Mathematics - 2014-03-13 (page 1 of 4)

12:14 AM

I have that lim((n+1/2)log(1+1/n)-1)=lim(1/(12n^2)) where both are limits as n tends to infinity. How can I use this to show that the sum from n=1 to infinty of ((n+1/2)log(1+1/n)-1) converges?

Pedro Tamaroff

@user112495 You there?

user112495

@PedroTamaroff yes

Pedro Tamaroff

@user112495 OK. You know that as $x\to 0$; $$\log(1+x)=x-\frac{x^2}{2}+o(x^2)$$ yes?

user112495

@PedroTamaroff Yes

Pedro Tamaroff

I think you need another term. At any rate, your n-th term goes like $$\frac{1}{n^2}$$

Alexander Gruber

12:26 AM

so, who likes commutative algebra?

Pedro Tamaroff

$$\eqalign{
& \mathop {\lim }\limits_{n \to \infty } \frac{{\left( {n + \frac{1}{2}} \right)\log \left( {1 + \frac{1}{n}} \right)}}{{{n^2}}} = \cr
& \mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{n} + \frac{1}{{2{n^2}}}} \right)\log \left( {1 + \frac{1}{n}} \right) = \cr
& \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\log \left( {1 + \frac{1}{n}} \right) + \mathop {\lim }\limits_{n \to \infty } \frac{1}{{2{n^2}}}\log \left( {1 + \frac{1}{n}} \right) = 1 \cr} $$

@AlexanderGruber I guess I do! =D

I know little about it.

@AlexanderGruber Wanna see something beautiful?

Alexander Gruber

@PedroTamaroff i think me too

sure

Pedro Tamaroff

Alexander Gruber

@PedroTamaroff that's a lotta Bourbaki.

Karl Kronenfeld

@AlexanderGruber hey yo

Pedro Tamaroff

12:29 AM

@AlexanderGruber Yeah. It's in my uni's library deposit.

In particular, I borrowed this:

user112495

@PedroTamaroff Thanks!

Alexander Gruber

@PedroTamaroff that looks fun

@KarlKronenfeld hi

Pedro Tamaroff

@AlexanderGruber Yeah. I will learn some French while I'm at it. =O

Karl Kronenfeld

@AlexanderGruber I'll answer your question with a question. Who doesn't like commutative algebra.

Alexander Gruber

@KarlKronenfeld it either blows my mind or makes me want to set myself on fire

on alternating days

@PedroTamaroff i'm learning German for my PhD

i'm trying to translate Huppert's Endliche Gruppen for it

Jasper Loy

12:34 AM

@AlexanderGruber Survey: What is your favourite ODE book?

Karl Kronenfeld

@AlexanderGruber It makes me warm and cozy inside.

Alexander Gruber

@JasperLoy i don't have one.

Jasper Loy

@PedroTamaroff Maybe you can borrow Dieudonne's Treatise on Analysis.

Pedro Tamaroff

@JasperLoy They a.s. have it.

Jasper Loy

Apostol's Calculus is very expensive...

qwr

12:37 AM

Can someone help me try to gain "insight"

I've been doing some contest problems

and I don't understand how people can draw conclusions so quickly

Jasper Loy

They draw with a pencil, lol.

Alexander Gruber

@KarlKronenfeld so i'm trying to find a counterexample to this when $M$ isn't flat.

i'm thinking i'm going to take $M=\mathbb{Z}/n\mathbb{Z}$

qwr

Like one problem "sum all a/b where a and b are divisors of 1000"

Karl Kronenfeld

@AlexanderGruber Yeah, that's where I'd start

qwr

And people are able to figure out it's actually (2^-3 + 2^-2 + ...)(5^-3 + 5^-2 ...)

how do people realize this

Alexander Gruber

12:40 AM

and then have $N_1$ and $N_2$ be $a\mathbb{Z}$ and $b\mathbb{Z}$, and $A=N=\mathbb{Z}$

but things are getting complicated

Jasper Loy

@qwr They do this by thinking deeply for a long time.

qwr

several hours are given in the contest

Jasper Loy

@qwr Not anyone can see it, don't be discouraged when you can't. Not everyone is like Terence Tao.

qwr

but the connections are so broad in scope??

Jasper Loy

Actually, the more problems you do, the more techniques you develop.

So the answer to your question is: keep doing problems, QED.

