Landau–Lifshitz model: Difference between revisions
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The concept of '''size homotopy group''' is analogous in [[size theory]] of the classical concept of [[homotopy group]]. In order to give its definition, let us assume that a [[size pair]] <math>(M,\varphi)\ </math> is given, where <math>M\ </math> is a [[closed manifold]] of class <math>C^0\ </math> and <math>\varphi:M\to \mathbb{R}^k\ </math> is a [[continuous function]]. Let us consider the [[partial order]] <math>\preceq\ </math> in <math>\mathbb{R}^k\ </math> defined by setting <math>(x_1,\ldots,x_k)\preceq(y_1,\ldots,y_k)\ </math> if and only if <math>x_1 \le y_1,\ldots, x_k \le y_k\ </math>. For every <math>Y\in\mathbb{R}^k\ </math> we set <math>M_{Y}=\{Z\in\mathbb{R}^k:Z\preceq Y\}\ </math>. | |||
Assume that <math>P\in M_X\ </math> and <math>X\preceq Y\ </math>. If <math>\alpha\ </math>, <math>\beta\ </math> are two paths from <math>P\ </math> to <math>P\ </math> and a [[homotopy]] from <math>\alpha\ </math> to <math>\beta\ </math>, based at <math>P\ </math>, exists in the [[topological space]] <math>M_{Y}\ </math>, then we write <math>\alpha \approx_{Y}\beta\ </math>. The '''first size homotopy group''' of the [[size pair]] <math>(M,\varphi)\ </math> computed at <math>(X,Y)\ </math> is defined to be the [[quotient set]] of the set of all [[path (topology)|path]]s from <math>P\ </math> to <math>P\ </math> in <math>M_X\ </math> with respect to the [[equivalence relation]] <math>\approx_{Y}\ </math>, endowed with the operation induced by the usual composition of based [[loop (topology)|loop]]s.<ref name="FroMu99">Patrizio Frosini, Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society – Simon Stevin, 6:455–464, 1999.</ref> | |||
In other words, the '''first size homotopy group''' of the [[size pair]] <math>(M,\varphi)\ </math> computed at <math>(X,Y)\ </math> and <math>P\ </math> is the image | |||
<math>h_{XY}(\pi_1(M_X,P))\ </math> | |||
of the first [[homotopy group]] <math>\pi_1(M_X,P)\ </math> with base point <math>P\ </math> of the [[topological space]] <math>M_X\ </math>, when <math>h_{XY}\ </math> is the [[homomorphism]] induced by the inclusion of <math>M_X\ </math> in <math>M_Y\ </math>. | |||
The <math>n\ </math>-th size homotopy group is obtained by substituting the loops based at <math>P\ </math> with the [[continuous function]]s <math>\alpha:S^n\to M\ </math> taking a fixed point of <math>S^n\ </math> to <math>P\ </math>, as happens when higher [[homotopy group]]s are defined. | |||
==References== | |||
{{reflist}} | |||
==See also== | |||
* [[Size function]] | |||
* [[Size functor]] | |||
* [[Size pair]] | |||
* [[Natural pseudodistance]] | |||
[[Category:Algebraic topology]] | |||
Latest revision as of 18:50, 30 July 2013
30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair is given, where is a closed manifold of class and is a continuous function. Let us consider the partial order in defined by setting if and only if . For every we set .
Assume that and . If , are two paths from to and a homotopy from to , based at , exists in the topological space , then we write . The first size homotopy group of the size pair computed at is defined to be the quotient set of the set of all paths from to in with respect to the equivalence relation , endowed with the operation induced by the usual composition of based loops.[1]
In other words, the first size homotopy group of the size pair computed at and is the image of the first homotopy group with base point of the topological space , when is the homomorphism induced by the inclusion of in .
The -th size homotopy group is obtained by substituting the loops based at with the continuous functions taking a fixed point of to , as happens when higher homotopy groups are defined.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
See also
- ↑ Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society – Simon Stevin, 6:455–464, 1999.