Symmetry Axis Research Articles

Experimental study on the rock breaking effect of different tooth shapes of inserted teeth in a hobbing-cone hybrid PDC bit under complex stress environment

The hobbing-cone hybrid PDC bit has emerged with the increasing exploitation of deep oil and gas resources. As an important component of new bits, studying the mechanism of toothed fracturing of deep rocks is crucial for improving rock-breaking efficiency. In this paper, the spherical teeth and conical teeth were used to intrude sandstone under different confining pressures, and the brittleness index (BIm), specific energy (SE), and average fragment size (dm) were used to analyze the rock-breaking effect. The 3D contour scanner quantitatively analyzed the crushing pit area. Finally, the in-situ CT scanning and discrete element numerical simulation were used to summarize the crack propagation law. The results show that with the increase of confining pressure, the BIm of inserted teeth increases continuously, and the growth rate of conical teeth increases, while that of spherical teeth decreases. The SE and dm exhibit a symmetrical distribution with a confining pressure of 10MPa as the axis of symmetry. When the difference in principal stress is 5MPa, the SE is the lowest point, the rock-breaking efficiency is the highest, and the corresponding dm is at the highest point, but the curve trend is the opposite. When the stress difference is 5MPa, it is conducive to propagating surface cracks, thus forming a larger crushing area and the highest rock-breaking efficiency. Based on CT results, the dense particle core is formed near the inserted teeth, and that of the spherical teeth is located below. In contrast, that of the conical teeth is located on the side, which is also the reason for the difference in crack propagation between different tooth shapes.

2D mechanical representation of superconducting magnets: A comparative study of plane stress and plane strain

When approaching the mechanical design of a superconducting magnet, whenever possible the starting model is a 2D approximation. If rotational symmetry (solenoid-like winding) is present, the 2D representation is unique and contains no approximations. If, on the other hand, a non-axisymmetric system is opted for, the 2D representation is not unique and there are main options available, as plane stress, plane stress with thickness, plane strain and generalized plane strain. Considering z as the direction normal to the 2D plane, the plane stress option defines a stress state in which no normal or shear stresses perpendicular to the xy plane can occur (σz=σxz=σyz=0). In this option, deformation can occur in the thickness direction of the element, which will become thinner when stretched and thicker when compressed; it is generally used for objects with limited depth (thin objects).In contrast, plane strain refers to the fact that deformation can only occur in plane, which means that no out-of-plane deformation will occur (ϵz=ϵxz=ϵyz=0). The plane strain option is generally appropriate for structures of nearly infinite length, relative to their cross section, that exhibit negligible length changes under load. The “generalized plane strain” option imposes the axial strain equal to a constant value; this condition does not reproduce the real operating condition of the magnet and is therefore excluded. The “plane stress with thickness” uses the same equations as the plane stress option but, output quantities are given per unit length defined with thickness; therefore, it is again excluded from the comparison. Superconducting magnets, such as dipoles, do not fit neatly into any of the above options: they are far from thin but deform longitudinally under load. This work reports a comparative study of plane stress and plane strain in the specific case study of a dipole magnet.

Effect of a longitudinal magnetic field on streamer propagation in air: Numerical simulation

Two-dimensional numerical simulation of the negative streamer propagation in external electric and magnetic fields is performed for the case when applied electric and magnetic field vectors are parallel to the symmetry axis of the problem. The calculations are performed for air (a mixture of 80% N2 and 20% O2) with a concentration of gas molecules equal to Loshmidt's number. It is shown that the presence of a longitudinal field leads to a noticeable increase in the streamer propagation velocity associated with a decrease in its radius, which agrees with the known analytical estimates.

