Symmetry doi: 10.3390/sym13081439

Authors: Igor A. Artyukov Nikolay L. Popov Alexander V. Vinogradov

Ptychography is a lensless imaging technology that is validated from hard X-rays to terahertz spectral range. It is most attractive for extreme ultraviolet (EUV) and X-rays as optical elements are expensive and often not available. Typically, the set up involves coherently illuminated object that directs the scattered radiation normally to detector which is parallel to the object plane. Computer processing of diffraction patterns obtained when scanning the object gives the image, more precisely, the distribution of intensity and phase on its surface. However, this scheme is inefficient for EUV and X-rays due to poor reflectivity and low penetration in all materials. Reflection mode ptychography solves the problem if illumination angles do not exceed the critical angle of object material. Changing the geometry of experiment changes physical and mathematical model of image formation. Including: diffraction integral describing beam propagation from object to detector, inverse problem, optimization of object illumination angle, position and orientation of detector, choosing size and grid of coordinate and frequency computer domains. This paper considers the wavefield scattered to detector by obliquely illuminated object and determines a domain for processing of obtained scans. Solution of inverse problem with phase retrieval and resulting numerical images will be presented in the next paper.

]]>Symmetry doi: 10.3390/sym13081438

Authors: Kejia Zhuang Zhenchuan Shi Yaobing Sun Zhongmei Gao Lei Wang

Accurate monitoring and prediction of tool wear conditions have an important influence on the cutting performance, thereby improving the machining precision of the workpiece and reducing the production cost. However, traditional methods cannot easily achieve exact supervision in real time because of the complexity and time-varying nature of the cutting process. A method based on Digital Twin (DT), which establish a symmetrical virtual tool system matching exactly the actual tool system, is presented herein to realize high precision in monitoring and predicting tool wear. Firstly, the framework of the cutting tool system DT is designed, and the components and operations rationale of the framework are detailed. Secondly, the key enabling technologies of the framework are elaborated. In terms of the cutting mechanism, a virtual cutting tool model is built to simulate the cutting process. The modifications and data fusion of the model are carried out to keep the symmetry between physical and virtual systems. Tool wear classification and prediction are presented based on the hybrid-driven method. With the technologies, the physical–virtual symmetry of the DT model is achieved to mapping the real-time status of tool wear accurately. Finally, a case study of the turning process is presented to verify the feasibility of the framework.

]]>Symmetry doi: 10.3390/sym13081437

Authors: Irina Volinsky

In the current paper, I research the influence of IL-2 therapy and I introduce the regulation by distributed feedback control with unbounded memory. The results of the stability analysis are presented. The proposed methodology in the article uses the properties of Cauchy matrix C(t,s), especially symmetry property, in order to study the behavior (stability) of the corresponding system of integro-differential equations.

]]>Symmetry doi: 10.3390/sym13081436

Authors: Filip Ujaković Nejc Šarabon

In this study, we investigated the association of magnitude and agreement in direction between asymmetries measured on single-joint (hip and trunk), complex movement (jumping), and skill (change of direction (CoD)) levels. The study sample comprised 43 junior- and senior-level (age = 20.5 ± 6.0 years; height = 194.5 ± 7.2 cm; body mass = 86.8 ± 10.1 kg) elite male basketball players. Both limbs/sides were tested in hip and trunk isometric strength; passive range of motion (RoM); unilateral, horizontal, and vertical jumping; and CoD tests, from which asymmetry indexes were calculated. The associations between asymmetry magnitudes were calculated with Spearman’s ρ correlation coefficient. The agreement between the direction of asymmetries on different levels was calculated with Cohen’’s Kappa (κ) coefficient. The average magnitude of asymmetry varied substantially (2.9–40.3%). Most associations between asymmetry magnitudes measured on different levels were small and statistically non-significant, with a few exceptions of moderate and large associations. Asymmetry in single-leg countermovement jump parameters was strongly associated with hip abduction maximal strength (ρ = 0.58 and 0.50, p &lt; 0.01). Agreement between asymmetry directions was slight to fair, with a few moderate exceptions. Results indicate that multiple tests are needed to obtain a comprehensive picture of athletes’ asymmetries and that universal thresholds and golden standard tests for return to play should be reconsidered and reinvestigated.

]]>Symmetry doi: 10.3390/sym13081435

Authors: Victor G. Yarzhemsky

Sr2RuO4 and Fe-pnictide superconductors belong to the same point group symmetry D4h. Many experimental data confirm odd pairs in Sr2RuO4 and even pairs in Fe-pnictides, but opposite conclusions also exist. Recent NMR results of Pustogow et al., which revealed even Cooper pairs in Sr2RuO4, require reconsideration of symmetry treatment of its SOP (superconducting order parameter). In the present work making use of the Mackey–Bradley theorem on symmetrized squares, a group theoretical investigation of possible pairing states in D4h symmetry is performed. It is obtained for I4/mmm , i.e., space group of Sr2RuO4, that triplet pairs with even spatial parts are possible in kz direction and in points M and Y. For the two latter cases pairing of equivalent electrons with nonzero total momentum is proposed. In P4/nmm space group of Fe- pnictides in point M, even and odd pairs are possible for singlet and triplet cases. It it shown that even and odd chiral states with angular momentum projection m=±1 have nodes in vertical planes, but Eg is nodal , whereas Eu is nodeless in the basal plane. It is also shown that the widely accepted assertion that the parity of angular momentum value is directly connected with the spatial parity of a pair is not valid in a space-group approach to the wavefunction of a Cooper pair.

]]>Symmetry doi: 10.3390/sym13081433

Authors: Martin Tamm

In this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these two cases. The main concern in this paper is about metrics in R3, which are close to being invariant under rotation. If we add a time-axis and let the metric start to rotate with time, it turns out that, in the case of (locally) Euclidean geometry, the (four-dimensional) scalar curvature will increase with the speed of rotation as expected. However, in the case of Lorentz geometry, the curvature will instead initially decrease. In other words, rotating metrics can, in this case, be said to be less curved than non-rotating ones. This phenomenon seems to be very general, but because of the enormous amount of computations required, it will only be proved for a class of metrics which are close to the flat one, and the main (symbolic) computations have been carried out on a computer. Although the results here are purely mathematical, there is also a connection to physics. In general, a deeper understanding of Lorentz geometry is of fundamental importance for many applied problems.

]]>Symmetry doi: 10.3390/sym13081434

Authors: Albert Feijoo Daniel Gazda Volodymyr Magas Àngels Ramos

We present a chiral K¯N interaction model that has been developed and optimized in order to account for the experimental data of inelastic K¯N reaction channels that open at higher energies. In particular, we study the effect of the higher partial waves, which originate directly from the chiral Lagrangian, as they could supersede the role of high-spin resonances employed in earlier phenomenological models to describe meson-baryon cross sections in the 2 GeV region. We present a detailed derivation of the partial wave amplitudes that emerge from the chiral SU(3) meson-baryon Lagrangian up to the d-waves and next-to-leading order in the chiral expansion. We implement a nonperturbative unitarization in coupled channels and optimize the model parameters to a large pool of experimental data in the relevant energy range where these new contributions are expected to be important. The obtained results are encouraging. They indicate the ability of the chiral higher partial waves to extend the description of the scattering data to higher energies and to account for structures in the reaction cross-sections that cannot be accommodated by theoretical models limited to the s-waves.

]]>Symmetry doi: 10.3390/sym13081432

Authors: Dmitry O. Chernyshov Andrei E. Egorov Vladimir A. Dogiel Alexei V. Ivlev

Recent observations of gamma rays with the Fermi Large Area Telescope (LAT) in the direction of the inner galaxy revealed a mysterious excess of GeV. Its intensity is significantly above predictions of the standard model of cosmic rays (CRs) generation and propagation with a peak in the spectrum around a few GeV. Popular interpretations of this excess are that it is due to either spherically distributed annihilating dark matter (DM) or an abnormal population of millisecond pulsars. We suggest an alternative explanation of the excess through the CR interactions with molecular clouds in the Galactic Center (GC) region. We assumed that the excess could be imitated by the emission of molecular clouds with depleted density of CRs with energies below ∼10 GeV inside. A novelty of our work is in detailed elaboration of the depletion mechanism of CRs with the mentioned energies through the “barrier” near the cloud edge formed by the self-excited MHD turbulence. This depletion of CRs inside the clouds may be a reason for the deficit of gamma rays from the Central Molecular Zone (CMZ) at energies below a few GeV. This in turn changes the ratio between various emission components at those energies and may potentially absorb the GeV excess by a simple renormalization of key components.

]]>Symmetry doi: 10.3390/sym13081431

Authors: Bilal Basti Nacereddine Hammami Imadeddine Berrabah Farid Nouioua Rabah Djemiat Noureddine Benhamidouche

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.

]]>Symmetry doi: 10.3390/sym13081430

Authors: Fernando Barbero Marc Basquens Valle Varo Eduardo J. S. Villaseñor

The Hamiltonian description of mechanical or field models defined by singular Lagrangians plays a central role in physics. A number of methods are known for this purpose, the most popular of them being the one developed by Dirac. Here, we discuss other approaches to this problem that rely on the direct use of the equations of motion (and the tangency requirements characteristic of the Gotay, Nester and Hinds method), or are formulated in the tangent bundle of the configuration space. Owing to its interesting relation with general relativity we use a concrete example as a test bed: an extension of the Pontryagin and Husain–Kuchař actions to four dimensional manifolds with boundary.

]]>Symmetry doi: 10.3390/sym13081429

Authors: Soubhagya Kumar Sahoo Muhammad Tariq Hijaz Ahmad Jamshed Nasir Hassen Aydi Aiman Mukheimer

Recently, fractional calculus has been the center of attraction for researchers in mathematical sciences because of its basic definitions, properties and applications in tackling real-life problems. The main purpose of this article is to present some fractional integral inequalities of Ostrowski type for a new class of convex mapping. Specifically, n–polynomial exponentially s–convex via fractional operator are established. Additionally, we present a new Hermite–Hadamard fractional integral inequality. Some special cases of the results are discussed as well. Due to the nature of convexity theory, there exists a strong relationship between convexity and symmetry. When working on either of the concepts, it can be applied to the other one as well. Integral inequalities concerned with convexity have a lot of applications in various fields of mathematics in which symmetry has a great part to play. Finally, in applications, some new limits for special means of positive real numbers and midpoint formula are given. These new outcomes yield a few generalizations of the earlier outcomes already published in the literature.

]]>Symmetry doi: 10.3390/sym13081428

Authors: Guang Wang Yixuan Shen Yujiao Jiang Jiahao Chen

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.

