Showing papers on "Trigonometric functions published in 2003"

A generic geometric transformation that unifies a wide range of natural and abstract shapes

Abstract: To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man-made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non-Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.

A Schur-Parlett Algorithm for Computing Matrix Functions

Abstract: An algorithm for computing matrix functions is presented. It employs a Schur decomposition with reordering and blocking followed by the block form of a recurrence of Parlett, with functions of the nontrivial diagonal blocks evaluated via a Taylor series. A parameter is used to balance the conflicting requirements of producing small diagonal blocks and keeping the separations of the blocks large. The algorithm is intended primarily for functions having a Taylor series with an infinite radius of convergence, but it can be adapted for certain other functions, such as the logarithm. Novel features introduced here include a convergence test that avoids premature termination of the Taylor series evaluation and an algorithm for reordering and blocking the Schur form. Numerical experiments show that the algorithm is competitive with existing special-purpose algorithms for the matrix exponential, logarithm, and cosine. Nevertheless, the algorithm can be numerically unstable with the default choice of its blocking parameter (or in certain cases for all choices), and we explain why determining the optimal parameter appears to be a very difficult problem. A MATLAB implementation is available that is much more reliable than the function funm in MATLAB 6.5 (R13).

The extended Jacobian elliptic function expansion method and its application in the generalized Hirota–Satsuma coupled KdV system

Abstract: In this paper an extended Jacobian elliptic function expansion method, which is a direct and more powerful method, is used to construct more new exact doubly periodic solutions of the generalized Hirota–Satsuma coupled KdV system by using symbolic computation. As a result, sixteen families of new doubly periodic solutions are obtained which shows that the method is more powerful. When the modulus of the Jacobian elliptic functions m →1 or 0, the corresponding six solitary wave solutions and six trigonometric function (singly periodic) solutions are also found. The method is also applied to other higher-dimensional nonlinear evolution equations in mathematical physics.

An introduction to basic Fourier series

Abstract: Foreword. Preface. 1: Introduction. 2: Basic Exponential and Trigonometric Functions. 3: Addition Theorems. 4: Some Expansions and Integrals. 5: Introduction of Basic Fourier Series. 6: Investigation of Basic Fourier Series. 7: Completeness of Basic Trigonometric Systems. 8: Improved Asymptotics of Zeros. 9: Some Expansions in Basic Fourier Series. 10: Basic Bernoulli and Euler Polynomials and Numbers and q-Zeta Function. 11: Numerical Investigation of Basic Fourier Series. 12: Suggestions for Further Work. Appendix A: Selected Summation and Transformation Formulas and Integrals. A.1. Basic Hypergeometric Series. A.2. Selected Summation Formulas. A.3. Selected Transformation Formulas. A.4. Some Basic Integrals. Appendix B: Some Theorems of Complex Analysis. B.1. Entire Functions. B.2. Lagrange Inversion Formula. B.3. Dirichlet Series. B.4. Asymptotics. Appendix C: Tables of Zeros of Basic Sine and Cosine Functions. Appendix D: Numerical Examples of Improved Asymptotics. Appendix E: Numerical Examples of Euler-Rayleigh Method. Appendix F: Numerical Examples of Lower and Upper Bounds. Bibliography. Index.

Optimal designs for estimating individual coefficients in Fourier regression models

Abstract: In the common trigonometric regression model, we investigate the optimal design problem for the estimation of the individual coefficients, where the explanatory variable varies in the interval $[-a,a]$, $0

Numerical Evaluation of Uniform Beam Modes

Abstract: The equation for calculating the normal modes of a uniform beam under transverse free vibration involves the hyperbolic sine and cosine functions. These functions are exponential growing without bound. Tables for the natural frequencies and the corresponding normal modes are available for the numerical evaluation up to the 16th mode. For modes higher than the 16th, the accuracy of the numerical evaluation will be lost due to the round-off errors in the floating-point math imposed by the digital computers. Also, it is found that the functions of beam modes commonly presented in the structural dynamics books are not suitable for numerical evaluation. In this paper, these functions are rearranged and expressed in a different form. With these new equations, one can calculate the normal modes accurately up to at least the 100th mode. \IMike’s Arbitrary Precision Math\N, an arbitrary precision math library, is used in the paper to verify the accuracy.