Electrodeposition of graphene oxide on titanium foil. Application field: Rechargeable batteries

The method of electrodepositing graphene oxide on titanium utilizes a thin grade 1 titanium foil with a thickness of 0.2 mm, a width of 10 mm, and a rectangular graphite plate with a parallelepiped shape with a thickness of 15 mm, a width of 40 mm, and a length of 100 mm. The titanium foil has a cut on one side, forming an isosceles triangle with a slightly inclined tip, creating an angle where the distance between the tip and the vertical symmetry axis of the foil is 3 mm. The titanium foil is positioned parallel to the graphite plate, with the tip as the only electrical contact. The two electrodes, titanium and graphite, assembled in this way, are immersed in an electrolytic solution composed of water and sulfuric acid. When a direct current voltage is applied between the two electrodes, a strong electric field is generated at the contact point, resulting in high temperature and the production of ultraviolet rays. Following a treatment described later in the so-called “preparation protocol,” reduced graphene oxide (rGO) is produced in suspension in the solution at the contact point between the tip of the titanium foil and the graphite plate, which, due to the electric field, tends to deposit on the titanium surface. A possible application found experimentally refers to rechargeable batteries.

Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration.

The deep unfolding approach has attracted significant attention in computer vision tasks, which well connects conventional image processing modeling manners with more recent deep learning techniques. Specifically, by establishing a direct correspondence between algorithm operators at each implementation step and network modules within each layer, one can rationally construct an almost "white box" network architecture with high interpretability. In this architecture, only the predefined component of the proximal operator, known as a proximal network, needs manual configuration, enabling the network to automatically extract intrinsic image priors in a data-driven manner. In current deep unfolding methods, such a proximal network is generally designed as a CNN architecture, whose necessity has been proven by a recent theory. That is, CNN structure substantially delivers the translational symmetry image prior, which is the most universally possessed structural prior across various types of images. However, standard CNN-based proximal networks have essential limitations in capturing the rotation symmetry prior, another universal structural prior underlying general images. This leaves a large room for further performance improvement in deep unfolding approaches. To address this issue, this study makes efforts to suggest a high-accuracy rotation equivariant proximal network that effectively embeds rotation symmetry priors into the deep unfolding framework. Especially, we deduce, for the first time, the theoretical equivariant error for such a designed proximal network with arbitrary layers under arbitrary rotation degrees. This analysis should be the most refined theoretical conclusion for such error evaluation to date and is also indispensable for supporting the rationale behind such networks with intrinsic interpretability requirements. Through experimental validation on different vision tasks, including blind image super-resolution, medical image reconstruction, and image de-raining, the proposed method is validated to be capable of directly replacing the proximal network in current deep unfolding architecture and readily enhancing their state-of-the-art performance. This indicates its potential usability in general vision tasks.

On Boundary Discontinuity in Angle Regression Based Arbitrary Oriented Object Detection.

With vigorous development e.g., in autonomous driving and remote sensing, oriented object detection has gradually been featured. The majority of existing methods directly perform regression on the rotation angle, which we argue has fundamental limitations of boundary discontinuity (even if using Gaussian or RotatedIoU-based losses). In this paper, a novel angle coder named phase-shifting coder (PSC) is proposed to address this issue. Different from another well-explored alternative i.e., angle classification, PSC achieves boundary-discontinuity-free in a continuous and differentiable manner and thus can work together with Gaussian or RotatedIoU-based methods to further boost their performance. Moreover, by rethinking the boundary discontinuity of elongated and square-like objects as rotational symmetry of different cycles, a dual-frequency version (PSCD) is proposed to accurately predict the orientation of both types of objects. Visual analysis and extensive experiments on several popular backbone detectors and datasets demonstrate the effectiveness and the potentiality of our approach. When facing scenarios requiring high-quality bounding boxes, the proposed methods are expected to give a competitive performance.

