Mathematics - 2021-04-17 (page 3 of 3)

copper.hat

16:00

i have forgotten your word Leslie for the nitpickynesss of pathematicianss

leslie townes

captious?

copper.hat

thank you. i need to writ it down

leslie townes

i would try a trigonometric substitution, y = 4 tan(t), and go from there.

i don't know what results but that would be the standard thing.

copper.hat

i am glad symbolic systems are around to do that stuff

geocalc33

If you always get the wrong answer on multiple choice tests are you smart?

copper.hat

16:02

smart is not a one dimensional characteristic

leslie townes

the same way that a system that predicts the result wrongly more than half of the time is smart.

don't do what donny don't does.

geocalc33

so if you can avoid the correct answer you must have knowledge of the correct answer

copper.hat

dang, my friend is not responding.

leslie townes

geocalc, i don't think so. it depends upon what the objective is. if the objective is 'avoid the right answer,' sure. if the objective is 'get the right answer,' no.

which isn't to say that something that more often than not gets the wrong answer can't usefully be used as an input to get the right answer.

copper.hat i guess you'll just have go out there biking with your demons.

copper.hat

that could result in a cosmic singularity

leslie townes

16:04

bike bike bike "feck" bike bike bike "arse"

copper.hat

:-) there is some sort of sonic shock wave because of the speed of words coming out of by mouth

leslie townes

one time in the fourth grade i was given a true false test and i couldn't decide on one, so i wrote a symbol which was a T, but also had the crosshatch of the F in the middle. my teacher circled it and wrote "clever" and marked it wrong. good lesson there.

geocalc33

lol

Matheus Sousa

guys, pls help me

copper.hat

is this homework?

Matheus Sousa

16:07

hey guys someone could help me? I don't know what rules I need for integrate this function: $\int \sqrt{(16+y^2)^3}$ \\ sorry, I repeat this message because I don't know how to reply this

leslie townes

matheus did you try my substitution? what did you get?

Matheus Sousa

I don't know what to do because the expression is contained in the root

leslie townes

it's (16 + y^2)^(3/2). no root now. did you try my substitution?

Matheus Sousa

not yet

leslie townes

sorry to be a broken record about this.

Matheus Sousa

16:09

I'll try this

leslie townes

one thing i miss about the radio are the times when you'd be listening and the record would begin to skip and you'd wait for the human to fix it.

doesn't happen anymore, it's all bytes now.

geocalc33

does anyone want to discuss the projective plane?

copper.hat

i loved tech when it gave me an advantage :-)

leslie townes

the classical station in the bay area in the 80s was the worst for this because they had a lot of really old records.

you loved yellen when she gave you and yours an advantage :)

geocalc33, think of how awful it would be if the answer were no.

copper.hat

a local pop station 92.7 turned religious recently, unbelievable the amount of stereotypical tea party evangelist rubbish they churn out

keep in mind my perspectives are more old school republican

leslie townes

16:12

oh no! that used to be energy, the go-to spot for mindless dance music.

copper.hat

yes, after i picked up our cheesesteaks last night i nearly had a heart attack

leslie townes

it was also KJAZ growing up, the only station my dad listened to other than KQED.

copper.hat

which is bad because that ids the one organ that works propely

i like a variety, not the modern focus on me stations

i like to hear different perspectives

speaking of projective planes...

leslie townes

someday i'm going to tell my daughter, "when i was growing up, we had a radio on our kitchen table. it was actually in the center of our table. we would gather around it and listen to the news, and also jazz."

that wasn't even an 80s thing, i think my parents just carried it over from their 1950s upbringing.

copper.hat

i used to lecture my kids about the dictionary i keep beside the kitchen table

leslie townes

16:15

it was a nice radio. it's hard to find a good tabletop radio these days

geocalc33

What do you think the reason for that is?

copper.hat

i gave my father in law an awesome grundig short wave radio when he went to live in china, but it seems to have disappeared :-(

leslie townes

has he seemed to have disappeared along with it? if not, count your blessings.

copper.hat

i used to love hunting for stations late at night when the ionosphere was doing its nightly thing

unfortunately he passed away last year. (all remaining grandparents passed away last year sadly)

leslie townes

i am sorry to hear that.

