User bsteinhurst - MathOverflow

Answer by BSteinhurst for How to define a differential form on a fractal?

2013-06-14T00:30:57Z

Alexander Teplyaev has been working on exactly this question this is a paper in which he and Michael Hinz show that the Navier-Stokes equation on a Sierpinski gasket is sensibly defined and that it has a solution. There is a cluster of papers on this topic that they have written, all are available on arXiv.

Answer by BSteinhurst for Is there any result concerning on the metric dimension of inverse limit?

2013-05-19T03:49:39Z

I really should avoid answering questions late at night. My original answer is muddled enough to not work. But here is what it should have been:

Let $X_0 = \{0,1\}$ and $X_i = X_{i-1} \times X_0$. Give each $X_i$ the $2-$adic ultrametric. That is the distance between two sequences of $0'$s and $1'$s is $2^{-n}$ where $n$ is the number of share initial symbols the two sequences share. The projections maps are given by truncation. The inverse limit is now $\mathbb{Z}_2$ which has box counting dimension equal to one.

Answer by BSteinhurst for Sierpinski Triangle and the Chaos Game

2013-04-21T21:42:31Z

Because the iterated function system that defines the Sierpinski gasket is a contraction mapping in the metric space of non-empty compact subsets of $\mathbb{R}^{2}$ with the Sierpinski gasket as its only fixed point. So if you start with any non-empty compact set it will "get closer" to the Sierpinski triangle each time you apply the iterated function system. Here the "getting closer" is in the sense of the Hausdorff metric on subset of $\mathbb{R}^{2}$. (Yes I know the metric is definable in considerably more generality but we only use this instance here.)

Answer by BSteinhurst for Fractal dimension of 1D set, what if logN vs log(e) is a polygonal chain?

2013-03-05T02:21:00Z

The canonical reference for this material (at least with the people I hang out with) is Ken Falconer's Fractal Geometry Mathematical Foundations and Applications.

For most sets that are fully self-similar with infinite levels of geometry such as the Cantor set the function $\log(N)/\log(e)$ will be a similar polygonal chain far away from 0. It will get smoother as you approach zero and there will be a non-zero slope there. What you have however is a finite set of points so for an $e$ below a certain threshold $N$ stays constant. That is the function will have slope zero and you have a set of fractal dimension zero. In the book there is some discussion of how to interpret situations like there where you are approximating something that will have a non-zero dimension. But that is not something I can say much about.

Answer by BSteinhurst for Hutchinson's formula for asymptotically homogeneous Cantor sets

2013-01-13T17:00:25Z

Just a quick note before having to move on, but when thinking about IFSs which approached self-similarity the papers of Igudesman have been useful. He has a notion of lacunary self-similar sets which (without checking details) looks like your set up. The main paper I am thinking of is the one referred to here.

Answer by BSteinhurst for What are integration on fractal?

2013-01-03T20:26:25Z

As Steve Huntsman points out in his comment this is just writing out the integral against the Hausdorff measure on the fractal domain in polar coordinates over the same fractal. Since the author does not specify what the fractal domain actually and merely asserts that it is distributed throughout $\mathbb{R}^{4}$ such that it is "regular and homogenous" with respect to the Hausdorff measure he is able to claim that for the right sort of fractal domain the formulas that we are used to when integrating over $\mathbb{R}^{d}$ hold even when $d$ is not an integer. That is what looks like the tricky part to me, just assuming the existence of such a fractal. The discussion of why the coefficient is what it is appears just after (2.6) and arises simply by analogy to the volume of the unit sphere that usually appears in polar coordinate transforms.

ADDENDUM (4 Jan 2013): The formula for the volume of the unit sphere that the author uses is wrong if you take it out of the context that the author takes for granted. And the author never makes the claim that it should. The logic of the paper is: given a fractal subset of $\mathbb{R}^{4}$ such that this formula for integration in polar coordinates and the volume of the unit sphere holds then the following physics can be done. It is not fair to the author to complain that when you take a fractal such that his assumptions don't hold that the formula is wrong. That being said, I still wish he had actually exhibited one of these fractals.

Answer by BSteinhurst for What is the "real osculating space" of a (minimal) immersion?

2012-11-28T14:38:40Z

I took a quick look around and Wolfram Mathworld has a great animation of an osculating circle here. Also the entry on osculating curves here has a nice definition. The intuition is that when the you look at $S^{2}$ immersed in $S^{n}$ you take a point $x$ and ask for another surface which also contains $x$ and both the immersion of $S^{2}$ and this new surface have the same $k$ derivatives at $x$. So a tangent plane is an osculating surface of order 1.

