Triangular Domain Research Articles

Triangular Bernstein moment-based identification of algebraic curves

Extending our previous results, in this paper we present a theoretical improvement of a strategy for the identification of binary images with algebraic boundaries. Such identification is obtained from few samples and it is based on a representation of the image shape in terms of non-separable bivariate Bernstein polynomials piecewisely defined over triangular domains.

Substrate‐Dependent Anisotropy and Damping in Epitaxial Bismuth Yttrium Iron Garnet Thin Films

AbstractIron garnets that combine robust perpendicular magnetic anisotropy (PMA) with low Gilbert damping are desirable for studies of magnetization dynamics as well as spintronic device development. This paper reports the magnetic properties of low‐damping bismuth‐substituted iron garnet thin films (Bi0.8Y2.2Fe5O12) grown on a series of single‐crystal gallium garnet substrates. The anisotropy is dominated by magnetoelastic and growth‐induced contributions. Both stripe and triangular domains form during field cycling of PMA films, with triangular domains evident in films with higher PMA. Ferromagnetic resonance measurements show damping as low as 1.3 × 10−4 with linewidths of 2.7 to 5.0 mT. The lower bound for the spin‐mixing conductance of BiYIG/Pt bilayers is similar to that of other iron garnet/Pt bilayers.

Relaxation effects in twisted bilayer molybdenum disulfide: structure, stability, and electronic properties

Manipulating the interlayer twist angle is a powerful tool to tailor the properties of layered two-dimensional crystals. The twist angle has a determinant impact on these systems’ atomistic structure and electronic properties. This includes the corrugation of individual layers, formation of stacking domains and other structural elements, and electronic structure changes due to the atomic reconstruction and superlattice effects. However, how these properties change with the twist angle, θ, is not yet well understood. Here, we monitor the change of twisted bilayer (tBL) MoS2 characteristics as a function of θ. We identify distinct structural regimes, each with particular structural and electronic properties. We employ a hierarchical approach ranging from a reactive force field through the density-functional-based tight-binding approach and density-functional theory. To obtain a comprehensive overview, we analyzed a large number of tBLs with twist angles in the range of . Some systems include up to half a million atoms, making structure optimization and electronic property calculation challenging. For , the structure is well-described by a moiré regime composed of two rigidly twisted monolayers. At small twist angles ( and ), a domain-soliton regime evolves, where the structure contains large triangular stacking domains, separated by a network of strain solitons and short-ranged high-energy nodes. The corrugation of the layers and the emerging superlattice of solitons and stacking domains affects the electronic structure. Emerging predominant characteristic features are Dirac cones at K and kagome bands. These features flatten for θ approaching 0∘ and 60∘. Our results show at which range of θ the characteristic features of the reconstruction, namely extended stacking domains, the soliton network, and superlattice, emerge and give rise to exciting electronics. We expect our findings also to be relevant for other tBL systems.

Interface Structure and Doping of Chemical Vapor Deposition‐Grown MoS2 on 4H–SiC by Microscopic Analyses and Ab Initio Calculations

The interface structure and electronic properties of monolayer (1L) MoS2 domains grown by chemical vapor deposition on 4H–SiC(0001) are investigated by microscopic/spectroscopic analyses combined with ab initio calculations. The triangular domains are epitaxially oriented on the (0001) basal plane, with the presence of a van der Waals (vdW) gap between 1L–MoS2 and the SiC terraces. The high crystalline quality of the domains is confirmed by photoluminescence emission. Furthermore, a very low tensile strain (ε ≈ 0.03%) of 1L–MoS2, consistent with the small in‐plane lattice mismatch, and a p‐type doping of (0.45 ± 0.11) × 1013 cm−2, is evaluated by Raman mapping. Density functional theory (DFT) calculations of the MoS2/4H–SiC(0001) system are also performed, considering different levels of refinement of the model: 1) the simple case of the junction between Si‐terminated SiC and MoS2, showing a covalent bond between the Si–S atoms and n‐type doping of MoS2; 2) the complete passivation of Si dangling bonds with a monolayer (1 ML) of oxygen atoms, resulting in a vdW bond with dSi–S ≈ 3.84 Å bond length and p‐type doping of MoS2; and 3) partial (¼ ML and ½ ML) oxygen coverages of the 4H–SiC surface, resulting in intermediate values of dSi–S and doping behavior.

