As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis...

This paper presents a piecewise cubic approximation method with approximate G1 continuity. For a given triangular mesh of points with arbitrary topology, one cubic triangular Bézier patch surface is constructed. The resulting surfaces have G1 continuity at the vertex points, but only requires approximate G1 continuity along the macro-patch boundaries so as to lower the patch degree. While our s...

We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of C. The first continuity result holds for any family of polynomial automorphisms of constant dynamical degree. The second result is the continuity of the upper exponent for families degenerating to a 1-dimensional map.

The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. We shall first observe that given α being smaller than the square root of the degree of polarization, the set of local Seshadri constants in the range (0, α] is finite. This in particular implies that the square root of the degree of polarization is the only possibl...

Beginning in 2004, the Department of Homeland Security (DHS) began to define and implement a national domestic all-hazards preparedness goal, intended to improve the nation’s preparedness for national catastrophes, including terrorist attacks. DHS’s approach was capabilities-based planning (CBP), adopted from the Department of Defense (DoD). This article illustrates several components important...

Let f(z) := ∑n ν=0 aνz ν be a polynomial of degree n having no zeros in the open unit disc, and suppose that max|z|=1 |f(z)| = 1. How small can max|z|=ρ |f(z)| be for any ρ ∈ [0 , 1)? This problem was considered and solved by Rivlin [4]. There are reasons to consider the same problem under the additional assumption that f ′(0) = 0. This was initiated by Govil [2] and followed up by the present ...

Given a Bézier curve of degree n, the problem of optimal multi-degree reduction (degree reduction of more than one degree) by a Bézier curve of degreem (m< n−1) with constraints of endpoints continuity is investigated. With respect to L2 norm, this paper presents one approximate method (MDR by L2) that gives an explicit solution to deal with it. The method has good properties of endpoints inter...

Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single parameter space. A general co...

This paper derives an approximation algorithm for multi-degree reduction of a degree n triangular Bézier surface with corners continuity in the normL2. The new idea is to use orthonormality of triangular Jacobi polynomials and the transformation relationship between bivariate Jacobi and Bernstein polynomials. This algorithm has a very simple and explicit expression in matrix form, i.e., the red...