Symmetric Elements Research Articles

Some commutativity criteria for prime rings with involution involving symmetric and skew symmetric elements

UDC 512.5 We study the Posner second theorem [Proc. Amer. Math. Soc., <strong>8</strong>, 1093–1100 (1957)] and strong com\-mu\-ta\-tivity preserving problem for symmetric and skew symmetric elements involving generalized derivations on prime rings with involution. The obtained results cover numerous known theorems. We also provide examples showing that the obtained results hold neither in the case of involution of the first kind, nor in the case where the ring is not prime.

Design and Validation of Symmetrical Elastic Elements in Series Elastic Actuator

Abstract In recent years, the demand for robot joint compliance increased with more complex human–robot interaction scenarios. Series elastic actuators (SEAs) are extensively utilized in multiple fields with abilities to provide accurate force control and energy storage. As the main flexible component in SEA, the accurate modeling of the stiffness of the elastic element is essential. However, the existing stiffness models based on classical Euler–Bernoulli beam theory contain large errors with the actual situation, which increases the difficulty of design. In this paper, a typical elastic element is analyzed by the finite element method to investigate its stiffness properties with different geometric parameters. A more accurate stiffness model is proposed for designing elastic elements. The stiffness model is validated by designed experiments, with a fitting accuracy of 98.27%, which significantly exceeds the stiffness model based on classical beam theory. The proposed stiffness model can be applied to design elastic elements that meet specific stiffness requirements.

A new adaptive compressible element for tunnel lining support in squeezing rock masses

The so-called “Non-Deformable Support System” and “Yielding Support System” have been used in tunnel ground support designs in squeezing rock conditions. According to the convergence confinement method, it serves as a pivotal role to use yielding elements in the primary support for expressway tunnels in squeezing rocks. The elements of yielding support are able to convert the radial displacement into circumferential tangential closure without affecting the support capacity of the lining system, which prevents from damage in the primary lining composed of mesh, steel arch and shotcrete. This paper introduces a new adaptive compressible element that significantly improve the deformability of support lining in squeezing rock. The new compressible element features by its straightforward mechanistic principle, compact design form and convenient construction and installation. The load–displacement curves of the compressible element were obtained by laboratory compression tests. Field tests were carried out in Qinghe tunnel of Zhaolu Expressway in Yunnan Province, China. The settlements of the tunnel in three sections, the contact pressure between the surrounding rock and the ground support and the stress in steel arches were monitored during the excavation of the tunnel. Based on the monitoring data, it is concluded that when two symmetrical and longitudinal compressible elements are installed in the lining, the support pressure required at the equilibrium is greatly reduced, and the support pressure at the rock support contact is more evenly distributed. The construction measures of the compressible elements were elaborated through the field application. Finally, the advantages and suggestions for the improvement of the compressible element were summarized.

Enumeration of symmetric arc diagrams

We give recurrence relations for the enumeration of symmetric elements within four classes of arc diagrams corresponding to certain involutions and set partitions whose blocks contain no consecutive integers. These arc diagrams are motivated by the study of RNA secondary structures. For example, classic RNA secondary structures correspond to 3412-avoiding involutions with no adjacent transpositions, and structures with base triples may be represented as partitions with crossings. Our results rely on combinatorial arguments. In particular, we use Motzkin paths to describe noncrossing arc diagrams that have no arc connecting two adjacent nodes, and we give an explicit bijection to ternary words whose length coincides with the sum of their digits. We also discuss the asymptotic behavior of some of the sequences considered here in order to quantify the extremely low probability of finding symmetric structures with a large number of nodes.

H κατοικία στην επικράτεια της Κορίνθου από την Αρχαϊκή έως και την Ελληνιστική περίοδο. Μια πρώτη κριτική προσέγγιση

H κατοικία στην επικράτεια της Κορίνθου από την Αρχαϊκή έως και την Ελληνιστική περίοδο. Μια πρώτη κριτική προσέγγιση

