Questions tagged [global-analysis]

In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector space bundles.

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6
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0answers
156 views

Fréchet manifold structure on space of sections

I know that the space $\mathsf{C}^\infty(M;N)$ of smooth maps from a closed (smooth) manifold $M$ to a (smooth) manifold $N$ is a Fréchet manifold. I have been looking for a more general version of ...
5
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2answers
52 views

Rigorous global optimization

The work by Thomas Hales (see enter link description here) before the formal proof solves a number of global optimization problems that need to be solved exactly. The strategy relies on following ...
5
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0answers
71 views

Has this functional been studied somewhere?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ Let $\M,\N$ be Riemannian manifolds, $f:\M \to \N$ smooth. ...
5
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0answers
47 views

Is the spectrum of a first order PDO always unbounded from both sides?

Let $E \to X$ be a smooth vector bundle over a compact Riemannian manifold $X$ and assume that $P:\Gamma(E) \to \Gamma(E)$ is a self-adjoint partial differential operator of order $1$. We think of ...
4
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0answers
108 views

Generalizations of Sard-Smale Theorem

Sard-Smale theorem holds for Fredholm maps $f:M\rightarrow B$ between separable Banach manifolds $M,N$. There are some constrains relating the Fredholm index $\operatorname{ind}(f)$ of $f$ to its ...
3
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0answers
126 views

Showing that $\bar\partial_J$ is a smooth section of Banach bundle

I am reading Chapter 3 (Moduli Spaces and Transversality) of "J-holomorphic curves and symplectic topology" by McDuff & Salamon. Fixing $k\in\mathbb N,p>1$ such that $kp>2$, they consider $\...
3
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0answers
106 views

How far can we push the Schwartz kernel theorem?

The Schwartz kernel theorem works for operators defined on $C_ {c}(\mathbb{R}^n,E)$, as long as $E$ is finite-dimensional and we introduce the right notion of a generalised section. In every ...
2
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1answer
255 views

Global Implicit Function Theorem

I encounter the problem that I would like to extend the implicit function theorem (for real numbers) to a global version. The classical implicit function theorem is given by the following: Assume $F: ...
2
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0answers
38 views

Let a compact group $G$ act on a manifold $M$. Why does G act as a group of automorphisms on $H^p (M, \mathbb{R} )?

Let $(\sigma , m) \rightarrow t_{\sigma}$ be this action. A definition for acting on a group of automorphisms I found here: Action via automorphism. t^{*}_{\sigma} seems to be this action and I can ...
2
votes
0answers
239 views

Friedrich extension of an elliptic operator

I'm working with elliptic operators on Riemannian manifolds. In the case where the metric $g$ on $(M,g)$ is not complete, I understand that, for example, the Hodge laplacian may be not essentially ...
2
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0answers
88 views

Condition for Dirichlet boundary conditions

Let D be a differential operator on a manifold with boundary. We consider the differential equation $Df = 0$ with Dirichlet-boundary-conditions $f|_{\partial M}= g$. Are there cases, where not every ...
2
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0answers
68 views

asking a way to prove an inequality

Assume $\Omega$ is a bounded smooth domain in $\mathbb R^N $ with $N \ge 5 $ and $u \in C^2(\Omega)$ . I want to proof $$\int_{\Omega}\frac{|\nabla u|^2}{|x|^2}d{x} \;\ge\; \left(\frac{N-4}{2}\...
1
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0answers
27 views

How to show that $\left\{x^*\in X \mid \limsup_{y \to x,\;\|y\|=1}\frac{\langle x^*,y-x\rangle}{\|y-x\|} \le 0\right\} = \mathbb R x$ if $\|x\|=1$

Let $X$ be a Banach space with topological dual $X^\star$. Given an extended value function $f:X \to \mathbb R \cup \{+\infty\}$, defined its Fréchet subdifferential at a point $x \in X$ as $$ \...
1
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0answers
65 views

Using the covariant derivative for a Riemannian Metric for $H^1([0,1]; S^2)$

I am currently researching gradient flows on Riemannian Hilbert manifolds and in this paper, Trombe considers the Hilbert space $((H^1([0,1]; \mathbb{R}^3), \langle \cdot, \cdot \rangle_{H})$ with ...
1
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0answers
45 views

Is the functional $w \mapsto \int_0^1 | \ \| w(t) \| - 1 | \ dt$ $C^1$ or even smooth?

