平面图的圈基内插性质

图Ｇ的一个圈基的长度是该自基中所有圈的长度之和．设Ｃ－、Ｃ－分别是Ｇ的最小、最大圈基长度．如果对任一自然数Ｃ，Ｃ－＜Ｃ＜Ｃ－，都存在Ｇ的一个长为Ｃ的圈基，则称Ｇ有圈基内插性质．本文证明了无三角形的外平面图没有圈基内插性质，并说明存在围长任意大且有圈基内插性质的平面图．     

Arc Length Estimation and the Convergence of Polynomial Curve Interpolation

When fitting parametric polynomial curves to sequences of points or derivatives we have to choose suitable parameter values at the interpolation points. This paper investigates the effect of the parameterization on the approximation order of the interpolation. We show that chord length parameter values yield full approximation order when the polynomial degree is at most three. We obtain full approximation order for arbitrary degree by developing an algorithm which generates more and more accurate approximations to arc length: the lengths of the segments of an interpolant of one degree provide parameter intervals for interpolants of degree two higher. The algorithm can also be used to estimate the length of a curve and its arc-length derivatives. AMS subject classification (2000) 65D05, 65D10     

Boundary interpolation in the ball

We solve a boundary interpolation problem in the reproducing kernel Hilbert space of functions analytic in the unit ball of with reproducing kernel 1/(1−∑₁^Nz_kw_k^*). We introduce the notion of Brune factor (or Blaschke–Potapov factor of the third kind) in this setting.     

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two‐ and three‐dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two‐level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two‐level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two‐ and three‐dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two‐ to three‐dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non‐symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Free surface Stokes flows obstructed by multiple obstacles

Gravity‐driven Stokes flow down an inclined plane over and around multiple obstacles is considered. The flow problem is formulated in terms of a boundary integral equation and solved using the boundary element method. A Hermitian radial basis function (RBF) is used for the interpolation of the free surface, generation of the unit normal and curvature, and to prescribe the far‐field conditions. For flow over an obstacle, hemispheres are taken. For flow around an obstacle, circular cylinders are modelled and the contact angle condition on the obstacle/free surface intersection specified using the RBF formulation. Explicit profiles are produced for flow over and around two obstacles placed in various locations relative to one another. Interaction due to two obstacles is given by comparisons made with the profiles for flow over and around individual obstacles. In general, when the obstacles are separated by a sufficiently large distance the flow profiles are identical to a single obstacle analysis. For flow over and around two obstacles in‐line with the incident flow, effects of the governing parameters are examined, with variations in plane inclination angle, Bond number, obstacle size, and in the case of obstacles intersecting the free surface, static contact angle is considered. Finally flows over and around three obstacles are modelled. Copyright © 2009 John Wiley & Sons, Ltd.     

Some considerations for high‐order ‘incremental remap’‐based transport schemes: edges,reconstructions, and area integration

The problem of two‐dimensional tracer advection on the sphere is extremely important in modeling of geophysical fluids and has been tackled using a variety of approaches. A class of popular approaches for tracer advection include ‘incremental remap’ or cell‐integrated semi‐Lagrangian‐type schemes. These schemes achieve high‐order accuracy without the need for multistage integration in time, are capable of large time steps, and tend to be more efficient than other high‐order transport schemes when applied to a large number of tracers over a single velocity field. In this paper, the simplified flux‐form implementation of the Conservative Semi‐LAgrangian Multi‐tracer scheme (CSLAM) is reformulated using quadratic curves to approximate the upstream flux volumes and Gaussian quadrature for integrating the edge flux. The high‐order treatment of edge fluxes is motivated because of poor accuracy of the CSLAM scheme in the presence of strong nonlinear shear, such as one might observe in the midlatitudes near an atmospheric jet. Without the quadratic treatment of upstream edges, we observe at most second‐order accuracy under convergence of grid resolution, which is returned to third‐order accuracy under the improved treatment. A shallow‐water barotropic instability also reveals clear evidence of grid imprinting without the quadratic correction. Consequently, these tests reveal a problem that might arise in tracer transport near nonlinearly sheared regions of the real atmosphere, particularly near cubed‐sphere panel edges. Although CSLAM is used as the foundation for this analysis, the conclusions of this paper are applicable to the general class of incremental remap schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

An algorithm for constrained convex optimization

This paper presents a feasible direction algorithm for the minimization of a pseudoconvex function over a smooth, compact, convex set. We establish that each cluster point of the generated sequence is an optimal solution of the problem without introducing anti-jamming procedures. Each iteration of the algorithm involves as subproblems only one line search for a zero of a continuously differentiable convex function and one univariate function minimization on a compact interval.     

Hilbert function spaces and the Nevanlinna-Pick problem on the polydisc

In [4], Agler and McCarthy (2005) used Hilbert function spaces to study the uniqueness properties of the Nevanlinna-Pick problem on the bidisc. In this work we give a geometric procedure for constructing a Nevanlinna-Pick problem on Dn with a specified set of uniqueness. On the way to establishing this procedure, we prove a result about Hilbert function spaces and partially answer a question posed by Agler and McCarthy.     

Support-type properties of generalized convex functions

Chebyshev systems induce in a natural way a concept of convexity. The functions convex in this sense behave in many aspects similarly to ordinary convex functions. In this paper support-type properties are investigated. Using osculatory interpolation, the existence of support-like functions is established for functions convex with respect to Chebyshev systems. Unique supports are determined. A characterization of the generalized convexity via support properties is presented.     

Coherent trace-class operators in Hilbert couples

In this paper it is proved that the ideal of trace-class operators acting in a couple of Hilbert spaces coincides with the ideal of coherent trace-class operators. A new formula is derived for theK-functional in the couple of algebras of all bounded linear operators acting in a Hilbert couple, and new interpolation theorems are proved for trace class operators.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 866–872, June, 1998.This research was partially supported by the International Science Foundation and the Russian Government under grant JD 7100.