Equilateral Research Articles

Nonlocal coherence harvesting from quantum vacuum

Abstract It is well known that nonlocal coherence reflects nonclassical correlations better than quantum entan-
glement. Here, we analyze nonlocal coherence harvesting from the quantum vacuum to particle detectors
adiabatically interacting with a quantum scalar field in Minkowski spacetime. We find that the harvesting-
achievable separation range of nonlocal coherence is larger than that of quantum entanglement. As the
energy gap grows sufficiently large, the detectors harvest less quantum coherence, while the detectors could
extract more quantum entanglement from the vacuum state. Compared with the linear configuration and the
scalene configuration, we should choose the model of equilateral triangle configuration to harvest tripar-
tite coherence from the vacuum. Finally, we find a monogamous relationship, which means that tripartite
l1-norm of coherence is essentially bipartite types.

Constructible Sandwich Cut

In mathematical measure theory, the “Ham-Sandwich” theorem states that any n objects in an n-dimensional Euclidean space can be simultaneously divided in half with a single cut by an (n-1)-dimensional hyperplane. While it guarantees its existence, the theorem does not provide a way of finding this halving hyperplane, as it is only an existence result. In this paper, we look at the problem in dimension 2, more in the style of Euclid and the antique Greeks, that is from a constructible point of view, with straight edge and compass. For two arbitrary regions in the plane, there is certainly no hope for constructing the halving line by straight edge and compass. At the opposite end, for two circles the problem is very easy, as the halving line is simply the line which passes through the centers of both circles. It turns out that for the case of two equilateral triangles the problem is already interesting and challenging from a constructible point of view, and this is the main goal of our paper. We will investigate the “deltoid” of a triangle, that is, three hyperbolas which form an “envelope” of all halving lines of a given triangle, and how to construct halving lines with certain given properties.

Perfectly Packing an Equilateral Triangle by Equilateral Triangles of Sidelengths $$n^{-1/2-\epsilon }$$

AbstractEquilateral triangles of sidelengths 1, $$2^{-t}$$ 2 - t , $$3^{-t}$$ 3 - t , $$4^{-t},\ldots \ $$ 4 - t , … can be packed perfectly into an equilateral triangle, provided that $$\ 1/2<t \le 37/72$$ 1 / 2 < t ≤ 37 / 72 . Moreover, for t slightly greater than 1/2, squares of sidelengths 1, $$2^{-t}$$ 2 - t , $$3^{-t}$$ 3 - t , $$4^{-t},\ldots \ $$ 4 - t , … can be packed perfectly into a square $$S_t$$ S t in such a way that some squares have a side parallel to a diagonal of $$S_t$$ S t and the remaining squares have a side parallel to a side of $$S_t$$ S t .

Investigation on laser-induced bubble collapse among triple particles based on high-frame-rate photography and the Kelvin impulse model

In fluid machinery, the concurrent presence of cavitation bubbles and particle clusters leads to considerably damage to material surfaces. This study investigates the dynamics of a bubble situated among triple particles based on the Kelvin impulse model and high-frame-rate photography, focusing on the impact of the dimensionless distance of particles and the bubble size. Specifically, the jet, bubble motion, and bubble interface evolution characteristics are quantitatively evaluated. The following conclusions are obtained: (1) The collapse shapes of the bubble can be divided into three typical cases: equilateral triangle shape, isosceles triangle shape, and arcuate shape. (2) Among the triple particles, four zero-Kelvin-impulse locations are present, around which the jet direction is extremely sensitive to the bubble initial position. As the bubble initial position moves along the central line, the bubble motion direction dramatically changes during its collapse. (3) The relative position of bubble and particles is the key parameter that affects the bubble dynamics. As the bubble–particle distance decreases, the non-uniformity of bubble collapse morphology and the bubble motion distance will become more significant.