A SVM-3D generalized algorithm for multilevel converters

Abstract: A novel three dimensional space vector algorithm of multilevel converters for compensating harmonics and homopolar component in system with neutral is presented. This generalized method provides an online computation of the nearest switching vectors sequence to the reference vector and calculates the on-state durations of the respective switching state vectors without involving trigonometric functions, look-up tables or coordinate system transformations which increase the computational load corresponding to the modulation of multilevel converters. The low computational cost of the proposed method is always the same and it is independent of the number of levels of the converter. The conventional 2D space vector algorithms are particular cases of the proposed generalized modulation algorithm. The algorithm provides the switching sequence that minimizes the total harmonic distortion and the commutation number of the semiconductor devices.

A trapezoidal Fourier p-element for membrane vibrations

Abstract: A trapezoidal Fourier p-element for the analysis of membrane transverse vibrations is investigated. Trigonometric functions are used as shape functions instead of polynomials to avoid ill-conditioning problems. The element matrices are analytically integrated in closed form. With the enrichment degrees of freedom in Fourier series, the accuracy of natural frequencies obtained is increased in a stable manner. One element can predict many modes accurately. Since a triangle can be divided into three trapezoidal elements, the range of application is wider than the previously derived rectangular Fourier p-element. The natural modes of a square membrane consisting of two trapezoidal elements are computed as test cases and convergence is very fast with an increasing number of trigonometric terms. Comparison of natural modes calculated by the trapezoidal Fourier p-element and the conventional finite elements is carried out. The results show that the trapezoidal Fourier p-element produces higher accurate natural frequencies than the conventional finite elements with the same number of degrees of freedom.

Subspaces with Equal Closure

Abstract: We take a new and unifying approach toward polynomial and trigonometric approximation in topological vector spaces used in analysis on R n . The idea is to show in considerable generality that in such a space a module, which is generated over the polynomials or trigonometric functions by some set, necessarily has the same closure as the module which is generated by this same set, but now over the compactly supported smooth functions. The particular properties of the ambient space or generating set are, to a large degree, irrelevant for these subspaces to have equal closure. This translation—which goes in fact beyond modules—allows us, by what is now essentially a straightforward check of a few properties, to replace many classical results in various spaces by more general statements of a hitherto unknown type. Even in the case of modules with one generator the resulting theorems on, e.g., completeness of polynomials are then significantly stronger than the classical statements. This extra precision stems from the use of quasi-analytic methods (in several variables) rather than holomorphic methods, combined with the classification of quasi-analytic weights. In one dimension this classification, which then involves the logarithmic integral, states that two well-known families of weights are essentially equal. As a side result we also obtain an integral criterion for the determinacy of multidimensional measures which is less stringent than the classical version. The approach can be formulated for Lie groups and this interpretation then shows that many classical approximation theorems are “actually” theorems on the unitary dual of R n , thus inviting to a change of paradigm. In this interpretation polynomials correspond to the universal enveloping algebra of R n and trigonometric functions correspond to the group algebra. It should be emphasized that the point of view, combined with the use of quasi-analytic methods, yields a rather general and precise ready-to-use tool, which can very easily be applied in new situations of interest which are not covered by this paper.

Investigation on Solutions of Cubic Equations with One Unknown

Abstract: In this paper, we investigated the cases of solutions of cubic equations with one unknown, and proved some formulas by using coefficients of equations, using trigonometric functions and using hyperbolic functions respectively, and we corrected some errors in references.

16 bit quadrature direct digital frequency synthesizer using interpolative angle rotation

Abstract: A direct digital synthesizer employs a trigonometric function generator utilizing decomposition of a larger angle into smaller sub-angles, interpolation of a desired sub-angle between two known angles and calculating the trigonometric function using complex arithmetic. The direct digital synthesizer has a phase accumulator to generate an angular increment signal of the output signal. A trigonometric function generator in communication with the phase accumulator receives the angle signal and from the angle signal creates the trigonometric function signal. An angle decomposing circuit is connected to receive the angle signal to separate the angle signal into sub-angles of the angular increment, a sum of the sub-angles equaling the angular increment. An interpolation circuit receives the smallest of the sub-angles to generate the trigonometric function for the smallest of the sub-angles by interpolating between the trigonometric function of two known angles. The direct digital synthesizer has a first angle trigonometric retaining for retaining the trigonometric functions of the known angles. At least one second angle trigonometric retaining circuit retains the trigonometric functions of for the remaining sub-angles. A complex arithmetic unit combines the interpolated trigonometric function and the second trigonometric function from each of the second angle trigonometric retaining circuits to create the trigonometric function.

