Timeline for Morse theory Vs degree theory
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2019 at 9:34 | vote | accept | Vrouvrou | ||
| Sep 12, 2014 at 15:32 | review | Close votes | |||
| Sep 13, 2014 at 10:38 | |||||
| Sep 12, 2014 at 10:37 | answer | added | Jean Van Schaftingen | timeline score: 3 | |
| Sep 12, 2014 at 9:45 | answer | added | Liviu Nicolaescu | timeline score: 5 | |
| Sep 12, 2014 at 6:44 | comment | added | Vrouvrou | @LiviuNicolaescu have you an idea about example, | |
| Sep 12, 2014 at 5:58 | comment | added | Vrouvrou | @ViditNanda please an example, i don't find, and what is the relation with the number of solutions ? please | |
| Sep 11, 2014 at 20:20 | comment | added | Vrouvrou | what is the relation with the problem @LiviuNicolaescu please ? | |
| Sep 11, 2014 at 20:18 | comment | added | Liviu Nicolaescu | Chang refers to variational equations. For a long time the variational methods were the only ones used to find solutions to (variational) equations. Judging by the number of publications, degree theory does not have as many applications as the variational techniques. | |
| Sep 11, 2014 at 20:09 | comment | added | Vrouvrou | @ViditNanda please have you a document where i can understand this please | |
| Sep 11, 2014 at 18:53 | comment | added | Vrouvrou | I don't realy understand can you give me an exaple please? and what about :"better estimate of the number of the solutions " thank you | |
| Sep 11, 2014 at 18:46 | comment | added | Vidit Nanda | Right, so the Leray Schauder index is the Euler characteristic of the critical groups. It is easy to come up with examples of spaces which have the same Euler characteristic but completely different homology: in that sense, the critical groups are more refined invariants. | |
| Sep 11, 2014 at 18:06 | comment | added | Vrouvrou | Iknow that the critical groups distiguishe between the different critical point | |
| Sep 11, 2014 at 18:04 | comment | added | Vrouvrou | I do not know many things about the topological degree, i know that the critical groups of a functionnal at an isolated critical point is given by $C_k(\varphi,p)=H_k(\varphi^c\cap U,\varphi^c\cap U\setminus \lbrace p\rbrace), ~k\in\mathbb{N},$ and i know the relation between the leray-schauder index and the critical groups for a compact perturbation of identity $i(\varphi',p)=\sum_{k=0}^{\infty}(-1)^k \dim C_k(\varphi,p)$ | |
| Sep 11, 2014 at 17:57 | comment | added | Vidit Nanda | It might help you get answers if you mention what you already know about critical groups, etc. For instance, it is clear that the topological degree is just an integer whereas a homology group has more algebraic structure. In general, given $f:X \to R$ you can associate to each critical point $c$ the relative homology of $(X^c,X^c\setminus \{c\})$ where $X^c$ is given by $x$ in $X$ with $f(x) \leq f(c)$. | |
| Sep 11, 2014 at 16:39 | history | asked | Vrouvrou | CC BY-SA 3.0 |