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.

Filter by
Sorted by
Tagged with
0
votes
0answers
22 views

Prove that function is convex, first order smoothness,second order smoothness

I have the following function: $$\mathbb{R}^n \ni (x_1, \cdots , x_n) \mapsto \ln( \sum_{k=1}^{n}{exp(x_k)})$$ I know how to prove if a function is convex, but I have trouble with this specific one. ...
0
votes
0answers
11 views

What's the smoothness of the 4-D spacetime?

If treating General Relativity's 4-D spacetime as a function $f(x,y,z,t)$. Now use it to describe the timeline of a ball, so $f(x,y,z,t)$ will return you a real number that represents how much ...
0
votes
1answer
20 views

approximate identity- common compact set

let us consider a sequence of smooth functions with compact support $\phi_n$, which approximate Dirac measure at zero, such that $\int_{\mathbb{R}}\phi_n(x)\,dx=1$. Okay, I can consider something like ...
2
votes
2answers
57 views

If $f$ is a continuous function and $\phi$ is a smooth function with compact support such that $\int\phi\,dx=1$, is $f*\phi$ Lipschitz continuous?

Let us take a continuous function $f$. Let $\phi$ be a smooth function with a compact support, such that $\int \phi(x)\,dx=1$. We consider the following convolution $(f*\phi)(x)=\int f(x-y)\phi(y)\,dy$...
1
vote
0answers
28 views

second derivative inequality

I was reading a paper and the autor did the following inequality, but I couldn't understand why: If $u:\mathbb{R}^n\to\mathbb{R}$ is smooth and bounded then $$|u(x+y)-u(x)-\nabla u(x)\cdot y|\leq C||D^...
2
votes
1answer
15 views

Composition of local diffeomorphisms is a local diffeomorphism

Let $F: M\rightarrow N$ , $G:N\rightarrow P$ be local diffeomorphisms, where $M,N,P$ are smooth manifolds. I would like to show that $G\circ F: M\rightarrow P$ is a local diffeomorphism. My attempt: ...
0
votes
0answers
30 views

Smooth maps between smooth manifolds Problem

Definition: Let $M$ and $N$ be smooth manifolds. Suppose $F: M\rightarrow N$ is any map. $F$ is said to be smooth at $p\in N$ if: There exists a chart $(V,\psi)$ about $F(p)$ in $N$ and there exists a ...
3
votes
2answers
51 views

Determining that a certain diffeomorphism of $\Bbb R^n-\{0\}$ is orientation preserving or not

Consider the diffeomorphism $f:\Bbb R^n-\{0\} \to \Bbb R^n-\{0\}$ (whose inverse is itself) given by $x\mapsto x/|x|^2$. How can we determine that $f$ is orientation preserving? For $n=1$ it is ...
3
votes
1answer
27 views

Finding a nonexact closed $1$-form on a surface embedded in $\Bbb R^3$

Consider the subset $S=\{(x,y,z):x^2-y^2-z^2+1=0\}$ of $\Bbb R^3$. Defining a function $f:\Bbb R^3\to \Bbb R$ by $f(x,y,z)=x^2-y^2-z^2+1$, it is easily seen that $0$ is a regular value of $f$, so it ...
3
votes
1answer
62 views

Smooth Sylvester's law of inertia

Let $Q(x)$ be a smooth symetric matrix with constant signature $(p,q,k)$ where $x$ belong in $\mathbb{R}^n$ and $p+q+k=m$. Question: Locally around $x_0$, does an invertible matrix $P(x)$ of size $m$...
-1
votes
1answer
41 views

Is there a Smooth, convex alternative to the Gamma fucntion.

Is the a function $f(x)$ other then the Gamma function with said properties. 1. $f(x)=x!$ when x is a non-negative integer. 2. $f(x)$ is smooth (infinitely differentiable.) 3. $f(x)$ is convex. 4. $f(...
17
votes
2answers
376 views

Is there a smooth, preferably analytic function that grows faster than any function in the sequence $e^x, e^{e^x}, e^{e^{e^x}}…$

Is there a smooth, preferably analytic function that grows faster than any function in the sequence $e^x, e^{e^x}, e^{e^{e^x}}$? Note: Here the answer is NOT required to be an elementary function, as ...
0
votes
0answers
33 views

What can we say about a not-regular level sets of a smooth function?