copper.hat

16:18

he was great to walk the streets in china, people would talk to me when he was with me

and he was unusually sociable

and put up with my peculiarities

i miss him

leslie townes

i had a grundig short wave radio for about a year, then my sister decided she would get into it and used the wrong power adapter to plug into it. the magic smoke that made the device work escaped.

copper.hat

aarrrgh

i liked electronics when i could look inside and guess what stuff did

now its all sdr and rubbish

leslie townes

even kits for children now, they hardly even let you mess with wires. it's like, connect string A to B. it's not the same. you don't even learn about how to make a proper connection between two pieces of metal. i was looking into this for my daughter. it's all rubbish now.

i'm going to teach my daughter how to wind a good electromagnet out of copper wire.

(not copper hats)

copper.hat

:-) my daughter can solder (sold-er, not sodder) at least

my 17 yo son wont even try my ebike

and he returned my emergency bivvy bag i gave him a few years ago.

never know when you might need one :-)

i am failing as a parent

leslie townes

my daughter wants to play my electric guitar. she's already broken two strings. i think she's going somewhere in life.

somewhere good? too soon to say. but somewhere

copper.hat

16:25

on alternate days, when my daughter was in the 2-5 range, she would greet me at the door with a hug and next day a sort of "oh. its you again" look.

i shared a room with two brothers who liked to listen to and play heavy metal

leslie townes

one time my daughter said "hi" to me, and then immediately turned back to a laptop and pressed the spacebar, which she learned around 2 unpauses the video

copper.hat

to this day heavy metal bothers me

leslie townes

i love metal but i don't play it, precisely because i live with people

copper.hat

@MatheusSousa you may need to do a few substitutions to get an answer. i might try something like $t^2 = 16+x^2$ to start with.

wondering if i should put my kid's savings into dogecoin :-)

leslie townes

you should really put them in townescoin. it's going to go through the roof.

i have a swiss bank account, you can just wire it there, you won't regret anything.

Shobhit

16:37

when I search "groups of order $12$ " in wolfram alpha, it gives $5$ groups, one of which is a dicyclic group of order $12$. I think what it means is $Z_{2} \times Z_{6}$. My question is "does dicyclic refers to something other than being non-cyclic? Like its external direct product of groups of the form $Z_{m}$ only?" Link :: wolframalpha.com/input/?i=groups+of+order+12

leslie townes

yes, the dicyclic group is not (generally) a cyclic group.

Dicyclic group

In group theory, a dicyclic group (notation Dicn or Q4n, ⟨n,2,2⟩) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension can be expressed as: 1 → C 2 n → Dic n → C...

that's actually not a very good discussion of the idea, although the presentation in terms of generators and relations is my own mental image of it.

it is not a direct product of abelian groups, and it is not abelian.

geocalc33

consider a project to supply 100 million postage stamps per year to the u.s. postal service for the next five years. you have an idle parcel of land available that cost $750,000 five years ago; if the the land were sold today, it would net you $1,125,000 aftertax. The land can be sold for $1,295,000$ after taxes in five years. You will need to install $5.1$ million in new manufacturing plant and equipment to actually produce the stamps; this plant and equipment will be depreciated straight-line

to zero over the project’s five-year life. The equipment can be sold for $450,000 at the end of the project. You will also need $425,000 in initial net working capital for the project, and an additional investment of $50,000 in every year thereafter. Your production costs are 0.38 cents per stamp, and you have fixed costs of $1,100,000 per year. If your tax rate is 23% and your required return on this project is 10% what bid price should you submit on the contract?

copper.hat

i need to take a group class to get over my fears.

geocalc33

I'm stuck on this word problem. Where should I even begin?

leslie townes

anything with straight-line depreciation requires a spreadsheet, not mathematicians.

Shobhit

16:41

but $Z_{2} \times Z_{6}$ is abelian. @leslietownes

geocalc33

okay I will boot up excel

leslie townes

this is fundamentally a problem in accounting. set up columns for each year, put in the 'straight line' stuff, etc.

@shobhit yes which is among the reasons why it is not the dicyclic group of order 12.

Thorgott

I've never heard the term cyclic before

Shobhit

so wolfram alpha is wrong?

