Sorry guys to disturb you.. I think this room is not for helping new learner. Those are not at your level.

But thank you for your clear me few confusions.

123: remember one thing: moment/torque is defined about a point just as a vector product. You want to know what it means by torque/moment about an axis?

@123 You cannot accuse us of that. You need to learn to respond to questions that we ask, not keep repeating yourself over and over.

I in fact did help you.

1drv.ms/u/s!AozWlUoG8z4tiHDRoD1BRhl8a0to

echo echo echo echo

1drv.ms/u/s!AozWlUoG8z4tiGUnvN-yQO7oSNuH

Can anyone please review my answer here? math.stackexchange.com/questions/4181981/… Thanks.

I learned more about power series today and this question came across and i tried to answer it.

@TedShifrin I admire you helped, Thanks for support.

I haven't read it carefully, @Koro, but if you're going to be pedantic, you might as well be pedantic by saying there is $N$ so that blah holds for $n>N$ (etc.). Do we know that $r^{1/n} \to 1$?

I'm not being pedantic and that's why I didn't say there exists N such that....

Yes we know that $r^\frac 1n\to 1$ because r is positive.

Well, I'll let someone else check the rest. I prefer a different solution, the one following yours.

@Koro I used to help the OP in that question outside of mse but they reacted a bit oddly when I replied indicating that I will be particularly busy for some period of time and will be slow in responding. So, I don't want to get involved.

Yeah, I have a list of OPs I no longer wish to respond to, @copper. The trouble is that I don't always remember their names.

But then I usually get reminded quickly.

I understand the urgency to solve a problem that bugs oneself, but follow up emails the next day reminding were a wee bit irritating.

Nothing untoward, just uncomfortable for me.

I got an email from someone on whose question I'd commented asking me for advice on where to apply to grad school. Agh.

Someone trying to read Griffiths/Harris's algebraic geometry book but not understanding the most basic thing in complex analysis (how you get from holomorphic to analytic).

no good deed something something

My teacher said the word: compact support in class today. He gave an example of a function $f(x)=e^{-\frac 1{1-x^2}}, |x|<1$ and $f(x)=0$ when |x|>=1.

I used to get a lot of stuff asking questions directly.

Some folks are very nice, however.

That looks troubling. Maybe the exponent should have a negative?

What's compact support? Where is it used? In topology, cohomology?

This is what you all have done to me.......

Everywhere. Analysis, topology, cohomology, but mostly analysis.

@Koro why is that troubling?

If your prof wanted the function to be continuous, you need to repair the exponent.

Oh, you did. Never mind.

@TedShifrin where? :P

I said troubling, @copper, not he.

@dc3rd I will add an answer shortly that illustrates that :-)

My whole plane ride I was trying to conceptualize all of the phenomena from how we rise to how the plane is able to stay on path, to anticipating turbulence......

@dc3rd the physics of flight are still surprisingly unsettled.

Oh, that's complicated stuff.

I've been corrupted and marveled at what humans have achieved

OK, I'm headed out for a quick brunch.

Enjoy!

greetings from rainy, damp, grey Liverpool by the way...

I don't know if it should be a smile or a frown.....so bloody wet

I have a close friend who is a prof at Liverpool.

came to visit my brother who just finished at Liverpool in the MBA

koro: partitions of unity often involve such functions. any time you want to apply a theorem that might only apply on [a,b] to an arbitrary function, you might consider such functions. there is nothing exotic about the notion. sometimes people don't even give it a name.

You know that folks from Liverpool are called scousers?

I'm a Man Utd fan....we just call them Scum thank you very much :)

that's the nicest thing we call them at our house.

@Koro the term support often means the closure of the points were the function is non zero. Why closure I have no idea.

it's just a more useful notion with the closure.

I thought if f is defined on closed and bounded set like [-1,1] to be something but 0 outside that then f is support.

x is in the support if f is not identically zero on an interval around x.

But i think dirac delta is also compact support.

uug

please don't bring dirac delta into this.

that is a distribution, different world

isn't the support the domain of a function?

the dirac delta is not a function.

it is the closure of where the function is non zero.

mathworld.wolfram.com/BumpFunction.html

dc3rd sometimes it is a subset tuned to exactly where the function is nonzero, but more generally "supported on bleh" can mean "identically zero outside of bleh," so yes. it depends on who is writing the definitions.

wolfram, cough, spit../

@copper.hat hence how the term is used in probability

Oh yes, dirac delta is not a function.