]]>Symmetry doi: 10.3390/sym13081427

Authors: Waheed Ahmad Khan Babir Ali Abdelghani Taouti

In this manuscript, we introduce and discuss the term bipolar picture fuzzy graphs along with some of its fundamental characteristics and applications. We also initiate the concepts of complete bipolar picture fuzzy graphs and strong bipolar picture fuzzy graphs. Firstly, we apply different types of operations to bipolar picture fuzzy graphs and then we introduce various products of bipolar picture fuzzy graphs. Several other terms such as order and size, path, neighbourhood degrees, busy values of vertices and edges of bipolar picture fuzzy graphs are also discussed. These terminologies also lay the foundations for the discussion about the regular bipolar picture fuzzy graphs. Moreover, we also discuss isomorphisms, weak and co-weak isomorphisms and automorphisms of bipolar picture fuzzy graphs. Finally, at the base of bipolar picture fuzzy graph we present the construction of a bipolar picture fuzzy acquaintanceship graph, which would be an important tool to measure the symmetry or asymmetry of acquaintanceship levels of social networks, computer networks etc.

]]>Symmetry doi: 10.3390/sym13081426

Authors: Assem Barakat Mohammad Shahidul Islam M. Ali Abdullah Mohammed Al-Majid Saeed Alshahrani Abdullah Saleh Alamary Sammer Yousuf M. Iqbal Choudhary

A series of new spiro-heterocycles engrafted spirooxindole/pyrrolidine/thiochromene scaffolds was synthesized by the three-component 1,3-dipolar cycloaddition reactions in a fully controlled regio- and stereo-selective fashion. Condensation of several substituted isatin derivatives with L-proline generated the azomethine ylides which subsequently reacted with chalcones based thiochromene scaffold, and finally afforded the target spiro-compounds. This simple protocol furnished a structurally complex, biologically relevant spiro-heterocycles in good yields through a one-pot process. All synthesized chalcone-based thiochromene, along with the spirooxindole/pyrrolidine/thiochromene scaffolds, were tested for their anticancer activity against four cancer cell lines (PC3, HeLa, MCF-7, and MDA-MB231). Toxicity of these compounds was also evaluated against human fibroblast BJ cell line, and they appeared to be not cytotoxic. For the prostate cancer (PC3) cell line, the most active hybrid, among synthesized series, was compound (7f, IC50 = 8.7 ± 0.7 µM). The most potent spirooxindole/pyrrolidine/thiochromene hybrid against cervical (HeLa) cancer cells was compound (7k, IC50 = 8.4 ± 0.5 µM) having chlorine and p-trifluoromethyl substituents attached to phenyl rings. Finally, against the MCF-7 and MDA-MB231 breast cancer cell lines, compound (7d) was the most active member of this series (IC50 = 7.36 ± 0.37, and 9.44 ± 0.32 µM, respectively).

]]>Symmetry doi: 10.3390/sym13081425

Authors: Yu-Zhu Chen Shi-Lin Li Yu-Jie Chen Wu-Sheng Dai

Gravitational waves are regarded as linear waves in the weak field approximation, which ignores the spacetime singularity. In this paper, we analyze singularities in exact gravitational wave solutions. We provide an exact general solution of the gravitational wave with cylindrical symmetry. The general solution includes some known cylindrical wave solutions as special cases. We show that there are two kinds of singularities in the cylindrical gravitational wave solution. The first kind of singularity corresponds to a singular source. The second kind of singularity corresponds to a resonance between different gravitational waves. When two gravitational waves coexist, the interference term in the source may vanish in the sense of time averaging.

]]>Symmetry doi: 10.3390/sym13081424

Authors: Xiuling Yin Xiulian Gao Yanqin Liu Yanfeng Shen Jinchan Wang

Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-path distributions and expected values of solutions are given. Then, structure preserving uncertain differential equations, uncertain Hamiltonian systems driven by Liu processes, which possess a kind of uncertain symplectic structures, are presented. A symplectic scheme with six-order accuracy and a Yao-Chen algorithm are applied to design an algorithm to solve uncertain Hamiltonian systems. At last, numerical experiments are given to investigate four uncertain Hamiltonian systems, which highlight the efficiency of our algorithm.

]]>Symmetry doi: 10.3390/sym13081423

Authors: Gianluca Rho Alejandro Luis Callara Giovanni Petri Mimma Nardelli Enzo Pasquale Scilingo Alberto Greco Vilfredo De Pascalis

Hypnotic susceptibility is a major factor influencing the study of the neural correlates of hypnosis using EEG. In this context, while its effects on the response to hypnotic suggestions are undisputed, less attention has been paid to “neutral hypnosis” (i.e., the hypnotic condition in absence of suggestions). Furthermore, although an influence of opened and closed eye condition onto hypnotizability has been reported, a systematic investigation is still missing. Here, we analyzed EEG signals from 34 healthy subjects with low (LS), medium (MS), and (HS) hypnotic susceptibility using power spectral measures (i.e., TPSD, PSD) and Lempel-Ziv-Complexity (i.e., LZC, fLZC). Indeed, LZC was found to be more suitable than other complexity measures for EEG analysis, while it has been never used in the study of hypnosis. Accordingly, for each measure, we investigated within-group differences between rest and neutral hypnosis, and between opened-eye/closed-eye conditions under both rest and neutral hypnosis. Then, we evaluated between-group differences for each experimental condition. We observed that, while power estimates did not reveal notable differences between groups, LZC and fLZC were able to distinguish between HS, MS, and LS. In particular, we found a left frontal difference between HS and LS during closed-eye rest. Moreover, we observed a symmetric pattern distinguishing HS and LS during closed-eye hypnosis. Our results suggest that LZC is better capable of discriminating subjects with different hypnotic susceptibility, as compared to standard power analysis.

]]>Symmetry doi: 10.3390/sym13081422

Authors: Antonio Masiello

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

]]>Symmetry doi: 10.3390/sym13081421

Authors: Susmit Bagchi

The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path-connected components of covering spaces establish a set of finite topological groups, maintaining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non-compact topological (C, R) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path-homotopy equivalence is admitted by multiple identity fibers if, and only if, the group homomorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path-homotopy equivalence in the discrete fundamental group. If the fiber is an identity-rigid variety, then the fiber-restricted finite and symmetric translations within the embedded covering space successfully admits path-homotopy equivalence involving kernel. The topological projections on a component and formation of 2-simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2-simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.

]]>Symmetry doi: 10.3390/sym13081420

Authors: Chuanfu Wang Yi Di Jianyu Tang Jing Shuai Yuchen Zhang Qi Lu

Dynamic degradation occurs when chaotic systems are implemented on digital devices, which seriously threatens the security of chaos-based pseudorandom sequence generators. The chaotic degradation shows complex periodic behavior, which is often ignored by designers and seldom analyzed in theory. Not knowing the exact period of the output sequence is the key problem that affects the application of chaos-based pseudorandom sequence generators. In this paper, two cubic chaotic maps are combined, which have symmetry and reconfigurable form in the digital circuit. The dynamic behavior of the cubic chaotic map and the corresponding digital cubic chaotic map are analyzed respectively, and the reasons for the complex period and weak randomness of output sequences are studied. On this basis, the digital cubic chaotic map is optimized, and the complex periodic behavior is improved. In addition, a reconfigurable pseudorandom sequence generator based on the digital cubic chaotic map is constructed from the point of saving consumption of logical resources. Through theoretical and numerical analysis, the pseudorandom sequence generator solves the complex period and weak randomness of the cubic chaotic map after digitization and makes the output sequence have better performance and less resource consumption, which lays the foundation for applying it to the field of secure communication.

]]>Symmetry doi: 10.3390/sym13081419

Authors: Guillermo Martínez-Flórez Sandra Vergara-Cardozo Roger Tovar-Falón

In this paper, an asymmetric regression model for censored non-negative data based on the centred exponentiated log-skew-normal and Bernoulli distributions mixture is introduced. To connect the discrete part with the continuous distribution, the logit link function is used. The parameters of the model are estimated by using the likelihood maximum method. The score function and the information matrix are shown in detail. Antibody data from a study of the measles vaccine are used to illustrate applicability of the proposed model, and it was found the best fit to the data with respect to an others models used in the literature.

]]>Symmetry doi: 10.3390/sym13081418

Authors: Serge Lawrencenko Abdulkarim M. Magomedov

Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P′(1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G=K2,2,2,2, in which specific case P(x)=x31. The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (12 in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q8 with the three imaginary quaternions i, j, k as generators.

]]>Symmetry doi: 10.3390/sym13081417

Authors: Marcelo Schiffer

It is a well-known fact that the Newtonian description of dynamics within Galaxies for its known matter content is in disagreement with the observations as the acceleration approaches a0≈1.2×10−10 m/s2 (slighter larger for clusters). Both the Dark Matter scenario and Modified Gravity Theories (MGT) fail to explain the existence of such an acceleration scale. Motivated by the closeness of the acceleration scale and the Hubble constant cH0≈10−9 h m/s2, we are led to analyze whether this coincidence might have a Cosmological origin for scalar-tensor and spinor-tensor theories by performing detailed calculations for perturbations that represent the local matter distribution on the top of the cosmological background. Then, we solve the field equations for these perturbations in a power series in the present value of the Hubble constant. As we shall see, for both theories, the power expansion contains only even powers in the Hubble constant, a fact that renders the cosmological expansion irrelevant for the local dynamics.

]]>Symmetry doi: 10.3390/sym13081416

Authors: Jesús S. Dehesa

In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, the Rényi and Shannon entropies, and their corresponding spreading lengths are analytically expressed in terms of the degree and the parameter(s) of the orthogonality weight function. These entropic quantities are closely related to the gradient functional (Fisher) and the Lq-norms (Rényi, Shannon) of the polynomials. In addition, the degree asymptotics for these entropy-like functionals of the three canonical families of HPOs (i.e., Hermite, Laguerre, and Jacobi polynomials) are given and briefly discussed. Finally, a number of open related issues are identified whose solutions are both physico-mathematically and computationally relevant.

]]>Symmetry doi: 10.3390/sym13081415

Authors: William P. Fisher

As part of his explication of the epistemological error made in separating thinking from its ecological context, Bateson distinguished counts from measurements. With no reference to Bateson, the measurement theory and practice of Benjamin Wright also recognizes that number and quantity are different logical types. Describing the confusion of counts and measures as schizophrenic, like Bateson, Wright, a physicist and certified psychoanalyst, showed mathematically that convergent stochastic processes informing counts are predictable in ways that facilitate methodical measurements. Wright’s methods experimentally evaluate the complex symmetries of nonlinear and stochastic numeric patterns as a basis for estimating interval quantities. These methods also retain connections with locally situated concrete expressions, mediating the data display by contextualizing it in relation to the abstractly communicable and navigable quantitative unit and its uncertainty. Decades of successful use of Wright’s methods in research and practice are augmented in recent collaborations of metrology engineers and psychometricians who are systematically distinguishing numeric counts from measured quantities in new classes of knowledge infrastructure. Situating Wright’s work in the context of Bateson’s ideas may be useful for infrastructuring new political, economic, and scientific outcomes.

