In his 1991 book Topological Invariants of Plane Curves and Caustics, Arnold asks the following question (which I paraphrase)..
Does every topological knot type in $ST^* \Bbb R^2$ arise as a Legendrian curve of an immersion $S^1 \to \Bbb R^2$ (in other words is every knot type parameterized b...
I forgot that avoiding $\infty - \infty$ doesn't require that only finitely many terms were of a certain sign, convergent series are a thing which exist
@Ted is there an easy mental picture that explains what Ricci curvature means? we never discussed it in class, only proved relations it satisfies on the psets
does it make any sense to pass as input to trigonometric functions degrees per second (or radian per second), where the point is to highlight the "per second"?
Let $h=(b-a)/3, x_1=a+h$ and $x_2=b$. Find the degree of precision of the quadrature formula, $\int_{a}^{b} f(x) dx$ approximately $\frac{9}{4}hf(x_1)+\frac{3}{4}hf(x_2}$
Let $h=(b-a)/3, x_1=a+h$ and $x_2=b$. Find the degree of precision of the quadrature formula, $\int_{a}^{b} f(x) dx$ approximately $\frac{9}{4}hf(x_1)+\frac{3}{4}hf(x_2)$
Hi, is it true to say that if $f'$ has no analytic continuation through $\partial\mathbb{D}$ (where $\mathbb{D}$ is the unit disc) then $f$ has no analytic continuation through $\partial\mathbb{D}$ (where $\mathbb{D}$?
Here's the problem: I have $N$ samples ($S_k$) from the set $\{1,2,\dots,n\}$. Using these, how can I determine an approximation of $n$? (Note that these can be repeating samples; if they were unique, then this would be equivalent to the German Tank Problem.)
What is the length of shortest path that begins at the point (-1,1) , touches the x axis and then ends at point on the parabola $(x-y)^2 =2(x+y-4)$ .
I put $x=x-y$
$y=x+y-4$
then the starting point would be $(-4,-2)$ .
can I proceed by this methos of changing the coordinates
After that I g...
It appears this thought process leads to the equation that is needed to show under what condition that continuity implies continuity of one sided derivatives (assume it exists)
The condition is that one can interchange the limits, and this is possible only when uniform convergence is obeyed:
Suppose I have $f:\mathbb{R}^2 \to \mathbb{R}$. What conditions do I need to say that
$$\lim_{x \to a} \lim_{y \to b} f(x,y) = \lim_{y \to b} \lim_{x \to a} f(x,y)$$
?
What about in a more general case, by taking $X,Y$ and $Z$ topological (Hausdorff) spaces and $f$ from $X \times Y$ to $Z$ ?...
That means, letting $D(y)=c$ be an ODE and $D_+(y)=c$, $D_-(y)$ be right and left ODEs, if $f$ is continuous within its domain $\textrm{Dom}(f)$ and satisfy e.g. $D_+(y)=c$, then $f$ does not necessary satisfy $D(y)=c$ unless it also satisfy $D_-(y)=c$
Therefore $\textrm{f solves $D(y)=c$}\iff \textrm{f solves $D_+(y)=c$ and $D_-(y)=c$}$ but $\textrm{f solves $D_+(y)=c$ or $D_-(y)=c$}=\not\implies \textrm{f solves $D(y)=c$ }$ in general
@Astyx Can't get it any closer than the range $n\in[26,30.77]$. Thus, I have a new theory: Instead of the set $\{1,2,\dots,n\}$, I'm instead sampling from $\{m,m+1,\dots,n\}$ for $m=5$ or so. Not sure how to find the exact bounds...
Quick question on polynomial interpolation: Given a polynomial function $f(x)$ and a point $(a,b)$, the modified function $f^\prime(x)=(f(x)-a)(x-b)+a$ will always cross through $(a,b)$, correct?
"The prison is deep and of stone; it's form, that of a perfect hemisphere, though the floor (also of stone) is somewhat less than a great circle, a fact which aggravates the feelings of oppression and of vastness"
Some ordinal property in a nutshell: Am ordered set. Adding stuff on the right sticks behind, but adding on the left and if it is not big enough, will be eaten up
Does anyone here have a solid experience with numerical computations? By solid I mean that you haven't just implemented one iterative algorithm in your life and that you actually know, at least, the main problems when implementing these algorithms.
What kind of numerical computation? There's a big difference between, say, a PDE solver or an MCMC sampler or a large integer factorization algorithm, even though all of those could be called "numerical computation".
@IlmariKaronen The reason I ask here it's because here there's usually a lot more active people than in other more specific communities, like the one you're pointing me too... but their chat is completely desertic...
True, not a lot of smaller sites have active chat rooms. But if you have a specific question about optimization algorithms, it might on-topic at the main site there.
because I once asked a question and they were very reluctant to answer, it was like, why don't you look up in a 600 pages-book? I mean, if I ask a question, sometimes, it's not just because I don't know currently how to solve a problem, but it's also because I do not have enough time to look up in a book, don't you think so?
$T$ is the period, $[-L,L]$ is the interval at which we consider the function @blue
Anonymous
@nbro You need to ask a question and wait for 1 or 2 days to get an answer on beta sites. On bigger sites you can expect answers in a few hours. Keep in mind that no one here gets paid to answer your question. People answer questions voluntarily to help other people. So don't lose your patience and go ahead and ask your question on the site which you feel is most suitable.
