A basic family of iteration functions for polynomial root finding and its characterizations

Let p(x) be a polynomial of degree n?2 with coefficients in a subfield K of the complex numbers. For each natural number m?2, let L_m(x) be the m×m lower triangular matrix whose diagonal entries are p(x) and for each j=1,…,m−1, its jth subdiagonal entries are . For i=1,2, let L_m^i)(x) be the matrix obtained from L_m(x) by deleting its first i rows and its last i columns. L₁⁽¹⁾(x)≡1. Then, the function B_m(x)=x−p(x) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, B_m(x) coincides with Newton's and Halley's, respectively. The function B_m(x) is a member of S(m,m+n−2), where for any M?m, S(m,M) is the set of all rational iteration functions g(x) ∈ K(x) such that for all roots θ of p(x), then g(x)=θ+∑_i=m^Mγ_i(x)(θ−x)ⁱ, with γ_i(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gⁱ(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is . For B_m(x) we obtain . If all roots of p(x) are simple, B_m(x) is the unique member of S(m,m + n − 2). By making use of the identity , we arrive at two recursive formulas for constructing iteration functions within the S(m,M) family. In particular, the family of B_m(x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schröder, whose mth order member belongs to S(m,mn), m>2. The iteration functions within S(m,M) can be extended to any arbitrary smooth function f, with the uniform replacement of p^(j) with f^(j) in g as well as in γ_m(θ).     

Randomized incremental construction of simple abstract Voronoi diagrams in 3-space

We introduce the simple abstract Voronoi diagram in 3-space as an abstraction of the usual Voronoi diagram. We show that the 3-dimensional simple abstract Voronoi diagram of n sites can be computed in O(n²) expected time using O(n²) expected space by a randomized algorithm. The algorithm is based on the randomized incremental construction technique of Clarkson and Shor (1989). We apply the algorithm to some concrete types of such diagrams: power diagrams, diagrams under ellipsoid convex distance functions, and diagrams under the Hausdorff distance for sites that are parallel segments all having the same length.     

A weak Brun—Titchmarsh theorem for multiplicative functions

An important theorem of Shiu gives a (precise) bound for the average of values of multiplicative functions, of a certain class, over ‘short’ intervals. Here we obtain, by simple means, the above result of same qualitative order.     

Replenishment-Depletion Urn in Equilibrium

An urn has balls of colors C1 and C2. It is replenished (R) by balls of both colors and then depleted by (D) the same number; this constitutes a cycle. When R = D, the system is closed and equilibrium will be reached after many cycles. The ultimate distribution is found only when the replenishment is the same for each color. Asymptotic normal and asymptotic binomial distributions arise when the parameters reach extreme values. For the multicolor urn an expression is given for the correlation between the number of balls of any two colors.     

ON GREEN＇S FUNCTION FOR HYPERBOLIC-PARABOLIC SYSTEMS

We study the Green＇s function for a general hyperbolic-parabolic system, including the Navier-Stokes equations for compressible fluids and the equations for magnetohydrodynamics. More generally, we consider general systems under the basic Kawashima- Shizuta type of conditions. The first result is to make precise the secondary waves with subscale structure, revealing the nature of coupling of waves pertaining to different characteristic families. The second result is on the continuous differentiability of the Green＇s function with respect to a small parameter when the coefficients of the system are smooth functions of that parameter. The results significantly improve previous results obtained by the authors.     

On a condition for α-starlikeness

In this paper we give a sufficient condition for function to be α-starlike function and some of its applications. We use the techniques of convolution and differential subordinations.     

On k-uniformly close-to-convex functions of complex order

In this paper, we define and study some subclasses of analytic functions by using fractional derivative operator. Some interesting properties, coefficients problems and inclusion results of these classes are investigated. It is also shown that these classes are closed under convolution with convex functions and some applications are given.