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Questions tagged [pushforward]

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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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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 $f_∗\mathcal{O}_X=\mathcal{O}_Y$ ...
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Interpretation of the pushforward of a vector under a flow.

Suppose $\theta_t(p)$ is the flow of some vector field. I don't really understand how to interpret the meaning of the pushforward $(\theta_t)_*X$ of some vector $X$. What is its intuitive meaning?
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Holomorphic Euler characteristic of a pushforward

It shouldn't be difficult to show that when $C$ is union of lines in generic position in $\mathbb{P}^2_k$, then its normalisation map $f:\tilde C \rightarrow C$ with $\tilde C$ being the disjoint ...
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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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Hartshorne, Exercise III 4.2 (a): A morphism $\mathcal{O}^r \to f_* \mathscr{M}$ that is an iso over the generic point.

I'm having some trouble with Exercise III 4.2 a) in Hartshorne's Algebraic geometry. It is Let $f: X \to Y$ be a finite surjective morphism of integral noetherian schemes. Show that there is a ...
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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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The pushforward of transition function of an orientable Riemannian manifold is in $SO(n)$?

Based on this notes last line, the pushforward of transition function of an orientable Riemannian manifold of dimension $2$ is in $SO(2)$. I wonder why it is true.
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Pushforward of covariant and contravariant tensor.

Le $F : M \rightarrow N$ be a map between manifolds. What is the pushforward of a covariant or contravariant tensor? I think that for a covariant tensor $T : T_pM \times...\times T_pM \rightarrow \...
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Why does $g\circ \exp(tv) \circ g^{-1}$ give the one-parameter group of diffeomorphisms generated by $g_* v$?

I have a question regarding the following proof from Cannas da Silva - Introduction to Symplectic and Hamiltonian Geometry: Let $(M, \omega)$ be a symplectic manifold, and let $\alpha$ be a 1-form ...
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Computing the push forward of vector field $X = y^2 \partial/\partial x$ using Jacobians

I am trying to solve the following problem. Let $M$ and $N$ be submanfiolds of $\mathbb{R}^2$ given by $M = \{(x,y) \in \mathbb{R}^2 : x > 0, x+y>0\}$ and $N = \{(u,v) \in \mathbb{R}^2 : u > ...
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Push-forward of inverse map

If I define the inverse map in a Lie group $G$ as, $$i: G \rightarrow G,\quad i(g) = g^{-1}, \forall g \in G \tag1$$ I think that the associated push-forward would be, $$i_*: T_gG \rightarrow T_{g^...
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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 ...
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Definition of pushforward and pullback for modules

Let $R$ be a commutative ring with unit and $\sigma:R\longrightarrow R$ an endomorphism of the ring $R$. Let $M$ be an $R$-module. What are the definitions of $\sigma^*M$ and $\sigma_* M$? How do ...
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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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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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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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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)_\#\...
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partial derivatives and chain rule of functions defined on manifold

For functions on $\mathbb{R}^{n}$, applying chain rule and taking partial derivatives is straight forward. I am a bit confused about how this concept can be extended to functions defined on manifolds. ...
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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 ...
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Blow up of $\mathbb{P}^2$ in a point and direct image sheaves

I am trying to understand better direct image sheaves. To do so, I want to start working in a particular and easy example. Let $\pi:X\rightarrow \mathbb{P}^2$ be the blow up of $\mathbb{P}^2$ in a ...
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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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Calculating push forwards of a vector field

For the transformation from spherical coordinates to cartesian $$F(r,\theta,\phi) = (r\cos\theta\sin\phi,r\sin\theta\sin\phi,r\cos\phi)$$ Calculate the push forward of the vector field $V = \frac{\...
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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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Basic question about related vector fields and pushforwards

Problem: Let $F \in C^{\infty}(M, N)$ be a diffeomorphism, $X,Y$ vector fields on the manifolds $M,N$ respectively. Then $X$ and $Y$ are $F$-related (i.e. $T_pF(X_p) = Y_{F(p)}$) if and only if $Y = ...
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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 $df_{F(p)} \in T_{F(p)}^*N$.
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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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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 $(\...
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Pushforward of vector field multiplied by real valued function

Let M and N be smooth manifolds: $$\mathbb{R}\xleftarrow{g} M \xrightarrow{h}N\xrightarrow{f}\mathbb{R}$$ and of course: $$TM\xrightarrow{h_*}TN$$ I need to show that: $$h_{*}(gX)f=(g\circ h^{-1})h_{*}...
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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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Push-forward of a probability measure is mixture linear?

Let $C$ be a compact metric space, $\mathcal P(C)$ be its set of Borel probability measures (with topology of weak convergence) and $\mathcal P^2(C)$ be the set of Borel probability measures on $\...
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Why is it necessary to talk about a pushforward measure?

I understand that a random variable $X$ and a probability measure $P$ on a space $(\Omega,\mathcal{A})$ induce the distribution $P_X$ on a space $(\Omega',\mathcal{A}')$. But is there an example ...
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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 ...
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Identity concerning push forward of two vector fields

How would you prove the identity $\displaystyle \frac{\partial}{\partial s}\Psi_{s^*} \mathbb{X} = (-1)L_{\mathbb{Y}}\Psi_{s^*}\mathbb{X}$ where $\Psi_{s}$ is the flow of $\mathbb{Y}$ and $\Psi_{s^*}...