Lagrangian Grassmannians, CKP Hierarchy and Hyperdeterminantal Relations

This work concerns the relation between the geometry of Lagrangian Grassmannians and the CKP integrable hierarchy. The Lagrange map from the Lagrangian Grassmannian of maximal isotropic (Lagrangian) subspaces of a finite dimensional symplectic vector space $V\oplus V^*$ into the projectivization of the exterior space $\Lambda V$ is defined by restricting the Pl\"ucker map on the full Grassmannian to the Lagrangian sub-Grassmannian and composing it with projection to the subspace of symmetric elements under dualization $V \leftrightarrow V^*$. In terms of the affine coordinate matrix on the big cell, this reduces to the principal minors map, whose image is cut out by the $2 \times 2 \times 2$ quartic {\em hyperdeterminantal} relations. To apply this to the CKP hierarchy, the Lagrangian Grassmannian framework is extended to infinite dimensions, with $V\oplus V^*$ replaced by a polarized Hilbert space $ {\mathcal H} ={\mathcal H}_+\oplus {\mathcal H}_-$, with symplectic form $\omega$. The image of the Plucker map in the fermionic Fock space ${\mathcal F}= \Lambda^{\infty/2}{\mathcal H}$ is identified and the infinite dimensional Lagrangian map is defined. The linear constraints defining reduction to the CKP hierarchy are expressed as a fermionic null condition and the infinite analogue of the hyperdeterminantal relations is deduced. A multiparametric family of such relations is shown to be satisfied by the evaluation of the $\tau$-function at translates of a point in the space of odd flow variables along the cubic lattices generated by power sums in the parameters.

Centrally Extended Jordan (∗)-Derivations Centralizing Symmetric or Skew Elements

Let A be a non-commutative prime ring with involution ∗, of characteristic ≠2(and3), with Z as the center of A and Π a mapping Π:A→A such that [Π(x),x]∈Z for all (skew) symmetric elements x∈A. If Π is a non-zero CE-Jordan derivation of A, then A satisfies s4, the standard polynomial of degree 4. If Π is a non-zero CE-Jordan ∗-derivation of A, then A satisfies s4 or Π(y)=λ(y−y*) for all y∈A, and some λ∈C, the extended centroid of A. Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results.

Contribution of higher-order structure to perception of mirror symmetry: Role of shapes and corners.

Visual mirror symmetry is a global feature that is dependent on specific low-level relationships between component elements. Initially conceptualized as virtual lines between paired elements, it has been suggested that higher-order structure between pairs of symmetric elements forming virtual four cornered shapes may also be important for strengthening the percept of mirror symmetry. We utilize corner elements, formed by joining two Gabor elements along a central midline creating vertices with variable internal angles, in a temporal integration paradigm. This allows us to specifically manipulate the presence and type of higher-order versus lower-order structure in patterns with symmetrically placed elements. We show a significant contribution of higher-order structure to the salience of visual symmetries compared with patterns with only lower-order structures. We also find that although we are more sensitive to patterns with higher-order structure, there is no difference in the temporal processing of higher-order versus lower-order patterns. These findings have important implications for existing spatial filter models of symmetry perception that rely on lower-order structures alone and reinforces the need for elaborated models that can more readily capture the complexities of real-world symmetries.

Unveiling the mechanism and origin of stereocontrol in dinuclear-zinc-catalyzed reductive desymmetrization of malonic esters

Mechanism of catalytic desymmetrization of malonic esters have been explored, which provides molecular-level insights on how the ligand-casted chiral scaffolding can be intentionally applied to differentiate symmetrical elements in desymmetrization.

Dual-Band MIMO Antenna with Enhanced Isolation for 5G NR Application.

A two-port multiple-input and multiple-output (MIMO) antenna with dual-band characteristics operating at the fifth-generation (5G) new radio (NR) sub-6 GHz n7/n38/n41/n79 bands is proposed. The proposed MIMO antenna is composed of two symmetric antenna elements and a defected ground plane. The antenna element consists of an incomplete circular patch with two L-shaped branches. By applying the defected ground structure and the slotted stub, the current distribution on the ground plane is changed to reduce the mutual coupling between the antenna elements. The measured -10 dB reflection coefficients cover 2.34-2.71 GHz and 3.72-5.10 GHz, while the measured isolation is larger than 20 dB at the whole operating frequency band. The paper has investigated different performance parameters in terms of the envelope correction coefficient (ECC), diversity gain (DG), radiation patterns, antenna gain, and efficiency. The proposed MIMO antenna is suitable for 5G applications.

Weakly symmetric functions on spaces of Lebesgue integrable functions

In this work, we present the notion of a weakly symmetric function. We show that the subset of all weakly symmetric elements of an arbitrary vector space of functions is a vector space itself. Moreover, the subset of all weakly symmetric elements of some algebra of functions is an algebra. Also we consider weakly symmetric functions on the complex Banach space $L_p[0,1]$ of all Lebesgue measurable complex-valued functions on $[0,1]$ for which the $p$th power of the absolute value is Lebesgue integrable. We show that every continuous linear functional on $L_p[0,1],$ where $p\in (1,+\infty),$ can be approximated by weakly symmetric continuous linear functionals.