]]>Symmetry doi: 10.3390/sym13081414

Authors: Lizhen Duan Shuhan Sun Jianlin Zhang Zhiyong Xu

Atmospheric turbulence significantly degrades image quality. A blind image deblurring algorithm is needed, and a favorable image prior is the key to solving this problem. However, the general sparse priors support blurry images instead of explicit images, so the details of the restored images are lost. The recently developed priors are non-convex, resulting in complex and heuristic optimization. To handle these problems, we first propose a convex image prior; namely, maximizing L1 regularization (ML1). Benefiting from the symmetrybetween ML1 and L1 regularization, the ML1 supports clear images and preserves the image edges better. Then, a novel soft suppression strategy is designed for the deblurring algorithm to inhibit artifacts. A coarse-to-fine scheme and a non-blind algorithm are also constructed. For qualitative comparison, a turbulent blur dataset is built. Experiments on this dataset and real images demonstrate that the proposed method is superior to other state-of-the-art methods in blindly recovering turbulent images.

]]>Symmetry doi: 10.3390/sym13081413

Authors: José F. Cariñena José Fernández-Núñez

We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given.

]]>Symmetry doi: 10.3390/sym13081412

Authors: Sergei D. Odintsov

With deep sadness, we announce that on 23 July 2021 our Editorial Board Member Steven Weinberg passed away [...]

]]>Symmetry doi: 10.3390/sym13081411

Authors: Magda Dettlaff Magdalena Lemańska Jerzy Topp

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

]]>Symmetry doi: 10.3390/sym13081410

Authors: Juan Liu Yilun Du Song Shi

In a one-flavor NJL model with a finite temperature, chemical potential, and external magnetic field, the self-energy of the quark propagator contains more condensates besides the vacuum condensate. We use Fierz identity to identify the self-energy and propose a self-consistent analysis to simplify it. It turns out that these condensates are related to the chiral separation effect and spin magnetic moment.

]]>Symmetry doi: 10.3390/sym13081409

Authors: Jiawen Li Yi Zhang Zhenghong Jin

In this paper, the Singular-Polynomial-Fuzzy-Model (SPFM) approach problem and impulse elimination are investigated based on sliding mode control for a class of nonlinear singular system (NSS) with impulses. Considering two numerical examples, the SPFM of the nonlinear singular system is calculated based on the compound function type and simple function type. According to the solvability and the steps of two numerical examples, the method of solving the SPFM form of the nonlinear singular system with (and without) impulse are extended to the more general case. By using the Heine–Borel finite covering theorem, it is proven that a class of nonlinear singular systems with bounded impulse-free item (BIFI) properties and separable impulse item (SII) properties can be approximated by SPFM with arbitrary accuracy. The linear switching function and sliding mode control law are designed to be applied to the impulse elimination of SPFM. Compared with some published works, a human posture inverted pendulum model example and Example 3.2 demonstrate that the approximation error is small enough and that both algorithms are effective. Example 3.3 is to illustrate that sliding mode control can effectively eliminate impulses of SPFM.

]]>Symmetry doi: 10.3390/sym13081408

Authors: Taichiro Kugo

In general coordinate invariant gravity theories whose Lagrangians contain arbitrarily high order derivative fields, the Noether currents for the global translation and for the Nakanishi’s IOSp(8|8) choral symmetry containing the BRS symmetry as its member are constructed. We generally show that for each of these Noether currents, a suitable linear combination of equations of motion can be brought into the form of a Maxwell-type field equation possessing the Noether current as its source term.

]]>Symmetry doi: 10.3390/sym13081407

Authors: Peyman Lahe Motlagh Adnan Kefal

Recently, topology optimization of structures with cracks becomes an important topic for avoiding manufacturing defects at the design stage. This paper presents a comprehensive comparative study of peridynamics-based topology optimization method (PD-TO) and classical finite element topology optimization approach (FEM-TO) for designing lightweight structures with/without cracks. Peridynamics (PD) is a robust and accurate non-local theory that can overcome various difficulties of classical continuum mechanics for dealing with crack modeling and its propagation analysis. To implement the PD-TO in this study, bond-based approach is coupled with optimality criteria method. This methodology is applicable to topology optimization of structures with any symmetric/asymmetric distribution of cracks under general boundary conditions. For comparison, optimality criteria approach is also employed in the FEM-TO process, and then topology optimization of four different structures with/without cracks are investigated. After that, strain energy and displacement results are compared between PD-TO and FEM-TO methods. For design domain without cracks, it is observed that PD and FEM algorithms provide very close optimum topologies with a negligibly small percent difference in the results. After this validation step, each case study is solved by integrating the cracks in the design domain as well. According to the simulation results, PD-TO always provides a lower strain energy than FEM-TO for optimum topology of cracked structures. In addition, the PD-TO methodology ensures a better design of stiffer supports in the areas of cracks as compared to FEM-TO. Furthermore, in the final case study, an intended crack with a symmetrically designed size and location is embedded in the design domain to minimize the strain energy of optimum topology through PD-TO analysis. It is demonstrated that hot-spot strain/stress regions of the pristine structure are the most effective areas to locate the designed cracks for effective redistribution of strain/stress during topology optimization.

]]>Symmetry doi: 10.3390/sym13081406

Authors: Rémi Bougault Bernard Borderie Abdelouahad Chbihi Quentin Fable John David Frankland Emmanuelle Galichet Tom Genard Diégo Gruyer Maxime Henri Marco La Commara Nicolas Le Neindre Ivano Lombardo Olivier Lopez Marian Pârlog Piotr Pawłowski Giuseppe Verde Emmanuel Vient Mariano Vigilante

Correlations and clustering are of great importance in the study of the Nuclear Equation of State. Information on these items/aspects can be obtained using heavy-ion reactions which are described by dynamical theories. We propose a dataset that will be useful for improving the description of light cluster production in transport model approaches. The dataset combines published and new data and is presented in a form that allows direct comparison of the experiment with theoretical predictions. The dataset is ranging in bombarding energy from 32 to 1930 A MeV. In constructing this dataset, we put in evidence the existence of a change in the light cluster production mechanism that corresponds to a peak in deuteron production.

]]>Symmetry doi: 10.3390/sym13081405

Authors: Feng Pan Yingwen He Lianrong Dai Chong Qi Jerry P. Draayer

A diagonalization scheme for the shell model mean-field plus isovector pairing Hamiltonian in the O(5) tensor product basis of the quasi-spin SUΛ(2) ⊗ SUI(2) chain is proposed. The advantage of the diagonalization scheme lies in the fact that not only can the isospin-conserved, charge-independent isovector pairing interaction be analyzed, but also the isospin symmetry breaking cases. More importantly, the number operator of the np-pairs can be realized in this neutron and proton quasi-spin basis, with which the np-pair occupation number and its fluctuation at the J = 0+ ground state of the model can be evaluated. As examples of the application, binding energies and low-lying J = 0+ excited states of the even–even and odd–odd N∼Z ds-shell nuclei are fit in the model with the charge-independent approximation, from which the neutron–proton pairing contribution to the binding energy in the ds-shell nuclei is estimated. It is observed that the decrease in the double binding-energy difference for the odd–odd nuclei is mainly due to the symmetry energy and Wigner energy contribution to the binding energy that alter the pairing staggering patten. The np-pair amplitudes in the np-pair stripping or picking-up process of these N = Z nuclei are also calculated.

]]>Symmetry doi: 10.3390/sym13081404

Authors: María Dolores García-Sanz José Carlos R. Alcantud

There is a substantial strand of literature about ranking the subsets of a set of alternatives, usually referred to as opportunity sets. We investigate a model that is dependent on the preference of a grand set of alternatives. In this framework, the indirect-utility criterion ranks the opportunity sets by the following rule: a subset A is weakly preferred to another subset B if and only if A contains at least one preference maximizing element from A∪B. This criterion leads to the indifference of each subset of alternatives to a singleton; symmetry appears at this stage, as the property holds true for any one of the maximizers in A. Conversely, suppose that a ranking of opportunity sets satisfies the property that each opportunity set is indifferent to a singleton contained within it. Then, we prove that such a ranking must use a generalized form of the indirect-utility criterion, where maximization is applied to a selection of the alternatives. Altogether, these results produce a characterization of the advised indirect-utility criterion for ranking opportunity sets.

]]>Symmetry doi: 10.3390/sym13081403

Authors: Ranjan Vepa M. Hasan Shaheed

Several researchers are considering the plausibility of being able to rapidly launch a mission to an asteroid, which would fly in close proximity of the asteroid to deliver an impulse in a particular direction so as to deflect the asteroid from its current orbit. Planetary motion, in general, and the motion of asteroids, in particular, are subject to planetary influences that are characterised by a kind of natural symmetry, which results in an asteroid orbiting in a stable and periodic or almost periodic orbit exhibiting a number of natural orbital symmetries. Tracking and following an asteroid, in close proximity, is the subject of this paper. In this paper, the problem of synthesizing an optimal trajectory to a NEO such as an asteroid is considered. A particular strategy involving the optimization of a co-planar trajectory segment that permits the satellite to approach and fly alongside the asteroid is chosen. Two different state space representations of the Hill–Clohessy–Wiltshire (HCW) linearized equations of relative motion are used to obtain optimal trajectories for a spacecraft approaching an asteroid. It is shown that by using a state space representation of HCW equations where the secular states are explicitly represented, the optimal trajectories are not only synthesized rapidly but also result in lower magnitudes of control inputs which must be applied continuously over extended periods of time. Thus, the solutions obtained are particularly suitable for low thrust control of the satellites orbit which can be realized by electric thrusters.

]]>Symmetry doi: 10.3390/sym13081402

Authors: Samad Noeiaghdam Denis Sidorov

Linear and nonlinear integral equations of the first and second kinds have many applications in engineering and real life problems [...]

]]>Symmetry doi: 10.3390/sym13081401

Authors: Alexei Uteshev Ivan Baravy Elizaveta Kalinina

We treat the interpolation problem {f(xj)=yj}j=1N for polynomial and rational functions. Developing the approach originated by C. Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences of special symmetric functions of the data set like {∑j=1Nxjkyj/W′(xj)}k∈N and {∑j=1Nxjk/(yjW′(xj))}k∈N; here, W(x)=∏j=1N(x−xj). We also review the results by Jacobi, Joachimsthal, Kronecker and Frobenius on the recursive procedure for computation of the sequence of Hankel polynomials. The problem of evaluation of the resultant of polynomials p(x) and q(x) given a set of values {p(xj)/q(xj)}j=1N is also tackled within the framework of this approach. An effective procedure is suggested for recomputation of rational interpolants in case of extension of the data set by an extra point.