Universal Long-Time Relaxation on Lattices of Classical Spins: Markovian Behavior on Non-Markovian Timescales

Abstract: The long-time behavior of certain fast-decaying infinite temperature correlation functions on one-, two-, and three-dimensional lattices of classical spins with various kinds of nearest-neighbor interactions is studied numerically, and evidence is presented that the functional form of this behavior is either simple exponential or exponential multiplied by cosine. Due to the fast characteristic timescale of the long-time decay, such a universality cannot be explained on the basis of conventional Markovian assumptions. It is suggested that this behavior is related to the chaotic properties of the spin dynamics.

Smooth local cosine based Galerkin method for scattering problems

Abstract: The smooth local trigonometric (SLT) functions are employed as the basis and testing functions in the Galerkin based method of moments (MoM), and sparse impedance matrices are obtained. The basic idea of SLT is to use smooth cutoff functions to split the function and to fold overlapping parts back into the intervals so that the orthogonality of the system is preserved. Moreover, by choosing the correct trigonometric basis, rapid convergence in the case of smooth functions is ensured. The SLT system is particularly suitable to handle electrically large scatterers, where the integral kernel behaves in a highly oscillatory manner. Numerical examples demonstrate the scattering of electromagnetic waves from two-dimensional objects with smooth contours as well as with sharp edges. A comparison of the new approach versus the traditional MoM and wavelet methods is provided.

Symbolic computation and observable effect for the (2 + 1)-dimensional symmetric regularized-long-wave equation from strongly magnetized cold-electron plasmas

Abstract: In this paper, with the aid of computerized symbolic computation, we use the generalized hyperbolic-function method to obtain new families of exact analytic solutions for the (2 + 1)-dimensional symmetric regularized-long-wave equation. This equation describes weakly nonlinear ion-acoustic and space-charge waves in strongly magnetized cold-electron plasmas. The families we obtain consist of solitary waves and trigonometric functions. We outline an observable (2 + 1)-dimensional effect that could be of interest to future experiments on space and laboratory plasma systems. The usage of the Wu elimination method has also been addressed.

Automorphic Orthogonal and Extremal Polynomials

Abstract: It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szego polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szego polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients etc. are derived.

Two-dimensional viscoelastic vibration by analytic Fourier p-elements

Abstract: Three new Fourier p -elements of rectangular, skew and trapezoidal shapes are given analytically for plane viscoelastic vibration problems. The natural frequencies of the plane viscoelastic structures with complex Young’s modulus are computed by a complex eigenvalue solver. With the additional Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly increased. Since trigonometric functions are used as enriching functions instead of polynomials in the proposal elements, the ill-conditioning problems associated with polynomials of higher degree in the traditional p -version finite element method are avoided. The two mapped plane coordinates in the Jacobian are uncoupled for trapezoidal elements whose element matrices can then be integrated analytically. A triangle can easily be divided into three trapezoids. Therefore, any plane viscoelastic problem with polygonal shape can be analyzed by a combination of rectangular and trapezoidal elements. Numerical examples show that the convergence of the present elements is very fast with respect to the number of trigonometric terms. The natural frequencies of several polygonal viscoelastic plates subject to in-plane vibration are presented.

A Fourier/Hopfield neural network for identification of nonlinear periodic systems

Abstract: This paper presents a method based on using a Hopfield neural network for the identification of nonlinear periodic systems. The system model is obtained by calculating the optimum coefficients of the expansion of the system over a set of Fourier basis functions. The identification process is accomplished using a "Fourier/Hopfield Neural Network". Fourier basis were chosen because they are best suited to analyzing periodic functions. Initially, the signals are expanded over their first harmonic Fourier base. A Hopfield neural network adapts the expansion coefficient till the relative error reaches a specified threshold value, at which point the neural network is approaching a local minimum. The global error is then computed, the number of Fourier basis is incremented by one, and the process is repeated till the global error becomes smaller than a desired minimum. The network would have then approached a global minimum. The Fourier/Hopfield Neural Network technique was applied to a nonlinear periodic function composed of sine and cosine waves of various powers. For a global and relative error threshold of 0.005, two basis functions were required with a final error of 0.004. Another simulation was run with a smaller error threshold of 10/sup -4/. Six basis functions were then needed to obtain a final error of the order of 10/sup -5/. In both simulations, the neural network converged to a global minimum, with a limited number of basis functions, showing thus the successful feasibility of using a Hopfield neural network in conjunction with Fourier analysis for the identification of nonlinear periodic systems.

Accelerating sine and cosine evaluation with compiler assistance

Abstract: Some software libraries add special entry points to enable both the sine and cosine to be evaluated with one call for performance purposes. We propose another method which does not involve new function names. By having the compiler front end recognize trigonometric function invocations, and replace them with a call to a common function which executes the code common to all the functions, followed by a short routine to produce the desired computation, it is possible to compute both the sine and cosine, when needed in about the same time as to compute only one of them.