Bivariate Fractal Interpolation Functions on Triangular Domain for Numerical Integration and Approximation

The primary objectives of this paper are to present the construction of bivariate fractal interpolation functions over triangular interpolating domain using the concept of vertex coloring and to propose a double integration formula for the constructed interpolation functions. Unlike the conventional constructions, each vertex in the partition of the triangular region has been assigned a color such that the chromatic number of the partition is 3. A new method for the partitioning of the triangle is proposed with a result concerning the chromatic number of its graph. Following the construction, a formula determining the vertical scaling factor is provided such that the actual double integral coincides with the integral value calculated using fractal theory. Convergence of the proposed method to the actual integral value is proven with sufficient lemmas and theorems. Adequate examples are presented to illustrate the method of construction and to verify the value of double integration.

Toward Two-Dimensional Tessellation through Halogen Bonding between Molecules and On-Surface-Synthesized Covalent Multimers.

The ability to engineer sophisticated two-dimensional tessellation organic nanoarchitectures based on triangular molecules and on-surface-synthesized covalent multimers is investigated using scanning tunneling microscopy. 1,3,5-Tris(3,5-dibromophenyl)benzene molecules are deposited on high-temperature Au(111) surfaces to trigger Ullmann coupling. The self-assembly into a semi-regular rhombitrihexagonal tiling superstructure not only depends on the synthesis of the required covalent building blocks but also depends on their ratio. The organic tessellation nanoarchitecture is achieved when the molecules are deposited on a Au(111) surface at 145 °C. This halogen-bonded structure is composed of triangular domains of intact molecules separated by rectangular rows of covalent dimers. The nearly hexagonal vertices are composed of covalent multimers. The experimental observations reveal that the perfect semi-regular rhombitrihexagonal tiling cannot be engineered because it requires, in addition to the dimers and intact molecules, the synthesis of covalent hexagons. This building block is only observed above 165 °C and does not coexist with the other required organic buildings blocks.

Temperature dependence Raman spectroscopy studies of CVD grown few layer MoS2 triangular domains

The effect of thermal perturbations on different Raman-active phonon modes was investigated using temperature-dependent Raman Spectroscopy of few layers MoS2 triangular domains (FMTD). The MoS2 triangular domains were prepared using the chemical vapour deposition (CVD) method and were characterised using optical microscopy, scanning electron microscopy (SEM), X-ray diffraction, and micro Raman spectroscopy. The micro Raman spectra of few FMTD recorded at higher temperatures (above room temperature) and were analysed using a theoretical framework developed by taking hot phonon anharmonicity into account. Temperature-dependent responses for the E12g, A1g, and 2LA(M) modes soften as the temperature rises from 25 °C to 300 °C. The first-order temperature coefficients of MoS2 triangular domains, E12g, A1g, and 2LA(M) modes are −0.018 cm-1K−1, −0.015 cm-1K−1, and −0.017 cm-1K−1, respectively. The experimental results revealed that phonon–phonon kinematics played a dominant role in broadening and peak shifts in the Raman mode, confirming the anharmonic effect's prevalence. Three or four phonon decay processes were used to explain the boarding and redshift in the peak position of temperature-dependent Raman mode. Moreover, thermal management is so crucial to the performance of low-dimensional devices, our findings pave the way for improved design strategies in this field.

TheLpconvergence of Fourier series on triangular domains

AbstractWe proveLpnorm convergence for (appropriate truncations of) the Fourier series arising from the Dirichlet Laplacian eigenfunctions on three types of triangular domains in$\mathbb{R}^2$: (i) the 45-90-45 triangle, (ii) the equilateral triangle and (iii) the hemiequilateral triangle (i.e. half an equilateral triangle cut along its height). The limitations of our argument to these three types are discussed in light of Lamé’s Theorem and the image method.

Significance of mixed convective heat transfer model in an equilateral triangular enclosure subjected to cylindrical heated objects inside

This analysis articulately presents the impact of heat transfer along with associated characteristics inside an equilateral triangular enclosure. Aiming to focus on the significance of heat transfer, four heaters (w1,w2,w3,w4) and three cold walls (C1,C2,C3) are brought into notice in equilateral enclosure. The bottom wall of the enclosure moves with uniform velocity (U0̄) in its own axis while the inclined and corner cold walls of the enclosure are fixed. The position of heaters is considered in four quadrants of cylindrical wall while the segments are placed at the corners of enclosure. Mathematical expressions are developed for heaters and cold walls. The computational model is considered in two-dimensional triangular domain. Water liquid is treated as a working fluid (Pr=6.9). Buoyancy effect is incorporated into Navier–Stokes equations using Boussinesq estimation. The governing equations are explored numerically by using Finite Volume Method based ANSYS-Fluent implicit solver software. Comparison has been made for the current results with published work in limiting case. Excellent matched has been seen. The findings show that the heat transfer rate is noted maximum in third quadrant of cylindrical wall (w4) when compared with other walls against Reynolds number. The temperature is noted maximum at the top corner near the segment C3 when Reynolds number increases. The results have it possible applications in thermal insulation of buildings, heating and cooling processes in machineries, furnaces and controlling of fire in buildings etc.

