Theorem 1. The nth derivative of the function 1/r along n constant (translation-invariant) vector fields in R coincides on the sphere with a spherical function of degree n. Any nonzero spherical function of degree n can be obtained by this construction from some n-tuple of nonzero vector fields. These n fields are uniquely defined by the function (up to multiplication by nonzero constants and p...

This study is devoted to the numerical solution of an inverse coefficient problem for a density dependent nonlinear reaction-diffusion equation. The method is based on approximating the unknown coefficient by polynomials. An optimal idea for solving the inverse problem is to minimize an error functional between the output data and the additional data. For this purpose, we find a polynomial of d...

We consider a sequence of independent and identicaly distributed (iid) random variables with absolutely continuous distribution function F(x) and probability density function (pdf) f(x). Let Rnl be the largest observation after observing nth record and R(ns) be the smallest observation after observing the nth record. Then we say Wnr = Rnl− R(ns), n > 1, as the nth record range. We will c...

We prove that if an nth degree rational Bézier curve has a singular point, then it belongs to the two (n− 1)th degree rational Bézier curves defined in the (n− 1)th step of the de Casteljau algorithm. Moreover, both curves are tangent at the singular point. A procedure to construct Bézier curves with singularities of any order is given.  2001 Elsevier Science B.V. All rights reserved.

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R0(2m) be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infmR0(2m). Define R1(m) to be the minimal root discriminant of totally real number fields of degree m and put α1 = lim infmR1(m). Assuming the Generalized Riemann Hypothesis, α...