@Goku no luck needed :-)

@ZaWarudo Somewhere there you need to assume that $f_n(t) \ne -1$ for all $t$.

can anyone help me how modularmultiplication works

Is mathjax in chat now inbuilt with \rm font?

$f(x)$ and $\rm f(x)$

hmm

Hey! How are you? We have the function $f_k(x)=\begin{cases}1-kx & x\in \left [0,\frac{1}{k}\right ] \\ 0 & x\in \left [\frac{1}{k}, 1\right ]\end{cases}, k=1, 2, \ldots$ . Is the supremum $\|f_k\|_{\infty}=\sup_{x\in [0,1]}|f_k(x)|$ equal to $1$ ?

We get the max value at x=0 which is 1, or not?

Does anyone know whether there is a polynomial $f(n)$ with integer coefficients (can also be constant) such that $$10^n+f(n)$$ is squarefree (negative squarefree values are allowed as awell) for every positive integer $n$ ?

@MaryStar Hi, I am fine (concerning health)

@MaryStar I would be tempted to say yes, it seems to me a segment from (0,1) to (1,0) for k=1 whose right end gradually moves towards (0,0) as k grows.. but maybe I am missing some catch, seems too simple

@MaryStar note that that the max value I was taking about is the max value of $f$ but I am not sure what you are indicating with | and || in your question. Is that a measure of some kind?

I made a comment on Disintegrating By Parts' answer, and I was wondering if someone could help me.

In the comment, I ask what is k and why is $T^k x \neq 0$?

Perhaps because of the assumption that $T^nx\ne0$ (I am assuming they meant $k\lt n$, but that is not explicitly stated)

Do mathematicians need integration? I heard it's needed more for engineers and such

@Ajay From the US

@TedShifrin I have solved my problem with using bernoulli's inequality. i could not have quite understood your argument but it looks like they are related. in any case, thank you.

to recall, i was trying to show that $\sum_{k=1}^\infty p^k = \frac{p}{1-p}$ with epsilons and Ns.

i have applied bernoulli's inequality: since $0<p<1$, i have $(1+(1/p - 1))^N \geq 1 + N(1/p - 1)$, therefore $p^N \leq \frac{p}{p+N(1-p)}$.

ilike: 'mathematicians' is pretty broad. also sometimes hard to say whether something 'needs' integration. certainly it is useful in a very large number of fields whether or not it is strictly necessary.

@leslietownes Alr, thx

Given that $g(x)=\int_{[x,1]}\frac {f(y)}yd\lambda (y)$ for x in (0,1], where f is integrable on (0,1], how to show that g is measurable?

koro: could you prove it for simple functions f?

Leslie: no, the y in denominator is a problem.

so not even for characteristic functions (indicator functions).

@oneofvalts Yes, I was thinking of Bernoulli for $(1+y)^n$, $y>0$, because that's more intuitive. Then set $1/p = 1+y$.

am I the only one who's experiencing this? $f(x)$ and $\rm f(x)$

i'm seeing rm too.

@TedShifrin ah, i see.

koro: OK. for f a characteristic function of a subset of (0,1], isn't g monotone?

yes, that's true!

monotone is mble.

For $x\gt y$: $g(x)-g(y)=\int_{[y,x]}\frac {1_E(t)}td\lambda(t)\ge 0$

Similarly for non negative simple functions.

thereofore for any non negative $f$ by monotone convergence theorem.

Greetings to Munchkin & Olivia.

And for general f, $f=f^+- f^-$.

the result follows :-). Leslie: is my understanding correct?

Are there any interesting properties which are true for Product space $X^{\alpha}$ but not for $X$?

when $X$ is a topological space or manifolds and $X^{\alpha}$ has product topology.

ted: livvy says meow.

Screech meows back. She's right by the screen.

koro: looks good to me.

thanks a lot :-).

This was such an amazing solution.

most of the times projection map transfers the properties to original space, but does it happen every time?

Look for properties that are not preserved by continuous, open maps?

koro: those basic approximation-by-simple-functions theorems are pretty useful. as is the fact that the set of measurable functions is closed under pointwise limits.

@robjohn Hmm...That is very confusing; it's not clear how $k$ arises. Also, if $k < n$ and $T^k x = 0$ but $T^{k+1}x = 0$, then won't this imply $k < n < k+1$, which is impossible?

