User mircea - MathOverflow

rigidity of isoradial graphs

2013-04-24T19:48:41Z

Suppose given a $1$-separated net $\Gamma\subset\mathbb R^2$. Is it true or false that there exists $\delta>0$ and a $\delta$-isoradial graph containing $\Gamma$ as a subset of its vertices?

(I am also interested in partial results in this direction.)

edit: a $\delta$-isoradial graph in $\mathbb R^2$ is a graph each of whose faces is a finite polygon inscribed in some circle of radius $\delta$.

edit2: A case where the answer is "yes": if we have a graph whose faces are $\delta$-rhombi then by considering only odd vertices gives us an isoradial graph. It will help to consider such graphs instead of $\delta$-isoradial ones. If $\Gamma$ is included in the graph of a continuous function $G_f:={(x,f(x)):x\in\mathbb R}$ then we may approximate $G_f$ by a polygonal $\delta$-path $P$ which comes from a graph of a function $\tilde f$ and contains $\Gamma$ as a subset of its odd vertices. Now extend $P$ to a path of $\delta$-rhombi and then extend to a $\delta$-rhombic tiling of $\mathbb R^2$ e.g. by periodicity.

Examples of loss of regularity by "creation of topology"

2010-05-21T15:57:12Z

I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered) fails, giving rise to interesting topological classifications. What comes to my mind are the following famous facts:

1) The condition for $C^\infty(\mathbb R^k, M^n)$ to be dense in the manifold-valued Sobolev spaces $W^{1,p}(R^k, M^n)$, is that the homotopy group $\pi_{[p]}(M)$ should be trivial.(Hang-Lin)

[this was kind of general, but gives the idea of what I'm looking for, maybe!]

2) A map $u$ in $W^{1,2}(B^3,S^2)$ is in the closure of $C^\infty(B^3, S^2)$ if and only if for any $2$-form $\omega$ on $S^2$ such that $\int_{S^2}\omega\neq 0$ one has $d(u^*\omega)=0$.(Bethuel-Coron-Demengel-Helein)

3) In $4$ dimensions, if the Yang-Mills functional is finite on a connection $A\in L^2(M^4)$, then the curvature $F_A$ of $A$ realizes an integral Chern class (i.e. the number $c_2(A):=1/(8\pi^2)\int_{M^4}Tr(F_A\wedge F_A)$ is an integer).(Uhlenbeck)

(Maybe I could also formulate the question differently, asking for mathematical situations having the "loss of differentiability" via "creation of new topology" analogous to the list above.)

When do there exist locally regular embeddings of regular graphs?

2012-07-09T13:56:48Z

I don't know how one can tackle the following kind of question, so any hint is welcome. I formulate a precise question in order to fix ideas, but it is to be considered as an example out of a more general class.

Example: Can one embed Petersen's graph in $\mathbb R^4$ in such a way that all edges are mapped to segments (of not necessarily equal lenght) and each pair of adjacent segments forms an angle of $2\pi/3$ ?

The question can be generalized in the following way, to give a wider class of questions:

instead of Petersen's graph, consider a given $k$-regular graph (we had $k=3$ above)
instead of $\mathbb R^4$ consider $\mathbb R^n, n>k$, or embeddings into the round sphere $S^n$ where the edges of the graph are mapped into geodesics
ask that the angles formed between any pair of adjacent edges are equal to the angle $VOV'$ made at which two vertices $V,V'$ of a regular euclidean $(k-1)$-simplex are seen from its barycenter $O$

I think that $1$-skeletons of spherical regular polytopes give such kind of graphs. I am interested in any partial answer which introduces essentially more ingredients than just constructions based on symmetry groups, or in obstructions which show negative answers.

nontrivial $\pi_2(Diff(M))$

2012-05-24T10:29:20Z

http://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups/8996#8996 I asked it also there, and I still don't know the answer, so I try again.

I would like to know a closed manifold (possibly of low dimension) such that $\pi_2(Diff(M))\neq 0$.

interval exchange maps and surfaces

2012-05-24T10:19:53Z

I apologize if the question is a well-known theorem, but I'm just starting to learn about laminations, so I don't know much.

The question is roughly, if interval exchange maps have an underlying closed smooth surface, or if not, what is known about conditions on that.

Now I try to be more precise.

Usual interval exchange functions are bijective functions $\mathbb R/\mathbb Z=S^1\to S^1$ which are piecewise translations where "piecewise" is defined using a finite partition of $S^1$ in segments $s_i$.

