@leslietownes thank you very much

novice, for a given subset $A$ of $[0,\infty)$, the integral $\int_A \frac{1}{\theta^2} x e^{-x/\theta} \, dx = \frac{1}{\theta^2} \cdot \int_A x e^{-x/\theta} \, dx$ is a measurable function of $\theta$ e.g. because it's the product of a continuous function of $\theta$ and a nondecreasing function of $\theta$

vrouvrou: thank robjohn's great answer (it seems like there's a link to one of those for almost every question) :D

Oh, thanks. Is that a standard analysis fact? I am not sure where I would find that

yes. your book or notes or some book somewhere should have a few fairly general theorems about properties of functions that ensure that those functions are measurable, and operations that preserve measurability

it's not too common to prove measurability from the definition, unless you're walking through a special case of a proof of a more general result

unless you're in a class or a part of a class where the whole point is to do that, i guess

Oh, I remember that sums, differences, products, max, min, etc. of measurable functions are measurable. I guess what you're talking about is just a less common member of that group. I'll see if I can find that somewhere

So I did the integral I mentioned above (as Riemann integrals), and I got a measure of about 5/6. It's pretty long and tedious though, so I won't show it

i thank you on behalf of the chat for sparing us :) but i suppose it is good hygiene to work out a definite integral every now and again

I'm trying to find the middle ground between showing that I'm making an effort, and not boring everyone to death

i'm doing 'heavy math' right now in the form of my taxes, which is actually just entering numbers into a piece of software.

i remember growing up when my dad did his taxes an actual calculator was involved. also, i think, a typewriter??? i don't know why

I'm just playing around on Desmos to see why that integral is a nondecreasing function of theta. What a slick answer. I wouldn't have thought of that. I wasn't thinking about the shape of the integrand at all.

yeah, there's nice pointwise behavior as t varies

for a more complicated family of functions you might have to use a more complicated general theorem. there oughta be a bag of them in your book. if not, demand a better book.

I'm using a book (near completion) written by my professor

not near enough completion, clearly

i'm kidding of course, anything involving foundations of measure theory is like: "works on all measure spaces, has a short proof, is useful: pick two"

I just want to try to get this straight. So $\mu_{x_1}$ is a measure from $\mathcal M_2$ to $[0, \infty]$. But for a fixed $A_2 \in \mathcal M_2$, $\mu_{x_1}(A_2)$ is a function from $\mathbb X_1$ to $[0, \infty]$. Does the theorem mean that I need this function to be $(\mathcal M_1, \mathscr B_{[0, \infty]})$ measurable, or $(\mathcal M_2, \mathscr B_{[0, \infty]})$ measurable?

the former, but i still advocate dropping all of those subscripts, it would be clearer

as much as people hate running into multiple letters, it's a lot easier to confuse M_1 and M_2 than E and F

this is my unsent note to the author of this book in progress

particularly having an exercise where "mu_1" immediately means two different things, yeesh

i guess a lot of this stuff gets caught once people read full drafts of chapters/the book

or, failing that, in the second edition, if there is one

I think I was confused because $\mathbb X_1 = [1, \infty)$ and $\mathbb X_2 = [0, \infty)$. I have trouble visualizing what's going on, so I thought maybe that missing bit from $\mathbb X_1$ was a problem somehow.

Re: notation, you might be right, but translating a theorem I barely understand into new notation introduces a new burden when I'm trying to solve an exercise. Maybe I should have done it, but I opted for (what seemed like) the path of least resistance

i don't think there's any abstract reason why you couldn't take theta smaller (maybe not 0, but at least anything positive), but who knows

whenever people talk about gamma distributions they might have actual applications in mind and maybe for the application theta >= 1 is the only thing of interest, or the only thing that matters

Gamma distribution can take any positive theta. (According to Wikipedia.)

