bio  website  math.ucla.edu/~tao 

location  Los Angeles  
age  39  
visits  member for  4 years, 10 months 
seen  28 mins ago  
stats  profile views  71,617 
Professor of Mathematics at UCLA
1d

awarded  Nice Answer 
Aug 23 
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Interpolation between $L_p$ and $B^s_{q,q}$
Generally the situation with $p \leq 1$ is quite pathological, particularly the nonconvex, nonlocally integrable case $p<1$. Things are better if one uses Hardy spaces instead of Lebesgue spaces though. 
Aug 22 
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Operator norm vs spectral radius for positive matrices
In fact, your weaker conjecture is true! Let $a_{ij}$ be the $ij$ component of $A$, then $A e_j \geq a_{ij} e_i$ and $A e_i \geq e_j$ using the product partial ordering on ${\bf R}^d$. Iterating, we have $A^{2n} e_j \geq a_{ij}^n e_j$ for any $n$; sending $n$ to infinity we conclude that $a_{ij} \leq \sigma(A)^2$, and so $\A\ \leq d \sigma(A)^2$ by Schur's test. 
Aug 22 
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Operator norm vs spectral radius for positive matrices
Gah, that example doesn't have all entries positive. I don't think that's an essential obstacle, but it does make finding a counterexample trickier. (For instance, the diagonal entries of A are bounded by the trace, which is bounded by d times the spectral radius.) It may be that one has to go to higher dimensions to find a counterexample. 
Aug 22 
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Operator norm vs spectral radius for positive matrices
The weaker conjecture is also false; for instance, conjugate $\hbox{diag}(1,2)$ by a large element of $SL_2({\bf Z})$, e.g. $\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$. Note again that ill conditioning is the culprit. For integer nonsingular matrices one has $\hbox{det}(A) \geq 1$, which doesn't give the bounds you want, but may give some other useful inequality for you. 
Aug 22 
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Operator norm vs spectral radius for positive matrices
(correction: "positive" should be "positive integer entry", and "positive semidefinite" should be "nonnegative real entry", and "taking limits" should be "rescaling and taking limits" (to first pass from integers to rationals).) 
Aug 22 
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Operator norm vs spectral radius for positive matrices
A slightly different way to view Mike's counterexample: if such a bound were true for positive matrices, then by taking limits it would also hold for positive semidefinite matrices, and then a nilpotent matrix would be a counterexample. But this also suggests that one can salvage some bound if one also controlled the condition number of the matrix. Indeed, since the product of the absolute values of the eigenvalues is equal to the product of the singular values (both are equal to $\hbox{det}(A)$), one can get some inequality involving the condition number. 
Aug 21 
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Small values of a polynomial evaluated at roots of unity
OK, got it. (I got confused and thought $f$ was the minimal polynomial for the sums of roots of unity.) I'm not seeing how the lemma follows from the theorem though  in the limit as $\delta \to 0$, wouldn't one need a very large Lipschitz constant for $\lambda$? 
Aug 21 
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Small values of a polynomial evaluated at roots of unity
Why is the Mahler measure of f bounded by k? I thought it was the logarithmic height that was controlled by k, which would make the Mahler measure exponential in k and n. 
Aug 21 
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Examples of unproven but likely true existential sentence (in the sense of incompleteness)
Number theory has plenty of unproven existential claims, e.g. (the negation of) en.wikipedia.org/wiki/… . 
Aug 20 
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Interpolation between $L_p$ and $B^s_{q,q}$
Ah, I see now. (I had misread the Wikipedia entry.) In that case, one probably has no choice but to use some CalderonZygmund theory (as is for instance implicit in the statement that $L^p = F^0_{p2}$) as otherwise one doesn't seem to be able to handle all the varying indices simultaneously. 
Aug 20 
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Interpolation between $L_p$ and $B^s_{q,q}$
I don't believe t=p is necessary. Bergh and Lofstrom (for instance) should have details; I don't have access to it here, but the Wikipedia page en.wikipedia.org/wiki/… asserts that the required interpolation result is in Theorem 6.4.5 of that text. 
Aug 20 
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Interpolation between $L_p$ and $B^s_{q,q}$
I haven't checked this carefully, but perhaps one could simply use the inclusions $B^0_{p,1} \subset L_p \subset B^0_{p,\infty}$ together with Besov space interpolation? 
Aug 18 
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Is it possible to find explicit formula for the product $\prod_{dn,\ d>1} (1\mu(d)/\varphi(d))^{\varphi(d)}$?
I think I made a number of sign errors in my previous comments and are no longer able to edit them to correct this, but the broad point of my comments are still valid even if the specific formulae should not be taken literally. 
Aug 18 
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Is it possible to find explicit formula for the product $\prod_{dn,\ d>1} (1\mu(d)/\varphi(d))^{\varphi(d)}$?
Asymptotics are more interesting; writing $(1  \frac{\mu(d)}{\phi(d)})^{\phi(d)}$ as $\exp( \mu(d)  \frac{\mu(d)^2}{2\phi(d)} ) (1 + O(\frac{1}{\phi(d)^2} ))$, the product becomes $\exp( \frac{1}{2}  \sum_{dn} \frac{\mu(d)^2}{2\phi(d)}) ( 1 + O( \sum_{dn; d>1} \frac{1}{\phi(d)^2} ))$ which can simplify to $\exp(\frac{1}{2}\frac{1}{2}\prod_{pn} (1+\frac{1}{2(p1)})) (1 + O( \sum_{pn} \frac{1}{p^2} )))$. One can be a bit more accurate about the contribution of the small primes $p$ to the error term if one wants more precise asymptotics. 
Aug 18 
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Is it possible to find explicit formula for the product $\prod_{dn,\ d>1} (1\mu(d)/\varphi(d))^{\varphi(d)}$?
If for instance $n=pq$, then the product is $(\frac{p2}{p1})^{p1} (\frac{q2}{q1})^{q1} (\frac{pqpq+2}{(p1)(q1)})^{(p1)(q1)}$ which has no evident cancellation or further factorisation, and only a small amount of like terms to collect. Doesn't seem like there is much else to be done here. 
Aug 18 
awarded  Populist 
Aug 17 
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Is there a generalization of Sobolev spaces for certain locally compact groups?
There is some literature on Sobolev spaces in arbitrary metric measure spaces: www2.pitt.edu/~hajlasz/OriginalPublications/… . For a locally compact group, one can use Haar measure for the measure, and if the group is second countable one can use BirkhoffKakutani to get a metric (but the choice of metric is not unique, and this can lead to different Sobolev spaces, e.g. a Riemannian metric gives different results to a CarnotCaratheodory metric). 
Aug 14 
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For integers $a \ge b > 1$ is $f(a,b) = a^b + b^a$ injective?
The problem looks at least as hard as the Catalan conjecture en.wikipedia.org/wiki/Catalan's_conjecture , which concerns the equation $a^b = c^d + 1$ instead of $a^b + b^a = c^d + d^c$, and an unconditional solution is likely out of reach of current methods. The 4variable version of the abc conjecture (see jstor.org/stable/2153551 ) may be able to imply the conjecture (or at least a large fraction of it), though. 
Aug 13 
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Geometric Interpretation of Trace
This property, together with linearity, determines the trace uniquely, and so one can view the trace as the linearised version of the dimensioncounting operator. (This is basically the "noncommutative probability" way of thinking about the trace.) 