bio | website | math.uvic.ca/faculty/aquas |
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location | University of Victoria | |
age | 46 | |
visits | member for | 3 years, 5 months |
seen | 7 hours ago | |
stats | profile views | 3,201 |
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...
Apr 15 |
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Is every sufficiently dense well mixed set an additive basis?
Another example that goes strongly against the spirit of the question is this: pick $n_1\ll n_2\ll n_3\ll n_4\ll \ldots$ and set $B=\mathbb N\setminus\bigcup_k [n_k,k\cdot n_k]$. For fast enough growing $n_k$ ($2^{k^2}$ is ample), it's easy to see that this set is not a basis of any order, as you can't make $k\cdot n_k$ using $k$ terms from the set. Its counting function exceeds $N^{1-\epsilon}$ for any $\epsilon$ and it's well mixed. The only issue is that it doesn't have an actual growth rate, as the counting function is between $N^{1-o(1)}$ and $N$. |
Apr 15 |
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Is every sufficiently dense well mixed set an additive basis?
Yes. I've fixed this. |
Apr 15 |
revised |
Is every sufficiently dense well mixed set an additive basis?
edited body |
Apr 15 |
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Is every sufficiently dense well mixed set an additive basis?
I'm not sure of the above claim any more. Below is the proof of a weaker statement (which still answers your question in the negative). |
Apr 15 |
answered | Is every sufficiently dense well mixed set an additive basis? |
Apr 15 |
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Is every sufficiently dense well mixed set an additive basis?
No. Using irrational rotations on tori, you can get a set with counting function $\sqrt N$ which is not a basis of any order. |
Apr 14 |
awarded | Informed |
Apr 14 |
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Non-hyperbolic fixed points in multidimensional systems
I don't think you're criterion in 1D is right. $\dot x=-x^2$ satisfies the criterion, but 0 is not Lyapunov stable. In fact, in one dimension, I think it's a necessary condition that the second derivative is 0. The third derivative then can give a sufficient condition for Lyapunov stability. |
Apr 2 |
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What areas of pure mathematics research are best for a post-PhD transition to industry?
I find this a bit of a stretch. |
Apr 2 |
awarded | ergodic-theory |
Apr 1 |
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conditional expectation under convex combinaison of probability measures(II)
I think you can say there's only one measure, namely $P$, and say that everything ($P_1$ and $P_2$) are abs. cts. with respect to it. |
Mar 30 |
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“Mixed” Waring bases of fixed order
It's customary (and basic politeness) to acknowledge responses to questions you ask. |
Mar 29 |
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Primitive, non-2-transitive groups with very large orbitals?
had a hard time understanding your example, but I see now. $x=1$ and you're showing the orbit of 3 under the stabilizer of $x$. |
Mar 29 |
revised |
“Mixed” Waring bases of fixed order
edited body |
Mar 29 |
answered | “Mixed” Waring bases of fixed order |
Mar 26 |
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An example of a homeomorphism on $[0,1]^2$ with constant Jacobian determinant $\pm1$
You're looking for area preserving diffeomorphisms of the square. There are old theorems guaranteeing that there are lots of these. I don't have a reference right now. |
Mar 24 |
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Example for a dynamical system which is not point-distal
If you do a rank one cutting and stacking construction, you can get something with this property if you add in more and more spacers all labelled with the same symbol. |
Mar 22 |
awarded | ds.dynamical-systems |
Mar 21 |
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Mixing property of first return map
Good answer. I didn't know your result (or the Friedman-Ornstein result). Sounds quite cool... |
Mar 21 |
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Mixing property of first return map
This doesn't change anything. Make the roof function 10 in most places; occasionally 9 and then next time around 11. The new system is conjugate to something with constant roof function, but is aperiodic. If you want unbounded, you can arrange that as well by deleting some parts of the base from $A$. |