# Questions tagged [smoothness]

The smoothness tag has no usage guidance.

113
questions

**5**

votes

**1**answer

553 views

### Is every variety an image of a smooth variety?

Let $X$ be a finite type scheme over a field $k$.
Is it true that there exists a surjective morphism $f : Y \rightarrow X$, where $Y$ is smooth over $k$?
In other words, is every such scheme a ...

**0**

votes

**0**answers

51 views

### (Linear combinations) Algebra of translated radial test functions [Radial basis functions]

Let $n \in \mathbb{N}$, $\Omega \subseteq \mathbb{R}^n$. Function $f \colon \Omega \to \mathbb{R}^n$ shall be called translated radial function if there exists $x \in \mathbb{R}^n$ and $g \colon [0, \...

**3**

votes

**0**answers

110 views

### A question regarding base change of a smooth algebra via completion

Let $(R,m)$ be an excellent Noetherian local ring. Let $S$ be a smooth (i.e. $R \rightarrow S$ is flat and has geometrically regular fibers) Noetherian $R$-algebra. Let $T$ be the $m S$-adic ...

**3**

votes

**0**answers

105 views

### Transversal intersection with linear subspaces

Let us work over an algebraically closed field $K$. If $X\subset \mathbb{P}^n$ is a closed subset of dimension $r$, then there should exist a linear subspace $L\subset \mathbb{P}^n$ of dimension $n-r$ ...

**4**

votes

**0**answers

177 views

### Effective bounds for a Bertini-type result

Suppose $X$ is a projective subvariety of $\mathbb{P}^n$ of codimension $r$ over $\mathbb{C}$, defined set-theoretically by $r$ homogeneous polynomials $P_1,\dots,P_r$ of degree at most $d$. By ...

**2**

votes

**0**answers

85 views

### etale locally infinitesimal lifting property

For a morphism $X\rightarrow Y$ of qcqs schemes, one has the usual notion of formal smoothness which says that for a pair $(R,I)$ with $I^2=0$, if there is a point $y\in Y(R)$ such that $y_{\vert R/I}$...

**3**

votes

**2**answers

155 views

### smooth functions on closed intervals with values in infinite-dimensional spaces

There are three ways to define when a ($\mathbb{R}$-valued) function on a closed interval is smooth:
$f$ can be extended to a smooth function on $(a - \epsilon, b + \epsilon)$ for some $\epsilon > ...

**1**

vote

**0**answers

129 views

### On smoothness and roughness of a number related to triangular numbers

Define $\triangle_n$ to be the $n$th triangular number.
Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$
Define $(\ell,k)$-smough numbers to be numbers that ...

**5**

votes

**1**answer

102 views

### Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?

We denote by $B_{p}^s(\mathbb{T}) := B_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$.
For $p>0$ fixed and ...

**6**

votes

**1**answer

330 views

### A smooth function $\mathbb{R}\to\mathbb{R}$ agrees with an analytic function on a bounded infinite set

Fix a smooth function $f:\mathbb{R}\to\mathbb{R}$. Do there exist real numbers $a<b$, an infinite set $S\subset (a, b)$ and an analytic function $g$ defined on $(a-\epsilon, b+\epsilon)$ for some $\...

**0**

votes

**0**answers

132 views

### How smooth can this be?

If $a$ is an even integer then how smooth can $a^2-1$ be?
Approximately how many integers in $a\in[0,t]$ are there such that $a^2-1$ is $k$-smooth?

**6**

votes

**1**answer

157 views

### On Glaeser's result for the square-root of a smooth non-negative function

One of the results due to Georges Glaeser is the following: there exists a non-negative $C^\infty$ function $f$ on the real line, flat at its zeroes, such that $\sqrt{f}$ is not $C^2$. On the other ...

**2**

votes

**1**answer

214 views

### On the proof of “Mapping space is a Chen space”

According to the page 5 in the paper Convenient Categories of Smooth Spaces https://arxiv.org/pdf/0807.1704.pdf by Baez and Hoffnung, Chen space is defined as follows:
(Note:I used different ...

**0**

votes

**0**answers

169 views

### Understanding Krantz's proof of Hefer's lemma in $\mathbb{C}^2$

Note: I initially phrased the question in a different way, and it did not receive much attention. In the hope to make it more interesting, I have included a (long) introduction to contextualize and ...

**3**

votes

**0**answers

82 views

### 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**

votes

**2**answers

293 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 $x=0$?
...

**8**

votes

**2**answers

291 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 $(x_1, \dots, x_n)$.
...

**2**

votes

**0**answers

118 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 ...

**4**

votes

**1**answer

153 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

149 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

50 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

90 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

96 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

99 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

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). ...

**0**

votes

**1**answer

111 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

309 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

148 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

156 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

272 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 ...

**14**

votes

**0**answers

234 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

597 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

296 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

263 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

243 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

105 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

102 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

230 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

60 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

183 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

239 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

90 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 ...

**20**

votes

**1**answer

856 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

107 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

388 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

417 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

261 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

208 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

920 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 - ...

**6**

votes

**1**answer

309 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) \, \,...