@TedShifrin yeah right. I meant something "interesting", but "interesting" is subjective.

the one taught in class goes something like this: Define $A=\{(x,y)\in (0,1]\times (0,1]: x\le y\le 1\}$ and define $F(x,y)= \frac {f(y)}y 1_E (x,y)$. Then show that $F$ is integrable (by Tonelli theorem on |F|) and then by Tonelli's the x section of |F| is measurable.

But I don't quite like this one.

Disintegrating By Parts' answer doesn't really make much sense (at least to me).

Well, can you answer my question, @PNDas? Are there any such properties?

A lot of important things are preserved by continuous functions.

@leslietownes yes, I agree.

Leslie: Axler's measure theory is also amazing :-).

I learnt Cantor function from there :-).

math.stackexchange.com/questions/3364/… properties preserved by continuous functions.

Image of simply connected space is not simply connected.

Yeah, but if $X$ is not simply connected, nor is $X\times X$.

Notice that I said continuous and open :)

@user193319 please see my comment.

@TedShifrin right, I didn't read that message correctly. Sorry

@Koro Ah, so actually $k \ge n$?

@user193319 I don't see why that implication holds. Perhaps I need to look at the question again.

@user193319 with the assumption in the post, yes.

@PNDas Note that you also have $X$ embedded as a closed subset of $X^\alpha$. So the property can't be inherited by (closed) subspaces, either.

Ah, okay...I think I see what's going on. Let me think about it for another minute.

@robjohn Thanks for correcting my sloppiness. I added a comment to yours this morning.

koro: i didn't realize he'd done a measure theory book, is it new?

yes. the book came out in 2020.

I think he's trying to outdo me and, then, Lang. :D

haha

i saw via twitter he is/was gradually releasing an updated version of the linear algebra book free on the web.

he previously had an edited-down free version of it

yes, fourth edition will be freely available.

what a communist

not like profiteer ted here, hoarding his multivariable mathematics millions and getting steadily more right wing as he ages

Right. Next I'll be buying out Musk's Twitter.

or bankrolling the 'ted news network'

Just like Ted Turner?

I don't think align works in here?

Oh, it does.

I still think looking at the reciprocal is most intuitive :P

@TedShifrin The addition of space between lines using [6pt] does not work in chat

@robjohn that's also nice, thanks (:

Ah, that makes sense, @robjohn. Thanks.

The links also work weird in a multiline comment

Did you see my addendum to your $Hz=\lambda\bar z$ answer?

Is there a difference between compiled-together lecture notes and a book? I would think lecture notes aren't written like books and just outline the most important things but that doesn't seem to be the case, lecture notes that are put up online can easily have 300+ pages and read just like a book

Depends how polished the lecture notes are. Often at the level of grad courses, one sticks an exercise here or there, but not as seriously as in a good text.

@TedShifrin I did. Looking at it.

My diff geo notes are a (short) book that I just didn't want to bother making a book. Besides, I no longer trust any publishers.

I should probably use lectures notes more often, thanks

To me, the exercises are the most important part of (particularly an undergraduate) text, but even with beginning graduate texts.

@TedShifrin Same, there are lecture notes that include exercises at the end of each chapter

nothing replaces an in person lecture

@copper There are some pretty horrid lecturers, however, who actually write semi-decently.

@TedShifrin this is true.

i am just moaning about the loss of in person contact due to covid

What's a person?

@TedShifrin But yeah, most lecture notes don't have that many problems as books (if any), so books will always be my go-to resource

@TedShifrin :-)

If you decide to learn undergraduate differential geometry, @ILike, you will find that I have zillions of exercises in my "notes," ranging in difficulty from routine computation to challenging.

This came from using those as a text (in progress) for at least the last ten years I taught the course.

@TedShifrin it is a big sad that after 62 years of human interaction, a couple of years of covid sets one's (my) social skills back a few decades

@copper You mean to say that as a typical math nerd, you had weak social skills to start with :D

@TedShifrin i was definitely a nerd (a 'swot' was the term back then), but i always enjoyed social interaction, even if my skills were 'evolving'

irish society was fairly tolerant of evolving social skills

not studying your catechism, well, that was different story

convex seems to have gone out of fashion, except for the odd geometry/polygon question.

it was very strange (wrong word) to meet up with my research advisor after a few decaces

Very odd that you ended up chauffeuring him to the doctor after all these decades ...

another of his students was supposed to do it but the appointment was changed, so i brought him for his appointments. still sharp as a razor. a holocaust survivor.

the first one was suppose to be a checkup, but ended up taking 5h. highland hospital is a nightmare.

meeting him reminded me of how little i have accomplished, always disturbing.

