Here's some random problem: if all humans on earth flipped coins for 1 year and it takes 10 seconds to flip them, what is the expected highest "run" of either heads or tails.

coins are magic 50% for either side

they still need to eat and sleep tho, but otherwise they can flip instead of work

mmh sounds like coinflipping should make a machine, it can throw like 100 coins at once :D

@robjohn ,thanks for the conversation yesterday. It is maybe obvious and it makes sense intuitively, but could you motivate this statement some more? How can one be sure about that what is left will be bounded or at least tend to $0$?. I'll be a bit slow in replying henceforth.

Isn't there a typo in this answer: math.stackexchange.com/a/4202578/425395

When they compare integral, they do not integrate $|M_f g|$, which is weird

@robjohn , what do you mean by "...with no detail about how it goes to $0$...?

@robjohn , if it interests you, there are or what I believe are lecture notes from Princeton University about density estimation available online (the first link when you google "princeton kernel density estimator") in which the expectation of the KDE is nicely derived using Big-O instead of little-o.

In the Taylor expansion of the probability density function, the Lagrange form of the remainder is used. How this remainder term is dealt with is very well motivated in the lecture notes, using absolute convergence and the triangle inequality for integrals I think.

Anyway, there seems to be no explicit counterpart to the remainder when using little-o and I guess having $f\in C^{m+1}$ (which is a requirement for using Big-O) would require changing the conditions on the kernel as given in my question. It feels like the assignment related to my question is written so as to use little-o.

@schn it could go very slowly to $0$ or quickly. Careful use of big-O will allow better estimates to be used and more of a calculus can be developed. "Bounded by a constant" is better for estimates than "vanishes".

@schn that would be a better exposition, I bet.

@schn why is $f\in C^{m+1}$ a requirement for big-O?

Let $g: \mathbb{R}^n \rightarrow R$, let $\vec a, \vec b \in \mathbb{R}^n$ and $a^*, b^* \in \mathbb{R}$ with $g(\vec a) = a^*, g(\vec b) = b^*$. It is not generally the case that $b^* - a^* = g'(\vec a)(\vec b - \vec a)$, right?

Not even $g’(\vec c)$.

argh I knew I was missing something

I assume by $g'$ you mean $\nabla g$

the linear map s.t. $\lim \limits_{\vec h \to \vec 0} \frac{f(\vec a +\vec h) - f(\vec a) - f'(\vec a)\vec h}{||h||} = \vec 0$

you can turn it into a one dimensional question by defining $h(t)=g(a+t(b-a))$

but $h'$ is not equal to $\nabla g$

not even the right dimension :-)

He applied it to (dotted with) $b-a$.

I'm reading a proof that if $G$ is a finite $p$-group, and $r_s$ is the number of subgroups of $G$ having order $p^s$ (where $\vert G\vert=p^n$, and $s<n$)), then $r_s$ = 1 mod p. There is one (probably trivial) step that I don't understand: let $K$ be a subgroup of order $p^{s+1}$, and let $H_1,\dots,H_b$ be its subgroups of order $p^s$. We then know that the $H_i$ are normal in $K$, and $H_1H_2=K$.

I didn't see the dot. $h'=\nabla g\cdot(b-a)$

Assume now that there is some $j$ such that $H_1\cap H_2$ is not contained in $H_j$. My book claims then that the only subgroup of order $p^s$ containing $H_1\cap H_2$ and $H_1\cap H_j$ is equal to $H_1$, which is equal to $(H_1\cap H_2)\cdot(H_1\cap H_j)$. I don't know why we have this equality.

So I was wrong. Didn’t pay attention to scalar values.

wait, isn't in this case $g'(\vec a)^T = \nabla g$?

The MVT should always be a bound, not an equality.

oh, right. the gradient has a definite norm while the linear map has a maximal norm

huh?

No, identical.

i'll meditate on your answers, thank you

(very much)

When you map to one dimension.

oh never mind

I see it

also hi Ted, bye Ted, I'm off!

here one minute, gone the next.

Hi @AkivaWeinberger

i don't merit a hi. that's fine.

You're always here so you defeat the purpose of a greeting. Hi, Lesie :P

namaste.

i am indeed frighteningly almost always here, although i did do quite a lot of work this morning. mostly before my daughter got up

Why frighteningly? It's a nice place to be

i am not here. i am with Magritte eating apples.