Thorgott

it's the non-trivial semidirect product of Z_3 and Z_4, I wager

leslie townes

16:43

i'm not disputing that Z_2 times Z_6 is an abelian group of order 12. i'm saying that Z_2 times Z_6 is not the dicyclic group of order 12. if wolfram alpha says it is, then yes, it is wrong.

Thorgott

@Shobhit no, you are wrong in assuming they mean Z_2xZ_6, they don't

Shobhit

oh ok. my mistake. thanks for spotting that @Thorgott and thank you for the extra info @leslietownes

leslie townes

"dicyclic" is its own thing. it doesn't mean direct product of two cyclic

although absolutely everybody is forgiven for suspecting that it might mean that.

there's probably some cute geometric realization of these groups, as there is with the dihedral groups, which have similar but not identical presentations. but geometry is forbidden so we will shut our eyes.

Shobhit

i was just trying to find subgroups of $Q_{8} \times Z_{3}$ (random group) of every order, but didn't now about groups of order 12

know*

leslie townes

there used to be a really good site, where for groups of small order you could see, automatically generated printouts of stuff like the orders of elements, conjugacy classes, subgroup lattices, and stuff. i can't remember the name. this would have been like 10 years ago. it may all have been a dream.

Shobhit

16:51

haha

Thorgott

you're probs thinking of the groupprops wiki

leslie townes

yeah. and maybe not subgroup lattices.

Rover

If the independent variable x is changed to y, then the differential equation $x\frac{d^2y}{dx^2} +\left(\frac{dy}{dx}\right)^3-\frac{dy}{dx}=0$ means what ?

Shobhit

i am bad at modern algebra, like i know how to find something in the group but it takes time, and my professor just says all the subgroups of a group like instantly.

leslie townes

if you work a lot with groups of small order you internalize it. i know a guy where anything under 100, except for some of the p-groups, he can just break it down very quickly.

i've only asked a few questions on math.SE but i really liked qiaochu yuan's answer to this one. math.stackexchange.com/questions/263885/…

exactly what i wanted in a few lines.

Shobhit

16:58

went over my small head

robjohn

@Rover is that the equation you meant?

Rover

Wow how that happened ?

@robjohn yes

leslie townes

if x is changed to y, are you also changing y to something? i'm not sure of what the desired transformation is.

if you swap the variables, the equation means whatever it means, but now in new variables.

Shobhit

So isomorphism is an equivalence relation, so it partitions the set into disjoint classes and the number of classes should be the number of groups upto isomorphism. so like a computer must be able to break every group, right? in sufficient time. Tell me if this is stupid.

leslie townes

sometimes when people say 'equivalence relation' they imply the existence of a set. because there's no set of all groups, such people wouldn't quite say that.

but yes, a computer could determine the number of isomorphism classes of groups of order n, for any n, given enough time.

oeis.org/wiki/Number_of_groups_of_order_n may be instructive, as may be oeis.org/A000001

wow, it's sequence A000001.

Shobhit

17:01

_

leslie townes

golem.ph.utexas.edu/category/2012/11/… is amusing.

there are a lot of p-groups. as n gets big, the number of groups of order p^n gets very very big.

Rover

@leslietownes Then question says the differential equation changes to $x\frac{d^2x}{dy^2} +\left(\frac{dx}{dy}\right)^2=k$

Shobhit

yes, i noticed that

2^6 is 267

Rover

Then we have to find k

Sorry it took much time :$}

Thorgott

that particular is easier, because any subgroup of Q_8xZ_3 is a product of a subgroup of Q_8 and a subgroup of Z_3

usually, subgropus of direct products get much more complicated

leslie townes

17:11

what accounts for the relative ease of this situation? the small order of Z_3?

some arithmetic relation between 8 and 3? smallness of numbers generally?

Thorgott

the fact that their orders are coprime

in general, all subgroups of a direct product being direct products of subgroups happens iff the two factors have coprime orders

leslie townes

this is the quality content i am here for.

Thorgott

a different answer would be that this happens, because they admit no common subquotient

there's a general, very technical description of all subgroups of a direct product and it trivializes in the absence of common subquotients

copper.hat

what @leslietownes, you don't want to hear my detailed economic gripes???

leslie townes

copper, only if you want to hear mine. yours are probably worse than mine, though, so even that isn't a fair trade for me.

