Questions tagged [smooth-functions]

Question 1

I have trouble understanding the difference in the following problem. I have the function F(x) which is not differentiable at x=a $$F(x):=\left\lbrace\begin{array}{cc} 0 & \text{ for $x<a$}\\ G(...

Question 2

Let $X$ be a finite CW complex. I know that $X$ has a topological embedding into $\mathbb{R}^N$ for some $N$. We can proceed inductively. Suppose that $X_k$, the union of the k-cells of $X$ are ...

Question 3

Suppose that $M$ is a manifold with boundary $\partial M$ and that there is a continuous map $f: M \rightarrow N$ (where $N$ is a smooth manifold). Whitney's approximation theorem states that $f$ can ...

Question 4

I'm working on a problem not from any homework set, and the following question came up. Tools or references from any field might help. Say we have a regular smooth function $f:\mathbb{D}\to\mathbb{R}^...

Question 5

I'm looking for a formula that would work to elevate my students' grades. What I'm trying to say is when the minimum score gotten by my student is $0$ and the maximum is $42$, I want to convert them ...

Question 6

Suppose $f$ is a differentiable real-valued function of a real variable. By linearisation, we can write $$f(x)=f(0)+xf'(0)+xh(x)$$ where $\lim_{x\to 0} h(x)=0$. If $f$ is twice-differentiable then we ...

Question 7

I've got a question in the context of smooth and convex optimization: Let $f_i\in C(\mathbb{R}^d)$ for all $i\in[N]$ be convex and Lipschitz-smooth. That means $||\nabla f_i(x)-\nabla f_i(y)\leq L||x-...

Question 8

I'm learning differential geometry properly for the first time and I'm having a hard time understanding how the notion of a tangent vector or a derivative in the context of smooth manifolds squares ...

Question 9

For didactical / illustrative pourposes I’m searching a real valued function with the following properties: $\mathcal{C}^\infty$ over all $\mathbb{R}$ or better analytic over the entire complex plane....

Question 10

This is a past analysis exam problem: Let $f \in C^{\infty}(\mathbb{R})$ be an infinitely differentiable real-valued function on $\mathbb{R}$ so that $f(x)=1$ for all $x \in[-1,1]$ and $f(x)=0$ for ...

Question 11

Let $f,g\in C^{\infty}(\mathbb{R})$ such that \begin{equation*} f(1/n)=1, g(1/n)=\frac{n}{1-n^2} \end{equation*} for $n=2,3,4,\dots$ What are the possibile values of $f(\pi)$ ? Do you have any ...

Question 12

I just start reading Introduction to Manifolds by Loring W.Tu, and on page 6 it states Lemma 1.4 (Taylor's theorem with remainder). Let $f$ be a $C^\infty$ function on an open subset U of $\mathbb{R}^...

Question 13

I am reading "An Introduction to Manifolds Second Edition" by Loring W. Tu. Problem 1.3.(b) Let $a,b$ be real numbers with $a < b$. Find a linear function $h: \mathopen]a,b\mathclose[ \...

Question 14

Let $K$ be a constant and $x$ be a variable. What is a smooth, monotonic function that is as close to $\min(K,x)$ as possible, but never exceed $\min(K,x)$? Also f(x)>=0 for x>=0 and f(0)=0 ...

Question 15

I’ve recently been introduced to sheafs and it made me wonder weather the following statement is true: $f:\mathbb R^n \rightarrow \mathbb R^m$ is $C^k$ iff it maps $C^k$ curves to curves $C^k$. One ...

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Questions tagged [smooth-functions]

are smoothstep transition function and hermite splines equally in this smoothing problem?

Smooth embeddings of CW complexes (on cell interiors)

Extending continuous maps to smooth maps

A regular map sending concentric circles to disjoint circles

The formula to convert any values from certain range to different range if minimum and maximum are known

Is there a differentiable function with a "slow" linearisation?

Search Direction in Variance Reduced Algorithm Well Defined

How does the differential geometry notion of a differential align with the standard notion of a derivative?

Function with incompatible properties?

If $f\in C^\infty(\mathbb R)$ and $C> 0$ then find $n \in \mathbb N$ and $\xi \in \mathbb R$ such that $|f^{(n)}(\xi)| > C$

Exercise on smooth functions

A Question Regarding Taylor's Remainder Theorem

Any two finite open intervals are diffeomorphic. ("An Introduction to Manifolds Second Edition" by Loring W. Tu.)

Is there a smooth function approximating the minimum of a constant and a variable?

$f$ is $C^k$ iff it maps $C^k$ curves to $C^k$ curves?

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