Uniqueness for the homogeneous Dirichlet Willmore boundary value problem

We give a sufficient condition for curves on a plane or on a sphere such that if these give the boundary of a Willmore surface touching tangentially along the boundary the plane or the sphere respectively, the surface is necessarily a piece of the plane or a piece of the sphere. The condition we require is that the curves bound a strictly star-shaped domain with respect to the Euclidean geometry in the plane and with respect to the spherical geometry in the sphere, respectively.     

A surface which contains planar geodesics

It is proved that a complete surface in E ³ is a sphere or a plane if it contains at least four geodesics through each point which are plane curves.     

Representation formula for discrete indefinite affine spheres

We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed from two discrete plane curves.     

A reformulation of Le's conjecture

We give a new proof of Le's conjecture on surface germs in ?³ having as link a topological sphere for the case of surface singularities containing a smooth curve. Our proof leads to a reformulation of the general case of the conjecture into a problem of plane curve singularities and their relative polar curves.     

Surfaces of positive curvature inE 3 whose characteristic curves form a Tchebychef net

In this paper, it is proved that the surfaces of positive curvature with no umbilical points in 3-dimensional Euclidean space whose characteristic curves form a Tchebychef net are translation surfaces and that the characteristic curves are represented on the unit sphere by a rhombic net. The determination of these surfaces depends on two elliptic integrals of the first kind. Furthermore, the case where these elliptic integrals reduce to elementary integrals is studied and it is shown that the surfaces corresponding to this case belong to one of the following two classes: (a) Translation surfaces of positive curvature with plane characteristic curves as generators lying in two planes intersecting each other under a constant angle. The special case where these planes are perpendicular gives an analogue of the Scherk's minimal surfaces of translation. (b) Translation surfaces of revolution of positive curvature with characteristic curves as generators which are circular helices.     

Closed-Loop Control of the Motion of a Sphere Rolling on a Moving Horizontal Plane

A uniform sphere is rolling without slipping on a horizontal plane. The motion of the sphere is controlled via the control of the acceleration of the plane. At the time t=0, the sphere and the plane are stationary and the center of the sphere is located at a point A in the plane. Given a time interval [0, t _f], the problem dealt with here is: Find a closed-loop strategy for the acceleration of the moving plane such that, at the time t=t _f, the plane and the sphere will be nearly at rest and the center of the sphere will be in a given neighborhood of the origin. By introducing the concept of path controllability, a closed-loop strategy for the solution of the above-mentioned problem is proposed and its efficiency is demonstrated by solving numerically some examples.     

First order hermite interpolation with spherical Pythagorean-hodograph curves

The general stereographic projection which maps a point on a sphere with arbitrary radius to a point on a plane stereographically and its inverse projection have the Pythagorean-hodograph (PH) preserving property in the sense that they map a PH curve to another PH curve. Upon this fact, for given spatialC ¹ Hermite data, we construct a spatial PH curve on a sphere that is aC ¹ Hermite interpolant of the given data as follows: First, we solveC ¹ Hermite interpolation problem for the stereographically projected planar data of the given data in ?³ with planar PH curves expressed in the complex representation. Second, we construct spherical PH curves which are interpolants for the given data in ?³ using the inverse general stereographic projection.     

Generating simple convex Venn diagrams

In this paper we are concerned with producing exhaustive lists of simple monotone Venn diagrams that have some symmetry (non-trivial isometry) when drawn on the sphere. A diagram is simple if at most two curves intersect at any point, and it is monotone if it has some embedding on the plane in which all curves are convex. We show that there are 23 such 7-Venn diagrams with a 7-fold rotational symmetry about the polar axis, and that 6 of these have an additional 2-fold rotational symmetry about an equatorial axis. In the case of simple monotone 6-Venn diagrams, we show that there are 39 020 non-isomorphic planar diagrams in total, and that 375 of them have a 2-fold symmetry by rotation about an equatorial axis, and amongst these we determine all those that have a richer isometry group on the sphere. Additionally, 270 of the 6-Venn diagrams also have the 2-fold symmetry induced by reflection about the center of the sphere.Since such exhaustive searches are prone to error, we have implemented the search in a couple of ways, and with independent programs. These distinct algorithms are described. We also prove that the Grünbaum encoding can be used to efficiently identify any monotone Venn diagram.     

On ordinary points in arrangements

We show that in an arrangement ofn curves in the plane (or on the sphere) there are at leastn/2 points where precisely 2 curves cross (ordinary points). Furthermore there are at least (4/3)n triangular regions in the complex determined by the arrangement. Triangular regions and ordinary vertices are both connected with boundary vertices of certain distinguished subcomplexes. By analogy with rectilinear planar polygons we distinguish concave and convex vertices of these subcomplexes. Our lower bounds arise from lower bounds for convex vertices in the distinguished subcomplexes.     

Classical motions of infinitesimal rotators on Mylar balloons

This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two-dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three-dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so-obtained results are also compared with those for the two-dimensional sphere embedded into the three-dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.     

Asymptotics of Maxwell Time in the Plate-Ball Problem

The problem on rolling of a sphere on a plane without slipping or twisting is considered. One should roll the sphere from one contact configuration to another so that the length of the curve traced by the contact point in the plane is the shortest possible. The asymptotics of Maxwell time for rolling of the sphere along small amplitude sinusoids is studied. A two-sided estimate for this asymptotics is obtained.     

Extremal problems for the central projection

We consider area minimizing problems for the image of a closed subset in the unit sphere under a projection from the center of the sphere to a tangent plane, the central projection. We show, for any closed subset in the sphere, the uniqueness of a tangent plane that minimizes the area, and then the minimality of the spherical discs among closed subsets with the same spherical area.     

