# Questions tagged [smoothness]

The smoothness tag has no usage guidance.

99
questions

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### When is the quotient of a manifold by a discrete group of diffeomorphisms a diffeological covering space?

I was reading An Introduction to Diffeology by Patrick Iglesias-Zemmour and he defines a diffeological covering space as a diffeological fiber bundle with discrete fiber.
My question: Consider a ...

**12**

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226 views

### Is the square root of a monotonic function whose all derivatives vanish smooth?

Let $g:[0,\infty] \to [0,\infty]$ be a smooth strictly increasing function satisfying $g(0)=0$ and $g^{(k)}(0)=0$ for every natural $k$.
Is $\sqrt g$ is infinitely (right) differentiable at $x=0$?
...

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votes

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268 views

### On Glaeser's Theorem for non-smooth functions

Glaeser's Theorem says that a $C^\infty$ function $F$ on $\mathbb R^n$ which is invariant under permutation of the variables is a smooth function of the symmetric polynomials of $(x_1, \dots, x_n)$.
...

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98 views

### Morphism between jet spaces smooth

In this article "Introduction to Jet Schemes and Arc Spaces" S. Ishii introduces the spaces of $m$-jets:
Let $X$ be a variety over algebraically closed field $k$. The space $X_m$ of $m$-jets ...

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votes

**1**answer

143 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 ...

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126 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 $X$....

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38 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 ...

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**1**answer

88 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)}(\...

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90 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 ...

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73 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(\overline{\...

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86 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). ...

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94 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?

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votes

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295 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 ...

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146 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?

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129 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 ...

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244 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 ...

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212 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 ...

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540 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 ...

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283 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 ...

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198 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?...

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223 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.
...

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95 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 ...

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96 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}_{...

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213 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 ...

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59 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, ...

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172 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-...

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vote

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185 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,\...

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84 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 ...

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751 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 ...

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106 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 $M$ (...

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344 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 ...

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389 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 $D^b(X) \cong D^b(Y)$.
...

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votes

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231 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 ...

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198 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 $g\geq2$. ...

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851 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 - ...

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287 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) \, \,...

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518 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 ...

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206 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) ...

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votes

**1**answer

341 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 ...

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1k 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(...

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232 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 ...

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285 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 $\...

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236 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 ...

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100 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 ...

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243 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 ...

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**1**answer

383 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 ...

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379 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:...

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319 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 ...

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votes

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125 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?

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111 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 $H$ ...