Asymptotics of random convex polygonal lines

The present paper deals with the limit shape of random plane convex polygonal lines whose edges are independent and identically distributed, with finite first moment. The smoothness of a limit curve depends on some properties of the distribution. The limit curve is determined by the projection of the distribution to the unit circle. This correspondence between limit curves and measures on the unit circle is proved to be a bijection. The emphasis is on limit distributions of deviations of random polygonal lines from a limit curve. Normed differences of Minkowski support functions converge to a Gaussian limit process. The covariance of this process can be found in terms of the initial distribution. In the case of uniform distribution on the unit circle, a formula for the covariance is found. The main result is that a.s. sample functions of the limit process have continuous first derivative satisfying the Hölder condition of order a, for any fixed a with 0相似文献   

A Measure of Non-convexity in the Plane and the Minkowski Sum

In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced. It is proved that for “not far from convex” regions this measure does not decrease under the Minkowski sum operation, and guarantees that the Minkowski sum has no “holes”.     

Toroidal circle planes that are not Minkowski planes

We construct first examples of circle planes on the torus that are no Minkowski planes, but satisfy the same axiom of joining as flat Minkowski planes. The circle planes constructed by us form a special class ofhyperbola structures (see [4]) or(B*)-Geometrien (see [2]).This research was supported by a Feodor Lynen Fellowship and an ARC International Research Fellowship.     

On the numerical approximation of the rotation number

The starting point of this paper is a polygonal approximation of an invariant curve of a map. Using this polygonal approximation an approximation for the circle map (the restriction of the map to the invariant curve) is obtained. The rotation number of the circle map is then approximated by the rotation number of the approximated circle map. The error in the obtained approximate rotation number is discussed, and related to the error in the polygonal approximation of the invariant curve. Simple algorithms for the approximation of the rotation number are described. A numerical example illustrates the theory.     

The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes

In their paper ``An Introduction to Finsler Geometry,' J. C. Alvarez and C. Duran asked if there are other Minkowski planes besides the Euclidean for which the ratio of the Minkowski length of the unit ``circle' to the Holmes-Thompson area of the unit disc equals 2. In this paper we show that this ratio is greater than 2, and that the ratio 2 is achieved only for Minkowski planes that are affine equivalent to the Euclidean plane. In other words, the ratio is 2 only when the unit ``circle' is an ellipse.

     

Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

We classify all helicoidal non-degenerate surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian circle, we prove that the only possibility is that the axis is spacelike and the center of the circle lies on the axis.     

Spherical containment and the Minkowski dimension of partial orders

The recent work on circle orders generalizes to higher dimensional spheres. As spherical containment is equivalent to causal precedence in Minkowski space, we define the Minkowski dimension of a poset to be the dimension of the minimal Minkowski space into which the poset can be embedded; this isd if the poset can be represented by containment with spheresS ^d–2 and of no lower dimension. Comparing this dimension with the standard dimension of partial orders we prove that they are identical in dimension two but not in higher dimensions, while their irreducible configurations are the same in dimensions two and three. Moreover, we show that there are irreducible configurations for arbitrarily large Minkowski dimension, thus providing a lower bound for the Minkowski dimension of partial orders.     

Flat Circle Planes as Integrals of Flat Affine Planes

In a previous article (Arch. Math. {64} (1995), 75–85) we showed that flat Laguerre planes can be constructed by'integrating' flat affine planes. It turns out that'most' of the known flat Laguerre planes arise in this manner. In this paper we show that similar constructions are also possible in the case of the other two kinds of flat circle planes, that is, the flat Möbius planes and the flat Minkowski planes. In particular, we show that many of the known flat Möbius planes can be constructed by integrating a closed strip taken from a flat affine plane. We also show how flat Minkowski planes arise as integrals of two flat affine planes. All known flat Minkowski planes can be constructed in this manner.     

On the Solution of Some Non-Local Problems

This paper deals with two types of non-local problems for the Poisson equation in the disc. The first of them deals with the situation when the function value on the circle is given as a combination of unknown function values in the disc. The other type deals with the situation when a combination of the value of the function and its derivative by radius on the circle are given as a combination of unknown function values in the disc. The existence and uniqueness of the classical solution of these problems is proved. The solutions are constructed in an explicit form.     

