44
votes
Applications of Rademacher's Theorem
I will mention seven different applications:
Characterization of almost everywhere differentiability.
The following result is a consequence of the Rademacher theorem:
Theorem (Stepanov). A function ...
23
votes
Accepted
Are functions of bounded variation a.e. differentiable?
No. Take a dense countable set $\{x_1,x_2,\dots\}$ in $\mathbb{R}^d$ and a sequence $(r_i)\subseteq\mathbb{R}^+$ such that $\sum_i r_i^{d-1}<\infty$. Then the function
$$f=1_{\bigcup_{i=1}^\infty ...
21
votes
Accepted
Why do almost all points in the unit interval have Kolmogorov complexity 1?
I'm not an expert on Kolmogorov complexity, but this does seem like a counting argument: for any fixed $\epsilon > 0$, there are only $\sum_{i = 1}^{(1-\epsilon)n} 2^i < 2^{(1-\epsilon)n+1}$ ...
20
votes
Generalized Stokes' theorem
When I wrote that Wikipedia paragraph, I think I had in mind Theorem 5.16 in Evans & Gariepy, Measure Theory and the Fine Properties of Functions, which essentially proves Stokes' theorem for ...
18
votes
Accepted
A gerrymandering problem - can you always turn a tie into a landslide victory?
Yes, the almost partition exists. Instead of letting $\mu(E)\geq\frac{\mu(\Omega)}{2}$, I let $\mu(E)\in(0,\mu(\Omega))$ be arbitrary and proved that you can divide $\Omega$ into $N$ open simply ...
17
votes
Accepted
How badly can the Lebesgue differentiation theorem fail?
Metafune has given an example of the limit failing to be $0$ at a particular point - namely for $n > 1$, the function $|x|^{-\alpha}$, with $1 \leq \alpha < n$ has that limit equal to $\infty$ ...
16
votes
Almgren's mimeographed lectures notes on varifolds
I just bumped into your post after doing some google search to find a bibtex entry for Almgren's notes.
I have a copy of them: I could copy it and send it to you (or maybe scan it and share it via ...
14
votes
Accepted
Can $C^1$ mappings with derivative of low rank be approximated by smooth maps?
There is a counterexample.
Example. There is $f\in C^1(\mathbb{R}^5,\mathbb{R}^5)$ with ${\rm rank}\, Df\leq 3$ that cannot be approximated in the supremum norm by mappings
$g\in C^2(\mathbb{R}^5,\...
14
votes
Continuous deformation of soap films
You are asking whether a least area surface depends continuously on its boundary curve, or in mathematical terms: whether for every curve $\gamma$ there exists a least area surface $S$ with $\partial ...
14
votes
Accepted
Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?
That is true. Caccioppoli sets are also known as sets of finite perimeter.
Theorem. Suppose $f\in L^1(\mathbb{R}^n)$ vanishes outside the unit cube $[0,1]^n$. For $i=1,2,\ldots,n$ consider the ...
14
votes
Accepted
Existence of subset with given Hausdorff dimension
First of all, $\dim_{H} (A) = \alpha$ iff $ H^k(A)=\infty$ for all $k<\beta$ and $H^k(A) = 0$ for all $k>\beta$. Then $H^\alpha(A) = \infty$ for all $\alpha \in (0,\beta)$.
If $A$ is closed ...
14
votes
Generalized Stokes' theorem
Harrison, Jenny, Stokes' theorem for nonsmooth chains.
Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 235–242.
This research announcement reports progress in developing a viable theory of integration ...
12
votes
Accepted
Almgren's mimeographed lectures notes on varifolds
Here is the story behind these notes, and a redirect to On the First Variation of a Varifold, W.K. Allard (1972).
a quote from: Selected Works of Frederick J. Almgren
12
votes
Accepted
Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms?
Such a measure cannot exist. Suppose to the contrary that we have an uncountable family of lines $\ell$ such that $\mu(\ell)>0$. Then there is $\epsilon>0$ and an infinite family of lines $\{\...
12
votes
Existence of subset with given Hausdorff dimension
The answer is yes under the additional assumption that the set is compact and I do not know what happens in the general case. The result is a consequence of the following one, see [1] and references ...
