Questions tagged [pushforward]

Question 1

Given a scheme $(X, \mathcal O_X)$ and $\mathcal F$ a sheaf of $\mathcal O_X$ -modules, I say that $\mathcal F$ is coherent when $X$ has an affine open cover by $U_i = \mathrm{Spec} A_i$ for $i \in I$ ...

Question 2

In my studies of category theory, I encountered the definition of a pullback. Given a category $\mathcal{C}$ and two morphisms $f\colon A \to C$ and $g\colon B \to C$, the limit of the diagram formed ...

Question 3

Let $\mu$ be a probability measure on $\mathbb{R}^n$ that is absolutely continuous w.r.t Lebesgue measure, and $T_1$ is a push-forward map such that $T_1\# \mu= \mu_1$. Let $\mu_2$ be any probability ...

Question 4

Consider a smooth map $F: M \rightarrow N$ between smooth manifolds with or without boundary. Choose smooth coordinate charts $(U, \varphi)$ for $M$ containing $p$ and $(V, \psi)$ for $N$ containing $...

Question 5

Consider a measurable function $f: \mathbb{R} \to \mathbb{R}$. I'd like to define a measure $g$ as the pushforward of the normal distribution via $f$: $$ g = f \ {\sharp}\ \mathcal{N}(0, 1) $$ Is ...

Question 6

Theorem on formal functions: Let $f:X\to Y$ be a projective morphism between noetherian schemes, $\mathcal{F}\in\textrm{Coh}(X)$, and $y\in Y$. Then the natural map $R^if_*(\mathcal{F})_y^\wedge\to\...

Question 7

I was trying to show that a push forward operation of a vector field $\phi$ in a density $p$ $( \phi \# p )$ preserves the quantiles. For example, I want to show that: \begin{equation} F_{p}(x) = \...

Question 8

This is Problem 3-12 from Introduction to Riemannian Manifolds by Lee. I know these facts: The hyperbolic metric $\breve{g}_R^4$ on the upper half plane $\mathbb{U}^n(R)$ is given in coordinates $(x^...

Question 9

I've been dealing with probability integral transforms a lot recently, and understand how to use them well enough at this point. For completeness, a probability integral transform pertaining to a ...

Question 10

I was reading a measure theory book (Fremlin, vol 4) where it states that given two probability spaces $(X,\Sigma_X,\mu),(Y,\Sigma_Y,\nu)$ and $f:X\rightarrow Y$ a measurable function such that $\nu$ ...

Question 11

Let $\mu$ be a signed measure on $X$ with Jordan decomposition $\mu^+-\mu^-$, $\pi:X\rightarrow Y$ a measurable map, $\pi_\#\mu$ the pushforward measure of $\mu$ with respect to $\pi$. If $f:Y\...

Question 12

Example 8.20 of Lee's "Smooth Manifolds" is as follows: Let $M$ and $N$ be the following open submanifolds of $\mathbb{R}^2:$ $$\begin{align*}M &= \{(x,y) : y > 0 \text{ and } x + y &...

Question 13

The Wasserstein-1 metric between two probability measures is defined as $$W_1(\mathbb{P}_X, \mathbb{P}_Y) = \inf_{\gamma \in \Gamma(\mathbb{P}_X, \mathbb{P}_Y)} \mathbb{E}_{(X,Y) \sim \gamma } \Vert X-...

Question 14

Consider a smooth map $f:X\to Y$ and let $E\to X$ and $E'\to Y$ be vector bundles such that $E=f^* E'$, and it admits a filtration $$0=E_0\subset E_1\subset\cdots\subset E_{n-1}\subset E_n\simeq E.$$ ...

Question 15

Consider two measures $\mu , \nu$ on $\mathbb{R}^n$. Now consider a map (a priori only measurable) $f$ and the push forward measures $f_\# \mu, f_\# \nu$. Is there any relation between the Wasserstein ...

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Questions tagged [pushforward]

Behaviour of coherent sheaves under pushforward and pullback.

Pushforward in Slice Categories

Decompose a joint distribution by push forward maps

Computing pushforward/differential in coordinates

Notation for pushforward of a normal distribution

Non-examples for Theorem on Formal Functions

Does the push forward operation of a vector field in a probability density preserve quantiles?

Show that for each $R>0,$ the hyperbolic metric on the upper half plane is left-invariant

Measure theoretic definition of the probability integral transform

Defining $T:L^\infty(\mu)\rightarrow L^\infty(\nu)$ with integrals

Is change of variables formula for signed measures valid also for functions with infinite integral?

Computing the pushforward of a vector field

Confusion regarding Wasserstein-1 distance between a probabilty measure and its push-forward

Filtration of Vector Bundles under Pullback and Pushforward

Wasserstein distance of push-forward measures

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