Questions tagged [smoothness]

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Smooth morphisms under base change, Qing Liu's proposition 4.3.38

I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
BernyPiffaro's user avatar
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1 answer
95 views

The sequence has a stationary accumulation point

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that (a) $\{x^k\}$ has an ...
Dat Ba Tran's user avatar
1 vote
1 answer
125 views

Smoothness of an equivalent norm

For an arbitrary set $\Gamma$, Day's norm on $c_0(\Gamma)$ is defined by $$ \Vert x \Vert = \sup \bigg \{ \bigg ( \sum_{k=1}^n 4^{-k} x^2(\gamma_k) \bigg )^{\frac{1}{2}} : (\gamma_1, \cdots, \gamma_n) ...
PPB's user avatar
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6 votes
1 answer
108 views

$C^{k, \alpha}$ gradients $\implies C^{k + 1, \alpha}$ level sets

Referring to the statement in the parentheses below (Regularity Theory for Elliptic PDE, Xavier Ros-Oton, p. 172) -- the authors go forward to make a bootstrapping argument that uses this result: ...
user43389's user avatar
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1 answer
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From relative convexity to modulus of continuity estimates for the dual gradient mapping

Let $F: \mathbf{R}^d \to \mathbf{R}$ be a convex function, let $m > 0$, and define $Q_m: \mathbf{R}^d \to \mathbf{R}$ to be the mapping $x \mapsto \frac{m}{2} \| x \|_2^2$. One says that $F$ is $m$-...
πr8's user avatar
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2 votes
0 answers
105 views

Admissible representations of an $\ell$-group are a (neutral) Tannakian category?

Let $G$ be an $\ell$-group in the sense of Bernstein/Zelevinsky (sometimes also called td-group), i.e. $G$ is a Hausdorff locally compact totally disconnected topological group. Prominent examples ...
Maty Mangoo's user avatar
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46 views

Approximation of function that has Lipschitz-continuous $n$-th derivative

Good afternoon. I'm trying to find in literature the solution for such a problem: for given function with $L_p$-Lipschitz continuous $p$-th derivative I need to find function $f_\varepsilon$ with $L_n$...
Dmitry Vilensky's user avatar
3 votes
2 answers
407 views

The real dimension of any real algebraic set equals the complex dimension of its complexification

I want to prove the following statement. Please help! Given any semialgebraic set $A$, consider its real Zariski closure $V_{\mathbb{R}}$ (which always has the same real dimension of $A$). Now ...
user86954's user avatar
1 vote
0 answers
178 views

Can we check smoothness of a morphism after base change to the algebraic closure?

I know that smoothness is fppf local on the base, but this is not enough because taking algebraic closures is not finitely presented. The reason I'm asking this is because I want an easy/quick ...
TCiur's user avatar
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1 answer
638 views

Is a Lipschitz continuous gradient equivalent to this condition?

I know if a function $f: \mathbb{R}^n \to \mathbb{R}$ is $L$-smooth, i.e. its gradient $\nabla f$ is $L$-Lipschitz continuous, then it satisfies the following inequality for any $x, x_0 \in \mathbb{R}^...
aest's user avatar
  • 143
12 votes
1 answer
379 views

Are algebras of smooth functions formally smooth?

Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$? If it helps, feel free to assume that $M$ is compact. (This is not a joke ...
Tobias Fritz's user avatar
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2 votes
1 answer
140 views

Smooth extension of functions at corners

Let $\mathbb{B}_1(0)\subseteq\mathbb{R}^n$ be the ball of radius $1$ in the Euclidean space, $n>1$. Suppose we have a cylinder $C=[0,1]\times \mathbb{B}_1(0)$ and suppose we are given smooth ...
D. Diddiero's user avatar
1 vote
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173 views

Why does Deligne's construction of the Galois representation attached to the new cuspidal forms require that the Kuga-Sato manifold be regular?

The origin of this question is related to the construction of Galois representations of Deligne attached to $f$ a new cuspidal form (of weight $k\geq 2$). To do this, we consider the fiber product $k$-...
Marsault Chabat's user avatar
8 votes
0 answers
338 views

Interpretation of $p$-adic 'smoothness'

real case: In the very first course of Calculus, one learns that a real function $f \colon \mathbb{R} \to \mathbb{R}$ is called smooth, if it is differentiable as many times as one pleases. So the ...
Maty Mangoo's user avatar
7 votes
1 answer
216 views

Is the Borel lemma projection a smooth principal bundle?

