Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.     

On the capacity of surfaces in manifolds with nonnegative scalar curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ³, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.     

带有圆柱形孔洞弹性半空间的表面效应分析

针对表面应力在纳米结构控制机械响应中的重要性,利用复变函数的基本方法,研究了含有圆柱形孔洞的弹性半空间的表面应力问题.将含有缺陷的接触问题分解为均匀介质的接触问题和无外载荷的非均匀介质的接触问题两部分进行分析.结果显示:接触表面的应力和位移有很强的尺寸依赖性,同时表面位移可以用表面应力函数表示.     

Stresses due to Heterogeneous Surface Tension in Solids

When surface‐active melt interacts with metal a zone with high dislocation density arises in the surface layer of metal. The interaction between melt and metal leads to a decrease of surface tension and to considerable stresses in the surface layer due to the surface tension gradient. The estimation shows that the stress distribution in the vicinity of the surface active melt perimeter is sufficient for nucleation and movement of dislocations in the surface layer.     

The mean curvature of the influence surface of wave equation with sources on a moving surface

The mean curvature of the influence surface of the space–time point ( x , t) appears in linear supersonic propeller noise theory and in the Kirchhoff formula for a supersonic surface. Both these problems are governed by the linear wave equation with sources on a moving surface. The influence surface is also called the Σ‐surface in the aeroacoustic literature. This surface is the locus, in a frame fixed to the quiescent medium, of all the points of a radiating surface f( x , t)=0 whose acoustic signals arrive simultaneously to an observer at position x and at the time t. Mathematically, the Σ‐surface is produced by the intersection of the characteristic conoid of the space–time point ( x , t) and the moving surface. In this paper, we derive the expression for the local mean curvature of the Σ‐surface of the space–time point ( x , t) for a moving rigid or deformable surface f( x , t)=0. This expression is a complicated function of the geometric and kinematic parameters of the surface f( x , t)=0. Using the results of this paper, the solution of the governing wave equation of high‐speed propeller noise radiation as well as the Kirchhoff formula for a supersonic surface can be written as very compact analytic expressions. Copyright © 1999 John Wiley & Sons, Ltd.     

分形插值曲面理论及其应用*

本文叙述了分形曲面的生成原理,给出了分形插值曲面的计算公式,证明了分形插值曲面迭代函数系唯一性定理,导出了分形插值曲面的维数定理,并应用实际数据进行了分形插值曲面的实例研究。     

Degenerations of rational Bézier surface with weights in the form of exponential function

Rational Bézier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bézier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bézier surfaces. In this paper, we study on the degenerations of the rational Bézier surface with weights in the exponential function and indicate the difference of our result and the work of Garc′?a-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bézier surface with weights in the exponential function is defined, which is just the limit of the surface.Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.     

Reducing variance in nonparametric surface estimation

We suggest a method for reducing variance in nonparametric surface estimation. The technique is applicable to a wide range of inferential problems, including both density estimation and regression, and to a wide variety of estimator types. It is based on estimating the contours of a surface by minimising deviations of elementary surface estimates along a quadratic curve. Once a contour estimate has been obtained, the final surface estimate is computed by averaging conventional surface estimates along a portion of the contour. Theoretical and numerical properties of the technique are discussed.     

A generalized dynamic model of nanoscale surface acoustic wave sensors and its applications in Love wave propagation and shear-horizontal vibration

A generalized dynamic model to depict the wave propagation properties in surface acoustic wave nano-devices is established based on the Hamilton's principle and variational approach. The surface effect, equivalent to additional thin films, is included with the aid of the surface elasticity, surface piezoelectricity and surface permittivity. It is demonstrated that this generalized dynamic model can be reduced into some classical cases, suitable for macro-scale and nano-scale, if some specific assumptions are utilized. In numerical simulations, Love wave propagation in a typical surface acoustic wave device composed of a piezoelectric ceramic transducer film and an aluminum substrate, as well as the shear-horizontal vibration of a piezoelectric plate, is investigated consequently to qualitatively and quantitatively analyze the surface effect. Correspondingly, a critical thickness that distinguishes surface effect from macro-mechanical behaviors is proposed, below which the size-dependent properties must be considered. Not limited as Love waves, the theoretical model will provide us a useful mathematical tool to analyze surface effect in nano-devices, which can be easily extended to other type of waves, such as Bleustein-Gulyaev waves and general Rayleigh waves.     