the exercise as written is just inviting a lot of notational confusion for at least some readers. eventually, suppressing the names of functions is unavoidable, particularly in applications where you eventually want to write things so the underlying measure space doesn't even expressly appear

but in a beginning exercise it seems like it's asking for trouble

when i see a function of one variable, i wanna call it x or t or something and not "x_1"

and subscripts on subscripts are just cringe

since multivariable ted hasn't yelled at me i assume he fully agrees with all of these statements

this chat is so underpopulated this weekend i wonder if there's been a stackexchange equivalent of an irc netsplit

@leslietownes I could excite you talking about all the geometrical wonders I'm seeing arise form studying factorial designs and how I've for the first time seen the use in group theory.........I know geometry is your thing so.....

also Balanced Incomplete Blocks and the such....own little field of study.......so overwhelmed...

balanced incomplete blocks sounds like something my daughter would try and fail to do with her toys

@leslietownes Thanks for your help. I wouldn't claim to be close to really understanding this stuff, but I think I got through this exercise, at least.

one step at a time. as dc3rd loves to hear, nobody will ever be close to really understanding anything :)

😭😭😭

and some of us will always be close to misunderstanding everything.

@leslietownes isn't that what one plays Jenga with?

wow, yeah

@leslietownes I was noticing that it had been quiet in here for over 8 hours on a Saturday.

somewhere, a bunch of people are having a great party that we're not invited to

@leslietownes They must be at Ted's place.

clearly

asked in Matter Modeling SE

I got covid 19

Can someone look at this problem and tell how to continue?Prove that $G=${$x\in R^m:\rho_A(x)<\alpha$} set is open.$\forall y\in G$ we need to show that $\exists B_r(y),B_r(y)\subset G$.$x \in B_{\frac{\varepsilon}{2}}(y)$, $|x-y|<\frac{\varepsilon}{2}$,$\rho_A(y)<\alpha.$ $inf|x-a|\leq|x-a|\leq|x-y|+|y-a| $ . I am stuck here because I have only for $\inf|y-a|<\alpha$

$\rho_A(x) = inf_{a \in A}|x-a|$

@love_sodam Get well soon.

im looking for a screenshot of a deleted post

math.stackexchange.com/questions/4407557/…

where can I find these type of practice problems? math.stackexchange.com/questions/252832/…

@unit1991 I think the notation here is confusing, this is really much simpler than it looks

Take any $x\in G$, then $B(x, \alpha)\subseteq G$

this works in any metric space, really

or like Kavi Rama Murthy mentioned in a comment to your question, you can prove that $\rho_A$ is Lipschitz continuous, and so $G = \rho_A^{-1}((-\infty, \alpha))$ is open as preimage of open set.

that solution is even neater

@Koro what are your prime interests? Just curious

@Jakobian $(-\infty,\alpha)$ is open in $R^m$?

in R

Problem is with $R^m$ it doesn't contradict itself?

$\rho_A:R^m\to R$

if you were to take preimage of a subset of $R^m$, that wouldn't be very smart, would it now

I see

@Jakobian Can you give some hints how show that it's Lipschitz continuous.

@unit1991 Given any $\epsilon\gt 0$, there exists an a' in A such that $r_A(x)-r_A(y)\le \inf_a|x-a|-|y-a'|+\epsilon$.

@Jakobian with reference to the practice problems that I mentioned that I'm looking for?

I have difficulty finding ideals of a quotient ring so I want to know where I can find some practice problems on that.

@Koro Thanks but I am not sure how this helps to prove

@unit1991: please ignore that. I didn't think it through.

@unit1991 $|a-y| \le |a-x| + |x-y|$ so take the $\inf$ on the left first and then the right to get $\rho_A(y) \le \rho_A(x) + |x-y|$. So if you choose $|x-y| < \alpha-\rho_A(x)$ then $\rho_A(y) < \alpha$.

@love_sodam Do you know how long ago you might have gotten it?

@copper.hat Thank you!

@dc3rd Block design is a huge field. The literature can be a little confusing because some of the early results were rediscovered by people who weren't aware of earlier work, so there are several different terminology / notation systems in common use. I assume you've encountered Kirkman's schoolgirl problem. There's a popular card game known as Dobble or Spot It!, based on block design.

