Terry Tao
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Registered User
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Professor of Mathematics at UCLA
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6h |
answered | Is there any proof that you feel you do not “understand”? |
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1d |
revised |
Why are Schur multipliers of finite simple groups so small? added 51 characters in body; added 21 characters in body |
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1d |
revised |
Why are Schur multipliers of finite simple groups so small? added 77 characters in body; added 20 characters in body; deleted 2 characters in body |
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1d |
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Why are Schur multipliers of finite simple groups so small? added 6 characters in body; edited body |
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1d |
asked | Why are Schur multipliers of finite simple groups so small? |
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2d |
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Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function. This question sounds like it should be covered by the literature on Fourier-based edge detection, see e.g. jstor.org/stable/27642530 |
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May 15 |
accepted | For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$? |
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May 14 |
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Awfully sophisticated proof for simple facts Vinogradov's theorem already suffices for this (but that theorem in turn relies on the prime number theorem, which certainly is stronger than Euclid's theorem). In any case Helfgott's argument uses effective estimates on the number of primes less than x which also gives Euclid's theorem. |
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May 14 |
awarded | ● Great Answer |
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May 14 |
answered | For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$? |
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May 9 |
awarded | ● Enlightened |
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May 9 |
accepted | Compactness in Sobolev spaces |
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May 6 |
comment |
Relating two characterizations of ${\mathfrak sl}_{n > 2}$ among simple Lie algebras Perhaps "perpendicular to" should be replaced by "perpendicular to all but"? |
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Apr 27 |
awarded | ● Nice Answer |
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Apr 23 |
accepted | How many distinct eigenvalues does a random graph have? |
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Apr 23 |
awarded | ● Nice Answer |
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Apr 23 |
answered | Compactness in Sobolev spaces |
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Apr 22 |
answered | How many distinct eigenvalues does a random graph have? |
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Apr 18 |
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Is rigour just a ritual that most mathematicians wish to get rid of if they could? I find it useful to distinguish between pre-rigorous thinking, rigorous thinking, and post-rigorous thinking (see my essay on this at terrytao.wordpress.com/career-advice/… ). It is desirable to transition from a rigorous mindset to a post-rigorous one, but it is not desirable to transition from a rigorous mindset to a pre-rigorous one. |
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Apr 18 |
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$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$ In mathoverflow.net/questions/105438 the explicit example of $f(x) := \sin^2(1/x) e^{-1/x} + e^{-2/x}$ (for $x>0$) and $f(x) := 0$ (for $x \leq 0$) is given for a smooth function vanishing to infinite order at the origin, such that the square root of $f$ fails to be smooth at the origin. |
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Apr 18 |
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$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$ For what it's worth, I have a writeup of the Joris argument at math.ucla.edu/~tao/preprints/Expository/… |
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Apr 16 |
awarded | ● Favorite Question |
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Apr 15 |
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Does the derivative of log have a Dirac delta term? Yes, there is a d/dx missing on the left-hand side, thanks for the correction! |
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Apr 15 |
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Does the derivative of log have a Dirac delta term? I would write Dirac's formula as $\log(x+i0^+)= p.v.\frac{1}{x}−i\pi \delta(x)$, and this is now a perfectly rigorous equation in the space of distributions (using whatever branch of the logarithm one wishes which does not cut ${\bf R}+i0^+$), indeed it is just the Plemelj formula given in the answers below, written in distributional form. |
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Apr 14 |
awarded | ● Good Question |
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Apr 13 |
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Intuition behind the spectral density of random matrices If M is upper-triangular, one can write $\sum_{j=1}^n |\lambda_j|^2$ as $\hbox{tr}(M \overline{M})$. For the general case, one can use the Schur decomposition, en.wikipedia.org/wiki/Schur_decomposition |
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Apr 13 |
awarded | ● Nice Answer |