@copper.hat Ahh, I see.

things that approximate dirac delta in a distributional sense might well be compactly supported. "approximate identities" these families are sometimes called.

@dc3rd probability, in particular the implications of independence, is tough on the mind.

how so copper?

as an engineer i have thrown delta functions (the delta durex function as i used to cal it in class) around with abandon.

@dc3rd i think it is a little like differentiability in the complex world, it carries far wider implications that are evident at first glance.

in fact, it took some time before mathematical objects that could be called independent were shown to exist.

Is f identically 0 compact supported?

i'm loosely quoting Kac

koro, yes.

the empty set is compact

^

Ohh, supported with no support.

zero function

This is probably why I haven't fully seen the depth of these objects yet, since I haven't done a measure theoretic level course yet......that's for next fall....

well, empty support

@dc3rd measurability is another one with far reaching implications.

thanks a lot :) i think I understand it now.

well just took a quick glance at delta functions are the derivative of the Heavside function which itself is complicated as it is...

interesting stuff.

i like to integrate dirac delta from minus infty to +infty.

please don't learn about delta functions from the internet.

lol...no no...just a quick picture of what is happening....i'll save learning about them to MSE....

it is useful to get some sort of intuition, but one has to realise that you are driving uninsured...

@leslietownes it's very, very small at one end, much, much bigger in the middle, and then small again at the other end.

i mean, in one sense it's fine, engineers and such have been using them forever and who cares. but if you get to the point where you're spewing engineering talk on mse, go back and re-evaluate your life choices.

i prefer to think of it as the sum of exponentials...

robjohn, hah

just like Brontosauruses

if one teaches such things to engineers, one should be able to back the technicalities up when asked...

i do remember the total look of disdain from the professor in a communications class when i mentioned the term measure zero.

mathematicians and engineers have different goals.

wow, i'm getting philosophical. time for lunch i think

Given a positive operator T, it doesn't have to have a unique square root. Right?

right. unique positive square root though.

example: on C^1, -1 is a square root of 1, but not a positive square root of 1.

and we can construct such square roots (not positive) by taking $Re_i=-\sqrt \lambda_ie_i$

$T=R^2$ and $\lambda_i$'s are eigenvalues of T.

loosely yes, but you are omitting much context...

$T$ is an operator defined on a finite dimensional vector space.

koro we understand although you are not quite writing all of it.

you can diagonalize the positive square root and sprinkle $\pm$ signs over the diagonal entries and get other, non-positive square roots, yes.

since the lawyer is not doing his job, the engineer must take over...

i just wonder where this is leading. no point digging too heavily into the setup if there's no question at the end

I know you understand as I've been discussing lot of my doubts with you :)

Nonetheless, I should have written the complete context.

speaking of which, the whole profusion of mini helicopter-like gadgets has got to be a legal wet dream...

@Koro ignore me, i am just in a pedantic mood

no, it's fine. The discussion may help some future visitor of this chat.

Thanks a lot for clearing my confusion on square roots. :)

Just don't start with the statement that $\sqrt 4 = \pm 2$.

i wonder how many folks are thrown off maths by sloppy teaching at the high school level?

i maintain that one of the small mercies for high school teaching: angle-side-side is not a valid congruence of traingles

This comes back to our problem with education majors being at the bottom of the heap. And crummy textbooks. But I don't agree that PhD mathematicians should be the arbiters.

HUH @Semiclassic

@TedShifrin indeed.

high school geometry teachers would have a lot more headaches if ASS was a valid answer :P

Geometry is almost entirely out of the curriculum now, anyhow. I hated hated hated the Statements/Reasons rote that it had been for decades.

koro for fun you might think in the fd case whether you get all conceivable square roots of a positive operator by sprinkling signs down the diagonal of a diagonalization of the operator. (spoiler alert: no)

Headaches in that noncongruent triangles would be congruent? I guess that's a headache.

ted, you're being too mature about this. headaches in the form of people saying ASS in class.

^

Oh. I didn't even think of that. Duh.

we all need to think on the level of my daughter about this. she's already grasped that some words are funnier than others. thanks, day care.

i am a little afraid to ask what ass is?

