Trigonometric Operations Research Articles

2D hybrid meshes for direct simulation Monte Carlo solvers

The efficiency of the direct simulation Monte Carlo (DSMC) method decreases considerably if gas is not rarefied. In order to extend the application range of the DSMC method towards non-rarefied gas regimes, the computational efficiency of the DSMC method should be increased further. One of the most time consuming parts of the DSMC method is to determine which DSMC molecules are in close proximity. If this information is calculated quickly, the efficiency of the DSMC method will be increased. Although some meshless methods are proposed, mostly structured or non-structured meshes are used to obtain this information. The simplest DSMC solvers are limited with the structured meshes. In these types of solvers, molecule indexing according to the positions can be handled very fast using simple arithmetic operations. But structured meshes are geometry dependent. Complicated geometries require the use of unstructured meshes. In this case, DSMC molecules are traced cell-by-cell. Different cell-by-cell tracing techniques exist. But, these techniques require complicated trigonometric operations or search algorithms. Both techniques are computationally expensive. In this study, a hybrid mesh structure is proposed. Hybrid meshes are both less dependent on the geometry like unstructured meshes and computationally efficient like structured meshes.

Norms of Eigenfunctions of Trigonometric KZB Operators

Norms of Eigenfunctions of Trigonometric KZB Operators

ON NORMS AND CERTAIN CHARACTERISTICS OF TRIGONOMETRIC APPROXIMATION BY BASKAKOV OPERATORS

The article covers actual question in the theory of approximations. It researches approximative opportunities of concrete approximating structures. In the article one of the actively studied in recent times types of approximating operators — Baskakov's trigonometric operators. Some characteristics of these operators are being investigated: norms and approximation constants, assessment and improved. In particular, the assessment of their difference is obtained.

Some studies on differential evolution variants for application to nuclear reactor core design

During the design of nuclear reactor cores, reactor cell parameters such as dimensions, enrichment and materials must be adjusted considering restrictions such as the average thermal flux, criticality and sub-moderation. Due to the complexity of this problem, trial-and-error methods do not necessarily guarantee acceptable solutions. Thus, the necessity of formulating the design of a nuclear reactor core as an optimization problem, where the decision variables are optimized taking constraints into account. Once the problem is formulated in such way, global search methods may be used in order to generate candidate solutions, that is, sets of core design parameters, to be analyzed by programs that simulate the neutron interactions in the reactor core. The Differential Evolution (DE) algorithm is one of those state-of-art global search methods, whose application has outperformed competitive algorithms in several optimization problems including the optimization problem of nuclear core design. Modifications in the DE such as the Trigonometric Mutation Operator and the Opposition-based learning have been reported in the literature, with considerable advances in the results of many problems. This performance motivated us to apply these two variants of differential evolution to the nuclear core design problem, which is the aim of this paper. The results are promising, demonstrating that the improvements are applicable in nuclear science and engineering applications.

Norms of Eigenfunctions of Trigonometric KZB Operators

Norms of Eigenfunctions of Trigonometric KZB Operators

Quantitative approximation by fractional smooth general singular operators

In this article we study the fractional smooth general singular integral operators on the real line, regarding their convergence to the unit operator with fractional rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the engaged function. Furthermore we produce a fractional Voronovskaya type result giving the fractional asymptotic expansion of the basic error of our approximation. We finish with applications to fractional trigonometric singular integral operators. Our operators are not in general positive.

Three Gorges Cascade Hydropower System Joint Simulative Optimal Operation Research in Power Market

The short-term joint optimal operation simulation of Three Gorges cascade hydropower system aiming at maximum power generation benefit is proposed. And a new method for optimizing cascade hydropower station based on Adaptive Genetic Algorithm (AGA) with trigonometric selective operators is presented. In this paper, the practical optimal operation in power market is described. The temporal-spatial variation of flow between cascade hydropower stations is considered, and time of use (TOU) power price is also taken into account. Moreover, a contrast between Tangent-roulette selection operator and traditional one is made. To a certain degree, the results of simulative optimal operation based on several representative hydrographs show that Tangent-roulette wheel selection operator can find a more excellent solution, because the Tangent-roulette one can overcome the fitness requirements of non-negative. The research achievements also have an important reference for the compilation of daily generation scheduling of Three Gorges cascade hydropower system in the environment of power market.