Thorgott

17:27

let me try to recall the proof. consider $G_1\times G_2$ with $|G_1|,|G_2|$ coprime and let $H\le G_1\times G_2$ be a subgroup. denote $H_1,H_2$ the projections of $H$ to $G_1,G_2$ respectively. obviously, $H\subseteq H_1\times H_2$. now take $(h_1,h_2)\in H_1\times H_2$. by definition, we can find $g_2\in G_2,g_1\in G_1$ such that $(h_1,g_2),(g_1,h_2)\in H$.
Now, by Bezout, we can find integers $x,y$ s.t. $x|G_1|+y|G_2|=1$, then $(h_1,g_2)^{y|G_2|}=(h_1,1)$ and $(g_1,h_2)^{x|G_1|}=(1,h_2)$, so that $(h_1,h_2)=(h_1,g_2)^{y|G_2|}(g_1,h_2)^{x|G_1|}\in H$.

robjohn

@Rover $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac1{\frac{\mathrm{dx}}{\mathrm{d}y}}\frac{\mathrm{d}}{\mathrm{d}y}\left(\frac1{\frac{\mathrm{dx}}{\mathrm{d}y}}\right)=-\frac{\frac{\mathrm{d}^2x}{\mathrm{d}y^2}}{\left(\frac{\mathrm{dx}}{\mathrm{d}y}\right)^3}$

copper.hat

i have a full spectrum of gripes. however you will be spared today as my friend has responded. hopefulky he won't be too energetic today, not really up for a 2h loop around the nbh.

robjohn

@copper.hat how about a full spectrum donut?

leslie townes

if you're bragging about having a friend within biking distance, you win. i don't.

that's a nice donut.

Thorgott

conversely, if $G_1,G_2$ do not have coprime order, there's a prime $p$ dividing both the order of $G_1$ and $G_2$. by Cauchy's theorem, we can find elements $x\in G_1,y\in G_2$, both of order $p$. then $(x,y)\in G_1\times G_2$ generates a subgroup of order $p$, but it cannot be a product of subgroups of $G_1$ and $G_2$ for then one factor would have to be trivial, but $x,y\neq1$

copper.hat

17:30

@robjohn that reminds me, i found the slow leak in my bike. is suspect that i am the only one left in the world who actually patches inner tubes :-)

Thorgott

ah, the joy of finite group theory

leslie townes

thorgott thanks.

robjohn

@leslietownes colored by curvature

copper.hat

i like the graphics! i;m in awe really

leslie townes

i noticed!

you are a magician with your computer algebra system of choice.

copper.hat

17:31

still use coloUring pencils

robjohn

@copper.hat I know people here who patch it with stuff you spray in, but most do have self-patching tires.

copper.hat

i still carry a little bottle of talcum powder to stop the patch sticking to the tyre

Thorgott

check out this nice answer by Tobias for the other thing I mentioned

leslie townes

that is a quality answer.

i'm looking over some of my answers from 10 years ago and can't even remember the person who wrote them. me, i guess.

Quin

18:08

@leslietownes I just read your answer about transitioning out of academia and I really appreciated the frankness of it. I gotta get out of pure math and into something applied before grad school is over as I def do not have what it takes to hack it in academia!

leslie townes

i think there are a large number of people who could hack it in academia, except the jobs aren't there. and sometimes the connections aren't there, and things are missing in one way or another.

i am an attorney now. it's not optimal, but it's fine.

shintuku

@leslietownes connections are important?

Quin

Yeah, plus hacking it seems to require a lot of moving around the country and massive work hours for little pay and little job security. As an older grad student, this is really not something I wanna do. Quality of life is real important to me at this stage

leslie townes

a little bit, particularly at the beginning. i don't mean to be cynical but it kind of doesn't matter how good you are if you don't have people with some amount of reputation who can vouch for you out of graduate school.

and ideally it's helpful of those people are still there a few years later when you're trying to move on from a postdoc.

Quin

My whole family are lawyers actually! That is something I have seriously considered. Maybe if the industry prospects go bust, I will try that haha

leslie townes

18:12

that was exactly my problem. lots of interstate moves, no money. i earned more in my first year of attorneying than i did in 3 years of postdocing.

