12:06 AM
we had channel 2 (fox), 4 (nbc), 5 (cbs), 7 (abc), and 9 (pbs).
you may not have had 2.
and after 1988, channel 3 for the vcr.

12:37 AM
You know, I want to generalize on CD a little bit for my app: math.stackexchange.com/questions/4460865/…
I don't think it should limit itself to DAG's for a commutative diagram (CD)
And non-multigraphs
I think it should be a functor from any finite category $A$ into your diagram's ambient category $C$
You can then associate to it and any choice of two of its objects $X,Y \in \text{Ob}(A)$ the set of all strings in your diagram $J: A \to C$ and regular language (or "regex") of equal compositions.

12:55 AM
@leslietownes was always curious as to how channel 3 ended up being the vcr channel, was the same at my home too....

good evening

hello, I want to ask the same comment @mmm made here (which was no longer responded to by the author of the answer)
https://math.stackexchange.com/questions/748792/r-primitive-root-of-prime-p-where-p-equiv-1-mod-4-prove-r-is-also-a

Why is it that just because a power of -r is congruent to r then -r must be a primitive root?
is sir @AndréNicolas here too?

1:11 AM
Haven't seen Andre Nicolas post in years...

Who is Andre Nicolas?

the one thing i know about that guy is that he hasn't posted in years

anyone who knows this?
Why is it that just because a power of -r is congruent to r then -r must be a primitive root?

1:54 AM
@D.C.theIII yeah. :(

"In a while, I will for some time not be able to answer questions I am asked about past posts (medical). Apologies!"
from his bio

2:13 AM
Wow, almost 6 years gone.

$A^{-t}$ in a 2009 annals paper annals.math.princeton.edu/2009/170-3/p09
that's the earliest I've seen this
(arxiv version here, submitted 2007)

Crazy. Applied guys from Switzerland, but the editors didn't complain, apparently.
That's why I was wondering if people in a certain field or from certain geographical locations did this commonly.
I've certainly not encountered it in any linear algebra text or differential (or algebraic) geometry paper that I noticed.

2:31 AM
I’m pretty sure this paper proving the existence of Euler fluids with compact support in time has no practical applications :P but yes I was floored back then too

@TedShifrin ... Oh he passed?...damn.. :(

@D.C.theIII i think it's just six years since he was active here
hasn't stopped his rep from growing at a steady rate

2:46 AM
his profile pic too changed, it used to be blue

3:24 AM
@copper.hat well im gonna guess its good news all the same so congrats!

4:13 AM
@D.C.theIII why would you think that?
@Semiclassical I think so too.

The way Ted phrased it... as "6 years gone"

+since he was last active here.

i went inactive for about that long, and now look at me.

Suppose that V is a vector space and U is a subspace of V such that V/U is finite dimensional then V is isomorphic (as vector spaces) to $U\times (V/U)$.
I can prove this.
But I'm having difficulty in proving a more general version of this math.stackexchange.com/questions/4446340/…
The theorem is cool. It also gives intuition why V/U (V over U) is the suitable symbol for factor space (as U seems to cancel U as per $V\cong U\times (V/U)$)
Although the answer is given there but it seems complicated in that it uses some heavy machinery I think.

5:03 AM
hmm. @Koro I gave back all my group theory.... but first step should be to verify that $G\mapsto H\times (G/H)$ given by $g\mapsto (\pi (g), [g])$ is indeed a homomorphism. Then you check the kernel is trivial. Just like the vector space case?

That is a homomorphism indeed.
But the problem comes in showing that the image of G under this map is indeed $H\times G/H$.
That is, in showing the surjectivity of the homomorphism.
@CalvinKhor

5:27 AM
I think a preimage of $(h,gH)\in H\times (G/H)$ is $v=g\pi(g)^{-1} h$ @Koro

$g\pi(g^{-1})h H=gh^{-1}\color{blue}{h\pi(g^{-1})h^{-1} }H=gh^{-1}H=gH$

well $\pi(g^{-1})\in H$ and $H$ is closed under multiplication

yeah, you're right! I was thinking since H is normal so aha^{-1} is in H but that's not required here.
Thanks a lot @CalvinKhor. That's a very neat solution :).

you're welcome, i'm surprised i managed to solve it lol...
i think the normality was only used to get the quotient group

you should post an answer there.
of course, the map is well defined should also be shown in the answer but that can be done./
@CalvinKhor yes :).

5:35 AM
lol haha maybe. I'll see if i can bother to write it up properly
i wonder if i have ever written an answer to a question

There's always a first time :).

i asked a question with tag group-theory lol

I was thinking: let K be the kernel of $\pi$ so that $G/K \cong H$ so that the question reduces to proving that G is isomorphic to $G/K \times G/H$. So the map $g\mapsto (gK,gH)$ has kernel {e} but I had difficulty showing surjectivity :(.
@CalvinKhor I am sure this is exactly what the question maker had this in mind.

