Triangle Approximation Research Articles

The Physical Application of a Triangle Approximation Model

The Physical Application of a Triangle Approximation Model

Specht Triangle Approximation of Large Bending Isometries

Specht Triangle Approximation of Large Bending Isometries

An attempt of identification of barium hydride molecular lines in sunspot umbral spectra

A high-resolution sunspot umbra spectrum recorded in National Solar Observatory, Kitt Peak in the visible and infrared wave number range 13, 600 – 25, 000cm−1 was taken in the present study for identifying the rotational lines of barium hydride (BaH) molecule. Number of chance coincidences was evaluated for the A 2Π1/2 – X 2Σ ((0,0), (1,1), (2,2), (1,0), (2,1)), A 2Π3/2 – X 2Σ ((0,0), (1,1), (2,2)), B 2Π1/2 – X 2Σ (0,0), B 2Π3/2 – X 2Σ (0,0), C 2Σ – X 2Σ ((1,1), (1,0), (2,2), (2,1), (3,2)) and D 2Σ – X 2Σ ((1,0), (2,0), (3,0), (4,0), (5,0), (8,0), (9,0)) band systems of BaH using line identification procedure. The obtained number of chance of coincidences was compared with I- parameter values. The highly resolved rotational lines were chosen to evaluate equivalent widths using triangle approximation method. The effective rotational temperatures were calculated for the bands (0,0), (1,1), (2,2) and (2,1) of A 2Π1/2 – X 2Σ, (0,0) (1,1) and (2,2) of A 2Π3/2 – X 2Σ, B 2Π1/2 – X 2Σ (0,0) and B 2Π3/2 – X 2Σ (0,0) of BaH molecule. The rotational temperature values calculated for these bands were found to be in the range 1185 – 3514 K. They were also compared with the already reported sunspot temperatures.

Comment on “Hearing the signal of dark sectors with gravitational wave detectors”

We revisit the calculation of the gravitational wave spectra generated in a classically scale-invariant $SU(2)$ gauge sector with a scalar field in the adjoint representation, as discussed by J. Jaeckel, et al. The finite-temperature potential at 1-loop level can induce a strong first-order phase transition, during which gravitational waves can be generated. With the accurate numerical computation of the on-shell Euclidean actions of the nucleation bubbles, we find that the triangle approximation employed by J. Jaeckel et al. strongly distorts the actual potential near its maximum and thus greatly underestimates the action values. As a result, the gravitational wave spectra predicted by J. Jaeckel et al. deviate significantly from the exact ones in peak frequencies and shapes.

Monte-Carlo ray tracing for axisymmetrical optical elements

A modification of the Monte-Carlo ray tracing procedure for axisymmetric surfaces is proposed. The main idea consists in the approximation of optical surfaces by truncated cones and the use of an unconventional k-d tree. The simulation results show high performance of the proposed method, with the ray tracing procedure working 3-12 times faster than conventional algorithms based on a triangle approximation.

Implicit Simulation for Three-Dimensional Spatial Morphology of Geological Body Based on Marching Cubes Algorithm

Due to the discrete and incomplete sample data for the geological body in geological investigation, it is difficult to construct the three-dimensional model of geological body. Therefore, based on data interpolation, we propose a method to implement the three-dimensional implicit simulation for the geological body by programming with marching cubes algorithm. Firstly, the spatial interpolation method is used to predict the distribution trend of geological body and the three-dimensional spatial regular data field for the geological body is established. Secondly, the three-dimensional isosurface of geological body is extracted from discrete points by the marching cubes algorithm. Finally, combining with three-dimensional visualization technology of OpenGL, three-dimensional spatial morphology of geological body is simulated by tiny triangle approximation in computer graphics. Taken the borehole data in exploration engineering as example, this method is applied to develop the three-dimensional implicit simulation for ore body. The result of implicit simulation is close to the delineation of explicit simulation by human-computer interaction in three-dimensional geological modeling software. It shows that this method is feasible and efficient for the three-dimensional simulation of geological body.

Adaptive simplification of huge sets of terrain grid data for geosciences applications

We propose and discuss a new Lepp-surface method able to produce a small triangular approximation of huge sets of terrain grid data by using a two-goal strategy that assures both small approximation error and well-shaped 3D triangles. This is a refinement method which starts with a coarse initial triangulation of the input data, and incrementally selects and adds data points into the mesh as follows: for the edge e having the highest error in the mesh, one or two points close to (one or two) terminal edges associated with e are inserted in the mesh. The edge error is computed by adding the triangle approximation errors of the two triangles that share e, while each L2-norm triangle error is computed by using a curvature tensor (a good approximation of the surface) at a representative point associated with both triangles. The method produces triangular approximations that capture well the relevant features of the terrain surface by naturally producing well-shaped triangles. We compare our method with a pure L2-norm optimization method.

