Does anyone know of some (preferably old so as to be kinder to me) papers that use modular forms quite a bit? I'm asking so as to become more proficient in their use
I'll restate the question, so on skimming through people don't think that you've answered it.
Does anyone know of some (preferably old so as to be kinder to me) papers that use modular forms quite a bit? I'm asking so as to become more proficient in their use.
You won't. There's some transformation stuff on double theta functions which relates a special sextic to Klein's quartic that we call Borchardt's transform.
It's mentioned at King's beyond the quartic. I think it's also in Elkies' lectures in eight-fold ways.
@barznjy I meant to reply earlier, sorry. They're the two linearly-independent solutions to a linear 2nd order DE. Most of the time M is analytic at the origin and U isn't; that's pretty common for those types of DEs in my experience.
@BalarkaSen I could never really tell the difference between the two. I have a blind spot for higher dimensions though.
@BalarkaSen Tate's thesis ties harmonic analysis with number theory, and the space of square-integrable functions on certain topological groups is often very important. Is that a good excuse?
@BalarkaSen I don't understand it either, I'm just throwing out examples. I'm certainly not an analyst, I dislike the taste. But nowadays, many different fields are inextricably linked to each other, but especially to number theory
I can't imagine any geometric interpretation of $x^2 + y^3 = z^7$, unlike the $5$ case which is a syzgy of the icosahedral invariants (fortunately i have studied that a little for the connection with quintics).
But if it's $X(7)$ then there must be Klein's quartic involved.
@BalarkaSen Not quite. I think you're thinking of the introductory algebraic topology stuff, but whenever you see the term 'covering', it's an analogue of the concept more applicable to algebraic geometry (the topology on varieties is no good for standard algebraic topology). And there are many, many different homology and cohomology theories.
@MikeMiller if you want to know about the $x^2 + y^3 = z^5$ case, in the case of an icosahedron inscribed inside a unit sphere, there are three main invariant polynomials left unchanged by the symmetries of the icosahedron, the vertex polynomial which vanishes at the vertices, edge polynomials which vanishes at the midpoint of the edges and face polynomials which vanished at the midpoint of the faces.
however, these polynomials satisfy some $X^2 + Y^3 + Z^5 = 0$ relation, which apparently resembles that generalized fermat equation. on the other hand a quintic can be algebraically transformed into a modular equation for $j$ at $n = 5$.
that is why I suspected that $x^2 + y^3 + z^5 = 0$ over $\Bbb CP^1$ might just be something like $X(5)$, which is confirmed by Elkies.
what's more, it turns out that $X(5)$ is a riemann sphere with 12 points removed, precisely the ones that form an icosahedron!
rather interesting, but most of it looks like coincidences to me. of course, they aren't.
oh, and i forgot to mention that quintics and icosahedron are intimately connected : the 5 tetrahedrons that can be canonically inscribed inside an icosahedron are representatives of the roots of a special quintic.
i don't know what kind of black magic that might be involved with that 7 case though.
question is, why is a $\sigma$-algebra defined using only countable unions and intersections?
the way i'm understanding it, the $\sigma$-algebra generated by an observer is the collection of events he can decide the truth and falsehood of.
there must be something that happens when we allow uncountable unions/intersections that is undesirable with respect to that interpretation, but i can't think of an example.
You're missing a bit of context, because what you're really asking about is the choice of $\sigma$-algebras over a different kind of structure without saying what the choice is for.
@AlexanderGruber Determine the value(s) of $h$ such that the matrix is augmented matrix of consistent linear system. $$\begin{bmatrix} 1 & h & 4\\ 3& 6 & 8 \end{bmatrix}$$
@usukidoll 'without assuming' in the sense you have to prove it first in order to derive your result with the aid of the theorem .. or is it in the sense .. do it without the theorem (ie independent solution) ?
so wait what.... so I have to prove the particular theorem first
wait wait... I don't have my number theory book with me... it's in the living room. I was going to attempt after dinner. It said something about without assuming theorem 2.1...something to do with positive integers.
oh wait I have the ebook version as well XD! Just realized it
studying it hai
Without assuming Theorem 2-1, prove that for each pair of integers j and k (k>0), there exists some integer q for which j-qk is positive
Theorem 2-1 (Euclid's Division Lemma) states that for any integers k (k>0) and j, there exist unique integers q and r such that $ 0 \leq r < k$ and j=qk+r
Has anyone ever read some book or seen some lecture where various integral forms have been denoted as $I_1$ to $I_{25}$ and special methods have been given to solve them?
Let x_n be a real number sequence and b = lim sup x_n. Let A_k = sup{x_n}_(n>=k) which ends up forming a sequence of supremums of the original sequence ( starting at gradually increasing indexes ) . What's the proof that for every e >0 , there exists w>=k such that b - e < x_w < A_k ?
@nerdy If you install bookmarklets from here, you would be able to use latex syntax in chat, which is more readable.
I suppose you do not have it installed, so you it makes no sense for me to write somethings like $b=\limsup x_n$ and $a\le b \le c$.
Anyway you should clarify your question a little bit.
You wrote b-e < x_w < A_k ($b-\varepsilon < x_w < A_k$).
What is k? (You prefaced this with "for each e>0 there exists w".)
Did you mean to say for each e>0 and for each k?
And I think that you should have non-strict inequality $x_w\le A_k$, otherwise it is not true. (Think about constant sequence.)
In any case, if you are given e>0 then there exist k0 such that for k>k0 you have A_k > b-e. (This is from the definition of limit.)
Since $A_k=\sup_{w\ge k} x_w$ you get existence of $w\ge k$ such that b-e<x_w. (From the definition of supremum you have that for any number $L<A_k$ there exists some element betwen L and $A_k$.)
I guess you should be able prove your claim (after clarifying what exactly you mean) by using arguments like that.
I'm on real analysis course at college we are seeing sequences ... i'm curious, is this whole deal of lim sup and lim inf used later ( in series, topology, functions, etc ) ?
I think it is useful to know about limsup, liminf. In many situation when you are trying to prove that some limit exists, you estimate limsup and limiinf separately.
I should post a question on main regarding a proof containing a being an odd integer that's divisible by 12 ... I know what the theorem is talking about. I just want to see if I'm on the right track
The older one is closed as a duplicate of a question of the type $|x-a|-|x-b|=a-b$. (I.e., difference instead of sum.) I am not sure that these two should be considered duplicates. (This remark is directed to @robjohn, who closed the question using mod-dupe-hammer.)
