Showing papers on "Trigonometric functions published in 2007"

Frequency analysis of functionally graded material cylindrical shells with various volume fraction laws

Abstract: In the current paper a frequency analysis is performed for functionally graded material (FGM) circular cylindrical shells. A comparative study of shell frequencies is given for algebraic polynomial, exponential, and trigonometric volume fraction laws. An FGM shell considered here is structured from two materials. Love's thin shell theory is utilized for strain-displacement and curvature-displacement relations. The Rayleigh-Ritz method is employed to derive the frequency equation in the form of eigenvalue problem. Natural frequencies are evaluated for a shell with simply supported edge conditions. The axial modal dependence is approximated by circular trigonometric functions. Theoretical results are compared with those available in the literature for the validity of the present methodology.

q-Extension of the Euler formula and trigonometric functions

Abstract: In this paper, we introduce a Daehee constant, the so-called q-extension of the Napier constant, and consider the Daehee formula associated with the q-extensions of trigonometric functions. That is, we derive the q-extensions of sine and cosine functions from our Daehee formula. Finally, we present the q-calculus related to the q-extensions of sine and cosine functions.

Local Discriminant Wavelet Packet Coordinates for Face Recognition

Abstract: Face recognition is a challenging problem due to variations in pose, illumination, and expression. Techniques that can provide effective feature representation with enhanced discriminability are crucial. Wavelets have played an important role in image processing for its ability to capture localized spatial-frequency information of images. In this paper, we propose a novel local discriminant coordinates method based on wavelet packet for face recognition to compensate for these variations. Traditional wavelet-based methods for face recognition select or operate on the most discriminant subband, and neglect the scattered characteristic of discriminant features. The proposed method selects the most discriminant coordinates uniformly from all spatial frequency subbands to overcome the deficiency of traditional wavelet-based methods. To measure the discriminability of coordinates, a new dilation invariant entropy and a maximum a posterior logistic model are put forward. Moreover, a new triangle square ratio criterion is used to improve classification using the Euclidean distance and the cosine criterion. Experimental results show that the proposed method is robust for face recognition under variations in illumination, pose and expression.

Synthesis of Bistable Compliant Four-Bar Mechanisms Using Polynomial Homotopy

Abstract: In this paper we formulate and solve the synthesis equations for a compliant four-bar linkage with three specified equilibrium configurations in the plane. The kinematic synthesis equations as for rigid-body mechanisms are combined with equilibrium constraints at the flexure pivots to form design equations. These equations are simplified by modeling the joint angle variables in the equilibrium equations using sine and cosine functions. Polynomial homotopy continuation is applied to compute all of the design candidates that satisfy these design equations, which are refined using a Newton-Raphson technique. A numerical example demonstrates design methodology in which the homotopy solver obtained eight real solutions. Two of them provide two stable and one unstable equilibrium, and hence, can be used as the prototype of bistable compliant mechanisms.

Further improved F-expansion method and new exact solutions of Kadomstev–Petviashvili equation

Abstract: In this paper, the generalized F-expansion method [Chen J, He HS, Yang KQ. Commun Theor Phys (Beijing, China) 2005;44:307] is improved and a further improved F-expansion method is proposed to seek more types of exact solutions of non-linear evolution equations. With the aid of symbolic computation, we choose the (3 + 1)-dimensional Kadomstev–Petviashvili equation to illustrate the validity and advantages of the proposed method. As a result, many new and more general solutions are obtained including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions. This method can also be applied to other non-linear evolution equations in mathematical physics.

Free vibration analysis of rotating tapered blades using Fourier-p superelement

Abstract: A numerically efficient superelement is proposed as a low degree of freedom model for dynamic analysis of rotating tapered beams. The element uses a combination of polynomials and trigonometric functions as shape functions in what is also called the Fourier-$p$ approach. Only a single element is needed to obtain good modal frequency prediction with the analysis and assembly time being considerably less than for conventional elements. The superelement also allows an easy incorporation of polynomial variations of mass and stiffness properties typically used to model helicopter and wind turbine blades. Comparable results are obtained using one superelement with only 14 degrees of freedom compared to 50 conventional finite elements with cubic shape functions with a total of 100 degrees of freedom for a rotating cantilever beam. Excellent agreement is also shown with results from the published literature for uniform and tapered beams with cantilever and hinged boundary conditions. The element developed in this work can be used to model rotating beam substructures as a part of complete finite element model of helicopters and wind turbines.

Comparison between the homotopy perturbation method and the sine-cosine wavelet method for solving linear integro-differential equations

Abstract: This paper compares the homotopy perturbation method with the sine-cosine wavelet method for solving linear integro-differential equations. From the computational viewpoint, the homotopy perturbation method is more efficient and easier than the sine-cosine wavelet method.

Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms

Abstract: Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family allows one to introduce symmetrized and antisymmetrized multivariate Fourier-like transforms involving the symmetric and antisymmetric multivariate sine and cosine functions.

Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method

Abstract: By using the extended hyperbolic auxiliary equation method, we present explicit exact solutions of the high-order nonlinear Schrodinger equation with the third-order and fourth-order dispersion and the cubic-quintic nonlinear terms, describing the propagation of extremely short pulses. These solutions include trigonometric function type and exact solitary wave solutions of hyperbolic function type. Among these solutions, some are found for the first time.

Hydroelastic vibrations of flexible rectangular tanks partially filled with liquid

Abstract: In this paper, the three-dimensional vibratory characteristics of flexible rectangular tanks partially filled with liquid are studied. The surface waves of the liquid are taken into account in the analysis. Both the bulging modes of the tank-wall vibration and the sloshing modes of the liquid oscillation are investigated. The vibrating modes of the liquid–tank system are divided into four distinct categories: double symmetric modes (SS); antisymmetric–symmetric modes (AS); symmetric–antisymmetric modes (SA) and double antisymmetric modes (AA). Each of these categories is separately investigated. The velocity potential of the liquid is analytically deduced by using a combination of the superposition method and the method of separation of variables. According to the liquid–tank interface conditions and the orthogonality of trigonometric functions, the coefficients in the solution of liquid velocity potential are expressed in the integral forms including the tank–wall dynamic deflection. A set of reasonable static beam functions is constructed as the admissible functions of the tank-wall vibration. The eigenfrequency equation of the liquid–tank system is derived by using a combination of the Rayleigh–Ritz method and the Galerkin method. Convergence study demonstrates the high accuracy and small computational cost of the proposed approach. Finally, some numerical results are presented for the first time. Copyright © 2006 John Wiley & Sons, Ltd.

Functions with derivatives given by polynomials in the function itself or a related function

Abstract: We construct the set of holomorphic functions S 1 = {f: Ωf ⊆ ℂ → ℂ} whose members have n-th order derivatives which are given by a polynomial of degree n+1 in the function itself. We also construct the set of holomorphic functions S 2 = {g: Ωg ⊆ ℂ → ℂ} whose members have n-th order derivatives which are given by the product of the function itself with a polynomial of degree n in an element of S 1. The union S 1 ∪ S 2 contains all the hyperbolic and trigonometric functions. We study the properties of the polynomials involved and derive explicit expressions for them. As particular results, we obtain explicit and elegant formulas for the n-th order derivatives of the hyperbolic functions tanh, sech, coth and csch and the trigonometric functions tan, sec, cot and csc.

Computational considerations in functional principal component analysis

Abstract: Computing estimates in functional principal component analysis (FPCA) from discrete data is usually based on the approximation of sample curves in terms of a basis (splines, wavelets, trigonometric functions, etc.) and a geometrical structure in the data space (L 2 spaces, Sobolev spaces, etc.). Until now, the computational efforts have been focused in developing ad hoc algorithms to approximate those estimates by previously selecting an efficient approximating technique and a convenient geometrical structure. The main goal of this paper consists of establishing a procedure to formulate the algorithm for computing estimates of FPCA under general settings. The resulting algorithm is based on the classic multivariate PCA of a certain random vector and can thus be implemented in the majority of statistical packages. In fact, it is derived from the analysis of the effects of modifying the norm in the space of coordinates. Finally, an application on real data will be developed to illustrate the so derived theoretic results.

Complex standard functions and their implementation in the CoStLy library

Abstract: The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function f and a given rectangular complex interval z, some interval w is computed that contains all function values of f in z. This is done by expressing the real and the imaginary part of f as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that w is the smallest rectangular interval containing the range of f.The algorithms presented in this article have been implemented in a Cpp class library called CoStLy (Complex Standard Functions License), which is distributed under the conditions of the GNU General Public License.

Floating-Point Trigonometric Functions for FPGAs

Abstract: Field-programmable circuits now have a capacity that allows them to accelerate floating-point computing, but are still missing core libraries for it. In particular, there is a need for an equivalent to the mathematical library (libm) available with every processor and providing implementations of standard elementary functions such as exponential, logarithm or sine. This is all the more important as FPGAs are able to outperform current processors for such elementary functions, for which no dedicated hardware exists in the processor. FPLibrary, freely available from www.ens-lyon.fr/LIP/Arenaire/, is a first attempt to address this need for a mathematical library for FPGAs. This article demonstrates the implementation, in this library, of high-quality operators for floating-point sine and cosine functions up to single-precision. Small size and high performance are obtained using a specific, hardware-oriented algorithm, and careful datapath optimisation and error analysis. Operators fully compatible with the standard software functions are first presented, followed by a study of several more cost-efficient variants.

The periodic wave solutions for the (2 + 1)-dimensional dispersive long water equations

Abstract: Periodic wave solutions expressed by Jacobi elliptic functions for the (2 + 1)-dimensional dispersive long water equations are obtained by using the extended F -expansion method. In the limit cases, the solitary wave solutions and the trigonometric function solutions for the equations are also obtained.