Approximation by families of linear trigonometric polynomial operators and smoothness properties of functions

AbstractThe error of approximation by families of linear trigonometric polynomial operators in the scale of Lp‐spaces of periodic functions with 0 < p ⩽ +∞ is characterized with the help of realization functionals associated with operators of multiplier type describing smoothness properties of functions. The results are formulated in terms of generators of the family and of the smoothness. Applications of the general scheme to approximation methods generated by classical kernels as well as some new constructions describing smoothness of odd orders via approximation are given. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

A gap between hyponormality and subnormality for block Toeplitz operators

This paper concerns a gap between hyponormality and subnormality for block Toeplitz operators. We show that there is no gap between 2-hyponormality and subnormality for a certain class of trigonometric block Toeplitz operators (e.g., its co-analytic outer coefficient is invertible). In addition we consider the extremal cases for the hyponormality of trigonometric block Toeplitz operators: in this case, hyponormality and normality coincide.

General theory of global smoothness and approximation by smooth singular operators

In this article we continue with the study of smooth general singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 ≤ p ≤ ∞ , by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of smoothness. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. We finish with applications to trigonometric singular integral operators.

Fast cell determination of the DSMC molecules in multi-stage turbo molecular pump design

In this study, an existing 2D parallel DSMC solver is modified, to analyze the multi-stage turbomolecular pumps more efficiently. Generally, molecule movements are traced cell-by-cell in DSMC solvers both in structured and unstructured meshes in order to determine which cell the DSMC molecule is positioned in. These calculations require time consuming trigonometric operations. If a nonrectangular physical domain can be converted into a rectangular computational domain using curvilinear coordinates, then it would be possible to calculate the DSMC molecule cell information not only in a very short time, but also with simple arithmetic operations. In this study, it is shown that the curvilinear coordinate technique is quite faster compared to cell-by-cell tracing technique. After that, the present 2D parallel DSMC solver is renewed to use implicit molecule indexing to shorten the calculation time even further. Thirdly, dynamically changing representative molecular ratios are used to decrease the statistical errors. Following that, molecule transfer method between computational domains is revised to employ different time steps and blade spacings. Finally, calculations are shown to be in close agreement with the previously published experimental results.

Global smoothness preservation and simultaneous approximation for multivariate general singular integral operators

In this article we continue with the study of multivariate smooth general singular integral operators over R N , N ≥ 1 , regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 ≤ p ≤ ∞ , by involving multivariate higher order moduli of smoothness. Also we study their multivariate simultaneous approximation to the unit operator with rates. The multivariate Jackson type inequalities obtained are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. In the uniform case of global smoothness we prove optimality. At the end we list the multivariate Picard, Gauss–Weierstrass, Poisson–Cauchy and Trigonometric singular integral operators as applicators of our general theory.

Trigonometric Solutions Using Sine Quadrant

In astronomical calculations, the sine quadrant is an important tool apart from the use of astrolabe (asturlab). In the Malay World, the use of the sine quadrant or also referred to as Rubuc Mujayyab is more significant compared to astrolabe. The sine quadrant is a quadrant-shaped plate, which contains grid lines of the scale of trigonometric functions, by which readings and solving trigonometric equations can be done. This equipment can also be used to measure the altitude of celestial and other landscape objects. Hence, the sine quadrant functions as a mathematical tool that can solve trigonometric calculations. The research presented in this paper relates to the influence of the use of mathematical tools of Islamic civilization in the Malay World, with special emphasis on mathematical calculations using the sine quadrant. With references to a number of manuscripts and printed material relating to pre modern astronomy, this study found that the steps in solving trigonometric calculation in astronomy are explained descriptively using the sine quadrant. The operations in this descriptive form are then matched with basic trigonometry in mathematics. Examples of solving trigonometric operations shown in this discussion, prove the effectiveness of the use of the sine quadrant.

Double Affine Hecke Algebras and Bispectral Quantum Knizhnik-Zamolodchikov Equations

We use the double affine Hecke algebra of type GL_N to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik's quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of q-difference equations acting on the central character of the principal series representations. We construct a meromorphic self-dual solution \Phi of BqKZ which, upon suitable specializations of the central character, reduces to symmetric self-dual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispectral problem for the Ruijsenaars' commuting trigonometric q-difference operators. Under this correspondence \Phi becomes a self-dual Harish-Chandra series solution \Phi^+ of the bispectral problem. Specializing the central character as above, we recover from \Phi^+ the symmetric self-dual Macdonald polynomials.