An evaluation algorithm for [formula omitted]-Bézier triangular patches formed by convex combinations

An extension to triangular domains of the univariate q-Bernstein basis functions is introduced and analysed. Some recurrence relations and properties such as partition of unity and degree elevation are proved for them. It is also proved that they form a basis for the space of polynomials of total degree less than or equal to n on a triangle. In addition, it is presented a de Casteljau type evaluation algorithm whose steps are all linear convex combinations.

Approximation by λ-Bernstein type operators on triangular domain

In this paper, a new type of ?-Bernstein operators (Bwm,?g)(w,z) and (Bzn ?g)(w,z), their Products Pmn,?g (w,z), Qnm,?g (w, z), and their Boolean sums Smn,?g (w,z), Tnm,?g (w, z) are constructed on triangle h with parameter ? [1,1]. Convergence theorem for Lipschitz type continuous functions and a Voronovskaja-type asymptotic formula are studied for these operators. Remainder terms for error evaluation by using the modulus of continuity are discussed. Graphical representations are added to demonstrate the consistency of theoretical findings for the operators approximating functions on the triangular domain. Also, we show that the parameter ? will provide flexibility in approximation; in some cases, the approximation will be better than its classical analogue.

Hamiltonian model for water waves in a triangular domain

We present a Hamiltonian model for free surface water waves in a 2-D symmetric triangular domain with sides that have a 45∘ slope. The construction uses a new amplitude expansion for the free surface, and a Lagrangian formulation of the water problem that is based on a formal mechanical/geometrical analogy of the fluid motion to geodesic motion in a potential force field. We define an approximate Lagrangian by choosing a physically plausible generalized coordinate and obtain the Hamiltonian by the Legendre transform. Hamilton’s equations are presented in spectral form using a known explicit construction of the linear normal modes for the domain considered. A key step is the approximation of a Dirichlet–Neumann operator appearing in the kinetic energy.

Moiré Potential, Lattice Relaxation, and Layer Polarization in Marginally Twisted MoS2 Bilayers.

Artificially twisted heterostructures of semiconducting transition-metal dichalcogenides (TMDs) offer unprecedented control over their electronic and optical properties via the spatial modulation of interlayer interactions and structural reconstruction. Here we study twisted MoS2 bilayers in a wide range of twist angles near 0° using scanning tunneling microscopy/spectroscopy. We investigate the twist angle dependence of the moiré pattern, which is dominated by lattice reconstruction for small angles (<2°), leading to large triangular domains with rhombohedral stacking. Local spectroscopy measurements reveal a large moiré-potential strength of 100-200 meV for angles <3°. In reconstructed regions, we see a bias-dependent asymmetry between neighboring triangular domains, which we relate to the vertical polarization that is intrinsic to rhombohedral stacked TMDs. This viewpoint is further supported by spectroscopy maps and ambient piezoresponse measurements. Our results provide a microscopic perspective of this new class of interfacial ferroelectrics and can offer clues for designing novel heterostructures that harness this effect.

Growth of ultrathin Bi2Se3 films by molecular beam epitaxy

Bi 2 Se 3 is a widely studied 3D topological insulator having potential applications in optics, electronics, and spintronics. When the thickness of these films decreases to less than approximately 6 nm, the top and bottom surface states couple, resulting in the opening of a small gap at the Dirac point. In the 2D limit, Bi2Se3 may exhibit quantum spin Hall states. However, growing coalesced ultrathin Bi2Se3 films with a controllable thickness and typical triangular domain morphology in the few nanometer range is challenging. Here, we explore the growth of Bi2Se3 films having thicknesses down to 4 nm on sapphire substrates using molecular beam epitaxy that were then characterized with Hall measurements, atomic force microscopy, and Raman imaging. We find that substrate pretreatment—growing and decomposing a few layers of Bi2Se3 before the actual deposition—is critical to obtaining a completely coalesced film. In addition, higher growth rates and lower substrate temperatures led to improvement in surface roughness, in contrast to what is observed for conventional epitaxy. Overall, coalesced ultrathin Bi2Se3 films with lower surface roughness enable thickness-dependent studies across the transition from a 3D-topological insulator to one with gapped surface states in the 2D regime.