Angle relations inside a bent waveguide for a wedge display

Owing to the physical limitations of the tools used to manufacture waveguides, a some amount of bending occurs. For a wedge display, such bending would cause problems for the final picture presented as the angles of light injection into the wedge are susceptible to changes in curvature. The right-angled triangle approximation shows that a 5° bending might be tolerable in the display, but significant distortion occurs for greater bending.

Consequences of Increasing Heart Rate on Deceleration Time, the Velocity–Time Integral, and E/A

The ascendancy of diastolic heart failure to "epidemic" proportions has increased the use of and reliance on Doppler echocardiography as a source for diagnosis and as the preferred method for determining indexes of diastolic function (DF). Current indexes are primarily derived from shape-based features of Doppler E and A waves, such as their amplitudes, slopes, durations, and areas. Load dependence and pathologic correlates of these indexes have been considered, but DF indexes are not routinely corrected for heart rate (HR). To determine the dependence of selected Doppler-derived indexes of DF on HR, transmitral Doppler flow velocities and electrocardiograms were simultaneously recorded during supine bicycle exercise in 21 young, healthy volunteers. Standard E- and A-wave shape-based indexes (acceleration time, deceleration time [DT], peak E, peak A) were measured using triangle approximation. Velocity-time integrals (VTIs) were calculated by trapezoidal and triangular approximations. A-wave peak velocity (A) was measured conventionally, relative to baseline, and also using 2 alternative methods: A*, measured relative to the E@A velocity, and Ac, relative to the E-wave deceleration value at peak A-wave velocity. E/A was calculated conventionally and by using A* and Ac. The results showed that DF indexes derived from individual E waves are essentially HR independent. DT showed a mere 20% decrease for a 100% increase in HR. A triangular approximation for the E-wave VTI and the corrected E/Ac were found to be nearly HR independent. In conclusion, on the basis of the established continuity of cardiac output as a function of increasing HR and the observed data, Doppler-derived indexes of DF (DT, VTIs, E/Ac) can be treated as essentially HR independent only if the VTI and A-wave peak are corrected for HR as described.

Money Metric, Consumer Surplus and Welfare Measurement

Abstract The paper provides a rigorous derivation of the `welfare triangle approximation' (WTA), which is at the centre of cost-benefit analysis. The result is generalized by showing that the WTA is one of two dual expressions, one of which approximates the change in real consumption, the other the change in the cost of living. The result is based on a correction of a proof attempted by Hicks. Many other derivations are also given, each based on a different definition of the theoretical functions to be approximated. The final result is the following: each of the empirical variations corresponds to a range of theoretical variations. The edges of the range are theoretical Laspeyres and Paasche variations which are approximated linearly; the interior region of the range is approximated quadratically; the centre of the range is replicated exactly by the empirical measures.

Money Metric, Consumer Surplus and Welfare Measurement

The paper provides a rigorous derivation of the welfare triangle approximation (WTA), which is at the center of cost benefit analysis. The result is generalized by showing that the WTA is one of two dual expressions, one of which approximates the change in real consumption, the other the change in the cost-of-living. The result is based on a correction of a proof attempted by Hicks. Many other derivations are also given, each based on a different definition of the theoretical functions to be approximated. The final result is the following: Each of the empirical variations corresponds to a range of theoretical variations. The edges of the range are theoretical Laspeyres and Paasche variations which are approximated linearly; the interior region of the range is approximated quadratically; the center of the range is replicated exactly by the empirical measures. Key Words: Cost-of-living, Consumer Surplus, Money Metric, Real Consumption, Welfare measurement

Cluster variation studies of the anisotropic exchange interaction model

The cluster variation method is applied to study critical properties of the Potts-like ferromagnetic anisotropic exchange interaction model. Phase transition temperatures, order parameter discontinuities and latent heats of the model on the triangular and the fcc lattices are determined by the triangle approximation; and those on the square and the sc lattices are determined by the square approximation.

Chiral order is a property of a directed bond

Abstract It is argued that the chiral order in a magnetic system is a property of a directed bond on a lattice, and the chiral order is discussed in terms of the distribution function for a pair of lattice sites. This fact is illustrated by the calculation of the chiral order in the triangle approximation in the cluster variation method, for the six- and twelve-clock models on the triangular lattice.