You could easily call it SSA.

really Leslie? I thought I would get atmost 2^n square roots by sprinkling $\pm$.

yes, you could. but teenagers would inevitably notice that the reverse is also valid

Nonsense, @leslie. You're the one who's taught her all the four-letter words.

In fact, when I've seen this discussed, it's been SSA.

she did learn the big one from me, but not a lot of the other ones.

@TedShifrin sure. but if you think that would stop teenage boys...

my kids learned their powers of bombastic expression from their father

i do remember that my high school geometry book's image showing that SSA wasn't a valid congruence was nonsense

it gave two triangles which visibly didn't satisfy the premises of SSA

which is silly, b/c giving counterexamples is easy

i'll bet that i will not find ass in coxeter

Really? That seems easy to fix with proofreading.

i think it is time i was banned again...

okay, this is a funny page:

mathwarehouse.com/geometry/congruent_triangles/…!

"The way that many people remember this fact is that the ASS postulate would be the name for a donkey! (ASS) And there is no postulate named after a donkey!"

Don't give me no SASS.

koro: try squaring $\begin{pmatrix} \pm 1 & 0 \\ c & \mp 1 \end{pmatrix}$ some time. doesn't matter what c is or how you choose the sign.

there's buridan's ass. i guess that's not a postulate, but it's named after an ass.

ahh, metastability....

I'll try that Leslie. Thanks :)

scourge of asynchronous design

yum, nilpotency

It's sorta like people also forget that all reflections square to the identity.

there you go again with your geometry

those people need to take a look in the mirror

Which reminds me of one of my favorite puzzles: Why do mirrors reverse left and right but not up and down?

i can't see myself in the mirror

That, along with the telling apart a magnetized bar and an unmagnetized bar, was given to me by a math friend at a conference zillions of years ago. I was stuck on both for embarrassingly long.

hmm

i only use a mirror in the dark

The mirror one is in the first section of my chapter on differential forms and integration.

i was going to say something about our eyes being placed horizontal, but mirrors still flip left-right with one eye closed

oh my...

the only thing i can see breaking the (rotational) symmetry is biology/anatomy tho

that's a nice puzzle. I knew the answer but I forgot.

looking at the answer to that: it illustrates the danger in accepting the premises of a question without thinking about it

If you cheated and found the answer, do not share it.

yep

I did give a hint by saying it was in my chapter on forms and integration (on manifolds).

i love those conundrums but can never remember them at parties...

another one that usually throws physicists is what volume of water you need to float a boat.

Do you get naked and yell "Eureka"?

I suspect that a math (or engineering) conference is more appropriate than a party.

pronounced 'ev-reeka' by Greeks :-)

@leslietownes an ass

But where's the triangle(s), @robjohn?

i went to mount bromo on an ass

wearing flip flops. pretty cold i might add.

I would be scared to wear flip flops in case I ended up having to hike miles.

it was an oversight on my part, i didn't want to bring my heavy hiking boots on that part of the journey.

Yeah, but still ...

I'm looking up Mount Bromo.

i don't own flip flops. i am too high-class.

I don't find it. I get a volcano Bromo.

i dont think your feet will thank you if you wear flip flops in this cold

i arrived into the Paharganj in Delhi (this was when it was a little wilder) and one of my flip flops broke.

@leslietownes That's pretty much all I wear, Mr Snobbyshoes.

Yes, flip flops do break.

Plus it's hard to brake going downhill in them.

@TedShifrin yup, that's it, in eastern java

@TedShifrin I didn't read back far enough to see that we were talking about triangles. I looked too far back.

of course, referring to this comment

@robjohn not often i use this, but lol

LOL, oh, @robjohn, that explains it, I suppose.

just doing my part to further topology

serge lang loved comparing things to a hole in the ground.

And he loved telling Chern he should learn some mathematics, too.

I was astounded the first time I heard that at a seminar at Berkeley.

what's wrong with that, everyone should learn some mathematics

the wonders of language

of human inference, i suppose

hmm, my daughter is finished term today, she should call me

if i had the chocie to go back three years in the past, i would have enrolled myself in the mathematics department instead physics.. completly my mathematical education, and then started physics. i was back then however ignorant and unaware... now i see the light..