Given a interval exchange function $\phi:S^1\to S^1$ one can consider its suspension $S_\phi=[0,1]\times S^1/\sim$ where the nontrivial identifications are $(0,x)\sim (1,\phi(x))$. Then one has a smooth 1-dimensional structure on $S_\phi$, given by the differentiable structure on $[0,1]$, which can be extended with continuity through the identifications. In the $S^1$-direction there is just some piecewise $C^1$-structure on the pieces $]0,1[\times s_i$, since $\phi$ is not even assumed to be continuous.

Then if I understood correctly one defines $\partial S_\phi=\cup_i [0,1]\times\partial s_i/\sim$, and this set is an union of (topological but not $C^1$, since there are cusps) copies of $S^1$. Then one obtains a surface by gluing some annuli to these circles.

The question is if there is some way of obtaining a smooth surface in this way. The case where the lenghts of $s_i$ are rational and $\phi$ is piecewise equal to translations by rational numbers is simple (one refined the segments and gets a longer, smooth, boundary), so I am interested if there is a result in the other case.

Moser regularity proof avoiding John-Nirenberg lemma

2012-03-22T07:55:23Z

I heard a rumor that there exists a proof by Moser-style iteration of the $C^{0,\alpha}$-regularity for $W^{1,2}$-solutions $u$ to elliptic equations with measurable coefficients which does not rely on the John-Nirenberg lemma.

I was wondering if somebody can point out a reference for that proof, or a reason why the John-Nirenberg lemma cannot be truly avoided (or both!).

iterated traces for sobolev functions

2012-03-19T13:47:38Z

It is well known that if $M$ is a smooth $(n-1)$-dimensional surface in $\mathbb R^n$ (e.g. a subspace) then there is a continuous trace operator $W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M)$. Now suppose that $M$ is a manifold with boundary, e.g. $M={0}\times\mathbb R^+\times \mathbb R^{n-2}\subset\mathbb R^n$, and in this case call the boundary $\Sigma:=\partial M={(0,0)}\times\mathbb R^{n-2}$.

The question (it's more a poll) is how to define the trace on $\Sigma$ (and I'd expect a function $W^{s,p}(\mathbb R^n)\to W^{s-2/p,p}(\Sigma)$) in such a way that the composed trace $$W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M)\to W^{s-2/p,p}(\Sigma)$$ is independent of the (smooth) manifold $M$.

Another (i.e. the more challenging) question is if you can (or prove it impossible to) find an example of two different compositions of linear continuous functionals as above (with two $M, M'$ smooth and with boundaries both equal to $\Sigma$, which by locality we suppose here to be orthogonal half-hyperplanes), without the requirement that the first functions $W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M\:or\:M')$ be the usual traces, but such that then composed with the traces within $M, M'$ they give the codimension $2$ trace on $\Sigma$ of the first question.

The question is inspired from a small book by Tartar ("An intorduction to Sobolev spaces and interpolation spaces", chapter 40) where (codimension one) trace spaces are identified with interpolation spaces, baiscally by parametrizing the "killed" spatial coordinate in $\mathbb R^n$ with the interpolation parameter. In the light of that construction it would appear that the reason why the first question should have a positive answer is because partial derivatives of linear functions commute: if no example could be done for the second question, it would (to me) indicate strongly that linearity is a die-hard property, so to speak.

candidates for "projection" of the trace of a set onto the associated perimeter minimizer

2012-02-02T13:54:34Z

Let $E$ have finite perimeter in $\mathbb R^n$. Consider the minimization of

$$P(B)-\int_{\mathbb R^n\setminus L}|D\chi_E|$$

among the sets $B$ of finite perimeter differing from $E$ only inside $L$ (the negative piece of the minimized quantity is not influent, since it does not depend on $B$, but it's there to stress the idea that we care just about the perimeter inside $L$). Let $\bar E$ be the minimizer, and that the minimum is not zero. Suppose also that $L$ has finite perimeter.

I'm looking for a geometrically inspired way for defining a function $\Psi$ which sends a piece (I would like to ask "all" but it might be unfeasible) of $\mathcal FL\cap E$ to $\mathcal F\bar E\cap L$ bijectively, is $H^{n-1}$-measurable and decreases (more precisely, does not increase) $(n-1)$-dimensional area.

The idea should be that $\Psi$ be some kind of projection.

In the case that $L$ is a ball and $E$ is a thin rectangle around its diameter it is clear that usual projections don't quite work.

On the other hand, such a map surely exists, by the consideration that the area of the target is smaller than the one of the domain, and both of them are rectifiable.

My question is if there is a famous geometric/constructive way of doing it, perhaps using some minimization principle, a flow, etc., or under more restrictive hypotheses on $L$.

is there a relation between the complex Hardy spaces and the Hardy spaces of harmonic analysis?

2012-01-18T17:13:37Z

Maybe my question is just a matter of knowing the right equivalent definition.