Suppose $F : M \to \mathbb{R}$ is a smooth function, $M$ a smooth manifold and $a \in \mathbb{R}$ is a critical value. Of course $N = F^{-1}(a)$ need not be an embedded submanifold of $M$. How could ...
-1
votes
0answers
34 views

Piecewise Smoothness of a Function

Is there a way to determine if the following function: is piecewise smooth or not? I've been told that since for both the function and it's derivative the improper integral $$\int_{-\infty}^\infty ...
1
vote
1answer
28 views

Determining the differential of a map defined on a submanifold of $\Bbb R^n$

Let $M=\{(x,y,z,w)\in \Bbb R^4:x^3+y^3+z^3-3xyz=1\}$ and consider the function $f:M\to \Bbb R^2$ defined by $f(x,y,z,w)=(x+y+z+w,w^3+w)$. It is easily checked that $1$ is a regular value of the ...
0
votes
0answers
11 views

A way to prove that a function is smooth except in zero

So i got this demonstration where i have to show that $E \in C^{\infty}(R^n-\{0\})$. If $\alpha$ is a multi index, $D_{\alpha}= \partial_{x_1}^{\alpha_1} \ldots \partial_{x_n}^{\alpha_n}) $, in my ...
0
votes
1answer
22 views

Show that $C^\infty(\overline{\Omega}) \subseteq C^{0,1} (\overline{\Omega})$

Let $\Omega$ be a bounded, connected, open domain in $\mathbb{R}^d$ with smooth boundary. Denote by $C^{0,1} (\overline{\Omega})$ the space of continuous functions $u$ on $\overline{\Omega}$ such ...
1
vote
1answer
16 views

Does null average against every smooth function implies independence?

Are these assertions equivalent? $f:\mathbb{S}^1\times \mathbb{S}^1\to\mathbb{C}$ is such that $$ \int_0^{2\pi}\int_0^{2\pi}f(x,y)\psi(y)dydx=0$$ for all $\psi\in C^{\infty}(\mathbb{S}^1).$ $f:\...
0
votes
1answer
48 views

Interesting examples of submersions that are not surjective

What are some "interesting" examples of submersions that are not surjective? Usually, the notion of submersion comes with a prefix of surjective. Most of the maps we come across when we do ...
0
votes
0answers
17 views

Can every line drawn on paper which “looks like” a smooth curve be described with a function?

There's a reason I've said "looks like" a smooth curve (I'm hesitant to say this but maybe one could define this as meaning it is a smooth curve using definitions of "geometric continuity") I'm ...
0
votes
0answers
13 views

Show a function is smooth

We have $w \in \mathbb{R}^d$, and functions $f, g:\mathbb{R}^d\to\mathbb{R}$. Additionally, we define $g(w) = \sum_{i=1}^d \frac{1}{\alpha}\ln(1 + \alpha|w_j|)$. Now define $h(w) = f(w) + g(w) - ||...
-2
votes
1answer
51 views

Where is $f(x) = |x^2(x+1)|$ differentiable? And where are they $C^1$ and $C^2$?

I'm a maths student taking a real-analysis paper and I'm currently working down my problem sheet. I've been asked the above question. First I define a piece-wise function to describe the absolute ...
0
votes
0answers
33 views

Ring of smooth functions proof

$C^{\infty}(U)$ is a ring. Where $U$ is open in $\mathbb{R}^n$. Not $D_jf(a)$ denote the partial derivative of $f$ in the direction of $e_j$ at a. My attempt: Abelian group: Let $f,g\in C^{\infty}(...
2
votes
1answer
42 views

Tangent space identification basis

Let $e_{i_p}$ denote the standard basis for $T_p(\mathbb{R}^n)$. Theres a vector space isomorphism between $T_p(\mathbb{R}^n)$ and $D_p(\mathbb{R}^n)$, where $D_p$ is the set of derivations at $p$, ...
0
votes
3answers
48 views

Where is $f(x) = |4x-4-x^2|$ differentiable and where is it $C^1$ and $C^2$.