Derivative Polynomials for tanh, tan, sech and sec in Explicit Form

Abstract: The derivative polynomials for the hyperbolic and trigonometric tangent, cotangent and secant are found in explicit form, where the coecients are given in terms of the Stirling numbers of the second kind. As an application we evaluate some integrals and also give the reflection formula for the Polygamma function in explicit form.

Algebraic approach to absorbing boundary conditions for the Helmholtz equation

Abstract: Recent work has shown that designing absorbing boundary conditions through algebraic approaches may be a nice alternative to the continuous approaches based on a Fourier analysis. In this paper, an original algebraic technique based on the computation of small patches is presented for the Helmholtz equation. This new technique is not directly linked to the continuous equations of the problem, nor to the numerical scheme. These properties make this technique very convenient to implement in a domain decomposition context. The proposed algebraic absorbing boundary conditions are used in a non-overlapping domain decomposition method and are defined on the interface between the subdomains. An additional coarse grid correction is then applied to ensure full scalability of the domain decomposition method upon the number of subdomains. This coarse grid correction involves trigonometric functions defined on the interface between the subdomains. Numerical experiments are presented and illustrate the robustness and parallel efficiency of the proposed method for acoustics applications.

An elementary proof of the Wallis product formula for pi

Abstract: We give an elementary proof of the Wallis product formula for pi. The proof does not require any integration or trigonometric functions

On the Error-Free Realization of a Scaled DCT Algorithm and Its VLSI Implementation

Abstract: This brief is concerned with the efficient and error-free implementation of the order-8 Linzer-Feig (L-F) scaled discrete cosine transform (sDCT). We present a novel 3-D algebraic integer encoding scheme which maps the transform basis functions (transcendental functions such as cosine and tangent) with integer values, so that the quantization errors can be minimized and the cross-multiplications (in the signal path) can be avoided. This scheme also allows the separable computation of a 2-D DCT in multiplication-free, fast, and efficient architectures with a process rate of 80 mega-samples/sec. The proposed scheme also reduces the latency and the power consumption compared to previously employed designs for DCT implementations.

Magnetic field angular sensor with a full angle detection

Abstract: An integrated angular magnetic sensor apparatus for determining a magnetic field angle within two axes of a plane is formed on a substrate onto which two anisotropic magneto-resistive sensing elements and at least one magneto-resistive sensing element are fabricated. The two anisotropic magneto-resistive sensing elements are oriented such that the output voltages of a first and second of the anisotropic magneto-resistive sensing elements are a function of a first and second trigonometric function (a sine function) of the magnetic field angle to a reference axis. The at least one magneto-resistive sensing element on the substrate and having a fixed reference magnetization oriented with respect to the reference axis such that an output voltage of the at least one magneto-resistive sensing element provides a quadrant indicator for the magnetic field angle with respect to the reference axis. The quadrant indicator is a trigonometric function such as a sine or cosine function.

On bi-orthogonal systems of trigonometric functions and quadrature formulas for periodic integrands

Abstract: In this paper, quadrature formulas with an arbitrary number of nodes and exactly integrating trigonometric polynomials up to degree as high as possible are constructed in order to approximate 2π-periodic weighted integrals. For this purpose, certain bi-orthogonal systems of trigonometric functions are introduced and their most relevant properties studied. Some illustrative numerical examples are also given. The paper completes the results previously given by Szegő in Magy Tud Akad Mat Kut Intez Kozl 8:255–273, 1963 and by some of the authors in Annales Mathematicae et Informaticae 32:5–44, 2005.

A simpler and faster method for SVM implementation

Abstract: Space vector modulation is perhaps the common technique mostly applied to drive three-phase voltage-source inverters. During every switching period it calculates three duty cycles in order to generate a suitable pulse sequence. This paper presents a new, faster and, most important, simpler method to compute these time values without using either trigonometric functions or even Clarke or Park transformations. The result is a light-weight algorithm easier to implement in small digital signal processors or microcontrollers. The relationship between SVM and PWM is also explained.

Exact solutions of the Gross-Pitaevskii equation in periodic potential in the presence of external source

Abstract: Exact periodic solutions, solitonlike solutions, singular solitary, and singular trigonometric wave solutions of the time-dependent Gross-Pitaevskii equation (GPE) with elliptic function potential in the presence of external source are analyzed. A simple approach that applies equally to both attractive and repulsive time-dependent GPE and allows one to find an extensive list of explicit periodic solutions of the GPE in terms of the Jacobian elliptic functions is developed. In the limit as the elliptic modulus tends to unity or to zero, the linear solutions, in either the Jacobian elliptic cosine or the Jacobian elliptic function of third order, give solitonlike solutions, while the rational solutions in these elliptic functions lead to singular solitary or trigonometric wave solutions. The stability of these solutions is investigated numerically.