:) its not because im good at this, i've probably seen the solution before
@Koro where is the finite dimensionality of the quotient used in the vector space one?

6:01 AM
@CalvinKhor So first I showed that if $v_i+U$, i from 1 to k is a basis of V/U then v_i’s are linearly independent.
So for every v in V, $v+U= \sum c_i v_i+U$ whence $v=\sum c_iv_i+u$ for some u.
Then I considered the map: $v\mapsto (u, v+U)$
this map has the right properties needed to prove the result.

i always find algebra (as in groups, category theory) very opaque. perhaps too abstract for my engineering mind...

same @copper.hat
@Koro maybe a hamel basis lets you drop this....? Or is there a way without talking about bases....?

6:17 AM
@CalvinKhor Perhaps using one of the isomorphism theorems will save using basis.
I’m not sure :(.

@CalvinKhor Much appreciated. Albeit my advice would be to become financially stable as early as possible :-). Long story.

I guess if someone gives you a vector space homomorphism \pi:V\to U you can do the same thing, just without a basis maybe not so easy to make $\pi$ yourself?

2 hours later…
8:16 AM
May I ask if anybody could take a look at my question regarding the non-existence of a diffeomorphism between a circular annulus of radii r and s, and an open disc of radius r, with the same centre?
1

There is one example in my script about an application of a differential $1$ form in proving some subsets of $\Bbb R^2$ aren't diffeomorphic. As far as I've understood the explanation, we used a $C^2$ diffeomorphism and I don't understand how it's non-existence implies the non-existaece of a $C^1... 3 hours later… 10:53 AM Proving$\sqrt[3]{2}$is irrational using FLT is circular argument lol I didn't know that 11:06 AM @onepotatotwopotato why? 422 Irrationality of$2^{1/n}$for$n\geq 3$: if$2^{1/n}=p/q$then$p^n = q^n+q^n$, contradicting Fermat's Last Theorem. Unfortunately FLT is not strong enough to prove$\sqrt{2}$irrational. I've forgotten who this one is due to, but it made me laugh. EDIT: Steve Huntsman's link credits it to W.... I am not sure. FLT for n=3 was proved by Fermat, and I am not sure that his proof relies on this fact (however, his proof is also by infinite descent). @copper.hat So you don't like Haskell either? 11:30 AM was hamburger derived from hamburg? 12:02 PM @KarlKroningfeld Thanks! 2 hours later… 2:00 PM @Yai0Phah I don't know the proof of FLT in detail but to use that argument maybe it should be note that the argument is not circular because one might suspect it's circular. 2 hours later… 3:44 PM @Yai0Phah I like Haskell, but some aspects are a bit challenging, like monads. 2 hours later… 5:56 PM @NotTfue hamburger is the derived functor of hamburg 2 hours later… 7:27 PM Does this quiz question make any sense at all? "The mean and standard deviation of a population are 200 and 20, respectively. Sample size is 25. What is the mean and standard deviation of the sample distribution respectively?" Choices are 200 and 4, 16 and 25, 8 and 50. 200 and 4 seems most plausible, but can I even answer this without seeing the sample? 8:05 PM The variance of the sample mean will be$ \operatorname{var} \overline{X} = { 1\over N} \operatorname{var} X$. That makes no sense to me, just as the question made no sense to me. 8:27 PM$\operatorname{var} ({1 \over N} \sum_k X_k)) = {1 \over N^2} \sum_k \operatorname{var} X_k = {1 \over n} \operatorname{var} X$, assuming iid. But why do we care about the variance of the mean? And to be specific, your sample size is$N$, so$1\le k\le N$. It is the basis of Monte Carlo integration. Huh? What does it have to do with the question that was asked? the sample mean will be the distribution mean, the$\sigma$of the mean of$N$samples will be${1 \over \sqrt{N}} \sigma$. Why are you saying the variance of the mean? You mean the variance relative to the mean? 8:34 PM you are losing me You're the one losing me. :) variance of the same mean to be more precise. Who is talking about the standard deviation or variance of the mean? You're using language I've never seen. Variance is computed of the population relative to the mean. Anyhow, I know essentially NO statistics. I did teach probability, but no statistics. Why do we expect the mean of any sample (no matter how small) to be the same as the mean as a giant population? linearity of expectation. Ah. 8:48 PM not to be confused with the sample variance which would be$s^2 = {1 \over N-1} \operatorname{var} X$(an unbiased estimator of$\sigma^2$). Yeah, that whole N vs. N-1 thing always bothered me. i get lost in the terminology. Someone once explained it to me so that I understood, but poof it's gone. Typical in health sciences reports is also the standard error (SE) which is the square root of the sample variance$s^2=\frac{1}{n-1}\sum^n_{j=1}(X_j-\overline{X})^2$, which is use primarily in HEOR from probabilistic sensitivity analysis in cost-effective analysis of new medical technologies. 