However perhaps the best source of intuition is what the word osculate means in latin, ``to kiss.''

As for what the modern algebraic geometry term might be, I did some quick Googling for ``osculating algebraic geometry" and the term still seems to be in use. Perhaps not very commonly.

Answer by BSteinhurst for transition probability convergence for Harris chains - Durrett.

2012-10-28T16:14:40Z

You already know that the random walk $S_m-T_m$ is recurrent. It sounds like the piece that you are missing is that recurrent random walks (especially on $\mathbb{R}$) not only visit their initial value infinitely often but any other finite value as well as long as it has a positive probability of reaching this other value in finite time. A similar example to consider is a simple random walk on $\mathbb{Z}$ started at $10$ and it's visits to $0$. This is short Borel-Cantelli argument.

Edit: Suppose that $S_0-T_0 = \beta >0$ for the moment. Then you need the process $Y$ to hit it's initial value $\beta$ times before $X$ does. Since the times of these events are all i.i.d. then it is a question of the probability of $\beta$ returns for $Y$ before 1 return for $X$. Which can be crudely lower bounded by the probability that the return times for $Y$ are less than 1 and the return time for $X$ is greater than $10$ which have a positive, albeit small, probability.

Answer by BSteinhurst for Background Reading for Proving Irrationality of Real Numbers

2012-10-26T12:48:41Z

I'd like to add the book for one of the two number courses I've taken which was an entire senior seminar (last year of undergraduate) on transcendental numbers. Making Transcendence Transparent by Burger and Tubbs was good read and stressed the basic structure of the proofs by using sidebars to explain the ideas next to the formal proof. To this day the book contains my favorite theorem: $\pi \neq \frac{22}{7}$. Because this book is aimed at American undergraduates it is more conversational than a good for graduate students or beyond.

As a bonus, it appears to be less expensive than I remembered.

Answer by BSteinhurst for Derivatives through random variables?

2012-10-17T18:20:14Z

It does not because $x$ is just some element of your state space. You could conceivably choose $x$ as your sample point for any or all of the $\theta$'s. So what it would make sense to differentiate with respect to $\theta$ is $$\mathbb{P}_{\theta}(X=x)$$ as long as you have a discrete distribution. For a continuous distribution you could substitute $x$ with some interval.

Answer by BSteinhurst for Infinite dimensional manifold

2012-10-07T03:46:15Z

As far as the QM goes, I don't know enough to really say, but books like Quantum Mechanics for Mathematicians would give a mathematical overview of the large pieces of QM. However, infinite dimensional manifolds are out there in the literature. A quick googling brought up what looks like a good place to is arXiv:math/9202206. This paper is appears to be a 20 year old survey of infinite dimensional manifolds. For something a tad more recent and superficial you can try the wiki article on Banach manifolds which does not have many references but it will provide you with the keywords to do more googling.

EDIT: As mentioned by Kofi, what you need to extend the definition of a smooth manifold to an infinite dimensional setting is a notion of a derivative in an infinite dimensional setting. There is one that goes by the name of ``Frechet derivative.'' Using this derivative you just start mimicking finite dimensional spaces. For QM you will probably only ever need the Hilbert manifold case (locally diffeomorphic to a Hilbert space) and not the more general Banach or Frechet spaces but the idea for those is that as long as a notion of derivative is available you can define a smooth structure.

Answer by BSteinhurst for Who uses keywords (and how)?

2012-08-29T03:15:01Z

I know that Mathematical Reviews use the keywords to assign articles to reviewers since the MSC codes are sometimes too broad to be of much use. This does help find reviewers who can actually say something meaningful about the article.

Answer by BSteinhurst for Asking for a Fourier inverse transform, which is related to stable laws

2012-03-15T14:02:14Z

I think you may want to look at this MSE question. Amusingly enough, this question was to prepare a lecture for when the asker was covering my class last fall. This kind of argument is mentioned in Durrett's Probability: Theory and Examples in Chapter 3.

Answer by BSteinhurst for Itô-like calculus for $\alpha$-stable processes $\alpha \neq 2$.

2012-02-15T04:59:29Z

David Applebaums' "Lévy processes and stochastic calculus"

It looks like integrating with respect to an $\alpha$-stable process is addressed directly on page 212 with the Ito formula to follow starting on page 218. Sadly the Ito formula section is not included in the Google preview.