Alexander Gruber

12:42 AM

@KarlKronenfeld do you think that's the right $N$ to use?

Karl Kronenfeld

yeah, one moment

qwr

I'm talking about the AIME, which is tomorrow (if I choose to take the 1st one)

Jessy Cat

Let's say that I have $C exp[k(||u||_{L^{n}}+||Du||_{L^{n}})^{\frac{n}{n-1}]$ and I need to get it to become $C exp(||u||_{L^{n}}^{n}+c||Du||_{L^{n}}^{n})$. How would I do this (i.e., what inequalities)?

qwr

I feel like I'm almost there, but always just off of the required

Jessy Cat

even if it's less than or equal to that

qwr

12:45 AM

solution

Karl Kronenfeld

@AlexanderGruber Is this a problem, or do you care if I tell you an answer?

Alexander Gruber

@KarlKronenfeld go for it.

Karl Kronenfeld

@AlexanderGruber It's not too crazy of a counterexmple. Maybe I'll stick to a hint. Note that $M\otimes A/\mathfrak a\cong M/\mathfrak aM$.

Jessy Cat

3

Q: Use of cut-off functions and partitions of unity

this is a simple problem, but I would still be very thankful if you could give me an advice on it. I'm trying to show that in a compact M-dimensional manifold, $ \int e^w \sqrt{g} dx \leq C \exp ( c ||Du||_{L^n}^n + ||u||_{L^n}^n ) $, where we used the standard Einstein notation. I know I sho...

I've posted a bunch of comments to help answer this question (the person who asked it is a person in my class at uni, and I actually am having trouble with the same problem), and I think my only trouble at this point is algebra/use of proper inequalities.

So, even if you don't know about manifolds, you might be able to help.

Karl Kronenfeld

@AlexanderGruber Oh, sorry, I didn't read this carefully. No, I don't like the choice of $N$ here.

Jessy Cat

1:06 AM

Please help if you can.

Mikhail

1:40 AM

Hey does anybody know how to show that taking the fourier transform twice flips the original signal? Writing down the integral isn't helping...

Karl Kronenfeld

2:25 AM

@AlexanderGruber I looked back at the example I had; it was bs, so I apologize. I dunno what would constitute a counterexample. (CA is certainly not making me feel warm and fuzzy today :D)

Ryker

3:52 AM

Does anyone know whether, in the definition of an inner product space, the $<f,f> = 0$ iff $f=0$ condition is to be taken strictly or up to a.e.?

sea turtles

@Ryker is this not something that can be simply looked up? are you asking the question in a specific, not general, context from which you might get the idea almost-everywhere is a relevant qualifier? I've never seen a.e. used near the definition, so your question strikes me as strange.

Ryker

No, it's not mentioned anywhere in the definitions I've checked.

usukidoll

hi hi

skullpatrol

hello hello

usukidoll

4:09 AM

Let R be a relation on a set S. Prove the following...
If R is relfexive and transitive, then $ R \circ R = R$... besides finding the definititons... I have no idea what's next

usukidoll

4:39 AM

0

Q: Modular Arithmetic - Finding the smallest possible length of the room in inches

I need to know if I've done this proof correctly. Question: A rectangular room is to be tiled with square tiles. Consider only the length of the room. The tiles are available in 9-inch, 10-inch, or 11-inch squares. If only 9-inch tiles are used, there is a 5 inch gap at one wall. If only 10-inch...

modular-arithmetic proof-verification

ooo bunny rabbit

Pedro Tamaroff

@ryker if you're working on Lp then you do want to take the quotient by 'a.e.' equality' if you want an inner/normed/metric space. Else you have 'semi' spaces, which is not that terrible.

Ryker

I am, yeah.

Weighted $L^2$, actually.

skullpatrol

5:03 AM

Congratulations on achieving 100,000 reputation points @robjohn

robjohn

@skullpatrol thanks... Jasper wants me to retire ;-)

skullpatrol

6:27 AM

@robjohn that would be a huge loss for us :(

skullpatrol

7:14 AM

0

Q: Congratulations robjohn for being the first active moderator in the 100k club.

We truly appreciate your constant presence and insightful hints in the Mathematics chat room. Your dedication to learning is inspiring. Thank you.

discussion specific-user celebration

usukidoll

8:00 AM

hi @mike

Mike

hi

i'm goni to bed

usukidoll

k night

robjohn

8:28 AM

@skullpatrol I didn't say I was even considering that.

