Self-Formation of the Spatial Planar Object. Topological Approach

2004 ◽  
Vol 97-98 ◽  
pp. 85-90
Author(s):  
Stepas Janušonis
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Anisotropic Medium ◽  
Topological Approach ◽  
Spatial Objects ◽  
General Approximation ◽  
Complementary Space ◽  
Self Formation ◽  
Homogeneous Anisotropic Medium

Eight-dimensional topological space providing an object evolution in time, including causes of evolution is presented. Part of Euclidean space separated by any close surface from complementary space, where any Euclidean point of space is juxtaposed with parameter, is being felt as an object. Coplanar approximation of flat planar devices is based on the flat, homogeneous, isotropic planar object and chaotic medium. The new, more general approximation of the topological space by equidistant surfaces, suitable for spatial planar objects, is presented. Selfformation of spatial objects (homogeneous, non-homogeneous, anisotropic), medium (chaotic, chaotic oriented, homogeneous oriented, structural) based on non-homeomorpheous mapping in peculiar points and evolution irreversibility, is discussed.

scholarly journals Singular homology algorithm for MA-spaces

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2139-2147
Author(s):  
Demir Unver
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Digital Geometry ◽  
Topological Approach ◽  
Homology Groups ◽  
And Topology ◽  
Singular Homology ◽  
Digital Spaces

The work on digitizing subspaces of the 2-D Euclidean space with a certain digital approach is an important discipline in both digital geometry and topology. The present work considers Marcus-Wyse topological approach which was established for studying 2-D digital spaces, ?2. We introduce the digital singular homology groups of MA-spaces (M-topological space with an M-adjacency), and we compute singular homology groups of some certain MA-spaces, we give a formula for singular homology groups of 2-D simple closed MA-curves, and an algorithm for determining homology groups of an arbitrary MA-space.

The wave front in a homogeneous anisotropic medium

1980 ◽  
Vol 70 (6) ◽  
pp. 2097-2101
Author(s):  
M. J. Yedlin
Keyword(s):  
Dispersion Relation ◽  
Wave Front ◽  
Group Velocity ◽  
Anisotropic Medium ◽  
Slowness Surface ◽  
Constant Frequency ◽  
Common Notion ◽  
The Common ◽  
Intuitive Method ◽  
Homogeneous Anisotropic Medium

abstract A simple geometric construction is derived for the shape of the wave front in a homogeneous anisotropic medium. It is shown to be equivalent to the intuitive method of constructing a wave front using Huygen's principle. Although this construction has been referred to and tersely described in the literature (Musgrave, 1970; Kraut, 1963; Duff, 1960), it is instructive to demonstrate its relationship to the common notion of the wave front obtained via consideration of the group velocity. The wave front is shown to be the polar reciprocal of the slowness surface (the dispersion relation at constant frequency). An appreciation of the pole-polar correspondence between the two surfaces allows quick inference of some of the important features of the wave front in a homogeneous anisotropic medium.

Thermoelastic Interaction Between Singularities and Interfaces in an Anisotropic Trimaterial

2007 ◽  
Vol 74 (6) ◽  
pp. 1285-1288
Author(s):  
Seung Tae Choi
Keyword(s):  
Analytic Continuation ◽  
Anisotropic Medium ◽  
Dissimilar Materials ◽  
Infinite Body ◽  
Series Form ◽  
Alternating Technique ◽  
Thermoelastic Interaction ◽  
General Plane ◽  
Homogeneous Anisotropic Medium ◽  
Plane Deformations

The method of analytic continuation and Schwarz-Neumann’s alternating technique were applied to the thermoelastic interaction problems of singularities and interfaces in an anisotropic “trimaterial,” which denotes an infinite body composed of three dissimilar materials bonded along two parallel interfaces. It was assumed that the linear thermoelastic materials are under general plane deformations in which the plane of deformation is perpendicular to the planes of the two parallel interfaces. The author then showed that by alternately applying the method of analytic continuation across two parallel interfaces the solution for the thermoelastic singularities in an anisotropic trimaterial can be obtained in a series form from a solution for the same singularities in a homogeneous anisotropic medium.

scholarly journals Topological Manifolds

2014 ◽  
Vol 22 (2) ◽  
pp. 179-186 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Interior Point ◽  
Cartesian Product ◽  
Closed Ball ◽  
Topological Manifold ◽  
Open Ball ◽  
Connected Component ◽  
Euclidean Spaces ◽  
Topological Manifolds

Summary Let us recall that a topological space M is a topological manifold if M is second-countable Hausdorff and locally Euclidean, i.e. each point has a neighborhood that is homeomorphic to an open ball of E n for some n. However, if we would like to consider a topological manifold with a boundary, we have to extend this definition. Therefore, we introduce here the concept of a locally Euclidean space that covers both cases (with and without a boundary), i.e. where each point has a neighborhood that is homeomorphic to a closed ball of En for some n. Our purpose is to prove, using the Mizar formalism, a number of properties of such locally Euclidean spaces and use them to demonstrate basic properties of a manifold. Let T be a locally Euclidean space. We prove that every interior point of T has a neighborhood homeomorphic to an open ball and that every boundary point of T has a neighborhood homeomorphic to a closed ball, where additionally this point is transformed into a point of the boundary of this ball. When T is n-dimensional, i.e. each point of T has a neighborhood that is homeomorphic to a closed ball of En, we show that the interior of T is a locally Euclidean space without boundary of dimension n and the boundary of T is a locally Euclidean space without boundary of dimension n − 1. Additionally, we show that every connected component of a compact locally Euclidean space is a locally Euclidean space of some dimension. We prove also that the Cartesian product of locally Euclidean spaces also forms a locally Euclidean space. We determine the interior and boundary of this product and show that its dimension is the sum of the dimensions of its factors. At the end, we present several consequences of these results for topological manifolds. This article is based on [14].

Almost Continuity of Mappings

1968 ◽  
Vol 11 (3) ◽  
pp. 453-455 ◽  
Author(s):  
Shwu-Yeng T. Lin
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Direct Proof ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Natural Context ◽  
Set Of Points

Let E be a metric Baire space and f a real valued function on E. Then the set of points of almost continuity in E is dense (everywhere) in E.Our purpose is to set this result in its most natural context, relax some very restricted hypotheses, and to supply a direct proof. More precisely, we shall prove that the metrizability of E in Theorem H may be removed, and that the range space may be generalized from the (Euclidean) space of real numbers to any topological space satisfying the second axiom of countability [2].

Two-dimensional invisibility anti-cloak structured by a homogeneous anisotropic medium

2016 ◽  
Vol 83 (9) ◽  
pp. 532
Author(s):  
Xuan Liu ◽  
Yicheng Wu ◽  
Chengdong He ◽  
Yuzhuo Wang ◽  
Xiaojia Wu ◽  
...  
Keyword(s):  
Anisotropic Medium ◽  
Two Dimensional ◽  
Homogeneous Anisotropic Medium

Polarization modulation of electromagnetic waves scattered from a quasi-homogeneous anisotropic medium

2012 ◽  
Vol 14 (12) ◽  
pp. 125705 ◽  
Author(s):  
Jia Li ◽  
Zhaoxia Shi ◽  
Hongliang Ren ◽  
Hao Wen ◽  
Jin Lu ◽  
...  
Keyword(s):  
Anisotropic Medium ◽  
Electromagnetic Waves ◽  
Polarization Modulation ◽  
Homogeneous Anisotropic Medium
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