I have not read the new version of this book but I did read the first edition in graduate school and I found it quite readable.

Answer by BSteinhurst for Any reference on Brownian Motion continuity

2012-02-06T23:20:52Z

This is a partial answer but shows the kind of subtlety that makes the continuity of Brownian motion non trivial. If you try and take the first three axioms of Brownian motion and try to prove that the process has continuous paths using a central limit theorem argument what you get is that on a probability space $(\Omega,\mathbb{P})$, that $\forall t > 0$

$\mathbb{P}(B_t\ is\ discontinuous\ at\ t) = 0 $

This means that there are null sets $\mathcal{N}_t \subset \Omega$ such that if $\omega \not\in \mathcal{N}_t$ then $B_t(\omega)$ is continuous at time $t$. And here is the delicate part. This does not imply that there exists any single $\omega \in \Omega$ such that $B_t(\omega)$ is continuous for all $t$. I have seen this property called stochastic continuity in some places.

What you usually want is a single null set $\mathcal{N}$ so that for $\omega \not \in \mathcal{N}$ $B_t(\omega)$ is continuous for all $t \ge 0$.

Of course the usual constructions of Brownian motion do take care of this subtlety but some times without mentioning it.

Answer by BSteinhurst for Using slides in math classroom

2011-11-04T16:02:19Z

If you intend to post your slides online after class then you run the risk of students not even taking notes/digesting the material on their own (I've had this feeling myself) or feeling that they don't have to attend class. This is obviously a con but the other side is that the students then have a good outline of what you talked about in class with your emphasis included.

I second Thierry's comments.

Answer by BSteinhurst for Is there a fair coin?

2011-11-01T12:45:02Z

I think a fairly good demonstration is Persi Diaconis' machine to toss a perfectly standard US quarter to a single predetermined side with something like 99% accuracy. I have heard it said that he could do it himself by hand years ago when he had practiced extensively. So the question may be more along the lines of "is there a fair coin tosser?" not "is there a fair coin to be tossed?" Your instructor may have been referring to a paper of his "Fair Dice." With J. Keller, Amer. Math. Mo., 96:337-339, 1989. (He has it freely available on his website). Just remember that a coin is really just a two sided die.

Answer by BSteinhurst for Non-existence of such a continuous stochastic process

2011-09-24T16:39:46Z

Suppose you had such a process that is not trivial. Suppose you have $W_s \neq W_0$. For $t >s$ we have assumed that $W_s$ and $W_t$ are independent, have mean zero, and the same distribution. Now choose a sequence $t_n \downarrow s$ such that $|W_{t_n} - W_s| > \epsilon$. This sequence exists for some $\epsilon >0$ since $W_t$ is independent of $W_s$, but has the same distribution as $W_s$. Once you has this sequence you can quickly see that the process cannot be continuous.

Answer by BSteinhurst for Historical basis and mathematical significance of Riemann surfaces

2011-09-15T12:30:52Z

From the wording of your question it is possible you are asking someone to write an entire historical overview for you. So instead what I did was spend a few minutes on Ye Olde Google and found this:

The Concept of a Riemann Surface by Hermann Weyl. It is cheap and your local library might have it already it.

Answer by BSteinhurst for A formal definition of Scaling Limits?

2011-09-05T15:39:20Z

I was listening to Lawler give a series of talks this summer and his sense of scaling limit is the same as is often used in defining Brownian motion as a scaling limit of simple random walks. In which both time and space have to be scaled so that the limit process doesn't become trapped at the origin. The difference between Brownian motion and SLE is that the random walks are no longer simple. The conformal invariance in SLE comes of the properties of the not-simple random walks if memory serves.

So if one wanted a rigorous definition of a scaling limit I would look for a text on Brownian motion (most any probability with measure theory book will do this construction in the continuous time chapter) or a more specialized book on continuous time processes. There should be plenty of these in most any language you'd want.

Answer by BSteinhurst for Fractal questions: Weierstraß-Mandelbrot

2011-07-13T11:31:01Z

A quick, partial answer to your second question about the definition of fractals. If a fractal is generated by an iterated function system with a scaling ratio less than one then you do get a Hausdorff dimension less than the inductive dimension. However it is not particularly difficult to create a set with Hausdorff dimension less than inductive dimension that should be a fractal that isn't self-similar. The idea is to choose between two iterated function systems aperiodically.

Answer by BSteinhurst for Riemann Zeta Function connection to Quantum Mechanics.