An In-depth Analysis of Symmetries and their Implications in Physics

The concept of symmetry is essential in understanding classical and quantum physics. Symmetry describes a system that remains unchanged in structure and behavior after undergoing a transformation. In this paper, I will describe the implications of symmetries (global and gauge) in Newton’s laws of mechanics, Maxwell’s electromagnetism equations, and quantum particle physics – in particular the Higgs mechanism – with the help of Noether’s Theorem. Purely based on symmetrical elements, this paper will then determine the isospin composition of pions given certain restraints. This solution will connect findings in preceding theories to current and future studies relevant to the subject of symmetry in physics.

Optimization of four-Chamber and eccentric ellipse mufflers hybridized with multi-Segment perforated/non-perforated tubes

Muffler with circular section and inner symmetric acoustical elements has been habitually used in industry. In order to improve the acoustical performance, an advanced eccentric muffler having ellipse cross-section, multiple chambers, and multi-segment perforated/non-perforated tubes is presented. Concerning about high order wave effect to the muffler, a finite element method (FEM) run on COMSOL package is adopted. In addition, the influence of sound transmission loss with respect to various geometric parameters has been explored. In order to simplify the optimization process, a simplified objective function established by an Artificial Neural Network (ANN) and FEM has been implemented and connected to a Genetic Algorithm (GA). Two case studies of muffler optimization for two kinds of design parameter sets (one of Lin and Lout and the other of Lin1 and Lout 1) are introduced. Two targeted frequencies (1000 Hz and 2500 Hz) are selected and applied in the two cases’ muffler optimization. Simulated results reveal that the acoustical influence relating to various geometric parameters is huge. The sound attenuation will increase in the low frequency region if the curvature ratio δ (ratio of long axis to short axis in ellipse) or the X (horizontal length of the muffler’s body) increases. Consequently, the study demonstrates a methodology in the optimization of a four-chamber ellipse and eccentric muffler internally inserted with multi-segment perforated/non-perforated tubes.

Energy norm analysis of exactly symmetric mixed finite elements for linear elasticity

We consider mixed finite element methods for linear elasticity for which the symmetry of the stress tensor is exactly satisfied. We derive a new quasi-optimal a priori error estimate uniformly valid with respect to the compressibility. For the a posteriori error analysis we consider the Prager-Synge hypercircle principle and introduce a new estimate uniformly valid in the incompressible limit. All estimates are validated by numerical examples.

Circular tooth based low profile 1x2 superlative MIMO antenna with high gain and isolation for fixed satellite service, and land mobile navigation applications

The 1 × 2 superlative MIMO antenna structure is proposed for multiband and broadband communication applications. The presented antenna is composed of two symmetrical circular-shaped patch elements. The circular-shaped Tooth is inserted in the outer periphery of the patch area. The Radius of the patch and Tooth varies to identify the behaviour of reflectance response, directivity, and total gain. The tooth-loaded circular patch attained a better gain of 8.44 dB, broader directivity of 163°, reflectance coefficients of −29.16 dB, and good isolation among transmittance and reflectance response. The comparison between simulated and measured results is included in the analysis. Performance of the presented design structure with previously published articles to identify performance improvement. Standard performance measurement parameters like TARC, ECC, and DG are also carried out. The low-profile material FR4 is used in the proposed design as a substrate. The presented structure is suitable for fixed satellite service and Land mobile navigation applications.

A novel approach to calculate the KIII factor for a crack on a sharp corner using Green's function

This paper presents a method for determining Green's function for calculating the KIII factor for a crack initiated at the vertex of a sharp corner and coming out at different angles when the crack length is small compared to other element dimensions. A new approach to find a Green's function by means of FEM for symmetric problems and adapting it for non-symmetric problems has been applied, using the hypothesis of free released energy in combination with a special technique of building non-symmetric shapes from symmetric elements. This approach obtained approximate solutions, so special emphasis was placed on estimating the error of the obtained solutions. It was found that the high precision of the obtained Green's function is sufficient for practical use in calculating the durability and strength of structural elements under an anti-plane strain state.

