Questions tagged [smooth-functions]

For questions about infinitely or arbitrarily differentiable (smooth) functions of one or several variables. To be used especially for real-valued functions; for complex-valued functions, the tag holomorphic-functions is more appropriate.

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Is a map which sends a $3\times 3$ symmetric tensor to an element of $SO(3)$ which diagonalizes it necessarily discontinuous?

For a $3\times 3$ symmetric matrix $Q$, one can construct a map to $SO(3)$ which sends $Q$ to a matrix which diagonalizes it. If $Q$ has distinct eigenvalues, there are three choices for rotation ...
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Smooth function that "dominates" the inverse of the derivatives of positive function in $\mathbb(R)^n$ that explodes at infinity

I know that probably the title is not that clear, but i couldn't find another way to name this problem. Let's say i have a strictly positive function $f \in C^2(\mathbb{R}^n)$ such that $\lim\limits_{|...
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0 votes
0 answers
10 views

Equivalent characterization of L-smooth function

If a function $f(x)$ is $L$-smooth, then is it equivalent to say that $$f(x) - \dfrac{1}{2L} || \nabla f(x)||^2 \geq 0 ?$$ Can someone help me prove this? I have the definition that a function $f(x)$ ...
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5 votes
0 answers
55 views

Smoothness of the function $\lambda \mapsto \inf_{x \in \mathcal{C}}\|\lambda x - y\|^2$

Let $y\in \mathbb{R}^d$ and suppose that $\mathcal{C}\subset \mathbb{R}^d$ is a compact convex set. I want to know if the function $\psi\colon\lambda \mapsto \inf_{x \in \mathcal{C}}\|\lambda x - y\|^...
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0 answers
61 views

Asymptotic expansion of non analytical function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real smooth (not necessarily analytical) function. Suppose I tell you that $f(t)$ admits a full asymptotic, at $t\rightarrow0$ expansion up to all orders ...
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1 vote
1 answer
30 views

Analytic tanh-like function with infinite radius of convergence.

Consider the Taylor expansion of $\tanh$ around $0$. The radius of convergence is finite ($\pi/2$). Define a $\tanh$-like function a function $f:\mathbb R\to\mathbb R$ such that: $f(0) = 0$; $\lim_{x\...
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  • 2,801
3 votes
1 answer
76 views

Can germs be defined as a quotient of vector spaces?

Summary: Let $M$ be a smooth manifold and $p\in M$. I know of two notions of germs of functions at $p$, the more restrictive of which can be written as a quotient vector space. I am curious whether ...
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  • 4,859
2 votes
1 answer
69 views

showing that a function is smooth [closed]

Here is the question I am trying to understand its answer: Here is the answer given to me at the back of the book: Here is the definition of a smooth map: Still I do not understand the solution ...
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  • 1,187
1 vote
0 answers
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Construction of a special diffeomorphism with some special properties

Let $x,y\in(a,b)$ be real numbers. I am trying to find a diffeomorphism $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying that $f(x)=y$ and $f(t)=t$ for all $t\notin(a,b)$. Here is my attempt. Let $g\in ...
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  • 179
8 votes
3 answers
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Why does a function has to be differentiable so many times to be considered smooth?

I'm studying "Smoothness". If a function is once differentiable for all x's, shouldn't it be considered smooth? Because it does "look smooth" for all f(x), there's no way it will ...
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Smooth function vs $L$-smooth function

Smooth function is a function that has derivatives in all domains, and $L$-smooth function is a function that satisfies $||\nabla f(x) - \nabla f(y)||_2 \leq L||x-y||_2$ for all $x,y$. What is the ...
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1 answer
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Extension of a Partial Section of a Fibre Bundle

I have to prove the following fact. Let $\pi :E\to M$ be a vector bundle of rank $k$, over a $n$-manifold $M$. If $S\subseteq M$ is closed and $s\colon S\to E$ is a partial section of $\pi$ (i.e. $\pi(...
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  • 179
4 votes
0 answers
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It is possible to find non trivial solutions $f(t) \in C_c^\infty$ to $\dot{f}(t)=2f(2t+1)-2f(2t-1),\,f(0)=1$ for the whole $\mathbb{R}$ domain?