Total torsion of curves in three-dimensional manifolds

A classical result in differential geometry assures that the total torsion of a closed spherical curve in the three-dimensional space vanishes. Besides, if a surface is such that the total torsion vanishes for all closed curves, it is part of a sphere or a plane. Here we extend these results to closed curves in three dimensional Riemannian manifolds with constant curvature. We also extend an interesting companion for the total torsion theorem, which was proved for surfaces in by L. A. Santaló, and some results involving the total torsion of lines of curvature. Dedicated to Professor Manfredo P. do Carmo on his 80th birthday.     

Shortest Inspection Curves for the Sphere

What is the form of the shortest curve C going outside the unit sphere S in ?³ such that passing along C we can see all points of S from outside? How will the form of C change if we require that C has one (or both) of its endpoints on S? A solution to the latter problem also answers the following question. You are in a half-space at a unit distance from the boundary plane P, but you do not know where P is. What is the shortest space curve C such that going along C you will certainly come to P? Geometric arguments suggest that the required curves should be looked for in certain classes depending on several parameters. A computer-aided analysis yields the best curves in the classes. Some other questions are solved in a similar way. Bibliography: 4 titles.     

Enumeration of loopless maps on the projective plane

In this paper we study the rooted loopless maps on the sphere and the projective plane with the valency of root-face and the number of edges as parameters. Explicit expression of enumerating function is obtained for such maps on the sphere and the projective plane. A parametric expression of the generating function is obtained for such maps on the projective plane, from which asymptotic evaluations are derived.     

On Rational Cuspidal Projective Plane Curves

In 2002, L. Nicolaescu and the fourth author formulated a verygeneral conjecture which relates the geometric genus of a Gorensteinsurface singularity with rational homology sphere link withthe Seiberg--Witten invariant (or one of its candidates) ofthe link. Recently, the last three authors found some counterexamplesusing superisolated singularities. The theory of superisolatedhypersurface singularities with rational homology sphere linkis equivalent with the theory of rational cuspidal projectiveplane curves. In the case when the corresponding curve has onlyone singular point one knows no counterexample. In fact, inthis case the above Seiberg--Witten conjecture led us to a veryinteresting and deep set of ‘compatibility properties’of these curves (generalising the Seiberg--Witten invariantconjecture, but sitting deeply in algebraic geometry) whichseems to generalise some other famous conjectures and propertiesas well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yauinequalities). Namely, we provide a set of ‘compatibilityconditions’ which conjecturally is satisfied by a localembedded topological type of a germ of plane curve singularityand an integer d if and only if the germ can be realized asthe unique singular point of a rational unicuspidal projectiveplane curve of degree d. The conjectured compatibility propertieshave a weaker version too, valid for any rational cuspidal curvewith more than one singular point. The goal of the present articleis to formulate these conjectured properties, and to verifythem in all the situations when the logarithmic Kodaira dimensionof the complement of the corresponding plane curves is strictlyless than 2. 2000 Mathematics Subject Classification 14B05,14J17, 32S25, 57M27, 57R57 (primary), 14E15, 32S45, 57M25 (secondary).     

Nonlinear Poisson-Boltzmann equation in a model of a scanning tunneling microscope

The nonlinear Poisson-Boltzmann equation is solved in the region between a sphere and a plane, which models the electrolyte solution interface between the tip and the substrate in a scanning tunneling microscope. A finite difference method is used with the domain transformed into bispherical coordinates. Picard iteration with relaxation is used to achieve convergence for this highly nonlinear problem. An adsorbed molecule on the substrate can also be modelled by a superposition of a perturbing potential in a small region of the plane. An approximate analytical solution using a superposition of individual solutions for plane, the adsorbed molecule, and the sphere is also attempted. Results for cases of different potential values on the boundary surfaces and different distances of the sphere from the plane are presented. The results of the numerical method, the approximate analytical method, as well as the previous solutions of the linearized equation are compared. © 1994 John Wiley & Sons, Inc.     

射影平面上的对偶无环不可分近三角剖分地图

本文研究了球面和射影平面上对偶无环不可分近三角剖分带根地图的以根面次和内面数为参数的计数问题,得到了这类地图在球面和射影平面上的计数函数满足的方程.还得到了射影平面上2连通地图一个参数的显示表达式和渐近估计式.     

The number of nonseparable maps on the projective plane

In this paper, we study the rooted nonseparable maps on the sphere and the projective plane with the valency of root-face and the number of edges as parameters. Explicit expression of enumerating functions are obtained for such maps on the sphere and the projective plane. A parametric expression of the generating function is obtained for such maps on the projective plane, from which asymptotic evaluations are derived. Moreover, if the number of edges is sufficiently large, then almost all nonseparable maps on the projective plane are not triangulation.     

Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres

This paper is devoted to the development of algorithms for finding unit distance graphs with chromatic number greater than 4, embedded in a two-dimensional sphere or plane. Such graphs provide a lower bound for the Hadwiger–Nelson problem on the chromatic number of the plane and its generalizations to the case of the sphere. A series of 5-chromatic unit distance graphs on 64513 vertices embedded into the plane is constructed. Unlike previously known examples, these graphs do not use the Moser spindle as the base element. The construction of 5-chromatic graphs embedded in a sphere at two values of the radius is given. Namely, the 5-chromatic unit distance graph on 372 vertices embedded into the circumsphere of an icosahedron with a unit edge length, and the 5-chromatic graph on 972 vertices embedded into the circumsphere of a great icosahedron are constructed.