Torque-driven superhydrophobic cylinders swim in circles

A superhydrophobic cylinder is a circular cylinder with a boundary pattern of alternating shear-free menisci and solid portions. This article derives analytical solutions for the torque-driven Stokes flow around a superhydrophobic cylinder that is free to move, or ‘swim’, subject to the constraint that it is free of net force. The mathematical approach is a natural generalization of one used to solve for transverse shear flow over a superhydrophobic surface comprising a periodic array of no-shear slots (Crowdy 2011 Phys. Fluids 23 (doi: 10.1063/1.3605575 )). First, the torque-driven Stokes flow around a superhydrophobic cylinder with two unequal no-shear menisci and two equal solid portions is solved in detail. It is then shown how that analysis generalizes, in a natural way, to a cylinder with M no-shear menisci between M solid portions with full exposition given of the case of M = 3 equal solid portions. A further generalization to flow around a superhydrophobic cylinder with an N -fold rotationally symmetric patterning with M menisci in each 2 π / N sector is made. Torque-driven superhydrophobic cylinders with a no-shear pattern devoid of any rotational symmetry are shown to swim on circular trajectories with formulas for the radius and cylinder angular velocity on these trajectories derived here as functions of the no-shear pattern geometry. The M = 1 case of a ‘Janus’ superhydrophobic cylinder having one meniscus and one solid portion can be solved completely explicitly.

Transport through a monolayer-tube junction: Sheet-to-tube spin current

We develop a method to calculate the electron flow between an arbitrary atomic monolayer sheet and an arbitrary tube by expressing the corresponding sheet-tube tunneling matrix elements with those between sheets. We use this method to calculate the spin current from a monolayer silicene sheet with sublattice-staggered current-induced spin polarization to a silicene tube. The calculated sheet-to-tube spin current exhibits an oscillation as a function of the tube circumferential length because the Fermi points in the tube cross the Fermi circle in the sheet. Furthermore, the spin current with spin in the out-of-plane direction, which is absent in the sheet-sheet junction (including twisted sheets) with C3 rotational symmetry, appears in an oscillating form in the tube-sheet junction due to the broken C3 rotational symmetry. This is an example of the symmetry manipulation which realizes switching a particular component of the spin current.

Застосування методу двобічних наближень до знаходження додатних аксіально-симетричних розв’язків крайових задач з монотонними нелінійностями

This paper considers the application of the method of two-sided approximations for finding positive solutions to boundary value problems for nonlinear elliptic differential equations with axial symmetry. The case of Dirichlet boundary conditions (first kind) is considered, with a power-type monotone nonlinearity, where the exponent ranges from 0 to 1. By transitioning to polar coordinates in the boundary value problem for the elliptic equation, due to the axial symmetry of the solution, the problem is reduced to a boundary value problem for an ordinary differential equation on an interval with respect to a function that depends only on the polar radius, thus eliminating the dependence on the angle. The pole of the polar coordinate system becomes a singular point, where a boundedness condition must be imposed. For the boundary value problem, a Green's function is constructed to further reduce the problem to a Hammerstein integral equation. The integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on the interval, partially ordered by a cone of non-negative continuous functions on this interval. The operator is examined for properties such as monotonicity (isotonicity), positivity, boundedness, and concavity. Next, the initial approximation is found as the endpoints of the invariant conical segment for the isotone operator in such a way as to ensure the highest convergence rate of the iterative process. The following iterative sequences of two-sided approximations are constructed: the first sequence, which is non-decreasing with respect to the cone, and the second sequence, which is non-increasing with respect to the cone. At each iteration, the arithmetic mean of the upper and lower approximations is taken as the next approximation. The iterative process is terminated when the error estimate of the solution satisfies the specified accuracy. The theoretical results obtained in this work were validated through a computational experiment. The dependence of the solution on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs

Pentagonal Symmetry in Aperiodic Cyclic Multiply Twinned Nano- and Mesodiamonds.

