The generation and classification of tile-k-transitive tilings of the Euclidean plane,the sphere and the hyperbolic plane

Algorithms based on the theory of Delaney-Dress symbols are discussed that can be used to recursively produce all possible equivariant types of tile-k-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane, for any (reasonable)k. A number of results can be obtained using computer implementations of the algorithms.     

A topological characterization of the ω-limit sets for analytic flows on the plane, the sphere and the projective plane

In this paper the ω-limit sets for analytic and polynomial differential equations on the plane are characterized up to homeomorphisms. The analogous problem is solved in full detail for analytic flows on the sphere and the projective plane. We also explain how to carry on the same program for analytic flows defined on open subsets of these surfaces.     

A characterization of the sphere

Summary. This paper gives a "nice" characterization of the sphere: Taking into account the well-known secant's law that the product of the segments of a secant through a point P in the sphere is independent of the secant but depends only on the point P, one can derive -- roughly speaking -- the following theorem: If you run along a closed curve in the interior of a sphere with constant speed, at any point you can measure the distance to the sphere 'in front of' and 'behind' you. Then the integral over the distance 'in front of you' along the curve equals the integral over the distance 'behind you'. Moreover, and this is the more interesting fact, this property characterizes the sphere among closed convex domains.     

A primer of substitution tilings of the Euclidean plane

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to represent the field as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.     

Solid circle-packings in the Euclidean plane

The main result of this paper is the proof of a conjecture of L. Fejes Tóth saying that the incircles of the Archimedean tiling (4, 8, 8) form a solid packing. To achieve this a new technique, the method of weighted densities, has been developed and applied, besides the case mentioned, to prove the solidity of a number of other circle-packings in the Euclidean plane.To L. Fejes Tóth on his seventy-fifth birthday     

Planar nearrings on the Euclidean plane

Planar near-rings are generalized rings which can serve as coordinate domains for geometric structures in which each pair of nonparallel lines has a unique point of intersection. It is known that all planar nearrings can be constructed from regular groups of automorphisms of groups which can be viewed as the “action groups” of the planar nearring. In this article, we study planar nearrings whose additive group is \({(\mathbb{R}^n,+)}\) , in particular, n = 1 and 2. It is natural to study topological planar nearrings in this context, following ideas of the late Kenneth D. Magill, Jr. In the case of n = 1, we characterize all topological planar nearrings by their action groups \({(\mathbb{R}^*, \cdot)}\) or \({(\mathbb{R}^+, \cdot)}\) . For n = 2, these action groups and the circle group \({(\mathbb{U}, \cdot)}\) seem to be the most interesting cases, but the last case can be excluded completely. As a consequence, we obtain characterizations of the semi-homogeneous continuous mappings from \({\mathbb{R}^n}\) to \({\mathbb{R}}\) for n = 1 and 2. Such a mapping f enjoys the property that f(f(u)v) = f(u)f(v) for all \({u,v \in \mathbb{R}^n}\) . When \({f(\mathbb{R}^n) = \mathbb{R}^+}\) , f is a positive homogeneous mapping of degree 1.     

Geometry of finite-dimensional normed spaces,and continuous functions on the Euclidean sphere

Let ℝⁿ be the n-dimensional Euclidean space, and let { · } be a norm in Rⁿ. Two lines ℓ₁ and ℓ₂ in ℝⁿ are said to be { · }-orthogonal if their { · }-unit direction vectors e ₁ and e ₂ satisfy {e ₁ + e ₂} = {e ₁ − e ₂}. It is proved that for any two norms { · } and { · }′ in ℝⁿ there are n lines ℓ₁, ..., ℓ_n that are { · }-and { · }′-orthogonal simultaneously. Let be a continuous function on the unit sphere with center O. It is proved that there exists an (n − 1)-cube C centered at O, inscribed in , and such that all sums of values of f at the vertices of (n − 3)-faces of C are pairwise equal. If the function f is even, then there exists an n-cube with the same properties. Furthermore, there exists an orthonormal basis e ₁, ..., e _n such that for 1 ≤ i ≤ j ≤ n we have . Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 107–117.     