Static and modal analysis of sandwich panels with rib-reinforced re-entrant honeycomb

The low stiffness resulting from substantial core porosity within the re-entrant honeycomb sandwich panel can be addressed by introducing rib-reinforcements. To investigate its static and vibration characteristics, an energy-based 2D equivalent-homogenization model (2D-EHM) was derived through asymptotic analysis of the leading terms in the governing equations considering the energy-related characteristics. The accuracy of the constructed model for predicting the elastic bending performance of the sandwich panel was verified by comparing the results of a three-point bending experiment with 3D-printed specimens. Comparative analysis with a 3D FE model demonstrates that the 2D-EHM exhibits maximum errors of only 4.65% and 6.46%, respectively, in analyzing static deformation and natural frequency while improving computational efficiencies by up to 59- and 173-folds. Parameter analysis reveals that the performance of SP-RRH is minimally affected by the reinforcement-to-honeycomb wall thickness ratio. However, its performance is compromised when the honeycomb walls align in an equilateral triangle with the reinforcing walls. Comparing specific stiffness as well as static and vibration characteristics of sandwich panels with different core configurations (including traditional and diamond reinforced re-entrant honeycombs) shows that inclusion of rib reinforcements enhances deformation resistance and natural frequencies through increased specific stiffness. These findings provide valuable insights for optimizing design of re-entrant honeycomb sandwich panels.

Hexagonal finite differences for the two-dimensional variable coefficient Poisson equation

For many years, finite differences in hexagonal grids have been developed to solve elliptic problems such as the Poisson and Helmholtz equations. However, these schemes are limited to constant coefficients, which reduces their usefulness in many applications. The main challenge is accurately approximating the diffusive term. This paper presents examples of both successful and unsuccessful attempts to obtain accurate finite differences based on a hexagonal stencil with equilateral triangles to approximate two-dimensional Poisson equations. Local truncation error analysis reveals that a second-order scheme can be achieved if the derivative of the diffusive coefficient is included. Finally, we provide numerical examples to verify the accuracy of the proposed methods.

3D auxetic linkage based on Kirigami

The structural design of 3D auxetic linkages is a burgeoning field in digital manufacturing. This article presents a novel algorithm for designing 3D auxetic linkage structures based on Kirigami principles to address existing limitations. The 3D input model is initially mapped to a 2D space using conformal mapping based on the BFF method. This is followed by 2D re-meshing using an equilateral triangle mesh. Subsequently, a 3D topological mesh of the auxetic linkage is calculated through inverse mapping based on directed area. We then introduce new basic rotating and non-rotating units, employing them as the initial structure of the 3D auxetic linkage in accordance with Kirigami techniques. Lastly, a deformation energy function is defined to optimize the shape of the rotating units. The vertex coordinates of the non-rotating units are updated according to the optimized positions of the rotating units, thereby generating an optimal 3D auxetic linkage structure. Experimental results validate the effectiveness and accuracy of our algorithm. Quantitative analyses of structural porosity and optimization accuracy, as well as comparisons with related works, indicate that our algorithm yields structures with smaller shape errors.

The tunneling splittings of the ground state and some excited vibrational states for the inversion motion in H3C− anion and H3Si[rad] radical

Splitting of the ground state and some excited symmetric bending vibrational states due to inversion tunneling in the H3C− anion and H3Si radical is analyzed by numerically solving the vibrational Schrödinger equation of restricted (2D) dimensionality. We used the following two vibrational coordinates for the H3X structure (X = C, Si): the distance of the X atom from the plane of a regular triangle formed by three hydrogen atoms (1) and a symmetry coordinate composed of three distances between chemically non-bonded hydrogen atoms (2). The kinetic energy operator in this case takes the simplest form. The 2D potential energy surface (PES) in the given coordinates was calculated for H3C− at the CCSD(T)/aug-cc-pVTZ, CCSD(T)/aug-cc-pVQZ, CCSD(T)/aug-cc-pV5Z, CCSD(T)/CBS(TQ5), CCSD(T)/d-aug-cc-pVTZ, CCSD(T)/t-aug-cc-pVTZ, and CCSD(T)/q-aug-cc-pVTZ levels of theory, based on recommendations from recently published work [M.C. Bowman, B. Zhang, W.J. Morgan, H.F. Schaefer III, Mol.Phys., 117 (2019) 1069–1077]. The same 2D PES for the H3Si radical was calculated at the CCSD(T)/aug-cc-pVDZ CCSD(T)/aug-cc-pVTZ, CCSD(T)/aug-cc-pVQZ, and CCSD(T)/CBS(D,T,Q) as well as at the CCSD(T)/d-aug-cc-pVTZ, CCSD(T)/un-aug-cc-pVTZ, CCSD(T)/un-aug-cc-pVQZ levels of theory. The tunneling splittings for the D3C− anion and D3Si radical were calculated as well. The results of calculations demonstrate good agreement with available experimental data on umbrella vibration frequencies and inversion splittings for the title molecules.

