Coloring vertices and faces of maps on surfaces

The vertex-face chromatic number of a map on a surface is the minimum integer m such that the vertices and faces of the map can be colored by m colors in such a way that adjacent or incident elements receive distinct colors. The vertex-face chromatic number of a surface is the maximal vertex-chromatic number for all maps on the surface. We give an upper bound on the vertex-face chromatic number of the surfaces of Euler genus ≥2. The upper bound is less (by 1) than Ringel’s upper bound on the 1-chromatic number of a surface for about 5/12 of all surfaces. We show that there are good grounds to suppose that the upper bound on the vertex-face chromatic number is tight.     

A degree bound for families of rational curves on surfaces

We give an upper bound for the degree of rational curves in a family that covers a given birationally ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We introduce an algorithm for constructing examples where the upper bound is tight. As an application of our methods we improve an inequality on lattice polygons.     

A lower bound on the performance of the quadratic discriminant function

The quadratic discriminant function is often used to separate two classes of points in a multidimensional space. When the two classes are normally distributed, this results in the optimum separation. In some cases however, the assumption of normality is a poor one and the classification error is increased. The current paper derives an upper bound for the classification error due to a quadratic decision surface. The bound is strict when the class means and covariances and the quadratic discriminant surface satisfy certain specified symmetry conditions.     

Isoperimetric inequalities and conjugate points on Lorentzian surfaces

Recently, an isoperimetric inequality for a sector on the Minkowski 2-spacetime has been derived by the method of parallels and the relativistic Gauss-Bonnet formula. In the present paper, we derive an isoperimetric inequality for a sector on a Lorentzian surface with curvatureK ≤ C. As a sector can be modeled by a geodesic variation of a timelike geodesic, our isoperimetric inequality gives an upper bound for the spacelike boundary of a sector. As an application of our results, we give an elementary proof of the existence of conjugate points on a Lorentzian surface with curvatureK ≤ C < 0 and we obtain an upper bound for the (timelike) diameter of a globally hyperbolic Lorentzian surface withK ≤ C < 0 by comparison of sectors.     

关于Finsler曲面的几何和拓扑

本文在Ｆｉｎｓｌｅｒ曲面上定义了一个新的不变量Ｈ。该不变量等于零刻画了Ｒｉｅｍａｎｎ流形。文章给出了Ｈ的一个上界并且构造了Ｈ为常值的非Ｒｉｅｍａｎｎ的Ｆｉｎｓｌｅｒ曲面。此外，本文还推广了Ｌａｎｄｓｂｅｒｇ曲面的Ｇａｕｓ－Ｂｏｎｎｅｔ－Ｃｈｅｒｎ定理并分类了非正曲率的Ｆｉｎｓｌｅｒ曲面。     

Andreev states in a quasi-one-dimensional superconductor on the surface of a topological insulator

Theoretical and Mathematical Physics - We study bound states in an s-wave superconducting strip on the surface of a topological superconductor with the perpendicular Zeeman field. We prove...     

A note on irregularity of two dimensional simple hyperbolic singularities

Deformation theory is an important aspect of the study about isolated singularities. The invariant called irregularity is very useful in the study on the deformation of isolated singularities. In this note we give an optimal upper bound for a class of surface singularities by the computation of cohomology. Moreover a sufficient condition is given for the positivity of irregularity of some simple hyperbolic surface singularities. Therefore a class of surface singularities with non-rigid deformation is constructed.     

On the number of cusps on cuspidal curves on Hirzebruch surfaces

In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the results in this new setting.     

可定向曲面上两个地图的交叉数

In this paper, we discuss the crossing numbers of two one-vertex maps on orientable surfaces. By using a reductive method, we give the crossing number of two one-vertex maps with one face on an orientable surface and the crossing number of a one-vertex map with one face and a one-vertex map with two faces on an orientable surface. This provides a lower bound for the crossing number of two general maps on an orientable surface.     