I'm a university student taking a real analysis paper. I'm currently working down my problem sheet and have arrived at a series of questions reguarding smoothness and the like. I'm still getting the ...
0
votes
1answer
17 views

Tangent space isomorphic to derivations Taylor series

$T_p(\mathbb{R}^n)\cong D_p(\mathbb{R}^n)$ Lemma: Let $f \in C^{\infty}(U)$ where $U$ is an open subset of euclidean space, star shaped with respect to a point $p=(p^1...,p^n)$ in $U$. Then there ...
0
votes
1answer
19 views

Restriction of smooth map to open domain calculus

Suppose $f: \mathbb{R}^n\rightarrow \mathbb{R}^m$ is $C^{\infty}$ at a (continuous and has continuous partial derivatives of all order). Let $U$ be an open set containing $a$. Then, $f|_U: U\...
2
votes
1answer
43 views

Can $\frac{\mathrm d}{\mathrm dx}$ increase the support of a function?

Let "support" mean the closed support. Can $\frac{\mathrm d}{\mathrm dx}$ increase the support of a function? That is, is there any $f=f(x)$ in one variable with $\operatorname{supp}(f)$ completely ...
2
votes
1answer
32 views

Proving that $f(n)=nlog(n)$ is a $b$-smooth function

First I start with the definition: a function $f:\mathbb{N} \rightarrow \mathbb{R}^{+}$ is b-smooth for an integer $b \geq 2$ if $f$ is eventually non decreasing and if $$ \exists c \in \mathbb{R}^{+} ...
0
votes
0answers
13 views

Spaces of functions which remain linear combination of themselves after being convolved by gaussian?

Say we have the following set of functions $$\mathcal S = \{f_1,f_2,\cdots\}$$ so that for any $n$ there exist uniquely a set of $ \{ c_k \}$: $$(f_n * g)(t) = \sum_{k\in \mathbb Z^+} c_k f_k(t)$$ ...
0
votes
0answers
22 views

Are all functions with Lipschitz continuous gradient also strongly convex?

I am sorry for asking probably well known question. But I am so confused. Are all functions, say $f$, with Lipschitz continuous gradient, say, $\nabla f$ is $L$-Lipschitz, also strongly convex ...
1
vote
0answers
23 views

Intuition behind the Boman's theorem

I stumbled upon this result called Boman's Theorem twice last weak while working with the group of diffeomorphisms of a manifold and in a discussion with a friend about a certain function space. ...
0
votes
1answer
14 views

Constructing a Transition Function Satisfying Additional Constraints

The problem: I'm looking to construct a smooth function $f(x)$ for $x\in[0, 1]$ that satisfies the following simple constraints $$f(0)=1$$ $$f(1)=0$$ $$f'(0)=f'(1)=0$$ as well as the following ...
0
votes
0answers
22 views

Does there exist a smooth concave function which starts linear and become strictly concave?

Fixe positive numbers $c>1,\lambda \in (0,1)$ such that $\lambda c <1$. I am trying to convince myself that there is a smooth function $\psi:[0,1] \to [0,1]$ which satisfies the following ...
0
votes
1answer
15 views

Smooth function that interpolates between constant functions

How can we construct a function that smoothly interpolates between constant $1$ and constant $0$? That is, find $h(x)$ smooth such that $h(x) = 1$ if $x < 1$ and $h(x) = 0$ if $x>2$. I know that ...
0
votes
1answer
24 views

Constructing $\sqrt{\sqrt{1+x^2}-1}$ to be smooth (cancelling the square)

$\DeclareMathOperator{\sign}{sign}$ Is there a way to rewrite $f(x)=\sign(x)\sqrt{\sqrt{1+x^2}-1}$ using (smooth) elementary functions? As far as I can see the function seems infinitely ...
2
votes
1answer
38 views

If a quotient of smooth functions can be continuously extended to a singularity, is the extension automatically smooth?

