@PedroTamaroff Well, first I'm not sure I understand what wrt stands for (I probably know, but can't figure it out just by looking at it right now). Second, that would be 2f(x), I think.
so I'm thinking about writing something for the blog, and I'm thinking of writing on evidence of Zipf's Law in MSE: which basically says that the terms used happen to follow a power law in distribution. Does that sound interesting?
@PedroTamaroff so, MSE has a blog, and I'm thinking of writing something for it.
There's this heuristic law called "Zipf's Law". Let's say that in MSE, the first most common word is "the", the second is "is", the third is "math", and so on
@PsychOPhobiA well, unless mixedmath is a lot more patient than I (entirely possible) someone here isn't likely to be willing to teach you all of calculus...
@mixedmath ah, I'm really only motivating the theorem rather than proving it (not possible, given length and background constraints), so I suppose I'll refrain from stating it for now
Find equation of a line tangent to the curve at the given point:
determine an equation of a line tangent to the curve at the given point: 9($x^2$ + $y^2$)$^2$ = 100x$y^2$; (1, 3)
How would I go about solving this. I know it requires implicit differentiation.
I have been having a lot of trouble...
@MikeMiller Interesting thing: if $G$ is a finite group and we form $k[G]$, the center of the algebra has dimension $[G:Z(G)]$, that number of conjugacy classes in $G$.
@DemCodeLines That's not the point. Someone can give you a nice colorful uppity answer, but you need to read your notes and learn the rules, and use the and reuse them till they are easy to remember and come off naturally.
@MikeMiller In fact a sum $\sum \alpha_x x$ is central if the $\alpha_x$ are constant over the conjugacy classes. That's basically the important observation.
@PedroTamaroff Yes, I understand that, but I am stuck on the last question and having to spend an hour or two mastering the concept before I can finish it is a little tedious. Plus, it's a little late in the night here too.
Not that I won't learn, I don't feel like failing tests, so i have to learn eventually
@PsychOPhobiA If you are trying to "learn calculus" for the first time, I might be so bold as to recommend a post I wrote for some of my students once on calculus at an introductory but math-literate level
@DemCodeLines anyway, your homework grade is probably miniscule, as compared to the effect of the tests. it's very important that you do the homework in order to pass the course, but the reason it's important is that it helps you understand the concepts. focus more on the reason you do things in your computations rather than the computations themselves.
From my experience in other chat rooms, the people in them usually talk about everything but the topic that the room was created for. I'm guessing the trend applies to this room too.
Prove that $\Bbb Q$ has no (nontrivial) subgroup of finite index.
That is, seen as a additive group.
In fact, prove that it does have nontrivial subgroups of finite index when seen as a multiplicative group.
Hint $\Bbb Q^\times \simeq (\Bbb Z/2\Bbb Z)\oplus \bigoplus\limits_{i\geqslant 1}\Bbb Z$
Hint $\Bbb Q^+$ is divisible.
user105491
@PedroTamaroff Let me prove the first one first. Suppose H is a subgroup of Q. Then, [Q:H]=n=|Q/H|. For every q\in Q, n(q+H)=H. Hence, nq+H=H, i.e., nq\in H. Thus, nQ\subseteq H. However, nQ=Q, hence Q=H.
@PedroTamaroff Probably not right now; it's almost 2200 here, and it's a weekday. I've got school tomorrow, so I hope it's OK if I chat tomorrow (possibly with a proof).
@PedroTamaroff Thanks for the hint; I'll try proving it tomorrow at school. Thanks for the enlightening conversation, but it's time for me to say goodbye. Until tomorrow, my fellow mathematician!
@MikeMiller yeah. I read Terry Tao, Tim Gowers, Jordan Ellenberg, the AMS grad student blog, and sometimes click randomly on Terry Tao's blogroll for posts
Gowers' nontechnical writing is great, but his technical stuff is always a bit too dense for me (technically self-contained, but clearly not meant for the combinatorial neophyte)
@MikeMiller I remember he wrote on Taylor's Theorem once, and I felt very surprised at how hard it was for me to really read something that I understand so well
it's not that I don't get it - I do - but the presentation is just so different than how I present it, and different than how I think of it
I once read someone on this site (perhaps Georges E.?) who said that the best sign of understanding something is that everyone else's presentation of it seems wrong to you
Man I clearly didn't properly learn Taylor's theorem! The only statement I remember is that an analytic function is locally expressed by its Taylor series
Scott Carnahan appears to have posted something a couple weeks ago I should read
I learned fairly recently (couple months ago) that a great way to learn some new math is to go to MO once a week and click the week tab - because of the number of upvotes, these will generally be applicable to/understandable by a wide audience, and usually will be really cool
@MikeMiller oh, that sounds like a good plan. I keep on intending to read more MO. I also feel like I should be spending more time asking/answering on MO, as I'm trying to become a research mathematician
usually I don't get a whole lot out of my weekly checks except for a few interesting new things (since so much I don't understand), but those new things are often great
of course, as a wee child, I have more time to check stuff like that
There is a new tag called precalculus. What is the purpose of this tag? Should it be synonym of algebra-precalculus? (AFAIK precalculus is an American term, so maybe some comments on this familiar with American education system could be useful.)
@MikeMiller I work in analytic number theory on automorphic forms, Maass forms, and mutliple Dirichlet series usually; recently, I happen to be working on explicit asymptotics arising from number theory in function fields
but my dissertation (assuming it goes along in the same direction it's currently going), is on GL(3) automorphic forms
and I'd be happy to explain anything more - just point me where
these are functions that have zeta functions (or L-functions) as data in some way, like as coefficients or as complex residues at poles. I study bounds on these functions, which translate to bounds on the data
if we were super duper and understood all powers of the bound I'm looking at, we'd have the Riemann Hypothesis and the Lindelhof Hypothesis
@robjohn it's interesting to see that Mma goes crazy when computing that series. Just look at the result of Sum[(Zeta[2] - HarmonicNumber[n, 2]) (Zeta[3] - HarmonicNumber[n, 3]), {n, 1, Infinity}]
Numerically it's about $-0.128523$. However, the partial sums are correctly computed.
@Chris'ssis To compute your series, use the Hurwitz Zeta function. Not that it can compute your series in closed form, but it gets the numerical value quickly.
@Chris'ssis This is weird... I think the plot should be smoother... DiscretePlot[n^2(Zeta[3]-HarmonicNumber[n, 3]), {n,1000,100000,1000}, ImageSize -> 1000]
@DanielFischer, Sir will you please help me with Homothetic functions?
I find several definitions, and some say that " A homothetic function is a monotonic transformation of a function which is homogeneous of degree 1. " while others do not require that "degree 1" restriction.
@DanielFischer, how does $\displaystyle \sum_k \binom r k x^k \sum_{n-k} \binom s {n - k} x^{n - k}=\displaystyle \sum_n \left({\sum_k \binom r k \binom s {n-k} }\right) x^n$ hold?
Find equation of a line tangent to the curve at the given point:
determine an equation of a line tangent to the curve at the given point: 9($x^2$ + $y^2$)$^2$ = 100x$y^2$; (1, 3)
How would I go about solving this. I know it requires implicit differentiation.
I have been having a lot of trouble...
