8,704 reputation
11849
bio website math.uvic.ca/faculty/aquas
location University of Victoria
age 46
visits member for 3 years, 10 months
seen 7 hours ago
The pic is the phase portrait of a simple "piecewise isometry". Define a map by sliding the two halves of the plane: the top half to the right; the lower half to the left and then rotate. Around each periodic point of the map there's a "periodic island". These are what are in the image...

17h
comment Generalization of the alternating sign test for convergence of a series?
You probably want to try summation by parts. It's an analog of integration by parts. This is the engine behind Dirichlet's test, but maybe you can get somewhere by using the underlying technique. Just as in integration by parts, you "differentiate" (difference) one term (here the $|a_n|$) while you "integrate" (sum) the other term. Even if the sum diverges, if the differences are smaller, you might be in business.
Sep
9
comment Concentration of weighted random chirp
@ChristianRemling: Oops: I didn't notice that the $x$ was scaled to have $L^2$ norm of 1.
Sep
9
comment Concentration of weighted random chirp
I don't think you should expect a result like this. You should probably expect that the $e^{2\pi i u k^2}$ behave like independently chosen points on the unit circle. Then the sum should have size something like $\sqrt n$?
Sep
9
comment $L^p$ norm means
I think @YemonChoi is exactly right here (for the asymptotics of large $n$). For the large $q$ asymptotics, I think the normals are useful again: probably it is something like $\sqrt{2\log n}$ as $q\to\infty$.
Sep
2
comment $L^2$ discrepancy bound for sequences in $[0,1)$
@NoamD.Elkies: Not sure how you deduce an $O(\sqrt{\log n})$ upper bound for $\min \|f_n\|_2$ from an $O(\log n)$ upper bound for $\min \|f_n\|_\infty$ (I do see how to get an $O(\log n)$ upper bound).
Aug
27
comment A Problem Concerning Odd Perfect Number
Don't you also need to look at {3,5,13}?
Aug
23
comment Aperiodic set of corner Wang Tile
So if that's what you're looking for, this should be made clear in your question. At this point, you should have enough information to find references yourself using google. The best edge type Wang tiling is the Kari-Culik tiling. Any literature on corner Wang tilings will certainly refer to that.
Aug
22
answered Aperiodic set of corner Wang Tile
Aug
22
comment Aperiodic set of corner Wang Tile
If you believe the answer to your previous question mathoverflow.net/questions/164400/… which you accepted, then an aperiodic edge-type Wang tile translates directly to an aperiodic corner-type Wang tile.
Aug
19
comment boundary density of the Von Koch flake
Right. Presumably the $\infty$ is a typo. All points have the same limit in the $\infty$ direction.
Aug
18
comment Integer valued polynomial through some points with rational coordinates
If you're looking for an integer-valued polynomial such that $y_i=P(x_i)$ for each $i$, you should certainly assume the $x_i$ are distinct(!)
Aug
17
comment Linear dependency of real numbers with integer coefficients adding up to zero
Your question is precisely whether the reals $x_2-x_1$,...,$x_n-x_1$ are linearly independent over the rationals.
Aug
12
answered Bounded martingales of infinite path length
Aug
12
awarded  Popular Question
Aug
5
comment Name of a generalized version of semi-continuity
The empty set is open. Do you mean $f^{-1}U$ contains a non-empty open set whenever $U$ is a non-empty open set?
Jul
31
comment Is any particular algebraic number known to have unbounded continued fraction coefficients?
Nice revision @StevenStadnicki! I didn't understand at all what the original question was asking.
Jul
30
comment Prove that …, f(x-2), f(x-1), f(x), f(x+1), f(x+2),… is algebraically linearly independent without the Fourier transform
Instead of changing the norm, I think it's probably nicer to say you know the space of translates is finite-dimensional, so pick a (finite) orthogonal basis. By the condition in this answer, the inner product with any basis element goes to 0. Hence the strong norm (of $f(x-k)$) goes to 0 as $k\to\infty$.
Jul
28
comment Double Markovity
Do you mean that you have three random variables $X$,$Y$ and $W$ all defined on the same probability space, such that $\mathbb P(Y=y|X=x,W=w)=\mathbb P(Y=y|W=w)$ and $\mathbb P(W=w|X=x,Y=y)=\mathbb P(W=w|Y=y)$?
Jul
26
comment Pairwise dependent random walk recurrent
Your summands have the property of 1-dependence: if $|i-j|>1$, then $X_i$ and $X_j$ are independent. There are CLT for 1-dependent random variables. The events that $\limsup S_n=\infty$ and $\liminf S_n=-\infty$ are tail events, and therefore have probability 0 or 1 by Kolmogorov's 0-1 law. It's not hard to see the probability is not 0.
Jul
25
comment The lonely molecule
If they're randomly placed, then they will almost surely cluster somewhere sooner or later. It's easiest to work on the torus, but completely equivalent because you can unfold $2^d$ copies of the cube to make 1 copy of the torus.