]]>Symmetry doi: 10.3390/sym13081400

Authors: Juan M. Torres-Rincon

Chiral symmetry represents a fundamental concept lying at the core of particle and nuclear physics. Its spontaneous breaking in vacuum can be exploited to distinguish chiral hadronic partners, whose masses differ. In fact, the features of this breaking serve as guiding principles for the construction of effective approaches of QCD at low energies, e.g., the chiral perturbation theory, the linear sigma model, the (Polyakov)–Nambu–Jona-Lasinio model, etc. At high temperatures/densities chiral symmetry can be restored bringing the chiral partners to be nearly degenerated in mass. At vanishing baryochemical potential, such restoration follows a smooth transition, and the chiral companions reach this degeneration above the transition temperature. In this work I review how different realizations of chiral partner degeneracy arise in different effective theories/models of QCD. I distinguish the cases where the chiral states are either fundamental degrees of freedom or (dynamically-generated) composed states. In particular, I discuss the intriguing case in which chiral symmetry restoration involves more than two chiral partners, recently addressed in the literature.

]]>Symmetry doi: 10.3390/sym13081399

Authors: Alexander M. Banaru Sergey M. Aksenov Sergey V. Krivovichev

Structural complexity measures based on Shannon information entropy are widely used for inorganic crystal structures. However, the application of these parameters for molecular crystals requires essential modification since atoms in inorganic compounds usually possess more degrees of freedom. In this work, a novel scheme for the calculation of complexity parameters (HmolNet, HmolNet,tot) for molecular crystals is proposed as a sum of the complexity of each molecule, the complexity of intermolecular contacts, and the combined complexity of both. This scheme is tested for several molecular crystal structures.

]]>Symmetry doi: 10.3390/sym13081398

Authors: Oleksij Fomin Alyona Lovska Juraj Gerlici Yuliia Fomina Ján Dižo Miroslav Blatnický

An articulated covered wagon design was developed. The wagon feature is that the body-bearing elements are made of circular pipes. This technical solution made it possible to reduce the tare weight of the wagon while ensuring the strength conditions. Mathematical simulation of the dynamic loading of the developed articulated covered wagon design was carried out under the main operating conditions. In the calculations of the observed quantities, an application of symmetry with regard to the longitudinal axis of the wagon was used. The accelerations, as the components of the dynamic load acting on the wagon, were determined. The dynamic loading computer simulation results of the developed wagon design are also presented. The strength analysis of the articulated covered wagon supporting structure made it possible to conclude that the strength indexes were within the allowed limits. The wagon bearing structure was analyzed for fatigue strength. The weld strength analysis results of the most loaded part of the wagon-bearing structure are presented. The results obtained for the desired quantities revealed their symmetrical distribution in the wagon structure. This research will contribute to improving the efficiency of railway transport operation.

]]>Symmetry doi: 10.3390/sym13081397

Authors: Hanan Alolaiyan Muhammad Haris Mateen Dragan Pamucar Muhammad Khalid Mahmmod Farrukh Arslan

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define (α,β)-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.

]]>Symmetry doi: 10.3390/sym13081396

Authors: Jin Hyuk Choi Hyunsoo Kim

In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations.

]]>Symmetry doi: 10.3390/sym13081395

Authors: Danila Kostarev Dmitri Klimushkin Pavel Mager

We consider the solutions of two integrodifferential equations in this work. These equations describe the ultra-low frequency waves in the dipol-like model of the magnetosphere in the gyrokinetic framework. The first one is reduced to the homogeneous, second kind Fredholm equation. This equation describes the structure of the parallel component of the magnetic field of drift-compression waves along the Earth’s magnetic field. The second equation is reduced to the inhomogeneous, second kind Fredholm equation. This equation describes the field-aligned structure of the parallel electric field potential of Alfvén waves. Both integral equations are solved numerically.

]]>Symmetry doi: 10.3390/sym13081394

Authors: Siwei Duo Taras I. Lakoba Yanzhi Zhang

We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian (−Δ)α/2. The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power α∈[1,2] but unstable for α∈(0,1). In the focusing case, they can be linearly unstable for any α∈(0,2]. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with earlier studies of solitary wave solutions of the fractional focusing NLS, we find that as α∈(1,2] decreases, the solution evolves towards an increasingly localized pulse existing on the background of a “sea” of small-amplitude dispersive waves. Such a highly localized pulse has a broad spectrum, most of whose modes are excited in the nonlinear stage of the pulse evolution and are not predicted by the linear stability analysis. For α≤1, we always find the solution to undergo collapse. We also show, for the first time to our knowledge, that for initial conditions with nonzero group velocities (traveling plane waves), an onset of collapse is delayed compared to that for a standing plane wave initial condition. For defocusing fractional NLS, even though we find traveling plane waves to be linearly unstable for α&lt;1, we have never observed collapse. As a by-product of our numerical studies, we derive a stability condition on the time step of the SSFS to guarantee that this method is free from numerical instabilities.

]]>Symmetry doi: 10.3390/sym13081393

Authors: Weichun Bu Tianqing An José Vanteler da C. Sousa Yongzhen Yun

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations a+b∫Ω×Ω|ξ(x)−ξ(y)|p|x−y|N+ps(x,y)dxdyp−1(−Δ)ps(·)ξ+λV(x)|ξ|p−2ξ=f(x,ξ),x∈Ω,ξ=0,x∈∂Ω, where Ω is a bounded Lipschitz domain in RN, 1&lt;p&lt;+∞, a,b&gt;0 are constants, s(·):RN×RN→(0,1) is a continuous and symmetric function with N&gt;s(x,y)p for all (x,y)∈Ω×Ω, λ&gt;0 is a parameter, (−Δ)ps(·) is a fractional p-Laplace operator with variable-order, V(x):Ω→R+ is a potential function, and f(x,ξ):Ω×RN→R is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.

]]>Symmetry doi: 10.3390/sym13081392

Authors: Ana Maria Acu Gülen Başcanbaz-Tunca Ioan Raşa

We consider positive linear operators having the same fundamental functions and different functionals in front of them. For differences involving such operators, we obtain Voronovskaja-type quantitative results. Applications illustrating the theoretical aspects are presented.

]]>Symmetry doi: 10.3390/sym13081391

Authors: Constantin Udriste Ionel Tevy

Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented.

]]>Symmetry doi: 10.3390/sym13081390

Authors: Zhenyong Hou Zhenghua Huang Lidong Xia Hui Fu Youqian Qi Dayang Liu Ning Tang

Solar Ultraviolet bursts (UBs) associated with flux emergence are expected to help understand the physical processes of the flux emergence itself. In the present study, we analyse imaging and spectroscopic observations of a special group of UBs (including twelve of them) occurring in the joint footpoint regions of multiple transition region loops above the flux emerging regions. Consistent with previous studies of common UBs, we found that the spectral characteristics of this group of UBs are varied. Our results show that the responses of UBs in Ni ii, NUV continuum, Mg ii h and O i are originated from locations differ from that emits Si iv. The imaging data show that UBs have connections with the dynamics in the transition region loops. Brightenings starting from UB-regions and propagating along loops can be seen in SJ 1400/1330 Å and AIA 304 Å images and the corresponding time-space images. The apparent velocities are tens of kilometers per second in AIA 304 Å. For symmetry, the brightenings can propagate from the UB-regions towards opposite directions with similar apparent velocities in some cases. Given that these UBs are magnetic reconnection phenomena, we suggest that the propagating brightenings are the signals of the plasma flows resulted from heatings in the UB-regions.

]]>Symmetry doi: 10.3390/sym13081389

Authors: Jong Min Kim Il Do Ha

A residual (r) control chart of asymmetrical and non-normal binary response variable with highly correlated explanatory variables is proposed in this research. To avoid multicollinearity between multiple explanatory variables, we employ and compare a neural network regression model and deep learning regression model using Bayesian variable selection (BVS), principal component analysis (PCA), nonlinear PCA (NLPCA) or whole multiple explanatory variables. The advantage of our r control chart is able to process both non-normal and correlated multivariate explanatory variables by employing a neural network model and deep learning model. We prove that the deep learning r control chart is relatively efficient to monitor the simulated and real binary response asymmetric data compared with r control chart of the generalized linear model (GLM) with probit and logit link functions and neural network r control chart.

]]>Symmetry doi: 10.3390/sym13081388

Authors: Mannque Rho

I discuss how the axial current coupling constant gA renormalized in scale symmetric chiral EFT defined at a chiral matching scale impacts on the axial current matrix elements on beta decays in nuclei with and without neutrinos. The “quenched” gA observed in nuclear superallowed Gamow–Teller transitions, a long-standing puzzle in nuclear physics, is shown to encode the emergence of chiral-scale symmetry hidden in QCD in the vacuum. This enables one to explore how trace-anomaly-induced scale symmetry breaking enters in the renormalized gA in nuclei applicable to certain non-unique forbidden processes involved in neutrinoless double beta decays. A parallel is made between the roles of chiral-scale symmetry in quenching gA in highly dense medium and in hadron–quark continuity in the EoS of dense matter in massive compact stars. A systematic chiral-scale EFT, presently lacking in nuclear theory and potentially crucial for the future progress, is suggested as a challenge in the field.

]]>Symmetry doi: 10.3390/sym13081387

Authors: Vladimir Gurvich Mariya Naumova

We show that the one-way ANOVA and Tukey–Kramer (TK) tests agree on any sample with two groups. This result is based on a simple identity connecting the Fisher–Snedecor and studentized probabilistic distributions and is proven without any additional assumptions; in particular, the standard ANOVA assumptions (independence, normality, and homoscedasticity (INAH)) are not needed. In contrast, it is known that for a sample with k&gt;2 groups of observations, even under the INAH assumptions, with the same significance level α, the above two tests may give opposite results: (i) ANOVA rejects its null hypothesis H0A:μ1=…=μk, while the TK one, H0TK(i,j):μi=μj, is not rejected for any pair i,j∈{1,…,k}; (ii) the TK test rejects H0TK(i,j) for a pair (i,j) (with i≠j), while ANOVA does not reject H0A. We construct two large infinite pseudo-random families of samples of both types satisfying INAH: in case (i) for any k≥3 and in case (ii) for some larger k. Furthermore, case (ii) ANOVA, being restricted to the pair of groups (i,j), may reject equality μi=μj with the same α. This is an obvious contradiction, since μ1=…=μk implies μi=μj for all i,j∈{1,…,k}. Such contradictions appear already in the symmetric case for k=3, or in other words, for three groups of d,d, and c observations with sample means +1,−1, and 0, respectively. We outline conditions necessary and sufficient for this phenomenon. Similar contradictory examples are constructed for the multivariable linear regression (MLR). However, for these constructions, it seems difficult to verify the Gauss–Markov assumptions, which are standardly required for MLR. Mathematics Subject Classification: 62 Statistics.

]]>Symmetry doi: 10.3390/sym13081386

Authors: Firdaus E. Udwadia

This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

]]>Symmetry doi: 10.3390/sym13081385

Authors: Vincenzo Bonifaci

The approach to equilibrium in certain dynamical systems can be usefully described in terms of information-theoretic functionals. Well-studied models of this kind are Markov processes, chemical reaction networks, and replicator dynamics, for all of which it can be proven, under suitable assumptions, that the relative entropy (informational divergence) of the state of the system with respect to an equilibrium is nonincreasing over time. This work reviews another recent result of this type, which emerged in the study of the network optimization dynamics of an acellular slime mold, Physarum polycephalum. In this setting, not only the relative entropy of the state is nonincreasing, but its evolution over time is crucial to the stability of the entire system, and the equilibrium towards which the dynamics is attracted proves to be a global minimizer of the cost of the network.