Sadly, the world does not see the beuaty of mathematics. i believe studying it has made me a better human, more logical, more compassionate, more critical. etc.

both have their beauty

Perhaps a simultaneous double major. Unless you want to be a theoretical physicist, much of a math degree may not be so "useful."

I am doing a double major now. : )

You could have studied philosophy and ethics and read classic literature to become a better human, more compassionate, more critical.

I do not know about that, it surely however happened to me while studying mathematics. the language of it forces one to think logically and connect statements correctly.

Yes, logical thinking for sure, although proper study of philosophy lends itself to that, too.

Did you see the link i just sent? :)

Yeah, I clicked on it, didn't listen.

Yea i am very intrigued about that why mathematics describe our universe. its so facsinating and wierd :)

But the problem with these questions is that one enters the realm of philosophy and exists the realm of experimental science... hard

So it is obvious philosohpy leads to that, since i think it is very closely intwined in it. (logical thinking)

I like to think about it as follows. if you have any kind of pheonemena.. you can ask "why" n times and stay in the realm of science.. however.. at n+1 "why" you just entered philosophy :=)

why are people so obsessed with meaning?

I don't know what you mean.

copper: pbhbhhthtbhbtht

hi chat

Salut, @Astyx.

Como estas ?

For example: why is this body hot? because its temperature is high, why is the temp high? because it was in the sun, why did staying in the sun made it high? because photons dropped on the surface transforming their energy into kinitic enrgy... and soooooo on, at one point you ask, why do atoms jiggle to make heat? ... and then why do atoms exist? and so on. at one point, the question of why. ceases to be answerable withen the realm of science and becomes more philosophical

У меня всë хорошо

aaa ) vi govorite po russki! .. ya tozhe mogo rozgovorivat na ruskom yazik )

I studied Russian for one year in college. That was a long time ago.

nil aon tintean mar a tintean fein

My father was a communist. and he spoke russian at home :) so i can speak a bit ^^

Do I write up stuff

or do I make pizza dough?

make up stuff

Pizza dough isn't hard.

I'm faced with such high stake dilemna

my pizza dough is hard

What will go on the pizza?

Yeast, copper, yeast.

anchovies hopefully

I made myself a salade niçoise (but without tuna) last night. A whole can of anchovies :)

mozzarella, tomato sauce mushrooms, olives and artichokes on the first one

i love anchovies, even the tinned in olive ones

it is practically impossible to get fresh anchovies now

What kind of olives, @Astyx? Sounds good, as long as not too much tomato sauce.

probably cream, goat cheese, walnuts, honey and possibly pine nuts on the second

yeah, too much tomato sauce makes the dough soggy

You can't do both walnuts and pine nuts.

black olives? I'm not an expert

@TedShifrin try me

What kind of black olives, silly?

hang on let me check

Niçois or kalamata?

I love both, but I might prefer the dry niçois-style olives on pizza.

the latter

there is also the pizza diavlo

Oh well.

Still good.

Have you ever made pissaladière? Same pizza dough. Lots of caramelized onion and anchovies.

only consumed, never created

same, my mom makes the best pissaladière

Never had a go at itmyself

unrelated, Olivetos in Rockridge is closing down

I've made it a few times. This discussion is reminding me I should do it again soon.

Good night Friends : = )

Same pizza dough, @Astyx, essentially. All you have to do is cook the onions low and slow for hours (like for onion soup).

Wait, it's getting very late for pizza dough making, @Astyx. Unless it's for resting overnight for tomorrow.

copper: that sucks. any reason why?

Gute Nacht, @MadSpaces.

@TedShifrin no one understands the mean square.

@TedShifrin it is!

@leslietownes i think retirement was the excuse

It was this or unfreezing premade dough

and I'd rather eat fresh dough

(who wouldn't)

@Astyx Cool. Did you make the premade dough?

I have frozen my pizza dough for a few months and it was great.

it's certainly been a while. i used to live a few blocks from there. i liked the downstairs for many years. good baked goods there in the morning.

No, these are from the store, for when I don't have time to do the dough

Oh, OK.

I make the dough in the Cuisinart. Super easy.

in the late 2000s they began cramming more tables into the downstairs and ruined it as a place to eat dinner. the upstairs was still OK. i always liked the food.

all this talk has me reaching for a frozen tj's Margherita on top of which i might add a tin of anchovies...

