# Sanchez

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I really appreciate it when you take time to answer my questions. Thanks!

 Nov30 comment Measure theory inquiry Your condition works. As for the Radon-Nikodym derivative, you need to figure out a function $f$ such that $\nu = \int_{[0,1]} f d \mu$, i.e. $m(X \cap B) = \int_{X \cap A} f$. Can you think about a natural $f$ that does just that? (Hint: some characteristic function would work.) Nov29 comment What do we know about the class group of cyclotomic fields over $\mathbb{Q}$? Ah, Kronecker-Weber, good point! If that's the case, is there any way to characterize the set of primes $p$ that splits completely in $H$, using only arithmetic data of the ground field? Nov28 comment A problem involving the Hahn-Banach Theorem 2. $\|T\|$ is the maximum of $|Tf|$ for $\|f\|_B \leq 1$. If $\|f\|_{B/S} \leq \|f\|_B$, it means that there are more $f$ with $B/S$-norm less than 1, which gives the result. Nov28 comment A problem involving the Hahn-Banach Theorem 1. It's just a typo: the map is $B \to B/S$, the standard quotient map. Nov28 comment proof regarding zeta function of a curve from Ireland and Rosen's “A Classical Introduction to Modern Number Theory” A point $[x,y,t]$ on $C$ in $\mathbb{P}^2$ with $t \neq 0$ corresponds bijectively to a solution $(x/t,y/t)$ to your given equation. There are 2 points on $C$ in $\mathbb{P}^2$ when $t = 0$. Nov26 comment Bound on $|f(x)|^2 + |f'(x)|^2$ @ParamanandSingh, was that used in robjohn's solution? As long as the function reaches its maximum, the derivative would be 0 at that point. Nov26 comment Why does this integral resolves to a delta function It depends on what your definitions are. Strictly speaking left hand side does not converge. However, you can interpret LHS by taking a uniform limit, or as distributions. In both interpretations, you would get the right hand side. Nov25 comment On conjugacy class size of finite groups. So are you saying that there is one conjugacy class of size 1, one conjugacy class of size 2, and so forth? If so, then because the size of conjugacy class must divide the order of the group, we have $n-1 | n(n+1)/2$. But $(n,n-1) = 1$, and $(n+1,n-1)$ is 1 or 2, this is impossible unless $n-1 = 1$ or 2. Nov25 comment Problem with tau function It's more conventional to write it as $\tau(n) = O(n^{\epsilon})$ for any $\epsilon > 0$. What have you tried? Nov25 comment Hartshorne Theorem III.5.2 (finite generation of cohomology for coherent sheaves on projective schemes over a noetherian ring) Your induction hypothesis shouldn't be on a specific $\mathcal{F}$, but all coherent sheaves at the same time. If you assume that $H^j$ is finitely generated for all $j > i$ for all coherent sheaves (in particular $\mathcal{F}$ and $\mathcal{R}$), then you can say something about $H^i(X,\mathcal{F})$, assuming that you already know the cohomology of $H^i(X,\mathcal{E})$, which amounts to cohomology of $O(n)$ on projective spaces. Nov25 comment For any real numbers $a,b,c$ show that $\displaystyle \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$ @some1.new4u, it's quadratic in $t$, completing squares would work. Nov25 comment proving inequality involving lebesgue measurable function on [0,1] Do you know Cauchy-Schwarz inequality? Nov20 comment Lower Limit Prime Gap The Hardy-Littlewood conjectures predict that there are infinitely many twin primes. If the twin prime conjectures fail, it means that Cramer's model (together with the modification of leading term by local consideration) fails, and we probably need to model primes in other ways. Nov19 comment Invertible sheaves Hint: For any $P$, pick a uniformizer $z_P$ at $P$, and a small enough neighborhood $U_P$ around $P$ so that $z_P$ is holomorphic on $U_P$. What does $L_D|{U_P}$ look like? Nov18 comment Finite generation of Tate cohomology groups @Jared, (just dropping in here) In that case they are equivalent. Nov18 comment Number of fixed points of automorphism on Riemann Surface Your $f$ is holomorphic away from $P$, which means you know quite well where $h$ has poles, and you have a bound on the degree of poles at $P$ and $F^{-1}(P)$. On the other hand, a fixed point of $F$ gives a zero of $h$, and degree of $Div(h)$ is 0. Can you proceed from here? Nov17 comment Order of subgroup on elliptic curve over $Z_p$ @JyrkiLahtonen, done :) Nov16 comment $\mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$? not exactly - $p = w^n u$ for some unit $u$. I'm not sure why you need that though. Nov16 comment $\mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$? $\mathcal{O}_K/(\omega^n)$ should look like polynomials in $\omega$ wih degree less than $n$. Can you then write down an explicit isomorphism with $\mathbb{F}_p[T]/(T^n)$? Nov16 comment Equicontinuity and Uniform Boundedness In the one-variable case, you can use mean value theorem to proceed. Can you see how that generalizes?