]]>Symmetry doi: 10.3390/sym13081384

Authors: Zengtai Gong Le Fan

With the increasing complexity of the human social environment, it is impossible to describe each object in detail with accurate numbers when solving multiple attribute decision-making (MADM) problems. Compared with the fuzzy set (FS), the intuitionistic fuzzy set (IFS) not only has obvious advantages in allocating ambiguous values to the object to be considered, but also takes into account the degree of membership and non-membership, so it is more suitable for decision makers (DMs) to deal with complex realistic problems. Therefore, it is of great significance to propose a MADM method under an intuitionistic fuzzy environment. Moreover, compared with the traditional 2WD, by putting forward the option of delay, the decision-making risk can be effectively reduced using three-way decision (3WD). In addition, the binary relations between objects in the decision-making process have been continuously generalized, such as equivalence relation which have symmetrical relationship, dominance relation and outranking relation, which are worthy of study. In this paper, we propose 3WD-MADM method based on IF environment and the objective IFS is calculated by using the information table. Then, the hybrid information table is used to solve the supplier selection problem to demonstrate the effectiveness of the proposed method.

]]>Symmetry doi: 10.3390/sym13081383

Authors: Sreenivasa Rao Jammalamadaka Emanuele Taufer Gyorgy H. Terdik

This paper provides a systematic and comprehensive treatment for obtaining general expressions of any order, for the moments and cumulants of spherically and elliptically symmetric multivariate distributions; results for the case of multivariate t-distribution and related skew-t distribution are discussed in some detail.

]]>Symmetry doi: 10.3390/sym13081382

Authors: Roger D. Maddux

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

]]>Symmetry doi: 10.3390/sym13081380

Authors: Sen Wang Xiaoming Sun Pengfei Liu Kaige Xu Weifeng Zhang Chenxu Wu

The purpose of image registration is to find the symmetry between the reference image and the image to be registered. In order to improve the registration effect of unmanned aerial vehicle (UAV) remote sensing imagery with a special texture background, this paper proposes an improved scale-invariant feature transform (SIFT) algorithm by combining image color and exposure information based on adaptive quantization strategy (AQCE-SIFT). By using the color and exposure information of the image, this method can enhance the contrast between the textures of the image with a special texture background, which allows easier feature extraction. The algorithm descriptor was constructed through an adaptive quantization strategy, so that remote sensing images with large geometric distortion or affine changes have a higher correct matching rate during registration. The experimental results showed that the AQCE-SIFT algorithm proposed in this paper was more reasonable in the distribution of the extracted feature points compared with the traditional SIFT algorithm. In the case of 0 degree, 30 degree, and 60 degree image geometric distortion, when the remote sensing image had a texture scarcity region, the number of matching points increased by 21.3%, 45.5%, and 28.6%, respectively and the correct matching rate increased by 0%, 6.0%, and 52.4%, respectively. When the remote sensing image had a large number of similar repetitive regions of texture, the number of matching points increased by 30.4%, 30.9%, and −11.1%, respectively and the correct matching rate increased by 1.2%, 0.8%, and 20.8% respectively. When processing remote sensing images with special texture backgrounds, the AQCE-SIFT algorithm also has more advantages than the existing common algorithms such as color SIFT (CSIFT), gradient location and orientation histogram (GLOH), and speeded-up robust features (SURF) in searching for the symmetry of features between images.

]]>Symmetry doi: 10.3390/sym13081381

Authors: Qiuyan Zhan

The aim of this paper is to investigate several operators on quantum B-algebras. At first, we introduce closure and interior operators on quantum B-algebras and consider their relations on bounded quantum B-algebras. Furthermore, we discuss very true operators on quantum B-algebras by three cases via the unit element, and present some similar conclusions and different results. Finally, by constructing a very true operator on a quotient very true perfect quantum B-algebra, we establish a homomorphism theorem on very true perfect quantum B-algebras.

]]>Symmetry doi: 10.3390/sym13081379

Authors: Dong-Hai Han Jing-Bo Zhao Guang-Jun Zhang Hong Yao

In order to solve the problem of low-frequency noise of aircraft cabins, this paper presents a new Helmholtz type phononic crystal with a two-dimensional symmetric structure. Under the condition of the lattice constant of 62 mm, the lower limit of the first band gap is about 12 Hz, and the width is more than 10 Hz, thus the symmetric structure has distinct sound insulation ability in the low-frequency range. Firstly, the cause of the low-frequency band gap is analyzed by using the sound pressure field, and the range of band gaps is calculated by using the finite element method and the spring-oscillator model. Although the research shows that the finite element calculation results are basically consistent with the theoretical calculation, there are still some errors, and the reasons for the errors are analyzed. Secondly, the finite element method and equivalent model method are used to explore the influence of parameters of the symmetric structure on the first band gap. The result shows that the upper limit of the first band gap decreases with the increase of the lattice constant and the wedge height and increases with the increase of the length of wedge base; the lower limit of the band gap decreases with the increase of the wedge height and length of wedge base and is independent of the change of lattice constant, which further reveals the essence of the band gap formation and verifies the accuracy of the equivalent model. This study provides some theoretical support for low-frequency noise control and broadens the design idea of symmetric phononic crystal.

]]>Symmetry doi: 10.3390/sym13081378

Authors: Yulan Wang Zhixia Yang Xiaomei Yang

In this paper, we propose a novel binary classification method called the kernel-free quadratic surface minimax probability machine (QSMPM), that makes use of the kernel-free techniques of the quadratic surface support vector machine (QSSVM) and inherits the advantage of the minimax probability machine (MPM) without any parameters. Specifically, it attempts to find a quadratic hypersurface that separates two classes of samples with maximum probability. However, the optimization problem derived directly was too difficult to solve. Therefore, a nonlinear transformation was introduced to change the quadratic function involved into a linear function. Through such processing, our optimization problem finally became a second-order cone programming problem, which was solved efficiently by an alternate iteration method. It should be pointed out that our method is both kernel-free and parameter-free, making it easy to use. In addition, the quadratic hypersurface obtained by our method was allowed to be any general form of quadratic hypersurface. It has better interpretability than the methods with the kernel function. Finally, in order to demonstrate the geometric interpretation of our QSMPM, five artificial datasets were implemented, including showing the ability to obtain a linear separating hyperplane. Furthermore, numerical experiments on benchmark datasets confirmed that the proposed method had better accuracy and less CPU time than corresponding methods.

]]>Symmetry doi: 10.3390/sym13081377

Authors: Ruba Abu Khurma Iman Almomani Ibrahim Aljarah

In the last decade, the devices and appliances utilizing the Internet of things (IoT) have expanded tremendously, which has led to revolutionary developments in the network industry. Smart homes and cities, wearable devices, traffic monitoring, health systems, and energy savings are typical IoT applications. The diversity in IoT standards, protocols, and computational resources makes them vulnerable to security attackers. Botnets are challenging security threats in IoT devices that cause severe Distributed Denial of Service (DDoS) attacks. Intrusion detection systems (IDS) are necessary for safeguarding Internet-connected frameworks and enhancing insufficient traditional security countermeasures, including authentication and encryption techniques. This paper proposes a wrapper feature selection model (SSA–ALO) by hybridizing the salp swarm algorithm (SSA) and ant lion optimization (ALO). The new model can be integrated with IDS components to handle the high-dimensional space problem and detect IoT attacks with superior efficiency. The experiments were performed using the N-BaIoT benchmark dataset, which was downloaded from the UCI repository. This dataset consists of nine datasets that represent real IoT traffic. The experimental results reveal the outperformance of SSA–ALO compared to existing related approaches using the following evaluation measures: TPR (true positive rate), FPR (false positive rate), G-mean, processing time, and convergence curves. Therefore, the proposed SSA–ALO model can serve IoT applications by detecting intrusions with high true positive rates that reach 99.9% and with a minimal delay even in imbalanced intrusion families.

]]>Symmetry doi: 10.3390/sym13081376

Authors: Hideki Kanda Wahyudiono Motonobu Goto

The Tolman length and interfacial tension of partially miscible symmetric binary Lennard–Jones (LJ) fluids (A, B) was revealed by performing a large-scale molecular dynamics (MD) simulation with a sufficient interfacial area and cutting distance. A unique phenomenon was observed in symmetric binary LJ fluids, where two surfaces of tension existed on both sides of an equimolar dividing surface. The range of interaction εAB between the different liquids and the temperature in which the two LJ fluids partially mixed was clarified, and the Tolman length exceeded 3 σ when εAB was strong at higher temperatures. The results show that as the temperature or εAB increases, the Tolman length increases and the interfacial tension decreases. This very long Tolman length indicates that one should be very careful when applying the concept of the liquid–liquid interface in the usual continuum approximation to nanoscale droplets and capillary phase separation in nanopores.

]]>Symmetry doi: 10.3390/sym13081375

Authors: Duan Zhu Zhende Zhu Cong Zhang Xinghua Xie

Fault-crossing tunnels are often severely damaged under seismic dynamics. Study of the dynamic response characteristics of tunnels crossing faults is thus of great engineering significance. Here, the Xianglushan Tunnel of the Central Yunnan Water Diversion Project was studied. A shaking table experimental device was used, and four sets of dynamic model tests of deep-buried tunnels with different fault inclination angles were conducted. Test schemes of model similarity ratio, similar material selection, model box design, and sine wave loading were introduced. The acceleration and strain data of the tunnel lining were monitored. Analysis of the acceleration data showed that when the input PGA was 0.6 g, compared with the ordinary tunnel, the acceleration increases by 117% when the inclination angle was 75°, 127% when the inclination angle was 45°, and 144% when the inclination angle was 30°. This indicates that the dynamic response of the cross-fault tunnel structure was stronger than that of the ordinary tunnel, and the effect was more obvious as the fault dip angle decreased. Analysis of the strain data showed that the strain response of the fault-crossing tunnels was more sensitive to the fault dip. The peak strain and increase in fault-crossing tunnels were much larger than those of ordinary tunnels, and smaller fault dips led to larger increases in the strain peak; consequently, the tunnel would reach the ultimate strain and break down when the input PGA was smaller. Generally, the influence of fault inclination on the dynamic response of the tunnel lining should receive increased consideration in the seismic design of tunnels.

]]>Symmetry doi: 10.3390/sym13081374

Authors: Umar Ali Hassan Raza Yasir Ahmed

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2n squares, denoted by Ln6,4,4. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of symmetric tri-diagonal matrices LA and LS of order 4n+1. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.

]]>Symmetry doi: 10.3390/sym13081371

Authors: Cheon-Seoung Ryoo

In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted (p,q)-Euler polynomials by using computer.

