a topology $\tau$ is a collection of subsets of $X$ closed under arbitrary unions and finite intersections. one can check that $\bigcap \tau_i$ is a topology if $\tau_i$ for all $i$ are. then, one can define the set $T$ of all topologies containing $\tau_1\cup\tau_2$ (the union need not itself be a topology); then $T$ is nonempty since it contains the discrete topology, and it has a minimum element (namely the intersection of all topologies in $T$)
@PeterTamaroff Sorry, didn't get your message. Modding is mostly easy stuff, but it makes it much harder to actually write answers on MSE when you get constant notifications to check. But things are rather peaceful.
@Chris'swisesister I found another solution: the limit is non-negative; then $$0\leqslant\lim_{n\to\infty}\frac{\sqrt[n^2]{(4^n+1)(4^n+2)\cdots (4^n+n)}}{2^n}\leqslant\lim_{n\to\infty}2^{-n}\prod^{n}_{k=1}\left(4^n+n\right)^{1/n^2}\\\leqslant\lim_{n\to\infty}2^{-n}\left(4^n+n\right)^{1/n}$$
@Chris'swisesister we now choose a constant $\xi$ such that $4^n + n\leqslant\xi^n$ for sufficiently large $n$. Then $$0\leqslant L\leqslant\lim_{n\to\infty}2^{-n}\left(4^n+n\right)^{1/n}\leqslant \lim_{n\to\infty}2^{-n}\left(\xi^{n}\right)^{1/n}=\lim_{n\to\infty}\frac{\xi}{2^n}=0.$$ Therefore, $L=0.$
@amWhy: I wanted to find a video of the bugs bunny toon where he is chased by the orange monster - in the beginning the mad scientist promises to reward the monster with goulash
In the end, there is one of my favorite moments in animation (if not film). The monster finally catches up to bugs, and bugs points at the screen (the viewer) and says "look! People!" and the monster runs away screaming
Hi, I have an off topic question. Are there a lot of infinite series which sum to 1? I'm looking for something other than 1/2 + 1/4 +1/8 +.. or any trivial sums which include a 1 and zeros only.
@TheSubstitute the number of summations which evaluate to a given limit is uncountable. they can be transformed into each other completely trivially: for any sum $\sum a_n=L$ when $L\ne 0$, just consider $\sum a_n/L$...
Say we have a path $\lambda$ which join $f$ with $g$. Then the candidate for homotopy is $G(x,t)=\lambda(t)(x)$. Now consider an open subset $B$ of $Y$. Pick $(x,t)$ in the inverse image of $B$ by $G$. One can use that $X$ is locally compact and Hausdorff to prove that there is a compact $x\in K^x$ such that $ t\in \lambda^{-1}(F(K^x,B))$ which is open. So it only remains to find an open in $X$, say $A$, such that $(x,t)\in A\times \lambda^{-1}(F(K^x,B))\subseteq G^{-1}(B)$
$$(r^5)^{-2}=\frac{1}{(r^5)^2}=\frac{1}{(r\cdot r\cdot r\cdot r\cdot r)^2}$$ $$=\frac{1}{(r\cdot r\cdot r\cdot r\cdot r)\cdot (r\cdot r\cdot r\cdot r\cdot r)}=\frac{1}{r^{10}}=r^{-10}$$ etc. Never forget the meaning of multiplication.
Awe, @anon if you were here I'd so pay you to tutor me! I'm taking the course by distance and I live in a small town... There are no math tutors here to my knowledge. :(
@BandeiraGustavo Yep, it helps a ton (understatement)! I'm actually trying to learn it not copy off of something else. I'm doing this math for teachers class post-BA to get into a Bachelor of Education program... So basically... I'd hate to teach kids how to do their math incorrectly!
@BandeiraGustavo Actually... Tell your friend to look into the psyc studies on sleep vs wakefulness and circadian rhythms on effectiveness of those who work long shift work... Such as that in nurses. If he does, he'd have a hard time saying it doesn't exist. ;)
science reporting leaves much to be desired regardless of which branch is being discussed, yet the public's perception of science is mainly through this and hearsay.
I wonder how much of the bs evolutionary psyc is just people who don't really know what they are talking and that at the expert level there is actually something there or whether the field is just all bs
If you do that more often, It'll give me a sense of equality. When you guys talk only about serious inteligent stuff, I feel dumb. Is that a social problem @Shayna ?
@BandeiraGustavo LOL I don't think it really qualifies as a social problem, although it may have something to do with one's sense of self... Possibly being a psyche problem where for one feels as if they are alienated by lack of ability to partake in all events. I'll let you call it a first world problem if you want though. ;)
in general, the terms will be $a_nb_mx^{n+m}$ so when the terms are collected the coefficient in front of $x^k$ is the sum of all $a_nb_m$ for which $n+m=k$
@BandeiraGustavo You can look at it combinatorially to get individual terms, say you want the coefficient on x^3 look at all the ways you can get x^3 through distributive property.
