Equilateral Research Articles

Propagation properties of ring Airy beams array in a nonlinear media

The control of lasers is one of the most important subjects in optics. In this paper, we investigate the propagation behavior of three ring Airy beams (RABs) with an equilateral triangle arrangement. The visualized intensity distribution of RABs array exhibits two types of autofocus: 1. sudden autofocus, which can be applied in producing laser bullets; 2. triangle autofocus, which can be used as laser-tweezers. The properties of laser bullets are investigated. The results reveal that the center intensity (position) of laser bullets increases (decreases) in a power-law way versus initial RABs' intensity. Moreover, we discuss the range of parameters to produce laser-tweezers with the best quality. Since the position and power of the RABs-bullet and RABs-tweezers can be controlled with high precision, it has important application value in many fields such as laser targeted therapy of cancers, manipulation of microscopic objects, photoresist technology in chips and so on.

Research on the mathematical model construction of art and ideology in visual communication design teaching

Abstract Aims To enhance the teaching quality in the field of visual communication design and make students to improve in all aspects of moral, intellectual and physical development, strengthen their ideological education, improve their ideological quality and strengthen their understanding of visual communication design art. Methods By using mathematical methods to construct models, and making appropriate selection of the literature method, practical teaching method and investigation method, the author made a comprehensive study on the artistic problems in art teaching from the aspects of definition, function, characteristics and students’ ideological quality, and discusses potential solutions. Results It had a positive effect on teachers and on their teaching methods, so that they have a better understanding of their teaching work, to motivate themselves towards their goals, make the classroom teaching vivid and improve the teaching efficiency. Conclusion The art of teaching visual communication has a close relationship with the strength and quality of people's thoughts; so, we must strengthen the artistic quality of visual communication design education, to improve students’ ideological quality and the art of teaching visual communication design. In our model, the motivation of teachers to create and sustain interest in students studying visual communication design is stronger when the imagination of ideological content is broad, and when the artistry is positive for right triangle and thoughtful for backward right triangle. The construction of the model is like a regular triangle and an inverted triangle combined to form a regular rectangle mathematical model. It will make visual communication design teaching like Mona Lisa so that people linger.

On bifurcations and stability of central configurations in the planar circular restricted four-body problem

The restricted four-body problem is considered. That is, we consider motion of an infinitely small body (particle) under the Newtonian gravitational attraction of three bodies (primaries). It is assumed that the primaries move in circular orbits, forming a stable equilateral Lagrange triangle. It is supposed that four bodies move in a plane. There exist relative equilibriums of the particle in the rotating with the primaries coordinate system. In such an equilibrium the particle forms a central configuration with the primaries. In the case of small mass ofa primary the bifurcations ofthe central configurations are investigated, as well as conditions oflinear stability for the central configurations is obtained. The study is performed analytically by using small parameter method.

Viscoplastic flow in a pipe of complex cross section with stick–slip at the wall

The Poiseuille flow of a Bingham fluid with the yield slip condition at the wall is examined. The governing equations are solved employing the accelerated augmented Lagrangian method for both the viscoplastic model and the yield slip equation. We solve numerically the flow in pipes with different cross sections: equilateral and isosceles triangles, L-shape, and hourglass-shape. The combined effects of viscoplasticity and slip on the flow have been investigated. The evolution of the unyielded regions has been studied. The computations cover the full range of the two main dimensionless numbers and exhibit complex flow patterns: all the different flow regimes are identified in ducts of various cross sections.

