advanced_tools:topology:homotopy
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
advanced_tools:topology:homotopy [2018/03/24 14:46] jakobadmin ↷ Links adapted because of a move operation |
advanced_tools:topology:homotopy [2018/04/12 15:33] (current) jakobadmin [Concrete] |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ====== Homotopy ====== | ====== Homotopy ====== | ||
| + | |||
| + | |||
| + | <tabbox Intuitive> | ||
| + | |||
| + | <note tip> | ||
| + | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
| + | </note> | ||
| + | | ||
| + | <tabbox Concrete> | ||
| + | |||
| + | ** Definition ** | ||
| + | Two maps $f$ and $g$ are called **homotopic**, denoted by $f \tilde g$ if we can continuously transform them into each other. (See the example below). | ||
| + | |||
| + | Source: page 14 in https://arxiv.org/pdf/hep-th/0403286.pdf | ||
| + | |||
| + | ---- | ||
| + | **Example ** | ||
| + | |||
| + | --> Equivalence of $f(x)=x$ and $g(x)=$const.# | ||
| + | |||
| + | The two maps | ||
| + | |||
| + | $$ \mathbb{R}^n \to \mathbb{R}^n : f(x)= x $$ | ||
| + | and | ||
| + | $$ \mathbb{R}^n \to \mathbb{R}^n : g(x)= x_0 = \text{const} $$ | ||
| + | |||
| + | are homotopic, because we can define | ||
| + | |||
| + | $$ F(x,t) = (1-t)x+tx_0 ,$$ | ||
| + | |||
| + | which continuously transforms $f(x)=F(x,0)$ into $g(x)=F(x,1)$. | ||
| + | |||
| + | |||
| + | <-- | ||
| + | |||
| + | <tabbox Abstract> | ||
| + | |||
| + | <note tip> | ||
| + | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| + | </note> | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| Line 41: | Line 81: | ||
| <cite>http://www.dartmouth.edu/~dbr/topdefects.pdf</cite> | <cite>http://www.dartmouth.edu/~dbr/topdefects.pdf</cite> | ||
| </blockquote> | </blockquote> | ||
| - | |||
| - | <tabbox Layman> | ||
| - | |||
| - | <note tip> | ||
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
| - | </note> | ||
| | | ||
| - | <tabbox Student> | ||
| - | ** Definition ** | ||
| - | Two maps $f$ and $g$ are called **homotopic**, denoted by $f \tilde g$ if we can continuously transform them into each other. (See the example below). | ||
| - | Source: page 14 in https://arxiv.org/pdf/hep-th/0403286.pdf | ||
| - | <tabbox Researcher> | ||
| - | |||
| - | <note tip> | ||
| - | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| - | </note> | ||
| - | |||
| - | | ||
| - | <tabbox Examples> | ||
| - | |||
| - | --> Equivalence of $f(x)=x$ and $g(x)=$const.# | ||
| - | |||
| - | The two maps | ||
| - | |||
| - | $$ \mathbb{R}^n \to \mathbb{R}^n : f(x)= x $$ | ||
| - | and | ||
| - | $$ \mathbb{R}^n \to \mathbb{R}^n : g(x)= x_0 = \text{const} $$ | ||
| - | |||
| - | are homotopic, because we can define | ||
| - | |||
| - | $$ F(x,t) = (1-t)x+tx_0 ,$$ | ||
| - | |||
| - | which continuously transforms $f(x)=F(x,0)$ into $g(x)=F(x,1)$. | ||
| - | |||
| - | |||
| - | <-- | ||
| - | |||
| - | |||
| - | |||
| - | <tabbox FAQ> | ||
| - | | ||
| - | <tabbox History> | ||
| </tabbox> | </tabbox> | ||
advanced_tools/topology/homotopy.1521899197.txt.gz · Last modified: 2018/03/24 13:46 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
