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.     

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.     

Leavitt path algebras are Bézout

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. Furthermore, using a theorem by Basmajian, we get a bound on the number of mapping class group orbits on a given hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the signature of the surface.The arguments we use are based on counting embeddings of ribbon graphs.     

On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus

We obtain an upper bound for the number of holomorphic mappings of a genus 3 Riemann surface onto a genus 2 Riemann surface in a series of cases. In particular, we establish that the number of holomorphic mappings of an arbitrary genus 3 Riemann surface onto an arbitrary genus 2 Riemann surface is at most 48. We show that this estimate is sharp and find pairs of Riemann surfaces for which it is attained.     

Periodic points for area-preserving birational maps of surfaces

It is a basic problem to count the number of periodic points of a surface mapping, since the growth rate of this number as the period tends to infinity is an important dynamical invariant. However, this problem becomes difficult when the map admits curves of periodic points. In this situation we give a precise estimate of the number of isolated periodic points for an area-preserving birational map of a projective complex surface.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

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.     

Crossing distances of surface-knots

A surface-knot is an embedded closed connected oriented surface in 4-space. A surface diagram is a projection of a surface-knot into 3-space with crossing information. In this paper we define a distance from a special surface diagram to a trivial diagram as the minimal number of special double cycles, where we can change the crossing information to obtain the trivial diagram. We estimate the distance using the number of 1-handles needed to obtain a trivial diagram.     

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.     

Unsteady Marangoni convection heat transfer of fractional Maxwell fluid with Cattaneo heat flux

Fractional shear stress and Cattaneo heat flux models are introduced in characterizing unsteady Marangoni convection heat transfer of viscoelastic Maxwell fluid over a flat surface. Governing equations and boundary condition are formulated firstly via the balance between the surface tension and shear stress. Numerical solutions are obtained by new developed numerical technique and some novel phenomena are found. Results shown that the fractional derivative parameters, Marangoni number and power law exponent have significant influence on characteristics velocity and temperature fields. As fractional derivative parameters increase, the temperature profiles rise remarkably and the viscoelastic effects of the fluid enhance with delayed response to surface tension, however the temperature profiles decline significantly with a thinner thickness of thermal boundary layer with the increase of Marangoni number. The average skin friction coefficient increases with the augment of Marangoni number, while the average Nusselt number decreases for larger values of power law exponent.     

Chromatic numbers of quadrangulations on closed surfaces

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N_k has chromatic number at least 4 if G has a cycle of odd length which cuts open N_k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N_k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100–114, 2001     

Bridge position and the representativity of spatial graphs

In this paper, we consider closed surfaces which contain spatial graphs. In the case that a closed surface is a 2-sphere, we show that the 2-sphere can be isotoped so that it intersects a bridge sphere for the spatial graph in a single loop. In the case that a closed surface is not a 2-sphere, we define an invariant of a spatial graph by counting the number of intersection of a compressing disk for the closed surface and the spatial graph. By using this invariant, we give a lower bound for the bridge number of a spatial graph.     

Inflection points and topology of surfaces in 4-space

We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.

     

Fundamental groups of elliptic fibrations and the invariance of the plurigenera for surfaces with odd first Betti number

We study the fundamental groups of elliptic fibrations and the close relationship between the Kodaira dimension of an elliptic surface with odd first Betti number and the geometry of its base orbifold. We show that the plurigenera and hence the Kodaira dimension of a complex surface with odd first Betti number are determined by the isomorphism type of the fundamental group.     

Instabilität einer von unten erwärmten Flüssigkeitsschicht bei gleichzeitiger Berücksichtigung von Oberflächenspannung und Auftriebskraft

Combined effects of surface tension and buoyancy force on the thermal instability in a horizontal liquid film heated from below, which is bounded by a rigid wall and a free surface, are considered under a generalized boundary condition for the temperature disturbance at the free surface, which is introduced by consideration of the continuity of heat flow through the free surface to a gas above the liquid film. The results show that the critical Rayleigh number varies with the Marangoni number and with the Nusselt number.

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Die Arbeit ist während eines Aufenthaltes als Dozentenstipendiat der Alexander-von-Humboldt-Stiftung bei Herrn Prof. Dr.H. Görtler im Institut für Angewandte Mathematik der Albert-Ludwigs-Universität, Freiburg i. Br. entstanden.     

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.

     

Tight upper bounds on the number of invariant components on translation surfaces

An abelian differential on a surface defines a flat metric and a vector field on the complement of a finite set of points. The vertical flow that can be defined on the surface has two kinds of invariant closed sets (i.e. invariant components) — periodic components and minimal components. We give upper bounds on the number of minimal components, on the number of periodic components and on the total number of invariant components in every stratum of abelian differentials. We also show that these bounds are tight in every stratum.     

Exceptional points in the elliptic-hyperelliptic locus

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.     

Transcendental lattices of some K 3‐surfaces

In a previous paper, [12], we described six families of K 3‐surfaces (over ?) with Picard‐number 19, and we identified surfaces with Picard‐number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover, we show that the surfaces with Picard‐number 19 are birational to a Kummer surface which is the quotient of a non‐product type abelian surface by an involution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical modelling for a three-dimensional hydrofoil with winglets operating beneath a free surface

The current study focuses on establishing a theoretical lifting surface model for predicting the hydrodynamic loads acting on the three-dimensional hydrofoil with winglets, which is considerably influenced by the proximity to the free surface through finding the three-dimensional Green’s function for the planar and vertical horseshoe vortices operating below a free surface. The hydrofoil surface is decomposed into a finite number of elements along the span direction and the chord directions, each of which can then be represented by a horseshoe vortex. The linearized free surface boundary condition is applied to analyze the influence of the free surface on the hydrofoil as well as the winglets. The thickness problem is considered using the source distribution among the hydrofoil and winglets surfaces and the analytical Green’s function that satisfies the linearized free surface boundary condition is used. As a sample application, numerical examples were conducted to show the performance of the hydrodynamic characteristics for the hydrofoil with winglets as a function of the Froude number. It was concluded that there are significant efficiency benefits from using winglets inside the free surface proximity effect. These results are substantiated by the comparison with the available published data.