The world unique triangular shaped Jonah High in the Levant Basin, eastern mediterranean – Tectonic setting, stratigraphy and exploration considerations

A clear magnetic anomaly is evident across the central part of the Southern Levant Basin, coinciding with prominent structural features known as the Jonah High and Jonah Ridge. Jonah High is a triangular shaped, horst-like structure, bounded at each of its three rims by segments of steep dips and is capped by a prominent seismic marker. An attempt to drill through the cap resulted in a loss of circulation in the Lower Miocene. The well was side-tracked to the east of the cap and penetrated a Lower Miocene and Upper Oligocene deep-water section, as found in other wells in the basin. Few models were suggested since the 1980's to explain the combined magnetic anomaly, gravity and oddly shaped structure; they include basement high, intrusive body and shallow volcanic edifice. The upper part of the high has been suggested to be a carbonate build-up of either Mesozoic or Cenozoic age or a shale structure. This study takes advantage of recently released 3D surveys acquired over the Jonah High and the results of the well. It is suggested that the Jonah High was intruded during the Lower to Middle Miocene by allochthonous bodies; these form an equilateral triangle, measuring 265 sq. km at 6.5 km depth, that might be world unique. The interior of the upper part of Jonah High, from about 5 km to 6.5 km, is suggested to be composed of a deep-water Oligocene and Lower Miocene section and probably contains the Tamar Sands, the main gas-bearing reservoir in the basin. The deep part exhibits folded and tilted high amplitude strata that may be partly composed of old magmatic material that upwelled during the Cenozoic by mobile sediment acting under the regional contractional regime; these might be associated with the observed magnetic and gravity anomalies. Further exploration by better imaging and drilling together with analogue cases from other basins are required in order to solve this enigmatic feature.

The effect of pore shape on the Poisson ratio of porous materials

A brief review is given of the effect of porosity on the Poisson ratio of a porous material. In contrast to elastic moduli such as K, G, or E, which always decrease with the addition of pores into a matrix, the Poisson ratio may increase, decrease, or remain the same, depending on the shape of the pores, and on the Poisson ratio of the matrix phase, . In general, for a given pore shape, there is a unique critical Poisson ratio, , such that the addition of pores into the matrix will cause the Poisson ratio to increase if , decrease if , and remain unchanged if . The critical Poisson ratio for spherical pores is 0.2, for prolate spheroidal pores is close to 0.2, and tends toward zero for thin cracks. For two-dimensional materials, for circular pores, 0.306 for squares, 0.227 for equilateral triangles, and again approaches 0 for thin cracks. The presence of a “trapped” fluid in the pore space tends to cause to increase, and for the range of parameters that may occur in rocks or concrete, this increase is more pronounced for thin crack-like pores than for equi-dimensional pores. Measurements of the Poisson ratio therefore may allow insight into pore geometry and pore fluid. If the matrix phase is strongly auxetic, small amounts of porosity will generally not cause the Poisson ratio to become positive.

Quantum billiards with correlated electrons confined in triangular transition metal dichalcogenide monolayer nanostructures

Forcing systems through fast non-equilibrium phase transitions offers the opportunity to study new states of quantum matter that self-assemble in their wake. Here we study the quantum interference effects of correlated electrons confined in monolayer quantum nanostructures, created by femtosecond laser-induced quench through a first-order polytype structural transition in a layered transition-metal dichalcogenide material. Scanning tunnelling microscopy of the electrons confined within equilateral triangles, whose dimensions are a few crystal unit cells on the side, reveals that the trajectories are strongly modified from free-electron states both by electronic correlations and confinement. Comparison of experiments with theoretical predictions of strongly correlated electron behaviour reveals that the confining geometry destabilizes the Wigner/Mott crystal ground state, resulting in mixed itinerant and correlation-localized states intertwined on a length scale of 1 nm. The work opens the path toward understanding the quantum transport of electrons confined in atomic-scale monolayer structures based on correlated-electron-materials.

Spectral Determinant on Euclidean Isosceles Triangle Envelopes

We study extremal properties of the determinant of Friederichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. Small-angle asymptotics show that the determinant grows without any bound as an angle of triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and find the critical value. Moreover, if the area of envelopes is not too large, then the determinant achieves its absolute minimum only on the equilateral triangle envelope and there are no other critical points, whereas for sufficiently large area the equilateral triangle envelope corresponds to a local maximum of the determinant.

