Mathematics - 2015-05-09 (page 4 of 4)

Remember me

20:00

We do think stuffs in n dim but why is it so difficult in the 4 dim

Balarka Sen

stop thinking about stuff you don't even know about, @Remember.

pjs36

Well that's a silly thing to say, Balarka; I wouldn't think about anything then :). But one interesting (in my opinion) fact about the $4$th dimension is that it has the richest variety of convex regular polytopes.

In dimensions $2, 3$ and $4$, there are $\infty, 5$, and $6$ regular polytopes, respectively. But then for dimensions $5$ or larger, there are only $3$.

That's obviously not what Mike was talking about, but just a fun factoid.

Remember me

@Balarka According to you 1/0 is not infinity right ?

Balarka Sen

I think it's pointless to comment on 4th dimension without even knowing what 4th dimension means. (btw, polytopes are not the point of the complexity of 4-th dimension)

@Rememberme no, it's not.

Remember me

then why do people write tan(90)= $\infty$

Its what my prof writes in the class

Balarka Sen

20:09

Well, it's not technically correct, but morally it's true.

anon

1/0 can be usefully interpreted as a point at infinity in many geometric applications

Mike Miller

@pjs36 This is interesting. Why are there only 3 for $n \geq 5$?

Balarka Sen

The whole point is that $1/x$ as $x$ goes to $0$, gets larger and larger.

Remember me

yes limits

Balarka Sen

So one might as well construct the extended reals and identify 1/0 with $\infty$

anon

20:10

or $\pm\infty$ as may be necessary

Balarka Sen

The extended reals are bad when you do arithmetic on them, however. They are only good when you're thinking about them order-theoretically.

pjs36

@MikeMiller You know... I don't really know! I just know that they're only the simplex, cube, and its dual, the cross-polytope (convex hull of $\{\pm e_i\}$). I also don't know why exactly the $4$th dimension gets exactly one polytope that isn't a $4$-dimensional analogue of one in the $3$rd dimension.

Balarka Sen

Or doing something seriously topological.

pjs36

But I'll try and find out, and report back some time.

Mike Miller

Cool.

anon

20:12

@BalarkaSen I am reading about farey diagrams in herstein's topology of numbers, and 1/0 is used in computations meaningfully, not just as an order-theoretic decoration.

Balarka Sen

oh?

Remember me

Everything ends at topology......

anon

@BalarkaSen yes when two rationals a/b and c/d are joined by an edge, they are conjoined with the rational (a+c)/(b+d) in between them. 1/0 is on the diagram and used in this (although you have to interpret it as -1/0 on the bottom semicircle)

same idea with linear fractional transformations and I presume projective geometry

Balarka Sen

sounds nice.

well, projective geometry is what i mean when i say "topology" :P

Remember me

Topology is called that?

Balarka Sen

20:16

why should I care about Farey diagrams, @anon?

nah, @Remember. but the Riemann sphere can be thought off as a one-pt compactified $\Bbb C$

anon

because of continued fractions, quadratic forms and class numbers

Balarka Sen

class numbers come into the play?

tell me about it

anon

yes, class numbers of quadratic fields are related to quadratic forms

in fact I think class groups are related to "composition" of forms

Balarka Sen

yes, i know, but how does all this relate to Farey diagrams?

anon

I haven't read yet how, not that far in his notes

Balarka Sen

20:20

that's Hatcher's, not Herstein's :p

anon

haha, oops

someone was talking about herstein yesterday and I think they melded in my broca's area

Balarka Sen

i knew "topology of numbers" sounded familiar. haven't read his notes on number theory, but am a fan of his algebraic topology textbook.

Seth Kitchen

Math makes me sad when I can't find treasure :(

Balarka Sen

oh. well, that's just a compactified version of the upper half farey diagram (i.e., the standard ones)

Mike Miller

woo 10k

4

Balarka Sen

20:24

congrats! :)

Karim Mansour

sorry @Rememberme the internet was off

hi @BalarkaSen

Remember me

@Mike Now i get it.....that log one was good and the use of the theorem you mentioned thanks!!!!!Now no more misconceptions and misinterpretations

Karim Mansour

10 k per for your TA @MikeMiller?