This Review provides the first comprehensive overview of the structure and properties of exotic sp3-crystalline aperiodic cyclic multiply twinned nano- (10-100 nm) and meso- (up to 1 mm) diamond particles (MTPs) exhibiting pentagonal symmetry. It spans their independent experimental discoveries (1963, 1964, 1972, and 1983) and theoretical structural insights (1993) to recent advancements. The Review focuses on high-symmetry MTPs formed by the fusion of multiple cubic diamond fragments through [111] facets. The sp3 diamond lattice of individual fragments offers a vast range of MTP varieties. These particles are shown to be a special case of aperiodic crystalline solids with limited dimensions and rotational symmetry, leading to a breakdown of translational invariance. Detailed mathematical analysis of the MTPs' lattices highlights the crucial role of central cores in determining the symmetry and effective dimensions of these structures. Both structural and kinetic aspects of the formation mechanisms of pentagonal diamond particles are considered, revealing the main role of embryo seeds (low fullerenes, polyhexacyclo[5.5.1.12,6.18,12.03,11.05,9]pentadecane (C15), and polyheptacyclo[5.5.1.12,7.19,14.03,13.06,11]octadecane (C18)) in determining the MTPs' symmetry and structure. The effective dimensions of multiply twinned diamonds are shown to be limited by structural stress caused by the mismatch of perfect pentagonal and tetrahedral dihedral angles. The extraordinary mechanical and electronic properties of multiply twinned diamonds are discussed, highlighting that hexagonal interfaces between cubic diamond fragments may determine exceptional ultrahard and quantum characteristics. The MTP X-ray diffraction spectra reveal clear pentagonal patterns with single (diamond decahedrons or dodecahedrons) or ten (diamond icosahedra) central 5-fold axes. A comparative analysis of experimental structural data and simulations at different theoretical levels demonstrates a perfect correspondence of theoretical models with crystalline lattices.

CONSTRUCTING QUANTUM ANGULAR MOMENTUM L3 IN SPECIFIC DIRECTION BY USING U(1) GROUP

The purpose of this work is to investigate the mathematical structure of finite quantum angular momentum in a specific direction L3 which can be constructed from the representation of the U(1) group. The angular momentum eigenstate is invariant under Abelian rotation symmetry. The character group of U(1) is constructed to show that there exists an additive unitary operator for the angular momentum eigenstate, and the rotation eigenstate is invariant under it. The Weyl relation is proved by showing that the angle and angular momentum L3 are the canonical conjugate pair of observables.

Remarks on Soft Ball Packings in Dimensions 2 and 3

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean 3-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with 3-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.

Small-Size Eight-Element MIMO Metamaterial Antenna with High Isolation Using Modal Significance Method.

This article presents a symmetrical reduced-size eight-element MIMO antenna array with high electromagnetic isolation among radiators. The array utilizes easy-to-build techniques to cover the n77 and n78 new radio (NR) bands. It is based on an octagonal double-negative metamaterial split-ring resonator (SRR), which enables a size reduction of over 50% for the radiators compared to a conventional disc monopole antenna by increasing the slow-wave factor. Additionally, due to the extreme proximity between the radiating elements in the array, the modal significance (MS) method was employed to identify which propagation modes had the most impact on the electromagnetic coupling among elements. This approach aimed to mitigate their effect by using an electromagnetic barrier, thereby enhancing electromagnetic isolation. The electromagnetic barriers, implemented with strip lines, achieved isolation values exceeding 20 dB for adjacent elements (<0.023 λ) and approaching 40 dB for opposite ones (<0.23 λ) after analyzing the surface current distribution by the MS method. The elements are arranged in axial symmetry, forming an octagon with each antenna port located on a side. The array occupies an area of 0.32 λ2 at 3.5 GHz, significantly smaller than previously published works. It exhibits excellent performance for MIMO applications, demonstrating an envelope correlation coefficient (ECC) below 0.0001, a total active reflection coefficient (TARC) lower than -10 dB for various incoming signals with random phases, and a diversity gain (DG) close to 20 dB.