@Koro no, in general

@PM2Ring oh, I was interested in block designs a little too, but I didn't know where to start learning them and gave up

I don't think I'll dedicate time to learn them now though

I got into block design via Spot It! / Dobble. Someone on the (now-defunct) XKCD forum asked about how to generate Spot It decks, and which combinations of parameters are valid. It's fairly easy to show which parameters are valid, but it's only easy to generate decks that correspond to a finite field.

Playing with that stuff can be a fun way to improve your finite field arithmetic skills. ;)

I find it interesting that a lot of work in block design wasn't done as an exercise in pure mathematics. It was driven by people who were trying to solve practical research problems: how to test the effects of varying a bunch of parameters efficiently. Agricultural experiments tend to take time, and require land to grow stuff on. So you want to use the time & land optimally.

Another application of block design is in organising tournaments. Here's an example from last year: math.stackexchange.com/q/4100315/207316

. o O ( blockheads ) ;-)

@Koro: This is very closely related to your quotient ring ideals issue. Two of my favorite problems: What are the quotient rings $\Bbb Z_6[x]/\langle 2x-3\rangle$, $\Bbb Z_6[x]/\langle 2x-1\rangle$?

morning, @robjohn

if you have a number of personalities you can use block design to decide which one to use each day, to maximize people guessing

@TedShifrin hello. We are supposed to get about 1.5" of rain here tomorrow.

@leslietownes Oooh, we like that! yes, we do!

huh, we're only supposed to get 0.5". wanna give me 0.5, so we're even?

We're supposed to get some, but I ain't done measured yet.

Is there an accepted pronoun for people who have more than a single personality? Are we stuck with just "we" and "us" and "our"?

forecasts are all lies, anyway. everybody knows the government uses block design to decide which plots of land get which rain

. o O ( blockheads )

you may have to ask the queen of england about that

the royal we

i hope the chat stays more active. yesterday i ended up doing a huge amount of work, not because i had to, but because i had few distractions.

Yes, it was very quiet and peaceful in here. No math. No politics.

Hardly any Ted.

even my daughter just wanted to color by herself, so i gave her about a 2'x5' stretch of butcher paper and she spent a lot of time filling it up with various media

She grew tired of abusing Olivia?

some mix of that, and olivia making very clear that attacks would follow.

Aha, she's finally wising up.

well, she still probably picked up livvy maybe 10-15 times yesterday. and another 5 this morning.

Interesting (?) when a totally false problem is posted.

hah, i was about to say 'helices??' and then saw the comments.

Indeed, generalized helices.

i remembered a correct version of a problem on generalized helices from my undergrad days. my memory works!

The correct result (even proved in my notes) is that $\tau/\kappa$ constant characterizes them.

There is a relatively new book with an alarming number of false problems in it. I don't know if that is the source of this one.

I've even emailed the author a few times after people posted questions on here.

@PM2Ring This is where I'm seeing it now. I'm taking a course in Experimental Design so the concepts have been introduced through that. I only discovered how vast the field is because I was looking for clarification on a topic..

dc3rd: if the fields weren't vast, you wouldn't need block design, would you? you could run a ton of lazily designed experiments in a single greenhouse.

Block design and finite fields are also useful in designing round-robin competitions (e.g., tennis).

or cockfighting tournaments

@leslietownes Little Jerry vs The Big Cock from Colombia.

I do hope you get the Seinfeld reference and this doesn't go off the rails....

ahaha, yes, i got it.

whew....

you'd be amused at how a bunch of geometry popped up in my reading about factorial and fractional factorial design @TedShifrin , but then again you probably already knew this....

cockfighting was a thing where i grew up. once or twice a year there'd be a story in the paper of some ring being broken up.

@dc3rd Nah, I don't even know what you're talking about.

alos involves group theory.......think of a bunch of cycles and generators......

