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.
767
questions
2
votes
0
answers
143
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Step in the proof of $f^{\epsilon}=\eta_{\epsilon}\ast f$ is $C^{\infty}$
I don't follow one step in the first proof of Theorem $7$ of Appendix $C$ of Partial Differential Equations by Lawrence Evans. This subsection goes through properties of mollifiers. The step I don't ...
3
votes
2
answers
349
views
Visualizing vector fields "multiplied" by a function
If $M$ is a smooth manifold,
how do I interpret an expression such as $f X$ where $f \in C^{\infty}(M)$ and $X$ is a vector field, i.e. a map $M \rightarrow TM$?
To my understanding, a vector field is ...
0
votes
2
answers
54
views
Is $f: S^1 \rightarrow \Bbb{R}$ an embedding, submersion, or immersion? Defined via $f(x,y)=y$.
Let
$$f: S^1 \rightarrow \Bbb{R}$$
be a smooth map given via
$$(x,y) \mapsto y$$
I can't it being surjective as large elements of $\Bbb{R}$ don't get "hit" and it cannot be $1-1$ as $(-1,0),(...
5
votes
1
answer
210
views
Why is it said that Laplacians "smooth solutions"?
Say you have some PDE for a function $f:[0,T]\times \mathbb{R}^n\to\mathbb C$
$$\frac{\partial f}{\partial t}=Lf,$$
for some differential operator $L$. If $L$ contains a Laplacian term $$\Delta f=\...
1
vote
1
answer
131
views
Smooth function satisfying $f^{-1}(x)=f(x+1/2)-1/2$
Let $f$ be a smooth (infinitely differentiable) function satisfying $f^{-1}(x)=f(x+1/2)-1/2$ for all real $x$ (where $f^{-1}$ is an inverse function).
I have strong feeling that it also must satisfy $...
1
vote
0
answers
111
views
Under what conditions can we push-forward the differential forms on a manifold $M$ to differential forms on a manifold $N$ by a smooth map $f$?
Let $M,N$ be two Complex Manifolds of the same (complex) dimension.
Let $f: M \longrightarrow N$ be a Smooth Map.
It is well-known that differential forms on the manifold $N$ can always be pulled-back ...
1
vote
0
answers
25
views
Find four smooth $C^{\infty}$ real functions satisfying a certain property around the origin
In the book "A Singular Mathematical Promenade", there's a Theorem of Kontsevich:
It's impossible to find polynomials $P_1(x)$, $P_2(x)$, $P_3(x)$, $P_4(x)$ satisfying the following two ...
0
votes
1
answer
67
views
$f,g \in C^\infty$ agree on a n-hood, yet are different?
Can $f,g : \mathbb{R} \rightarrow \mathbb{R}$ be smooth functions that agree on a neighborhood, yet are not equal everywhere ?
0
votes
0
answers
40
views
The Support of Differential Forms defined by Pulling-Back on Complex Manifolds of the Same Dimension
Let $M,N$ be two complex manifolds of the same (complex) dimension.
Let $\omega$ be a differential form on the manifold $N$ whose support denoted by $S$.
Let $f: M \longrightarrow N$ be a smooth map.
...
0
votes
0
answers
95
views
Formal properties of Gaussian kernel smoother?
Consider a continuous but not differentiable function $f: \mathbb R \to \mathbb R$ and $Z$ a standard normal random variable. Define $g: \mathbb R \times \mathbb R_{>0} \to \mathbb R$ as the ...
1
vote
1
answer
123
views
Every ring homomorphism $C^\infty(M)\rightarrow \mathbb{R}$ is an $\mathbb{R}$-algebra homomorphism.
In this answer to a question in Milnor, is says that if $e:C^\infty(M,\mathbb{R})\rightarrow \mathbb{R}$ is any ring homomorphism, then it is in fact an $\mathbb{R}$-algebra homomorphism, because $e(c)...
1
vote
0
answers
42
views
Does integration give a morphism out of the diffeological space of smooth maps?
The Cartesian closed structure on the category of diffeological spaces in particular gives, for any smooth manifolds (or even diffeological spaces) $M,M'$ a diffeological space $\mathrm{SmthMaps}(M,M')...
1
vote
1
answer
48
views
Why this function appears to have a singularity at each positive integer?
For fixed $l\in \mathbb Z_+$ and $\alpha\in \mathbb R$, define a function
$$p_l(\alpha) = \sum_{k=0}^\infty \binom \alpha k \binom{-\alpha}{k+l} e^{-2k}.$$
Due to the exponential suppression $e^{-2k}$,...
1
vote
1
answer
105
views
Smooth flattening of a function near zero with control over derivatives
I have a smooth function, say $f: \mathbb R \rightarrow \mathbb R$, such that $f' \rightarrow 1$ as $x \rightarrow \infty$ but there is a neighbourhood of zero in which $f' < 0$, possibly with a ...
1
vote
0
answers
70
views
Maximum of derivative of a smooth cutoff function
Let $a,b\in\mathbb{R}$ with $a<b$, and let $h:\mathbb{R}\to\mathbb{R}$ be the smooth cutoff function defined by
\begin{align}
h(x):=\frac{f(b-x)}{f(b-x)+f(x-a)}
\end{align}
where $f:\mathbb{R}\to\...