Consider the Fréchet spaces $C^\infty(\mathbb{R},\mathbb{R})$ and $\mathbb{R}^\infty$, and the continuous linear map $$ J\colon C^\infty(\mathbb{R},\mathbb{R}) \to \mathbb{R}^\infty $$ returning the ...
David Roberts's user avatar
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4 votes
1 answer
386 views

Smooth approximation of the $\max\{0,x\}$ function with controlled derivatives

Motivation/Hand-Wavy Question: In this post, it was asked what the best local approximation of $f(x):=\max\{0,x\}$ is by a polynomial of a given degree; with the answer provided by Chebyshev's ...
ABIM's user avatar
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5 votes
1 answer
650 views

A regular, geometrically reduced but non-smooth curve

Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ? Of course, the base field should be imperfect. In Exercise 4.3.22 of Qing Liu's book Algebraic ...
Yong Hu's user avatar
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0 answers
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$H^1 \cap C^0$ boundary, smooth $H^1$ extension

Assume we have a $u \in H^1(\Omega; \mathbb{R}^n) \cap C^0$ where $\Omega$ is a bounded open Set with smooth boundary. Also $u\vert_{\partial \Omega} \in H^1(\partial \Omega; \mathbb{R}^n) \cap C^0$. ...
Kilian Koch's user avatar
2 votes
0 answers
108 views

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^\...
dohmatob's user avatar
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1 vote
0 answers
137 views

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 $k$-...
cos_dm_math21's user avatar
3 votes
1 answer
175 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 ...
Benjamin Bauer's user avatar
3 votes
0 answers
134 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) ...
ABIM's user avatar
  • 4,989
3 votes
2 answers
300 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 ...
Overflowian's user avatar
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2 votes
0 answers
104 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)&...
Sam Benner's user avatar
2 votes
1 answer
252 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 $$ p_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\}, $$ ...
dohmatob's user avatar
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7 votes
2 answers
587 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 $(\...
John Pardon's user avatar
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6 votes
2 answers
1k 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 ...
Jacob Lu's user avatar
  • 903
2 votes
0 answers
84 views

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 } ...
ABIM's user avatar
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1 vote
0 answers
241 views

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(...
Antoine Balan's user avatar
2 votes
1 answer
100 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,...
Krishnarjun's user avatar
6 votes
2 answers
321 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 ...
Bazin's user avatar
  • 15k
5 votes
4 answers
581 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 ...
Leonid Positselski's user avatar
6 votes
1 answer
901 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 ...
P. Grabowski's user avatar
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0 answers
62 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, \...
Kacper Kurowski's user avatar
3 votes
0 answers
133 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 ...
Neil Epstein's user avatar
  • 1,752
3 votes
0 answers
242 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 $n-r$ ...
Jérémy Blanc's user avatar
4 votes
0 answers
203 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 ...
Zeyu's user avatar
  • 537
2 votes
0 answers
141 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 $y_{\vert R/I}$...
prochet's user avatar
  • 3,432
4 votes
2 answers
653 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 > ...
Carlos Esparza's user avatar
1 vote
0 answers
149 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 ...
VS.'s user avatar
  • 1,816
5 votes
1 answer
161 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 ...
Goulifet's user avatar
  • 2,174
6 votes
1 answer
449 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 $\...
user avatar
0 votes
0 answers
178 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?
VS.'s user avatar
  • 1,816
6 votes
1 answer
285 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 ...
Bazin's user avatar
  • 15k
2 votes
1 answer
232 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 ...
Adittya Chaudhuri's user avatar
0 votes
0 answers
202 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 ...
fram's user avatar
  • 11
4 votes
0 answers
178 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 ...
Hugo's user avatar
  • 384
12 votes
2 answers
644 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 $x=0$? ...
Asaf Shachar's user avatar
  • 6,611
8 votes
2 answers
333 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 $(x_1, \dots, x_n)$. ...
Bazin's user avatar
  • 15k
2 votes
0 answers
177 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 ...
user267839's user avatar
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