Questions tagged [smoothness]
The smoothness tag has no usage guidance.
145
questions
5
votes
1
answer
263
views
Smoothness of the radius of convergence
Let $(x\mapsto a_n(x))_n$ be a sequence of smooth functions defined on some fixed interval $I$. Consider the power series $\sum_{n\geq 0}a_n(x)t^n$ and denote by $R(x)$ its radius of convergence. Does ...
3
votes
0
answers
215
views
Lifting a Frobenius endomorphism under an étale morphism
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
1
vote
0
answers
185
views
Definition of Morphisms of algebraic stacks smooth of relative dimension n
There is a notion of smooth morphism of algebraic stacks e.g. Tag 075U and a notion of
relative dimension of a locally of finite type morphism $T\to \mathcal{X}$ from an algebraic space into an ...
4
votes
1
answer
115
views
Notion of a "smooth function of the order two" (Yakubovich, "Index Transforms")
In the book "Index Transforms" by S.B. Yakubovich (World Scientific, 1996), I encountered the notion of a "smooth function of the order two" with compact support on $\mathbb R_+$, denoted $C^{(2)}(\...
2
votes
0
answers
133
views
Formally smooth maps of schemes and factorization systems
I am thinking about how formally smooth maps of schemes relate to factorization systems.
Let $C$ be the category of schemes. Let $E$ be the class of morphisms of schemes consisting of closed ...
3
votes
0
answers
110
views
Look for a suitable cut-function: from Pierre Grisvard "Elliptic Problems in Nonsmooth Domains": (Theorem 1.4.2.4)
From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains":
Theorem[Theorem 1.4.2.1]
Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a continuous boundary, then $C_c^\infty(\...
1
vote
0
answers
90
views
Factorizations of closed embeddings of smooth schemes
All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). ...
0
votes
1
answer
222
views
Existence and smoothness for viscous Burgers equation?
What do we currently know about (references please!) the existence and smoothness of solutions to the viscous Burgers equation, in 1D, 2D, and 3D?
8
votes
2
answers
349
views
Can smoothness of curves into a convenient locally convex vector space be tested with just a dense subspace of the dual?
Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given ...
0
votes
0
answers
154
views
The stalk local nature of formal smoothness
Under finite presentation hypothesis on $X/S$, I believe it should be possible to define formal smoothness using only artin local rings as the test rings. What would a reference for this fact be?
3
votes
1
answer
231
views
Smoothness of a curve vs. smoothness of the squared distance from the curve to points on Riemann manifolds
I know that the squared distance function from a point $p$ on a Riemann manifold $M$ is smooth in a n-hood of $p$. Therefore for a smooth curve $c:\mathbb{R}\to M$ the concatenation $d(p,\cdot)^2\circ ...
6
votes
2
answers
314
views
Path algebras are formally smooth
In Ginzburg's notes Lectures on Noncommutative Geometry, he claim that the path algebra of a quiver is formally smooth (See Section 19.2).
I have two questions. First, how to show this claim and ...
15
votes
0
answers
310
views
How much smoothness does the tennis ball theorem need?
The tennis ball theorem states that a smooth-enough curve that bisects the surface area of a sphere must have at least four inflection points. There are plenty of sources on this but most of them are ...
7
votes
1
answer
817
views
Bertini's theorem over non-algebraically closed field
Let $K$ be a non-algebraically closed (infinite) field of characteristic $0$ and $X$ a smooth, projective $K$-variety. Does there exist an ample invertible sheaf $\mathcal{L}$ on $X$ such that a ...
8
votes
0
answers
328
views
Criterion for smooth functions [duplicate]
Let $f:\mathbb{R}→\mathbb{R}$ a real-valued function and $m,n∈\mathbb{N}^∗$ coprime, i.e. greatest common divisor of m and n is 1, and define $f^m:=f\cdot f\cdot\ldots\cdot f.$
Show that
$$f^m,f^n\in ...
8
votes
0
answers
458
views
How should I try to imagine exotic smoothness in R4?
I am trying to wrap my mind around the concept of exotic smoothness in (and only in) $\mathbb{R}^4$.
I have some mathematical literature, but can anyone point to a semi-intuitive, semi-visual example?...
1
vote
2
answers
267
views
Uniqueness of tangent space given local injectivity of orthogonal projection onto it
Definition. Let $X\subset \mathbb R^n$ be a locally Euclidean subset. Say it has a tangent space at $p\in X$ if there exists a linear subspace $V\leq \mathbb R^n$ satisfying the following conditions.