Surfaces of positive curvature whose characteristic lines constitute two families of geodesic parallels

In this paper, by means of characteristic lines a sufficient condition for a surface of positive curvature in 3-dimensional euclidean space to be isometric with a rotation surface is given, i.e. it is shown that if the characteristic lines of such a surface constitute two families of geodesic parallels, then the surface is isometric with a rotation surface. Furthermore, it is proved that the lines of curvature on such surfaces are isothermal-conjugate or, equivalently, the characteristic lines and the lines of curvature form a hexagonal 4-web.     

Intrinsic equations for a relaxed elastic line on an oriented surface

Hilbert and Cohn-Vossen [2, p. 221] incorrectly suggested a flexible knitting needle, constrained to conform to a surface, as one model for a geodesic on a surface. This model actually gives a relaxed elastic line on the surface, and is not generally a geodesic unless the surface lies in a plane or on a sphere.In this paper we derive the intrinsic equations for a relaxed elastic line on an oriented surface. This formulation should give a more direct and more geometric approach to questions concerning relaxed elastic lines on a surface. We apply this formulation to give alternate proofs of some results of [3] found by the less direct method of Lagrange multipliers and to give additional results about relaxed elastic lines on various surfaces. For further considerations of a relaxed elastic line on a surface as a model of the DNA molecule, see [3].Partially supported by NIH grant GM 36284-01.     

Propagation of Surface Waves at High Frequencies

An asymptotic theory is presented for the analysis of surfacewave propagation at high frequencies. The theory is developedfor scalar surface waves satisfying an impedance boundary conditionon a surface, which may be curved and, whose impedance may bevariable. A surface eikonal equation is derived for the phaseof the surface wave field, and it is shown that the wave fieldpropagates over the surface along the surface rays, which arethe characteristics of the surface eikonal equation. The wavefield in space is found by solving certain eikonal and transportequations with the aid of complex rays. The theory is then appliedto several examples: axial waves on a circular cylinder, sphericallysymmetric waves on a sphere, waves on a circular cone with avariable impedance, and waves on the plane boundary of an inhomogeneousmedium. In each case it is found that the asymptotic expansionof the exact solution agrees with the asymptotic solution.     

基于Delaunay三角剖分的地表平均温度模拟

利用Delaunay三角网对目标区域进行剖分,在对地表温度进行高度插值后,运用二重积分的思想建立了基于Delaunayr三角剖分的地表平均温度测量模型.同时以南极地表平均温度的测量为例,将67个自动气象地表台站、46个气象地表台站以及56个高空气象观测站的加权平均温度与地表平均温度的数据进行分析,得到南极2015全年地表平均温度均在-8℃以下,最低温约为-20℃,符合南极大陆地表温度的实际情况.     

The effect of surface elasticity and residual surface stress in an elastic body with an elliptic nanohole

The deformation of an elastic plane with an elliptic hole in a uniform stress field is considered, taking into account the surface elasticity and the residual surface tension. The solution of the problem, based on the use of the linearized Gurtin–Murdoch surface elasticity relations and the complex Goursat–Kolosov potentials, is reduced to a singular integrodifferential equation. Using the example of a circular hole, for which an exact solution of the equation is obtained in closed form, the effect of the residual surface tension and the surface elasticity on the stress state close to and on the boundary of a nanohole is analysed for uniaxial tension. It is shown that the effect of the residual surface stress and the surface tension, due to deformation of the body, depends on the elastic properties of the surface, the value of the stretching load and the dimensions of the hole.     

Bicanonical pencil of a determinantal Barlow surface

In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.

     

The zero surface tension limit two‐dimensional water waves

We consider two‐dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases. © 2005 Wiley Periodicals, Inc.     

Surfaces in 4-manifolds and their mapping class groups

A surface in a smooth 4-manifold is called flexible if, for any diffeomorphism ? on the surface, there is a diffeomorphism on the 4-manifold whose restriction on the surface is ? and which is isotopic to the identity. We investigate a sufficient condition for a smooth 4-manifold M to include flexible knotted surfaces, and introduce a local operation in simply connected 4-manifolds for obtaining a flexible knotted surface from any knotted surface.     

F-Willmore曲面的间隙现象

对于空间形式中的2维曲面,定义了F-Willmore泛函,此泛函包括经典的Willmore泛函作为特殊情形.F-Willmore泛函的临界点称为F-Willmore曲面.推导了第1变分公式并由此构造了F-Willmore曲面的典型例子.利用自伴算子作用于特殊的实验函数,得到了Simons类积分不等式,讨论了F-Willmore曲面的间隙现象,定出了间隙端点对应的特殊曲面.     

Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian co...     

Bounded automorphisms of surfaces

The equivalence among bounded homeomorphisms of a surface, reducible homeomorphisms of the surface, bounded automorphisms of the surface group, induced or exchange automorphisms of the surface group, and the stabilizer of the translation length function of the tree action is established.