]]>Symmetry doi: 10.3390/sym13081373

Authors: Louis H. Kauffman

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

]]>Symmetry doi: 10.3390/sym13081372

Authors: Hongqiu Li Jinhui Jiang M Shadi Mohamed

Dynamic load identification is an inverse problem concerned with finding the load applied on a structure when the dynamic characteristics and the response of the structure are known. In engineering applications, some of the structure parameters such as the mass or the stiffness may be unknown and/or may change in time. In this paper, an online dynamic load identification algorithm based on an extended Kalman filter is proposed. The algorithm not only identifies the load by measuring the structural response but also identifies the unknown structure parameters and tracks their changes. We discuss the proposed algorithm for the cases when the unknown parameters are the stiffness or the mass coefficients. Furthermore, for a system with many degrees of freedom and to achieve online computations, we implement the model reduction theory. Thus, we reduce the number of degrees of freedom in the resulting symmetric system before applying the proposed extended Kalman filter algorithm. The algorithm is used to recover the dynamic loads in three numerical examples. It is also used to identify the dynamic load in a lab experiment for a structure with varying parameters. The simulations and the experimental results show that the proposed algorithm is effective and can simultaneously identify the parameters and any changes in them as well as the applied dynamic load.

]]>Symmetry doi: 10.3390/sym13081370

Authors: Eisa Zarei Samad Noeiaghdam

The aim of this paper is to apply the Taylor expansion method to solve the first and second kinds Volterra integral equations with Abel kernel. This study focuses on two main arithmetics: the FPA and the DSA. In order to apply the DSA, we use the CESTAC method and the CADNA library. Using this method, we can find the optimal step of the method, the optimal approximation, the optimal error, and some of numerical instabilities. They are the main novelties of the DSA in comparison with the FPA. The error analysis of the method is proved. Furthermore, the main theorem of the CESTAC method is presented. Using this theorem we can apply a new termination criterion instead of the traditional absolute error. Several examples are approximated based on the FPA and the DSA. The numerical results show the applications and advantages of the DSA than the FPA.

]]>Symmetry doi: 10.3390/sym13081369

Authors: Ming-Chin Chuang Chia-Cheng Yen

In the Internet of things (IoT) environment, many applications access services through remote methods. In this paper, we designed a new geometric authentication mechanism to enhance security. The solution is based on geometric characteristics to achieve rapid authentication at low computational cost. In addition, we use the user’s biometrics to improve the security level of the system. Our solution meets the following security features: anonymity, resistance to forgery attacks and replay attacks, fast error detection, resistance to offline password guessing attacks, resistance to server overload attacks, mutual authentication, session key agreement, and flexibility in users choosing and changing their passwords easily.

]]>Symmetry doi: 10.3390/sym13081368

Authors: David L. Andrews

Optical vortices are beams of laser light with screw symmetry in their wavefront. With a corresponding azimuthal dependence in optical phase, they convey orbital angular momentum, and their methods of production and applications have become one of the most rapidly accelerating areas in optical physics and technology. It has been established that the quantum nature of electromagnetic radiation extends to properties conveyed by each individual photon in such beams. It is therefore of interest to identify and characterize the symmetry aspects of the quantized fields of vortex radiation that relate to the beam and become manifest in its interactions with matter. Chirality is a prominent example of one such aspect; many other facets also invite attention. Fundamental CPT symmetry is satisfied throughout the field of optics, and it plays significantly into manifestations of chirality where spatial parity is broken; duality symmetry between electric and magnetic fields is also involved in the detailed representation. From more specific considerations of spatial inversion, amongst which it emerges that the topological charge has the character of a pseudoscalar, other elements of spatial symmetry, beyond simple parity inversion, prove to repay additional scrutiny. A photon-based perspective on these features enables regard to be given to the salient quantum operators, paying heed to quantum uncertainty limits of observables. The analysis supports a persistence in features of significance for the material interactions of vortex beams, which may indicate further scope for suitably tailored experimental design.

]]>Symmetry doi: 10.3390/sym13081367

Authors: Alexander Melnikov

The overview discusses development of the unique fusion plasma diagnostics—Heavy Ion Beam Probing (HIBP) in application to toroidal magnetic plasma devices. The basis of the HIBP measurements of the plasma electric potential and processing of experimental data are considered. Diagnostic systems for probing plasma in tokamaks TM-4, TJ-1, TUMAN-3M and T-10, stellarators WEGA, TJ-II and Uragan-2M are presented. Promising results of the HIBP projects for various existing modern machines, such as TCV, TCABR, MAST, COMPASS, GLOBUS-M2, T-15 MD and W7-X and the international fusion tokamak reactor ITER are given. Results from two machines with similar size and plasma parameters, but with different types of the magnetic con-figuration: axisymmetric tokamak T-10 and helically symmetric stellarator TJ-II are compared. The results of studies of stationary potential profiles and oscillations in the form of quasimonochromatic and broadband fluctuations, turbulent particle flux, fluctuations of density and poloidal magnetic field are presented. The properties of symmetric structures—zonal flows and geodesic acoustic modes of plasma oscillations as well as Alfvén Eigenmodes excited by fast particles from neutral beam injection heating are described. General trends in the behavior of electric potential and turbulence in magnetized fusion plasmas are revealed.

]]>Symmetry doi: 10.3390/sym13081365

Authors: Philipp Honegger Natalie Gajic Alexander Prado-Roller Michael Widhalm

To date, only a few instances of S4-symmetric organic molecules exist. In principle, spirobi-(dinaphthoazepin)ium cations can achieve this highly symmetric point group. Heating racemic 2,2′-bis(bromomethyl)-1,1′binaphthyl with aqueous ammonia afforded a mixture of rac- and meso-3,3′,5,5′-tetrahydro-4,4′-spirobi[dinaphtho[2,1-c:1′,2′-e]azepin]-4-ium bromide which was separated by fractional crystallisation. Both stereoisomers were characterised spectroscopically, and their crystal structures were determined and compared. The rac crystal structure differs significantly from the known enantiopure one. The meso molecules display a near-perfect S4 symmetry. Upon treatment with KOtBu, both isomers undergo Stevens rearrangement.

]]>Symmetry doi: 10.3390/sym13081363

Authors: Damilare Peter Oyinloye Je Sen Teh Norziana Jamil Moatsum Alawida

Blockchain networks are based on cryptographic notions that include asymmetric-key encryption, hash functions and consensus protocols. Despite their popularity, mainstream protocols, such as Proof of Work or Proof of Stake still have drawbacks. Efforts to enhance these protocols led to the birth of alternative consensus protocols, catering to specific areas, such as medicine or transportation. These protocols remain relatively unknown despite having unique merits worth investigating. Although past reviews have been published on popular blockchain consensus protocols, they do not include most of these lesser-known protocols. Highlighting these alternative consensus protocols contributes toward the advancement of the state of the art, as they have design features that may be useful to academics, blockchain practitioners and researchers. In this paper, we bridge this gap by providing an overview of alternative consensus protocols proposed within the past 3 years. We evaluate their overall performance based on metrics such as throughput, scalability, security, energy consumption, and finality. In our review, we examine the trade-offs that these consensus protocols have made in their attempts to optimize scalability and performance. To the best of our knowledge, this is the first paper that focuses on these alternative protocols, highlighting their unique features that can be used to develop future consensus protocols.

]]>Symmetry doi: 10.3390/sym13081366

Authors: Shanie A. L. Jayasinghe Candice Maenza David C. Good Robert L. Sainburg

Typical upper limb-mediated activities of daily living involve coordination of both arms, often requiring distributed contributions to mechanically coupled tasks, such as stabilizing a loaf of bread with one hand while slicing with the other. We sought to examine whether mild paresis in one arm results in deficits in performance on a bilateral mechanically coupled task. We designed a virtual reality-based task requiring one hand to stabilize against a spring load that varies with displacement of the other arm. We recruited 15 chronic stroke survivors with mild hemiparesis and 7 age-matched neurologically intact adults. We found that stroke survivors produced less linear reaching movements and larger initial direction errors compared to controls (p &lt; 0.05), and that contralesional hand performance was less linear than that of ipsilesional hand. We found a hand × group interaction (p &lt; 0.05) for peak acceleration of the stabilizing hand, such that the dominant right hand of controls stabilized less effectively than the nondominant left hand while stroke survivors showed no differences between the hands. Our results indicate that chronic stroke survivors with mild hemiparesis show significant deficits in reaching aspects of bilateral coordination, but no deficits in stabilizing against a movement-dependent spring load in this task.

]]>Symmetry doi: 10.3390/sym13081364

Authors: Antonios Mitsopoulos Michael Tsamparlis Genly Leon Andronikos Paliathanasis

The derivation of conservation laws and invariant functions is an essential procedure for the investigation of nonlinear dynamical systems. In this study, we consider a two-field cosmological model with scalar fields defined in the Jordan frame. In particular, we consider a Brans–Dicke scalar field theory and for the second scalar field we consider a quintessence scalar field minimally coupled to gravity. For this cosmological model, we apply for the first time a new technique for the derivation of conservation laws without the application of variational symmetries. The results are applied for the derivation of new exact solutions. The stability properties of the scaling solutions are investigated and criteria for the nature of the second field according to the stability of these solutions are determined.

]]>Symmetry doi: 10.3390/sym13081362

Authors: Ana Moita António Moreira José Pereira

Nowadays, the thermal management of electronic components, devices and systems is one of the most important challenges of this technological field. The ever-increasing miniaturization also entails the pressing need for the dissipation of higher power energy under the form of heat per unit of surface area by the cooling systems. The current work briefly describes the use on those cooling systems of the novel heat transfer fluids named nanofluids. Although not intensively applied in our daily use of electronic devices and appliances, the nanofluids have merited an in-depth research and investigative focus, with several recently published papers on the subject. The development of this cooling approach should give a sustained foothold to go on to further studies and developments on continuous miniaturization, together with more energy-efficient cooling systems and devices. Indeed, the superior thermophysical properties of the nanofluids, which are highlighted in this review, make those innovative fluids very promising for the aforementioned purpose. Moreover, the present work intends to contribute to the knowledge of the nanofluids and its most prominent results from the typical nanoparticles/base fluid mixtures used and combined in technical and functional solutions, based on fluid-surface interfacial flows.

]]>Symmetry doi: 10.3390/sym13081361

Authors: José Tito Mendonça

This work considers the creation of electron-positron pairs from intense electric fields in vacuum, for arbitrary temporal field variations. These processes can be useful to study quantum vacuum effects with ultra-intense lasers. We use the quantized Dirac field to explore the temporal Klein model. This model is based on a vector potential discontinuity in time, in contrast with the traditional model based on a scalar potential discontinuity in space. We also extend the model by introducing a finite time-scale for potential variations. This allows us to study the transition from a singular electric field spike, with infinitesimal duration, to the opposite case of a static field where the Schwinger formula would apply. The present results are intrinsically non-perturbative. Explicit expressions for pair-creation as a function of the potential time-scales are derived. This work explores the spacetime symmetry associated with pair creation in vacuum: the space symmetry breaking of the old Klein paradox model, in contrast with the time symmetry breaking of the temporal Klein model.