Mike Miller

no lol

10k rep

Balarka Sen

haha

Remember me

20:25

TA??

Karim Mansour

oh I see

lol

Balarka Sen

:p

Karim Mansour

my brain keep relating things together :D out of memory xD

Seth Kitchen

@MikeMiller do you have any papers online? I'd like to read

Mike Miller

no

pjs36

20:26

I was curious when you'd hit 10k, @MikeMiller. You'll have to let me know how sweet those moderator tools are ... as long as you promise not to seek out my woefully wrong deleted answers :P

Seth Kitchen

You're not about that life?

Karim Mansour

hi @TedShifrin

Ted Shifrin

hi @Karim

Mike Miller

as far as I can tell I don't actually have delete votes

ah, yes I do

Ted Shifrin

I don't think I've ever voted to delete

Karim Mansour

20:29

delete votes

can you even do thhat

Ted Shifrin

Yup

Remember me

@Mike you are a mod?

anon

10k+ers get limited mod tools

Karim Mansour

I see

@anon you have like 50 k !

what special things do you have ?

Remember me

yes thats what i was about to say

Ted Shifrin

20:30

I'm nearing another milestone, @anon, but I hardly answer things any more, so I've slowed to a snail's pace.

Karim Mansour

haha

anon

you get the honorific of "trusted user" at 20k (although it can still captcha you after that, I found out, lol) and a congratulations thread on meta at 100k.

Karim Mansour

oh

who reached 100 k

Ted Shifrin

I'll be dead before I get to 100k ... or perhaps the site will be.

anon

multiple people have

Remember me

20:32

lol

anon

getting 100k for some is simply a matter of lowering your standards and having oodles of free time devoted to MSE

Mike Miller

@anon died?

Ted Shifrin

I didn't say that, @Mike. Watch your tautologies.

anon

yes, multiple people have be dead

Mike Miller

thought so

Karim Mansour

20:33

I am on mse to learn though not and discuss math not to like get points :D

anon

k

Ted Shifrin

I just found out that a mathematician I knew in the 80s died in 2001 ... very young.

Remember me

right!!!!

Karim Mansour

why

why did he die @TedShifrin

Remember me

Thats what is going to happen to me....lol

Ted Shifrin

20:34

I have no idea. Perhaps I knew this years ago, but I have no recollection.

Mike Miller

well, we're all going to die one day, barring me

I don't intend to die

Ted Shifrin

better not cross any streets, @Mike

Balarka Sen

RIP, @Mike.

Remember me

2

Q: $\sum\limits_{n=1}^{10000000000000000} \frac{1}{n}$

How does wolfram alpha solve $$\sum\limits_{n=1}^{10000000000000000} \frac{1}{n}\approx 37.4186$$so quickly? It solved it in like 3 seconds is there a equation or something

calculus

How does it really do it so quickly??

Balarka Sen

there are really good estimates of the harmonic series

anon

20:40

the Euler-Maclaurin formula yields a numerical approximation for partial sums of p-series

although it probably needs to be finagled because the expansion I'm familiar with doesn't work at s=1

Remember me

But till that large a number

Mike Miller

it's going to be actively easier for large bounds

Ted Shifrin

Or it could just use $\ln (n)-\gamma$ or whatever, @anon.

Balarka Sen

@Rememberme plug in log(10000000000000000)

Mike Miller

yeah

anon

20:41

the approximation can be to within ${\rm mild~number}/n^{\rm fairly-sized~exponent}$ quite quickly

Balarka Sen

you'll be surprised at the close numerics.

Remember me

36 something

anon

@TedShifrin that's only good to so many digits

Ted Shifrin

for large $n$, I imagine it's quite good :)

but I haven't thought about it too carefully.

anon

although I think it's good to about 1/(2n) which I guess is good enough here

Balarka Sen

20:42

it is

@Rememberme now add 1/2 or something.

Mike Miller

lol 1/2 or something

Balarka Sen

you'll get really close to what W|A gives

anon

aka 0.577

Remember me

lol

anon

remember me strikes me the alternate-universe version of skullpatrol which does math instead of trolling

Ted Shifrin

20:44

barely, @anon :P

Balarka Sen

haha

Remember me

what??

