A Special Basis of the Space of Meromorphic Sections on Riemann Surfaces

A special choice of basis for meromorphic sections of line bundles, in which all poles lie at the punctures, allows the decomposition of field operators (which are sections of bundles) into modes analogous to the standard decomposition on the sphere. Many of the calculational techniques used on the sphere can be reproduced for higher genus surfaces in this basis.Using this technique, in this paper, we compute a basis of K (the space of meromorphic sections on a Riemann surface, holomorphic away from two fixed points). This basis consists of the sections which have the expected zero or pole order at the two points.AMS Subject Classification (1991): 14H55     

四面体的十二点二次曲面

垂心四面体中四条高的垂足,四个面的重心及从各顶点与四面体的垂心连线的三等分点,共十二个点共球.试图把垂心改为四面体内的任意点,相应地把四条高线改换为过该点与每个顶点连线的共点直线组时,则将把垂心四面体的十二点球有趣地推广为四面体的十二点二次曲面.     

The rectilinear class Steiner tree problem for intervals on two parallel lines

We consider a generalization of the Rectilinear Steiner Tree problem, where our input is classes of required points instead of simple required points. Our task is to find a minimum rectilinear tree connecting at least one point from each class. We prove that the version, where all required points lie on two parallel lines, called Rectilinear Class Steiner Tree (channel) problem, is NP-hard. But we give a linear time algorithm for the case where the points of each required class are clustered, and the classes consist of non overlapping intervals of points.Part of this research was conducted while the author was attending a research initiative at the Leonardo Fibonacci Institute, Povo, Italy.     

Extreme points of the distance function on convex surfaces

We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that ``almost all" points admit a single farthest point on the surface.

     

On Two Conjectures of Steinhaus

We disprove two conjectures of H. Steinhaus by showing that: (1) there is a convex surface S such that for any point x on S and any point y in the set F _x of farthest points from x, there are at most two segments from x to y; (2) the properties and do not characterize the sphere.     

On cyclic <Emphasis Type="Italic">p</Emphasis>-gonal Riemann surfaces with several <Emphasis Type="Italic">p</Emphasis>-gonal morphisms

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)² the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)² which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.     

Computing the minimum distance between two Bézier curves

A sweeping sphere clipping method is presented for computing the minimum distance between two Bézier curves. The sweeping sphere is constructed by rolling a sphere with its center point along a curve. The initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve. The nearest point on a curve must be contained in the sweeping sphere along the other curve, and all of the parts outside the sweeping sphere can be eliminated. A simple sufficient condition when the nearest point is one of the two end points of a curve is provided, which turns the curve/curve case into a point/curve case and leads to higher efficiency. Examples are shown to illustrate efficiency and robustness of the new method.     

The Analytic Caratheodory Conjecture

The aim of this article is to provide the reader with a real possibility of becoming confident that the index of an isolated umbilic point of an analytic surface is never greater than one. For a surface homeomorphic to a sphere, this means in particular that on the surface there necessarily exist at least two umbilic points as it was conjectured by Caratheodory.     

Identification of objects in an acoustic wave guide inversion II: Robin–Dirichlet conditions

In this paper we investigate the unknown body problem in a wave guide where one boundary has a pressure release condition and the other an impedance condition. The method used in the paper for solving the unknown body inverse problem is the intersection canonical body approximation (ICBA). The ICBA is based on the Rayleigh conjecture, which states that every point on an illuminated body radiates sound from that point as if the point lies on its tangent sphere. The ICBA method requires that an analytical solution be known exterior to a canonical body in the wave guide. We use the sphere of arbitrary centre and radius in the wave guide as our canonical body. We are lead then to analytically computing the exterior solution for a sphere between two parallel plates. We use the ICBA to construct solutions at points ranging over the suspected surface of the unknown object to reconstruct the unknown object using a least‐squares matching of computed, acoustic field against the measured, scattered field. Copyright © 2005 John Wiley & Sons, Ltd.     

On convex bodies close to ellipsoids

It is known that the ellipsoids ind-dimensional Euclidean space, ford 3, are characterized among all convex bodies by the property that the intersection points of anyd pairwise orthogonal supporting hyperplanes lie on a fixed sphere. The subject of the present note is a quantitative improvement of this uniqueness theorem in the form of a stability result.Herrn Oswald Giering zum 60. Geburtstag gewidmet     

Analytically exact spiral scheme for generating uniformly distributed points on the unit sphere

The problem of constructing a set of uniformly-distributed points on the surface of a sphere, also known as the Thomson problem, has a long and interesting history, which dates back to J.J. Thomson in 1904. A particular variant of the Thomson problem that is of great importance to biomedical imaging is that of generating a nearly uniform distribution of points on the sphere via a deterministic scheme. Although the point set generated through the minimization of electrostatic potential is the gold standard, minimizing the electrostatic potential of one thousand points (or charges) or more remains a formidable task. Therefore, a deterministic scheme capable of generating efficiently and accurately a set of uniformly-distributed points on the sphere has an important role to play in many scientific and engineering applications, not the least of which is to serve as an initial solution (with random perturbation) for the electrostatic repulsion scheme. In the work, we will present an analytically exact spiral scheme for generating a highly uniform distribution of points on the unit sphere.     