Suppose that $f(x)$ and $g(x)$ are real-valued functions that are infinitely differentiable at the point $x=x_0$. Suppose that the limit $$ L_0=\lim_{x\to x_0} \frac{f(x)}{g(x)} $$ exists. Thus, if ...
4
votes
1answer
51 views

Sign of $df_x$ is locally constant

This question is about the book Topology from the Differentiable Viewpoint of Milnor. Let $M$ and $N$ be oriented $n$-manifolds without boundary, and assume $M$ is compact and $N$ is connected. Let $...
0
votes
0answers
25 views

Dimension intersection two manifolds

let $X$ and $Y$ be two $(n-1)$ dimensional manifolds in $\mathbb{R}^n$ of class $C^1$ and $C^\infty$ respectively, and denotes by $\mu^{(1)}$ and $\mu^{(2)}$ the two outer unit normal vectors to $X$ ...
0
votes
1answer
14 views

If $F: M\rightarrow N$ is smooth then its coordinate representation w.r.t any smooth charts $(U,\phi),~ (V,\psi)$ s.t. $F(U)\subseteq V$ is smooth

I want to prove that if $F: M\rightarrow N$ is smooth, where $M,N$ are smooth manifolds, then its coordinate representation w.r.t any smooth charts $(U,\phi),~ (V,\psi)$ such that $F(U)\subseteq V$ is ...
0
votes
0answers
21 views

Proving $\prod_{k=0}^n f\left(\frac{t-k}{T}\right)=f(t/T)^n + O(1/T)$ and better understanding Big-O notation

I am trying to taylor expand the term $$\prod_{k=0}^n f\left(\frac{t-k}{T}\right)$$ at the point $t/T$, where $f:\mathbf{R} \rightarrow \mathbf{R}$ is a smooth function. I am a little confused with ...
0
votes
1answer
34 views

Find a smooth function solving this derivative equation

Let $g$, $h \in C^\infty(\mathbb{R})$ be smooth functions on $\mathbb{R}$. Can I now always find $f \in C^\infty(\mathbb{R})$ such that $g^\prime \circ h^\prime - h^\prime \circ g^\prime = f^\prime$ ...
1
vote
0answers
25 views

Given a smooth, non analytic curve, obtain a non analytic smooth function.

I managed to prove (by methods of complex analysis) that the curve $$f:[0,1]\to \mathbb{C};\\ x\mapsto e^{2\pi ix}+\sum_{n=5}^\infty \frac{\left(e^{2\pi ix}\right)^{(2^n)}}{n!}$$ is a smooth, non ...
0
votes
0answers
17 views

Characterization of relative smoothness

A well-known equivalent condition of the Lipschitz smoothness is : $$ \|\nabla f(x)-\nabla f(y)\|\leq L\|x-y\| \iff L\|x\|^2/2 - f(x) \text{ is convex }.$$ As a generalization of the Lipschitz ...
0
votes
0answers
10 views

How to formally state and show that, for a smooth function from $R^2$ to $R$, the intersection of a contour line with itself is a saddle point?

I am referring to the line from https://en.wikipedia.org/wiki/Saddle_point (second paragraph, last sentence): In terms of contour lines, a saddle point in two dimensions gives rise to a contour ...
0
votes
1answer
26 views

How to remove singularity from the solution?

Suppose we have $f(x)=\frac{1}{sin(x-x0)cos(x-x0)}$, we know that $f(x)$ is undefined at $x=x0$. Is there any possibility to remove this singularity so that $f(x)$ would be a smooth ...
1
vote
1answer
34 views

Manifold definition of $C^1$ vs $p \mapsto dF_p$ continuous

I don't know much about differential geometry but as I understand it for a map between manifolds $F : M \to N$ to be $C^1$ around $p$ means that there exists charts $(U,\phi)$ containing $p$ and $(V,\...
0
votes
1answer
39 views

function smoothing with exponent

I found this statement in a proof and thought it is clear, thinking of the absolute value function as an example, but I can not give a proof. Can someone give me a hint? There is a function $f: R^6 \...
3
votes
1answer
58 views

Monotone interpolation with prescribed derivatives at endpoints

Let $(u_k)_{k\geq 0}$ and $(v_k)_{k\geq 0}$ be two sequences with $u_0 \lt v_0$ and $u_1\gt 0,v_1\gt 0$. My question : is there an $f\in{\mathcal C}^{\infty}([0,1],{\mathbb R})$ with $f^{(k)}(0)=u_k$ ...
5
votes
2answers
68 views

Can continuous functions be made smooth by changing the smooth structure on the domain?

Suppose $M$ is a smooth manifold and $f : M \to \mathbb R$ is a continuous function. The function $f$ may not be smooth, but does there exist another smooth structure $M'$ (on the same topological ...

1
2 3 4 5
7