9:11 PM Regarding Bessel ODE. This answer can-t-see-that-an-ode-is-equivalent-to-a-bessel-equation gives transformation which when applied to modified bessel ODE allows one to find possible solution to second order ode in terms of modified bessel functions if solution exist. Any one knows where this is published? I am looking for similar transformation but that works on regular bessel ODE also. The transformation in the above answer$\eta=\frac{y}{x^\alpha}, \quad \xi=\beta x^\gamma$did not work for standard Bessel ODE. It does work for the modified Bessel ODE. 10:12 PM ted: it is to remove bias. empirical reasons for using an unbiased estimator are not always present, but people remove bias as if it were the plague. I've just never truly understood how$N\rightsquigarrow N-1$magically does that. nobody truly understands it. they just do it. it is our heritage. munchkin was running in a fabric store when she should not have been, got her foot stuck on something, and tripped into a metal shelf. we went to urgent care to see if she needed stitches (no, but they thanked us for bringing her in). she looks like she was in a pretty epic fight. bloody cut on the bridge of her nose and another on the eyebrow. ouch ... could have been way worser! This is what happens when you shouldn't be out and about ... yeah, exactly. i have a scar on my face from a similar situation but it is visible only if i shave. Tests came back negative? 10:21 PM munchkin got a special lunch of chinese takeout because it was the nearest thing to the doctor's office. the cat dined on beef that the munchkin did not eat and seems to be carrying herself with the swagger of one who has recently eaten beef. still waiting on the test. I never feed my cats human food. we don't feed her anything that has been seasoned, but sometimes minimally cooked and unsalted protein. she's also been known to graze our dishes if we have to get up from the table before we can clear them (e.g. because munchkin needs to go to the bathroom). i've seen livvy clean up a bowl that had vegan cauliflower soup in it. Screech hasn't pulled that sort of thing on me. once he does, it's all over. She :P 10:26 PM oh right, i forgot. now livvy's hunting a spider. she keeps letting it get away so she can chase it, and knock over what it's hiding under. How very feline. Our bugs mostly perch up at the ceiling, but they still drive Screech nuts at night when the light illuminates them. cockroaches and crickets were a near-constant presence in our old house. we loved her for that although we tried to prevent her from eating her prey. A physicist who was perusing my multivariable math book and pestered me is now sending me math issues he's stuck on in his research. He's got a convergent improper integral he's trying to differentiate ... and for reasons I don't understand, he's trying to differentiate it and second-differentiate it at the endpoint where the function blows up. that's where the magic happens. as copper would say, -1/12. Well, if cats can eat mice and birds, why can't they eat cockroaches? Except for the issue that the cockroach might have ingested poison beforehand. 10:29 PM i have the same concern with all three, that the prey has been in contact with poison. yeah. poison is nasty stuff. people set it out without thinking about this. I think that one of my cats in GA might have died for exactly that reason. It was either that or the coyotes. one reason a mouse or bird might be limping around and easier to catch as prey is that it's dying. it's a nasty way to die, too. a lot of them prevent blood from clotting in small doses, and in large doses you end up with one enormous internal injury. my daughter's day care switched to mechanical traps after i pointed out the effect on local birds to them. they were really cool about it. i think literally one week after i sent the email i saw mechanical traps instead of poison. which makes me think maybe they have too much money. they should have complained about not having the money to switch over. You could have endowed a murder or two. 10:47 PM @TedShifrin If I understand correctly, this is a purely algebraic fact. Let$X$be a probability space,$A:=\mathbb R$and$R$the$A$-algebra of real-valued measurable functions on$X$("probability variables"). Then the expectation$\mathbb E\colon R\to A$is an$A$-linear augmentation of the structure map$A\to R$, and the covariance$\operatorname{Cov}(f,g):=\mathbb E(fg)-(\mathbb Ef)(\mathbb Eg)$is a bilinear form which measures how$\mathbb E$fails to be a ring morphism. Now consider$S:=R^{\otimes n}$and for every$f\in R$, let$f_i\in S$denote$1\otimes\cdots\otimes f\otimes 1\otimes\cdots$(where$f$lies in the$i$-th factor), and$\mathbb E$extends to$S\to A$. My initial response is — You've got to be kidding. Note that the variance$\operatorname{Var}(f)=\operatorname{Cov}(f,f)$. Yes, from teaching probability I know the definitions of covariance and variance ... :) Why are we looking at$R^{\otimes n}$? Oh, if your population has$n$entries. It is then direct to check that$\operatorname{Var}(f)=\operatorname{Var}(\frac1{n-1}\sum_{k=1}^n(f_k-n^{-1}\sum_{j=1}^nf_j)^2)\$.

So how does this explain leslie's comment about bias/unbiased?
I should go back and look at the probability book(s) I perused when teaching.

11:07 PM
oh, that it fixes the bias just falls out of the formula. i don't know the intuition behind it.
you hear weird bs about 'degrees of freedom' which does not seem to be well understood but partially explains the appearance of N-1's.

If I remember correct, the probability book that I read just phrases this as a theorem and gives a proof. I tend to think that it is an algebraic fact.