Olivetos was good for business lunches and the like

when i had an expense account

The guy running Olivetos is 74. He gets to retire.

I think I may have eaten there once years ago when I visited old friends (in Orinda) after I had long since left Berkeley.

i liked Cafe Venezia on University.

Hi @Ted

thanks to my hoarding of old credit card statements, i last paid for dinner there in 2009.

i think my wife and i had lunch with someone later than that, with someone else paying.

i have records going back to my time in berkeley

i really need to get rid of them.

Hi, a @Balarka!

I certainly don't have records going back to the 70s, copper. When I left GA, I pretty much destroyed everything except my taxes from the last years.

my bank has been electronic since the early 2000s, so i just have a folder full of PDF cc statements

you'll be screwed when the machines take over

Oh, I haven't kept bank statements. I can go back on line for numbers of years. Credit card bills I download every month.

i tried automating the downloading but it was more work updating the security on my scraper so i just do it manually now.

@dc3rd this is what happens when an engineer learns a little formal mathematics and then tries to solve a simple problem math.stackexchange.com/a/4324036/27978

I disagree with you. Lagrange multipliers is almost always easier than the algebra you get with substituting constraints and eliminating variables.

Proving that you have local/global extrema is not usually taught, although I taught my multivariable students how to approach such things, and the bordered Hessian test for constrained local extrema is an exercise in my book :P

This started off as a bizarre question. (The definition was only included after we pestered the OP.)

I made a prime counting function but then I realized that you need knowledge of the prime numbers in order to use it

that's a very broad question title

@TedShifrin I never meant to imply otherwise, my comment was that folks often do not check that an extremum exists.

I didn't even pick on the title, yes.

the dough is resting for now

don't make too much noise

@copper Usually a heuristic argument suffices, but one can rig a compact space outside of which the extremum cannot occur and then use the max value theorem. I didn't read what you wrote.

@Astyx Pizza dough is not like a soufflé.

WHAT DID YOU SAY, ASTYX?

what do you mean @TedShifrin ?

@leslietownes D:

You said not to make much noise.

Yeast doughs are quite robust until you punch them.

Soufflés fall if you step too hard on the kitchen floor or jerk the oven door open/closed.

oh really? I knew soufflé was hard to make, but not that it was so sensitive

punching dough is one of my favorite passtime

i remember thinking that the souffle thing was something invented for comedic purposes. then i saw someone make one.

spoiler alert, it's real

I have made lots of soufflés, but one does have to be a bit careful.

The heat makes the aerated egg whites rise.

If $V$ is a $2m$-dimensional real inner product space, $Cliff(V)$ apparently has a an irrep in a $2^m$-dimensional complex inner product space. What is this object?

Ugh @ math mode Cliff.

I cant be bothered

Oh, this stuff is in Lawson/Michelsohn, which I no longer possess.

But you should have it (or access to it).

Yeah but I'm too lazy to look for this in that tome

So asking if someone here already knows

Else I'll try to figure it out

Well, I no longer remember it, and I'm leaving for a walk.

tell me when you figure it out

My guess is its probably going to be an exterior algebra. Choose an orthobasis $e_1, \cdots, e_m, f_1, \cdots, f_m$ for $V$. Try $\Lambda^*(e_1 + i f_1, \cdots, e_m + i f_m)$, or something of this sort?

That's a $2^m$-dim complex vector space

I dont see the obvious action.

$e_1$ has to act on $\Lambda^*(e_1 + if_1, \cdots, e_m + if_m)$ in a way that $e_1^2$ acts by $-1$

How do I accomplish that

Probably multiplication by $i$

What's $\Lambda^*$?

exterior alg

Ah ok

Lets do $m = 1$. Cliff_2 is generated by e, f with e^2 = f^2 = -1, ef + fe = 0. This is like quaternions

so it embeds in $M_2(\Bbb C)$

thats the 2^1 dim complex rep

Seems like I can arrange some matrices in $M_{2^m}(\Bbb C)$ in general

Just send e_1 to the block 2^m x 2^m matrix with 2x2 block (i 0|0 -i) on top, f_1 to similar but 2x2 block (0 -1|1 0), then e_2, f_2 etc with the block below it and so on

but thats a 2m x 2m matrix