It is possible to find non trivial solutions $f(t) \in C_c^\infty$ to $\dot{f}(t)=2f(2t+1)-2f(2t-1),\,f(0)=1$ for the whole $\mathbb{R}$ domain? I am trying to find examples of solutions of finite ...
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  • 819
0 votes
0 answers
31 views

Extending smooth function on subset to manifold

I want to prove this proposition: Proposition: Let $M$ be a smooth manifold that is Hausdorff and $2^{nd}$ countable. Let $U \subseteq M$ be open and let $p \in U$ be some point. If $f : U \...
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1 answer
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For a smooth integral curve, $c$, if $\lim_{t \rightarrow \inf} c(t) = p$ does $\lim_{t \rightarrow \inf} c'(t) = 0$

If a smooth integral curve has a limit point $p$ and it takes infinite time to reach that point. Does that imply that its velocity approaches $0$ at that point? Or in other words, Updated based on ...
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  • 701
1 vote
2 answers
77 views

Why do we need tangent vector unequal to zero for smoothness of a vector function?

]1 My textbook gives this definition of smoothness of a $\vec r(t)$ on an interval $I$ of $t$. Why do we need $\vec r'(t) \neq\vec0$ on $I$?
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  • 332
3 votes
1 answer
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Applications of Partition of Unity

Theorem that I have to prove: Let $M$ be a smooth manifold, $f\colon M\to\mathbb{R}^n$ be a continuous map and let $S\subseteq M$ be a closed subset of $M$ such that the restriction $f_{|S}\colon S\to\...
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  • 179
0 votes
0 answers
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Let $U \subset \mathbb{R}^{m}$ be a open subset. Does there exists $f \in C_c^\infty(\mathbb{R}^{m})$ such that $U=f^{-1}\{(0,\infty)\}$?

Let $U \subset \mathbb{R}^{m}$ be a open subset. Does there exists $f \in C_c^\infty(\mathbb{R}^{m})$ such that $U=f^{-1}\{(0,\infty)\}$? Let $F=\mathbb{R}^{m}\setminus U$. I tried to use the results ...
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  • 2,064
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0 answers
27 views

Confusion on notion of $L$-smoothness

I have recently been reading about optimization and have come across two (seemingly different) notions of "L-smoothness". Let me record them now; let $H$ be a separable Hilbert space and $f:...
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  • 364
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2 answers
43 views

differentials of smoothly varying family of maps

Let $F:N\times M \to M'$ be a smooth map, which we interpret as a "smooth" family of maps $M \to M'$, parametrized by N, so we have a map $F(y,\cdot):M \to M' \: \forall y \in N$. Show that ...
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  • 330
0 votes
0 answers
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Construction of flat functions

The popular function f(x) = e^(-1/x^2) when x is nonzero = 0 when x = 0 defines one that is flat at 0 (all derivatives vanish there). My question is: How do I construct a (differentiable) ...
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1 vote
0 answers
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Construction of a immersion

It's possible construct a $C^\infty$ immersion $f$ of an open interval as in the next picture? Where the image curve has a part contained in the y-axis betwen -1 and 1, and self-intersections in the ...
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  • 252
1 vote
2 answers
52 views

How to prove product and quotient of smooth functions is smooth

I'm trying to prove the following problem: Let $A\subset\mathbb{R}^n$ be open. If $f, g: A \rightarrow \mathbb{R}$ are smooth, show that $fg$ and $f/g$ is smooth. (For the quotient case, $g$ is ...
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  • 197
1 vote
0 answers
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Smooth Atlas of a Smooth Manifold

In Lee's Introduction to Smooth Manifold (2nd Ed.) the Proposition 2.5 gives two equivalent characterizations of smoothness of a map $F \colon M \to N$ between smooth manifolds $M$ and $N$, the second ...
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  • 113
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0 answers
39 views

Mathematical Definition of "Smoothness" : Requirement of Differentiable Functions to be Smooth?

I have often heard the following arguments made about the requirement for differentiable functions to be smooth: Fractal Functions are nowhere smooth Differentiable functions need to be smooth ...
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  • 1,694
1 vote
1 answer
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For a Lie-group G and embedded Lie-subgroups K < H < G, prove that $\pi:G/K \rightarrow G/H$ is a surjective smooth submersion

The question is already in the title. For a Lie-group G and embedded Lie-subgroups $K < H < G$, prove that $\pi:G/K \rightarrow G/H$ is a surjective smooth submersion. Where it is meant that $K$ ...
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  • 330
0 votes
0 answers
39 views

Cartesian product of smooth functions is smooth.