Quin

wow. thats very compelling already

leslie townes

law is similar to academia in that it's very much a system in which a small portion of people get most of the opportunities. it's just a larger field so there are quantitatively more opportunities. and a lot of people who go into law are less prepared for thinking than math majors are.

i don't recommend chasing money as a habit, it's toxic and self-defeating. but there's some aspect of 21st century academia that's, let's say, not practicable unless you have a certain amount of money to begin with, which i didn't.

so now i write nasty letters to people and act like an asshole on the telephone and somehow that's more valuable to society than educating engineers about linear algebra.

shrugs.

if the bridges fall down, don't blame me. i tried. they chose not to listen.

Quin

lmao I feel that!

I still got a lot to try to figure out re jobs and all that. something i dont like thinking about!

anyway, i gotta run for a bit! bbl

Thorgott

@leslietownes one is a lost cause, the other isn't

leslie townes

that is a fair assessment of the situation.

shintuku

18:20

listen

here's how to fix everything

join your local social-democrat organization

leslie townes

i follow my locals on twitter and have donated to two political campaigns. is that enough?

i don't want to join. i don't want speeches or really to meet anybody.

shintuku

make sure they're social-democrat otherwise they won't seek academia reform, because journals make lotsa money

yeah that's fine too

bless you for that

leslie townes

these people might be further to the left than what you're thinking of. i can't wait for the next quarter, when this all gets released to the public.

leslie is trying to burn the system down, that will be the headline.

shintuku

my theory of academia is that it's all structured by the economic pressures from academic journals

you fix that, you get french university system

croissant, yes, but free university, and no one can tell me french academics are low tier

leslie townes

my experience, mostly vicariously through my wife for about the last 10 years, is that academia in the early 21st century thrives upon people having enough self-doubt to commit themselves to unsustainable systems, including, for example, academic journals, but also everything else with academia.

which is not the fault of academics. i looked at some numbers on the administrative costs for the school i went to in the late 1990s. they've quadrupled. i don't think that benefited students.

the best life would have been to be born about, let's say, somewhere between 1950 and 1965, and then drifted into university administration, to completely ride this wave before it crashes on the beach.

Ted Shifrin

19:06

Yes, at a lunch meeting with the president of UGA back in the 90s, I pulled out data from the university's own Fact Book to make the point that student population had gone up by something like 20% since I was hired, administrators had gone up by (I don't remember, but something like) over 50%, and faculty had shrunk.

They said, "We'll get back to you" when they were questioning my "facts." Ah, harbinger of alternative facts. Guess what ...

leslie townes

hahaha

Ted Shifrin

I had asked the President's administrative secretary who invited me if she really wanted me to be there. She knew me from other committees where I had not been exactly "silent." She said yes, but then surely regretted it.

robjohn

@TedShifrin Over the last 4 years, the meaning of "facts" seem to have changed

Ted Shifrin

Well, I'm wondering if there were pre-Trompers in that administration. The President surely was.

I mean. My "facts" came from the university's own FACT Book.

shrug

They had some bull about how the administration numbers included support staff; to which I say, so what?

leslie townes

what are they supporting

robjohn

19:13

@TedShifrin Yeah, but facts have to be filtered by the proper statistical jargon before they're palatable to some.

Ted Shifrin

The junior and senior level administration, of course, @leslie.

leslie townes

i withdraw my objections.

Ted Shifrin

Yes, to make them unfacts.

robjohn

Now you're talking

Ted Shifrin

@leslie: I'm pretty sure it did not include departmental secretaries, but right now I can't swear to it. At any rate, those numbers didn't change much, I can say empirically.

leslie townes

19:14

it included assistants to assitants to higher-ups. not anybody who does anything.

Ted Shifrin

We've truly moved into Orwell's 1984, just a few years late.

By the way, @robjohn, your double kerchief is not the unique way of (geometrically) making that topological object. Of course we could bend both sets of opposite corners up, just distorting the geometry slightly to make the square a rhombus so the diagonals have different lengths.

Something else for you and Mathematica to work on :P

robjohn

@TedShifrin maybe...

Ted Shifrin

LOL

robjohn

Did you see the spectral donut?

Ted Shifrin

I've seen lots of those over the years. Did you ever see my picture of the torus with the plane cutting through bitangent and giving two congruent circles as the cross-section?

I had that as one of the pictures on my class webpage for years.

I had never known about Villecearnu circles until 20 years ago or so.