Well, any time there's symmetries to consider, ...

that's all I rememebr from doing a Gorup THeory course.....WAAAY before I should've been doing it...... sigh....the wasted efforts...😪😪

speaking of symmetries.......did you get my ping about what P-set the question about showing that you can switch variables when taking partial derivatives of symmetric functions?

HI can anyone take a look please math.stackexchange.com/questions/4413963/…

Did you just look through the problem sets, @dc3rd? It's not that hard to do so.

The max/min stuff is the last few assignments of the (first) course.

I did....well the ones I think pertain to it. I think I know which question it is. You may have just not explicitly said the word "symmetric" function that's all.

@monoidal That problem is totally wrong.

The point was that even when the function is symmetric, the critical points need not be on the diagonal.

Most of the time (in typical applied problems) they are, and so students naturally assume ...

@TedShifrin Yeah how was the party?

@monoidal What you've written down is $\Bbb RP^{2n+2}$. Nothing to do with $\Bbb CP^n$.

@robjohn No party.

Yes I understood what you were saying about it, I just wanted to do the exercise myself.

@TedShifrin Oh, come on... that was the best explanation for the solitude here yesterday.

@monoidal The problem is total garbage. It should have been written as $S^{2n+1}$, first of all, but you have to take the quotient by $S^1$, not $\{\pm 1\}$.

@robjohn I am about to have my fourth extraction and implant done, so no parties for me for quite a while.

@TedShifrin ouch! Is that tomorrow?

Yeah, it should be $\mathbb{S}^{2n+1}$

@robjohn Wednesday.

$\mathbb{C}P^n = \mathbb{S}^{2n+1}/ v\sim -v$ right?

Well, the first part of it. Then 3-4 months later, the second part. Then 3-4 months later, the crown. Such fun.

Ack! so it drags on for months.

No, @monoidal. You should know better. Quotients of spheres by $\pm 1$ are real projective spaces. Just look at dimensions.

i remember eating a lot of soups.

I bought yogurt and a few cans of soup today. I'm also making a Chinese/Pakistani chicken corn soup.

Off to the physical bookstore to look for actual printed books.

@robjohn WTH are you talking about??!!

I think the last bookstore I went to was in Palo Alto, about 4 years ago.

No, that's wrong. I've been to one here a few times.

i went to a barnes and noble in michigan in 2011 or 2012. that's the last one i remember.

Sad comment on our lives …

hey there friends

long time no see

Heya Lucas

how you're doing Ted?

Bumbling along. You less overloaded, I hope?

yeah, thank God

oh, I'm finally on my last year :)

I hope grad school isn't as boring as taking the barchelor's degree

too much bureaucratic subjects

Would Math.SE be the right place to ask a question about math.dartmouth.edu/~carlp/PDF/paper88.pdf?

Assuming I don't have a particularly solid understanding of the math.

@forest yes

OK. Because I'm mostly wondering if there are any newer papers (that one is from '93) which can prove a stronger upper bound for the inequality than $\forall k > 2:p_{k,1} \le k^2 4^{2-\sqrt{k}}$. I don't know if that would count as a reference request.

Ugh, I keep forgetting mathjax doesn't work in chat.

is that against the guidelines? I see a lot of reference requests around

I don't know. I've never been on Math.SE before. I think it is on some sites.

MathJax does work, peek the description of this chat room

i don't know how a post on the network would fare as a pure reference request (although isn't there a tag for it?) but it's certainly fine on the chat.

LESLIE

hi!

> Access Denied - Sucuri Website Firewall

Can't access the tinyurl. :(

that is so not my field, however. i don't know of any more recent papers. i hadn't even heard of that one.

hello lucas. congrats on finally being on your last year.

thanks! hopefully it will, in fact, be my last year lol

@leslietownes OK. I've asked a similar question on Crypto.SE at crypto.stackexchange.com/q/99310/54184, but if that's not the right place for it, hopefully someone here would know.

math overflow might be a good site for this. it strikes me as the kind of place where if it wasn't instantly closed for some reason (i'm not clear on any site's guidelines and do recommend reading them), it would probably be answered quickly.

or perhaps, in a delicious twist of fate, answered quickly in a comment by someone who closes the question.