2011-05-17T14:33:51Z

This may be buried in one of the references above, but for those don't wish to go through them all...

The zeta function can arise as the trace of Hamiltonians governing physical systems. For example in an experiment to measure the Casimir effect (two perfectly conducting plates placed very close to each other) the force they exert on each other has a formula that involved the the derivative of the Riemann zeta function evaluated at $-\frac{1}{2}$. This has been experimentally validated to a reasonable amount of precision.

This may not get you zeros, but it gets you certain values.

Answer by BSteinhurst for What does progressively measurable actually entail?

2011-04-19T18:24:02Z

The formula

$$\mathbb{E} \int_{R+} \phi^{2}(t,\omega)dt$$

is a double integral a la Fubini-Tonelli. And if you did back there is probably a condition on the filtration saying that $\mathcal{F}_t \subset \mathcal{A}$ for all $t \ge 0$. So that progressive measurability does imply that $\phi^{2}(t,\omega)$ is $\mathcal{B}(\mathbb{R}^{+}) \otimes \mathcal{A}$ measurable. But making this integrand measurable isn't the main purpose of the progressive measurability condition. The main point is so that something like $f(t,X_t)$ where $X_t$ is an adapted process is again an adapted process. The integrability condition is to give $\mathcal{M}^{2}$ a Hilbert space structure.

Answer by BSteinhurst for What is the high-concept explanation on why real numbers are useful in number theory?

2011-04-14T12:20:43Z

I'll admit I'm not number theorist but here is my take on why the field has to embrace the completions of $\mathbb{Q}$. It is a simple reason: $e$. Other branches of math have run across interesting numbers that aren't algebraic which deserve studying in their own right. So not only should number theory as a whole (not necessarily every practitioner) should take the reals as valid objects of study.

Answer by BSteinhurst for Weierstrass' function and Brownian motion

2011-03-21T19:20:50Z

Some quick Googling brought me to this paper. The idea is to take the coefficients in the summation to be suitable independent random variables according to a suggestion of Mandelbrot. I can't actually access the paper right now so I can't say if the authors were able to include the case of standard Brownian motion in their results.

``Convergence of the Weierstrass-Mandelbrot process to Fractional Brownian Motion'' Murad Taqqu and Vladas Pipiras. Fractals. 8 (2000) 369-384.

Answer by BSteinhurst for Graphs Embedded on Fractals

2011-02-22T14:16:44Z

The Sierpinski gasket is not a good example for this because of the bound you saw on the degree of the graph. I'd venture to say that this is because the gasket falls into a class of fractals called post-critically finite. It is the fact that in pcf fractals when level n cells intersect there are a uniformly bounded number of them. You might be interested in looking at finitely ramified but not post-critically finite fractals. See this for some nice pictures of the Diamond fractal. Since the number of level n-cells intersecting is unbounded in n you can embed graphs without a bound on degree.

Answer by BSteinhurst for Why semigroups could be important?

2011-02-18T19:51:29Z

Semigroups of bounded $L^{2}$ operators are very important in probability. They in fact provide one of the main ways show the very close connection between a self-adjoint operator and a `nice' Markov process (nice can be taken to mean strong Markov, cadlag, and quasi-left continuous.) So how does one get this semigroup from a Markov process? If $X_t$ is your process let $\mu_{t}(x,A)$ be measure with mass $\le 1$ with value $P(X_t \in A | X_0=x)$. Then $\int f(y) \mu_t(x,dy) = T_tf(x)$ gives a semigroup of bounded $L^{2}$ operators.

Why are such constructions important and natural? If $f$ is your initial distribution of something (heat for example) and $X_t$ is Brownian motion. Then $T_t$ acts by letting the heat distribution $f$ diffuse the way heat should. This then gives a nice way to connect PDE and probability theory. I'd end by offering that semigroups are important, in part, because they do arise is so many places and can bridge between disciplines. There are other reasons as well.

Answer by BSteinhurst for Why is the Laplacian ubiquitous?

2011-02-10T15:00:39Z

To build on what Andrey has mensioned about Brownian motion. The Laplacian is one of the points of connection between stochastic processes and analysis. The Laplacian appears as the infinitesimal generator of Brownian motion and conversely a self-adjoint operator that has some of the properties of the Laplacian can be used to define a `Brownian motion' on spaces other than $\mathbb{R}^{d}$. For example one of the early proofs of the existence of a Brownian motion on the Sierpinski carpet centers on first creating a Laplacian on the Spierpinski carpet.