-What's the perfect guy/girl to marry? -1: create an acc on MSE, 2: make some question and earn some rep, 3: Go to the chat and find anon. He's the guy/girl.
Proposed Q&A site for people seeking answers to questions about dating, long term relationships, love, marriage or other commitments, and everything else typically considered a "relationship".
@BandeiraGustavo no, a polynomial in the variable $x$ is of the form $a_0+a_1x+\cdots$ with coefficients $a_i$. a multivariable polynomial has more than one variable.
@BandeiraGustavo Hahaha I haven't done this kind of math in... Well, I graduated from high school in '07... I took math 12 in '05. Most of this stuff hasn't been touched on since 9th grade math (if that LOL!). So I probably haven't even seen it in 11 years or more. :/
basically, a manifold is the mathematical gadget that describes curved spaces, where locally around every point it looks topologically like euclidean space
e.g. surface of a torus is a 2-dimensional manifold
@anon What's the meaning of algebra in that context? I mean, I've studied college algebra, boolean algebra and a little of abstract algebra. It seems to be about systems of rules for dealing with certain objects. Is that the case?
an algebra, in the context of universal algebra, is a set with various n-ary operations possibly with extra properties satisfied by them (e.g. distributivity). A lie algebra is a vector space with a bilinear operation called the lie bracket [-,-] satisfying the jacobi rule. the lie algebra is essentially a "linear approximation" of a lie group at its identity element, and the lie bracket corresponds to conjugation in the group
as with the case of finite groups, the classification of all simple lie groups essentially boils down to a handful of parametrized families of them (like {A1,A2,A3,...}, {B1,B2,B3,...}), but there are a handful of exceptional cases that do not belong to any families and stand on their own. one of these is E8.
it is the one with the highest number of dimensions
I'm trying to follow cambridge and oxford syllabi, I'm trying to learn analysis, abstract algebra, coordinate geometry, calculus and linear algebra.
But I also want to read some stuff that is accessible to a undergrad student.
That naive lie theory seems really cool.
But it starts with De Moivre's Formula and I still don't know much about that.
I'll read also about trigonometry.
I'm a little stuck in the proof of the binomial theorem - that is a pre-requisite in the oxford syllabus - but I still didn't do it just because I didn't try, went to read about calculus.
The more I learn about linear algebra, the more I think there should be more focus placed on it, it seems it has almost immediate applications to just about every other field (outside and inside of math). (Well my linear algebra class was crappy, at least the first one was)
@BandeiraGustavo At least in my college (and I think in most) your first three semesters of calculus will be calculation based/ memorize formula (you mentioned you hated math classes in school, it probably will be more of that)
As for why linear algebra is so usefull, i don't know exactly but fields are usefull and linear stuff is easy to calculate so that is part of the reason
I'm looking first to obtain the mechanical way of doing calculus, then I'll focus hard on spivak's. It's requested at the preface.
@Bageer I guess you can answer one question of mine: Sometimes topology seems to be a real thing like geometry. But sometimes it seems to do with sets and a lot of abstract stuff. What is this dicotomy?
Not sure if I am that knowledgeable, just got to the topology section of the book today :)
But it is basically they wanted topology that way
From what I understand most of the motivation from topology came from analysis and geometry and they wanted topology to generalize concepts like continuous functions to more abstract settings (that probably come up at some point)
true for me too. of course there can be a lot of intersection, I think one of the main problems is that the borders are really fuzzy and you can't really completly seperate them
it might be better to think of it as more of a continuum, there are no seperate fields but it is usefull to say that they are seperate because it gets a vague idea of what they are about
Its like if you ask them what you study and you say "I study number theory" to any one that knows math that is about as descriptive as saying "I study mathematics"
Next spring I think I will be studying from humphrey's lie algebra book (not sure though, the teacher seems pretty sure though) maybe we can be lie algebra buddies :D
Can anyone suggest some further readings on the topics covered in the WP article on the partition function, and in particular the section I linked to? Unfortunately the article doesn't contain relevant references.
@Ian Mateus Do you like my solution? $$\lim_{n\to\infty}\frac{\sqrt[n^2]{(4^n+1)(4^n+2)\cdots (4^n+n)}}{2^n}=\lim_{n\to\infty}\frac{\sqrt[n^2]{ 2^n 4^{n^2}}}{2^n}=0$$
@JoeStavitsky There are two of those episodes. One of them has the bald scientist with the "Evil Scientist (Boo)" neon on his castle, while the other one has the guy with thick lips. Anyway, the best scenes there are the ones where Bugs pretends to be the beautician. :D
I've gotten two strange downvotes today. One was while I was trying to ascertain what the OP was asking. The question they were asking was completely different than I think they thought, so I deleted my answer. The other is a good answer, and I cannot figure out why it was downvoted.
@Chris'swisesister The terms tend to $\infty$ like $2^n$.
@0x4A4D I saw that it looked ASCII, and I know what 4A and 4D are, I had just never put it together. I responded to you in the TL and didn't realize it was you.
