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Questions tagged [pushforward]

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2
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1answer
28 views

Can a probability measure with connected and compact support be realized as the pushforward of the uniform?

Suppose that $\mu$ is a Borel probability measure such that $\text{supp}(\mu) \subseteq \mathbb{R}^n$ is (locally) connected. Does there exist a continuous function $f:[0, 1]^n \to \mathbb{R}^n$ such ...
0
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1answer
39 views

Push-forward of a measure from $\mathbb{R}^d$ to $\mathbb{R}^d\times\mathbb{R}^d$

I'm having some problems in showing that, given a probability measure $\mu$ on $\mathbb{R}^d$, if $s,t:\mathbb{R}^d\to\mathbb{R}^d$ are such that $(\textrm{id}\times s)_\#\mu=(\textrm{id}\times t)_\#\...
3
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0answers
58 views

Transforming vector field on a manifold into canonical field

Let $M$ be a $C^{k}$ differentiable manifold of dimension $n$, and let $X$ be a $C^{k}$ vector field on $M$. Let $p$ be a point of $M$ such that $X(p) \neq 0$: is there a local parametrization $(\...
2
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0answers
37 views

When does $\mathcal{O}_Y = f_* \mathcal{O}_X$ hold?

In a comment to this question about Stein factorization, Tabes Bridges writes Moduli technicalities (particularly in positive characteristic), the condition $f_∗\mathcal{O}_X=\mathcal{O}_Y$ ...
2
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0answers
125 views

On the $\pi$-induced pushforward of a tensor field on $TM$

A short informal premise. As far as I understand, given a smooth map between manifolds $f: M \to N$ and a smooth vector field $X: M \to TM$, the pushforward $f_* X$ only defines in general a vector ...
2
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0answers
107 views

Is it wrong to say “pushforward” and “pullback” for general functors?

Is it wrong ot use the terms pushforward (resp. pullback) about the induced map of general covariant (resp. contravariant) functors and denote them $\varphi_*$ (resp. $\varphi^*$)? The terminology is ...
1
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0answers
23 views

Pushforward measure

Good evening, Let $\mu$ and $\nu$ two measures on $X$ and $Y$. Do you know when it exists a measurable function $h : X \rightarrow Y$ such as $ \nu = h\text{#}\mu$ with $ h\text{#}\mu(B) = \mu(h^{-1}...
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0answers
39 views

Computation of the push forward of vectors

I am trying to understand the push forward of a vector field by going through a specific calculation. Consider $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ given by $f(x,y,z) = (x+y+7, z-x-5)$ and ...
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0answers
62 views

Local construction of a map on a manifold

Assume $(M,J)$ is a smooth manifold equipped with an almost complex structure. Let $p$ be a point of $M$, and define the linear map $\Theta_p \colon T_pM \to T_pM$ so that $$\Theta_p = -\frac{1}{2}\...
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0answers
41 views

Pushforward-change-of-variable with quantile function

I’ve been dealing with an issue about change-of-variable formula. Let $\mu$ be a probability measure on $\mathbf R_+$. Let $F(x) = μ([0,p])$ and $Q$ its quantile function, ie $Q(p) = \inf \{q \in \...
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0answers
41 views

Pushforward formula for Lebesgue Stieltjes Measure

I got to have this problem in my hand. Problem Let $F: \mathbb{R}\rightarrow \mathbb{R}$ be an increasing, right continuous function, and let $\phi :\mathbb{R}\rightarrow \mathbb{R}$ be a ...
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0answers
149 views

Finding the components of a pushforward vector in local coordinates

Let $\phi:M\rightarrow N$ be a smooth map between smooth manifolds, $v \in \operatorname{Vect}(M)$. Let $\{x^\mu\}$ and $\{y^i\}$ be local coordinates on $M$ and $N$ respectively. How can I show that ...
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0answers
60 views

Operation on Vector fields

I am analyzing a program which transforms vector fields by action of diffeomorphisms and feedbacks. Here are the operation that I don't understand (it's Mathematica code) ...
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0answers
33 views

Showing that $\psi^{-1}_{*(x_o,g_o)}(v_1,v_2) = (\sigma_{g_o})_{*s(x_o)}(s_{*x_o}(v_1)) + A^\#(s(x_o).g_o)$

I am stuck in one exercise I found in the book : Topology, geometry and gauge fields. Here it is : Where : $\sigma_g$ is the right action $A^\#$ is the fundamental vector field associated at A. ...
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0answers
22 views

Prove that $(\mathbf g\circ\mathbf f)_*=\mathbf g_*\circ\mathbf f_*$ and $(\mathbf g\circ\mathbf f)^*=\mathbf g^*\circ\mathbf f^*$

Let $\mathbf f:\mathbf R^n\rightarrow\mathbf R^m$ and $\mathbf g:\mathbf R^m\rightarrow\mathbf R^k$. I figured out the pull-back part by finding $$ (\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,...
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0answers
80 views

Push-forward and Category theory

There obviously seems to be a connection between the push-forward and the pull-back of a smooth function $f:M \to N$ between smooth manifolds, and the Hom-maps from category theory $f^*=Mor_C(f,\...
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0answers
48 views

Equality Proof of Pushforward and Pullback

Trying to prove $df_{F(p)}(F_*(v_p)) = F^*(df_{F(p)})(v_p)$. It is given that F is a smooth function between manifolds M and N, p $\in$ M, $v_p \in T_pM$ and $df_{F(p)} \in T_{F(p)}^*N$.