Everything I see on MO is so far over my head that I doubt I'd last a minute there.

hey leslie, do you know any algebraic/differential geometry researchers around here?

it just has that built in community of people who actually read math papers for a living and have a good grasp on improvements on stuff and the state of the art.

@forest yeah, that's probably not the right place

math.SE has some of those folks but not a critical mass of them.

lucas: the / in "algebraic/differential" is covering a lot of ground. if here means the chat, i know ted.

we do have some AG folks that pop in but i forget their names because i haven't seen them lately.

@leslietownes yeah hahaha i'm actually into AG but the subject i'm gonna study keeps showing up in differential geometry so either of the subjects should have something to say

Regarding what, Lucas?

Would asking a question about whether or not there is a stronger (but still general) upper bound than $p_{k,1} \le k^2 4^{2-\sqrt{k}}$ be considered cross-posting for the above-linked Crypto question, or is it specific enough that it would count on its own?

I have to study the Lefschetz hyperplane section theorem which talks about cohomology of certain hyperplane sections of alg varieties. but every single resource I've seen talks about Morse theory and smooth (in the usual, metric sense) and that just blows my mind

Morse theory and Lefschetz pencils are the natural way to see the stuff topologically. But you can find a straightforward proof in Griffiths/Harris.

BTW, that is not differential geometry at all. It's algebraic/differential topology.

my bad. in fact it's not diff geometry

I get annoyed when everyone says differential geometry every time a manifold lurks around a corner.

@TedShifrin hmm. I've seen Griffiths and Harris but the whole differential forms part gave me anxiety...

ted, do you know where i can get exhaust manifolds and headers for a 1968 volkswagen beetle? i heard you do differential geometry

hahahahahahahahahaha

Well, get over it. Differential forms are central in algebraic geometry.

@leslie I already explained in here ages ago that I first bought Spivak's Calculus on Manifolds thinking it was about exhaust manifolds.

Anyhow, the Lefschetz Hyperplane Theorem and the Kodaira Vanishing Theorem were proved to be equivalent.

I see

uhh, pretend I'm a first-year student since I feel dumb to ask this. how can you use a theory about functions from the interval (which is like T1 T2 T3 T whatyouwant) to a manifold to study varieties whose topology is something as weird as Zariski?

Because in complex algebraic geometry you're not using the Zariski topology. You're using the usual topology.

You should read about GAGA. :)

oooh.

my advisor should have told me that :p

You can of course google and find all sorts of sources. Griffiths has some nice lectures on GAGA in the Princeton yellow series. Algebraic and Analytic Geometry, Griffiths and Adams.

my last scientific research which will also be my final paper will be made in two parts: basic AG (hartshorne chap 1) and lefschetz (since I already had algebraic topology). I'll guess the topics will be strongly disconnected

Doesn't seem like the cleverest combination.

Although chapter 1 of Hartshorne is lots of concrete examples.

You might also look at Harris's introductory Algebraic Geometry.

@TedShifrin that's actually not the first time I've heard about GAGA, but it'll be the first time I'll mix the topologies

The whole point of GAGA is mixing the topologies.

@TedShifrin so unfortunately I'll stick to the most algebraic approach to Lefschetz

I don't know what that means. The proof I presented when I taught it was the proof in Griffiths/Harris, based on sheaf cohomology and Kodaira vanishing.

I've seen a quite old book called "Homology theory on algebraic varieties" by Wallace. I think that's the most algebraic ( = forget about metric!) approach I've seen..

If you mean algebraic topology when you say algebraic, that's very confusing.

I mean algebraic in a totally incorrect sense. In the sense that doesn't know about GAGA

Milnor's book is the usual reference for the Morse theory proof.

I really don't know how to approach this. Sincerely it's just trying too hard to avoid talking about GAGA. I'll schedule another meeting w/ my advisor to change the plan somehow.