Laguerre and Minkowski planes with neighbor relation

In [7] we have introduced the notion of a Möbius plane with neighbor relation as a generalization of ordinary Möbius planes. In this paper we present two other classes of circle geometries which are locally affine Klingenberg planes: Laguerre and Minkowski planes with neighbor relation.Research supported by IWONL grant no-840037     

Minimal distance calculation between the industrial robot and its workspace based on circle/polygon-slices representation

The real-time and reliable collision detection is the vital basis for the robotic motion control in real applications. In this work, the minimal distance calculation between the industrial robot and its workspace is presented. This method first uses circular/polygonal slices to represent robotic sub-structures with curved edges more accurately, and uses polygonal planes to construct its workspace. Being the basis of the distance calculation, the shortest distance between the spatial circle/polygon and the polygonal plane, as well as that between the circular/polygonal plane and the polygonal plane are first modelled through spatial relation analysis. The simulation validation indicates that the proposed algorithm is more efficient and reliable than both the point-clouds-based and bounding-volume-hierarchies-based algorithms. And the influence of the refinement size on the performance of the proposed model is revealed. Finally, the application example shows that this work is a significant contribution for the collision-free motion control of the industrial robot. The proposed algorithm can also be applied to the distance calculation of other complex objects because of the generality of the proposed slices representation way.     

Cassini Curves in Normed Planes

We extend the concept of Cassini curves from the Euclidean plane to normed (or Minkowski) planes and show that geometric properties of (Minkowskian) Cassini curves are closely related to geometric properties of the unit disc determining the underlying normed plane.     

Computing families of Cohen-Macaulay and Gorenstein rings

We characterize Cohen-Macaulay and Gorenstein rings obtained from certain types of convex body semigroups. Algorithmic methods to check if a polygonal or circle semigroup is Cohen-Macaulay/Gorenstein are given. We also provide some families of Cohen-Macaulay and Gorenstein rings.     

Harmonic mappings onto stars

A general version of the Radó-Kneser-Choquet theorem implies that a piecewise constant sense-preserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.     

A new construction of Radon curves and related topics

We present a new construction of Radon curves which only uses convexity methods. In other words, it does not rely on an auxiliary Euclidean background metric (as in the classical works of J. Radon, W. Blaschke, G. Birkhoff, and M. M. Day), and also it does not use typical methods from plane Minkowski Geometry (as proposed by H. Martini and K. J. Swanepoel). We also discuss some properties of normed planes whose unit circle is a Radon curve and give characterizations of Radon curves only in terms of Convex Geometry.     

Complete Upper Angle Around a Point on the Minkowski Plane

The dependence of the complete upper angle in the sense of A. D. Aleksandrov about a point on the Minkowski plane on the form of the “unit circle” (the centrally symmetric convex curve Φ determining the Minkowski metric ρ_Φ) is studied.The complete upper angle is computed in three cases: if Φ is a square, a “cut circle,” or a “rounded rhombus.” Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 42–53.     

Splitting and pinching for convex sets

We consider noncompact, closed and convex sets with nonvoid interior in Euclidean space. It is shown that if such a set has one curvature measure sufficiently close to the boundary measure, then it is congruent to a product of a vector space and a compact convex body. Related stability and characterization theorems for orthogonal disc cylinders are proved. Our arguments are based on the Steiner-Schwarz symmetrization processes and generalized Minkowski integral formulas.     

Key distribution patterns using Minkowski planes

A key distribution pattern is a combinatorial structure which provides a secure method of distributing secret keys among a number of participants in a cryptographic scheme. Inversive and Laguerre planes have been used to construct key distribution patterns with storage requirements lower than the trivial distribution system. In this note we construct key distribution patterns from Minkowski planes, the third of the so-calledcircle geometries.The author acknowledges the support of the Australian Research Council     

Finite Blaschke product interpolation on the closed unit disc

We show how to construct all finite Blaschke product solutions and the minimal scaled Blaschke product solution to the Nevanlinna-Pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data. Based on a result of Jones and Ruscheweyh we note that there always exists a finite Blaschke product of degree at most n−1 that maps n distinct points in the closed unit disc, of which at least one is on the unit circle, into n arbitrary points in the closed unit disc, provided that the points inside the unit circle form a positive semi-definite Pick matrix of full rank. Finally, we discuss a numerical limiting procedure.     

Quermass-interaction process with convex compact grains

The paper concerns an extension of random disc Quermass-interaction process, i.e. the model of discs with mutual interactions, to the process of interacting objects of more general shapes. Based on the results for the random disc process and the process with polygonal grains, theoretical results for the generalized process are derived. Further, a simulation method, its advantages and the corresponding complications are described, and some examples are introduced. Finally, a short comparison to the random disc process is given.