The question is whether there is some relation between

$ H^p(D^2)$, defined as the space made of the analytic functions on the $2$-disk for which

$$\sup_{r\in]0,1[}\int|f(re^{i\theta})|^pd\theta<\infty$$

and the space $H^p(\mathbb R^2)$ which is defined for example as containing the functions such that $\sup_{t>0}|P_t*f|(x)\in L^p$, where is the Poisson kernel or some similar (but you can take $P_t(x)=t^{-2}P(x/t)$ and $P(x)$ some Schwartz function with nonzero mean).

The confusion of notation is just in this post, since (apparently) the books/people treating one kind of spaces never care about the other kind: are there any exceptions to this rule?

Thanks for the help!

Defining the slowest divergent series

2011-12-15T10:31:31Z

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is:

I know that a method of slowing a divergent series of positive reals is to replace the $n$-th term by it divided by the first $n$ terms. In this way the series obtained stays divergent, but it decreases infinitely faster.

Now consider the class $\Sigma$ of divergent series made of positive reals, where $(a_i)<(b_i)$ means that $\lim_{i\to\infty}a_i/b_i=0$. Consider now a decreasing sequence of series.

The question is if there is a notion of convergence fit to this order (most plausibly in a weak/generalized sense), and if there is an extension of $\Sigma$ where one could give some meaning to "the slowest divergent series".

I suspect that the/some answer would have to do with something like nonstandard analysis: one might then reframe even the definition of the order relation, in the natural way.. I would highly appreciate other speculations about the statement of the problem too.

criterion for being a curvature

2011-12-02T10:38:01Z

1) Is there a criterion for telling if a Lie-algebra-valued $2$-form (for example on a $SU(2)$-bundle) is a curvature, without taking derivatives? For example, using Bianchi's identity is not allowed.

2) Also partial criteria are welcome (i.e. ones in which just a necessary/sufficient condition, formulated without reference to derivatives of the "candidate" form, are given).

3) For example, is it true, for the trivial* $SU(2)$-bundle case, that having zero integer second chern class integral on all contractible domains implies that our form is a curvature? [*:correction suggested by David Speyer]

What would the best treatment of Gehring's lemma look like?

2011-07-18T10:37:04Z

In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a bit the regularity of a function, given the knowledge that such function already satisfies local bounds via another more integrable function (for example one bounds the integral of $f^2$ on any ball via a power of $f^{2-\epsilon}$ on the ball of double radius).

Even though I saw a proof of this (by Giaquinta-Modica) and I "believe" the result intuitively, I have the feeling that I didn't catch yet the "juice" of it.. for example I don't have the intuition of the following 1) how far it can be extended, or 2) if the more extended/general versions are meaningful to teach (e.g. because one can find a clearer proof, or a more natural one, or because they can show connections with other subjects..).

So I would like to ask you if you know how to put this result in a larger context, or about its connections with topics different than elliptic regularity. Also references to nice expositions of it are quite welcome!

solvability of an elementary functional equation

2011-04-05T10:16:52Z

Is there some other way to characterize the functions $f:\mathbb Z\times \mathbb Z\to \mathbb Z$ which are expressible as $$f(x,y)=g(x)+g(y)-g(x+y)$$ for some $g:\mathbb Z\to\mathbb Z$?

Easy facts: (1) $f$ must satisfy $f(x,y)=f(y,x)$ and $f(x,0)=g(0)$ for all $x$. (2) Not all functions $f$ are expressable, since for $x,y\in{1,\ldots,n}$ the number of choices of $f(x,y)$ the left grows quadratically in $n$ and the number of possible values of $g$ on which they depend, grows just linearly.

Trace space and Neumann boundary condition

2010-10-07T16:14:04Z

In which sense is it possible to solve $\Delta u=0$, $\partial_\nu u=\phi$, for $\int\phi=0$ on a closed domain, say a ball $B^3\subset\mathbb R^3$?

For example would a $\phi\in L^p(\partial B^3)$, $1< p<2$ make sense?

In other words, is $W^{1,p}$ really the right trace space, or else, which is?

Where can I find this kind of results?

Thanks!

Are there good product rules on the $k$-sphere?

2010-07-23T12:10:49Z

I have heard sometimes that the only dimensions $k$ for which there exists a "good" smooth product $P:S^k\times S^k\to S^k$ are $k = 0,1,3,7$ (the above products corresponding to $\mathbb Z_2, U(1)\subset\mathbb C$, the product of unit quaternions and of unit Cayley numbers).

I would like to ask for references about such a result.

More precisely, I am interested in finding out how is it that one can define "good" so that the above is true (with proofs, preferably).