...
1
vote
1
answer
136
views
Estimate for de Bruijn function with small fixed smoothness bound
Let $\Psi(x,B)$ denote the number of $B$-smooth numbers less than $x$. Wikipedia gives the following "good estimate" for small, fixed $B$:
$$\Psi(x,B) \sim \frac{1}{\pi(B)!} \prod_{p\le B}\frac{\log ...
0
votes
1
answer
119
views
Elegant / Canonical way to Extend Integer Iterates of a Function to a Real Parameter
Any map $f \colon \mathbb{R} \to \mathbb{R}$ induces a "composition map"
$$f^\circ\colon \mathbb{R} \times \mathbb{N} \to \mathbb{R},$$
where
$$f^{\circ n}(x) = \underbrace{f \circ \dotsb \circ f}_{...
3
votes
0
answers
273
views
Formal smoothness implies local freeness of the sheaf of relative differentials
What is the least restrictive finiteness assumptions guaranteeing that for a formally smooth morphism of schemes $f:X\rightarrow Y$, the sheaf of relative differentials $\Omega _{X/Y}$ would be ...
1
vote
0
answers
67
views
Smoothing in linear hyperbolic equations
This is a bit fuzzy, but I've somewhere read or heard something like:
"For linear hyperbolic equations smoothing in time leads to smoothing in space"
Is this in any sense true?
References, ...
6
votes
0
answers
230
views
Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces
I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...
2
votes
1
answer
309
views
Gevrey estimate of derivatives
Let $\phi$ be the $C^\infty$ function defined on $\mathbb R_+^*$ by $e^{-t^{-2}}$ and by $0$ on $\mathbb R_-$.
Question: I think that there exists $\rho>0$ such that
$$
\forall t\in \mathbb R,\...
1
vote
0
answers
106
views
Smoothness of the solution of 1D diffusion equation
How do I show that the solution of 1D diffusion equation is smooth for all t>0? I do know that in order to show a nonlinear PDE, for example Burger's equation, develops corners (instead of smooth ...
21
votes
1
answer
1k
views
A differentiable isometry is smooth?
I posted this question in MSE but got no response (even after giving a bounty), so I am trying here.
Let $M,N$ be smooth $d$-dimensional Riemannian manifolds.
Suppose $f:M \to N$ is a differentiable ...
4
votes
0
answers
117
views
Smoothness in von Neumann algebra of measurable functions
Let $A=L^{\infty}(M)$ be an algebra of essentially bounded measurable function on manifold $M$. Let $D$ be a first order elliptic differential operator acting on some hermitian bundle $S$ over $M$ (...
4
votes
1
answer
541
views
Isomorphic algebras determine diffeomorphic manifolds
It is a kind of folklore but I would like to see the proof of the following fact: given two smooth manifolds $M$ and $N$ if we assume that the algebras $C^{\infty}_0(M)$ and $C^{\infty}_0(N)$ are ...
7
votes
1
answer
530
views
Smoothness of a projective variety via the derived category
Let $X$ be a smooth projective integral variety over an algebraically closed field $k$. Let $Y$ be a (not necessarily smooth) projective integral variety over $k$.
Assume that $D^b(X) \cong D^b(Y)$.
...
3
votes
1
answer
389
views
Continuation of a smooth function, whose every derivative is strictly monotonic
Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where ...
2
votes
0
answers
249
views
Holomorphic symplectic form on the moduli space of Higgs bundles
I have the following problem: consider the moduli space $\mathcal{M}:=\mathcal{M}_X(n, 0)$ of semistable Higgs bundles of rank $n$ and degree $0$ on a compact Riemann surface $X$ of genus $g\geq2$. ...
5
votes
2
answers
1k
views
Derivatives of $C^{\infty}$ non analytic function
Question: Given $f\in C^{\infty}$ which is not analytic on a bounded domain $\Omega \subseteq \mathbb{R}$. What can we say about the sequence $\lbrace f^{(m)} \rbrace _{m=1}^{\infty} $? Specifically - ...
7
votes
1
answer
367
views
Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
11
votes
3
answers
609
views
smooth functional to detect whether a function has a zero
Does there exist a function $F : C^\infty(\mathbb{R}, [0, \infty)) \to \mathbb{R}$ with the following properties:
$F(f) = 0$ if and only if there exists an $x \in [0,1]$ such that $f(x) = 0$.