@robjohn :D

@r9m You here?

@Sawarnik ya

@skullpatrol The clubhouse is getting more and more crowded :-D

Sawarnik

@r9m Very rarely I have seen a mathematician interested in simple geometry problems.

eXtremiity

8:35 AM

Hey guys, please have a look at my question. This link will direct you.
http://math.stackexchange.com/questions/710253/my-proof-for-the-cardinality-of-a-particular-binary-distribution .
@Mike , if you want - post your alternative solution that we discussed earlier during the week and I'll Tick it if my brain understands it.

robjohn

@Sawarnik I think most of simple geometry has been beaten to death and there is little new research going on there.

Sawarnik

@robjohn Forum Geometricorum.

forumgeom.fau.edu

One, I know, maybe there are others.

r9m

@robjohn 'beaten to death' ? :( ..

robjohn

@Sawarnik However, I have added an extension to The Soddy-Gosset Theorem that governs the placement of the centers of the spheres.

@Sawarnik If anyone even remembers Soddy's Theorem regarding the radii of $n+2$ mutually tangent spheres in $\mathbb{R}^n$.

Sawarnik

@robjohn I dunno these things :/

skullpatrol

8:41 AM

@r9m Euclid's Elements began the "beat down" ;-)

Sawarnik

@r9m You are like a mathematician to me, just like Balarka :)

robjohn

@Sawarnik It says that $n\sum\limits_{j=1}^{n+2}\kappa^2=\left(\sum\limits_{j=1}^{n+2}\kappa\right)^2$ where $\kappa_j$ is the reciprocal of the radius of sphere $j$.

r9m

@robjohn so its like descartes theorem but in n-dim ?

robjohn

@r9m indeed

my addition says that $\frac{\sum\limits_{j=1}^{n+2}\kappa_j\vec{c}_j}{\sum\limits_{j=1}^{n+2}\kappa_j‌} =\frac{\sum\limits_{j=1}^{n+2}\kappa_j^2\vec{c}_j}{\sum\limits_{j=1}^{n+2}\kappa‌_j^2}$

which, if the centers are not all coplanar, allows one to compute the center of sphere $n+2$ given the other centers.

usukidoll

Let $S =[a,b,c]$. Give examples of
relations $R_1$ and $R_2$ on $S$ such that $R_2 \circ R_1 = R_1 \circ R_2$
relations $R_1$ and $R_2$ on $S$ such that $R_2 \circ R_1 \neq R_1 \circ R_2$

Definition 6.3.9 states that we let $R_1$ and $R_2$ be relations on a set $S$. The composition of $R_2$ with $R_1$ is the relation $R_2 \circ R_1 = [(x,y) \in S \times S :( \exists y \in S)[(x,v) \in R_1 \land (v,y) \in R_2]$

for a. let $ R_1 = [a,a]$ and $R_2 = [a,a]$
If we take the composition of $ R_2 \circ R_1$, then the result is $[a,a]$

skullpatrol

8:48 AM

@robjohn btw the answer is
inductive reasoning . . . deductive reasoning

robjohn

@skullpatrol which answer?

skullpatrol

@robjohn In mathematics, __________ contributes to the discovery of new ideas and information, while __________ is used to prove with logical certainty that an idea is true or false.

Sawarnik

@skullpatrol No its cats and dogs.

robjohn

@Sawarnik is that what the cats was about?

r9m

@robjohn wow .. that looks beautiful :) ..

robjohn

8:53 AM

@r9m The proof actually follows from the proof of Soddy's Theorem, but paying attention to what is usually ignored in the proof :-)

r9m

@robjohn can I see your proof ? (if it uses advanced geometry I wouldn't understand it though) :)

robjohn

@r9m it is more linear algebra... I will put it on Dropbox

@r9m Try here

r9m

@robjohn Thank you very much :) I got it

usukidoll

9:13 AM

0

Q: Composite Relations - Give Examples of relations $R_1$ and $R_2$ such that $R_2 \circ R_1 = R_1 \circ R_2$ and $R_2 \circ R_1 \neq R_1 \circ R_2$

