@PVAL @MikeM: Interestingly, there's a guy trying to read Chern's Lectures on Differential Geometry and posting various questions on Pfaffian systems and their implications.
Oh, right. Well, Chern is not the most pedantic writer on earth. People don't realize that he just "knew" what artillery of forms to write down to make his original proof of the Gauss-Bonnet-Chern theorem work. Amazing intuitive command of all the Cartan machinery.
At any rate, I did try to suggest to him that he needed to understand what an equation like $dx=0$ means. I think he's making progress.
@Ted: If you care about the question I asked about analytic regularity of mildly nonlinear PDE, one of the faculty members here told me he's rather confident it's true, but didn't have a reference off the top of his head.
I believe you're referring to a version of Cauchy-Kovalevskaya, which is an existence theorem for simple analytic PDE. I believe Levy disproved this in the smooth case.
In any case, my question is really about the distinction between equations that are linear and only mostly linear. I do have the same analytic regularity in the linear case.
You can then tell them what the connection $1$-form measures and use the change-of-frame formula to see how the index of the vector field comes out. (This is done somewhat tersely in my on-line notes. I already gave you permission to steal.)
@Ted I've been having trouble with one problem. The problem is: "From a cylindrical cake with chocolate icing on top, cut successive slices of a fixed angle x. Each slice is then inverted and inserted back into the cake. Find all angles x, such that after finitely many iterations, all the icing is back on top of the cake."
Thanks, Ted. The answer is all angles, but I don't know how to prove it. In other news, I learned what the $\frac{A}{\sin(\alpha)}$ means in the law of sines.
Well... as you've only managed to say "It's stupid." I hope you won't blame me for settling on the conclusion that it's simply a petty, egotistical reaction to the non-standard notation.
@MikeMiller You just told me what it was (at least modulo grading). I mean one doesn't prove something like that using something like Sarkar-Wang. One needs something extra about the invariant which allows it to be easily computed (pretty sure in this case its just the surgery exact triangle).
@MikeMiller Apparently, your 340 was a closer guess. The upper bound is alphabetsize-1, which would have been 255. $|255 - 17| = 238$, $|255 - 340| = 85$
I'm actually surprised it can get that big. Must be for that character that gets used once a year.
I gave him a set of lecture notes by Rolfsen for the orderable groups stuff. I want to see how forcing the student to find the source works out (much of the most useful sources for my own research I found myself).
@PVAL: I suspect the Eliashberg-Thurston paper/book begs to differ.
I think there are some places and situations in which it's better to tell someone to find the right sources. But sometimes one will find a source that's bad and since it's a source they'll stick with it.
Maybe he can prove that every 3-manifold with a left-orderable fundamental group admits a taut foliation and vice versa without ever touching these nonsensical modern homology theories...
@MikeMiller Hopefully I can tell him if a source is bad.
I wonder what the space of foliations looks like. It's Thurston's theorem that the inclusion of the space of codim 1 foliarions on a 3-manifold into the space of plane fields is a surjection on pi_0.
@Ted: Try to prove it! Actually for 3-manifolds it's not hard, but the general case for codim 1 distributions on n-manifolds was probably the theorem that people consider Thurston as solving foliations.
It should be something like write a surgery presentation for the 3-manifold, and make a foliation of S^3 such that that components in the link are all reeb components.
Ya thats what it looks like Wood does here. jstor.org/stable/1970673?seq=8#page_scan_tab_contents Maybe it's a little different but I am pretty sure he gets a reeb component for each component of the surgered link.
can someone explain to me the classifications for linear equations? I don't understand what the terms mean. Consistent, inconsistent, and dependent and how a system of equations can be both.
@Integral More precisely, rational representations are comodules over the coordinate algebra, but representations (not necessarily rational) are modules over the group algebra
@Integral In some sense, yes, but only in a very vague sense
If it is the coordinate algebra, then it is actually a quotient of the polynomial algebra in four indeterminates, mod a suitable ideal
If $S$ is the set of one-to-one mappings of $S$ onto itself, and if $x_0 \in S$, what is meant by $H(x_0) = \left\{\phi \in A(S): x_0 \phi = x_0\right\}$? The element $\phi$ in $A(S)$ that maps $x_0$ onto itself?
@Huy, it's intentional. I remember him saying something about algebraists preferring it that way in the earlier chapter. Yes, group theory (Topics in Algebra - Herstein).
This is a "reverse" question of finding the asymptote of a function
Recently, I am interested in doing some sort of modelling which involve equations of the form
$$@(t)=1-f(t)$$
where $f(t)$ is required to satisfy the following properties
$$1^*:\left\{\begin{matrix}f(0)=0 \\ \lim_{t\rightarro...
Does it stand that $$\lim_{x\rightarrow \infty}\frac{(1+x)^p}{1+x^p}=1$$ for $p>0$ because we look only at the terms of highest order of the numerator and the denominator?
@Secret Not really... Sorry...
@robjohn do you have an idea about my question above?
I think the x can be pulled out as usual $$\lim_{x\rightarrow \infty}\frac{(1+x)^p}{1+x^p}$$ $$=\lim_{x\rightarrow \infty}\frac{(x(\frac{1}{x}+1))^p}{x^p(\frac{1}{x^p}+1)}$$ $$=\lim_{x\rightarrow \infty}\frac{x^p(\frac{1}{x}+1)^p}{x^p(\frac{1}{x^p}+1)}$$ $$=\lim_{x\rightarrow \infty}\frac{(\frac{1}{x}+1)^p}{(\frac{1}{x^p}+1)}$$ $$=\frac{1^p}{1}=1$$
since p is real thus it is distributive in multiplication
A recent question asked about polynomial solutions to the functional equation $p(x^2)=p(x)p(x+1)$. Subsequently, Robert Israel posted an answer showing that solutions are necessarily of the form $p(x)=x^m(x-1)^m$.