(I guess, since even historically the result was crafted using topologies other than Zariski)

When people working with the Zariski topology do algebraic topology with varieties it's called étale cohomology, etc.

@TedShifrin I just hide when a manifold lurks around a corner..

oh, forgot to say. thanks leslie and Ted for all the help. :)

i don't know how i helped, but i'm sure it was more than ted.

@TedShifrin have you seen the symplectic proof of the hyperplane theorem?

buena

im assuming that means good in Spanish?

no, buena (or bueno) means good in spanish. that means that in english.

$\mathbb{R}P^2$ has a cell decomposition into one 0 cells, two 1 cells and one 2 cells, right?

Two 1 cells?

Do you mean 1? The one I am imagining is 1 in each.

maybe there's some goofy way of doing it. the textbook example is one in each.

is there anything non obvious one can say about when something doesn't have a cell decomposition with A 0 cells, B 1 cells, etc? with simplicial complexes there's less overlap and you have stuff like the euler characteristic. does that generalize?

i remember not liking cell decompositions very much

@leslietownes well if it has no 0-cells, then it must be the empty CW complex. :P

now you're going to tell me how many 10-cells there are in a cell decomposition of the circle

I like the idea of CW complexes, but I think I only have an intuitive definition. Kind of like simplicial complexes. This is probably going to kick me in the butt some day.

@geocalc33 what's up :)

what is the main application area of topology? Is it physics / engineering?

You guys are worse off than pure number theorists I'm thinking... -_-

Everyone stop studying math lol.

@PurpleHaze what's up

Dude, I'm in AZ now

we moved

I have a $20/hr construction job working for a friend of fam. Right now will be doing misc. tasks and bringing the concrete or mud in wheel barrels. Should lose weight fast doing that

P/T

@geocalc33 what is your job like?

I also biked to/from work. Extra fat loss regimen

@geocalc33 want to code with me over text? I'm on a temporary macbook pro, but have an unconnected desktop nearby

@PurpleHaze I metro to and from work. My job is not too far from Joe Biden's house

Is that a type of train?

it's like a subway system in a city

Cool! Underground tunnels are safer than above ground

I started to study from Lang about proving the standard complex sats $d^2 = 0$, but each time I start that problem, I get the urge to code my way around it in a generalized way. I want to make type of CAS + proof assistant + textbook creator (or something like that). You code your stuff in this C++-like language called D, and it will generate futuristic textbook content, or alternatively games for learning math eventually.

Stuff gets formally verified. It's a long-term project

Coding things in a host language such as D, which already exists and is well-maintained, beats coming up with a new language, which is very difficult for me as a coder

Eventually I want to do English - to - D code translation (just for the math application), so you'll be able to say things like "Let $M$ be an $R$-module." and it will automatically perform some kind of translation, but into the host language D and use the structures (classes) that I've coded. I'm not doing a type-theory based thing, which is already done by many projects. I'm working at a higher level. This might be called "implementing ZFC axioms" or something.

But really I'm just implementing whatever is needed for my system to understand human-designed math

I've already got the Markdown + KaTeX (faster than MathJax) viewer 50% done, with Qt + C++, and it has an instant previewer, which is Horizontal mind you (referring to MSE's bad design).

Eventually you'll be able to use Markdown + KaTeX in paragraphs, together with commutative diagrams in a vertically-oriented layout, so the user is given a few tools for working with that stuff at a high level, but they also have the option of coding things at the lower level (D programming).

So it's a "LowCode" platform for creating math learning software.

I believe the bottleneck for most people lies in learning the math, not in the resources available. So I'm trying to solve that problem.

I say I'm going to finish a reading in 3 hrs....the Math Gods say 6 hrs, I say I'll understand the whole chapter of concepts in one go....the Math Gods say "ha", I say I want to be able to have the time to go to the gym...the Math Gods look down in contempt and say "nay"..........oh wise sages what do I have to do to temper the wrath of the Math Gods?................