Place cells and geometry lead to a flexible grid pattern.

Place cells and grid cells are important neurons involved in spatial navigation in the mammalian brain. Grid cells are believed to play an important role in forming a cognitive map of the environment. Experimental observations in recent years showed that the grid pattern is not invariant but is influenced by the shape of the spatial environment. However, the cause of this deformation remains elusive. Here, we focused on the functional interactions between place cells and grid cells, utilizing the information of location relationships between the firing fields of place cells to optimize the previous grid cell feedforward generation model and expand its application to more complex environmental scenarios. Not only was the regular equilateral triangle periodic firing field structure of the grid cells reproduced, but the expected results were consistent with the experiment for the environment with various complex boundary shapes and environmental deformation. Even in the field of three-dimensional spatial grid patterns, forward-looking predictions have been made. This provides a possible model explanation for how the coupling of grid cells and place cells adapt to the diversity of the external environment to deepen our understanding of the neural basis for constructing cognitive maps.

Algorithms to generate saturated random sequential adsorption packings built of rounded polygons.

We present the algorithm for generating strictly saturated random sequential adsorption packings built of rounded polygons. It can be used in studying various properties of such packings built of a wide variety of different shapes, and in modeling monolayers obtained during irreversible adsorption processes of complex molecules. Here, we apply the algorithm to study the densities of packings built of rounded regular polygons. Contrary to packings built of regular polygons, where the packing fraction grows with an increasing number of polygon sides, here the packing fraction reaches its maximum for packings built of rounded regular triangles. With a growing number of polygon sides and increasing rounding radius, the packing fractions tend to the limit given by a packing built of disks. However, they are still slightly higher, even for the rounded 25-gon, which is the highest-sided regular polygon studied here.

Domes over Curves

Abstract A closed piecewise linear curve is called integral if it is composed of unit intervals. Kenyon’s problem asks whether for every integral curve $\gamma $ in ${\mathbb{R}}^3$, there is a dome over $\gamma $, that is, whether $\gamma $ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when $\gamma $ is a quadrilateral, thus giving a negative solution to Kenyon’s problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular $n$-gons.

Transverse plane ultrasound-guided caudal epidural injections: sonographic anatomy and stepwise technique.

The present study aims to provide a step-by-step procedural and anatomical familiarization guide for transverse plane ultrasound (US)-guided caudal epidural (CE) injection. The study cohort consisted of 23 chronic low back pain patients (23-67years old) previously unresponsive to conservative management. A transverse plane US-guided CE injection was performed, with each procedure step documenting and emphasizing sonographic anatomy. Several Thiel's method fixed cadaveric specimen dissections were also performed to demonstrate relevant CE injection-related anatomy. The sacral hiatus location can be estimated by visually forming an equilateral triangle between the posterior superior iliac spines and the sacral apex (trigonum sacrale). Follow-up palpation locates the sacral cornua, guiding transducer placement visualizing over the paired cornua 'bull frog's eye's appearance, with the epidural space visualized as a hypoechoic line, between the eyes. Then, 2-3ml of 1% lidocaine is injected subcutaneously at the mid-point between the sacral cornua and superficial to the posterior sacrococcygeal ligament (SCL). Although keeping the cornua, superficial posterior SCL and epidural space in view, the needle is slowly advanced to the epidural space at around a 20 degree cephalad angle till the tip becomes visible. Expansion of the epidural space is monitored under the transverse sacral ligament as the injectant is slowly introduced. The present study demonstrated the anatomical landmarks necessary for the transverse ultrasound caudal epidural technique and that the cornua, superficial posterior SCL, CE space, and other relevant sacral hiatal anatomy are well visualized with this technique.