]]>Symmetry doi: 10.3390/sym13081360

Authors: Maria Giovanna Buonomenna

Nowadays, membranes are key components in various relevant fields [...]

]]>Symmetry doi: 10.3390/sym13081359

Authors: Marin Marin Dumitru Băleanu Sorin Vlase

The formalism of multibody systems offers a means of computer-assisted algorithmic analysis and a means of simulating and optimizing an arbitrary movement of a possible high number of elastic bodies in the connection [...]

]]>Symmetry doi: 10.3390/sym13081358

Authors: Yiannis Contoyiannis Michael P. Hanias Pericles Papadopoulos Stavros G. Stavrinides Myron Kampitakis Stelios M. Potirakis Georgios Balasis

This paper presents our study of the presence of the unstable critical point in spontaneous symmetry breaking (SSB) in the framework of Ginzburg–Landau (G-L) free energy. Through a 3D Ising spin lattice simulation, we found a zone of hysteresis where the unstable critical point continued to exist, despite the system having entered the broken symmetry phase. Within the hysteresis zone, the presence of the kink–antikink SSB solitons expands and, therefore, these can be observed. In scalar field theories, such as Higgs fields, the mass of this soliton inside the hysteresis zone could behave as a tachyon mass, namely as an imaginary quantity. Due to the fact that groups Ζ(2) and SU(2) belong to the same universality class, one expects that, in future experiments of ultra-relativistic nuclear collisions, in addition to the expected bosons condensations, structures of tachyon fields could appear.

]]>Symmetry doi: 10.3390/sym13081357

Authors: Linxiao Cong Jianchao Mu Qian Liu Hao Wang Linlin Wang Yonggui Li Congfeng Qiao

The space gravitational wave detection and drag free control requires the micro-thruster to have ultra-low thrust noise within 0.1 mHz–0.1 Hz, which brings a great challenge to calibration on the ground because it is impossible to shield any spurious couplings due to the asymmetry of torsion balance. Most thrusters dissipate heat during the test, making the rotation axis tilt and components undergo thermal drift, which is hysteretic and asymmetric for micro-Newton thrust measurement. With reference to LISA’s research and coming up with ideas inspired from proportional-integral-derivative (PID) control and multi-timescale (MTS), this paper proposes to expand the state space of temperature to be applied on the thrust prediction based on fine tree regression (FTR) and to subtract the thermal noise filtered by transfer function fitted with z-domain vector fitting (ZDVF). The results show that thrust variation of diurnal asymmetry in temperature is decoupled from 24 μN/Hz1/2 to 4.9 μN/Hz1/2 at 0.11 mHz. Additionally, 1 μN square wave modulation of electrostatic force is extracted from the ambiguous thermal drift background of positive temperature coefficient (PTC) heater. The PID-FTR validation is performed with experimental data in thermal noise decoupling, which can guide the design of thermal control and be extended to other physical quantities for noise decoupling.

]]>Symmetry doi: 10.3390/sym13081356

Authors: Haden L. Scott Kristen B. Kennison Thais A. Enoki Milka Doktorova Jacob J. Kinnun Frederick A. Heberle John Katsaras

It is well known that the lipid distribution in the bilayer leaflets of mammalian plasma membranes (PMs) is not symmetric. Despite this, model membrane studies have largely relied on chemically symmetric model membranes for the study of lipid–lipid and lipid–protein interactions. This is primarily due to the difficulty in preparing stable, asymmetric model membranes that are amenable to biophysical studies. However, in the last 20 years, efforts have been made in producing more biologically faithful model membranes. Here, we review several recently developed experimental and computational techniques for the robust generation of asymmetric model membranes and highlight a new and particularly promising technique to study membrane asymmetry.

]]>Symmetry doi: 10.3390/sym13081355

Authors: Evgenii S. Baranovskii Natalya V. Burmasheva Evgenii Yu. Prosviryakov

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

]]>Symmetry doi: 10.3390/sym13081354

Authors: Jean-Pierre Antoine

The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.

]]>Symmetry doi: 10.3390/sym13081353

Authors: Marco Chianese Damiano F. G. Fiorillo Gianpiero Mangano Gennaro Miele Stefano Morisi Ofelia Pisanti

The symmetry of the theory of relativity under diffeomorphisms strongly depends on the equivalence principle. Violation of Equivalence Principle (VEP) can be tested by looking for deviations from the standard framework of neutrino oscillations. In recent works, it has been shown that strong constraints on the VEP parameter space can be placed by means of the atmospheric neutrinos observed by the IceCube neutrino telescope. In this paper, we focus on the KM3NeT neutrino telescope and perform a forecast analysis to assess its capacity to probe VEP. Most importantly, we examine the crucial role played by systematic uncertainties affecting the neutrino observations. We find that KM3NeT will constrain VEP parameters times the local gravitational potential at the level of 10−27. Due to the systematic-dominated regime, independent analyses from different neutrino telescopes are fundamental for robustly testing the equivalence principle.

]]>Symmetry doi: 10.3390/sym13081352

Authors: Semih Yavuzkilic Abdulkadir Sengur Zahid Akhtar Kamran Siddique

Deepfake is one of the applications that is deemed harmful. Deepfakes are a sort of image or video manipulation in which a person’s image is changed or swapped with that of another person’s face using artificial neural networks. Deepfake manipulations may be done with a variety of techniques and applications. A quintessential countermeasure against deepfake or face manipulation is deepfake detection method. Most of the existing detection methods perform well under symmetric data distributions, but are still not robust to asymmetric datasets variations and novel deepfake/manipulation types. In this paper, for the identification of fake faces in videos, a new multistream deep learning algorithm is developed, where three streams are merged at the feature level using the fusion layer. After the fusion layer, the fully connected, Softmax, and classification layers are used to classify the data. The pre-trained VGG16 model is adopted for transferred CNN1stream. In transfer learning, the weights of the pre-trained CNN model are further used for training the new classification problem. In the second stream (transferred CNN2), the pre-trained VGG19 model is used. Whereas, in the third stream, the pre-trained ResNet18 model is considered. In this paper, a new large-scale dataset (i.e., World Politicians Deepfake Dataset (WPDD)) is introduced to improve deepfake detection systems. The dataset was created by downloading videos of 20 different politicians from YouTube. Over 320,000 frames were retrieved after dividing the downloaded movie into little sections and extracting the frames. Finally, various manipulations were performed to these frames, resulting in seven separate manipulation classes for men and women. In the experiments, three fake face detection scenarios are investigated. First, fake and real face discrimination is studied. Second, seven face manipulations are performed, including age, beard, face swap, glasses, hair color, hairstyle, smiling, and genuine face discrimination. Third, performance of deepfake detection system under novel type of face manipulation is analyzed. The proposed strategy outperforms the prior existing methods. The calculated performance metrics are over 99%.

]]>Symmetry doi: 10.3390/sym13081351

Authors: Maopeng Li Guoxiong Zhou Weiwei Cai Jiayong Li Mingxuan Li Mingfang He Yahui Hu Liujun Li

Aiming at solving the problems of high background complexity of some butterfly images and the difficulty in identifying them caused by their small inter-class variance, we propose a new fine-grained butterfly classification architecture, called Network based on Multi-rate Dilated Attention Mechanism and Multi-granularity Feature Sharer (MRDA-MGFSNet). First, in this network, in order to effectively identify similar patterns between butterflies and suppress the information that is similar to the butterfly’s features in the background but is invalid, a Multi-rate Dilated Attention Mechanism (MRDA) with a symmetrical structure which assigns different weights to channel and spatial features is designed. Second, fusing the multi-scale receptive field module with the depthwise separable convolution module, a Multi-granularity Feature Sharer (MGFS), which can better solve the recognition problem of a small inter-class variance and reduce the increase in parameters caused by multi-scale receptive fields, is proposed. In order to verify the feasibility and effectiveness of the model in a complex environment, compared with the existing methods, our proposed method obtained a mAP of 96.64%, and an F1 value of 95.44%, which showed that the method proposed in this paper has a good effect on the fine-grained classification of butterflies.

]]>Symmetry doi: 10.3390/sym13081350

Authors: Mujahid Abbas Rizwan Anjum Vasile Berinde

The purpose of this paper is to introduce the class of enriched multivalued contraction mappings. Both single-valued and multivalued enriched contractions are defined by means of symmetric inequalities. Our main result extends and generalizes the recent result of Berinde and Păcurar (Approximating fixed points of enriched contractions in Banach spaces, Journal of Fixed Point Theory and Applications, 22 (2), 1–10, 2020). We also study a data dependence problem of the fixed point set and Ulam–Hyers stability of the fixed point problem for enriched multivalued contraction mappings. Applications of the results obtained to the problem of the existence of a solution of differential inclusions and dynamic programming are presented.

]]>Symmetry doi: 10.3390/sym13081347

Authors: Sultan Zeybek Duc Truong Pham Ebubekir Koç Aydın Seçer

Recurrent neural networks (RNNs) are powerful tools for learning information from temporal sequences. Designing an optimum deep RNN is difficult due to configuration and training issues, such as vanishing and exploding gradients. In this paper, a novel metaheuristic optimisation approach is proposed for training deep RNNs for the sentiment classification task. The approach employs an enhanced Ternary Bees Algorithm (BA-3+), which operates for large dataset classification problems by considering only three individual solutions in each iteration. BA-3+ combines the collaborative search of three bees to find the optimal set of trainable parameters of the proposed deep recurrent learning architecture. Local learning with exploitative search utilises the greedy selection strategy. Stochastic gradient descent (SGD) learning with singular value decomposition (SVD) aims to handle vanishing and exploding gradients of the decision parameters with the stabilisation strategy of SVD. Global learning with explorative search achieves faster convergence without getting trapped at local optima to find the optimal set of trainable parameters of the proposed deep recurrent learning architecture. BA-3+ has been tested on the sentiment classification task to classify symmetric and asymmetric distribution of the datasets from different domains, including Twitter, product reviews, and movie reviews. Comparative results have been obtained for advanced deep language models and Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms. BA-3+ converged to the global minimum faster than the DE and PSO algorithms, and it outperformed the SGD, DE, and PSO algorithms for the Turkish and English datasets. The accuracy value and F1 measure have improved at least with a 30–40% improvement than the standard SGD algorithm for all classification datasets. Accuracy rates in the RNN model trained with BA-3+ ranged from 80% to 90%, while the RNN trained with SGD was able to achieve between 50% and 60% for most datasets. The performance of the RNN model with BA-3+ has as good as for Tree-LSTMs and Recursive Neural Tensor Networks (RNTNs) language models, which achieved accuracy results of up to 90% for some datasets. The improved accuracy and convergence results show that BA-3+ is an efficient, stable algorithm for the complex classification task, and it can handle the vanishing and exploding gradients problem of deep RNNs.