12
votes
Accepted
Unknown work of Nöbeling on topological/Hausdorff dimension
So, the sought for paper is:
Nöbeling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931).
And here is a ``translation" (to English and to modern ...
12
votes
Accepted
On functions with strict Lipschitz constant
I guess it suffices to give an example for $n = 1$. If $f: \mathbb{R} \to \mathbb{R}$ is an example then $g(x_1, \ldots, x_n) = f(x_1)$ will be an example for any $n \geq 1$.
All we need is a ...
12
votes
Accepted
If a function $f$ is $(1+\varepsilon)$-times Lebesgue differentiable everywhere, is $f$ a constant function?
I realise I'm bumping into you again and already gave you an answer elsewhere after you posted this, but I thought I'd post my answer here for others to see. The answer is yes, $f$ has to be constant ...
12
votes
Accepted
Does every differentiable a.e. function admit a maximally differentiable representative?
$\DeclareMathOperator*\appliminf{app-liminf}\DeclareMathOperator*\applimsup{app-limsup}\DeclareMathOperator*\applim{app-lim}\DeclareMathOperator*\essliminf{ess liminf}\DeclareMathOperator*\esslimsup{...
11
votes
Accepted
Proper homotopy
The following counterexample is stolen from page 1 of this paper by Thomas Rot. The map $[0,1]\times\mathbb R\to\mathbb R$ given by $(t,x)\to (1-t)x^2+x$ is not proper, e.g., the preimage of $\{0\}$ ...
11
votes
Accepted
Metric measure spaces: in what sense is analysis on these spaces "non-smooth"
I highly recommend the survey article "Nonsmooth Calculus" by Juha Heinonen, available here.
The beginning of the introduction reads:
"The word nonsmooth in the title refers both to functions and ...
11
votes
Accepted
A Besicovitch-type Covering Theorem
This trivial counterexample in $\mathbb R^2$ should have taken me five minutes. Instead, I spent almost two days. The moral is the usual one: after 50 you'd better give up on mathematics.
Let $y,z$ ...
11
votes
Accepted
Is the composition of two nowhere differentiable functions still nowhere differentiable?
The composition may have points of differentiability.
Let $f_0(x)=x$ for $x\geq 0$ and $f_0(x)=2x$ for $x<0$. Let $g_0(x)=2x$ for $x\geq 0$ and $g_0(x)=x$ for $x<0$. Then none of them is ...
11
votes
Accepted
Tiling the plane with finitely many congruent pieces
Write $A_i=T_i(A)$ for $A=A_1$, where each $T_i$ is a rigid motion. For each $i$, we have $|A_i\cap B(r)|=|T_i(A\cap T_i^{-1}(B(r))|=|A\cap T_i^{-1}(B(r))|$. The symmetric difference between this set ...
11
votes
Accepted
If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Yes, $\mathbb{R}^n\setminus E$ has to be path-connected.
Let $x,y\in\mathbb{R}^n\setminus E$, we will prove that there are many paths going from $x$ to $y$ inside $\mathbb{R}^n\setminus E$. We can ...
10
votes
Fractals of dimension zero
One example: the set of
Liouville numbers has Hausdorff dimension zero.
In number theory, a Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, ...
10
votes
Accepted
Ergodicity and mixing of geodesic and horocyclic flows
The algebraic approach is just one of many ways to deal with the geodesic and horocycle flows. Also your quick summary does not really pay tribute to the many ways to deal with it using "...
10
votes
Accepted
A better version of Weyl's Law or uniform estimates of Laplacian higher eigenvalues
It seems unlikely that such a bound holds except in very special cases. For instance, it fails for round spheres, which have very large multiplicity of eigenvalues. In fact, for spheres the eigenvalue ...
10
votes
Accepted
Comparison of Information and Wasserstein Topologies
It is not the case that the Fisher-Rao distance dominates the Wasserstein distance. For instance, it fails for univariate normal distributions $\mathcal{N}(\mu,\sigma)$. In particular, the Wasserstein ...
10
votes
Accepted
Background for Varifold theory
The general prerequisites are almost the same as for currents, mainly a strong understanding of measure theory and a bit of geometrical intuition.
There is an aspect of multilinear algebra and some ...
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