Geometric imagination of differential forms

2010-06-03T17:34:31Z

In order to explain to non-experts what is a vectorfield, one usually describes an assignemnt of an arrow to each point of space, and this works quite well, also when moving to manifolds (where a generalized arrow will be a tangent vector).

My question is: What are similar objects that can help imagining differential forms?

Positive qualities for such object would be (for example) that it helps justifying easily change of coordinate formulas and formulas for pullbacks via functions, or that it "easily drawable", or that it helps understanding more complicated differential-form-based concepts (e.g. connections, cohomology groups, etc.).

Criterion for being a simple vector

2010-04-26T10:49:15Z

1) I was wondering if there exists a criterion (of (a) combinatorial or (b) geometric nature) for a sum of simple vectors $V\in\wedge^k(\mathbb R^n)$,

$V=e_{a_{11}}\wedge\cdots\wedge e_{a_{1k}} + \cdots + e_{a_{m1}}\wedge\cdots\wedge e_{a_{mk}}$, where $e_1,\cdots, e_n$ is a basis of $\mathbb R^n$

to be a simple vector: in other words, I want to know what should one verify in order to be sure that $V$ can be written as $V=v_1\wedge\cdots\wedge v_k$ for some vectors $v_i\in\mathbb R^n$.

2) What happens if we allow noninteger coefficients? In other words, how does the answer change if we consider question 1) for

$V=c_1e_{a_{11}}\wedge\cdots\wedge e_{a_{1k}} + \cdots + c_me_{a_{m1}}\wedge\cdots\wedge e_{a_{mk}}$, where now $c_i\in\mathbb R$.

3) any reference or hint about some (name of) branch of mathematics dealing with this is also welcome!

Comment by Mircea

2012-09-19T15:05:30Z

what is the difference between the case with the metal ring and the ring-less uni-knot here: mouches.free.fr/pagesus/articles/knots.html ?

Comment by Mircea

2012-07-09T14:33:57Z

Thanks, I didn't notice this criterion. If there will not be more answers (for example on $S^n$ or for other $k$) within 2 weeks, I will accept this one.

Comment by Mircea

2012-06-12T14:34:56Z

Like that one gets exactly all conformal maps of $S^n$ having a fixed point (the one corresponding to infinity), which also are all of those given by the above formula, as pointed out by Ryan Budney. Is there a more general formula for all the conformal diffeomorphisms?

Comment by Mircea

2012-06-12T09:45:25Z

actually the set that you call $B^4$ is usually called $B^5$

Comment by Mircea

2012-05-27T09:27:41Z

Thanks, I think I bothered you long enough. I will see in the book if the smoothness can be preserved in the irrational case.

Comment by Mircea

2012-05-26T08:41:58Z

Thanks! So one can say that for any even $n$ there exists $M_n$ such that $\pi_n(Diff(M_n))\otimes Q\neq 0$, and for odd $n>1$ we can achieve $\pi_n(Diff(M_n))\neq 0$. In the realm of elementary statements just examples of $\pi_{2n+1}(Diff(M))\otimes Q\neq 0$ seem to be missing. Or did I miss something?

Comment by Mircea

2012-05-26T08:35:34Z

I think some confusion was in my head regarding the precise meaning of the word "foliation". Is there more hope if one just looks for laminations?

Comment by Mircea

2012-05-25T05:52:42Z

Thanks again, now the answer is perfectly transparent! So not even by changing the surface construction can one avoid the singularity in the foliation? If the exchange map had rational parameters in the sense that I described in the question, one has some hope of avoiding the prongs by complicating the surface.

Comment by Mircea

2012-05-24T23:52:06Z

Thanks for the detailed description. I can follow it until "is removable" in the end. What does that precisely mean? Does the removal preserve the smoothness of the foliation? Maybe there is some well-known fact here which I don't know.

Comment by Mircea

2012-05-24T12:21:02Z

is there a similar reference for questions on the higher $\pi_k(Diff(M))$?

Comment by Mircea

2012-05-24T12:03:39Z

thanks! this is very clear.

Comment by Mircea

2012-05-24T12:03:34Z

Thanks! so what is a guess of a suitable n?

Comment by Mircea

2012-03-25T19:39:34Z

@Anton Petruin, thank you, I see now how one can do it in the case of perimeter.

Comment by Mircea

2012-03-25T18:27:39Z

..was that what you were referring to?

Comment by Mircea

2012-03-25T18:25:07Z

Thank you, this is a nice proof indeed, it is due to P. Tilli, I think ("Remarks on the Hölder continuity of solutions to elliptic equations in divergence form",Calculus of Variations and Partial Differential Equations, Vol. 25, Number 3, 395-401, DOI: 10.1007/s00526-005-0348-3)