@morphic: Given any funxtion between sets, if $A$ is a subset of the domain, then $f(A)$ is the set of $y$ in the coodomain with $f(a) = y$ for some $a\in A$. That is, it's the image of $A$. In particular it makes sense for $\varnothing$, and you always have $f(\varnothing) = \varnothing$.
ok because the question is to give an example of where $f(A \cap B) \neq f(A) \cap f(B)$ but we do have that $f^{-1}(V \cap W) = f^{-1}(V) \cap f^{-1}(W)$ ($f: X \to Y$ is a function of sets and $A, B \subset X$ and $V, W \subset Y$)
so far i could only find cases where $f(A \cap B) \neq f(A) \cap f(B)$ when $A \cap B = \varnothing$
To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}^2$ with $f=\left\{\begin{matrix} \frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\ 0 & , (x,y)=(0,0) \end{matrix}\right.$ is continuous on $(0,0)$ we have to show that $|f(x,y)-f(x_0,y_0)| \leq L ||(x,y)-(x_0,y_0)||$ so we have to show that $\left |\frac{x^3-y^3}{x^2+y^2}\right | \leq L \sqrt{x^2+y^2}$.
To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}^2$ with $f=\left\{\begin{matrix}
\frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\
0 & , (x,y)=(0,0)
\end{matrix}\right.$ is continuous on $(0,0)$ we have to show that $|f(x,y)-f(x_0,y_0)| \leq L ||(x,y)-(x_0,y_0)||$ so we have to s...
@MikeMiller: Yep, that is basically what the exam is in my case (ATC). It seems a bit different, though. Here we just present a current research paper related to our research, present it publicly, then answer questions about it and related topics to a PhD committee.
Hello, I came across this question today - "Can you make 50 with these numbers: 2, 2, 4, 6, 9?" The rules were anything goes except for putting two numbers together (for example you can put 2 and 2 together to make 22)
There are multiple solutions to this problem. I was wondering if there was a way to calculate the number of possible solutions.
You know, the interesting thing about having told us your real name, Dr. Shifrin, is that I can just google you to figure out exactly what you look/sound like
@TedShifrin I sat in the first differential geometry lecture today and they were reviewing smooth and topological manifolds and next lecture I think was supposed to be tangent spaces/vector bundles so I was like "nope nope nope" and will never go back
You need undergraduate differential geometry first, mr eyeglasses. The graduate course assumes at the very least both point-set topology and a solid mastery of multivariable analysis.
The downside of having friends that are too intelligent is that they think something will be do-able for me because it was easy for them and they suggest that I try it but good thing I didn't listen to them otherwise I would've gotten destroyed taking differential geometry..they told me that I could probably work through it just because they learned this stuff back in high school
@TedShifrin I don't understand why this school has this differential geometry course if they don't have a multivariate analysis or curves/surfaces course
Does anyone have a favorite them in beamer? I'm not sure it makes a big difference, but if there is an "industry standard" that I ought to adhere to, I probably ought to learn about it now... :)
I'm not the best student haha. I feel outclassed just about everywhere (here included). I'll be happy if I can just finish and get a PhD by the end of the program...has anyone else felt that way?
@Chris'ssistheartist WOw gosh your cool..... Can you show me the steps on how you found the part $$\int_0^{\infty} \frac{(\pi+1)\sin x}{(x-\pi)x^{3/2}}$$
This "answer" should be deleted. The author didn't even understand the question. But what's worse it got 11 upvotes while completely missing the point. Not even close, this one.
@RudytheReindeer The way I see it, the OP correctly identified the cycles, so the answerer then focused on how to write a cycle as a product of transpositions (which is also a part of the original question).
So I do not think that it should be deleted.
In any case, if you think it should be, you can always vote to delete. And there is also a separate chatroom related to deletion/undeletion, closing/reopening etc.
hi - off topic algebra. Is there another way to see that the exponent of the group $G=(\mathbb{Z}/ 77\mathbb{Z})^{\times}$ is 30 other than using the fact that $G\cong \mathbb{Z}/10\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ (where the latter is a product of additive groups)?
ok, redo : can you give me an example of an affine algebraic variety which cannot be open covered by Zariski open sets, all isomorphic to some $\Bbb A_k^n$?
yeah. just take a 1-dimensional subvariety of A_k^2!
what i am trying to prove is that for any affine variety $V$, there is a surjective regular map $V \to \Bbb A_k^n$ which has finite fibers. i.e., every affine variety appears as a branched cover of $\Bbb A_k^n$.
now if every reasonably nice variety can be covered by Zariski open sets, all isomorphic to some $\Bbb A_k^n$, then pick a finite subclass of the cover (possible, because compactness), and build the map $V \to \Bbb A_k^n$ by mapping each of the open set in the cover by the isomorphism. this is clearly surjective, and has finite fibers too (as the subclass we chose was finite)
no, it's a speculation of mine. Mike said it's morally a good definition of variety, and something along that line is given in Milne (apparently it's something about ringed spaces, though)
@TobiasKildetoft Algebraically, what I am trying to say is about the same as the statement that if $V$ is an affine variety, $x \in V$, then there is some Zariski neighborhood $U$ of $x$ such that the localization $k[V]_U$ is the polynomial ring $k[x_1, \cdots, x_n]$ where $x_i \in k[V]_U$ are $k$-algebraically independent.
(and at some point you will need to use that $k$ is algebraically closed or you might only get a polynomial ring of the form $K[x_1,\dots,x_n]$ for some extension field $K$ of $k$)
I had seen this result a while back in a Numberphile video:
$1+2+3+\cdots = -\frac{1}{12}$
I was trying to prove the same result using a different method when I accidently proved that the sum was 0!!
