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.
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?
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
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 $U\times (V/U)$.
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.
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?
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.
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?
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?
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...
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).
@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.
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?
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?
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.
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.
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.
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.
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.
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.
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.
@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 $S\to A$.
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.
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.