Let $S =[a,b,c]$. Give examples of a. relations $R_1$ and $R_2$ on $S$ such that $R_2 \circ R_1 = R_1 \circ R_2$ b. relations $R_1$ and $R_2$ on $S$ such that $R_2 \circ R_1 \neq R_1 \circ R_2$ My attempt: Definition 6.3.9 states that we let $R_1$ and $R_2$ be relations on a set $S$. The com...

homework proof-writing relations

r9m

@robjohn thats a beautiful derivation :) .. simply loved it :)

robjohn

@r9m Glad you enjoyed it :-)

r9m

@robjohn I am going through /nebula/math .. I'm enjoying that too :D

robjohn

@r9m the server is rather old and prone to going down. If you notice it not responding, let me know and I will restart it.

r9m

@robjohn thank you :)

robjohn

9:26 AM

@r9m those are mainly posts or support material for articles on sci.math

@r9m I am moving as much as I can to MSE when the opportunity arises.

or putting the PDFs on Dropbox

r9m

@robjohn thats awesome :) ..

N3buchadnezzar

when is $ |a b| < |a||b|$ ?

skullpatrol

when a=0 or b=0

Karl Kronenfeld

Only instance I can think of is with groups, $|a|$ meaning order of $a$.

usukidoll

anyone know the probability of buying 7 boxes and all of the boxes have a same character? there are 12 characters to get ^^

r9m

9:39 AM

@N3buchadnezzar that suddenly reminded me of a story .. my nuresry teacher taught in class that the inequality sign is an alligator that goes for the bigger meal :P .. but I dont see how putting $a$ and $b$ in different packets will make it a bigger meal :P

skullpatrol

also when |a|=1 or |b|=1

Sawarnik

@r9m You have a good memory, really.

r9m

@Sawarnik :P I remember it coz its funny:P

N3buchadnezzar

@r9m well $|a+b|\leq|a|+|b|$ ;)

Karl Kronenfeld

Right, tropical semirings, @N3buchadnezzar

Sawarnik

9:47 AM

@r9m Is there some kind of problems that you can get from me, that you will like?

r9m

@N3buchadnezzar hehe .. Indeed :D

@Sawarnik you want to give me a problem ? :)

N3buchadnezzar

@KarlKronenfeld Stop eating all those abelian grapes Karl!

robjohn

10:08 AM

@N3buchadnezzar never?

skullpatrol

10:33 AM

Hi @DanielFischer

Daniel Fischer

Hi @skullpatrol.

skullpatrol

@DanielFischer I found another nightmare question for high schoolers :(

Daniel Fischer

@skullpatrol Which?

skullpatrol

@DanielFischer The Creative Arts Fund must raise 2,500 next year by investing 30,000 in federal notes paying 9% interest and municipal bonds paying 8% interest. The treasurer wants to invest as much as possible in the bonds, even though they pay less, because the bonds are for projects in the local area. How much should the treasurer invest in bonds to ensure they reach their goal of $2,500?

The information that "The treasurer wants to invest as much as possible in the bonds, even though they pay less, because the bonds are for projects in the local area." makes no difference to the answer.

Daniel Fischer

I don't understand the words, but doesn't that translate to "maximise $x$ so that $(30000 - x) \cdot \frac{9}{100} + x\cdot \frac{8}{100} \geqslant 2500$"?

skullpatrol

10:40 AM

But x=$20,000.

Daniel Fischer

@skullpatrol But it isn't forbidden that they make more than $2500$ a priori, isn't it?

And then the goal of investing in bonds makes a difference.

Karl Kronenfeld

After trying off and on for days now, I finally beat that 2048 game.

Daniel Fischer

34 years early, @KarlKronenfeld?

Karl Kronenfeld

Yes, they didn't predict that someone like me would be so addicted to actually try to beat it when they released the game.

Daniel Fischer

What sort of game is it, @Karl?

Karl Kronenfeld

10:48 AM

@DanielFischer It's very minimalistic. gabrielecirulli.github.io/2048

2

I fathomed for a little bit that it was impossible.

Daniel Fischer

Aha. Planlessly typing arrow keys, I may have got an idea how the game works. I didn't understand it from the description.

Karl Kronenfeld

Yeah, the description sucks

skullpatrol

@KarlKronenfeld Have you seen this game?

