scholarly journals A sufficient condition for an extreme covering of n-space by spheres

1968 ◽  
Vol 8 (1) ◽  
pp. 56-62 ◽  
Author(s):  
T. J. Dickson
Keyword(s):  
Euclidean Space ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition

The problem of finding the most economical coverings of n-dimensional Euclidean space by equal spheres whose centres form a lattice, which is equivalent to a problem concerning the inhomogeneous minima of positive definite quadratic forms, has been discussed recently by Barnes and Dickson [1]. The reader is referred to [1] for a complete background on the problem. Terms and notations used will be as in that paper.

scholarly journals A characterisation of the ellipsoid

1970 ◽  
Vol 11 (4) ◽  
pp. 385-394 ◽  
Author(s):  
P. W. Aitchison
Keyword(s):  
Euclidean Space ◽  
Quadratic Forms ◽  
Convex Bodies ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
Major Result

The ellipsoid is characterised among all convex bodies in n-dimensional Euclidean space, Rn, by many different properties. In this paper we give a characterisation which generalizes a number of previous results mentioned in [2], p. 142. The major result will be used, in a paper yet to be published, to prove some results concerning generalizations of the Minkowski theory of reduction of positive definite quadratic forms.

scholarly journals Two finiteness theorems in the Minkowski theory of reduction

1972 ◽  
Vol 14 (3) ◽  
pp. 336-351 ◽  
Author(s):  
P. W. Aitchison
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Reduction Theory ◽  
Positive Definite Quadratic Form ◽  
Finiteness Theorems

Minkowski proved two important finiteness theorems concerning the reduction theory of positive definite quadratic forms (see [6], p. 285 or [7], §8 and §10). A positive definite quadratic form in n variables may be considered as an ellipsoid in n-dimensional Euclidean space, Rn, and then the two results can be investigated more generally by replacing the ellipsoid by any symmetric convex body in Rn. We show here that when n≧3 the two finiteness theorems hold only in the case of the ellipsoid. This is equivalent to showing that Minkowski's results do not hold in a general Minkowski space, namely in a euclidean space where the unit ball is a general symmetric convex body instead of the sphere or ellipsoid.

scholarly journals Canal surfaces in 4-dimensional Euclidean space

2016 ◽  
Vol 7 (1) ◽  
pp. 83-89 ◽  
Author(s):  
Betül Bulca ◽  
Kadri Arslan ◽  
Bengü Bayram ◽  
Günay Öztürk
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Four Dimensions ◽  
Canal Surfaces ◽  
Curvature Ellipse ◽  
Normal Vectors ◽  
Necessary And Sufficient

In the present study, we consider canal surfaces imbedded in an Euclidean space of four dimensions. The curvature properties of these surface are investigated with respect to the variation of the normal vectors and curvature ellipse. We also give some special examples of canal surfaces in E^4. Further, we give necessary and sufficient condition for canal surfaces in E^4 to become superconformal. Finally, the visualization of the projections of canal surfaces in E^3 are presented.

scholarly journals Framed slant helices in Euclidean 3-space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osman Zeki Okuyucu ◽  
Mevlüt Canbirdi
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Slant Helix ◽  
Spherical Images ◽  
Necessary And Sufficient

AbstractIn this paper, we define framed slant helices and give a necessary and sufficient condition for them in three-dimensional Euclidean space. Then, we introduce the spherical images of a framed curve. Also, we examine the relations between a framed slant helix and its spherical images. Moreover, we give an example of a framed slant helix and its spherical images with figures.

scholarly journals Holonomy Groups Of Hypersurfaces

1956 ◽  
Vol 10 ◽  
pp. 8-14 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Orthogonal Group ◽  
Closed Subgroup ◽  
Compact Riemannian Manifold ◽  
Holonomy Group ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition ◽  
Compact Hypersurface ◽  
Proper Orthogonal

The restricted homogeneous holonomy group of an n–dimensional Riemannian manifold is a connected closed subgroup of the proper orthogonal group SO(n) [1]. In this note we shall prove that the restricted homogeneous holonomy group of an n-dimensional compact hypersurface in the Euclidean space is actually the proper orthogonal group SO(n) itself. This gives a necessary (of course, not sufficient) condition for the imbedding of an n-dimensional compact Riemannian manifold into the (n +1)–dimensional Euclidean space.

scholarly journals Characterizations of the sphere by the mean II-curvature

1981 ◽  
Vol 23 (2) ◽  
pp. 249-253 ◽  
Author(s):  
George Stamou
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Gaussian Curvature ◽  
Convex Surface ◽  
Three Dimensional ◽  
Second Fundamental Form ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
The Second Fundamental Form ◽  
The Mean

The notion of “mean II-curvature” of a C4-surface (without parabolic points) in the three-dimensional Euclidean space has been introduced by Ekkehart Glässner. The aim of this note is to give some global characterizations of the sphere related to the above notion.In the three-dimensional Euclidean space E3 we consider a sufficiently smooth ovaloid S (closed convex surface) with Gaussian curvature K > 0 . The ovaloid S possesses a positive definite second fundamental form II, if appropriately oriented. During the last years several authors have been concerned with the problem of characterizations of the sphere by the curvature of the second fundamental form of S. In this paper we give some characterizations of the sphere using the concept of the mean II-curvatureHII (of S), defined by Ekkehart Glässner.

scholarly journals Enclosure Theorems for Eigenvalues of Elliptic Operators

1970 ◽  
Vol 13 (1) ◽  
pp. 1-7 ◽  
Author(s):  
John C. Clements
Keyword(s):  
Differential Operator ◽  
Euclidean Space ◽  
Partial Differential Operator ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
Partial Differential ◽  
Partial Differentiation ◽  
The Matrix ◽  
Image Position ◽  
Uniformly Continuous

Let L be the linear, elliptic, self-adjoint partial differential operator given by where Dj denotes partial differentiation with respect to xj, 1 ≤ j ≤ n, b is a positive, continuous real-valued function of x = (x1,…,xn) in n-dimensional Euclidean space En, the aij are real-valued functions possessing uniformly continuous first partial derivatives in En and the matrix {aij} is everywhere positive definite. A solution u of Lu = 0 is assumed to be of class C1.

scholarly journals On generating points of a lattice in the region

1964 ◽  
Vol 6 (3) ◽  
pp. 141-155 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Image Position ◽  
Necessary And Sufficient

A lattice An in n-dimensional Euclidean space En consists of the aggregate of all points with coordinates (xx,…, xn), wherefor some real ars (r, s = 1,…, n), subject to the condition ∥ αrs ∥nn ╪ 0. The determinant Δn of Λn, is denned by the relation , the sign being chosen to ensure that Δn > 0.If A1…, An are the n points of Λn having coordinates (a11, a21…, anl),…, (a1n, a2n,…, ann), respectively, then every point of Λn may be expressed in the formand Ai,…, An, together with the origin O, are said to generate Λn. This particular set of generating points is not unique; it may be proved that a necessary and sufficient condition that n points of Λn should generate the lattice is that the n × n determinant formed by their x coordinates should be ±Δn, or, equivalently, that the n×n determinant formed by their corresponding u-coordinates should be ±1.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

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