Quadrilateral Geometry Research Articles

Competencies in using Sketchpad in Geometry Teaching and Learning: Experiences of preservice teachers

The subject of teacher competencies has been a key issue in mathematics education reform as the quality of an education system is fundamentally defined by the quality of its teachers. The study reported in this article attempted to identify and analyse displayed preservice teacher competencies and challenges encountered in early experiences with a Sketchpad-mediated learning of geometry. The instrumental approach to technology integration in the classroom as articulated by Trouche, Artigue and others was used as an analytical framework together with the van Hiele theory of geometrical thought development. A qualitative research approach was used to investigate preservice teachers' emerging competencies. Data were gathered through worksheet productions, lesson observations, open-ended questionnaires and an exit focus group interview. Twenty third-year mathematics major preservice teachers participated in workshop and microteaching sessions involving the use of the Geometer's Sketchpad dynamic geometry software in the teaching and learning of the geometry of quadrilaterals. The competencies displayed by the participants were described in terms of the instrumental theory for technology use and the van Hiele theory for geometric thought development. The competency levels showed that the preservice teachers had difficulties with computer hardware and software usage and the associated classroom organisation and management initially but gradually improved and gained confidence. The participants' own evaluations of their competencies affirmed that early experiences were unpleasant for many and barriers could be overcome by early exposure to computer environments. More computer resources need to be availed for a day-to-day integration to be sustainable.

1D chlorine bridged mercury(II) complex containing nitronyl nitroxide radicals: Synthesis, crystal structures, and magnetic properties

A novel 1D chlorine bridged mercury(II) complex containing nitronyl nitroxide radicals, [Hg2(NIT-1′-MeBzIm)Cl4] n (NIT-1′-MeBzIm = 2-{2′-[(1′-methyl)benzimidazolyl]}-4,4,5,5-tetrameth-ylimidazoline-1-oxyl-3-oxide), has been prepared and characterized by magnetic and single-crystal X-ray diffraction studies. The complex crystallizes in triclinic, space group P $\bar 1$ , Z = 2. Crystal data: C15H19C14Hg2N4O2, M = 830.32, a = 7.4454(16), b = 9.463(2), c = 16.180(3) A, α = 94.245(2)°, β = 100.747(2)°, γ = 102.523(2)°, V = 1085.5(4) A3, ρ c = 2.540 g/cm3, μ(MoKα) = 14.636 mm−1, F(000) = 762, R = 0.0312 and wR = 0.0779 for 3990 observed reflections with I > 2σ(I). The X-ray analysis reveals that the Hg2+ ions shows two coordination geometries of five-coordinated with a distorted square pyramidal coordination geometry and quadrilateral geometry. The units of [Hg2(NIT-1′-MeBzIm)Cl4] n were connected as 1D chain structure by the chlorine bridges. Magnetic investigation indicates the existence of weak intramolecular interactions is antiferromagnetic with J = −1.23 cm−1, where the spin Hamitonian is defined as H = $ - 2\sum\nolimits_{i,j} {J_{i,j} S_i S} $ within the complex.

求解不可压缩Navier-Stokes方程的GLS平面八节点单元

给出了求解不可压缩Navier-Stokes方程的GLS平面八节点单元的“单元长度”的定义和相应的逆估计常数，由此可计算出该单元的GLS稳定因子。给出的“单元长度”只依赖单元形状且易于计算；各种形状的直边四边形单元的逆估计常数的最大值由遗传算法得出。对Reynolds数为20,000的方腔上盖板流的数值模拟表明，本文所提出的单元长度和逆估计常数的正确性。 The definition of the element length and the inverse estimate constant of straight-edged quadrilateral element with eight nodes for solving incompressible Navier-Stokes equations was proposed, based on this, the stabilization factor was readily calculated. The element length was only dependent on the quadrilateral geometry and easy to compute, and the largest inverse estimate constant among various quadrilateral was obtained by means of the optimization method of genetic algorithm (GA). The numerical solutions of the classical lid-driven cavity flow problem were ob- tained for Reynolds number of 20,000 which demonstrated the validity of the element length and the inverse estimate constant proposed.

Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics

AbstractWe study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

Slurry Supplying Method for Large Quartz Glass Substrate Polishing

Chemical mechanical polishing (CMP) has been used in polishing a photomask substrate in flat-panel display (FPD) manufacture. Moreover, the quadrilateral geometry of quartz glass used as the photomask substrate has been enlarged. However, the slurry cannot flow throughout the glass surface evenly owing to the enlarged substrate covering the center of the platen in the polishing process. In this work, we verified the beneficial of spreading slurry into a non-inflow region by the reinforced suction of the slurry supplied to the polishing pad through a hole at center of the platen. A fluorescent agent was used instead of the slurry for flow visualization.