I'd better check the spelling on that.

Oh, wow. I got it right.

robjohn

19:19

@TedShifrin I have an idea, I don't think I've seen that.

Time to take my son to get his covid inoculation

Ted Shifrin

The exercise a student gave me was to find a third family of circles on a standard doughnut, the obvious two families being dismissed.

Ah, congrats to your son!

leslie townes

well done! my first is on monday.

shintuku

Show $n\geq 0$; $n\mathbb{Z} = \{nk | k \in \mathbb{Z} \}$ is closed under $\mathbb{Z}$ multiplication.

Proof. Let $k1, k2 \in \mathbb{Z}$ Then, $n_1k_1, n_2k_2 \in n\mathbb{Z}$. Notice $n_1k_1 \cdot n_2k_2 = (n_1n_2) \cdot (k_1k_2)$.

$n_1n_2 \geq 0$, and $k_1k_2 \in \mathbb{Z}$, therefore $n_1n_2k_1k_2 \in n\mathbb{Z}$. $\blacksquare$

am I comprehensible

Thorgott

19:42

what's $n_1,n_2$

leslie townes

i co-sign thor's question.

shintuku

$n_1, n_2 \in \mathbb{Z}$

forgot that

leslie townes

to show that a set is closed under multiplication we must show that if a is in the set and if b is in the set then ab is in the set. here, our set is [blank]. so if a is in the set, there is k_1 in z such that blah, and if b is in the set, there is k_2 in z such that blah. from algebraic computation we conclude that ab = blah = bleh = blergh = blagghgghgh.

because this vomit has the form blehghghghgh, it is in the set. since a and b were arbitrary, ahgghghhghg.

for more on this innovative method of mathematical pedagogy, consult my pamphlets, "vomit math." by leslie townes.

shintuku

@leslietownes link pls

leslie townes

available in all airport bookstores next to the 300 copies of "the secret."

i'm sorry, i'm in a bad mood. my stepbrother attempted to commit suicide about two weeks ago and i was just on a call about how to deal with that. i'm an attorney and people want me to handle things i can't handle.

it's very helpful to use the definition of the set, but to plug it into a template proof form.

to show that S is closed under multiplication, pick a, b in S and show that ab in in S. because a is in S there is blah. because b is in S there is bleh. it follows that ab = (chain of equalities) = crap = that seems like it's in S.

that's the vibe.

shintuku

19:48

thanks!

leslie townes

in one line (nk)(mk) = (nmk) k would seem to do the job. but you need to dress it up with english and quantifiers.

Ted Shifrin

Yes, but you're wrong.

What have you done wrong?

leslie townes

ted has taken the unusual step of actually reading what the questioner asked.

strange strategy, let's see what happens.

shintuku

$n_1k$ and $n_2k$ aren't necessarily elements of the same $n\mathbb{Z}$

...right?

leslie townes

you're the one who defined n_1 and n_2, you tell us.

or, as we would say in the legal industry, purported to define n_1 and n_2.

you're on videotape and audiorecording right now.

Ted Shifrin

19:58

Hint: $n\ge 0$ is fixed, although your "show ..." made things confusing.

Thorgott

it seems you are confused on what you actually have to demonstrate

shintuku

I just realized. $n$ is fixed, but I defined $n_1,n_2 \in \mathbb{Z}$ which don't necessarily have any link to whatever $n \geq 0$ is

the conclusion doesn't follow from the premise, then

i think...

thanks!

Ted Shifrin

You have to read carefully and think carefully.

@leslietownes I'll try to avoid this in the future.

shintuku

you cannot understand the extent of my gratefulness

Ted Shifrin

Sarcasm duly noted, @shintuku.

shintuku

20:06

what!

I'm fully serious

but internet chats aren't typical environments conducive to the expression of gratefulness

it's just, the fact of contemplating that a stranger helps me write a proof

Ted Shifrin

LOL, I suspected you were serious. No problem :)

Most of us are happy to help people who are trying hard. I do get annoyed when people repeatedly don't read/analyze what I tell them. Then I stop helping.

leslie townes

i've found that i get secondarily annoyed by people who don't listen to ted. it's something like a virus.

i'm waiting for the vaccine.

Ted Shifrin

No, @leslie. You mostly get annoyed by listening to ted.

