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comment Necessary condition for decouplings for surfaces in $\mathbb{R}^4$
Evaluate (or more precisely, ascertain the order of magnitude of) $Ef$ and $E_\Delta f$ at a small ball around the origin (of radius $c$ for some small absolute constant $c>0$.)
Feb
3
comment Why are quantum groups so called?
Interesting how the meaning of "quantum" has now dramatically evolved twice, from "discretised into quanta" to "relating to quantum mechanics" and finally to "being less commutative than before".
Feb
3
answered Is there an alternate name for the symplectic convolution?
Feb
3
comment Is there an alternate name for the symplectic convolution?
In the harmonic analysis literature this is known as "twisted convolution", and can be viewed as a component of convolution on the Heisenberg group. The last few chapters of Stein's "Harmonic analysis" has some discussion of this operator, if I recall correctly. One may also wish to check Folland's "Harmonic analysis on phase space".
Jan
30
answered Decoupling in mixed norm spaces
Jan
29
awarded  Notable Question
Jan
28
comment Pointwise convergence of Fourier series, Fefferman's article
In the case that $\omega$ is centred at the origin, one can simply test $T_p$ with the function $f = 1_I$, the point being that the phase $e^{in(x) y}$ is close to 1. The general case is a frequency modulation of this general case, so one needs to multiply $f$ by an appropriate phase. Also, the double $I^*$ of an interval $I$ usually refers to the interval with the same centre as $I$, but twice the length.
Jan
27
awarded  Nice Answer
Jan
20
awarded  Good Answer
Jan
13
comment Does quantum mechanics ever really quantize classical mechanics?
Sure, you could do this if you want to see classical probability emerge as a semiclassical limit of quantum probability. But I was speaking to what I believed was the point of your previous remark, in that the presence of superpositions in one's mechanics is not incompatible with observers viewing the world classically, since probabilistic classical mechanics already exists as a perfectly consistent theory. (This is not to say that the measurement problem in quantum mechanics is trivial to solve, but it isn't a priori impossible to resolve either.)
Jan
13
awarded  Enlightened
Jan
13
awarded  Nice Answer
Jan
11
comment Does quantum mechanics ever really quantize classical mechanics?
Well, in any of the above four models of mechanics, the macroscopic observers will never "see" any superpositions directly; they are only visible from a "God's eye perspective" external to the mechanics. An analogy is with probability theory: one can model the universe probabilistically, but an observer within the universe can never actually "see" any random variables or probability distributions, as such observers only can access one of the multiple outcomes in the sample space. The distributions are only visible externally (or approximately visible by empirical sampling).
Jan
11
revised Does quantum mechanics ever really quantize classical mechanics?
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Jan
10
revised Euler product for sum of multiplicative function times log
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Jan
10
revised Euler product for sum of multiplicative function times log
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Jan
10
revised Euler product for sum of multiplicative function times log
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Jan
10
answered Euler product for sum of multiplicative function times log
Jan
10
revised How do estimates on $N_\chi(\alpha,T)$ lead to the Dirichlet prime number theorem for arithmetic sequences?
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Jan
10
awarded  Necromancer