Questions tagged [smoothness]

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5
votes
1answer
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 ...
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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2answers
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 > ...
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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
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1answer
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
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1answer
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
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0answers
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
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1answer
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
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1answer
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 ...
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0answers
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
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0answers
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
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2answers
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
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2answers
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
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0answers
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
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1answer
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
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0answers
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 $X$....
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0answers
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
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1answer
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)}(\...
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0answers
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
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0answers
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(\...
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0answers
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
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1answer
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
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2answers
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
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0answers
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
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1answer
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
2answers
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
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0answers
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
1answer
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
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0answers
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
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0answers
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
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2answers
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
1answer
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 ...
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1answer
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
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0answers
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
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0answers
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
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0answers
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
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1answer
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,\...
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0answers
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
1answer
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
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0answers
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
1answer
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
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1answer
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
1answer
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
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0answers
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
2answers
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
1answer
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) \, \,...