The reason I'm asking is because pointed homotopic maps are easily seen to induce the same map on the fundamental groups at the respective basepoints, but I do not know whether maps that are just homotopic do this as well. In particular, pointed homotopy-equivalent spaces obviously have isomorphic fundamental groups, but I can't actually convince myself that homotopy-equivalent spaces have isomorphic fundamental groups (assume spaces are path-connected, so that fundamental group makes sense up to isomorphism type without choice of basepoint).
So if we look at those weird examples of contractible $Y$ that does not strongly deformation retract on any point and pick as $X$ a single point, does that work?
Yes.. In odd order group. I have above theorem. Which required odd order element in odd order group. What is the guarantee there is always odd order element in odd order group?
The problem I am struggling with is
"Find an example of a finite group G of order n where n is odd and each element of G is a square".
In my understanding this means
"Find a finite group with odd number of elements where each element can be expressed in ( )^2 form".
Am I understanding the qu...
@Alessandro The point is that I want to say if $C_n$ is decreasing to $C$ and $C\subseteq U$ for some open $U$, then $C_n\subseteq U$ for large enough $n$, which sounds true, but I haven't been able to prove it
Aahh okay. It means it holds for every group. My fault. but again it required odd order element of any G. What is the guarantee $G$ has element if odd order.
@Alessandro In particular, if I try contradiction, I obtain a sequence $(x_n)$ with $x_n\in C_n\cap U^c$ and if we assume sequential compactness cause we don't like nets, then we may assume $x_n\rightarrow x$. Obviously, $x\in U^c$. I'd like to say $x\in C$, then we get a contradiction and it sounds really plausible, but I can't tell whether it's true.
the order of a group is the cardinality of the group. it is different from the order of an element. in eithe rcase, where did you get negative orders???
for any integer we can define x^m. the order of an element as any group text will tell you is the SMALLEST POSITIVE INTEGER SUCH THAT x^m = e. i have seen Ted tell you this so I'm not sure what it going on here.
@TedShifrin a student of mine wants to extrapolate climate data. he collected data from some website e.g. for August 2020 in the format "average sea temperature, latitude, longitude". there is also data for August 2019, August 2018, etc.
do you know what would be a reasonable way to interpolate and then extrapolate so many data points? I only remember basics on how to interpolate data sets of the form (x, y) with different methods.
it doesn't need to be extremely super scientific, of course. just somewhat reasonable and not simply linear between two points
@TedShifrin is this the least square thing? with the normal equations?
I only remember some parts from numerical analysis because I haven't applied them since my exams in uni - but if I know what I have to look up again, I'll do it of course
@TedShifrin if it is directly related to what I need, yes, otherwise, later. we have to "solve" this before Christmas because he was too shy to ask me before, when he would still have had more time to work on it...
@TedShifrin I was thinking about extremely simplifying the process and just compute the average temperature of the 1 million data points for each time point
but I'm not sure if this will still lead to a useful extrapolation
Is there any general term of 'angle displacement: If $S$ is oriented and $\gamma$ is of unit speed, then the angle displacement along $\gamma$ is $\Delta\theta = \theta(b)-\theta(b)$ where $\theta$ is an angle function of $\gamma$ and $\gamma:[a,b]\to S$ is a regular curve'
I asked people from our geography faculty for help. none of them has any idea about maths or statistics or any possible mathematical model for the data :(
@TedShifrin stupid question (cause I don't do statistics) - how does one do this? look at the graph and "guess"? or is there some more mathematical way to?
if this is a school project i might split the surface into 'squares' so i am not working on S^2 and compute a 'flat earth' model. this will have issues at the boundary but might work for you.
Yeah, one can fit with splines, which I know nothing about. Looking at the graph of the (localized?) average would suggest a shape and then maybe a logarithmic best fit would be appropriate.
Okay. So I have a 4x4 matrix over $\Bbb{Q}$ with characteristic polynomial $p_A(x) = x^4 + x^2$. I am trying to determine whether it has Jordan form over $\Bbb{Q}$. I don't think it does, because the only eigenvalue of $A$ over $\Bbb{Q}$ is $0$, and its multiplicity in $p_{A}(x)$ is $2$, which means, if it had a Jordan form, the Jordan block corresponding to $0$ would be 2x2. But the dimensions don't add up, so it can't possibly have a Jordan form over $\Bbb{Q}$.
Because Theorem of every subgroup of cyclic group is cyclic using normal order finding method (Euclid division algorithm), which is used to find the order of element of any non-cyclic and cyclic group. There is no special method for proving subgroup is cyclic.
Pls see attached THEOREM-2.11. Which is used to proof order of element in any group.
THEOREM-2.19: used the same analogy for proving subgroup of cyclic group is cyclic. as in attached. I don't see any special thing which proves subgroup is cyclic or normal.
@123 the operation is not the cross product. The operation is similar to the cross product in the sense that $ij=-ji$ and $ij=k$, but $i^2=j^2=k^2=-1$ is different from the cross product.
That is, products are not anti-symmetric; for example: $1^2=1$.
@user193319 $$\begin{bmatrix}i&0&0&0\\0&-i&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}$$ and $$\begin{bmatrix}i&0&0&0\\0&-i&0&0\\0&0&0&1\\0&0&0&0\end{bmatrix}$$ are possibilities
@copper.hat I can't say. I have not been here the whole time.
@copper Some of our denizens require a lot of patience. Some too much for me sometimes.
Um, well, I'm still here, @robjohn. I do see more people wearing masks. Maybe the fact that SoCal is about to be locked down seriously is getting their attention.
I don't know how the medical profession is hanging in there.
@TedShifrin My wife and I walk in the morning and she gets irate at the people without masks and exercising in the park. It feels unsafe to be near them.
@TedShifrin around here, the hospitals are overrun
Yup, I get it. But when I walk (pretty much daily) now I encounter fewer unmasked people, so that's progress. But the ignorance and selfishness that have been inculcated in our society by you-know-whom is astonishing and infuriating.
There are people, that I think are fairly intelligent except in politics, that support him that I deal with in the park on a daily basis. I make sure we no longer discuss politics.
I've never picked up on whatever charisma he has. Everything he says and does just makes me either roll my eyes or recoil in outrage/disgust. But he apparently has some kind of charisma.
My wife and I watched a webinar from the University of California this afternoon which dealt with the legal status of the Indigenous peoples' rights (my wife is an attorney), and they had much to say about the bad things that happened in the last 4 years and were looking forward to Biden/Harris.
Of course, when I was on the university admissions committee, I could usually tell when students had had help writing their essays. It didn't do them much good.
And, at least at UGA, the essay only mattered for borderline people in the middle. The very top kids and very bottom kids got in based on grades, test scores, and teacher recs.
Good for her. Tell your daughter I'd like to meet her :)
The problem I am struggling with is
"Find an example of a finite group G of order n where n is odd and each element of G is a square".
In my understanding this means
"Find a finite group with odd number of elements where each element can be expressed in ( )^2 form".
Am I understanding the qu...