Analysis of Reachable Workspace for Spatial Three-Cable Under-Constrained Suspended Cable-Driven Parallel Robots

Abstract The workspace is an important reference for designs of cable-driven parallel robots (CDPRs). Most current researches focus on calculating the workspace of redundant CDPRs. However, few literature study the workspaces of under-constrained CDPRs. In this paper, the static equilibrium reachable workspaces (SERWs) of spatial 3-cable under-constrained CDPRs are solved numerically since expressions describing workspace boundaries cannot be obtained in closed forms. The steps to solve the SERWs are as follows. First, expressions that describe the SERWs and their boundaries are proposed. Next, these expressions are instantiated through a novel anchor points model composed of linear equations, quadratic equations, and tension limits in cables. Then, based on the reformulated linearization technique (RLT), the constraints system is transformed into a system containing only linear equality constraints and linear inequality constraints. Finally, the framework of the branch-and-prune (BP) algorithm is adopted to solve this system. The effect of the algorithm is verified by two examples. One is searching the SERWs boundaries of a special three-cable-driven parallel robot (CDPR) whose anchor points layouts both on the moving platform (MP) and on the base are equilateral triangles, followed by a method to extract the SERW boundary where cables do not interfere with each other. The other is searching the SERWs boundaries of a general three-cable CDPR with randomly selected geometry arrangement. The presented method in this paper is universal for calculating the SERWs of spatial three-cable CDPRs with arbitrary geometry parameters.

New Geometric Constants in Banach Spaces Related to the Inscribed Equilateral Triangles of Unit Balls

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of JL(X) and YJ(X) as well as the values of JL(X) and YJ(X) for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for JL(X), YJ(X) and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established.

Steady slip flow of Newtonian fluids through tangential polygonal microchannels

Abstract The concern in this paper is the problem of finding—or, at least, approximating—functions, defined within and on the boundary of a tangential polygon, functions whose Laplacian is $-1$ and which satisfy a homogeneous Robin boundary condition on the boundary. The parameter in the Robin condition is denoted by $\beta $. The integral of the solution over the interior, denoted by $Q$, is, in the context of flows in a microchannel, the volume flow rate. A variational estimate of the dependence of $Q$ on $\beta $ and the polygon’s geometry is studied. Classes of tangential polygons treated include regular polygons and triangles, especially isosceles: the variational estimate $R(\beta )$ is a rational function which approximates $Q(\beta )$ closely.

A Novel Approach To Predict Gas Flow in Entire Knudsen Number Regime through Nanochannels with Various Geometries

Summary Various unified gas flow (UGF) and apparent permeability models have been proposed to characterize the complex gas transport mechanisms in shale formations. However, such models are typically expressed as combinations of multiple gas flow mechanisms so that they cannot predict gas velocity profile. In this study, we develop a novel approach to predict the gas velocity profile in the entire Knudsen number (Kn) regime for circular and noncircular (i.e., square, rectangular, triangular and elliptical) nanochannels and investigate the effects of cross-sectional geometry on gas transport in nanochannels. To this end, a new UGF model is proposed to describe the gas flow behaviors in the entire Kn regime, considering the effects of gas slippage, bulk diffusion, Knudsen diffusion, surface diffusion, and cross-sectional geometry of flow channel. In addition, the boundary condition of the semianalytical second-order slip model applicable to various cross-sectional geometries is modified by adjusting the slip coefficients through the comparison between the proposed UGF model and the Navier-Stokes (N-S) equation with second-order slip boundary condition. As a result, the velocity profile of free gas in the entire Kn regime for the nanochannel with a specific cross section can be determined by solving the second-order slip model with adjusted slip coefficients via the finite element method. The results indicate that the geometry of the cross section has a significant influence on the mass flow rate and gas velocity profile in nanochannels. The predicted mass flow rates for the nanochannels with identical hydraulic diameter decrease with the cross-sectional geometry in the sequence as ellipse > equilateral triangle > rectangle > square > circle. However, the ranking of velocity profiles for such nanochannels, which is governed by the cross-sectional geometry, also varies with Kn. These findings indicate that the developed approach can predict the synergetic gas transport (i.e., gas slippage, bulk diffusion, Knudsen diffusion, and surface diffusion) and gas velocity profile in nanochannels with different cross-sectional geometries for a wide range of Kn, which gives insight into the characterization of gas flow behaviors in nanoporous shale.