I wonder how many people would get that reference, copper

then she got up and ruled the house for about 90 minutes. running around and avoiding being put in clothes for her day care. very bad behavior this morning. at one point she was whacking my wife with a cat feather toy.

we are a very cultured group, balarka.

this is like a 19th century salon. all of the important intellectual matters are open for consideration.

Indeed so. And your daughter is quite the energetic kid.

she was running around naked like a gremlin and forcing us to put her in clothing. it's possible for her to just put it on, but these days she decides not to do that.

some of us have a more diverse practice. copper defrauds large corporations for a living, and i defend large corporations from allegations of fraud for a living.

my accumulation of culture (little that there was) stopped as my kids finished high school.

i introduce design bugs into consumer electronics in the most insidious ways.

My father used to read a lot but as he grew old he got bored with fiction or art. Age has something to do with an increased curiosity in historical reality.

He's more interested in facts and tidbits now

that's interesting. my dad was a newspaper man, mostly reporting and features. he has not read any fiction in 50 years and you can not get him to read a newspaper unless he is on the clock and being paid for it.

he's functionally illiterate.

That's a counterexample to my claim! Interesting

he doesn't care about made up stuff or whatever just happened. he's more interested in other things.

i can sometimes get him to watch a tv show, but i have never been able to get him to read a book.

he's amazing with my daughter, although that is entirely phone and zoom. she is his only grandchild. they can talk for an hour about ducks (a recent obsession of my daughter).

i've also made my best friend talk to my daughter for about an hour about ducks.

i might not be the best person to phone, is what i'm saying.

Exactly

surprisingly, there isn't an hour of conversation about ducks. it's mostly the same things over and over, and then weird detours into imaginary scenarios when ducks are doing things that ducks don't do.

my best friend put up with it, though. i bought her a lavender plant for her birthday and she sends me photos of how it's doing. those can grow like weeds.

we have some basil in a pot outside and it went insane. we have more than we can ever use.

Hmm, I have never seen a lavender plant or rather do not consciously remember identifying one as such

it was the cheapest plant that wasn't cut flowers that i could get from the flower service. that is how i selected it because that is the kind of friend i am.

My dad does a bunch of indoor gardening; on the balcony and such. I don't know what half the things are but it looks like a frigging forest

we are terrible at it but are somehow having good luck with the basil.

everything else is about to die

Sigh. I can count how many times I have even went to the balcony in the last 1.5 years. I just crawl in my damp room.

Oh yeah stuff dying is part of the process I feel. Lots of things just don't survive.

Hi @vitamind

i have a balcony that i don't use, too. it's right there, literally a few feet from my bed. i can't be bothered.

it's a nice view of some trees and a power plant.

There's a park staring at us on my end. All full of weeds and grass at human height.

A few more years and it'll be be a post apocalyptic marshland

one of my favorite memories as a kid was playing in our backyard where we had 3-4' grass when i was less than 4'. the cat loved it too.

we almost bought a house at sea level. thankfully, sanity prevailed.

If I get a patch of land in the near future I'd probably grow the grass until the whole house is gone.

Hopefully my family isn't leaving me some for fortune.

we have this weird contractual arrangement where the same landscapers come and do our entire neighborhood so it all looks the same. i like that i don't have to do it and that neighbors are not growing grass until their houses are gone. but it's a little weird. the contract says they can do the work whenever.

sometimes it's 6 in the morning.

sometimes it's 6 in the evening. we'll go out and get the mail and see someone doing the grass.

Hate curation. Unless they're cleaning the drains.

Cuz unlike grass that shit stinks

In my undergrad I lived in the college campus which was literally a giant forest

@leslietownes by "until their houses are gone" do you mean "until their houses are hidden"?

yes, i think so.

Swallowed by the grass rather

@leslietownes I had thought of getting a retirement place on Balboa Island. It is sinking even without global warming. Now we are considering Mammoth Lakes (you don't need to remind me that the area is a super-volcano).

yeah i would not insure anything on balboa right now. mammoth makes a lot more sense

volcanic stuff doesn't necessarily correlate with all of this other stuff.

i love balboa though. the place we were looking at was in seal beach but it floods literally every time there's heavy rain and i have to drive around it.

we're on a hill where we are now. probably got another 50 years, easy. and then i'm dead and nobody cares.