@blue it's not just a matter of waiting for the answer for a longer time, it's also a matter of how they first react to a question, it's like, I'm not going to answer this question because "reasons"
maybe they do not even know what I'm talking about
@nbro No one is obliged to answer to your question. You need to be patient. Questions which show more research efforts receive faster answers. It is quite possible that they don't know the answer. At the same time it is possible that your question doesn't interest them. If you want immediate positive responses to your question, then this is not the correct site for you.
I have been struggling with this question. Let a_n be a diverging sequence. Then can we find a sequence {x_n} such that \sum |x_{n}| converges but the series \sum |a_n x_n| diverges ?
I've been here on Stack Exchange websites for a sufficient time to say that, except maybe in this community, people often do not even read completely the question or do not even know what the question is about, but they like to start saying "Create a minimal verifiable, etc, example" or, again, "there are plenty of examples around", yes? where? and things like this. This only makes me think more than most of the people are stupid.
yeah and they also downvote your question if you don't have any idea of how to solve a problem
Anonymous
@nbro Yes, I agree there are such problems as you describe. It's more like if you are lucky you will get an answer. If you're not, you won't. But such is life. No promises.
But then again, you can't expect everyone to read all questions fully, as a LOT of questions either have not been thought about by the poster, or are incomprehensible for anyone except the poster, which leads to people not wanting to "waste time" on questions
@nbro To be honest, I find this remark quite rude. You claim not to have enough time to look something up, but you expect others to take their time to answer a question for you?
Also, if you use the index or table of contents well, it doesn't matter that much if a book has 50 pages or 600 pages.
@SteamyRoot If someone knows the answer, it just loses the time to put thoughts together and write it; OTOH, if someone doesn't know the answer, needs first to find the right resource, read it, which may take a lot of time, read it again to understand it better, etc... I mean, I don't know what you want to demonstrate, but you seem to be the type of person that thinks rules exist in an absolute sense, and that's sad
While I can said a few issues about numerical calculation in general (such as numerical instability, eigenvalues too small, nonconvergence or very slow convergence, well posedness of a problem, curse of dimensionality etc.) I do nto have suffiicient experience on the issues for the 3 specific algorithms in question
Anonymous
Sometimes I feel that Stack Exchange should be made a paid service so that people start valuing the time others spend on answering their questions.
@blue people are already impatient now when it's about people answering out of their generosity, if you make it paid then people will more rightfully demand instant and full responses to every homework question ever.
@Semi In office hours I saw my professor's chalkboard had the term "Polynomial ham sandwich theorem"
I don't know what the discussion was about but I find this chat is a more helpful community than the SE. And not helpful in the sense that "we'll do your homework" (which is not helpful!).
But of course if 5 million people invaded this room that would break everything.
So maybe that's just because it's a small, stable community.
hello. i have a quick question. If i have a graded ring A=\bigoplus_{i=0}^\infty A_i, then each A_i is an abelian group. But 0 is homogeneous degree 0, and hence shouldn't be in each A_i but for A_0
@alggeo So in your case you've got $A=\bigoplus_{i=0}^\infty A_i$, and you want to know where the zero of each abelian group is
Seems like it might be helpful to take a concrete case, e.g. $A=k[t_1,\ldots,t_n]$ is graded by degree (direct sum of homogeneous polynomials of degree $l$).
my guess would be that 0 is an exceptional case. if you add two nontrivial homogeneous polynomials of degree n you in general get another homogeneous polynomial of degree n.
but if you add 0 to any polynomial, the degree doesn't change.
I think the way to look at is that 0 is an element of A, not of one of the A_i's
And the wikipedia article specifically says: "$A_0$ is a subring of $A$; in particular, the additive identity 0 and the multiplicative identity 1 are homogeneous elements of degree zero."
Which makes it sound like your statement more than mine.
I'm working through Vakil's notes, and he defines a $\Bbb{Z}^{\ge0}$-graded ring
$$S_{\bullet}=\oplus_{n\ge0} S_n$$
and $X:=\operatorname{Proj}S_{\bullet}$ to be the collection of homogeneous prime ideals of $S_{\bullet}$ not containing $S_+:=\oplus_{n>0}S_n$.
My question is:
If $S_{\bulle...
In his 1991 book Topological Invariants of Plane Curves and Caustics, Arnold asks the following question (which I paraphrase)..
Does every topological knot type in $ST^* \Bbb R^2$ arise as a Legendrian curve of an immersion $S^1 \to \Bbb R^2$ (in other words is every knot type parameterized b...
What is the length of shortest path that begins at the point (-1,1) , touches the x axis and then ends at point on the parabola $(x-y)^2 =2(x+y-4)$ .
I put $x=x-y$
$y=x+y-4$
then the starting point would be $(-4,-2)$ .
can I proceed by this methos of changing the coordinates
After that I g...
Suppose I have $f:\mathbb{R}^2 \to \mathbb{R}$. What conditions do I need to say that
$$\lim_{x \to a} \lim_{y \to b} f(x,y) = \lim_{y \to b} \lim_{x \to a} f(x,y)$$
?
What about in a more general case, by taking $X,Y$ and $Z$ topological (Hausdorff) spaces and $f$ from $X \times Y$ to $Z$ ?...
I'm working through Vakil's notes, and he defines a $\Bbb{Z}^{\ge0}$-graded ring
$$S_{\bullet}=\oplus_{n\ge0} S_n$$
and $X:=\operatorname{Proj}S_{\bullet}$ to be the collection of homogeneous prime ideals of $S_{\bullet}$ not containing $S_+:=\oplus_{n>0}S_n$.
My question is:
If $S_{\bulle...