Kinetics of crystal growth on the solid-on-solid model

The growth kinetics on the (001) face of a hexagonal crystal from vapor is investigated on the solid-on-solid (SOS) model by dynamical cluster approximations. First the analysis is made in the layerwise treatment using the triangle approximation of the path-probability method (PPM) proposed by Kikuchi. Compared with the results of the point and the pair approximations, the region of the unphysical metastable states for small driving force L ( L is the chemical potential difference Δμ between vapor and solid divided by k B T) is reduced considerably in the present approximation, and the results are in good quantitative agreement with the Monte Carlo simulation data at temperatures slightly above the roughening point. For large driving forces, the present results deviate a little from the Monte Carlo simulation data. To improve this point, an assembly of columns is treated directly from the beginning, as a complementary method to the layerwise method. The results of the column method agree well with the Monte Carlo simulation data for the range of large driving forces. The behavior of the system at lower temperatures than the roughening point is also examined and the exponential dependence of the growth rate characteristic of nucleation is observed in the triangle approximation.

Reformulation of the path probability method and its application to crystal growth models

The microscopic master equation of a system is derived within the framework of the path probability method (PPM). Then, by extending Morita's method in equilibrium statistical mechanics, the path probability function constructed microscopically can be systematically decomposed to result in the conventional path probability function of cluster approximation when correlations larger than the chosen basic cluster are neglected. In order to critically compare the master equation method with the PPM, the triangle approximation is treated by both methods for crystal growth models. It is found that the PPM gives physically satisfactory kinetic equations, while the master equation (supplemented with a cluster probability in the superposition approximation) does not. The triangle PPM calculation considerably improves the result of the pair approximation for crystal growth velocity in the solid-on-solid model, and compares well with Monte Carlo results.

Derivation of the master equation from the path probability method

The master equation for a ferromagnetic Ising model with N spins is derived in the triangle approximation by the use of the path probability method (PPM) proposed by Kikuchi. It is shown that an equilibrium solution of the master equation gives a distribution function expected from the cluster variation method (CVM) of equilibrium statistical mechanics in the same stage of approximation. The path integral form for the two-gate probability is also derived for the present model and is compared with that of Kikuchi.

The measurement of gas residence time distribution in a pressurized fluidized-bed combustor using 41Ar as radiotracer

Applying 41Ar as tracer the residence time distributions of gas in a pressurized fluidized-bed combustor operating at up to 8.4 bar and 980°C have been measured. The tracer amount necessary for a measurement was estimated using a triangle approximation of the mentioned distribution. Good reproducibility was obtained for mean residence time t ̄ and somewhat less good for the variance of the residence time distribution [ σ 2( θ)]. Both t ̄ and 1 σ 2(θ) decreased with increasing pressure and bed content. At 8 bar the behaviour of the combustor approaches that of a perfect mixer.

Time evolution of fluctuations in the path-probability method. II.

Comparison of the path-probability method (PPM) with the master-equation method (MEM) is extended to a higher degree of approximation (triangle approximation) for the kinetics of Ising ferromagnetism. The evolution equations of the averages and of the fluctuations around them are derived analytically in this high degree of approximation for the first time for both methods. Contrary to the case of pair approximation presented in Ishikawa et al. (the preceding paper), it was not possible to bring the evolution equations derived by the MEM into a complete agreement with those by the PPM even with the use of the most reasonable closure relations.

Time evolution of fluctuations in the path-probability method. I.

The path-probability method (PPM) is applied for the first time to the time development of the fluctuation from the most probable path, where the kinetic Ising ferromagnetism in the pair approximation is treated as an example. The evolution equations of the variance as well as the average motion of order parameters are derived and compared critically with the results obtained by the master-equation method (MEM). In the pair approximation, the PPM results agree completely with those of the MEM. This agreement is to be compared with the disagreement in the triangle approximation presented in Wada et al. (the following paper).

Theory of atomic ordering in copper-platinum alloys

An expression is obtained for the energy of a binary system with an hcp lattice using a many-particle model. This expression is used for calculating regions of relative stability of modifications of the copper-platinum alloy. The effect of a many-particle interatomic interaction on the order-disorder phase equilibrium diagram and on order parameters is studied. Agreement between theory and experiment has been achieved for the first time for the phase diagram as well as for short-range order parameters in the two spheres of copper-platinum alloy. It is shown, within the framework of the model used, that such a comprehensive agreement between theory and experiment is possible only if many-particle interaction and the trigonal character of L11 phase are taken into account. Calculations were performed with the help of the multicluster version of the Kikuchi method in the tetrahedron + triangle approximation. Results obtained clarify significantly the Understanding of the interatomic interaction in copper-platinum alloy.