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looks like I started extremum
(I wanted minimum not to be gray at higher dimensional Chern-Simons theory…)
I added an introductory bit.
The typical setup in nonlinear analysis and optimization (which is the math of extremal values) is in fact not globally smooth as the entry asserts. A more typical setup is the analysis of convex functions, piecewise convexity etc. In smooth setup one emphasizes on critical points. In particular, the saddle point is critical, but not even locally extremal, unlike what the title of the section suggests.
I don’t think that anything there asserts that the typical setup is globally smooth. The section that Urs wrote, of course, focusses on the $C^2$-functions, which is enough to emphasise critical points, justifying a brief mention of saddle points. But I would regard that section as just a small part of what ought to be at that page.
In idea section: ” these terms are typically used in an analytic context”
In idea section: ” these terms are typically used in an analytic context”
What is meant is analysis not analytic functions (as the hyperlink also shows). But let’s add clarification.
I changed it to say ‘typically used in analysis’, which is better language anyway.
Thanks, but I still disagree mathematically and terminologically with the second section “Local extrema…”. First, saddle point is a critical point and not a local extremum at all, hence it should not be listed as a case of local extremum (by the definition in the entry!), but as a case of a critical point. Second, the entry says that we say that something is a strict local maximum if Hessian is negative definite. First, this is not a definition, but a result and the result is not quite right as stated, namely it may be that the Hessian is zero but still one has strict local maximum, for that one looks at higher derivatives. Thus the Hessian being negative definite is a sufficient condition only, and this is so not by the definition, but by a theorem of sufficiency. Do we agree ?
Do we agree ?
Sure. I would still keep the saddle point there for completeness of that list, and just add a sentence saying that this is not regarded as a local extremum.
It is not about regarding case by case but about fullfilling the general definition which is above.
- if $H_x(f)$ is an indefinite form, then $x$ is called a nondegenerate saddle point.
This is not true if indefinite = not definite. Hessian can be identical to zero and that we still have the local extremum. For that we need to look at higher derivatives. If however we define (strictly?) indefinite as having both strictly positive and strictly negative eigenvalues then it is true. I do not know what are the conventions on “indefinite” ?
Zoran, just go ahead and make the changes that you find necessary.
If you decide to remove any mentioning of saddle points, I will put them back in under “related concepts” or something.
I do NOT know what are the conventions on INDEFINITE (see 11), that is why I asked, and can not supply information about which I do not know – a person who used it should supply his/her definition in this discussion as we disagree. What is the use of nForum if it is not to build the knowledge by consensus but to say “do it alone” ?? Should I sign out from nForum ? Thanks for discouraging dicussion.
Hey, I didn’t understand that you are actually asking about what the right definition is. I thought you kept complaining that we mention saddle points. I didn’t even see #11 when posting my previous one.
No saddle point should be mentioned, but the facts should be made straight. And my opinion was we need to discuss to make things straight, it is not far from where we get. P.S. the entry is partly corrected but the indefinite part is open. I also do not quite understand what is the meaning of “nondegenerate” when referring to a saddle point.
Zoran and Urs, I tried to do some rewriting here. I hope I got everything right and that this is satisfactory, but let me know if not. In particular, the mention of saddle points is relegated to a parenthetical comment, just to round out discussion of nondegenerate critical points, as Urs wanted, but no more.
Oh great, your careful degeneracy treatment has solved the remaning questions in my view. I should have known this, but forgot.
Thanks, Todd, for taking the time.
If however we define indefinite as having both strictly positive and strictly negative eigenvalues
This is the meaning as I understand it. Compare at inner product (where the general definition is incomplete but the case for an inner product valued in an ordered field is covered).
I moved the definition of a strict extremum out of the discussion of $C^2$-functions and into the introduction, since it too applies generally.
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