Quadrature in Besov spaces on the Euclidean sphere

Let q1 be an integer, denote the unit sphere embedded in the Euclidean space , and μ_q be its Lebesgue surface measure. We establish upper and lower bounds for

where is the unit ball of a suitable Besov space on the sphere. The upper bounds are obtained for choices of x_k and w_k that admit exact quadrature for spherical polynomials of a given degree, and satisfy a certain continuity condition; the lower bounds are obtained for the infimum of the above quantity over all choices of x_k and w_k. Since the upper and lower bounds agree with respect to order, the complexity of quadrature in Besov spaces on the sphere is thereby established.     

On-line steiner trees in the Euclidean plane

Suppose we are given a sequence ofn points in the Euclidean plane, and our objective is to construct, on-line, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. The points appear one at a time, and at each step the on-line algorithm must construct a connected graph that contains all current points by connecting the new point to the previously constructed graph. This can be done by joining the new point (not necessarily by a straight line) to any point of the previous graph (not necessarily one of the given points). The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences of points, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points. There are known on-line algorithms whose competitive ratio isO(logn) even for all metric spaces, but the only lower bound known is of [IW] for some contrived discrete metric space. Moreover, for the plane, on-line algorithms could have been more powerful and achieve a better competitive ratio, and no nontrivial lower bounds for the best possible competitive ratio were known. Here we prove an almost tight lower bound of Ω(logn/log logn) for the competitive ratio of any on-line algorithm. The lower bound holds for deterministic algorithms as well as for randomized ones, and obviously holds in any Euclidean space of dimension greater than 2 as well. Noga Alon was supported in part by a USA-Israeli BSF grant.     

Hypersurfaces of the Euclidean sphere with nonnegative Ricci curvature

In this paper we prove that a compact oriented hypersurface of a Euclidean sphere with nonnegative Ricci curvature and infinite fundamental group is isometric to an H(r)-torus with constant mean curvature. Furthermore, we generalize, whithout any hypothesis about the mean curvature, a characterization of Clifford torus due to Hasanis and Vlachos. Received: 19 March 2002     

Pedal curves of frontals in the Euclidean plane

In this paper, we will give the definition of the pedal curves of frontals and investigate the geometric properties of these curves in the Euclidean plane. We obtain that pedal curves of frontals in the Euclidean plane are also frontals. We further discuss the connections between singular points of the pedal curves and inflexion points of frontals in the Euclidean plane.     

The equidistant involution of the hyperbolic plane and two models of the Euclidean plane geometry

In this paper we define an involution of the hyperbolic plane corresponding to an equidistant curve and a point of its base line that keeps a certain subset of the equidistant curves invariant. Based on this mapping we present two models of the Euclidean geometry in the hyperbolic plane.     

On the correlations of directions in the Euclidean plane

Let denote the repartition of the -level correlation measure of the finite set of directions , where is the fixed point and is an integer lattice point in the square . We show that the average of the pair correlation repartition over in a fixed disc converges as . More precisely we prove, for every and , the estimate

We also prove that for each individual point , the -level correlation diverges at any point as , and we give an explicit lower bound for the rate of divergence.

     

On extremal point distributions in the Euclidean plane

We ask for the maximum σ _n ^γ of Σ _i,j=1 ⁿ ‖x _i-x _j‖^γ, where x ₁,χ,x _n are points in the Euclidean plane R ² with ‖x_i-x_j‖ ≦1 for all 1≦ i,j ≦ n and where ‖.‖^γ denotes the γ-th power of the Euclidean norm, γ ≧ 1. (For γ =1 this question was stated by L. Fejes Tóth in [1].) We calculate the exact value of σ _n ^γ for all γ γ 1,0758χ and give the distributions which attain the maximum σ _n ^γ . Moreover we prove upper bounds for σ _n ^γ for all γ ≧ 1 and calculate the exact value of σ ₄ ^γ for all γ ≧ 1. This revised version was published online in June 2006 with corrections to the Cover Date.