The region of analytic functions for the convergence of trigonometric interpolation

For r > 0 let A P ( D r ) denote the set of 2 π -periodic functions which are analytic on the closed rectangle D r = { z ∈ C : 0 ≤ Re z ≤ 2 π , | Im z | ≤ r } , and let A P [ 0 , 2 π ] = A P ( D 0 ) . For a positive integer n let Z n = { t n , 1 , t n , 2 , … , t n , n } be a set of nodes in the interval [ 0 , 2 π ) such that t n , 1 < t n , 2 < ⋯ < t n , n , and let T n Z f denote the trigonometric interpolation operator which interpolates f ∈ A P [ 0 , 2 π ] on the set Z n . Finally, set (∗) h = lim sup n → ∞ ‖ ∏ j = 1 n sin t − t n , j 2 ‖ n , where ‖ f ‖ denotes the maximum norm of continuous 2 π -periodic function f on the interval [ 0 , 2 π ] . In this paper, we define a function r : [ 1 2 , 1 ] → [ 0 , ∞ ) by r ( h ) to be the infimum of all numbers r > 0 such that the sequence ( T n Z f ) converges to f uniformly on [ 0 , 2 π ] for every f ∈ A P ( D r ) and every sequence of nodal sets Z n which satisfy equation (∗). The main results are summarized as ln ( 4 h − 1 ) ≤ r ( h ) ≤ min { 2 ln ( h + 1 + h 2 ) , ln ( 4 h 2 + 16 h 4 − 1 ) } .

Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.

Design of a Mathematical Unit in FPGA for the Implementation of the Control of a Magnetic Levitation System

This paper presents the design and implementation of an automatically generated mathematical unit, from a program developed in Java that describes the VHDL circuit, ready to be synthesized with the Xilinx ISE tool. The core contains diverse complex operations such as mathematical functions including sine and cosine, among others. The proposed unit is used to synthesize a sliding mode controller for a magnetic levitation system. This kind of systems is used in industrial applications requiring high level of mathematical calculations in small time periods. The core is designed to calculate trigonometric and arithmetic operations in such a way that each function is performed in a clock cycle. In this paper, the results of the mathematical core are shown in terms of implementation, utilization, and application to control a magnetic levitation system.

Fast Equal-Area Mapping of the (Hemi)Sphere using SIMD

We present a fast vectorized implementation of a transform that maps points in the unit square to the surface of the sphere, while preserving fractional area. The mapping uses the octahedral map combined with an equal-area parameterization and has many desirable features such as low distortion, straightforward interpolation, and fast inverse and forward transforms. Our SIMD implementation completely avoids branching and uses polynomial approximations for the trigonometric operations, as well as other tricks. This results in up to 9 times speed-up over a traditional scalar implementation. Source code is available online.

Huygens' principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators

Let a be an Euclidean vector space ofdimension N, and let k = (kα)α∈R be a multiplicity function related to a root system R. Let ∆(k )b e the trigonometric Dunkl-Cherednik differential-difference Laplacian. For (a, t) ∈ exp(a) × R, denote by uk(a, t) the solution to the wave equation ∆(k)uk(a, t )= ∂ttuk(a, t), where the initial data are supported inside a ball ofradius R about the origin. We prove that uk has support in the shell {(a, t) ∈ exp(a) × R || t |− R ≤ � log a �≤| t| + R} ifand only ifthe root system R is reduced, kα ∈ N for all α ∈ R, and N is odd starting from 3.

Accelerating rotation of high-resolution images

Real-time image rotation is an essential operation in many application areas such as image processing, computer graphics and pattern recognition. Existing architectures that rely on CORDIC computations for trigonometric operations cause a severe bottleneck in high-throughput applications, especially where high-resolution images are involved. A novel hierarchical method that exploits the symmetrical characteristics of the image to accelerate the rotation of high-resolution images is presented. Investigations based on a 512×512 image show that the proposed method yields a speedup of ∼20× for a mere 3% increase in area cost when compared with existing techniques. Moreover, the effect of hierarchy on the computational efficiency has been evaluated to provide for area–time flexibility. The proposed technique is highly scalable and significant performance gains are evident for very high-resolution images.