Thanks @DanielFischer for the suggestion :-)

Karl Kronenfeld

@skullpatrol Now I have

skullpatrol

@KarlKronenfeld They play a lot of it in the ELU room

Nick

11:01 AM

in The Root of Math, 1 min ago, by Nick

What does $\sideset{_1^2 }{_3^4}\sum $ mean?

skullpatrol

It means something is missing ;-)

Nick

@skullpatrol: Care to demonstrate?

eXtremiity

A countably infinite set has a countably infinite subset, correct ?

Of course it does .....ugh - I have no faith .

Karl Kronenfeld

Trivially yes. Also a countably infinite set has a countably infinite proper subset.

eXtremiity

Yes - thank you ^_^

skullpatrol

11:11 AM

@Nick Sums are taken from an initial value to a final value. I have never seen that notation.

So, I'm missing something :-)

Nick

Oh! XD

Karl Kronenfeld

$\require{cancel}\cancel{\sum}\sideset{_1^2 }{_3^4}\sum$

eXtremiity

My finance tutor wrote out summation notation as $\sum_{n}^{t=1}$ . . .

I couldn't hack it.

Nick

O_O From n till t = 1 , sounds conditiony

eXtremiity

It sounds wrong :p .

Nick

11:16 AM

@skullpatrol: Would what eXtremity wrote be the summation $\sum_{t=1}^{n} t$ backwards?

skullpatrol

yes

it is backwards

1 + 2 + 3 + ... + n

Nick

$$n! = n \times (n-1) \times (n-2) \times\dots\times 3 \times 2 \times 1 = \prod_{n}^{x=1}x$$

@skullpatrol: Is that right?

skullpatrol

:D

n + (n-1) + ... + 3 + 2 + 1

Nick

@skullpatrol: What is that? A sumtorial? A factation? XD

N3buchadnezzar

@robjohn, do you know a link to a proof of the following statement?

skullpatrol

11:21 AM

@Nick It could be thought of that way...

Nick

@skullpatrol: It's not wrong to do that thing is it? switching initial and final. At the end isn't it all the same thing?

N3buchadnezzar

Let $P$ and $Q$ be polynomials such that $\deg(P) + 2 \leq \deg(Q)$.
If $Q(x)$ has no zeros on the real line, and $P(x)$ is analytic in the upper half plane then
\begin{align*}
\int_{-\infty}^{\infty} \frac{P(x)}{Q(x)} \mathrm{dx}
= 2 \pi i \sum_{k=1}^{m} \mathrm{Res}\left[ \frac{P(x)}{Q(x)} , z_k \right]
\end{align*}
where $z_k$, are the singularites to $P/Q$ in the upper half plane.

skullpatrol

@Nick Gauss did it at a very young age, so the legend goes...

eXtremiity

Amazing.

Nick

@skullpatrol: Who would have gaussed that

4

eXtremiity

11:26 AM

+1

skullpatrol

:D

Sawarnik

11:42 AM

@r9m Sorry for leaving without trace. You know my internet connection...

skullpatrol

@Sawarnik Are you on Wifi?

Sawarnik

@r9m So I was asking whether is there some field of maths you like, and I know it.

@skullpatrol Yes.

Daniel Fischer

@N3buchadnezzar Should be proved in the chapter/section on applications of the residue theorem in any introduction to complex analysis.

skullpatrol

So @DanielFischer you see nothing wrong with the question, given that the students have not been introduced to expressions involving inequalities?

N3buchadnezzar

@DanielFischer Not in Gamelin

I do have rudin lying around, but not here

Daniel Fischer

11:49 AM

@skullpatrol You didn't tell me that the pupils didn't know inequalities (but if they don't know inequalities, I can't imagine they could understand percentages).

skullpatrol

@DanielFischer Good point.

Daniel Fischer

@N3buchadnezzar Oh, Rudin doesn't have a section on applications of the residue theorem. But if you're reading Rudin, it is expected that you figure out such things on your own ;)

N3buchadnezzar

I can see why the countour goes to zero,but I had some problems on the discussion on the convergence of the real integral

Daniel Fischer

Anyway, @N3buchadnezzar The condition on the degrees guarantees a) that the integral over the real line exists, and b) that the integral over the semicircle tends to $0$.

N3buchadnezzar

Also why sometimes the condition is $\deg P + 1 \leq Q$, does not seem like this integral only converges conditionally

Daniel Fischer

11:56 AM

For a), the absence of poles on the real line is of course relevant too.

@N3buchadnezzar For $\deg P = \deg Q -1$, you need some further ingredients.