$F$ is ...
7
votes
0
answers
288
views
Smooth dependence on parameters of invariant manifolds
This may be considered a followup question to smooth dependence of stable manifold on parameters. In particular, I would like to know in more detail how smooth parameter dependence of a (generalized) ...
5
votes
1
answer
395
views
Confusion with formally unramified = immersion and formally smooth = submersion
From this MO question I learned to tentatively think of formally unramified arrows as immersions and of formally smooth arrows as submersions.
I'm trying to semi-formally handwave myself into ...
20
votes
5
answers
2k
views
Smoothness of the closest point on a submanifold
Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold.
Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
1
vote
0
answers
435
views
Quotient of two smooth functions extension
Assume we are given smooth functions $f, g: U \to \mathbb{C}$, where $0 \in U \subset \mathbb{R}^n$ is open and $0 \in g^{-1}(0) \subset \{x_n = 0\}$. Furthermore, suppose that $\nabla g \neq 0$ on ...
4
votes
1
answer
513
views
Uniqueness of smooth compactification upto a smooth morphism
By a $k$-variety, we will mean a separated scheme of finte type over a field $k$. Let $k$ be of characteristic 0. Given a smooth quasi-projective $k$-variety $X$, there is a projective $k$-variety $\...
5
votes
1
answer
343
views
Extension of functions from geodesically convex compact sets in a Riemannian manifold
In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...
1
vote
0
answers
103
views
Smoothness of the incicence correspondence associated to the join of two varieties
Let $Y\subseteq X\subsetneq\mathbb{P}^{N}$ be smooth projective varieties, and let
$$
S_{X,Y}=\overline{\{(x,y,z)\in X\times Y\times \mathbb{P}^{N}:x\neq y, z\in\langle x,y\rangle\}}.
$$
Can we ...
0
votes
0
answers
313
views
Unit sphere of a norm is a submanifold implies the norm is smooth?
Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map.
Suppose the unit sphere of a norm $\| \cdot \|$ is an ...
1
vote
1
answer
477
views
Composition algebra of Gevrey function for $s<1$
Let $g,f$ be real-valued functions defined on the real line. Let $s$ be a real number.
Assuming that $g,f$ are both in the Gevrey class $G^{s}$, it is true that $g\circ f$ belongs to $G^{s}$ if $s\ge ...
1
vote
1
answer
437
views
Bertini-type theorem in positive characteristic [closed]
Let $f:X \to Y$ be a morphism of finite type of irreducible schemes over an algebraically closed field of characteristic $0$. Assume that $Y$ is non-singular. Let $x \in X$ be a closed point and $T_xf:...
2
votes
0
answers
382
views
Blow up along a section of a smooth morphism
Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...
0
votes
2
answers
147
views
Smoothness of a power of smooth non-negative function [closed]
Let $f$ be a non-negative infinitely smooth function on the real line. Is it true that for any constant $\alpha$ the function $f^\alpha$ is infinitely smooth?
1
vote
0
answers
123
views
Set of smooth curves on the Hilbert scheme is open. H
Let $H = Hilb_{d,g,r}$ be the Hilbert scheme of genus $g$ curves of degree $d$ in proyective space $\mathbb{P}^r$, over an algebraically closed field $k$.
Is it true that the set of points of $H$ ...
1
vote
0
answers
339
views
Separability and smoothness
Let $A \subseteq B$ be commutative noetherian rings.
I have found the following claim: "Separability implies smoothness" with the following explanation:
"The natural thing is to prove that a separable ...
11
votes
2
answers
624
views
Do locally convex topological vector spaces embed into diffeological spaces?
The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results ...
5
votes
2
answers
181
views
Differentiability of polytope shadow areas
Let $P$ be an opaque convex polyhedron containing the origin in $\mathbb{R}^3$,
and let $S$ be an origin-centered sphere strictly containing $P$: $S \supset P$.
For a point $x$ on $S$, let $\sigma(x)$ ...
2
votes
0
answers
56
views
Smooth bivariate functions identifiable under permutations
Consider a "smooth" $f : [0,1]^2 \to [0,1]$ and let $\pi : [n] \to [n]$ be a permutation of $[n] =\{1,\dots,n\}$. Let us define
$$
A^{f,\pi} := \Big( f\Big(\frac{\pi(i)}{n},\frac{\pi(j)}{n}\Big) \...