is there someone who is willing to take me as a minor collaborator for solving some problem that is not too advanced and preferably belongs to number theory?
@Jonas: whoa not that kind of advanced! I have a reasonable grounding in elementary number theory, statistics and a very rudimentary smattering of analysis.
do we have some nice results regarding distribution of quadratic and higher power residues modulo a given n? I did some google search, but mostly they turned out to be analytic results giving some lower/upper limit on least quadratic residue/nonresidue. Terence Tao has recently asked a similar question in MO I guess. My question is whether we have some exact results? are we in some sense able to determine the precise location of higher power residues?
The resides are scattered everywhere (and in some sense, they are random or are thought to be random, I think). Even to point out which ones are (quadratic) residues, I will need to give you a list of around $n/2$ numbers.
Ok, fine. What about higher powers? I guess the situation should improve (not monotonically, of course) with increasing power since the no. of distinct residues modulo n for a given power a are phi(n)/gcd(n,a); i.e. no. of residues decreases; so the pattern should somehow be not random?
Anyone willing to only suggest a problem for me to work on? it should be preferably related to number theory, and not very advanced-i.e. algebra and analysis requirements should not be much if at all while some statistics is ok?
@JonasTeuwen Some results on normal numbers can be shown using Strong law of large numbers. I guess probability results might be useful in study of number theoretic densities. Probabilistic number theory. I would call it using probability, but perhaps the line between probability theory and statistics is not that clear...
The solution given in this problem could be called statistics, even though the problem is finding sum of a series: sosmath.com/CBB/viewtopic.php?t=28258
The problem is to find $\lim\limits_{n\to\infty}e^{-n}\sum_{k=0}^n\frac{n^k}{k!}$.
Well: I know how to write n-k. The only thing I do not know how to do is that upside down T. I'll try ^\perp - it does not work, since in needs something in front of it. So I'll write $U^\perp$.
I thought that catcode is that thing that says whether it is relation, operation etc. (and thus it defines spacing in math mode), but I remembered it wrong: en.wikibooks.org/wiki/TeX/catcode
I think that \makeatletter is equivalent to changing catcode...
@MartinSleziak Thanks. Strange; I see such a thing in some of the comments.
What's the policy on Project Euler problems? I know about Jonas M.'s meta post (and his exchange with Project Euler on posting their problems here), but was some course of action agreed upon?
When I view transcipt without applying bookmarklet, it's without scrollbar, it gets there only after I use it. So probably in the main chat people who do not have bookmarklet don't get that scrollbar either.
(Some silliness.) I once had a Thermodynamics textbook. It had all solutions at the back, except for every third problem, for which it mentioned only the answer and not the full solution. All the problems for which the solutions were available -- they were quite routine. But it turns out that each of the skipped problems was a monster. I used to struggle for hours to find some approach to the problem. And at the end of it, my answer will invariably be wrong. =)
It's strange thing that in Russian you write Физика (fizika), and other languages have the letter Y there. But maybe it's because it would change the pronunciation.
I don't follow you at all. I do not know the proof off the top of my head. Without seeing the proof, I cannot of course tell if it uses other theorems that aren't proved.
Sorry, All I meant was that in this complex analysis stuff it seems like I am meant to take a lot of things for granted and the proofs of most important are often omitted because they are 'out of scope'
basically would like to find a development of complex analysis that proves every result and doesn't skip anything. I don't need exercises or examples or anything like that (the sort of things which would go well in an undergrad class).
I think the books and notes I find are targeted at the wrong audience
@Srivatsan If ever you wondered why the Springer symbol looks the way it looks: Springer (literally: "jumper") is the German name for the chess piece knight.
@Srivatsan Yes, it's an abbreviation. No it's not referring to anyone specific. The story is this: A friend of mine invited a guy who is somewhat nuts to a dinner. That guy asked can I bring someone? My friend answered, yes, sure. The guy wrote on a piece of paper "ask s.o.". They talked a few minutes more and then the guy crossed out "ask s.o." and wrote "ask s.o.e." instead...
@tb It was never explained to me. But then again, I don't think I ever grasped that subtlety in the definition of Lebesgue-measurable when I was studying measure theory...
@Matt It doesn't make sense to speak of the completion of a $\sigma$-algebra without a measure (or measure class) present. The completion of the Borel $\sigma$-algebra with respect to a point measure is the power set.
The 5-adjoint characterisation of Set is quite interesting. I had seen it mentioned on the Categories list recently, but I didn't bother looking it up.
@ZhenLin I looked it up a few minutes ago but I decided that I would read it when I'm more in a categorical mood.
The morphisms in an abelian category example reminded me of my favorite homological algebra exercise: Compute the right derived functor of $\ker: \mathcal{A}^\to \to \mathcal{A}$.
Having been told that the third adjoint is the (co)kernel, I'm at a loss as to what the intermediate ones might be...
Also, a remark about using $\mathscr{P}\mathbf{C}$ to denote the presheaf category $[\mathbf{C}^\textrm{op}, \textbf{Set}]$: the resemblance power sets goes deeper than notation!
Marcelo Fiore pointed out to me that the presheaf category is a generalisation of the power set.
(Hint: Recall that a 0-category is a set, and a (-1)-category is a truth value...)
Interesting: `No jsMath TeX fonts found -- using image fonts instead. These may be slow and might not print well. Use the jsMath control panel to get additional information.`
@tb Thanks :). I don't have specific questions at the moment, but I've decided my math skills are deplorable for where they could be. I understand a lot of it, I just can't put it into practice easily.
I'm taking on some of the project euler problems, right now they're basic sums of arithmetic series, but as I move further down I don't want to rely on me using my programming skills when there's an elegant solution in math.
For instance, not brute-force run through all the numbers 1..999 when I could do (1+999)*999/2. I know about this solution, but as I move on it will be less obvious to me.
@Srivatsan the easiest way to do it is to go to the user page and do the search from there. The user:xxxx is then automatically added. Otherwise you need to remember the number xxxx of the user
@robjohn Good point. See David's comment. Bill has explained in some posts that the numerator does not matter. [This variation of L'Hospital isn't that well-known apparently.]
Q: While covering a distance of 24 Kilometers , a man noticed that after walking for 1 hour and 40 minutes , the distance covered by him was 5/7 of the remaining distance. What was his speed?
@robjohn Well, as I said earlier, I do not fully remember the constraints of the theorem, but I feel this solution might be correct after all. Let me find the relevant BD's posts and then we'll see if the solution can be redeemed. =)
@FreakEnum It sounds to me as if he has already covered 24 km, but I can see that it might also be thought that the distance travelled is 5/12 of 24 km
@tb Bill and I never meshed on sci.math. He never liked my answers and would link to lots of his posts as references.
@FreakEnum yes, so he has travelled 5/12 of the 24 km, so the distance he has left to go is 7/12 of the 24 km, that means that the distance he has travelled is (5/12)/(7/12) or 5/7 of the remaining distance.
@AsafKaragila If I want an injection $$ f: fin (\omega) \hookrightarrow \omega$$ what do you think of $$ \{ a_1 , \dots , a_n \} \mapsto p_1^{a_1} \dots p_n^{a_n}$$ for $p_i$ prime?
@robjohn Nice, thank you! Btw: the other day when I asked the question about a sequence that converges to zero and is not in any $\ell^p$ for $0< p < \infty$, do you know a better example? I can't do $\frac{1}{\log n}$ in my head so I was wondering if there was something simpler.