Investigation on Slurry Flow and Temperature in Polishing Process of Quartz Glass Substrate

A fluorescent agent was used instead of slurry to visualize the flow of slurry, the polishing liquid solution supplied in Chemical Mechanical Polishing (CMP). Under ultraviolet irradiation, the fluorescent agent emits light, which performs the flow of the slurry under the quartz glass, which is applied as a photomask substrate in Flat-Panel Display (FPD) manufacturing. The emitted light was observed with a digital camera. Compared to a round shape of quartz glass substrate, a large amount of fluorescent agent supplied was taken away by the straight edges of the quadrilateral geometry of the substrate and moved to the outer edge of a polishing pad placed on a platen. The fluorescent agent that was taken away by the quadrilateral geometry also remained as several uneven spiral patterns on the polishing pad when the revolution speeds both of the quadrilateral substrate and the platen, which are practically set at same or almost revolution speed, were not sufficient. Next, when the quadrilateral substrate rotated at high revolution speed (60 min-1), the fluorescent agent flowed most evenly and spread throughout the entire substrate area in the shortest time. Finally, a combination of thermocouples and a wireless signal transmission system is introduced to investigate the temperature in glass substrate CMP process.

Toric geometry of convex quadrilaterals

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kahler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kahler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kahler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.

Synthesis, Crystal Structure and Theoretical Calculation of Ni(II) Complex with N,N′‐1,2‐Cyclohexenylenebis (salicylideneimine)

A complex [Ni(C20H18N2O2)] [C20H18N2O2: N,N′‐1,2‐cyclohexenylenebis(salicylid‐eneimine)] has been synthesized and characterized by IR, elemental analysis and X‐ray diffraction single crystal analysis. The crystal crystallizes in monoclinic, space group P2(1)/c with a=10.998(2) Å, b=12.774(3) Å, c=12.370(3) Å, α=90°, β=95.31(3)°, γ=90°, V=1.7304(8)nm3, Z=4, F(000)=832, S=1.272, Dc=1.447 g · cm−3, R 1= 0.1027, wR 2=0.2136, μ=1.136 mm−1. The Ni(II) is four coordinated by two nitrogen and two oxygen atoms from the Schiff base ligand to furnish a planar quadrilateral geometry. The molecules are connected by hydrogen bonds to form a three‐dimensional network structure. The theoretical studies of the title complex were carried out by density functional theory(DFT) B3LYP method & 8226; CCDC: 287808

Theoretical study of electronic structures and spectroscopic properties of Ga 3Sn, GaSn 3, and their ions

Ground and excited states of mixed gallium stannide tetramers (Ga 3Sn, Ga 3Sn +, Ga 3Sn −, GaSn 3, GaSn 3 +, and GaSn 3 −) are investigated employing the complete active space self-consistent-field (CASSCF), density function theory (DFT), and the coupled-cluster single and double substitution (including triple excitations) (CCSD(T)) methods. The ground states of Ga 3Sn, Ga 3Sn +, and Ga 3Sn − are found to be the 2A 1, 3B 1, and 1A 1 states in C 2v symmetry with a planar quadrilateral geometry, respectively. The ground states of GaSn 3 and GaSn 3 − is predicted to be the 2A 1 and 1A 1 states in C 2v point group with a planar quadrilateral structure, respectively, while the ground state of GaSn 3 + is the 1A 1 state with ideal triangular pyramid C 3v geometry. Equilibrium geometries, vibrational frequencies, binding energies, electron affinities, ionization energies, and other properties of Ga 3Sn and GaSn 3 are computed and discussed. The anion photoelectron spectra of Ga 3Sn − and GaSn 3 − are also predicted. It is interesting to find that the amount of charge transfer between Ga and Sn 2 atoms in the 1A 1 state of GaSn 3 + greatly increases upon electron ionization from the 2A 1 state of GaSn 3, which may be caused by large geometry change. On the other hand, the results of the low-lying states of Ga 3Sn and GaSn 3 are compared with those of Ga 3Si and GaSi 3.