How many different cascades on a surface can have coinciding hyperbolic attractors?

It is shown that the number of essentially nonconjugate (i.e., not being iterations of topologically conjugate) diffeomorphisms of a surface having homeomorphic one-dimensional hyperbolic attractors can be arbitrarily large, provided that the genus of the surface is large enough. A lower bound for this number depending on the surface genus is given. The corresponding result for pseudo-Anosov homeomorphisms is stated.     

Further variation diminishing properties of Bernstein polynomials on triangles

A definition is given of the “variation” of a surface which generalizes those considered previously. It is then shown that the variation of a Bernstein polynomial on a triangle is bounded by that of its Bézier net and conditions are derived under which the bound is attained. A bound is also given for the variation of the Bézier net in terms of the variation of the function itself. Finally, it is mentioned how these results lead to variation diminishing properties of certain approximation operators involving polyhedral splines.     

Volume, surface area and inward injectivity radius

We show some relations among the volumes of domains in Euclidean spaces, their surface areas and the inward injectivity radii from their boundaries. In particular, we give an estimate for the upper bound of the ratios of their surface areas and volumes by means of inward injectivity radii. The upper bound seems to depend on their topological structures.

     

A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points in R³ may be quadratic in the worst case, we show in this paper that it is only linear when the points are distributed on a fixed set of well-sampled facets of R³ (e.g. the planar polygons in a polyhedron). Our bound is deterministic and the constants are explicitly given.     

Tight and 0-tight polyhedral embeddings of surfaces

In [1] T. Banchoff has studied the problem of high condimensional tight polyhedral embeddings of closed surfaces into Euclidean space. He gave an upper bound for the essential codimension depending on the Euler characteristic of the surface. In [1] and [5] he proved that this bound is attained in some cases and that it is not attained for the Klein bottle. In the present paper we show that this bound is sharp in any case (except the Klein bottle) and that for each surface there exist tight substantial embeddings into Euclidean space of arbitrary dimension up to the Banchoff upper bound. The proof depends essentially on the Heawood map color theorem proved by G. Ringel and J.W.T. Youngs. In addition we get similar results for tight and 0-tight embeddings of surfaces with boundary where it may be remarkable that in case of tight surfaces with one boundary component the Banchoff upper bound can be improved.     

The number of additional singular points in the Riemann-Hilbert problem on a Riemann surface

We present an upper bound for the number of additional singular points that are sufficient to construct a system of linear equations with given regular singular points and a given monodromy on a Riemann surface.     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.     

A gradient bound for free boundary graphs

We prove an analogue for a one‐phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation. It follows, in particular, that every energy‐minimizing free boundary that is a graph is also smooth. The method we use also leads to a new proof of the classical minimal surface gradient bound. © 2010 Wiley Periodicals, Inc.     

On the approximation of a smooth surface with a triangulated mesh

We approximate the normals and the area of a smooth surface with the normals and the area of a triangulated mesh whose vertices belong to the smooth surface. Both approximations only depend on the triangulated mesh (which is supposed to be known), on an upper bound on the smooth surface's curvature, on an upper bound on its reach (which is linked to the local feature size) and on an upper bound on the Hausdorff distance between both surfaces.

We show in particular that the upper bound on the error of the normals is better when triangles are right-angled (even if there are small angles). We do not need every angle to be quite large. We just need each triangle of the triangulated mesh to contain at least one angle whose sinus is large enough.     

Bounded geometry for Kleinian groups

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations. Oblatum 31-VII-2000 & 9-V-2001?Published online: 20 July 2001     

A Note on the Mackey topology of a von Neumann algebra

Some of the properties of the upper bound of the spectrum of a quasilinear eigenvalue problem, subject to a positivity requirement, are derived. It is shown that, as a function of the surface heat-transfer coefficient, this parameter is a continuous, monotonic increasing function and is bounded above.