# 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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1answer
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### How can I smoothly transform one periodic function into another if the period time is allowed to differ?

How can I smoothly transform one periodic function into another if the period time is allowed to differ? Let us visit the very simplest example I can think of : Two audio signals pure sine waves. Note ...
2answers
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### How can we construct a compactly supported function which is equal to $1$ on a given interval?

I know that for any open cover $(\Omega_i)_{i\in I}$ of $\mathbb R$ we can find a $C^\infty$-partition of unity subordinated to $(\Omega_i)_{i\in I}$. Moreover, if $\eta$ is a mollifying kernel$^1$ on ...
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### Smoothing function to combine a polynomial and a constant

click to see the image I have below function: $y=0$ for $x<=0.12$ and $y=80((x/0.12)^8-1)$ for $x>0.12$ I need to smooth out the transition between these two functions. Can I find out the best ...
0answers
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### On analytic test functions

In what follows, ${\cal C}^\infty_c(U)$ is intended to be the set of all smooth functions with compact support defined on a non-empty open subset $U \subseteq \mathbb R^n$. Wikipedia: There $[\dots]$ ...
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### Surjective, open and submersion all have in common: image has non-empty interior. What about injective and immersion?

From these: Does open map imply dimension of domain is greater than or equal to dimension of range? (please don't use sard's theorem) Dimension of domain is greater than/less than/equal to ...
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### Name for Set Of Functions Smoothly Interpolating Between Points

Suppose I have a continuous differentiable surface such as the following: $$f(x,z)=\sin(e^{2\pi+x^2+z^2}), \space\space\space x,z\in\mathbb{R}$$ As $x$ and $z$ become large, the function oscillates ...
0answers
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### Extending smooth functions on submanifolds

If $M$ is a smooth manifold and $S\subset M$ is an embedded submanifold that is not closed in $M$, then can we construct a smooth function $f\in C^{\infty}(M)$ that is non-zero everywhere on $S$ and ...
2answers
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### Does open map imply dimension of domain is greater than or equal to dimension of range? (please don't use sard's theorem)

Note 1: same question as this, but please don't use sard's theorem or anything similar. preferably, no measure theory. Note 2: Didier's comment: I think another proof can use the constant rank ...
1answer
66 views

### Does open map imply dimension of domain is greater than or equal to dimension of range?

Update: Answer given uses Sard's Theorem. For no Sard's Theorem, see here: Does open map imply dimension of domain is greater than or equal to dimension of range? (don't use sard's theorem) ...
1answer
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### Doubt about pushforwards

Given the smooth function $\psi:M\to N$, where $M$ and $N$ are two differentiable manifolds, we have introduced the pushforward map $\psi_{*,p}:T_pM\to T_{\psi(p)}N$ with $p\in M$ and the image of ...
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### A weak notion of local convexity of the objective function for feasable points satisfying suficient condition

Let us suppose that does hold the sufficient condition for the minimization problem \begin{equation*} \begin{array}{c c} \text{minimize}_{x\in\mathbb{R}^n} & f(x) \\ \text{subject to} & \...
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### Prove that $\left\vert u\left( x,t\right) \right\vert \leq\frac{C}{t}$

Let $u$ be a solution to \begin{align*} u_{tt} & =\Delta u\text{, }x\in\mathbb{R}^{3}\text{, }t>0,\\ u\left( x,0\right) & =g\left( x\right) ,u_{t}\left( x,0\right) =h\left( x\right)...
1answer
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### Proving that inverse of a smooth function is smooth

Suppose I have a smooth function $g: \mathbb{R}^n \to \mathbb{R}^t$ and write the variables as $(x,y)$ where $x \in \mathbb{R}^t$. Suppose the Jacobian matrix of $g(\cdot, y)$ is invertible at $y = 0$ ...
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### Finite group acting on a smooth manifold $M$ by diffeomorphisms. Fixed Point set is Smooth Manifold [duplicate]

Let $M$ be a smooth manifold and let $G$ be a finite group acting on $M$ by diffeomorphisms. Show that the set of fixed points $M^G := \{m \in M \mid g . m = m, \forall g \in G\}$ is a smooth manifold ...
0answers
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### Question regarding “affinely radial test functions”

We shall call the function $f \colon \mathbb{R}^n \to \mathbb{R}$ affinely radial, if there exists $x \in \mathbb{R}^n$ such that the value of $f$ at a given point depends solely on the distance of ...
1answer
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### Smooth approximations to the ramp function

I am looking for $C^1$ and $C^2$ continuous approximations to the ramp function, f(x), that satisfy the condition $f(x)=0$, $x\leq0$ (essentially smoothing out the discontinuity in the first ...
1answer
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### How to smooth out 2 corners in a piecewise function?

I have been experimenting with this a lot, but it eventually proves itself to be trickier than I expected. Let \begin{equation} f(z) = \begin{cases} 1, &z<0 \\ \frac{1}{2}...
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### Invariance of rank under diffeomorphism

So I am kind of confused by this: We have $U \subset \mathrm{R}^m$ and $V \subset \mathrm{R}^n$ as open sets. If $f: U \rightarrow V$ is a diffeomorphism then we essentially have $m=n$. Same holds ...
1answer
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### Embedding Real Projective Space in Sphere

Let $n > 1$ be an integer and set $N =\frac{n(n+1) }{2}$. Let $F : R^n \setminus \{0\} → R^N$ be defined by $(x_i)_i \to (x_ix_j)_{i \leq j}$ , where the $x_i x_j$ are ordered lexicographically. (...
1answer
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### Smooth Map between the smooth manifolds $GL_n(\mathbb{R})$ and $S_n(\mathbb{R})$

First of all I am sorry for the bad notation, but I shall denote the smooth manifold of real symmetric $n \times n$ matrices by $S_n(\mathbb{R})$. I shall denote the transpose of a matrix $A$ by $A^T$ ...
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### Is the set $S$ of all smooth functions on $B_1$ whose normal derivative vanishes along the boundary $\partial B_1$ dense in $C^0 (B_1, \Bbb C)\$?

Let $B_1 : = \{z \in \Bbb C\ |\ |z| \leq 1 \},$ and let $C^0(B_1,\Bbb C)$ be the space of continuous complex valued functions on $B_1$ equipped with the uniform convergence topology. Let $S$ be the ...
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### Prove that an injective immersion of closed manifolds is an embedding

Definition: An embedding of manifolds is a differentiable map $f:N^n\hookrightarrow M^m$ which has an injective pushforward. An immersion of manifolds $f:N^n\looparrowright M^m$ is a differentiable ...
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### Show that if $f$ is $L$-smooth, then $g(x) := f(x) - \frac{m}{2} \Vert x \Vert^2$ is $(L-m)$-smooth

A continuously differentiable function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is $L$-smooth if $\nabla f$ is $L$-Lipschitz, i.e., for all $x,y \in \mathbf{dom}\,f$,  \Vert \nabla f(x) - \nabla f(...
1answer
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### Lagrange Remainder the taylor expansion of $e^{-1/x^2}$

I know $e^{-1/x^2}$ and $f(x) = 0$ if $x = 0$, is an example of a smooth function that cannot be represented all the coefficients are $0$, but I am curious about the Lagrange remainder. Doesn't the ...
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### A smooth action of a Lie group on a smooth manifold. Circle Example

I want to check that I understand this definition correctly and that my example makes sense. Let $\varphi:G\to Diff(M)$ be a action of a Lie group $G$ on a smooth manifold $M$. We say that $\varphi$ ...
1answer
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