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Jan
7
comment A quantitative Kronecker theorem
Done. Feel free to ask questions if any step gives you trouble :-)
Jan
7
answered A quantitative Kronecker theorem
Jan
7
comment A quantitative Kronecker theorem
And what exactly is your difficulty with the standard proof via trigonometric sums? I tried to make a back of envelope computation and it seemed to work just fine but I surmise you've tried it too, so am I missing some subtlety?
Jan
4
comment Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals
@kodlu What do you mean? Each number between $1$ and $n$ can be represented in only one way as $ab$. And I have never said anything about the sum $\sum_a a^{-1}\log a$. In fact, I find the whole idea to split the logarithm $\log(n/a)$ into the difference $\log n-\log a$ rather misleading...
Jan
3
comment Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals
@codlu Or, if you ask where that came from, I just summed over $b$ first on the left and summed up all reciprocals of the numbers between $1$ and $n$ on the right.
Jan
3
comment Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals
@kodlu It is not the difference but the ratio that matters: as long as $a\le a_0$, the extra factor $\log(n/a)$ is at least $\sqrt{\log n}$ and even if you add the inverses of all numbers between $a_0$ and $n$, you still have at most $\sqrt{\log n}$ total.
Jan
2
answered Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals
Jan
2
comment Finite blowup time for a simple ODE
Well, the existence is obvious: for large $C$ the blow-up occurs before $1$, for small $C$ there is no blow-up, there is the trivial monotonicity, and the set of $C$ for which there is no blow-up on the closed interval $[0,T]$ is open for every fixed $T>0$.
Jan
2
comment Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$
"I don't know yet about 13×13×7 and 13×13×13" Any update after 2 months of "running the programs"? :-)
Jan
2
comment Finite blowup time for a simple ODE
To characterize in what terms? I hope you don't expect anyone to provide an explicit elementary formula $C=F(y(0))$ for the least possible $C$..
Dec
25
awarded  Good Answer
Dec
25
awarded  Good Answer
Dec
22
awarded  Enlightened
Dec
22
awarded  Nice Answer
Dec
22
comment Deceptively simple inequality involving expectations of products of functions of just one variable
I guess you meant $E[X^aY^a(X-Y)(\log X-\log Y)]$.
Dec
22
comment Deceptively simple inequality involving expectations of products of functions of just one variable
Erm... Are you really sure that $(**)$ is the same as $(*)$? My attempt to open the parentheses resulted in something quite different.
Dec
22
awarded  Nice Answer
Dec
22
awarded  Nice Answer
Dec
22
comment On the positive definiteness of RBF kernel with DTW distance
It all does not matter: the absence of the triangle inequality implies the absence of the positive definiteness of $K$ for sufficiently large $\sigma$ (or sufficiently small $d$) just because if $c>a+b$, then $a^2b(a+b)+b^2a(a+b)-c^2ab<0$. I hope that this remark is no more cryptic than the original post ;-)
Dec
22
answered Deceptively simple inequality involving expectations of products of functions of just one variable