BSteinhurst
  • Member for 11 years, 1 month
  • Last seen more than a month ago
  • Westminster, MD
Can this informal argument (for the fact that almost all reals in the unit interval are irrational) be saved?
12 votes

Short answer: The countable product of probability measures is a well-defined object so the countability of periodic sequences is enough to conclude that the probability of drawing a periodic sequence ...

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Who uses keywords (and how)?
12 votes

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 ...

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Why semigroups could be important?
7 votes

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 ...

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How to define a differential form on a fractal?
6 votes

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 ...

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What are integration on fractal?
6 votes

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 ...

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Using slides in math classroom
6 votes

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'...

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Historical basis and mathematical significance of Riemann surfaces
6 votes

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: ...

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Fractal questions: Weierstraß-Mandelbrot
6 votes

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 ...

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What does progressively measurable actually entail?
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6 votes

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 \...

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Why is the Laplacian ubiquitous?
6 votes

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 ...

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Why were matrix determinants once such a big deal?
5 votes

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 ...

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Can Hausdorff dimension make sets into a Tropical Semiring?
4 votes

Hatano gives on page 19 of this article "Notes on Hausdorff Dimensions of Cartesian Product Sets" HIROSHIMA MATH. J.1 (1971), 17-25 as class of generalized symmetric Cantor sets whose products have ...

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Any reference on Brownian Motion continuity
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4 votes

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 ...

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Is there a fair coin?
4 votes

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 ...

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A formal definition of Scaling Limits?
4 votes

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 ...

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Riemann Zeta Function connection to Quantum Mechanics.
4 votes

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 ...

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Itô-like calculus for $\alpha$-stable processes $\alpha \neq 2$.
3 votes

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 ...

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What is the high-concept explanation on why real numbers are useful in number theory?
3 votes

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 ...

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Weierstrass' function and Brownian motion
3 votes

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 ...

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notable inductive proofs relating to fractals
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2 votes

Spectral decimation is an inductive process where the eigenvalues of a natural Laplacian on "nice" fractals is computed inductively. The idea is that by using a sequence of finite graphs to ...

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Fractal dimension of 1D set, what if logN vs log(e) is a polygonal chain?
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2 votes

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-...

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Non-existence of such a continuous stochastic process
2 votes

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 ...

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Is there any result concerning on the metric dimension of inverse limit?
1 votes

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 $X_i = X_{i-1} \times X_0$. ...

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Sierpinski Triangle and the Chaos Game
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1 votes

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 ...

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What is the "real osculating space" of a (minimal) immersion?
1 votes

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 ...

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Background reading for proving irrationality of real numbers
1 votes

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 ...

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Derivatives through random variables?
Accepted answer
1 votes

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 ...

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Infinite dimensional manifold
1 votes

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 ...

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Asking for a Fourier inverse transform, which is related to stable laws
1 votes

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 ...

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Graphs Embedded on Fractals
1 votes

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 ...

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