Spherical time-encoded radiation imaging simulations

Radiation source localization is important for nuclear nonproliferation and can be obtained using time-encoded imaging systems with unsegmented detectors. A scintillation crystal can be used with a moving coded-aperture mask to vary the detected count rate produced from radiation sources in the far field. The modulation of observed counts over time can be used to reconstruct an image with the known coded-aperture mask pattern. Current time-encoded imaging systems incorporate cylindrical coded-aperture masks and have limits to their fully coded imaging field-of-view. This work focuses on expanding the field-of-view to 4π by using a novel spherical coded-aperture mask. A regular icosahedron is used to approximate a spherical mask. This icosahedron consists of 20 equilateral triangles; the faces of which are each subdivided into four equilateral triangle-shaped voxels which are then projected onto a spherical surface, creating an 80-voxel coded-aperture mask. These polygonal voxels can be made from high-Z materials for gamma-ray modulation and/or low-Z materials for neutron modulation. In this work, we present Monte Carlo N-Particle (MCNP) simulations and simple models programmed in Mathematica to explore image reconstruction capabilities of this 80-voxel coded-aperture mask.

Metasurface-inspired circularly polarized patch antenna using DGS for sub 6-GHz applications

A novel circularly polarized metasurface inspired regular triangle patch antenna (RTPA) with enhanced gain and bandwidth is proposed in this paper. The antenna is composed of a triangular patch-driven element sandwiched between a 3×3 sub-wavelength square patch metasurface array and the ground plane. Initially, the RTPA is designed, and the ground plane is etched with fractal slots to produce degenerate modes for circular polarization. A metasurface array is used as a superstrate to improve gain and bandwidth in terms of impedance and axial ratio (AR). Another advantage of using a metasurface is the reduction of cross-polar radiation, particularly when employing a triangular-shaped patch antenna. The asymmetry of this structure leads to cross-polar components contributed by dominant modes rather than higher-order modes, and the use of a metasurface markedly diminishes these components. The proposed antenna's experimental results show a 10 dB return loss (|S11|) with a metasurface demonstrates a measured impedance bandwidth and axial ratio bandwidth of 760 and 350 MHz, respectively. The antenna exhibits right-hand circular polarization with a measured gain of 5.87 dBic at the operating frequency.

METHOD OF CREATING A MINIMAL SPANNING TREE ON AN ARBITRARY SUBSET OF VERTICES OF A WEIGHTED UNDIRECTED GRAPH

Context. The relevance of the article is determined by the need for further development of models for optimal restoration of the connectivity of network objects that have undergone fragmentation due to emergency situations of various origins. The method proposed in this article solves the problematic situation of minimizing the amount of restoration work (total financial costs) when promptly restoring the connectivity of a selected subset of elements of a network object after its fragmentation.
 The purpose of the study is to develop a method for creating a minimal spanning tree on an arbitrary subset of vertices of a weighted undirected graph to minimize the amount of restoration work and/or total financial costs when promptly restoring the connectivity of elements that have a higher level of importance in the structure of a fragmented network object.
 Method. The developed method is based on the idea of searching for local minima in the structure of a model undirected graph using graph vertices that are not included in the list of base vertices to be united by a minimal spanning tree. When searching for local minima, the concept of an equilateral triangle and a radial structure in such a triangle is used. In this case, there are four types of substructures that provide local minima: first, those with one common base vertex; second, those with two common base vertices; third, those with three common base vertices; fourth, those without common base vertices, located in different parts of the model graph. Those vertices that are not included in the list of basic ones, but through which local minima are ensured, are added to the basic ones. Other vertices (non-basic) along with their incident edges are removed from the structure of the model graph. Then, using one of the well-known methods of forming spanning trees, a minimal spanning tree is formed on the structure obtained in this way, which combines the set of base vertices.
 Results. 1) A method for creating a minimal spanning tree on an arbitrary subset of vertices of a weighted undirected graph has been developed. 2) A set of criteria for determining local minima in the structure of the model graph is proposed. 3) The method has been verified on test problems.
 Conclusions. The theoretical studies and several experiments confirm the efficiency of the developed method. The solutions developed using the developed method are accurate, which makes it possible to recommend it for practical use in determining strategies for restoring the connectivity of fragmented network objects.