Surfaces which contain helical geodesics

LetM be a complete smooth surface of constant mean curvature in the Euclidean 3-space. If there exists two helical geodesics onM through each point ofM, thenM is either a plane, a sphere, or a circular cylinder.     

On the geometry of spheres in L-spaces

It is shown that any two points on the surface of the unit ball ofL ¹(μ), where the measureμ is non-atomic, may be joined in the surface by a curve whose length is equal to the straight-line distance between its endpoints. This property is contrasted with the metric properties of the unit sphere in other L-spaces. This work was supported in part by NSF grant GP-19126.     

Limiting points of stochastic flows

It is known that if the flow of a stochastic differential equation on a compact manifold has only negative Lyapunov exponents, then its limiting behavior can be described by a moving random set σof n points. We study the properties of σ and the associated domains of attraction. For example, we will show that it is supported by a compact set on which the induced flow has only one limiting point. On a d-dimensional sphere, if the support of the stochastic flow contains all the isometries, then n=1 or 2, and in the latter case, the two random points and their domains of attractions are antipodal symmetric     

Convex surfaces whose geodesics are planar

It is shown that ifC is a convex surface, in euclidean space of dimension at least 3, having the property that all shortest paths onC between pairs of its points are planar, thenC is a sphere, a hyperplane or the boundary of an intersection of two half-spaces. No smoothness assumptions are made.     

球面上的曲面插值

<正>1引言随着现代工业生产的飞速发展,航空、气象、环境监测等领域需要研究解决限制在曲面上的四维数据插值问题,即由有限个位置处的信息推测其它若干位置点的信息.例如,地球上某个地区的温度分布、降雨量分布、大气层的"温室效应"等;飞行器(飞机、火箭、导弹等)表面压力分布规律、肿瘤的生长规律等.这些在数学上都可归结为限制在曲面上的曲面插值与逼近问题.这个问题自Barnhill提出以后,人们针对限制在球面上     

An estimate of the deformation of a closed convex surface with a change in its curvature

We consider closed convex surfaces ℱ of the space R³ containing a fixed point 0 in the interior. A central projection from 0 enables us to transfer the curvature ω(u) of the surface ℱ, regarded as a function of a set uɛℱ, onto a sphere with center 0. A. D. Aleksandrov established the fact that the surface ℱ is determined (moreover, uniquely) to·within a homothetic transformation with center 0 by prescribing the curvature transferred in this way onto the sphere. In this paper we give an estimate of the variation of the distances τ _F (B) of points of the surface from 0 as a function of the variation of the curvature transferred onto the sphere. The derivation of this estimate relies substantially on nondegeneracy of the surface ℱ; as a measure of nondegeneracy we take the ratio R/ζ, of the radii ℱ of balls with center ℱ, circumscribed and inscribed, respectively, about 0. Also, in this paper, we introduce and study those characteristics ℒ _F and τ _F of the curvature of the surface ℱ, which make it possible to estimate R/ζ from above and, by the same token, to obtain an estimate of how τ _F (B) varies in terms only of the curvature of the surface and its variation. An analytical treatment shows that basically our result yields an estimate of the maximum of the modulus of the change in the solution of a Monge—Ampere type equation on a sphere in terms of the change in its right-hand side in some integral norm, while the estimate of R/ζ, yields an a priori estimate of the modulus of the solution of this equation. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 83–110, 1974.     

Thin Sets and Boundary Behavior of Solutions of the Helmholtz Equation

The Martin boundary for positive solutions of the Helmholtz equation in n-dimensional Euclidean space may be identified with the unit sphere. Let v denote the solution that is represented by Lebesgue surface measure on the sphere. We define a notion of thin set at the boundary and prove that for each positive solution of the Helmholtz equation, u, there is a thin set such that u/v has a limit at Lebesgue almost every point of the sphere if boundary points are approached with respect to the Martin topology outside this thin set. We deduce a limit result for u/v in the spirit of Nagel–Stein (1984).     

On the non-round points of the boundary of the numerical range ∗

Let A be a bounded linear operator acting on a Hilbert space. It is well known (Donoghue, 1957) that comer points of the numerical range W(A) are eigenvalues of A. Recently (1995), this result was generalized by Hiibner who showed that points of infinite curvature on the boundary of W(A) lie in the spectrum of A. Hübner also conjectured that all such points are either corner points or lie in the essential spectrum of A. In this paper, we give a short proof of this conjecture.     

Weierstrass Points and Regular Maps

It was shown by G. A. Jones and the first author in [8] that underlying any map on a compact orientable surface S there is a natural complex structure making S into a Riemann surface. In this paper we consider regular maps and enquire about the Weierstrass points on the underlying Riemann surface. We are particularly interested to know when these are geometric, i.e. whether they lie at vertices, face-centres or edge-centres of the map.