Ted Shifrin

what became of skull?

Remember me

I didnt even get what you are saying @anon....

anon

I forgot what his new name is, but hasn't been around for a bit

Balarka Sen

20:45

last time he was IceBoy

anon

ah yeah

Remember me

skull patrol..who is he/she/it

Ted Shifrin

I wish people had better things to do with their time than switch names constantly ... oh, haven't seen Jasper in ages ...

Mike Miller

last time I saw him he was editing an old answer about what mathematics is

Remember me

@TedShifrin correct yourself....please delete me :p

Mike Miller

20:46

originally it was "mathematics is what mathematicians do"

now it's "mathematics is what mathematicians do and the language with which god created the universe"

Remember me

i feel like this is some quote by someone@Mike

Balarka Sen

@TedShifrin I think I should change my name into "Grotesque Gorillas of Guinea"

Ted Shifrin

why is that, Balarka?

oh, it's past your bedtime again

Remember me

its wayyy past

Balarka Sen

inspired by pizza, i think.

Remember me

20:48

@anon you are talking about this guy: math.stackexchange.com/users/43115/skullpatrol

Mike Miller

I would probably stop talking to you.

Remember me

me?

Ted Shifrin

You stopped talking to me for a while, @Mike ...

Remember me

please dont....

Balarka Sen

@MikeMiller lol, why's that?

Mike Miller

20:49

Outraged by your lack of a working sense of humor.

Remember me

@Balarka you are having problems with chem?

Balarka Sen

@MikeMiller now you're gonna make me cry. at least I am trying :P

Remember me

lol

Mike Miller

That's a feature, not a bug.

Balarka Sen

What is?

anon

20:53

your tears apparently

Remember me

How much can one do maths....??

anon

about 52 miles, remember

Remember me

52 'miles'.....

hi@Hippalectryon

user2692669

MATLAB: Should I prefer a low level function for drawing a line or the common one? (for draing animation oriented things)

Balarka Sen

@anon oh. growls

user2692669

20:54

drawing*

Hippalectryon

$\color{red} {\huge{300\text{ points}}}$ for a lhf, anyone ?

@Rememberme hello

Ted Shifrin

ne prête aucune attention à @Hippa

Hippalectryon

@TedShifrin :c

@TedShifrin It's not even school maths

Ted Shifrin

too many memes, @Hippa.

Hippalectryon

@TedShifrin Only one

Still too many, replied Ted

Ted Shifrin

20:58

someone found a page with a whole bunch

Remember me

lol where

Ted Shifrin

even Mike thought they were too much ...

Balarka Sen

@TedShifrin haha, link me

Hippalectryon

@TedShifrin I only made one though q_q

Ted Shifrin

I have no idea where it was ... it's somewhere in the chat records.

Chris's sis

20:59

@Hippalectryon I proved this formula in an amazing way and then I developed the version in 3 dimensions that probably no one did so far (no sources available on internet)

mathworld.wolfram.com/HadjicostassFormula.html

Hippalectryon

@Chris'ssis :D

Chris's sis

@Hippalectryon see if any professor in France can do that without pen and paper. I did it like that either you believe it or not.

Remember me

hi@Chris'ssis

Hippalectryon

@Chris'ssis How would I know :-) I know very few professors

Mike Miller

@Hippalectryon really?

Chris's sis

21:01

@Hippalectryon ;)

Balarka Sen

hahaha

Hippalectryon

@MikeMiller e_e I forgot about those

@MikeMiller I didn't make ALL of those though

Remember me

lol...hahahhahahhhahahah

Hippalectryon

Like, this one (wut ??)

Mike Miller

most of them are very mean-spirited.

Hippalectryon

21:02

@MikeMiller Not really

Remember me

@TedShifrin I feel really sorry for you

Hippalectryon

@MikeMiller You need the context

Mike Miller

I was there for the context.

Hippalectryon

The only one that could really be 'mean' would be the Satan one, but in its context it's fine

Balarka Sen

I just burst into fits of laughter, without meaning to.