Association of two square difference identity to regular polygons and circles

GeoGebra is a dynamic software that is frequently used and of increasing importance in mathematics teaching processes in our digital age. Accordingly, in this study a new perspective has been brought to the proofs of the “two square difference identity” expressed for the square, which is a flat polygon, made with different approaches. With side lengths a, b, and a>b, it has been shown that the identity given by the equation (difference of area) a<sup>2</sup>-b<sup>2</sup>=(a-b)(a+b) is true for other regular polygons as well. In the study, direct proof method was used within the framework of the principle of conservation of measure, which is one of the basic principles of geometry teaching. GeoGebra program, which is a dynamic geometry software, was preferred for drawing geometric shapes used in proofs. In order to generalize the number n, a different fragmentation technique was preferred to the proofs made using different drawings for equilateral triangle and square, which are the simplest regular polygons. It has also been shown that this identity is true for circles viewed as polygons with an infinite number of sides.

Ultrafast Imaging of Jahn-Teller Distortion and the Correlated Proton Migration in Photoionized Cyclopropane.

The Jahn-Teller (JT) distortion is one of the fundamental processes in molecules and condensed phase matters. For photoionized organic molecules with high symmetry, the JT effect leads to geometric instability in certain electron configurations and thus has a significant effect on the subsequent isomerization and proton migration processes. Utilizing the femtosecond pump-probe Coulomb explosion method, we probe the isomerization dynamics process of a monovalent cyclopropane cation (C3H6+) caused by proton migration and reveal the relationship between proton migration and JT distortion. We found that the C3H6+ cation evolves from the D3h symmetric equilateral triangle geometry either to the acute triangle via two elongated C-C bonds (JT1) or to the obtuse triangle via a single elongated C-C bond (JT2). The JT1 pathway does not involve proton migration, while the JT2 pathway drives proton migration and can be mapped into the indirect dissociation channel of Coulomb explosion. The time-resolved experiment indicates that the delay time between those two JT pathways can be as large as ∼600 fs. After the JT distortion, the cyclopropane cation undergoes a subsequent structural evolution, which brings a greater variety of dissociation channels.

Effects of cross-sectional shape on flow and heat transfer of the laminar flow of supercritical carbon dioxide inside horizontal microchannels

Supercritical carbon dioxide (sCO2) flow at the micro scale provides an excellent heat transfer solutions for a range of microelectronics cooling applications. The cross-sectional shape of the microchannel (in combination with the abruptly changing thermophysical properties of sCO2) affects the heat transfer and flow characteristics, but these effects are not well understood at constant wall temperatures as most of the previous studies were performed at constant surface heat fluxes. Current study numerically investigated, using CFD, the effects of different cross-sectional shapes, such as circle, equilateral triangle, semi-circle, and square, in zero gravity (space) and in terrestrial conditions, at constant wall temperatures.In all cases, the hydraulic diameter of the microchannel was 100 μm, the total length was 27 mm, and the heated length was 25 mm. A total of 216 cases were analyzed to predict the effects of surface temperature ranging from 310 to 330 K, pressures ranging from 8 to 10 MPa, and Reynolds numbers from 100 to 500 on the heat transfer and flow. It was revealed that the microchannel with triangular cross-section has the highest heat transfer rate, highest mass flow rate, and lowest pressure drop both in zero gravity and in terrestrial conditions. The total surface heat transfer rate for the triangular microchannel was about 49% higher than the circular microchannel at a pressure of 8 MPa, around 53% higher at 9 MPa, and about 55% higher at 10 MPa. However, the average surface heat transfer coefficient displayed a complex behavior, and it depends on the combination of surface temperature, Reynolds number, and operating pressure. The surface heat transfer rate and pressure drop were increased for all shapes at terrestrial conditions, emphasizing that the effects of buoyancy cannot be neglected. Overall, this study revealed a complex behavior of the flow of sCO2 inside microchannels of different cross-sectional shapes, and the triangular cross-section performed better than other shapes.

