Mark Lewko
Reputation
6,896
Next privilege 10,000 Rep.
Access moderator tools
Badges
1 27 45
Newest
Impact
~123k people reached

• 0 helpful flags
• 242 votes cast

# 428 Actions

 Apr 26 comment problems from the scottish book Dan Mauldin edited a revised edition of the Scottish Book with updated commentary on many problems which was published last fall (2015). See: amazon.com/Scottish-Book-Mathematics-Selected-Problems/dp/… Apr 22 revised Two questions on Elias Stein paper (1976) added 10 characters in body Apr 22 revised Two questions on Elias Stein paper (1976) added 954 characters in body Apr 21 answered Two questions on Elias Stein paper (1976) Apr 10 revised Brownian motion, quadratic variation, existence of partitions? added 4 characters in body Apr 10 comment Wrongful conviction Bayesian argument in need of integral-solving talent I would love to hear how estimating this integral implies that "a certain piece of forensic evidence has no inculpatory value"... Mar 6 comment Is the number of representations as the sum of two elements of a polynomial sequence always small? I'm not sure. I thought about that briefly but didn't get too far. This quickly leads to questions outside of my expertise. Mar 6 comment Is the number of representations as the sum of two elements of a polynomial sequence always small? I had also observed this after posting the question. It certainly seems that this can't be generalized in any straightforward way to sums since there are polynomials such that f(x)+f(y) does not have linear factors. Mar 3 comment Is the number of representations as the sum of two elements of a polynomial sequence always small? I'm curious if an improvement on the divisor bound is known for, say, $f(x)=x^5$? Mar 3 comment Is the number of representations as the sum of two elements of a polynomial sequence always small? @GH: if you can prove that, I'll accept it as an answer :) Mar 3 comment The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$? Thus, recalling the definition of convolution, $f(x)$ is upper-bounded by essentially the average of $f(x)$ over a translate of $B_{0}$ centered at $x$. In just about any harmonic analysis estimate this observation leads to the same conclusion that would be obtained if $f$ was in fact a step function constant on a latices of boxes of the shape $B_{0}$. Mar 3 comment The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$? Now what is formally true is that if $\hat{\eta}$ is a a smooth bump function equal to $1$ on $B$ and rapidly decaying away form B, then $f(x) = f * \eta$. In addition, it isn't hard to see that $\eta$ is now a function that is nearly constant on the $b_{0}$ and decays rapidly away from it. By nearly constant we mean bounded above and below by universal constants. Mar 3 comment The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$? The idea in all of these "essentially constant" arguments (commonly referred to as "the uncertainty principle," as popularized by Tao) is as follows: Assume that $f$ is a function on $R^d$ such that $\hat{f}$ is supported on a box B. Then the essentially (but not formally) true claim is that $f$ is essentially constant on translations of the polar of $B$, say $B_{0}$. In the $1$-d case if $B$ is an interval of length $L$ then $B_{0}$ is an interval of length $1/L$. Mar 2 awarded Nice Question Mar 2 comment Is the number of representations as the sum of two elements of a polynomial sequence always small? @Bobby, yes the constant can depend on $f$. Mar 2 revised Is the number of representations as the sum of two elements of a polynomial sequence always small? deleted 13 characters in body Mar 2 revised Is the number of representations as the sum of two elements of a polynomial sequence always small? deleted 4 characters in body Mar 2 revised Is the number of representations as the sum of two elements of a polynomial sequence always small? deleted 13 characters in body Mar 2 revised Is the number of representations as the sum of two elements of a polynomial sequence always small? added 18 characters in body Mar 2 revised Is the number of representations as the sum of two elements of a polynomial sequence always small? edited tags