Let $f_i: U_i \subseteq \mathbb{R}^{m_i}\to \mathbb{R}^{n_i}$ be functions that are smooth at $p_i \in U_i$, where $U_i$ is an open subset of $\mathbb{R}^{m_i}$, for $i=1,2$. I want to show that the ...
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2 votes
1 answer
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What is the domain of $g^* \circ f^*$?

Here is the question I am trying to prove: If $M$ is a smooth manifold, let $C^{\infty}(M)$ be the set of smooth functions $M \to \mathbb R.$ The set $ C^{\infty}(M) $ is an $\mathbb R$-algebra under ...
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  • 1,187
0 votes
1 answer
64 views

Regarding Theorem 1.9, Conway, complex Analysis

I have a doubt in equation (1.11) in Chapter 4, Section 1 of J.B Conway’s, functions of one complex variable. I know that the estimation of the integral in equation 1.10, comes from the previous ...
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0 answers
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Smoothness of a function after integration w.r.t. one variable

Let $f:[0,1]\times [0,1]\to\mathbb{R}$ be a continuous function. Suppose that for any $t_0\in[0,1]$, $s\mapsto f(t_0,s)$ is a smooth function of $s\in[0,1]$, and that for any $s_0\in[0,1]$, $t\mapsto ...
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  • 177
0 votes
2 answers
40 views

Proving that $\lambda$ is not analytic in differential topology.

Let $\lambda : \mathbb{R} \to \mathbb{R}$ be defined by $\lambda(t) =\begin{cases} 0, & \text{for } t \leq 0 \\ e^{-1/t}, & \text{for } t > 0. \end{cases}$ This is a smooth function with ...
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  • 1,187
0 votes
0 answers
29 views

Generalization of differentiability/smoothness class "$C^n$" notation

Is there any notion, in the literature, of generalizing the notation $C^n$ where $n \in \mathbb{z}$, which represents smoothness (or, more precisely, existence and continuity of the $n$th derivative), ...
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2 votes
2 answers
300 views

Is the function $f(x)=\frac{\left(1-x^2+\sqrt{\left(1-x^2\right)^2}\right)}{2}e^{-\frac{x^2}{1-x^2}}$ a bump-function $\in C_c^\infty$? Diff. Eq.?

Is the function $f(x)=\frac{\left(1-x^2+\sqrt{\left(1-x^2\right)^2}\right)}{2}e^{-\frac{x^2}{1-x^2}}$ a bump-function $\in C_c^\infty$? Which autonomous differential equation it fulfill? (note it is ...
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1 vote
1 answer
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Do test functions have smooth extensions?

Let $\Omega\subset\mathbb R^d$ be open. A smooth function $\phi\colon \Omega\to\mathbb R$ is called a test function on $\Omega$ if its support is compact and inside $\Omega$. I expect that the ...
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  • 1,740
2 votes
1 answer
35 views

$\psi' \circ F \circ (\varphi')^{-1}$ is $C^\infty$ at $\varphi'(p)$.

Suppose $F: N \to M$ is $C^\infty$ at $p \in N$. Show that if $(U', \varphi')$ is any chart containing $p$ in the atlas of $N$ and $(V', \psi')$ is any chart containing $F(p)$ in the atlas of $M$, ...
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  • 95
1 vote
0 answers
44 views

Determining whether the evaluation mapping $p \mapsto \frac{d}{dx}\bigg|_p$ is smooth

For some reason, this quetion gives me a mental block: Suppose that we have the mapping $f(a, p) = a\cdot \frac{d}{dx}\bigg|_p$ for $a, p \in \mathbb{R}$, and we'd like to argue why this is a smooth ...
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  • 1,755
2 votes
1 answer
71 views

Discretization of the Optimization Problem

Say that $g:[0,1] \rightarrow \mathbb{R}$ is a smooth function that has a unique maximizer. Let $x_1,\cdots,x_n$ be a set of equally spaced points on $[0,1]$. Question: How close does the discretized ...
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1 vote
1 answer
34 views

