Partitioning Euclidean space

There is a partition of Euclideann-space into countably many sets none of which contains the vertices of a regularn-simplex.     

Rep-tiling Euclidean space

Summary Arep-tiling is a self replicating, lattice tiling ofR ⁿ .Lattice tiling means a tiling by translates of a single compact tile by the points of a lattice, andself-replicating means that there is a non-singular linear mapø: R ⁿ Rⁿ such that, for eachT , the imageø(T) is, in turn, tiled by . This topic has recently come under investigation, not only because of its recreational appeal, but because of its application to the theory of wavelets and to computer addressing. The paper presents an exposition of some recent results on rep-tiling, including a construction of essentially all rep-tilings of Euclidean space. The construction is based on radix representation of points of a lattice. One particular radix representation, called thegeneralized balanced ternary, is singled out as an example because of its relevance to the field of computer vision.     

Pohlmeyer transformation in Euclidean space

Journal of Mathematical Sciences - The Pohlmeyer reduction is generalized for σ-models [Kahler, ℂp n, O(3)] in a twodimensional Euclidean space, possessing instanton solutions. The...     

欧氏空间中具有常典则支撑函数的子流形

本文证明了下述结论：欧氏空间中完备连通且具有常典则支撑函数的子流形必为球面子流形或通过原点的线性子空间，从而改进了关于欧氏空间子流形的一个经典结果。     

Semi-parallel surfaces in Euclidean space

Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersions. Using results of Backes on Euclidean Jordan triple systems, the totally geodesic immersions are shown to be the only minimal semi-parallel immersions into a Euclidean space. Semi-parallel immersions of surfaces into E^m are studied and a classification of semi-parallel immersions with pointwise planar normal sections of surfaces in E^m is given.Research Assistant of the National Fund of Scientific Research     

Embedding metric spaces in Euclidean space

In this paper, we give necessary and sufficient conditions for embedding a given metric space in Euclidean space. We shall introduce the notions of flatness and dimension for metric spaces and prove that a metric space can be embedded in Euclidean n-space if and only if the metric space is flat and of dimension less than or equal to n.     

Stationary rotating surfaces in Euclidean space

A stationary rotating surface is a compact surface in Euclidean space whose mean curvature H at each point x satisfies 2H(x) = a r(x)² + b, where r(x) denotes the distance from x to a fixed straight-line L, and a and b are constants. These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about L with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary.     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

Spherical functions on Euclidean space

We study spherical functions on Euclidean spaces from the viewpoint of Riemannian symmetric spaces. Here the Euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)—spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in En.     

Brownian functionals on hypersurfaces in Euclidean space

Using the first exit time for Brownian motion from a smoothly bounded domain in Euclidean space, we define two natural functionals on the space of embedded, compact, oriented, unparametrized hypersurfaces in Euclidean space. We develop explicit formulas for the first variation of each of the functionals and characterize the critical points.

     

Parabolic Muckenhoupt weights in the Euclidean space

We introduce a parabolic analogue of Muckenhoupt?s Ap class. We show that these weights satisfy a reverse Hölder type inequality and also prove a John-Nirenberg inequality for a related BMO type class. Both results are multidimensional generalizations of known one-dimensional results. As an application of our methods, we extend a two-dimensional weighted norm inequality of A. Lerner and S. Ombrosi to dimensions n?3.     

Biharmonic hypersurfaces in Euclidean space E5

In this paper, we study biharmonic hypersurfaces in E⁵. We prove that every biharmonic hypersurface in Euclidean space E⁵ must be minimal.