Remember me

21:04

loel @BalarkaSen.....hahah

Balarka Sen

The original meme was ok, @Hippa. I share Mike and Ted's opinion about having too many of them.

Hippalectryon

Oct 10 '14 at 14:52, by The Game

^ that's the context iirc

Remember me

@Hippa there are too many of them....

Ted Shifrin

I believe I have a pretty good sense of humor and can take a lot of abuse (will be getting a whole bunch at my retirement party), but I find this offensive ... and I'm seriously gone.

Hippalectryon

@BalarkaSen @Rememberme There's only one that has been reposted several times here. The others were made for special occasions and never reused later. I didn't even remember they existed

Remember me

21:07

@Hipa @Ted is not mean why in the first place do youhave to make meme's

Hippalectryon

@Rememberme Memes are not mean imo

Balarka Sen

So essentially you saved the spare ones for reusing appropriately, @Hippa? :p

Remember me

These were...

Hippalectryon

@BalarkaSen e_e that's not what I mean

@Rememberme I honestly don't see what's mean, when they're placed in their context

Balarka Sen

@Hippalectryon I don't know : I don't think I'll feel too good if a bunch of weird memes of mine were flying around the internet.

Hippalectryon

21:09

I wouldn't have made them otherwise @Rememberme

Remember me

@Hippa balarka has a point

pjs36

Hippa... everyone who's weighed in has disagreed with "not a big deal". I'd just man up and apologize, rather than make excuses. There's plenty more I could say.

banach-c

hi

i have an other problem related to this math.stackexchange.com/questions/1270940/… topic

if the question is "now the field has 10x15 boxes. how

affects this on the mobility/flexibility of the cuboid?

in my opinion the problem is imprecise

Remember me

21:28

@Balarka any linear transformation is orthogonality preserving right..

Balarka Sen

Depends on what you mean by orthogonality preservation.

Remember me

i mean that if $x\cdot y=0$ then $T(x)\cdot T(y)=0$

Balarka Sen

no, not true.

Remember me

Currently i can think of two rotation and reflection

oh i got a counterexample

Projection is that right

Balarka Sen

You're looking for linear transformation which aren't orthogonal. There are a buckload of those.

Correct :)

Remember me

21:32

Thanks ...

@Balarka Hoffmann kunze goes with polynomials and all after linear transformations is that required

Balarka Sen

How come you're doing orthogonality without knowing basechange stuff?

Remember me

Orthogonality is in my textbook school one...i just extended it to linear transformations...simple :)

Balarka Sen

No, it's absolutely not simple. How do you define orthogonality of vectors of arbitrary vector spaces?

Joe Stavitsky

anything obviously wrong here?

Balarka Sen

@Remember These stuff are all about inner products. That is where the "real" linear algebra come to play.

Chris's sis

21:40

$$\int_0^1 \int_0^1 \frac{\ dx \ dy}{x^2+y^2+1}=\frac{1}{32} \left(32 \Im\left(\text{Li}_2\left(-i \left(-3+2 \sqrt{2}\right)\right)\right)+\pi \cosh ^{-1}(665857)-32 G\right)$$ It's $(5)$ from here mathworld.wolfram.com/UnitSquareIntegral.html

Joe Stavitsky

er sorry goo.gl/j9NTIL

user2941942

Hello, Can I ask a simple question, that probably not worth posting in crossvalidated.I am confused. I was taught this formula Var(resid)=Var(y)-Var(yhat). And then to get unbiased s^2=(N/N-k)*Var(resid). But what I discovered is that this formula provides us with Var(resid)=Var(y)-Var(yhat) sample variance.
Shouldn't the formula for s^2 then be s^2=(N-1)/(N-k)*Var(resid) ?
I mean for this particular case, should I modify the formula of s^2 or not?

Chris's sis

22:42

Here is a nice integral to compute in the spirit of the art $$\int_0^{\pi/2} \frac{\arctan(\cos(x))}{\cos(x)} \ dx$$

But you know what?

Math Man

approximately 1.3844583929236 but what's special about it?

Chris's sis

@MathMan You shouldn't be told which things are fascinating, it's up to you to decide that.