Computational modeling of cracking in cortical bone microstructure using the mesh fragmentation technique

AbstractThe cortical bone is a hierarchical composite material that, at the microscale, is segmented in an interstitial matrix, cement line, osteons, and Haversian canals. The cracking of the structure at this scale directly influences the macro behavior, and, in this context, the cement line has a protagonist role. In this sense, this work aims to simulate the crack initiation and propagation processes via cortical bone microstructure modeling with a two-dimensional mesh fragmentation technique that captures the mechanical relevance of its constituents. In this approach, high aspect ratio elements are inserted between the regular constant strain triangle finite elements to define potential crack paths a priori. The crack behavior is described using a composed damage model with two scalar damage variables, which is integrated by an implicit-explicit (Impl-Ex) scheme to avoid convergence problems usually found in numerical simulations involving multiple cracks. The approach’s capability of modeling the failure process in cortical bone microstructure is investigated by simulating four conceptual problems and one example based on a digital image of an experimental test. The results obtained in terms of crack pattern and failure mechanisms agree with those described in the literature, demonstrating that the numerical tool is promising to simulate the complex failure mechanisms in cortical bone, considering the properties of its distinct phases.

Analysis of aerodynamic forces and flow development with the size of square cylinders arranged in equilateral triangle

A two-dimensional in-house code is used to scrutinize the influence of the arrangement of three cylinders and their corresponding dimensions on the behavior of flow around square cylinders arranged in four different combinations. Three square cylinders, C1, C2 and C3, are arranged in an equilateral triangle with the first cylinder C1 positioned upstream while the two remaining cylinders are placed side-by-side downstream. The wake flow patterns, aerodynamic forces and Strouhal number (St) are determined and analyzed for the four-cylinder arrangements in terms of streamlines, iso-vortices and drag and lift coefficients evolutions for steady and unsteady regimes. The numerical outcomes demonstrate that Re values, the different equilateral-triangular arrangements of cylinders, and the dimension variations of each square cylinder have a prominent influence on the flow characteristics.

Vibration characteristics of a downstream cylinder with rounded corners in an equilateral triangular arrangement at Reynolds number of 180

In this study, we numerically investigate the vibration response of a cylinder arranged in an equilateral triangle with two stationary upstream cylinders at a low Reynolds number of Re = 180. We concentrate on three different rounded corner arrangements: four rounded corners (FRCs), upwind rounded corners (URCs), and leeward rounded corners (LRCs). In addition, we examine three dimensionless rounded corner radii: 0.1, 0.3, and 0.5. The results indicate that as the dimensionless radius increases, the influence of the corner position on the dimensionless amplitude and frequency becomes more pronounced. Furthermore, the dimensionless amplitude and dimensionless frequency curves exhibit significant variations under the different corner arrangements. Within the reduced velocity range (Vr) of 2–5, rounded corners are observed to reduce both the lift and drag coefficients. An increase in the dimensionless radius for a given corner configuration acts to decrease both the lift and drag coefficients. Frequency-domain analysis of the lift coefficient and dimensionless displacement indicates that the observed frequency peaks caused by mutual interference are produced by irregular vortex shedding due to the mutual extrusion and merging of shear layers. At Vr = 4, the FRC, URC, and LRC configurations with a corner radius of 0.5 all exhibit the 2S vortex shedding mode. As Vr increases, the vortex shedding mode for LRCs with a corner radius of 0.5 progressively becomes irregular. At Vr = 16, the vortex shedding mode for LRCs with a corner radius of 0.5 is the regular 2P + 2S mode.

A Low-Profile Circularly Polarized Millimeter-Wave Broadband Antenna Analyzed with a Link Budget for IoT Applications in an Indoor Scenario.