Let $H:= H(I;\mathbb{R}^3)$ be the space of $L^2$ + absolutely continuous functions with $L^2$ derivative. For $w \in H$ consider the functional $$\psi(w) = \int_0^1 | \ \| w(t) \|^2 - 1 | \ dt$$ ...
1
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0answers
18 views

What are examples of a second order operational tangent vector on an infinite dimensional Hilbert space.

In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space. http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf ...
1
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0answers
40 views

A question regarding mollifers on Sobolev spaces on closed manifolds

Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
1
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0answers
24 views

Difference between Homogeneous first order operator and Homogeneous first order differential operator

By definition, a Homogeneous first order operator on the algebra of differentiable functions is an operator like $$ Df = \sum \phi_i \frac{\partial f}{\partial x_i} $$ where $\phi_i$ are ...
1
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0answers
40 views

Clarification in the definition of presheaf

I need a clarification in the definition of presheaf. If $X$ is a topological space then we define presheaf on $X$ to be an assignment to every open subset $U \subset X$ and to every pair of open sets ...
1
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0answers
46 views

What are the differences among different notions of harmonic maps?

Given smooth (compact, if needed) Riemannian manifolds $M$ and $N$. There are at least 3 different notions of harmonic maps (shortly, HM): weakly HM. stationary HM. minimizing HM. It is well-known ...
1
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0answers
156 views

Apparently meaningless computation with the Hodge star operator

In the last lecture we started speaking about hodge star operator. Let $E$ be a $n$ dimensional vector space with a non degenerate bilinear form $g$. $\mathcal{O}$ the orientation line of $E$, i.e. ...
0
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0answers
17 views

Differential Operator and Trivialization Change in Lawson-Michelsohn

I am currently reading the excelent book by Lawson and Michelsohn on Spin Geometry. In chapter 3 paragraph 1, they define a differntial operator of order $m$ to be a $\mathbb{C}$-linear map of $\...
0
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0answers
24 views

How to estimate the probability of the type I error for the Bonferroni and Fisher tests?

I have the sample of 100 from Poisson distribution. I have 1000 of similar hypothesis and the meta problem of testing the global hypothesis $$\bigcap H_{0i}: E(X_i) = 5 \quad \text{vs.} \quad \bigcap ...
0
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0answers
77 views

geodesic balls in Riemannian manifolds with bounded geometry

Let $(M,g)$ be an open (:=complete, non-compact) Riemannian manifold with bounded geometry, in the sense that in some atlas of charts of radius $r_0>0$, the metric and all its derivatives are ...
0
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0answers
45 views

Generalisation of the Hopf–Rinow theorem

Is their a generalisation of the Hopf–Rinow theorem to infinite dimensional manifolds (Frechet, Banach or Hilbert)? I have actually never seen the prove of the Hopf–Rinow theorem so i don't know if ...
0
votes
1answer
106 views

Diffeomorphisms and their derivatives

If a $C^k$ map $f: X\to Y$ where $X, Y$ are Banach spaces is a diffeomorphism. What can we say about $d_xf$, the differential of $f$ at $x\in X$? It is true that we have $$ \inf_{x\in X}\big|\det [...
0
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0answers
527 views

Compute the first distributional derivative of a function

Consider $\Omega = (0,2) \subseteq \mathbb{R} $ to be a support of the function $f(x)$ defined as $$ f(x) = \begin{cases} x, &0 < x \leq 1\\ 2, &1< x \leq 2 \end{cases}$$ I need to ...
0
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0answers
56 views

Differentiable structure on the gauge group?

In this paper I have come across a formulation involving differentiation in the gauge group of a principal bundle which I do not understand (found at the very top of p. 369). Let $P\rightarrow M$ be ...