Balarka Sen

@Ted: Can you briefly remind me how the 2nd fundamental form of a surface embedded in R^n goes?

leslie townes

that's how i get primarily annoyed.

Balarka Sen

20:13

Oh, is it literally a package of 2nd fundamental forms, for each of the n - 2 normals? Aka, a normal bundle valued thing?

Ted Shifrin

Yes, precisely.

Balarka Sen

Thanks!

Ted Shifrin

That was easy :P

I'm leaving but will be back in several hours.

Balarka Sen

See you, Ted.

Shobhit

am I correct, " if a group has only one element of a given order, then it must be in the center group"?

Thorgott

20:22

what is the "center group"

copper.hat

$Z(G)$ presumably?

Thorgott

oh, I read over the "in"

Shobhit

the center of group

Thorgott

this hypothesis can only be satisfied if the order in question is 2, in which case the answer is yes

Shobhit

i was thinking, if $a$ is the only element of order $p$, then for every $x \in G$, order of $xax^{-1}$ is also $p$ but $a$ is the only element of order $p$, so $a = xax^{-1}$ and therefore commutes. Correct if wrong

Thorgott

20:38

this is correct

Shobhit

then why do you say order must be 2.

copper.hat

if $g^n=e$ then $g^{-n} = e$ and so $g=g^{-1}$ and so $g^2 = e$.

Thorgott

oh, that's even more elegant than what I had in mind, but yes

Shobhit

got it. thanks.

copper.hat

wow, i answered part of a group question!

dummit & foote is finally sinking in.

Shobhit

20:44

XD

can you also give me an idea as to how the order of $GL(n , Z_{p})$ is calculated. (general linear group)

Thorgott

by counting ordered bases

copper.hat

i'm not at Part III yet :-)

Shobhit

ok. i'll think.

Myprofileissaved

21:08

Can someone assist me with math.stackexchange.com/questions/1462020/… , I am stuck in the same place and can't understand the equality

robjohn

21:39

@TedShifrin It was an eventful trip (that is in the sense of "may your trip be uneventful").

Pedro Urday

Hi! Is there an article on Internet that explains how to solve quadratic congruences? I don't fint anything. I, only, find this calculator online: alpertron.com.ar/CUADMOD.HTM

For example: -x^2 - 6x + 178 ☰ 0 (mod 6) gives me x= 4 and x= 2 But i dont know how to reach that result

alpertron.com.ar/QUADMOD.HTM

PM 2Ring

22:05

@PedroUrday You may find this helpful: math.gordon.edu/ntic/ntic/section-quadratic-congruences.html

BTW, Alpertron has a page on solving Diophantine quadratics, but it's not easy to find. alpertron.com.ar/METHODS.HTM

shintuku

ah

any row of the table of a cyclic group is a shift from the row above

that's a nice thing

Pedro Urday

@PM2Ring Thank you!

Andrew Micallef

22:35

@shintuku I feel like I can appreciate that, I feel the same way.

@TedShifrin Someone I work with has a passive aggressive sign in his work bay that reads: I can explain it to you, but I can't understand it for you. Once you take away the sas this is how I have felt latley. But it's all good now, the part that I didn't understand is resolved! And I want to add an apology, I'm not sure if you mentioned I could swap the order of the partial derivatives, but I suspect you may have hinted as much.

leslie townes

23:22

that is a very passive aggressive sign. very aggressive, in terms of workplace branding.

the janitor at my elementary school had an aggressive sign in his broom closet. it was two clicks away from "go f*ck yourself." but he was the janitor.

we loved him because he would make his car backfire in the parking lot.

he'd also give people he liked little gifts on chinese new year. i just googled him and he died 20 years ago. :(

i'd bring him plums from my backyard tree.

Andrew Micallef

:/

The guy who owns this sign is less lovable, though he is human none the less, and deep down not a terrible person :P

leslie townes

bring him plums and see if he likes them.

Andrew Micallef

I got no plums to bring. But we talk about motorcycles, so that works too. (Also he sees me doing maths in the stairwells and suspects I'm building a h-bomb, so he is pretty chill with me :P)
hey leslie, if I have an integral that is of the form $\int{UVW'}dx$

On reflection I probably shouldn't say stupid things like that on public record

leslie townes

a single-variable integral? oh, how boring. integrate over the unitary group if you want my attention. just kidding.

