Trigonometric Type Research Articles

Quadrature Formulas of Quasi-Interpolation Type for Singular Integrals with Hilbert Kernel

In this paper, we first establish quadrature formulas of trigonometric interpolation type for proper integrals of periodic functions with periodic weight, then we use the method of separation of singularities to derive those for corresponding singular integrals with Hilbert kernel. The trigonometric precision, the type, the estimate of the remainder, and the convergence of each quadrature formula derived here are also established.

First order analytic difference equations and integrable quantum systems

We present a new solution method for a class of first order analytic difference equations. The method yields explicit “minimal” solutions that are essentially unique. Special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic, trigonometric and elliptic type—Euler’s gamma function being of rational type. We study these generalized gamma functions in considerable detail. The scattering and weight functions (u- and w-functions) associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions. We obtain detailed information on these u- and w-functions, exploiting the difference equations they satisfy.

Some new solutions of the Yang-Baxter equation

We have found some new solutions of both rational and trigonometric types by rewriting the Yang-Baxter equation as a triple product equation in a vector space of matrices.

A CLASSICAL COMPLETELY INTEGRABLE EXTENDED SINH-CORDON SYSTEM ARISING FROM NON CONFORMALLY CONSTRAINED SL(2,R) WZNW MODEL (I)

By imposing conformally broken constraints on SL(2,R) WZNW model, we obtain a new classical completely integrable system which contains the famous sinh-Gordon system as a special case. The integrability of this extended sink-Gordon system is exhibited by the existence of the Lax pair with spectral parameters. And its funda-mental .Poisson-Lie brackets can be ultralocalized, corresponding classical r-matrix depends on two spetral parameters and satisfies the classical Yang-Baxter equation. After a suitable automorphic transformation of the loop algebra on which the above Lax pair takes value, such an ultralocalized r-matrix can be classified to the so-cal-led trigonometric type of r-matrices depending essentially on one spectral parameter.

A New Representation of Birman-Wenzl Algebra and Solution of Yang-Baxter Equation11The project partially supported by National Natural Science Foundation of China and the Local Natural Science Foundation of Xinjiang.

Corresponding to a new braid group representation, a representation of Birman-Wenzl algebra is provided. The trigonometric type solutions of spectral-dependent Yang-Baxter equation are obtained in the framework of the Yang-Baxterization, and the related vertex models are also mentioned.

Analytical solutions to classes of trigonometric and hyperbolic potentials with the Jacobi Polynomials

We have presented a new class of analytical solvable trigonometric and hyperbolic type potentials, related to the Jacobi Polynomials, using the previously unconsidered equations. It should be remarked that several quasi-exactly solvable potentials having restricted parameters have similar energy eigenvalue expressions.

Trigonometric series regression estimators with an application to partially linear models

Let μ be a function defined on an interval [ a, b] of finite length. Suppose that y 1, …, y n are uncorrelated observations satisfying E( y j ) = μ( t j ) and var( y j ) = σ 2, j = 1, …, n, where the t j 's are fixed design points. Asymptotic (as n → ∞) approximations of the integrated mean squared error and the partial integrated mean squared error of trigonometric series type estimators of μ are obtained. Our integrated squared bias approximations closely parallel those of Hall in the setting of density estimation. Estimators that utilize only cosines are shown to be competitive with the so-called cut-and-normalized kernel estimators. Our results for the cosine series estimator are applied to the problem of estimating the linear part of a partially linear model. An efficient estimator of the regression coefficient in this model is derived without undersmoothing the estimate of the nonparametric component. This differs from the result of Rice whose nonparametric estimator was a partial spline.

Some triple trigonometric series

SynopsisIn this paper the authors investigate two classes of triple series equations of trigonometric type. Closed form solutions are given for eight triple series in each class.

せん断形切りくず生成による工具振動の励起

In machining β-brass, it is well known that chip is produced discontinuously in a periodical manner and this type of chip formation often induces the forced chatter vibration of machine-tool. When the cutting system is rigid enough, it is found experimentally in orthogonal cutting that chip size and cutting force are not changed with variation of cutting speed, while the size and the maximum of the periodical force are proportional to undeformed chip thickness though the minimum force is kept unchanged. Assuming these characteristics to be applicable to the cutting with weaker tool system, where undeformed chip thickness and cutting speed vary dynamically in accordance with the tool deflection, it becomes possible to predict dynamic behavior of the system and occurrence of the forced chatter through a numerical analysis of vibration equation with usual lumped parameters which have to be measured separately. The analysis shows that frequency of the tool vibration is decreased by lowering the tool stiffness, and strong chatter of tool's natural frequency takes place and continues above a certain cutting speed. It is also shown that wave forms of the tool vibration and the finished surface change from saw-tooth type to irregular type, and further to trigonometric function type of the chatter vibration as cutting speed is increased. These results are all verified experimentally, however, the quantitative agreement is not satisfactory enough due to over-simplifications involved in the analysis.

Asymptotic Solutions of Integral Equations with Convolution Kernels

SummaryThe equations considered are Fredholm integral equations of the second kind with regular kernels, whose argument depends only on the difference of the variables. Approximate solutions are sought for a given finite range of the eigenvalues, and for large values of the range of integration. Certain special conditions are imposed on the general form of the Fourier transforms of the kernel. Then it is shown that approximate solutions may be obtained in terms of the solutions of the corresponding (singular) Wiener-Hopf equations. Approximations to the eigenvalues are also found. It is shown that the eigenfunctions are unique, and that except possibly near the end points of the range, the solutions are of trigonometric type with the zeros of successive solutions interlacing.

The Use of a Mechanical Synthesizer to Solve Trigonometric and Certain Types of Transcendental Equations, and for the Double Summations Involved in Patterson Contours"

The Use of a Mechanical Synthesizer to Solve Trigonometric and Certain Types of Transcendental Equations, and for the Double Summations Involved in Patterson Contours"

The Use of a Mechanical Synthesizer to Solve Trigonometric and Certain Types of Transcendental Equations, and for the Double Summations Involved in Patterson Contours

A method is shown whereby equations in trigonometric form and types of transcendental equations, which can be expanded into rapidly converging series, may be solved by a mechanical and graphical process. A mechanical and graphical method is also shown which effects the double summation of Fourier series used in the Patterson method for the determination of interatomic distances in crystals. Graphs are given to illustrate each of the processes discussed.