Axially symmetric solutions in Ricci-inverse modified gravity

We investigate a family of axial symmetry solution constructed in general relativity (GR) within the framework of Ricci-inverse (RI) gravity theory. In GR, these solutions admitted closed time-like curves at an instant of time from an initial spacelike hypersurface in a causally well-behaved manner, thus, violates the causality condition. Our aim is to examine these axial symmetry solutions within the context of Ricci-inverse gravity theory to determine whether closed time-like curves still appear in this new gravity theory. We consider two Classes of RI-gravity models: (i) Class-II models defined by a function f=f(R,AμνAμν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=f({\\mathcal {R}}, A^{\\mu \ u }\\,A_{\\mu \ u })$$\\end{document} gravity and (ii) Class-III models defined by f=f(R,A,AμνAμν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=f({\\mathcal {R}},{\\mathcal {A}}, A^{\\mu \ u }\\,A_{\\mu \ u })$$\\end{document}, where Aμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\mu \ u }$$\\end{document} is the anti-curvature tensor, A=gμνAμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}=g_{\\mu \ u }\\,A^{\\mu \ u }$$\\end{document} as its scalar, and Rμν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{\\mu \ u }$$\\end{document} is the Ricci tensor. We are able solved the modified field equations considering these axial symmetry solutions as background in RI-gravity with null radiation as the matter content and the cosmological constant. This confirms that the chosen family of axial symmetry solutions are valid solutions in RI-gravity theory and, consequently, closed time-like curves is still form, analogous to their formation in GR.

Energy of gravitational radiation and the background energy of the space-time

We address the issue of gravitational radiation in the context of the Bondi-Sachs space-time, and consider the expression for the gravitational energy of the radiation obtained in the realm of the teleparallel equivalent of general relativity (TEGR). This expression is independent of the radial distance (i.e., of powers of 1/r) and depends exclusively on the functions c(u,θ,ϕ) and d(u,θ,ϕ), which yield the news functions (u is the retarded time, u=t−r). We investigate the mathematical and physical features of this energy expression in the simpler framework of axial symmetry. Once a burst of gravitational radiation takes place in a self gravitating system, that leads to a loss of the Bondi mass, gravitational radiation is emitted throughout the whole space-time. The existence and presence of this radiation in the background structure of the space-time is consistent with the analysis developed by Papapetrou, and Hallidy and Janis, who found no proof that a gravitational system that emits a burst of gravitational radiation is preceded and followed by two stationary gravitational field configurations, namely, it seems that it is impossible for a gravitational system, which is initially stationary, to return to a stationary state after emitting a burst of axially symmetric gravitational radiation, in which case the space-time is not even asymptotically stationary. Therefore, it is plausible that the gravitational energy of radiation is present in the background structure of the space-time, and this is the energy predicted in the TEGR. This analysis leads us to conjecture that the noise detected in the large terrestrial gravitational wave observatories is intrinsically related to the background gravitational radiation.

Lattice paths and the Prouhet-Thue-Morse sequence

Using Flajolet's combinatorial theory of continued fractions and axial symmetry of lattice paths, we show that the enumeration of certain lattice paths modulo two yields the ubiquitous Prouhet-Thue-Morse sequence. This answers an open problem of Berstel et al.

Computing properties of subdivision schemes using small real Fourier indexed matrices

The quality of a subdivision scheme in the vicinity of a vertex or a face-centre is related to the eigenstructure of the subdivision matrix. When the scheme has the appropriate symmetries, a common technique, based on discrete Fourier transform, builds small complex matrices that ease the numerical analysis of the eigenelements using in particular their Fourier index. But the numerical analysis of the eigenelements remains difficult when matrix entries involve complex numbers and unknowns, for example, in cases where we are tuning a scheme. We present techniques to build similar small matrices, still associated with a Fourier index and whose eigenstructure is simply related to the full matrix, but which are real. They extend the known techniques to schemes which rotate the lattice and with vertices which do not lie topologically on symmetry axes of the studied vicinity of vertex or face centre. Our techniques make it easier to tune these subdivision schemes. We illustrate it with the analysis of the so-called Simplest Scheme at the centre of an n-sided face.

