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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34 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 $f_∗\mathcal{O}_X=\mathcal{O}_Y$ ...
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1answer
27 views
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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1answer
61 views
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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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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1answer
137 views
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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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
$$ (\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,...
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1answer
22 views
The pushforward of transition function of an orientable Riemannian manifold is in $SO(n)$?
Based on this notes last line, the pushforward of transition function of an orientable Riemannian manifold of dimension $2$ is in $SO(2)$. I wonder why it is true.
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1answer
48 views
Pushforward of covariant and contravariant tensor.
Le $F : M \rightarrow N$ be a map between manifolds.
What is the pushforward of a covariant or contravariant tensor?
I think that for a covariant tensor $T : T_pM \times...\times T_pM \rightarrow \...
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1answer
46 views
Why does $g\circ \exp(tv) \circ g^{-1}$ give the one-parameter group of diffeomorphisms generated by $g_* v$?
I have a question regarding the following proof from Cannas da Silva - Introduction to Symplectic and Hamiltonian Geometry:
Let $(M, \omega)$ be a symplectic manifold, and let $\alpha$ be a 1-form ...
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2answers
60 views
Computing the push forward of vector field $X = y^2 \partial/\partial x$ using Jacobians
I am trying to solve the following problem. Let $M$ and $N$ be submanfiolds of $\mathbb{R}^2$ given by $M = \{(x,y) \in \mathbb{R}^2 : x > 0, x+y>0\}$ and $N = \{(u,v) \in \mathbb{R}^2 : u > ...
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2answers
77 views
Push-forward of inverse map
If I define the inverse map in a Lie group $G$ as,
$$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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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 ...
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1answer
101 views
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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39 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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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)_\#\...
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1answer
109 views
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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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 ...
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1answer
58 views
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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1answer
62 views
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)$$
Calculate the push forward of the vector field $V = \frac{\...
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148 views
Finding the components of a pushforward vector in local coordinates
Let $\phi:M\rightarrow N$ be a smooth map between smooth manifolds, $v \in \operatorname{Vect}(M)$. Let $\{x^\mu\}$ and $\{y^i\}$ be local coordinates on $M$ and $N$ respectively.
How can I show that ...
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1answer
78 views
Basic question about related vector fields and pushforwards
Problem: Let $F \in C^{\infty}(M, N)$ be a diffeomorphism, $X,Y$ vector fields on the manifolds $M,N$ respectively. Then $X$ and $Y$ are $F$-related (i.e. $T_pF(X_p) = Y_{F(p)}$) if and only if $Y = ...
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47 views
Equality Proof of Pushforward and Pullback
Trying to prove $df_{F(p)}(F_*(v_p)) = F^*(df_{F(p)})(v_p)$. It is given that F is a smooth function between manifolds M and N, p $\in$ M, $v_p \in T_pM$ and $df_{F(p)} \in T_{F(p)}^*N$.
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Operation on Vector fields
I am analyzing a program which transforms vector fields by action of diffeomorphisms and feedbacks.
Here are the operation that I don't understand (it's Mathematica code)
...
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58 views
Transforming vector field on a manifold into canonical field
Let $M$ be a $C^{k}$ differentiable manifold of dimension $n$, and let $X$ be a $C^{k}$ vector field on $M$. Let $p$ be a point of $M$ such that $X(p) \neq 0$: is there a local parametrization $(\...
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1answer
136 views
Pushforward of vector field multiplied by real valued function
Let M and N be smooth manifolds:
$$\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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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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1answer
121 views
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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1answer
151 views
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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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 ...
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1answer
120 views
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^*}...