Andrew Micallef

I am very much not building a H-bomb

so, I didn't finish, here $U,V,W$ are all functions of $x$, can I lump $U$ and $V$ together as one function and do integration by parts here?

leslie townes

23:31

if lumping them together means treating UV(x) as a single matrix valued function of a single variable, sure.

the mechanics of integration by parts may need checking, but it's not an insane thing to try to do.

Andrew Micallef

wait matricies?

that sounds complicated

maybe it isn't what I want to do

leslie townes

oh are U and V scalar valued functions?

sorry when i see U and V written like that I assume matrix. i have been conditioned that way.

Andrew Micallef

yeah, my bad for writtting in caps

leslie townes

sure, use UV(x) as a single function.

Andrew Micallef

all good, I see you opining on linear algebra, I should have figured

The thing that bugs me about the sign is that, that isn't how communication works If you explain something and the person you are talking to doesn't understand then you probably didn't do a very good job at explaining

leslie townes

23:34

for me, U is a unitary operator on a hilbert space. V is the same when you've already used U. then, you guessed it, W. you never get to X.

that's why it's an aggressive sign. it's basically, "if you aren't happy with me, go f*ck yourself"

it's my elementary school janitor's sign.

i wish i could remember the precise wording. it was in broken english.

i just texted my sister, "do you remember what mr. toy's sign said in his closet? in huge capital letters?" we are dredging up memories from 30+ years ago.

Andrew Micallef

(also possible that the person wasn't reading your responses, in which case it would be fair enough to loose patience)

leslie townes

his family was from guangzhou. he had one daughter. i'm just dredging up memories until i remember the sign.

Andrew Micallef

It sorta works that way huh

do you recall walking to the closet?

can you walk the route in your mind?

leslie townes

it's funny you mention that, i remember exactly where it was. it was halfway down the main hall of the main building.

our principal was this really frazzled WW2 veteran who would discipline bad children, mostly boys, by boring them to death by having them sit in his office all afternoon.

they'd have to watch him do work and he'd tell them about it.

Andrew Micallef

This makes me wonder what happened to my school janitor (though i only know his first name (Erni), would have no hope of finding him)

Ouch, sounds painful

leslie townes

23:41

when he died later i learned he had battled in the pacific theater and earned a silver star. you never heard about that in his office.

Andrew Micallef

Jeez, the pacific was rough (I listen to Dan Carlin's Hardcore History, the current series is all on the pacific and the real nastiness of it)

Avra

Hello,

leslie townes

hola

i don't wanna know what you'd need to do to get a silver star in the pacific.

Avra

@leslietownes. I hope you are doing fine. My friend asked me about a series,

$\frac{1}{2} \sum_{n=0}^{\infty} (z^n)+ \frac{-1}{2} \sum_{n=0}^{\infty}(z^n)$

I found that the sum should be

$\frac{n}{2}$.

But the answer is $z{2n}$

The trick here is to factor terms based on polarity of $n$ to even and to odd of,
$\frac{1}{2} \sum_{n=0}^{\infty} (z^n)+ \frac{-1}{2} \sum_{n=0}^{\infty}(z^n)$

The answer is $z^{2n}$

not as I answered, which is $\frac{n}{2}$

If we sum the

$\frac{1}{2} \sum_{n=0}^{\infty} (z^n)+ \frac{-1}{2} \sum_{n=0}^{\infty}(z^n)$

We would have

$\sum_{n=0}^{\infty}\frac{1+(-1)^n}{2} (z^n)$

Any idea if my answer $\frac{n}{2}z^n$ is correct or the book answer $z^{2n}$?

leslie townes

23:56

call me simple minded but when i see $\frac{1}{2} A + \frac{-1}{2} A$ i only think one thing. what is going on here?

Avra

@leslietownes. Hahah, sorry

I forgot to raise $n$ to $(-1)$

$\frac{1}{2} \sum_{n=0}^{\infty} (z^n)+ \sum_{n=0}^{\infty}\frac{(-1)^n}{2}(z^n)$

leslie townes

now i'm wondering how much of the stuff after the + is within parentheses, and if so, why n is used again in the second sum.

ok.

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$Mathematics$ Mathematics