Compact 8 × 8 MIMO Antenna Design for 5G Terminals

In this paper, a compact 8 × 8 MIMO antenna design for 5G terminals is proposed. The 8 × 8 MIMO antenna consists of two quad-element antenna pairs, each of which includes two symmetrical T-shaped monopole mode elements and two symmetrical edge-coupled fed dipole mode elements. The size of the quad-element antenna is 38 × 7 × 0.8 mm3. T-shaped monopoles are decoupled by parasitic elements, and dipoles are decoupled by grounding strips. Meanwhile, both T-shaped monopoles and dipoles are also decoupled by the orthogonal mode. The results show that the operating frequency band of each antenna element meets the requirement of 3.4–3.6 GHz, the reflection coefficient is less than −6 dB, and the isolation between any antenna element is more than 10 dB. The antenna radiation efficiency is over 50% in the entire operating frequency band for the 8 × 8 MIMO system.

Broadband Dual-Polarized Metal Lens With Theoretically Arbitrarily Variable Focal Diameter Ratio Using 3-D Printing Technology

A wideband dual-polarized metal lens is proposed in this article for millimeter-wave (mm-wave) applications. Compared with the traditional lens, the focal length of the proposed novel metal lens can be of any value in theory; in other words, the focal diameter ratio ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f/D$ </tex-math></inline-formula> ) can be arbitrarily adjusted. The metal 3-D printed technology is used to reduce the fabrication cost. The unit cell of the lens is a square open-ended waveguide. Since the phase velocity of the electromagnetic wave in the metal waveguide is faster than that in the air, the phase compensation can be controlled by adjusting the height of each unit of the lens. It helps to realize the conversion from the quasi-spherical wave of the feed source to the plane wave radiation of the lens. Due to the use of symmetrical elements, the lens can support different dual-polarized radiation (dual-linear polarization and dual-circular polarization). Considering the processing cost and design difficulty, a metal lens with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9\times $ </tex-math></inline-formula> 9 unit cells was processed to verify our design. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f/D$ </tex-math></inline-formula> is selected as 1. Simulated results for the lens antenna show a peak aperture efficiency (AE) of 43.8% and 38% for linearly polarized (LP) and circularly polarized (CP) radiation, respectively. The antenna shows a stable pattern and wide 3 dB gain bandwidth (26–40 GHz), which covers the entire Ka-band. The measured results of the lens demonstrate a peak gain of 24 dBi and 23 dBic for LP and CP radiation with an aperture of about <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$7.5\,\,\lambda \times 7.5\,\,\lambda $ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> is the free space wavelength at 30 GHz). The proposed lens features a wide gain bandwidth with dual polarization, low cost, and good radiation performance, which is well suited for mm-wave wireless applications.

A Difference Subgridding Method for Solving Multiscale Electro-Thermal Problems

Because of less memory costs and time consumption, a finite difference subgrid technique can effectively deal with multiscale problems in electromagnetic fields. When used in Maxwell equation, symmetric elements of the matrix are required; otherwise, the algorithm will be unstable. Usually, the electro-thermal problem also contains multiscale structures. However, the coefficient matrix of the heat transfer equation is asymmetric because the parameters of the equation vary with temperature and the Robbin boundary condition is used as well. In this paper, a three-dimensional (3D) finite difference subgridding method is proposed to simulate the electro-thermal coupling process of the multiscale circuits. The stability condition of the algorithm is deduced with a matrix method. And the efficiency and the effectiveness of the proposed subgridding approach are verified through square- and n-shaped resistances. Compared with the results of the COMSOL software and the traditional finite difference method (FDM), the proposed subgridding method has less unknowns and faster speed.

On Wire-Grid Representation for Modeling Symmetrical Antenna Elements

This paper focuses on the combination of the method of moments and the wire-grid approximation as an effective computational technique for modeling symmetrical antennas with low computational cost and accurate results. The criteria and conditions for the use of wire-grid surface approximation from various sources are presented together with new recommendations for modeling symmetrical antenna structures using the wire-grid approximation. These recommendations are used to calculate the characteristics of biconical and horn antennas at different frequencies. The results obtained using different grid and mesh settings are compared to those obtained analytically. Moreover, the results are compared to those obtained using the finite difference time domain numerical method, as well as the measured ones. All results are shown to be in a good agreement. The recommendations used for building a symmetrical wire-grid of those symmetrical antenna elements provided the most advantageous parameters of the grid and mesh settings and the wire radius, which are able to give quite accurate results with low computational cost. Additionally, the known equal area rule was modified for a rectangular grid form. The obtained radiation patterns of a conductive plate using both the original rule and the modified one are compared with the electrodynamic analysis results. It is shown that the use of the modified rule is more accurate when using a rectangle grid form.