In this context the eigenvalues of the Laplacian can be used to determine the properties of the heat kernel associated to the Brownian motion, like continuity or Gaussian/sub-Gaussian/or other bounds.

I hope this gives you some part of what you asked for.

Answer by BSteinhurst for Why were matrix determinants once such a big deal?

2010-12-22T21:33:10Z

From the realm of probability there are determinental and permanental processes. Terry Tao has a nice post about determinental processes here. For instance

"Examples of processes known to be determinantal include non-intersecting random walks, spectra of random matrix ensembles such as GUE, and zeroes of polynomials with gaussian coefficients."

I've not worked with these processes myself but I've heard enough seminar talks using them to say there is plenty of interest out there.

Answer by BSteinhurst for What advanced Area of Mathematics can be delved into with only basic Calculus and Linear Algebra

2010-12-16T15:46:58Z

Perhaps if you are analytically inclined then analysis on fractals is a good place to look. A good portion of the work done is with second finite difference equations leading towards a limit definition of a ``second derivative'' and uses some linear algebra such as inverting small matrices. Strichartz's Differential Equations on Fractals is a good place to start, especially the first few chapters where spectral decimation is discussed. As an aside signifigant number of papers in this area have come out of REU programs.

In general focusing on your courses is important but that should still leave you with some time to think about other topics as well. This kind of curiosity will help you see what's out there and give you a sensible way to choose a specialty when the time comes.

Best of luck.

Comment by BSteinhurst

2013-05-21T04:09:48Z

@Bingbing Liang, it is not clear to me right now whether the nonzero lower box counting dimension always holds for any compatible sequence of metrics. There are lots of metrics out there and I am not certain enough to rule out a counter example.

Comment by BSteinhurst

2013-05-19T05:42:19Z

@Misha thank you for making me think a bit longer about this. @Bingbing Liang yes, the idea is to do that and then take the limit. Z_2 is just the easiest example of this to write down.

Comment by BSteinhurst

2013-04-28T13:55:34Z

@Stefan, the same objection could be used against any physical science. Yet, as a community, we do not give up hope of modeling in the face of other very complicated systems like particle physics or the global climate.

Comment by BSteinhurst

2013-04-21T23:52:40Z

If you choose some initial point $x_0$ and then applying some sequence of the three maps is equivalent to picking one of the points in the image of the IFS which by the Hausdorff convergence must approaching the Sierpinski gasket. So my original answer was not quite complete.

Comment by BSteinhurst

2013-02-07T17:55:18Z

Perhaps the Cameron-Martin theorem is what you are looking for?

Comment by BSteinhurst

2013-01-14T04:23:27Z

Nikita, in the quick reading of your question that I had time for this morning it looked to me like your perturbed self-similar sets could be treated as lacunary self-similar sets. If that is not the case then such things happen and nothing but electrons have been wasted.

Comment by BSteinhurst

2012-11-30T01:31:22Z

All the better for when you do prove the fundamental theorem, I think.

Comment by BSteinhurst

2012-11-29T19:47:09Z

Thank Joel, I dashed off this answer before running off to give a lecture. I'll edit the answer for what it is worth now.

Comment by BSteinhurst

2012-11-29T13:30:06Z

@Davide: Alas, we were typing at the same time!

Comment by BSteinhurst

2012-11-20T03:21:39Z

How is S defined? That is if really is a continuous function of BM then there is a lot you can say. So, more detail please.

Comment by BSteinhurst

2012-11-09T15:42:15Z

This very much feels like homework. And muna, you can use LaTeX when writing your questions.

Comment by BSteinhurst

2012-11-03T16:35:59Z

I added the reference-request tag since this question asking for just that.

Comment by BSteinhurst

2012-10-08T22:40:09Z

Dear Jorg, in some sense it is. But one can create a non-trivial self-similar fractal with integer Hausdorff dimension as well. So dimensionality alone isn't a good definition for what a fractal is.

Comment by BSteinhurst

2012-09-26T14:11:10Z

The edit brought forth what I knew you were saying but in a less provocative style. I happen to agree with the main point of your answer, I just got stuck on one particular phrase.

Comment by BSteinhurst

2012-09-25T02:03:38Z

The idea that "everyone knows that Brownian Motion is the limit of simple random walks. They don't feel the need to make this rigorous, it is just self evident." really bothers me. Just because there is standard machinery that is cited to show the limit exists does not mean that no one feels the need for rigor.