2
votes
1
answer
81
views
Properties of the $L^2$ dual realizing test functions as distributions
Let $U \subset \mathbb{R}^n$ be open; let $C^\infty_c(U)$ denote the smooth real/complex compactly-supported functions on $U$, with the "Canonical LF" topology, i.e. the limit topology from ...
1
vote
0
answers
39
views
Are common spaces of test functions "diffeomorphism-invariant"?
Let $U,V \subset \mathbb{R}^n$ be open subsets, and suppose there is a diffeomorphism $\phi : U \rightarrow V$.
Then, $\phi$ should induce a linear isomorphism between $C^\infty_c(U)$ and $C^\infty_c(...
0
votes
0
answers
84
views
Non-trivial smooth involutions of the reals
I'm interested in certain involutions of the real numbers, i.e functions $f$ such that $f\circ f = \text{id}_{\mathbb{R}}$.
It has been shown here that $\text{id}_{\mathbb{R}}$ is the only increasing ...
3
votes
0
answers
83
views
Realizing singular homology boundaries as boundary restrictions of manifold maps, in smooth case
Let $X$ be a topological space.
Let $\rho = \sum_j \sigma_j$, where the sum is finite, be a chain of singular $k$-simplices in $X$ (in the $k$th singular chain group of $X$, with $\mathbb{Z}$ ...
1
vote
0
answers
73
views
Showing convolution improves smoothness
I want to show that for the $2\pi$-peroidc, integrable function, the convolution operation improves the smoothness, the convolution for my cases is only defined over the complex unit circle, that is
$$...
1
vote
0
answers
20
views
Two questions about convolution
Good day to everyone, I have the following two questions about convolutions of smooth functions with compact suppport ($C_c^k(\mathbb{R}^d)$ or $C_c^\infty(\mathbb{R^d})$).
If $f$ from $C_c^m(\mathbb{...
2
votes
0
answers
75
views
Let $F:M\to M$ is a smooth map where $M$ is compact connected smooth manifold and $F\circ F=F$ then show that $F(M)$ is a submanifold of $M$. [duplicate]
Given here composition of $F$ with $F$ gives $F$ then to show $F(M)$ is a submanifold of $M$. I was thinking in the way that $F(M)$ should be inverse image of some regular value of some smooth ...
1
vote
0
answers
149
views
Bandwidth in Gaussian kernel smoothing
I am trying to implement a Gaussian kernel smoother. The equation is
$$\begin{split}f(x)&=\sum_{i=1}^N w_i(x)y_i\\
w_i(x)&=\frac{\kappa_h(x-x_i)}{\sum_{i'=1}^N \kappa_h(x-x_{i'})}\end{split}$$
...
1
vote
0
answers
27
views
Let $\mathcal{C}$ be a graph given by $x=by^a$ where $b\in \mathbb{R}$ is just a constant and $a \in \mathbb{Q}_+^*$
Let $\mathcal{C}$ be a graph given by $$x=by^a$$ where $b\in \mathbb{R}$ is just a constant and $a \in \mathbb{Q}_+^*$.
Under the assumptions above, Is it possible to say that the graph $\mathcal{C}$ ...
1
vote
1
answer
92
views
How to represent this one-dimentional movement in math?
I'm struggling trying to convert this movement behavior into an equation.
For future reference, this is as a result of searching for a solution for a previous question of mine.
Thank you @eyeballfrog ...
0
votes
1
answer
132
views
Constrained smooth transition between points.
Since I'm not very skilled at math, I'll try to explain my problem with apples.
At a certain train station, every train driver has a specific route to cover everyday.
A pair of brothers, who happen to ...
0
votes
0
answers
38
views
Diffeomorphisms imply bijections between the set of smooth functions?
This interesting answer raised a question in my mind:
Suppose that $M$ and $N$ are smooth manifolds and $f:$ $M \rightarrow N$ is a diffeomorphism. Can we conclude that $C^{\infty} (M)$ and $C^{\infty}...
0
votes
0
answers
35
views
For closed $F\subset M$, there exists $\varphi : M\to \mathbb R$ s.t. $\varphi$ is smooth and $F=\varphi^{-1}(\{ 0 \})$.
Let $M$ be a manifold.
Show that for any closed set $F\subset M$, there exists $\varphi : M\to \mathbb R$ s.t. $\varphi$ is smooth and $F=\varphi^{-1}(\{ 0 \})$.
I think the function like this seems ...
4
votes
1
answer
127
views
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 ...
5
votes
0
answers
76
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\|^...
0
votes
0
answers
102
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 ...
1
vote
1
answer
163
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\...
3
votes
1
answer
150
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 ...
1
vote
0
answers
11
views
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 ...
8
votes
3
answers
1k
views
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 ...
0
votes
1
answer
147
views
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(...
4
votes
1
answer
194
views
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 ...
0
votes
0
answers
54
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 \...
0
votes
1
answer
101
views
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 ...
1
vote
2
answers
659
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 $I$?
3
votes
1
answer
236
views
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\...
0
votes
0
answers
58
views
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 ...
0
votes
0
answers
159
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:...
0
votes
2
answers
93
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 ...
0
votes
0
answers
88
views
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) ...
2
votes
2
answers
329
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 ...
2
votes
1
answer
370
views
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 ...
0
votes
0
answers
175
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
...
1
vote
1
answer
67
views
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 $K$ ...
2
votes
1
answer
38
views
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 ...