one gets the first actual formula for 7r that mankind ever discovered, dating from 1593 and due to Frangois Viete (1540-1603), whose Latinized name is Vieta. (Was any notice taken of the formula's 400th anniversary, perhaps by the issue of a postage stamp?) From the samples of a function t(x) at equally spaced points x", n E z, one can reconstruct the complete function with the aid of sin x/x, ...

Elementary complex analysis and Hilbert space methods show that a union of at most n arcs on the circle is uniquely determined by the nth Fourier partial sum of its characteristic function. The endpoints of the arcs can be recovered from the coefficients appearing in the partial sum by solving two polynomial equations.

In this short note, we focus on the use of the generalized Kullback–Leibler (KL) divergence in the problem of non-negative matrix factorization (NMF). We will show that when using the generalized KL divergence as cost function for NMF, the row sums and the column sums of the original matrix are preserved in the approximation. We will use this special characteristic in several approximation prob...

An algorithm is considered, and shown to lead to various unusual and unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also partially characterized.

In this work, the authors develop new methods for the accurate and efficient solution of the two-dimensional, incompressible Euler equations in the vorticity form. Here, the velocity is recovered directly from the Biot-Savart relation with vorticity, and the vorticity is evolved through its transport equation. Using a generalized Poisson summation formula, the full asymptotic error expansion is...

In this paper we have studied the role of varying sampling fractions on relative bias of conventional ratio estimator and also for the linear combination of ratio and PPS estimators to the first and second degree of approximations for a wide variety of populations. It will give the survey practitioners an idea whether it is worthwhile to ignore the expressions of mean sum of squares to the orde...

The purpose of the paper is to develop a new method for obtaining explicit formulas for the error of approximation of bivariate functions by sums of univariate functions. It should be remarked that formulas of this type have been known only for functions defined on a rectangle with sides parallel to coordinate axes. Our method, based on a maximization process over closed bolts, allows the consi...

Approximations of 1/x by sums of exponentials are well studied for finite intervals. Here the error decreases like O(exp(−ck)) with the order k of the exponential sum. In this paper we investigate approximations of 1/x on the interval [1,∞). We prove estimates of the error by O(exp(−c√k)) and confirm this asymptotic estimate by numerical results. Numerical results lead to the conjecture that th...