New periodic solutions to a generalized Hirota-Satsuma coupled KdV system

Abstract: Using expansions in terms of the Jacobi elliptic cosine function and third Jacobi elliptic function, some new periodic solutions to the generalized Hirota-Satsuma coupled KdV system are obtained with the help of the algorithm Mathematica. These periodic solutions are also reduced to the bell-shaped solitary wave solutions and kink-shape solitary solutions. As special cases, we obtain new periodic solution, bell-shaped and kink-shaped solitary solutions to the well-known Hirota-Satsuma equations.

A black box identification in frequency domain

Abstract: Harmonic oscillations as integer multiples of the fundamental frequency in a power system are caused by nonlinear physical effects such as switching or saturation Modelling and detection of these harmonics are crucial for power system control and protection The present paper proposes the use of wavelet networks with smooth local trigonometric functions as activation functions A new algorithm is proposed, together with the use of the Cross Entropy function as a tool for evaluating the model quality The algorithm consists of recursive dual iterations with biorthogonal smooth local sine and cosine wavelet packets in order to calculate the adjustable parameters related to the activation functions The algorithm efficiently minimizes the Shannon Entropy function by adoptively choosing the best time-frequency cells on the wavelet packet tree During every loop the Cross Entropy function between estimated outputs and target outputs is checked A procedure by using trigonometric wavelet packets is proposed as an effective tool for disturbance detection, power quality analysis and non-linear harmonic circuit modelling Simulations of a converter bridge for traction drives are included to illustrate the effectiveness of the algorithm and the choice of the activation function

Fast Computation of Wigner-Ville Distribution

Abstract: The paper proposes a new method for computation of the Wigner-Ville distribution(WVD) taking account of the conjugate symmetry of the WVD kernel function and the periodicity and symmetry of the trigonometric function. The method transfers the computation of WVD into real field from complex field to remove the redundancies in the fast Fourier transform(FFT) computation. The realization of fast cosine operation and fast sine operation considerably reduces the computation cost. Theoretical analysis shows that the algorithm provided in the paper is much more advantageous than the existed ones.

Fast multi-dimensional scattered data approximation with Neumann boundary conditions

Abstract: An important problem in applications is the approximation of a function $f$ from a finite set of randomly scattered data $f(x_j)$. A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials $\{e^{2\pi i kx}\}$. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials $\cos (\pi kx)$ as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem.

New Trigonometric Identities and Generalized Dedekind Sums

Abstract: We obtain new trigonometric identities. We show that the coefficients of Laurent expansions of the identities give rise to the relation between special values of Hurwitz zeta function and Bernoulli numbers. Then we look into in detail the parameterized cotangent sums appearing in the identities.

Shading by Spherical Linear Interpolation using De Moivre's Formula

Abstract: In the classical shading algorithm according to Phong, the normal is interpolated across the scanline, requiring a computationally expensive normalization in the inner loop. In the simplified and faster method by Gouraud, the intensity is interpolated instead, leading to faster but less accurate shading. In this paper we use a third way of doing the interpolation, namely spherical linear interpolation of the normals across the scanline. This has been explored before, however, the shading computation requires the evaluation of a cosine in the inner loop and this is too expensive to be efficient. By reformulating the original approach in a suitable way, De Moivre’s formula can b e used d irectly for computing the intensity so that no no rmalization is needed. Hence, no trigonometric functions, divisions or square roots are necessary to compute in the inner loop. Unfortunately the setup for each scanline will be rather slow unless some efficient reformulation of the necessary trigonometric calculations can be found. We suggest this problem for future research.

Eisenstein Series in Complexified Clifford Analysis

Abstract: In this paper we deal with generalizations of the classical Eisenstein series within the framework of complexified Clifford analysis. In particular, we discuss generalizations of the trigonometric functions, the elliptic functions and modular forms within this function theory. In this paper we study their fundamental properties and focus on construction methods.

The new doubly-periodic solutions for nonlinear coupled scalar field equations(II)

Abstract: In this paper, we obtained five kinds of new doubly-periodic solutions by using sinh-Gordon equation expansion method for nonlinear coupled scalar field equations which show physical significance and application value. In degeneration, we can obtain the new solitary wave solutions and trigonometric function solutions. The results showed that the equation has plenty of constructions of solutions, the physical phenomena of the physical model which the equation described, will be the object many physicists and mathematicians further explore.