Broadband antennas with a low-profile generating circular polarization are always in demand for handheld/ portable devices as CP antennas counter multipath and misalignment issues. Therefore, a compact millimeter-wave antenna is proposed in this article. The proposed antenna structure comprises two circular rings and a circular patch at the center. This structure is further embedded with four equilateral triangles at a 90° orientation. The current entering the radiator is divided into left and right circular directions. The equilateral triangles provide the return path for current at the differential phase of ±90°, generating circular polarization. Structural development and analysis were initially performed through the characteristic mode theory. It showed that Modes 1 to 4 generated good impedance matching from 20 to 30 GHz and Modes 1 to 5, from 30 to 40 GHz. It also demonstrated the summation of orthogonal modes leading to circular polarization. The antenna-measured reflection coefficient |S11| > 10 dB was 19 GHz (23-42 GHz), and the axial ratio at -3 dB was 4.2 GHz (36-40.2 GHz). The antenna gain ranged from 4 to 6.2 dBi. The proposed antenna was tested for link margin estimation for IoT indoor conditions with line-of-sight (LOS) and non-line-of-sight (NLOS) conditions. The communication reliability with co- and cross-polarization was also studied under these conditions, and the results proved to be satisfactory.

On the computation of some code sets of the added Sierpinski triangle

In recent years, the intrinsic metrics have been formulated on the classical fractals. In particular, Sierpinski-like triangles such as equilateral, isosceles, scalene, added and mod$-3$ Sierpinski triangle have been considered in many different studies. The intrinsic metrics can be defined in different ways. One of the methods applied to obtain the intrinsic metric formulas is to use the code representations of the points on these self-similar sets. To define the intrinsic metrics via the code representations of the points on fractals makes also possible to investigate different geometrical, topological properties and geodesics of these sets. In this paper, we investigate some circles and closed sets of the added Sierpinski triangle and express them as the code sets by using its intrinsic metric.

Biomechanical Research of Three Parallel Cannulated Compression Screws in Oblique Triangle Configuration for Fixation of Femoral Neck Unstable Fractures.

Surgical treatment with internal fixation, specifically percutaneous fixation with three cannulated compression screws (CCSs), is the preferred choice for young and middle-aged patients. The mechanical advantage of the optimal spatial configuration with three screws provides maximum dispersion and cortical support. We suspect that the spatial proportion of the oblique triangle configuration (OTC) in the cross-section of the femoral neck isthmus (FNI) may significantly improve shear and fatigue resistance of the fixed structure, thereby stabilizing the internal fixation system in femoral neck fracture (FNF). This study aims to explore the mechanical features of OTC and provide a mechanical basis for its clinical application. Twenty Sawbone femurs were prepared as Pauwels type III FNF models and divided equally into two fixation groups: OTC and inverted equilateral triangle configuration (IETC). Three 7.3 mm diameter cannulated compression screws (CCSs) were used for fixation. The specimens of FNF after screw internal fixation were subjected to static loading and cyclic loading tests, respectively, with five specimens for each test. Axial stiffness, 5 mm failure load, ultimate load, shear displacement, and frontal rotational angle of two fragments were evaluated. In the cyclic loading test, the load sizes were 700 N, 1400 N, and 2100 N, respectively, and the fracture end displacement was recorded. Results were presented as means ± SD. Data with normal distributions were compared by the Student's t test. In the static loading test, the axial stiffness, ultimate load, shear displacement, and frontal rotational angle of two fragments were (738.64 vs. 620.74) N/mm, (2957.61 vs. 2643.06) N, (4.67 vs. 5.39) mm, and (4.01 vs. 5.52)° (p < 0.05), respectively. Comparison between the femoral head displacement after 10,000 cycles of 700N cyclic loading and total displacement after 20,000 cycles of 700-1400N cyclic loading showed the OTC group was less than the IETC group (p < 0.05). A comparison of femoral head displacement after 10,000 cycles of 1400N and 2100N cycles and total displacement after 30,000 cycles of 700-2100N cycles showed the OTC group was less than another group, but the difference was not significant (p > 0.05). When three CCSs are inserted in parallel to fix FNF, the OTC of three screws has obvious biomechanical advantages, especially in shear resistance and early postoperative weight-bearing, which provides a mechanical basis for clinical selection of ideal spatial configuration for unstable FNF.