Nematic Superconductivity and Its Critical Vestigial Phases in the Quasicrystal.

We propose a general mechanism to realize nematic superconductivity (SC) and reveal its exotic vestigial phases in the quasicrystal (QC). Starting from a Penrose-Hubbard model, our microscopic studies suggest that the Kohn-Luttinger mechanism driven SC in the QC is usually gapless due to violation of Anderson's theorem, rendering that both chiral and nematic SCs are common. The nematic SC in the QC can support novel vestigial phases driven by pairing phase fluctuations above its T_{c}. Our combined renormalization group and MonteCarlo studies provide a phase diagram in which, besides the conventional charge-4e SC, two critical vestigial phases emerge, i.e., the quasinematic (QN) SC and QN metal. In the two QN phases, discrete lattice rotation symmetry is counterintuitively "quasibroken" with power-law decaying orientation correlation. They separate the phase diagram into various phases connected via Berezinskii-Kosterlitz-Thouless (BKT) transitions. These remarkable critical vestigial phases, which resemble the intermediate BKT phase in the q state (q≥5) clock model, are a consequence of the fivefold (or higher) anisotropy field brought about by the unique QC symmetry, which are absent in conventional crystalline materials.

Phononic Weyl nodal lines and Weyl pairs in van der Waals heavy fermion material CeSiI

Abstract Topological phonon is a new frontier in the field of topological materials. Different from electronic structures, phonons are bosons and most topological phonons are metallic form. Previous studies about topological phonon states were focused on threedimensional (3D) materials. Owing to the lack of material candidates, two-dimensional (2D) and van der Waals f-electron-related topological phonons were rarely reported. Based on first-principles calculations, we investigated the topological phononic state in the heavy fermion material CeSiI with a layered structure. Both three-dimensional (3D) bulk and two-dimensional (2D) monolayers have topological nontrivial states in the rarely seen f-electron dominated van der Waals metal. Owing to the PT and C 3z symmetries, Weyl nodal lines with non-zero Chern numbers exist on the hinge of the Brillouin zone (BZ). Protected by C 3z rotation symmetry, three pairs of Weyl points with ±π Berry phase exist at point K near the Frequency of 8 and 10 THz. In addition to the bulk topological charges, corresponding surface/edge states were also systematically analyzed, which gives a consistent understanding. Our results proposed another interesting point in the newly discovered rear earth heavy fermion material CeSiI and are helpful for future experimental study of its topological phonons.

The influence of the bounding surface on the structural ordering of short chains of oligoetherimides.

In this study, we have conducted a comparative analysis of the structural ordering of short oligoetherimide chains (dimers) near the bounding surface, depending on the structure of that surface. In order to clarify the possibility of oligoetherimide ordering along the symmetry axes of graphene, two types of bounding surfaces were considered: graphene, with a regular discrete position of interaction centers (carbon atoms), and a smooth, structureless impermeable wall. The chemical structures of the considered dimers consist of two repeating units of BPDA-P3, ODPA-P3, or aBPDA-P3 thermoplastic polyetherimides. Using all-atom molecular dynamics simulations, the process of structural ordering of the dimers near the surface of the graphene or wall was established. The ODPA-P3 and BPDA-P3 dimers form an ordered state near the graphene surface, while the aBPDA-P3 dimers do not demonstrate structural ordering. The simulation results confirmed that the ordering direction of the BPDA-P3 and ODPA-P3 dimers near the graphene surface is chosen randomly. Comparison of the oligoetherimide structure formed near the attracting wall without a symmetrical location of the interaction centers shows the similarity of the ordering of dimers near the graphene surface and the wall. As in the case of the graphene surface, the ordering of oligoetherimide molecules near the structureless wall demonstrates one direction of ordering. Therefore, we confirmed that the key factor for the onset of ordering is the presence of a confining surface, rather than the symmetrical arrangement of interaction centers in the substrate structure.