Existence of specific curves necessary to construct $T_p(M)$

Let $M$ be a manifold of dimension $d$, and $p$ some fixed point on $M$. Define the tangent space $$T_p(M):=\{v_{\gamma,p}:\gamma\text{ is a curve in $M$ passing through $p$}\}.$$ We want to construct ...
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  • 1,506
1 vote
0 answers
38 views

Smoothing function details in analytic number theory paper

My question: How does the bound \[ g_c(l)\ll 1/(xl)^2\hspace {10mm}\text {for }x^{1-2\epsilon }l>c^2\hspace {10mm}(1)\] towards the bottom of page 277 of http://matwbn.icm.edu.pl/ksiazki/aa/aa61/...
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  • 1,323
0 votes
2 answers
51 views

Scale a cutoff $\psi \in C^\infty(0, \infty)$ so that $\int^\infty_0 (\psi'_n)^2 r dr \to 0$ as $n \to \infty$.

Let $\psi \in C^\infty((0,\infty); [0,1])$ be supported away from zero and identically one near $[1, \infty]$. Is it possible to make some nonlinear scaling $\psi_n = \psi \circ f_n(r)$ of $\psi$ to ...
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  • 4,426
2 votes
2 answers
72 views

Prove that a map beween manifolds that is smooth when composed by an embedding is smooth.

I am studying differential geometry, and I am having problems with the following exercise: Let $M,N,W$ be smooth manifolds, $F:M\to N$ a smooth map and $i:W\to N$ a smooth embedding such that there ...
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  • 1,383
0 votes
0 answers
35 views

locally extending smooth functions on embedded submanifolds

I was wondering, if $S\subseteq M$ is an embedded submanifold and $f\in C^{\infty}(S)$ then for each $p\in S$, we may choose an adapted open set $U_p$ on which $U_p\cap S$ is defined by the vanishing ...
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2 votes
1 answer
26 views

Show that the multiplication of two complex numbers is a $C^{\infty}$ map

Let $$f: \mathbb{R}^{2} \times \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \quad f(z, w)=z w$$ be the map induced by the multiplication of complex numbers. Check whether it is a $C^{\infty}$-map. By ...
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  • 57
4 votes
2 answers
156 views

Smoothing in analytic number theory

Often we are interested in the sum \[ \sum _{n\leq x}a_n\] for some number theoretic sequence $a_n$ and often the study of the "smooth" sum \[ \sum _{n=1}^\infty a_n\phi _x(n)\] if simpler (...
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  • 1,323
0 votes
1 answer
38 views

What is the simplest formula for smooth step function (Or smooth transition function)?

I am working on designing a smooth step function, which is changing from 0 to 1 while x changes from a to b, and this f(x) are expected can equal to 0 or 1 exactly at point a or at point b. I am using ...
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0 votes
0 answers
53 views

Write non-differential equation as a smooth optimization problem

Edit: the previous title was "Re-writing a function so it becomes differentiable"- I see now that this is not exactly possible. The actual question is to "show that this (problem below) ...
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  • 11
0 votes
1 answer
50 views

Relationship between smoothness and Lipschitzness

Given a convex function $f:\Omega\mapsto\mathbb{R}$ has function value bounded $|f(x)|\leq B$, diameter of the convex domain bounded $\|x-y\|_2\leq D, x,y\in \Omega$ and $\beta$ smoothness: $\|\nabla ...
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1 vote
0 answers
37 views

Is the mapping from an empty set to an empty set smooth?

For a function/mapping of the form $f(x) = x^2$, we can perform differention to check if it is smooth. But for a mapping from empty set to empty set, what is the corresponding function $f(x)$ and how ...
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  • 627
3 votes
2 answers
126 views

Smooth function passing through a countable number of points

Given a sequence $(y_n)_n$ of real numbers, can we find a smooth ($C^1$, $C^2$ or even $C^\infty$) real function $\phi$ such that $\phi(2^n)=y_n$ for all $n\in\mathbb{N}$ ? It's clear if we want a ...
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1 vote
0 answers
50 views

Is $C^\infty_0(\mathbb{R}\!\setminus\!\{0\})$ dense in $L^2(\mathbb{R})$ with respect to the $L^2$-norm?

Why should that matter? I'd like to prove whether or not the operator $X^{-n}:D({X}^{-n})\subset L^2(\mathbb{R})\to L^2(\mathbb{R})$, with $n\in\mathbb{N}$, defined as $X^{-n} :f\mapsto{x^{-n}}f$ ...
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