Electronic states of Ga 3Si, GaSi 3, and their ions

The equilibrium geometries and electronic states of Ga 3Si, GaSi 3, and their ions are investigated using the complete active space self-consistent-field (CASSCF) and DFT(B3LYP)/CCSD(T) techniques. The 2B 1, 3B 1, and 1A 1 states in C 2v symmetry with a planar quadrilateral geometry are found to be the ground states of Ga 3Si, Ga 3Si +, and Ga 3Si −, respectively. On the other hand, the ground states of GaSi 3, GaSi 3 − are also predicted to undergo Jahn–Teller distortion to the 2A″ and 1A′ states in C s with a distorted triangular pyramid geometry, respectively, whereas that of GaSi 3 + is found to be the 1A 1 state in C 3v with symmetric triangular pyramid structure. Binding energies, electron affinities, ionization energies of Ga 3Si and GaSi 3 are computed at the CCSD(T)/QCISD(T) level and discussed.

639 構造及び非構造格子での多次元有理関数を用いる移流スキーム(S04-1 ここまで来たCIP法とその将来(1),S04 ここまで来たCIP法とその将来)

This paper presents rational function based fully multi-dimensional advection schemes for structured and unstructured grids of a quadrilateral geometry. For the structured grid, a fully 2D rational interpolation function is constructed by combining two 1D sweeps based on the 1D rational function with 3 degrees of freedoms, and is convexity preserving. For the unstructured, the interpolation function is constructed according to the same procedure as in the structured grid after each control volume is mapped to a rectangle. Numerical tests of advection transport show that the constructed rational interpolation functions effectively eliminate the numerical oscillations and give well-regulated solutions.

Bounded geometry of quadrilaterals and variation of multipliers for rational maps

Bounded geometry of quadrilaterals and variation of multipliers for rational maps

Some analytical integration formulas for trapezoidal shape quadrilateral elements of Lagrange family

This paper concentrates on some analytical integration formulas of rational functions having higher order polynomial numerators with linear univariate denominators. These integrals arise in finite element formulation of second-order partial differential equations provided that a trapezoidal shape quadrilateral element is normalized w.r.t. 2-square in a local parametric two-dimensional space isoparametrically or subparametrically. Explicit evaluation of these double integrals can be expressed as a product of two single variate integrals which produce simple analytical finite element relations. An efficient and simple method is also proposed to find the global derivatives and their products. In addition, some formulas of the rational integrated expressions are tabulated for the computational purpose which can be easily coded and save much computational time. The performance is demonstrated by means of two numerical examples, the first example considers a discretization made up of a single trapezium-shaped element and the other example considers an arbitrary quadrilateral geometry which is further discretized by means of three trapezium-shaped quadrilateral elements. Copyright © 2000 John Wiley & Sons. Ltd.

An analysis of four-quark energies in SU(2) lattice Monte Carlo using the flux-tube symmetry

Energies of four-quark systems calculated by the static quenched SU(2) lattice Monte Carlo method are analyzed in 2 × 2 bases for square, rectangle, tilted rectangle, linear and quadrilateral geometry configurations and in 3 × 3 bases for a non-planar geometry configuration. For small interquark distances, a lattice effect is taken into account by considering perimeter dependent terms which are characterized by the cubic symmetry. It is then found that a parameter f - that can be identified as a gluon field overlap factor — is rather well described by the form exp(−[ b s EA+√ b s FP]), where A and P are the area and perimeter mainly defined by the positions of the four quarks, b s is the string constant in the 2-quark potentials and E,F are constants.

Static and dynamic deflections of plates of arbitrary geometry by a new finite difference approach

Finite difference solutions for the static and dynamic displacements of a plate undergoing vibration are presented. The approach presented differs from the conventional methods in which the derivatives are expressed by their difference equivalents. Here the difference equations are obtained as solutions to the fourth order biharmonic equation. A single space varying drive number is found which varies from node to node and characterizes the true mode shape of the plate at a node. The technique presented can be applied to finite elements of triangular, rectangular or quadrilateral geometry without any restriction. The Marcus difference solution is seen to be a particular case of the general difference solution derived in this paper to give static deflection. The analysis is confined to small deflections of plate but the technique can also be applied to appropriate differential equations for large deflections. The method has applications in the determination of the displacements of structures having arbitrary boundary conditions. Evaluation of complicated modal response associated with the traditional approach is thus avoided.

Analytical integral formulae related to convex quadrilateral finite elements

Analytical quadrature formulae for rational functions, integrated over finite elements with quadrilateral geometry, are presented definitively. For second order differential equations, these formulae produce algebraic finite element relations associated with the isoparametrical or subparametrical rectangular interpolations.