THE TOPOLOGICAL AND DYNAMICAL PROPERTIES OF NONORIENTABLE SURFACES

Four topological and dynamical properties of nonorientable surfaces are proved.Thefirst is that for every continuous flow defined on any nonorientable closed surface,thereexist periodic or singular closed orbits.In the case of the projective plane,it confirms aconjecture of professor Ye Yian-qian in his lecture notes“dynamical systems on surfaces”.Secondly,the author gives an exact upper bound of the number of closed curves onnonorientable surfaces,which do not intersect each other and the complement of their sumis still connected.The third is concerned with the upper and lower bound of the number ofthe periodic or singular closed orbits with certain properties.The last is related to theconnectedness of the complement of a lifting curve on two-fold covering space.The firstproperty may be considered as a generalization of Kneser theorem from Klein bottle togeneral nonorientable surfaces and the second as a generalization of[4]Theorem 9.3.6from orientable surfaces to nonorientable surfaces.     

Laguerre Geometry of Surfaces in R^3

Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.     

Surface Reconstruction of 3D Scattered Data with Radial Basis Functions

We use Radial Basis Functions （RBFs） to reconstruct smooth surfaces from 3D scattered data. An object＇s surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improvements on the methods of surface reconstruction with radial basis functions. A sparse approximation set of scattered data is constructed by reducing the number of interpolating points on the surface. We present an adaptive method for finding the off-surface normal points. The order of the equation decreases greatly as the number of the off-surface constraints reduces gradually. Experimental results are provided to illustrate that the proposed method is robust and may draw beautiful graphics.     

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...     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

Dimension splitting method for the three dimensional rotating Navier-Stokes equations

In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces and is decomposed by a series of surfaces ■i into several sub-domains, which are called the layers of the flow. Every interface i between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate (S-coordinate) system based on ■i , Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on ■i , another one is called the bending operator taking value in the normal space on ■i . Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface ■i is obtained, which is called the two-dimensional three-component (2D-3C) Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D-3C Navier-Stokes equations is provided, and some approximate methods for solving 2D-3C Navier-Stokes equations are presented.     

Numerical comparison of three stochastic methods for nonlinear PN junction problems

We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.     

Incompressible pairwise incompressible surfaces in link complements

The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S~3 - L. We discuss the properties that the surface F intersects with 2-spheres in S~3 - L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves (R-move and S~2-move), and define the characteristic number of the topological graph for F∩S~2±. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating (or almost alternating) link complements. We prove that the genus of the surface equals zero if the component number of F∩S~2+(or F∩S~2-) is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #(F) ≤8.     

Laguerre minimal surfaces in ℝ3

Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.     

A Fast Numerical Method for Integral Equations of the First Kind with Logarithmic Kernel Using Mesh Grading

The aim of this paper is to develop a fast numerical method for two-dimensional boundary integral equations of the first kind with logarithm kernels when the boundary of the domain is smooth and closed. In this case, the use of the conventional boundary element methods gives linear systems with dense matrix. In this paper, we demonstrate that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. It will be demonstrated that this technique can increase the numerical efficiency significantly.     

THE MINIMAX DIRECTION FOR THE DIRECT PRODUCT OF A CONVEX CONE WITH ITS APPLICATION TO TESTING PROBLEMS

Let C be a closed convex cone in R~k and let C~p be the p-th direct product of C.Thispaper gives some results of the minimax direction with respect to C~p and an innerproduct based on(?),where is a k×k diagonal matrix with positive diagonalelements,is a p×p positive definite matrix and is the Kronecker product ofand.It is also shown that the results may be applied to test the homogeneity of knormal mean vectors where the mean vectors are restricted by a given partial order.     

A RIEMANN-HILBERT PROBLEM IN A RIEMANN SURFACE

One of the inspirations behind Peter Lax’s interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda lattice in different regimes has been able to shed light on some of von Neumann’s conjectures concerning the validity of the approximation of PDEs by dispersive systems of ODEs. Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints (an "equilibrium measure" problem). Later, it was found that the asymptotic method of Deift and Zhou (analysis of the associated Riemann-Hilbert factorization problem in the complex plane) could apply to previously intractable problems and also produce more detailed information. Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solutions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background. Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift-Zhou "nonlinear stationary phase method" to surfaces of nonzero genus. Second, we illustrate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!     

