scholarly journals Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem

2019 ◽  
Vol 24 (4) ◽  
pp. 88
Author(s):  
Miyuki Koiso
Keyword(s):  
Energy Density ◽  
Surface Energy ◽  
Euclidean Space ◽  
Mean Curvature Flow ◽  
Three Dimensional ◽  
Uniqueness Result ◽  
Dimensional Euclidean Space ◽  
Anisotropic Surface ◽  
Capillary Problem ◽  
Anisotropic Surface Energy

We study a variational problem for hypersurfaces in the Euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered hypersurface, which was introduced to model the surface tension of a small crystal. The purpose of this paper is two-fold. First, we give uniqueness and nonuniqueness results for closed equilibria under weaker assumptions on the regularity of both considered hypersurfaces and the anisotropic surface energy density than previous works and apply the results to the anisotropic mean curvature flow. This part is an announcement of two forthcoming papers by the author. Second, we give a new uniqueness result for stable anisotropic capillary surfaces in a wedge in the three-dimensional Euclidean space.

scholarly journals On a Question of Bourgain about Geometric Incidences

2008 ◽  
Vol 17 (4) ◽  
pp. 619-625 ◽  
Author(s):  
JÓZSEF SOLYMOSI ◽  
CSABA D. TÓTH
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Geometric Incidences

Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.

scholarly journals Orthogonal Isomorphic Representations Of Free Groups

1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Euclidean Space ◽  
Free Groups ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Orthogonal Matrices ◽  
Abelian Subgroups ◽  
Proper Orthogonal

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.

Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space

Robotica ◽  
2015 ◽  
Vol 34 (11) ◽  
pp. 2610-2628 ◽  
Author(s):  
Davood Naderi ◽  
Mehdi Tale-Masouleh ◽  
Payam Varshovi-Jaghargh
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Dimensional Euclidean Space ◽  
Grobner Basis ◽  
Kinematic Space ◽  
Forward Kinematic Problem ◽  
Forward Kinematic

SUMMARYIn this paper, the forward kinematic analysis of 3-degree-of-freedom planar parallel robots with identical limb structures is presented. The proposed algorithm is based on Study's kinematic mapping (E. Study, “von den Bewegungen und Umlegungen,” Math. Ann.39, 441–565 (1891)), resultant method, and the Gröbner basis in seven-dimensional kinematic space. The obtained solution in seven-dimensional kinematic space of the forward kinematic problem is mapped into three-dimensional Euclidean space. An alternative solution of the forward kinematic problem is obtained using resultant method in three-dimensional Euclidean space, and the result is compared with the obtained mapping result from seven-dimensional kinematic space. Both approaches lead to the same maximum number of solutions: 2, 6, 6, 6, 2, 2, 2, 6, 2, and 2 for the forward kinematic problem of planar parallel robots; 3-RPR, 3-RPR, 3-RRR, 3-RRR, 3-RRP, 3-RPP, 3-RPP, 3-PRR, 3-PRR, and 3-PRP, respectively.

The Evaluation of Moments of Regions With Piecewise-Linearly Approximated Boundaries via Line Integration

1989 ◽  
Author(s):  
J. Angeles ◽  
M. J. Al-Daccak
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Inertia Tensor ◽  
Divergence Theorem ◽  
Dimensional Euclidean Space ◽  
Repeated Application ◽  
The Subject ◽  
First Moment

Abstract The subject of this paper is the computation of the first three moments of bounded regions imbedded in the three-dimensional Euclidean space. The method adopted here is based upon a repeated application of Gauss’s Divergence Theorem to reduce the computation of the said moments — volume, vector first moment and inertia tensor — to line integration. Explicit, readily implementable formulae are developed to evaluate the said moments for arbitrary solids, given their piecewise-linearly approximated boundary. An example is included that illustrates the applicability of the formulae.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

scholarly journals Isometric Deformation of (m,n)-Type Helicoidal Surface in the Three Dimensional Euclidean Space

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 226 ◽  
Author(s):  
Erhan Güler
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Beltrami Operator ◽  
Dimensional Euclidean Space ◽  
Natural Numbers ◽  
Rotational Surface ◽  
Isometric Deformation ◽  
Helicoidal Surface

We consider a new kind of helicoidal surface for natural numbers ( m , n ) in the three-dimensional Euclidean space. We study a helicoidal surface of value ( m , n ) , which is locally isometric to a rotational surface of value ( m , n ) . In addition, we calculate the Laplace–Beltrami operator of the rotational surface of value ( 0 , 1 ) .

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