Equiareal Parameterization of Triangular Bézier Surfaces

Parameterization is the key property of a parametric surface and significantly affects many kinds of applications. To improve the quality of parameterization, equiareal parameterization minimizes the equiareal energy, which is presented as a measure to describe the uniformity of iso-parametric curves. With the help of the binary Möbius transformation, the equiareal parameterization is extended to the triangular Bézier surface on the triangular domain for the first time. The solution of the corresponding nonlinear minimization problem can be equivalently converted into solving a system of bivariate polynomial equations with an order of three. All the exact solutions of the equations can be obtained, and one of them is chosen as the global optimal solution of the minimization problem. Particularly, the coefficients in the system of equations can be explicitly formulated from the control points. Equiareal parameterization keeps the degree, control points, and shape of the triangular Bézier surface unchanged. It improves the distribution of iso-parametric curves only. The iso-parametric curves from the new expression are more uniform than the original one, which is displayed by numerical examples.

Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices

We have analysed the recently extended series for the number of self-avoiding walks (SAWs) that cross an L × L square between diagonally opposed corners. The number of such walks is known to grow as We have made more precise the estimate of based on additional series coefficients provided by several authors, and refined analysis techniques. We estimate that We have also studied the subdominant behaviour, and conjecture that CL(1)∼λSL2+bL+c⋅Lg, where and We implemented a very efficient algorithm for enumerating paths on the square and hexagonal lattices making use of a minimal perfect hash function and in-place memory updating of the arrays for the counts of the number of paths. Using this algorithm we extended and then analysed series for SAWs spanning the square lattice and self-avoiding polygons (SAPs) crossing the square lattice. These are known to also grow as The sub-dominant term λ b is found to be the same as for SAWs crossing the square, while the exponent for spanning SAWs and for SAPs. We have also studied the analogous problems on the hexagonal lattice, and generated series for a number of geometries. In particular, we study SAWs and SAPs crossing rhomboidal, triangular and square domains on the hexagonal lattice, as well as SAWs spanning a rhombus. We estimate that the analogous growth constant so an even more precise estimate than found for the square lattice. We also give estimates of the sub-dominant terms.

Iterates of q-Bernstein operators on triangular domain with all curved sides

Abstract In this article, Phillips-type Bernstein operators ( ℬ m , q t F ) ( t , s ) ({{\mathcal{ {\mathcal B} }}}_{m,q}^{t}F)\left(t,s) and ( ℬ n , q s F ) ( t , s ) ({{\mathcal{ {\mathcal B} }}}_{n,q}^{s}F)\left(t,s) , their products, and Boolean sum based on q-integer have been studied on a triangle with all curved sides. Furthermore, convergence of iterates of these operators have been analyzed using the weakly Picard operators technique and the contraction principle.

Magnetic triangular bubble lattices in bismuth-doped yttrium iron garnet

Magnetic bubbles have again become a subject of significant attention following the experimental observation of topologically nontrivial magnetic skyrmions. In recent work, tailoring the shape of the bubbles is considered a key factor for their dynamics in spintronic devices. In addition to the reported circular, elliptical, and square bubbles, here we observe triangular bubble domains in bismuth-doped yttrium iron garnet (Bi-YIG) using Kerr microscopy. The bubble domains evolve from discrete circular to latticed triangular and hexagonal shapes. Further, the orientation of the triangular bubbles in the hexagonal lattices can be flipped by decreasing the magnetic field. The sixfold in-plane magnetic anisotropy of Bi-YIG(111) crystal, which is presumably the mechanism underlying the triangular shape of the bubbles, is measured as 1179 erg/cm3. The study of the morphologies of topologically trivial bubbles in YIG offers insight into nontrivial spin textures, which is appealing for future spintronic applications.

A Numerical Algorithm for a Coupled Hyperbolic Boundary Value Problem

This paper develops an algorithm to find the quadruple [Formula: see text] solution to a double Goursat–Cauchy boundary value problem in a triangular domain. The functions [Formula: see text] and [Formula: see text] are coupled through the function [Formula: see text]. Numerical examples are provided to illustrate the algorithm. This work finds applicability in inverse spectral problems.

Exact sphere representations over Platonic solids based on rational multisided Bézier patches

The paper presents several possibilities for exact representation of the sphere obtained by composing rational multisided Bézier patches known as S-patches. We first derive a general method that utilizes the (inverse) stereographic projection and enables exact representation of a sphere section in terms of an S-patch over a regular polygon. Then, we apply the method to the faces of all five Platonic solids inscribed into the sphere. Depending on the Platonic solid, the obtained S-patches are defined over triangular, square or pentagonal domains and unify two previously known constructions based on triangular and tensor product Bézier patches.