The classification of bi-quintic parametric polynomial minimal surfaces

Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces represented in parametric polynomial form. Recently,Ganchev presented the canonical principal parameters for minimal surfaces. The normal curvature of a minimal surface expressed in these parameters determines completely the surface up to a position in the space. Based on this result, in this paper, we study the bi-quintic isothermal minimal surfaces. According to the condition that any minimal isothermal surface is harmonic,we can acquire the relationship of some control points must satisfy. Follow up, we obtain two holomorphic functions f(z) and g(z) which give the Weierstrass representation of the minimal surface. Under the constrains that the minimal surface is bi-quintic, f(z) and g(z) can be divided into two cases. One case is that f(z) is a constant and g(z) is a quadratic polynomial, and another case is that the degree of f(z) and g(z) are 2 and 1 respectively. For these two cases,we transfer the isothermal parameter to canonical principal parameter, and then compute their normal curvatures and analyze the properties of the corresponding minimal surfaces. Moreover,we study some geometric properties of the bi-quintic harmonic surfaces based on the B′ezier representation. Finally, some numerical examples are demonstrated to verify our results.     

几乎交错环链补中的不可压缩曲面

In this paper,we discuss mainly the properties of incompressible pairwies incom- prcssiblc surfaccs in almost altcrnating link complcmcnts. Lct L bca almost link and lct F be an incompressible palrwise incompressible surface in S~3-L.First,we give the properties that the surface F intersects with 2-spheres in S~3-L.The intersection consisting of a collection of circles and saddle-shaped discs is called a topological graph.One can compute the Euler Characteristic number of the surface by calculating the characteristic number of the graph.Next,we prove that if the graph is special simple,then the genus of the surface is zero.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

ON THE AREAS OF THE MINIMAL TRIANGLES IN VEECH SURFACES

Smillie and Weiss proved that the set of the areas of the minimal triangles of Veech surfaces with area 1 can be arranged as a strictly decreasing sequence {an}. And each an in the sequence corresponds to finitely many affine equivalent classes of Veech surfaces with area 1. In this article, we give an algorithm for calculating the area of the minimal triangles in a Veech surface and prove that the first element of {a_n} which corresponds to non arithmetic√Veech surfaces is(5-5~(1/2))/20, which is uniquely realized by the area of the minimal triangles of the normalized golden L-shaped translation surface up to affine equivalence.     

Enumerating reflexible 2-cell embeddings of connected graphs

Two 2-cell embeddings:X → S and j:X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h on S and a graph automorphism γ of X such that h = γj.A 2-cell embedding:X → S of a graph X into a closed orientable surface S is described combinatorially by a pair(X;ρ) called a map,where ρ is a product of disjoint cycle permutations each of which is the permutation of the darts of X initiated at the same vertex following the orientation of S.The mirror image of a map(X;ρ) is the map(X;ρ 1),and one of the corresponding embeddings is called the mirror image of the other.A 2-cell embedding of X is reflexible if it is congruent to its mirror image.Mull et al.[Proc Amer Math Soc,1988,103:321-330] developed an approach for enumerating the congruence classes of 2-cell embeddings of graphs into closed orientable surfaces.In this paper we introduce a method for enumerating the congruence classes of reflexible 2-cell embeddings of graphs into closed orientable surfaces,and apply it to the complete graphs,the bouquets of circles,the dipoles and the wheel graphs to count their congruence classes of reflexible or nonreflexible(called chiral) embeddings.     

Addendum to:Exotic structures arising from fake projective planes(Sci China Math,2013,56:43-54)

<正>The goal of this addendum is to generalize the argument of the original article[5]to handle the case of Cartwright-Steger surface,and to correct misprints in the tables in[5].It is known that 3 is the smallest Euler number achievable by a smooth surface of general type.Moreover,smooth surfaces of general type with c_2=3 are complex ball quotients B_C~2/Ⅱconsisting of 100     

ON DIRECT METHOD OF SOLUTION FOR A CLASS OF SINGULAR INTEGRAL EQUATIONS

In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.     

A vanishing result for Donaldson Thomas invariants of ℙ1 scroll

Let S be a smooth algebraic surface and let L be a line bundle on S.Suppose there is a holomorphic two form over S with zero loci to be a curve C.We show that the DonaldsonThomas invariant of the P^1 scroll X = P（L＋Бs） vanishes unless the curves being enumerated lie in D = P（L︱C＋БC）.Our method is cosection localization of Y.-H.Kiem and J.Li.