The corrugated system is connected from both sides to two flat graphene sheets. The rippled structure unit is modeled by upward and downward curved surfaces. The cooperative effect of N units connected together (the superlattice) on the transmission of electrons that incident at the arbitrary angles on the superlattice is considered. The set of optimal angles and corresponding numbers of N units that yield the robust spin inverter phenomenon are found. Full article

It is noted that some uncertain variables are independent while others are not. In general, there is a symmetrical relationship between independence and dependence among uncertain variables. The utilization of conditional uncertain measures as well as conditional uncertainty distributions proves highly efficacious in resolving uncertainties pertaining to an event subsequent to the acquisition of knowledge about other events. In this paper, the theorem about the conditional uncertainty distribution of two uncertain variables is proposed. It is demonstrated that the theorem holds regardless of whether the two variables are independent or not. In addition, it is also found that uncertainty distribution possesses an inherent inverse function when it is a regular uncertainty distribution within the framework of Uncertainty Theory; therefore, this paper delves into investigating the conditional inverse uncertainty distribution, including specific cases of the conditional inverse uncertainty distributions. Meanwhile, illustrative examples are applied to clarify the findings. Full article

The traditional theory of chaotic intermittency developed for return maps hypothesizes a uniform density of reinjected points from the chaotic zone to the laminar one. In the past few years, we have described how the reinjection probability density function (RPD) can be generalized as a power law function. Here, we introduce a broad and general analytical approach to determine the RPD function and other statistical variables, such as the characteristic relation traditionally utilized to characterize the chaotic intermittency type. The proposed theoretical methodology is simple to implement and includes previous studies as particular cases. It is compared with numerical data, the M function methodology, and the Perron–Frobenius technique, showing high accuracy between them. Full article

The rate of force development scaling factor (RFD-SF) has been used to assess neuromuscular quickness. However, the common protocols are lengthy. This study evaluated the validity and reliability of the reduced protocol to assess the RFD-SF and its validity in detecting inter-limb asymmetries. Eighteen participants (five females and thirteen males; mean age = 20.8 ± 0.6 years) performed the common and reduced RFD-SF protocols (five isometric pulse knee extensions at 30 and 70% of maximal voluntary contraction). A repeat measure design was employed including one test session of the common protocol and two test sessions of the reduced protocol. Correlation analysis was conducted to investigate the association between the two protocols, while a paired-sample t-test and a Bland–Altman plot assessed whether the reduced protocol provided valid results. The between-day reliability was assessed using an intra-class correlation coefficient, coefficient of variation, typical error of measurement, and paired-sample t-test. The validity to detect asymmetries was checked with the paired-sample t-test. The correlation between RFD-SF obtained using two protocols was significant (p < 0.001) and very large for the dominant (r = 0.71) and non-dominant (r = 0.80) legs. No significant difference occurred between protocols in the RFD-SF for the dominant (p = 0.480, d = 0.17) and non-dominant legs (p = 0.213, d = 0.31). The reliability was acceptable for both legs, with no between-day difference for the dominant (p = 0.393) and non-dominant legs (p = 0.436). No significant difference between the two protocols (p = 0.415, d = 0.19) was found in the detection of inter-limb asymmetries. The results of this study suggest that the reduced protocol could be used as a valid and reliable alternative to the common protocol, as well as to identify interlimb asymmetries. Full article

Based upon the Pascal distribution series ${\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +{\displaystyle \sum _{j=m+1}^{\infty }}\left(\genfrac{}{}{0pt}{}{j+r-2}{r-1}\right)\left[1+\lambda (j-1)\right]q^{j-1}{(1-q)}^r{a}_j{\zeta }^j$, we can obtain a set of fuzzy differential subordinations in this investigation. We also newly obtain class ${\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ of univalent analytic functions defined by the operator ${\mathcal{N}}_{q,\lambda }^{r,m}$, give certain properties for the class ${\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ and also obtain some applications connected with a special case for the operator. New research directions can be taken on fuzzy differential subordinations associated with symmetry operators. Full article