]]>Symmetry doi: 10.3390/sym13081349

Authors: Chiara Lucafò Daniele Marzoli Caterina Padulo Stefano Troiano Lucia Pelosi Zazzerini Gianluca Malatesta Ilaria Amodeo Luca Tommasi

Both static and dynamic ambiguous stimuli representing human bodies that perform unimanual or unipedal movements are usually interpreted as right-limbed rather than left-limbed, suggesting that human observers attend to the right side of others more than the left one. Moreover, such a bias is stronger when static human silhouettes are presented in the RVF (right visual field) than in the LVF (left visual field), which might represent a particular instance of embodiment. On the other hand, hemispheric-specific rotational biases, combined with the well-known bias to perceive forward-facing figures, could represent a confounding factor when accounting for such findings. Therefore, we investigated whether the lateralized presentation of an ambiguous rotating human body would affect its perceived handedness/footedness (implying a role of motor representations), its perceived spinning direction (implying a role of visual representations), or both. To this aim, we required participants to indicate the perceived spinning direction (which also unveils the perceived handedness/footedness) of ambiguous stimuli depicting humans with an arm or a leg outstretched. Results indicated that the lateralized presentation of the stimuli affected both their perceived limb laterality (a larger number of figures being interpreted as right-limbed in the RVF than in the LVF) and their perceived spinning direction (a larger number of figures being interpreted as spinning clockwise in the LVF than in the RVF). However, the hemifield of presentation showed a larger effect size on the perceived spinning direction than on the perceived limb laterality. Therefore, as we already proposed, the implicit representation of others’ handedness seems to be affected more by visual than by motor processes during the perception of ambiguous human silhouettes.

]]>Symmetry doi: 10.3390/sym13081346

Authors: Yeva Fadhilah Ashari A.N.M. Salman Rinovia Simanjuntak

A graph G admits an H-covering if every edge of G belongs to a subgraph isomorphic to a given graph H. G is said to be H-magic if there exists a bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that wf(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(e) is a constant, for every subgraph H′ isomorphic to H. In particular, G is said to be H-supermagic if f(V(G))={1,2,…,|V(G)|}. When H is isomorphic to a complete graph K2, an H-(super)magic labeling is an edge-(super)magic labeling. Suppose that G admits an F-covering and H-covering for two given graphs F and H. We define G to be (F,H)-sim-(super)magic if there exists a bijection f′ that is simultaneously F-(super)magic and H-(super)magic. In this paper, we consider (K2,H)-sim-(super)magic where H is isomorphic to three classes of graphs with varied symmetry: a cycle which is symmetric (both vertex-transitive and edge-transitive), a star which is edge-transitive but not vertex-transitive, and a path which is neither vertex-transitive nor edge-transitive. We discover forbidden subgraphs for the existence of (K2,H)-sim-(super)magic graphs and classify classes of (K2,H)-sim-(super)magic graphs. We also derive sufficient conditions for edge-(super)magic graphs to be (K2,H)-sim-(super)magic and utilize such conditions to characterize some (K2,H)-sim-(super)magic graphs.

]]>Symmetry doi: 10.3390/sym13081348

Authors: Cristian Duarte-Leiva Sebastián Lorca Exequiel Mallea-Zepeda

Micropolar fluids are fluids with microstructure and belong to a class of fluids with asymmetric stress tensor that called Polar fluids, and include, as a special case, the well-established Navier–Stokes model. In this work we study a 3D micropolar fluids model with Navier boundary conditions without friction for the velocity field and homogeneous Dirichlet boundary conditions for the angular velocity. Using the Galerkin method, we prove the existence of weak solutions and establish a Prodi–Serrin regularity type result which allow us to obtain global-in-time strong solutions at finite time.

]]>Symmetry doi: 10.3390/sym13081345

Authors: Qun-Ying Xie Qi-Ming Fu Tao-Tao Sui Li Zhao Yi Zhong

In this paper, we investigate thick branes generated by a scalar field in mimetic gravity theory, which is inspired by considering the conformal symmetry under the conformal transformation of an auxiliary metric. By introducing two auxiliary super-potentials, we transform the second-order field equations of the system into a set of first-order equations. With this first-order formalism, several types of analytical thick brane solutions are obtained. Then, tensor and scalar perturbations are analyzed. We find that both kinds of perturbations are stable. The effective potentials for the tensor and scalar perturbations are dual to each other. The tensor zero mode can be localized on the brane while the scalar zero mode cannot. Thus, the four-dimensional Newtonian potential can be recovered on the brane.

]]>Symmetry doi: 10.3390/sym13081344

Authors: Arjun Magotra Juntae Kim

The plastic modifications in synaptic connectivity is primarily from changes triggered by neuromodulated dopamine signals. These activities are controlled by neuromodulation, which is itself under the control of the brain. The subjective brain’s self-modifying abilities play an essential role in learning and adaptation. The artificial neural networks with neuromodulated plasticity are used to implement transfer learning in the image classification domain. In particular, this has application in image detection, image segmentation, and transfer of learning parameters with significant results. This paper proposes a novel approach to enhance transfer learning accuracy in a heterogeneous source and target, using the neuromodulation of the Hebbian learning principle, called NDHTL (Neuromodulated Dopamine Hebbian Transfer Learning). Neuromodulation of plasticity offers a powerful new technique with applications in training neural networks implementing asymmetric backpropagation using Hebbian principles in transfer learning motivated CNNs (Convolutional neural networks). Biologically motivated concomitant learning, where connected brain cells activate positively, enhances the synaptic connection strength between the network neurons. Using the NDHTL algorithm, the percentage of change of the plasticity between the neurons of the CNN layer is directly managed by the dopamine signal’s value. The discriminative nature of transfer learning fits well with the technique. The learned model’s connection weights must adapt to unseen target datasets with the least cost and effort in transfer learning. Using distinctive learning principles such as dopamine Hebbian learning in transfer learning for asymmetric gradient weights update is a novel approach. The paper emphasizes the NDHTL algorithmic technique as synaptic plasticity controlled by dopamine signals in transfer learning to classify images using source-target datasets. The standard transfer learning using gradient backpropagation is a symmetric framework. Experimental results using CIFAR-10 and CIFAR-100 datasets show that the proposed NDHTL algorithm can enhance transfer learning efficiency compared to existing methods.

]]>Symmetry doi: 10.3390/sym13081343

Authors: Abbas Mahdavi Vahid Amirzadeh Ahad Jamalizadeh Tsung-I Lin

Multivariate skew-symmetric-normal (MSSN) distributions have been recognized as an appealing tool for modeling data with non-normal features such as asymmetry and heavy tails, rendering them suitable for applications in diverse areas. We introduce a richer class of MSSN distributions based on a scale-shape mixture of (multivariate) flexible skew-symmetric normal distributions, called the SSMFSSN distributions. This very general class of SSMFSSN distributions can capture various shapes of multimodality, skewness, and leptokurtic behavior in the data. We investigate some of its probabilistic characterizations and distributional properties which are useful for further methodological developments. An efficient EM-type algorithm designed under the selection mechanism is advocated to compute the maximum likelihood (ML) estimates of parameters. Simulation studies as well as applications to a real dataset are employed to illustrate the usefulness of the presented methods. Numerical results show the superiority of our proposed model in comparison to several existing competitors.

]]>Symmetry doi: 10.3390/sym13081342

Authors: Dong Yang Jiajian Lin

Explosively formed projectiles (EFPs) are widely used in civil applications and the military field for their excellent impact performance. How to give full play to the energy accumulation effect of explosives and improve the penetration performance has become the main problem of EFP design. The aim of the present study was to investigate the effect of liner structure on EFP formation and its penetration behavior. In order to achieve this, a finite element (FE) model was first established on the basis of the Lagrange and ALE method. Then, formation and penetration performance tests of EFP were performed to verify the validity and feasibility of the proposed FE model, where the configuration, velocity of EFP, and penetration diameter left on the target plate were compared. Finally, by using the proposed FE model, the entire process of the formation and penetration behavior of EFP with axial symmetrical variable thickness liners were analyzed, where spherical-segment liners with uniform and non-uniform thickness were developed. The results were drawn as follows: the numerical simulation error of EFP velocity was less than 5%, and the simulated penetration diameter was compared to the 8.6% error obtained from the experimental method. It demonstrated that the proposed FE model had higher prediction precision. After the explosive was detonated, a forward-folding EFP was formed by the liner with a thin edge thickness, while the EFP formed by the liner with uniform thickness had a backward-folded configuration. It was also found that the liner with a thin edge thickness gave the largest steady velocity of EFP, and it was the lowest by using the liner with uniform thickness. There were two types of loads generated after the formation of an EFP, those were shock wave loading and an EFP, both causing damage in the target plate during the process of an EFP’s penetration into it. The shock wave induced by liners with non-uniform thickness caused higher damage in the target plate, the maximum value of stress was reached at about 4.0 GPa. The forward-folding EFP formed by the liner with the thinnest edge thickness had the largest penetration ability. The backward-folded EFP, owing to the hollow structure, had the worst penetration ability, which failed to penetrate the target plate.

]]>Symmetry doi: 10.3390/sym13081341

Authors: Surang Sitho Sina Etemad Brahim Tellab Shahram Rezapour Sotiris K. Ntouyas Jessada Tariboon

In this paper, we establish several necessary conditions to confirm the uniqueness-existence of solutions to an extended multi-order finite-term fractional differential equation with double-order integral boundary conditions with respect to asymmetric operators by relying on the Banach’s fixed-point criterion. We validate our study by implementing two numerical schemes to handle some Riemann–Liouville fractional boundary value problems and obtain approximate series solutions that converge to the exact ones. In particular, we present several examples that illustrate the closeness of the approximate solutions to the exact solutions.

]]>Symmetry doi: 10.3390/sym13081340

Authors: Wei Chien Chien-Ching Chiu Po-Hsiang Chen Yu-Ting Cheng Eng Hock Lim Yue-Li Liang Jia-Rui Wang

Multiple objective function with beamforming techniques by algorithms have been studied for the Simultaneous Wireless Information and Power Transfer (SWIPT) technology at millimeter wave. Using the feed length to adjust the phase for different objects of SWIPT with Bit Error Rate (BER) and Harvesting Power (HP) are investigated in the broadband communication. Symmetrical antenna array is useful for omni bearing beamforming adjustment with multiple receivers. Self-Adaptive Dynamic Differential Evolution (SADDE) and Asynchronous Particle Swarm Optimization (APSO) are used to optimize the feed length of the antenna array. Two different object functions are proposed in the paper. The first one is the weighting factor multiplying the constraint BER and HP plus HP. The second one is the constraint BER multiplying HP. Simulations show that the first object function is capable of optimizing the total harvesting power under the BER constraint and APSO can quickly converges quicker than SADDE. However, the weighting for the final object function requires a pretest in advance, whereas the second object function does not need to set the weighting case by case and the searching is more efficient than the first one. From the numerical results, the proposed criterion can achieve the SWIPT requirement. Thus, we can use the novel proposed criterion (the second criterion) to optimize the SWIPT problem without testing the weighting case by case.

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