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bio website math.ucla.edu/~tao
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Professor of Mathematics at UCLA

1d
comment Intuition for Integral Transforms
en.wikipedia.org/wiki/…
Jul
18
awarded  Nice Answer
Jul
10
comment Kloosterman-like sum with inverse to different moduli
The expressions you write are currently ambiguous. For instance, $\bar{s}^{(r)}$ is only determined up to a multiple of $r$, and this affects the first exponential. Are you going to take the representative of this residue class in $\{0,1,\dots,r-1\}$? If so, the sum is going to get rather non-algebraic, and it will be significantly more difficult to use the usual methods to handle these sums.
Jul
2
revised Polynomial representing all nonnegative integers
added 405 characters in body
Jul
2
revised Polynomial representing all nonnegative integers
added 2399 characters in body
Jul
2
comment $b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
It looks like the $b_n$ are in fact bounded in $L^\infty$, because the $u_n$ are. In any case, from the uniform boundedness principle, the $b_n(t)$ have to be bounded in $L^q$ if one is to have weak convergence, so such a bound must already appear somewhere in the argument.
Jul
2
awarded  Curious
Jun
30
comment $b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
I think you are omitting an important hypothesis (mentioned in the paper), namely that the $b_n(t)$ are uniformly bounded in $L^q(\Omega)$. With this uniform bound, one can obtain weak convergence through testing against $C^\infty_c(\Omega)$ functions, at which point one can use the $C^0$ strong convergence to conclude.
Jun
26
comment Are there any nontrivial near-isometries of the $n$-dimensional cube?
It occurs to me that inverse Littlewood-Offord theory (see e.g. Ch7 of my book with Van Vu) is likely to be helpful. Note that for most i, the distribution of $(Ox)_i$, with $x$ drawn uniformly from the discrete cube, is concentrated at $\pm 1$, which by ILO theory gives strong constraints on the coefficients of the i^th row of O (most of the entries must lie very close to a generalised arithmetic progression). By using multidimensional ILO and working with several i simultaneously, one should be able to cut down the possibilities further.
Jun
26
comment Are there any nontrivial near-isometries of the $n$-dimensional cube?
Nice question! A model question: if $Ox$ lies exactly in the unit cube $\{-1,+1\}^n$ for $n^{-100}$ of the $x$'s, does this force $O$ to be an approximate cube symmetry? Your situation is close to this, except that rather than exactly lying in the unit cube, all but $O(\log n)$ of the coefficients of $Ox$ lie within $0.0001$ of $\{+1,-1\}$ (together with some similar assertions of this nature). It's possible that some additive combinatorics or some Blum-Luby-Rubinfeld type arguments can be used to improve this structure, though $\{-1,+1\}^n$ isn't additively closed, which is a problem.
Jun
19
comment In what ways is physical intuition about mathematical objects non-rigorous?
In any event, how certain could one be that failure of claim i and failure of claim j could not possibly be (completely or partially) statistically independent? I doubt that one's certainty on this issue could be anywhere close to $1 - 10^{-10^{100}}$.
Jun
19
comment In what ways is physical intuition about mathematical objects non-rigorous?
One does not need full statistical independence here; even if only a tiny portion (say $10^{-10^{100}}$) of the event of failure for $i$ is independent of the event of failure for $j$, then there is still a strong likelihood of breakdown of the argument once $n$ exceeds, say, $10^{2 \times 10^{100}}$. For instance, an argument that somehow proceeds through an induction over all possible ensembles of an N-particle system, appealing to the second law of thermodynamics for each such ensemble, would be subject to this problem and be mathematically dubious.
Jun
18
comment In what ways is physical intuition about mathematical objects non-rigorous?
If $n$ is large (e.g. larger than $10^{10^{100}}$), then it becomes essential in mathematics that each of the claims $P(i) \implies P(i+1)$ are known with $100\%$ certainty (again, in the internal sense of the mathematical reasoning model), as even a $10^{-10^{100}}$ failure rate for each of these steps will throw the derivation of $P(n)$ into serious doubt. This requirement of absolute certainty is particularly in effect for mathematical arguments involving countable or uncountable infinities; one simply cannot afford any failure probability at all, no matter how small.
Jun
18
comment In what ways is physical intuition about mathematical objects non-rigorous?
I should perhaps clarify that in my post above, terms such as "certainty" and "probability" were meant to be interpreted internally within one's accepted mode of reasoning (whether physical or mathematical), and not with regards to the external degree of accuracy to which one can actually apply this reasoning. For instance, a common technique in mathematical reasoning is mathematical induction, in which a claim P(n) is deduced from an initial claim P(0) and the claims $P(i) \implies P(i+1)$ for $i=0,\dots,n-1$. (cont.)
Jun
17
revised Why differential forms are important?
corrected spelling of "Stokes"
Jun
16
comment References for LWP of a NLS Equation
"NLS" and "inverse square potential". (I had also tried NLS and "Coulomb potential" but this gave less relevant hits.)
Jun
16
comment References for LWP of a NLS Equation
A google search turns up this article tandfonline.com/doi/full/10.1080/… , although I don't have full access to it at present.
Jun
16
comment A question on the proof of Cheeger-Colding's second paper
From a quick glance at the paper, it appears that the second inequality is not a consequence of the first, but is simply a direct consequence of (the contrapositive of) Lemma 3.1, applied to the $\epsilon+\psi$-neighbourhood of $E_{\eta,i}$.
Jun
15
comment A sumset inequality
I believe Ruzsa's survey article link.springer.com/chapter/10.1007%2F978-1-4612-2418-1_21 should contain a similar counterexample, though I can't access it currently. (The existence of such a counterexample is Exercise 2.4.9 of my book with Van, but we didn't give a hint.)
Jun
14
comment A question on the proof of Cheeger-Colding's second paper
The paper in your link does not have a Theorem 3.7 or a Lemma 3.1. I suggest re-editing your question to add significantly more detail, and to quote relevant parts the paper (by screenshot if necessary).