Questions tagged [semicontinuity]
The semicontinuity tag has no usage guidance.
16
questions with no upvoted or accepted answers
8
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Lower semicontinuity of naive fiber size
I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
3
votes
0
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192
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Semicontinuity of Exts
Let $Z$ be a proper noetherian scheme over an algebraically closed field $k$, and $S$ a smooth scheme over $k$. Let $\cal E$, $\cal F$ be coherent sheaves on $Z\times S$, flat over $S$. Is it true ...
3
votes
1
answer
586
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Lower semicontinuous and convex envelope
L.Ambrosio, in paper [1] writes:
Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)
for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of ...
2
votes
0
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63
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Conditions ensuring that a compact set can be approximated by approximating its distance function
Let $\emptyset\neq K\subset Y$ be a closed subset of a compact metric space $(X,d)$ such that $K$ has at-least two points and such that
$$
d(Y,K):=\sup_{y\in Y}\,\inf_{k\in K}\,d(k,y)=:r>0.
$$
...
2
votes
0
answers
116
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Semicontinuity of length for coherent sheaves
Given a coherent sheaf F over a noetherian scheme Y, a classical result in algebraic geometry states the upper-semocontinuity of the function sending any point $y \in Y$ to $\mathrm{dim}_{k(y)}(F \...
2
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0
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37
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Weak relaxation of a strongly lower semi-continuous functional
Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak ...
2
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0
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81
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lower semicontinuity of the number of extreme points
Do you know the reference for the following fact:
the number of extreme points of a compact convex
subset of a locally compact space varies lower semicontinuously when we endow the space of compact ...
2
votes
0
answers
471
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When will the upper regularization of a bounded function not defined?
Suppose $E$ is a compact metric space.
A function $f :E \rightarrow \mathbb{R}$ is upper semicontinous if for all $c \in \mathbb{R}$, $f^{-1}(-\infty, c)$ is open in $E.$
For any real-valued ...
1
vote
0
answers
21
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Weakening compacity hypothesis in multifunctions intersection
Let $X,Y$ be metric spaces, $x^*\in X$
We define two multifunctions $F_1:X\rightrightarrows Y$,$F_2:X\rightrightarrows Y$.
We recall the upper-semi-continuity in Berge's sense :
A multifunction $F:X\...
1
vote
0
answers
60
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Lower semicontinuity of the number of attracting periodic points of a holomorphic family of rational maps?
Recently I have been reading the book Mathematical Tools for One-Dimensional Dynamics.
In the proof of the theorem 5.4.2, authors use the following fact that the number of attracting periodic points ...
1
vote
0
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33
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Semi-continuity extends to dual
Say $f \colon X \to \mathbb{R}$ is a lower semi-continuous on a compact space $X$. Let $\mathcal{P}(X)$ denote the space of Borel probability measures on $X$, and let $f^* \colon \mathcal{P}(X) \to \...
1
vote
0
answers
951
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Approximation of semicontinuous functions by continuous (or smooth) functions with closed form
I'm looking for a sequence $(f_{\epsilon})_{\epsilon>0}$ of continuous (or smooth) functions approximating a semicontinuous function $f$.
Here, for approximation, pointwise convergence is fine.
For ...
1
vote
0
answers
66
views
Upper semicontinuity in C(X)-algebras. Quotient norm question
upper semicontinuity in C(X)-algebras
In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x (a)||_A(x)$
By the definition of the quotient ...
0
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117
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From convergence pointwise to convergence of the supremum for semicontinuous functions
Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
0
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267
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Flatness of affine cone due to semicontinuity theorem
I would like to clarify an important aspect from the discussion in this question.
The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic
Geometry Chap. III page ...
0
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122
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Help showing F is weakly lower semicontinuous
Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...