Well, since my adviser was one of the math superstars of the 20th century (and likewise my co-adviser), I could never amount to much by comparison. I took pride in a few good papers and lots of good teaching (and books that leslie disparages) :D

It's a bit sad that your adviser has no friends around at this stage who can help out.

I had not heard of Highland Hospital. Google tells me it's Alameda's health system flagship.

i met someone there on the first visit who gave me her number. i called her before the second visit and she had a wheelchair lined up for us which transformed the experience. there are good people out there.

i'm not religious, but certainly like the idea of karma :-)

Indeed there are. Sadly, due to insurance crap and Covid, so many of the good people among health providers are giving it up.

yep, when the obamacare started we tried it for a while (because united healthcare decided to end the insurance), but our doctors all dropped up after 6 mos.

@robjohn I am not sure whether it is a good idea to use MathJax to achieve some rendering effect of non mathematical texts.

What does that even mean, @Yai0?

obama was suppose to take away the employment tax break for health insurance (probably the single biggest distortion) but he capitulated.

I was referring to $\text{(1):}$

Well, don't fret. In January, the thugs will take away Medicare, Social Security, and Medicaid. Because they really care about the well-being of the citizens here.

I don't see what's wrong with robjohn's typical style of answer/explanations. I don't do it, but he loves itl

I sometimes (in answers on main) will put a ($\star$) or something over an equals sign and then explain it below. But not on multiple, multiple steps.

maybe i can go on my daughter's insurance :-)

@robjohn's answers are works of art.

Well, his style is singular. I'll say that we never have to wonder who writes these solutions :)

I am not talking about putting things on equal signs. I was referring to something in pure textual environments, just like using that to mimic enumeration.

But here it's the annotations on displayed equations, not enumeration. shrug

Suppose $T_g\in\mathcal D'(\Omega)$ is defined by $T_g(\phi)=\int f\phi$ for $\phi\in\mathcal D(\Omega)$. We know that for any distribution $T$, we can define a distribution $T\ast S$ and a smooth function $T\ast \phi$ where $\phi\in\mathcal D(\Omega)$. Suppose $\rho_{\varepsilon}$ be mollifiers.

We also know that $T\ast T_{\rho_{\varepsilon}}\to T$. But what about $T_{T\ast \rho_{\varepsilon}}\to T$?

From what I have done, after taking limit, I get something like: $T(\lim \rho_{\varepsilon})$. And $\lim \rho_{\varepsilon}\not\in\mathcal D(\Omega)$.

The first should have been $T_f$ instead of $T_g$?

Yes, now I can't edit it.

What is $f$?

I mean, what sort of input do you allow?

a locally integrable function.

OK, so formally, $T_{(-)}$ is a map $L_{\mathrm{loc}}^1(\Omega)\to\mathcal D'(\Omega)$? Is it continuous?

I don't know. I don't even know the topology on $L^1_{\text{loc}}$.

The TVS $L_{\mathrm{loc}}^1(\Omega)$ is the (projective) limit of $L^1(K)$ along restrictions, where $K$ runs through compact subsets of $\Omega$, I think. In particular, it is Fréchet.

Since $T\ast\phi\in\mathcal E(\Omega)$, then I think I can take the map from $\mathcal E(\Omega)\to \mathcal D'(\Omega)$. I know that in the topology on $\mathcal E(\Omega)$, linear maps are continuous if they are sequential continuous. @Yai0Phah

oh boy, highland (just catching up). my mom sometimes worked there. for a while she worked at a registry that would assign nurses to temporary stints at hospitals due to need.

i would expect it to have a pretty good gunshot trauma ward and who knows as to what else.

what does it mean if $(X_n)$ is an adapted process at $0$

@Overtherainbow That $X_0$ is $\mathcal{F}_0$-measurable? Would be my guess

I really have no idea. But I haven't thought about yours yet

it could be correct

well, here is the question I was talking about. if anyone has ideas, it would be great: math.stackexchange.com/questions/4547149/… thanks!!

@Yai0Phah That is a mathematical text. To match the tag fonts, which are definitely part of the math, one needs to use \text in MathJax.

$\text{(1a)}$ uses different fonts than what is gotten using non-MathJax

Why doesn't chat have mathjax support?

it does. see the sidebar with the chat info.

Please read the room description.