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Apr 12 |
awarded | ● Popular Question |
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Apr 11 |
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At what point would an elementary generalization of Bertrand’s Postulate be interesting? Ah, I found the reference now, in this survey of Diamond: projecteuclid.org/… . In section 9 he discusses how, for each k, there is an elementary proof of Chebyshev type of a prime between $kx$ and $(k+1)x$ for large enough x. Unfortunately, the proof that the elementary proof exists (!) itself depends on the PNT! |
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Apr 11 |
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At what point would an elementary generalization of Bertrand’s Postulate be interesting? From the explicit formula linking primes and zeroes, the above assertion for a given $k$ is morally equivalent to the absence of a zero on the line $\{ Re(s)=1 \}$ of imaginary part $O(k)$. I vaguely recall reading some discussion in which this equivalence could be made more precise, in that such a zero-free region could be converted to an elementary Ramanujan-style result (somewhat analogously to how the non-vanishing of $L(1,\chi)$ for all $\chi$ of period $q$ can be converted to an elementary proof of Dirichlet's theorem mod $q$) but I don't remember the details. |
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Apr 11 |
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At what point would an elementary generalization of Bertrand’s Postulate be interesting? From the Erdos-Selberg argument, it is not too difficult to see (basically by iterating the Selberg symmetry formula) that if one can get $\gg_k x/\log x$ primes between $kx$ and $(k+1)x$ for all $k$ and all sufficiently large $x$, then the prime number theorem follows from elementary means (of course, this is a somewhat vacuous statement since the entire Erdos-Selberg proof of PNT is already considered elementary, but the derivation here is simpler than that of full Erdos-Selberg). |
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Apr 10 |
revised |
Intuition behind the spectral density of random matrices added 22 characters in body; edited body |
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Apr 10 |
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Iterative argument for ‘tempered’ NLS / NLW I would look at the H^1 argument for energy-critical NLS in these notes of Killip and Visan: math.ucla.edu/~visan/ClayLectureNotes.pdf . For s>1 the later paper arxiv.org/abs/0812.2084 of these authors has a good discussion (but there is a limit as to how large s can get when one is in very high dimension). For NLW there is a bit more room with respect to the derivatives and the H^1 theory should be fairly straightforward, though again the higher H^s theory can get very difficult in high dimensino. |
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Apr 10 |
accepted | Intuition behind the spectral density of random matrices |
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Apr 10 |
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Intuition behind the spectral density of random matrices Corrected, thanks! |
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Apr 10 |
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Intuition behind the spectral density of random matrices edited body; added 12 characters in body; added 142 characters in body |
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Apr 10 |
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Intuition behind the spectral density of random matrices added 150 characters in body |
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Apr 10 |
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Intuition behind the spectral density of random matrices added 121 characters in body |
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Apr 10 |
answered | Intuition behind the spectral density of random matrices |
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Apr 8 |
awarded | ● Nice Answer |
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Apr 4 |
answered | Ordinary Generating Function for Mobius |
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Apr 1 |
awarded | ● Good Answer |
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Apr 1 |
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Does this matrix shape have a name? This doesn't directly answer your question, but many types of combinatorial designs (en.wikipedia.org/wiki/Combinatorial_design) come with adjacency matrices $A$ whose square $A^2$ (or $AA^T$ or $A^T A$, depending on the design) are of the required form. |
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Mar 26 |
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The “interplay” between additive and multiplicative structure in a field Strictly speaking, Jean, Nets, and myself only established the above bound in the regime $p^\delta < |A| < p^{1-\delta}$. The extension to the full range $|A| < p^{1-\delta}$ was done shortly afterwards by Bourgain, Glibichuk, and Konyagin. |
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Mar 25 |
awarded | ● Popular Question |
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Mar 23 |
awarded | ● Necromancer |
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Mar 22 |
revised |
Good uses of Siegel zeros? added 14 characters in body; added 112 characters in body |
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Mar 22 |
answered | Good uses of Siegel zeros? |
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Mar 22 |
awarded | ● ca.analysis-and-odes |
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Mar 21 |
revised |
Applications of Rademacher’s Theorem added 46 characters in body |
