Robert Israel
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397/400 score
 Apr 28 comment Continuity, algebraic and rational numbers I think you misunderstand what I had in mind, but it'll be better to edit the answer rather than trying to clarify with further comments. I don't have time right now, but maybe later tonight... Apr 27 comment Orthonormal basis for Hilbert space Hint: $$\left\|\sum_{j=n+1}^m b_j/j\right\|^2 = \sum_{j=n+1}^m 1/j^2 < 1/n$$ Apr 27 comment How do I know the probability for me to be ranked in the 2nd place or last 2nd place? You don't know them, and can't know them unless you assume a particular probability model for the scores. Apr 27 answered Orthonormal basis for Hilbert space Apr 27 comment Continuity, algebraic and rational numbers If $f(w) = z \in \mathbb Q[i]$ with multiplicity $m$, take $r > 0$ such that $w$ is the only zero of $f - z$ within distance $r$ of $w$. By Rouché's theorem there exists $N$ such that for $n > N$, $F_n - z$ has exactly $m$ zeros (counted by multiplicity) within distance $r$ of $w$. But by construction, such zeros (if $n$ is large enough) are in $\mathbb A$ and will also be zeros of $F_k - z$ for all $k > n$, and therefore of $f - z$: thus they must be $w$, and $w \in \mathbb A$. Apr 27 answered Example of a bounded space which is not totally bounded Apr 27 answered Minimum and maximum with lagrange multiplier Apr 27 comment Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$ If you write $1/z_n = \pi/2 + k \pi + s$, $\cos(1/z_n) = (-1)^{k+1} \sin(s)$, and this will be $1/t$ if $s = (-1)^{k+1} \arcsin(1/t)$. As $n \to \infty$ you have $t = (n+1/2) \pi \to \infty$ so $1/t \to 0$ and $s \to 0$, i.e. $1/z_n \to \pi/2 + k \pi$ and $z_n \to 1/(\pi/2 + k\pi)$. Apr 27 comment Continuity, algebraic and rational numbers I think we can still have $f^{-1}(\mathbb Q[i]) = \mathbb A$ by modifying the construction, ensuring at stage $j$ that any $w$ with $|w| \le j$ for which $F_j(w)$ is of the first $j$ members of $\mathbb Q[i]$ (in a fixed enumeration) is in $\mathbb A$, and making $f_k(w) = 0$ for $k > j$. Apr 27 revised Continuity, algebraic and rational numbers added 1780 characters in body Apr 27 comment Continuity, algebraic and rational numbers Bijective on $\mathbb R$, I hope you mean. There aren't too many entire functions bijective on $\mathbb C$. Apr 27 answered Continuity, algebraic and rational numbers Apr 27 comment Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$ $\sin(t) = \pm 1$ if $t$ is an odd multiple of $\pi/2$, say $(n+1/2) \pi$. Then you want $t = 1/\cos(1/z_n)$, with $1/z_n$ near $\pi/2 + k \pi$... Apr 27 answered Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$ Apr 27 comment Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$? The ratio test works well here. Apr 27 comment Multiplying binomials to come up with $y^8 - 256$ Right to left may be easiest. Apr 27 answered Numerical solution of an ODE system of equations using RK4 Apr 27 answered Markov Chain that isn't Irreducible Apr 27 answered How to find an unknown matrix which when multiplied with a vector gives the cross product of 2 vectors Apr 27 awarded functional-equations