@Goku Chat does not have MathJax support by default because (1) SE doesn't really care about chat and (2) only a small number of the SE sites use MathJax, but chat serves everyone; MathJax induces overhead which a lot of users don't want or need.

math.ucla.edu/~robjohn/math/mathjax.html

But, as others have noted, there is a user script / bookmarklet which can be used to render mathematics.

Out of curiosity how much GPU power does it even take to render mathematics from a simple piece of code?

Surely not something by 3090 can't handle

(1) Rendering of mathematics is likely not done by the GPU, but by the CPU. (2) MathJax is built out of JavaScript, which is not a machine-optimized language. Pages with MathJax can take an extra half-second (or more, if there is a lot of stuff to render) to load. This extra half second is felt by people, hence SE made the decision not to enable it everywhere.

i notice it on my (admittedly old) mobile phone, it makes the site virtually unusable

lucky for everyone i also have a desktop

no tablet?

works fine on my $250 n10

tablets are for addicts

:(

(⁠╯⁠°⁠□⁠°⁠）⁠╯⁠︵⁠ ⁠┻⁠━⁠┻

@leslietownes Yeah, so lucky.

@user4539917 is that rage emoticon?

flipping tables, the universal symbol of rage

or tablets

What happened to flipping birds?

flipping the bird is so 70s

i do that too

i stick with old fashioned middle finger

sometimes both in concert

maybe with a little shocker motion, just for emphasis

i used the double birdie yesterday to illustrate a point

some thug on bart gave me the bird a few weeks ago. i gave an artistic rendering of the bird in return. he made some ineffectual threatening gestures.

i should probably up my life insurance. i can't quite back up such malarky anymore.

my daughter hasn't picked it up yet, thankfully

she did once come home from day care and say "people don't like it when you do this. [birdie] why?"

but she didn't get that from me

i'm sure i have recounted the "fck, fck, f*ck" story from my 3yo daughter when she repeated my commentary on missing bart.

i have never heard my mother utter anything resembling a curse.

@copper.hat I believe that my daughter was four years old the first time she called me an "assy-hole".

wow!

I was really quite adorable.

my two are fairly conservative language wise, but when they are in heated discussions with friends they seem to channel their inner dad.

my daughter's first f-bomb was in her late 2s, when she dropped a toy

it was both shocking and strangely rewarding at the same time.

i like the concept of late 2s

the days of washing their mouths out with soap are long gone

that was still used in schools when i was evolving

carbolic soap, none of the delicate ivory crap

Did it work?

personally i think the threat of a little physical action is a good motivator when applied appropriately.

i managed to escape the soap treatment.

peeling potatoes & kitchen duties were among the punishments.

slapping with sticks & rulers.

stings a bit, particularly in winter.

but only in one of the three elementary schools i attended

Nowadays law suits would be filed.

one 1/2nd grade teacher was a little heavy with one of the kids. the next monday morning he showed up at school with black eyes and a few bruises. different world.

we had 2 grades/classroom in that school

the most serious of threats was to contact parents

Sanity check: when someone sends you a message which starts with "I'm not judging you," then goes on to explain some action you took, they are judging you, right?

as a rule, yes

Yeah, that's what I thought.

yes

(⁠┛⁠◉⁠Д⁠◉⁠)⁠┛⁠彡⁠┻⁠━⁠┻

looks like instructions from a lego set

written by an ikea engineer

Pages with MathJax barely take any longer to load on my phone

Although mine is fairly new (iPhone 13) so that might be why

you must have a nokia e51

@Goku Good for you.

Not everyone has access to the same technology as you.

i need to put the hand crank in to see the graphics

And because Safari has exceptional JavaScript performance I believe

i think it is called tripescript now.

Wonder if my toaster can render this

another abomination of a language, joining perl in the bowels of linguistic hell

my toejam can render it

actually, i think my raspberry pi v1 can render it

@copper.hat Oh, I have one of those somewhere!

I've noticed my frame rate in game be lower when I have this site running in the background

I wonder where it is...

Especially in a 64-player multiplayer map that is a CPU hog

@XanderHenderson hell, my son's pinewood derby car has a micro in it instructables.com/Pinewood-Derby-with-programmable-LEDs

that was during a slow consulting period of my life

Ironically, back when the silicon shortage was at its peak, car manufacturers were affected the most

i hate that cars nowadays are so silicon/compute dependent

i do have another less techy instructable draft, basically a small scale sundial for estimating latitude. it was for the kids in elementary school, but i got more consulting hours then so never completed the writeup.

inspired by jantar mantar in new delhi