Questions tagged [smoothness]

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Smoothness of Radon transform

Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by $$ R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
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Two definitions for smoothness

I'm currently reading Sarah Witherspoon's book on Hochschild Cohomology. At the beginning of the fourth chapter it is given the following definition: Definition 1. If $k$ is a field and $A$ is a $k$-...
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3 votes
1 answer
119 views

Smoothness of ruled surface (asymptotic) parameterisations

A ruled surface $S$ shall be defined as surface consisting of straight line segments. It is commonly known (cf. [BER, p.362] or [STR, p.93] - bibliography at the end) that a ruled surface allows for a ...
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4 votes
0 answers
61 views

Covering number $C^k$-balls in $C(\mathbb{R}^n)$

Fix a positive integer $n$ and and an non-negative integer $k$. The Arzela-Ascoli theorem guarantees that for a given positive integer $k$ and a given $L>0$ the set $$ Ball_{C^{k,1}([0,1]^n)}(0,L) ...
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3 votes
2 answers
281 views

Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$

$\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$. I would like ...
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2 votes
0 answers
78 views

Can a smooth function always fit between two non-smooth functions? [closed]

Suppose I have two continuous functions, $f$ and $g$, with $f(x)<g(x)$ for all $x$ in some closed domain. Is it always possible to find a (piecewise, if needed) smooth function $h$ such that $f(x)&...
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2 votes
1 answer
98 views

On the Lipschitz continuity of $x \mapsto \arg\min_{c \in C}d(x,c)$ w.r.t Hausdorff distance

Let $C$ be a (nonempty) compact subset of euclidean $\mathbb R^n$, and consider the set-valued map $p_C:\mathbb R^n \to 2^C$ defined by $$ p_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\}, $$ ...
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7 votes
2 answers
467 views

Is $(x^2y,xy^2)$ log smooth?

Consider the map $$f:\mathbb C^2\to\mathbb C^2$$ $$(x,y)\mapsto(x^2y,xy^2)$$ We can view $f$ as induced by the map of monoids $g:\mathbb Z^2_{\geq 0}\to\mathbb Z^2_{\geq 0}$ given by the matrix $(\...
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29 views

Smoothness of an extension of the length function of gradient trajectories

Let $M$ be a closed manifold, $(f,g)$ a Morse-Smale pair, and $\varphi:\mathbb{R}\times M\to M$ the flow generated by $-\mathrm{grad}_gf$. Let $c_0,c_1$ and $c_2$ be critical points of $f$ with $l_i:=...
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Is these some transformation that transforms an arbitrary $\mathcal C^1$ function into an $L$-smooth function?

Let $\mathcal X\subset \mathbb R^d$. We say a function $f:\mathcal X\to \mathbb R$ is $L$-smooth if it is continuously differentiable and $\|\nabla f(x)-\nabla f(y)\|\le \|x-y\|$ holds for all $x,y\in ...
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  • 541
4 votes
2 answers
430 views

Zero points of a smooth function on $\mathbb{R}$

Assume $f(x)$ is a smooth function on $\mathbb{R}$ and $f$ does not vanish on any interval. In other words, $f$ can have zero points but we cannot find any interval $(a, b)$ such that $f(x)=0$ for all ...
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Functions with smooth projections on finite-dimensional subspaces

Let $E,F$ be Banach spaces and $F$ be finite-dimensional and $E$ be strictly convex. Let $f\in C(F,E)$ have the property that: $$ \text{For every finite-dimensional subspace $E'\subseteq E$ we have } ...
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1 vote
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A characterization of the integral

Let $I(f)$ be an endomorphism of the smooth functions with zero value in zero such that: $$\ln[1+I(f)]=I\left(\frac{f}{1+I(f)}\right). $$ Then, does it exist $g$ smooth such that: $$I(f)(x)=\int_0^x f(...
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2 votes
1 answer
68 views

The notion of smoothness in the local situation

I am reading Bump's book on Automorphic forms and Representations and I am able to draw a lot of parallels between the theory of $GL(2, \mathbb{R})$ which is the infinite place and the theory of $GL(2,...
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6 votes
2 answers
205 views

A smooth function such that the second derivative of its absolute value is a distribution of positive order

Let $f\in C^\infty(\mathbb R;\mathbb R)$ and let us define $g(x)=\vert f(x)\vert$. It is easy to verify that $g$ is locally Lipschitz-continuous function, but I would like to find an example of a ...
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5 votes
4 answers
482 views

Relative version of Hilbert syzygy theorem

I presume that answers to the following questions are likely to exist in the literature; so this question is mostly a reference request (but failing that, I would be certainly interested in learning a ...
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6 votes
1 answer
721 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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(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, \...
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3 votes
0 answers
127 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 ...
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  • 1,628
3 votes
0 answers
139 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 $n-r$ ...
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4 votes
0 answers
191 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 ...
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  • 527
2 votes
0 answers
102 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 $y_{\vert R/I}$...
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  • 3,151
3 votes
2 answers
333 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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1 vote
0 answers
132 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 ...
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  • 1,698
5 votes
1 answer
117 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 ...
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  • 2,010
6 votes
1 answer
343 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 $\...
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0 votes
0 answers
140 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?
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  • 1,698
6 votes
1 answer
200 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 ...
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2 votes
1 answer
219 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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0 votes
0 answers
175 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 ...
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  • 11
4 votes
0 answers
97 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 ...
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  • 374
12 votes
2 answers
408 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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  • 6,286
8 votes
2 answers
307 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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2 votes
0 answers
144 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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  • 479
4 votes
1 answer
191 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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  • 1,679
3 votes
0 answers
161 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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1 vote
0 answers
75 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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4 votes
1 answer
104 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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2 votes
0 answers
102 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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3 votes
0 answers
107 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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  • 720
1 vote
0 answers
88 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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  • 555
0 votes
1 answer
146 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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8 votes
2 answers
329 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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0 votes
0 answers
152 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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  • 6,782
3 votes
1 answer
204 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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  • 41
6 votes
2 answers
284 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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15 votes
0 answers
262 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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7 votes
1 answer
657 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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  • 1,899
8 votes
0 answers
309 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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  • 81
8 votes
0 answers
334 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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