# Tagged Questions

The tag has no usage guidance.

36 views

### Polynomial with subset of critical points and values prescribed

Motivated by this question I am motivated to pose the following question: Given $k$ points $(x_1, y_1), \ldots (x_k, y_k)$ with (WLOG) $x_i < x_{i+1}$, can we find a polynomial $p(x)\in\Bbb R[x]$ ...
189 views

49 views

### Estimation of the number of local extrema

I have a question about a simple proposition, I suppose that this is something well-known or a special case of something well-known: Let $D\subset\mathbb{R}^{2}$ be the closed unit disk in the plane ...
308 views

### Lusternik-Schnirelmann Theorem

In various paper i found this: But i don't find this theorem of Lusternik-Schnirelmann, have you an idea where i can find this theorem, the condition? Thank you.
248 views

### Set of critical values is compact [closed]

Let $M\subseteq \mathbb{R}^n$ be a compact manifold with $\partial M=\emptyset$ and let $f: M\rightarrow S^p$ be a smooth map. I want to unerstand the proof of the following lemma which occurs in ...
148 views

Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow \... 0answers 126 views ### Tubular neighborhoods in the proof of the Morse homology theorem I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here: http://www.mtm.ufsc.br/... 0answers 103 views ### Categorizing saddle points of real multivariate polynomials I have a multivariate polynomial function of N variables$f(x_1,x_2,…,x_N) = x_1 x_2 x_3 .. x_N \left( 1 + \sum_i^N (a_i x_i^2 - x_i) \right)$, where$a_i > 0$are real positive numbers. By ... 1answer 466 views ### Failure of Palais-Smale Condition C and the Mini-Max Principle To get a thorough analysis of the critical point structure of a smooth function$f:M\to\mathbb{R}$on a smooth Hilbert manifold$M$, a compactness assumption gets us far. That assumption is Condition ... 1answer 108 views ### Continuity of critical points with respect to a parameterisation. Hello. I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in ... 0answers 292 views ### What is the analog of “monotonic” for scalar functions on surfaces? "monotonic" is well defined for functions$f(x)$, where e.g.$x\in[0,1]$and$f(x)\in\mathbb{R}$. The quality I particularly care about is that if$f(x)$is monotonic then it will not have any local ... 1answer 530 views ### How to explain the condition (C) in critical point theory? Condition (C). The closure of any nonempty subset S of H on which f is bounded but on which$\|\nabla f\|$is not bounded away from zero, contains a critical point of f. How to see the meaning of "$\|\...
Let $f$ be a $C^1$ functional on a Hilbert space $X$, and $Y$ a closed subspace of $X$. Suppose the restriction of $f$ on $Y$ has a critical point $x_0 \in Y$. Q: Is $x_0$ a critical point of $f$?
### Do the focal points of a submanifold $M$ in $\mathbb{R}^k$ form a closed subset?
Let $M$ be a submanifold in an euclidean space $\mathbb{R}^k$, and $\nu(M)$ the normal bundle to $M$, let us denote $\phi$ the restriction to $\nu(M)$ of the exponential map for $\mathbb{R}^k$. A ...