A Four-Layer Odd Function Symmetrical Dual-Polarization Equilateral Right Triangle Slot Printed Broadband Directional Antenna for Wireless Lab Measurement Applications

A four-layer odd function symmetrical dual-polarization coplanar waveguide- (CPW-) fed equilateral right triangle slot printed broadband directional antenna for wireless lab measurement applications is presented. The proposed antenna consists of two stacked perpendicular odd function symmetrical equilateral right triangle slot antenna structures with two stacked reflector ground planes. It is miniaturized, simple, and easy to be fabricated. The measured results demonstrate that the proposed antenna can achieve wide impedance bandwidth that is almost 520% for |S11| and |S22|; VSWR ≦ 3 which has implemented the operating band from 1.37 to 7.12 GHz for Global Positioning System (GPS, 1.575 GHz), Wireless Local Area Networks band (2.4 GHz and 5.8 GHz), 6 GHz unlicensed spectrum (5.925 GHz∼7.125 GHz), Long Term Evolution Upper Band (LTE, 1710–2690 MHz), and sub-6 5 G band (3.5 GHz and 4.7 GHz) applications. The measured in-band isolation performance between the two input ports |S12| ≦ −17 dB is significantly achieved, and meanwhile, the antenna radiation pattern, peak gain, and efficiency of the proposed antenna are measured as well. In the end, the radiation pattern data are compared and analyzed with simulation results.

Precise Assembly and Supramolecular Catalysis of Tetragonal- and Trigonal-Elongated Octahedral Coordination Containers

The construction of distorted or irregular coordination polyhedrons with specific shapes and functionalities is highly challenging. Here, we demonstrate a viable strategy for attaining a severely d...

Stationary Points on Potential Energy Surface of Cyclic C3H3 with Coupled-Cluster Approaches and Density Functional Theory.

Cyclic C3H3 is the simplest cyclic hydrocarbon, but its configuration is complicated. The ground 2E″ state at the equilateral triangle geometry with D3h symmetry undergoes both Jahn-Teller (JT) distortion and pseudo-Jahn-Teller (PJT) distortion to structures with C3v, C2v, Cs, or C2 symmetry. Previous works using complete active space self-consistent field (CASSCF) or multiconfiguration SCF (MCSCF) differ on the characteristics of these structures. To clarify the characteristics of these stationary points on the potential energy surface (PES) of this radical, coupled-cluster methods CCSD, CCSD(T), and EOMEA/IP-CCSD as well as density functional theory are employed to calculate their geometries, harmonic frequencies, and relative energies. Deformations between these stationary points due to JT and PJT effects are analyzed in detail. Most of the results with these methods are consistent with each other, except for the b2 mode of the 2B1 state with C2v symmetry. CCSD and CCSD(T) provide an imaginary frequency for this vibrational mode, while it is calculated to be real with the other methods as well as with EOM-CCSDT-3. This may be related to unstable reference in CCSD and CCSD(T) calculations. On the other hand, most of the employed exchange-correlation functionals provide reliable results on the characteristics of these stationary points. Our results show that the 2A' state with Cs symmetry is the only minimum structure on the PES of cyclic C3H3. The 2A2 and 2B1 states of the C2v structure are second-order saddle points, while both the 2A state of the C2 structure and the 2A″ state of the Cs structure are transition states connecting the global minimum 2A' state.

Metric approximations of spectral triples on the Sierpiński gasket and other fractal curves

Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the Sierpiński gasket is the limit of finite graphs consisting of various affine images of an equilateral triangle. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise C1-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications.