Questions tagged [critical-point-theory]
The critical-point-theory tag has no usage guidance.
50 questions
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About the number of critical points of a function
Suppose that $f$ is a totally monotone function on $(0,\infty)$, so that $(-1)^n f^{(n)}\ge0$ for all $n=0,1,\dots$, $f(0+)\in(0,\infty)$, and $f(t)\sim\frac{1}{t^{\frac{3}{2}}}$ as $t\to\infty$. Can ...
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Characterization of critical point of an integral operator
I have an integral operator and I wonder how I can characterize the critical point.
I'll give a simplified example so maybe people can comment on and I can maybe generalize in another question.
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Sufficient conditions for a polynomial function to have the same critical points as its symmetrized version
Are there any sufficient conditions known for a polynomial function (of many variables) to have the same critical points as its symmetrized version (with respect to all variables)?
This question has ...
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How to distinguish birth and death bifurcations?
Let $f : \mathbb{R} \to \mathbb{R}$ have a degenerate critical point at $x = 0 \, ($ie, $f(0) = f'(0) = f''(0) = 0)$.
Perturbing $f$ locally around $0$ may cause multiple scenarios:
Birth: the ...
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Regularity requirements for Sard's Theorem
The most common formulation of Sard's Theorem is that for $f\in C^{n-m+1}(\mathbb R^n, \mathbb R^m)$ with $n\ge m$, the set $f(C_f)$ has Lebesgue measure 0, where $C_f=\{x, df(x)=0\}$.
Question. Is it ...
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Locality and restriction properties for self-avoiding and loop-erasing random walks
This question has been cross-posted from math.stackexchange.com : https://math.stackexchange.com/questions/4742746/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa
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Morse theory for compact sets bounded by hypersurfaces in euclidian space
I am having trouble understanding precisely how some part of Morse Theory works.
More precisely, take $X$ to be a compact set of $\mathbb{R}^d$ such that $\partial X$ (topological boundary) is a ...
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Diagrams for critical points [closed]
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) pages 13 and 15 we have :
for case "d&...
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configurations of three saddles on one level [duplicate]
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) page 13 we have :
There are sixteen ...
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Gap to fill in the Aubin–Ekeland proof of the mountain-pass theorem
Working through the proof of the mountain-pass theorem given in Applied Nonlinear Analysis by Aubin & Ekeland, at what seems to be a critical point of the proof (the top of page 274) they refer to ...
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Is a function looking like a cubic cusp globally equivalent to the cubic cusp?
Let's consider a family of smooth odd functions $\phi_v(u)\colon \mathbb{R}^2\to\mathbb{R}$, which ,looks like' a family of functions $f_y(x)=x^3-yx$ in the vicinity of $(0,0)$: $\phi_v(u)$ has no ...
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Geometry of critical points of holomorphic maps in projective space
Let $f:\mathbb{CP}^n\to\mathbb{CP}^n$ be a holomorphic map; I am interested in what the subvariety of critical points could be.
More specifically, let $J=\{p\in \mathbb{CP}^n\ :\ \det\mathrm{Jac}(f)=0\...
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what are definitions of born or die (birth-death point) and crossing point?
in this paper we have :
A presentation for the mapping class group of a closed orientable surface.by Hatcher.W.Thurston
...(a) $f_{t_{0}}$ has exactly one degenerate critical point, of the form $f_{t}(...
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Definition of Euler-Lagrange equation and properties, where can I find?
I'm studying a paper and in the introduction appears the following:
It is well known that existence of critical points and solvability of Euler-Lagrange equations are related, and there is and ...
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Critical points of the area functional restricted to CMC embeddings
For fixed closed smooth manifolds $M^n$ and $N^{n+1}$, two $C^{k,\alpha}$ embeddings $f, f' : M \to N$ are said to be equivalent if there exists $\varphi \in \operatorname{Diff}(M)$ such that $f' = f \...
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Optimal configurations on the flat torus
I'm studying (with my colleagues R.Piergallini and S.Isola) configurations of points on the flat torus which minimize an attractive or repulsive potential depending on the distance. Two model cases ...
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Number of critical points of sum of two functions
I ran into the following "simple" question and I am wondering whether there are any references, which might help me. I am coming from statistics, so I am not so aware which branch of math ...
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Is there a flaw in this proof of the validity of the Palais-Smale condition?
In Chapter 3 of his monograph (available on Researchgate), Kavian applies the Mountain Pass Theorem to a semilinear elliptic equation. To this aim, he needs to check that a functional satisfies the ...
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Minimizing sequence $\implies$ Palais–Smale sequence
Set $F:\mathbb{R}^n\rightarrow \mathbb{R}$ a $C^2$-function that is bounded from below. Set $x_n$ a minimizing sequence, i.e., $F(x_n)\to \alpha = \inf F$. I want to prove that under the assumption of ...
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Laplacian variational problem with asymptotically quadratic term
Consider the functional
$$J= \int_\Omega |\nabla u|^2 - \int_\Omega F(u),$$
where $\Omega$ is a bounded smooth domain.
The problem has been solved for example if $F$ is (1) subquadratic, or (2) ...
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Differences among various index theories in critical point theory
Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones?
the ...
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Generalized Sard's lemma
Let $f: X \to \mathbb{R}$ be a $C^{1,1}$ (that is $C^1$ with Lipschitz differential) function on a manifold $X$. Suppose that $f$ is smooth at all points of a subset $C \subset \text{Crit}f$ of ...
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Is there an upper bound on the number of critical points of a spherical harmonic on a local scale?
Take a spherical harmonic $y_d$ of degree $d$ on the sphere $\mathbb{S}^2$ and a spherical disk of radius $\frac{1}{d^2}$ centered at any point (let's say the north pole).
Is there an upper bound, ...
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Pullback of Morse form satisfies Palais Smale
Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector ...
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Smoothing a periodic function of two variables
Let $F \colon \mathbb{R}^2 \to \mathbb{R}^n$ be a $C^{1}$-function 1-periodic in each variable, so it can be considered as a function on the flat torus $\mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2$. We ...
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Critical values of analytic functions of several variables
Let $f:\mathbb{R}^d\to \mathbb{R}$ be real analytic. Define $S=\{x\in\mathbb{R}^d, \nabla f (x)=0\} $. Is it true that for any compact set $K\subset \mathbb{R}^d$, $f(S\cap K)$ is a finite set ?
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Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$
$\newcommand{\id}{\operatorname{Id}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\...
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Finding the critical points of a degree $5$ Blaschke product
The derivative of a degree $5$ polynomial $p\in\mathbb{C}[z]$ is a degree four polynomial $p'\in\mathbb{C}[z]$, and as such, the zeros of $p'$ may be found explicitly using the quartic formulae.
One ...
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C^1 fractals in statistical mechanics
It is well-known--even famous--that the Schramm-Loewner curves appear as domain boundaries between phases at second-order critical points like the critical Ising model or percolation in two dimensions....
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Bijection of critical points on two manifolds
Suppose that $f$ and $g$ are two smooth functions defined on $R^n$. Assume that $(a-\epsilon, b+\epsilon)$ contains no critical point of $g$. Then $g^{-1}[a,b]$ it homomorphic to $g^{-1}(a)\times [a,b]...
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Applying min-max to find a critical point in a ball
Let $\mathbb B^n$ be an open unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\frac{1}{2}\mathbb B^n\subset \mathbb B^n$ be an open ball or radius $\frac{1}{2}$. Let $\mathbb B^...
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Polynomial with subset of critical points and values prescribed
Motivated by this question I wish 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 $p(x)\in\Bbb R[x]$ ...
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When is a critical value of a map contained in the interior of the image?
Let $M^n$ be a compact manifold, and $F\colon M \to \mathbb{R}^n$ a smooth map. The inverse function theorem implies that every regular value of $F$ lies in the interior of $F(M)$, hence every point ...
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Measure of the Attractor of Critical Points of a Manifold
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...
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Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities
This question is an expansion of another question that I asked over at Math Stack Exchange.
In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ [...
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Height function on 2-torus with only 3 critical points
It is well-known that a Morse function on $T^2$ has at least $4$ critical points, but also that there exist functions $f\colon T^2\to\mathbb R$ with only 3 critical points (the least possible number ...
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Existence of at least one positive solution for semilinear biharmonic equation with critical exponent
Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & \mathrm{in}\hspace{...
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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 ...
2
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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.
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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 ...
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Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
The Gauss--Lucas Theorem states that all zeros of derivative of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros ...
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Does this squared distance functional have a unique critical point on geodesically convex manifolds?
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 \...
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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/...
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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 ...
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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 ...
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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 ...
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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 " $\|\...
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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 ...
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Critical points in Hilbert space [closed]
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 $f$?
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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 ...
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