What I had hoped to do myself was provide a solution by interpreting $p(x)$ as a g...
@robjohn any algorithms or systematic approaches you know of can handle system of limit equations? (I am not even sure if there's a fiel of study for them cause I rarely see limits appearing in systems of equations)
This is a "reverse" question of finding the asymptote of a function
Recently, I am interested in doing some sort of modelling which involve equations of the form
$$@(t)=1-f(t)$$
where $f(t)$ is required to satisfy the following properties
$$1^*:\left\{\begin{matrix}f(0)=0 \\ \lim_{t\rightarro...
Not sure if it is concrete enough ,but I am interested in given some asymptotes, find some functions that basically describes the decay of a population\
That is, I am interested in given a starting value, find as many functions of t such at at the limit of infinite time, it decays asymototically to zero
and then I am wondering for more complicated systems of equations where the assymptotes are functons of t themselves
So for the population example in the question, there happens to be an easy graphical solution. However I am not really sure if finding that solution can be expressed algebriacally
I continue to see questions that are "homework" question with little effort. May I add a comment to those closed question referring them to algebra.com, a free math tutor site? I post there to practice my HS math teaching, but am otherwise not affiliated. Asking here, because I don't want it being considered spamming. (It's just one every couple days, no intention of going back to old ones) Simple questions are welcome there and for these questions, it would help the OP.
@Secret I think that this gives to vague a condition in most cases to present a unique solution. That is, there are too many solutions to give a good approach.
@JoeTaxpayer Perhaps get them to put a blurb in the sidebars. They will have to contact the StackExchange community managers to get information about that.
That would be better than a user advertising for the site.
@robjohn If there is a matrix $W$ such that $W={ A \in M : A_ij =0 i>j}$ where $M$ is an $m$ by $n$ matrix, then I can say that $W$ is a subspace of $M$ right?
@robjohn kind of, although I have to prove that the direct sum of two matrices, an upper triangular and a lower triangular is equal to the matrix $M$. for direct sum, the triangular matrices have to be a subspace of $M$ right?
@Paradox101 usually one speaks of a space being the direct sum of two disjoint subspaces, not a single element being a direct sum of two other elements.
@Paradox101 of course, diagonal matrices are both upper and lower triangular.
@robjohn yes. In the book I'm reading a direct sum is described as follows: A vector $V$ is called the direct sum of $W_1$ and $W_2$ if they are subspaces of $V$ such that intersection is ${0}$ and $W_1+W_2=V$. That's why I was asking about the intersection. since 'the only matrix which is upper triangular and strictly lower triangular is the zero matrix' then it follows the definition of the direct sum
I thought we needed two condition for $H \subset G$ to be a subgroup of $G$. (i) $a,b \in H \implies ab \in H$, and (ii) $g \in H \implies g^{-1} \in H$. However, my tutor said we need a third condition; (iii) $e \in H$. Isn't this contained in (i) and (ii), i.e. $ g \in H \implies g^{-1} \in H$ thus $gg^{-1} \in H$?
I'm not paying much attention because your claim has no counterexample. If a subset is closed under product and inverse and is nonempty, it's a subgroup.
I mean, I guess that's the real point of (iii). It guarantees your subset isn't empty.
Uh, that's clearly not a counterexample, anyway, because it's not even a little bit closed under product.
Hi everyone, I'm a gambler and I start with $x$ money, i put down a dollar and get 2 dolalrs back with p=0.5, and lose my dollar with p=0.5. I am interested in how long it will take me to either A) go broke or B) reach N dollars. Take T to be this stopping time.
I showed $(X_n^2-n)$ is a martingale and that I reach N dollars with probability $x/(N+x)$, but now how do I get the EV of how long it takes to finish the game?
I used the optional stopping theorem to say that E[X_T^2-T]=E[X_0^2-0]=x^2, so that E[T]=E[X_T^2]-x^2=x/(N+x)*(N+x)^2-x^2=Nx
I need to find the equation in cylindrical coordinates $(r, \theta, z)$ for the sphere of radius 2 centered at the origin. I said that $r=2, 0 \leq \theta \leq 2\pi, -2 \leq z \leq 2$. Apparently r = 2 is wrong, but i don't understand why. Is it actually wrong?
i don't understand because in rectangular coordinates, then it is $\lange 2\cos\theta, 2\sin\theta \rangle$, so in cylindrical it is $r = \sqrt{x^2 + y^2}$, so that is 2
From cylindrical coordinates you always have r^2 = x^2 + y^2 and the sphere equation in cartesian coordinates is x^2 + y^2 + z^2 = 4. Then just substitute.
@Evinda cosets are either something to do with groups (so you need a group) or something in the dual category of the category of sets (this is not what you mean I think)
This is a "reverse" question of finding the asymptote of a function
Recently, I am interested in doing some sort of modelling which involve equations of the form
$$@(t)=1-f(t)$$
where $f(t)$ is required to satisfy the following properties
$$1^*:\left\{\begin{matrix}f(0)=0 \\ \lim_{t\rightarro...