Math Man

I guess I think everything is fascinating. I am trying to collect fascinating numbers, in fact.

Chris's sis

@MathMan That sounds better.

Math Man

My collection is on extraordinarynumbers.wordpress.com It is not very big yet but growing.

anon

22:58

"two is the only even prime" - well, for every prime p, the only prime which is a multiple of p is p itself

so that doesn't actually make 2 unique (although there are I think deeper motivations for considering it unique for this reason, which I can't summon at the moment)

Chris's sis

@MathMan Thanks. There is a book you might like to read too, Mathematical constants by Finch.

Math Man

@anon you have a good point. "How about 2 is the smallest prime." I lean towards features of numbers which make them more unique. Consequently "Only" generally takes precedence over "largest" or "smallest".

@Chris'ssis Thanks I looked up that book. He is way ahead of me in terms of compiling interesting numbers.

Mike Miller

@anon that $1 = -1$ in $\Bbb Z/2\Bbb Z$ frequently causes issues in a variety of fields, whence some of the consternations that it's such an odd prime, I think

anon

yes

there was a mathoverflow question I believe

Nickolas

Hey

Mike Miller

23:09

it also frequently causes amusement, i suppose

Math Man

So what do all of you think is the most unique thing about the number 2. To be more specific I am looking for things that make it important in mathematics and number theory - pure mathematics.

Mike Miller

that fact makes it easy to do plenty of things in topology when you don't want to worry about orientations; just do it mod 2

robjohn

@Rememberme Look up Euler-Maclaurin Sum Formula

Nickolas

Quick question, can articles be published at arxiv?

Math Man

@MikeMiller I haven't studied topology much. Do you have a good link on that that would be interesting reading.

Mike Miller

23:13

nothing concise

Math Man

@anon I updated my site based on your suggestion.

anon

@robjohn das what I said two comments down!

@Nickolas you mean like the history and overview section? sure, people do expository and lay stuff sometimes.

pff, there's no way more people understood Qiaochu's answer than mine

pjs36

@MathMan You'll also probably like to read about some of John Baez's Favorite Numbers

Mike Miller

i upvoted his because i liked his

sry bae

pjs36

23:29

@anon Group...poid? :P

anon

groups are to groupoids as humans are to humanoids

Mike Miller

like a group, but you can't necessarily multiply any two elements

pjs36

Hmm, I see. Explain it to me as if I were a diehard Bladerunner fan, and I think I'll get it

anon

think of it as a bunch of cities with bidirectional travel paths between them. one can of course compose paths to get new travel paths, as long as the destinations and starting points match up correctly.

Mike Miller

and you can always invert any one of your travel paths: just go backwards

anon

23:33

(and I guess going a->b and then the same path back b->a cancel out, but going a different path b->a back doesn't cancel)

Mike Miller

so while you can't multiply any two elements, there is an 'identity element' e at each city that's the identity against everything you can multiply it with, and an inverse for each element that gets you back to the identity element of the city that elt started at

the actual definition is motivated by the categorical definition of a group: it's just a category with one object s.t. every morphism is invertible. so a groupoid is a category such that every morphism is invertible

the 'one object' bit allows you to multiply any two elements; aka compose any two morphisms

in the general case you can't do that

pjs36

You know, this all started off as a joke because I wanted to write "poid"

But I think I'm learning things nonetheless.

Mike Miller

poid

pjs36

See? It's a funny word!

So I took topology, which was cool. Then I leave, and my former advisor teaches algebraic topology... which is a shame.

Math Man

@pjs36 that is a great article and I can see why he likes the numbers 24 and 12. My favorite thing about 12 is that it is the smallest abundant number.

pjs36

23:38

So homotopies et al were crammed in at the end of the class I took, and I haven't had the will-power to make it far in Hatcher. Such a shame.

Mary Star

Mazzy Star - Fade Into You

►

Mike Miller

homotopy is good

pjs36

I know! I don't even think there were any homework assignments on it, it was like the last 3 days of class...

Mary Star

Do you like the song?

anon

I await the day porton invents vocaloids.

that is my niche joke of the day

Mike Miller

23:52

unfortunately I get the joke

Transcript for

$Mathematics$ Mathematics