I am unable to find out what mistake I have made. Or is it that perhaps the sum is not reall...
@BalarkaSen A typical example of a singular space is a curve with a cusp, since the "point" has tangent space of dimension $2$, and the curve has dimension 1
@TobiasKildetoft heh. I figured it's a bit tricky.
a cool thing I heard recently is that you can tell which point is singular or not by checking whether the stalk of the structure sheaf at that point is regular.
@TobiasKildetoft Out of curiosity, are tangent lines defined classically just as hyperplanes isomorphic to A^1? i am not sure if that'd be good. how to define "tangentiality"? I don't think you can distinguish between traversal and tangential intersection of two subvarities of a generat variety, can you?
@TheArtist I answered under whacka. The manipulations are hiding the fact that they are deleting 0s from divergent series, which is not generally valid under summability methods. It is however valid under zeta regularization, which is how 1+2+3+...=-1/12 is achieved, but under that regime there is no means of simplifying the subtraction at the very end.
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@JackD'Aurizio I don't see how that question was a duplicate of 1+2+3+...=-1/12. The OP had their own calculation of 1+2+3+..=0 and specifically requested an explanation of what they did wrong, an explanation not present in the old one.
@MartinSleziak I think the question (including the title) asks specifically for disjoint cycles. It is a very common question about permutation groups. Writing a permutation as product of transpositions on the other hand -- I have never seen before. Of course the answer won't be deleted, no matter how beside the point it is. Wrong answers never get deleted by mods. But I can still utter my thoughts here and they are still that this answer is bad enough to be purged from the site.
@RudytheReindeer I don't understand why you think the answer is wrong or beside the point. Yes, writing permutations as products of disjoint cycles is a common question, and the OP did that successfully. OP is also tasked with writing the permutation as a product of transpositions - which is a common textbook problem - and in order to do that, it suffices to know how to write cycles as products of transpositions, since the OP already knows how to express the permutation as a product of cycles.
amWhy gave two methods for writing a cycle as a product of transpositions (didn't really explain them, but gave illustrative examples for the reader to extract the pattern from). She also noted that being odd/even and having odd/even order are distinct concepts, which was important to put to the OP.
Hi. I added a bounty tohttp://mathoverflow.net/questions/208020/probability-of-many-overlapping-zero-inner-products-on-a-circle but no comments or answers!
Is it just something that math knows nothing about?
empirically, to test if math knows something about a problem, one puts the problem to a sufficient sample of experts. whether or not enough people have read the question (and been interested in it enough to think about it) to say "math knows nothing about it" is shaky.
@Balarka: You don't want to cover it by opens isomorphic to affine space. You want to be able to cover it by opens isomorphic to an open subset of affine space.
Complex geometry analogue: you need your charts to be open subsets of $\Bbb C^n$, otherwise you can't make the open unit disc into a complex manifold!
And it is true that you can cover by open subsets of affine space.
Okay here goes my proof : So we have $x_1e_1+ x_2e_2 +x_3e_3 \cdots +x_ne_n$. I have to get scalars $x_1,x_2\cdots x_n$ such that they equal $\{a_1,a_2,\cdots a_n\}$ where $a_i\in \Bbb{Z}$ . So now if I let $x_1=a_1,x_2=a_2, \cdots ,x_n=a_n$ (I can do this elements of $\Bbb{Z}$ are scalars), the linear combination with the standard basis vectors will give me $\{a_1,a_2,\cdots a_n\}$ . So a linear combination of $e_i$'s will give me the n tuple of integers . Hence proved@Balarka
@Balarka A question in informal spirit: Can we say stuff about $\mathcal{E}(A,B)$ where $\mathcal{E}(A,B)$ means the set of all exact sequence between two groups $A$ and $B$ where the short exact sequence should be of the form $A\to N \to B$ where $N$ is an abelian group
@Ted well I am trying to show that $\Bbb{Z}^n$ is generated by the basis vectors in order to find a homomorphism between $\Bbb{Z}^n$ and some abelian group $A$
Yes, right, because $\Bbb Z^n$ is free, the short exact sequence splits. Have you proved that splitting is equivalent to getting a direct sum decomposition?
Connect sum is a weird notion to try to make sense of in the holomorphic category, @MikeM. I mean, I don't even know how to define it.
Well, I think he's just missing a basic point-set point there, @MikeM, although, to be honest, as I said in my answer, I think it's a case where Guillemin screwed up his hint. I'm proud of my spiral example to understand what's going on.
How would we prove this result by real methods ?
$$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 \sqrt{2}\right)+2 \text{Si}(4 \pi ) \right)$$
As you can easily see, Fresnel integrals are involved. What are yo...
To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}^2$ with $f=\left\{\begin{matrix}
\frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\
0 & , (x,y)=(0,0)
\end{matrix}\right.$ is continuous on $(0,0)$ we have to show that $|f(x,y)-f(x_0,y_0)| \leq L ||(x,y)-(x_0,y_0)||$ so we have to s...
I had seen this result a while back in a Numberphile video:
$1+2+3+\cdots = -\frac{1}{12}$
I was trying to prove the same result using a different method when I accidently proved that the sum was 0!!
I am unable to find out what mistake I have made. Or is it that perhaps the sum is not reall...
How would we prove this result by real methods ?
$$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 \sqrt{2}\right)+2 \text{Si}(4 \pi ) \right)$$
As you can easily see, Fresnel integrals are involved. What are yo...