The principal value of the integral exists, but the integral over the semicircle doesn't tend to $0$ then.

skullpatrol

12:13 PM

@KarlKronenfeld they just found the 2048 game in the ELU room :D

Sawarnik

@skullpatrol Can you give the link again?

I wanna try it now.

skullpatrol

Here

1 hour ago, by skullpatrol

@KarlKronenfeld Have you seen this game?

Sawarnik

Ok, ok.

Matt N.

12:45 PM

One can but wonder why this person bumped an old thread of mine. But for some reason they did so anonymously...

Karl Kronenfeld

@MattN Someone posted the same question today here.

John Doe

1:38 PM

Does this follow in anyway, if u_n converges weakly * to u in W^{1,infty} then we can show that ||Du||_{L^{infty}} <= C, it follows then that |u(z)-u(y)|/|z-y| <= C.

It is also given that (u_n)_n is Lipschitz on the domain.

Pedro Tamaroff

@KarlKronenfeld

Karl Kronenfeld

1:55 PM

@PedroTamaroff ey

Matt N.

2:07 PM

@KarlKronenfeld Sure but that's no reason to bump old stuff.

But whatever.

robjohn

@N3buchadnezzar This looks just like the Residue Theorem which is a corollary of Cauchy's Theorem.

Pedro Tamaroff

@KarlKronenfeld What's up? @seaturtles

@Kasper I have no idea what you're getting at with this.

How does $\Bbb R\times 0$ make no sense...?

Anush

hi

Pedro Tamaroff

There are things that you might not understand fully, but there is not such thing as something that makes no sense! =D

Except well, if written by a pothead or something.

Anush

I was wondering why this question has so few votes and interest math.stackexchange.com/questions/706848/… ?

I added a large reward as it seems very interesting to me

but maybe there is something wrong with it?

Pedro Tamaroff

2:16 PM

@Anush Some people just don't care, or don't know.-

Anush

@Pedro OK. So there is nothing obviously wrong the question?

in the old days people used to upvote things :)

or maybe it's just really really hard

who knows

@Pedro did you upvote it? :)

IvanMatala

2:40 PM

Hi, lets say we have an infinite series.. how can we write a close expression that zeros those odd terms?

Pedro Tamaroff

@IvanMatala ??

IvanMatala

say let x(n) = (1/2) ^n for n = 0,2,4....... and 0 otherwise

Pedro Tamaroff

@IvanMatala Oh.

IvanMatala

is it possible to create a close form of that

Pedro Tamaroff

$\dfrac{1+(-1)^n}2x_n$

IvanMatala

2:42 PM

lemme try that :)

it works

Pedro Tamaroff

Of course it does!

IvanMatala

amazing.

@PedroTamaroff how do we go learn this tricks?

Pedro Tamaroff

@IvanMatala Well, it is not a trick. $(-1)^n$ is $1$ if $n$ is even and $-1$ if $n$ is odd.

Then $1+(-1)^n$ is $2$ if $n$ is even and $0$ if $n$ is odd.

Kasper

@PedroTamaroff $\BbbR \times 0$ does make sense to me, as is. I don't know what about the thing that didn't make sense to me, don't make sense to you ? :P

Pedro Tamaroff

What does "make sense" mean to you?

Kasper

2:46 PM

defining $\mathbb{R}:=\mathbb{R} \times 0$ doesn't make sense to me.

Pedro Tamaroff

@Kasper We're not doing that.

We're not defining something using that very same thing.

It would be circular.

We are embedding the reals in $\Bbb R^2$.

Kasper

You maybe not, but I was doing that (for myself). To fix my problems I had.

Pedro Tamaroff

@Kasper Well, but then you were doing something incorrect. That doesn't mean some mathematical concept "makes no sense."

We're identifying the real numbers we already had with complex numbers with zero imaginary part, that is, complex numbers of the form $(a,0)$.

Kasper

True. Did you read my post ? :-) I was saying that what I was doing, didn't make sense. But the confusion came from that $\mathbb{R}$ is not a subset of $\mathbb{C}$. I was trying to fix that for myself.

Pedro Tamaroff

The point is the map $\eta:\Bbb R\to\Bbb C$ with $\eta(x)=(x,0)$ is not only an injection, but it is also a ring homomorphism.

So we're truly embedding $\Bbb R$ along with all its structure inside the complexes.

@Kasper Well, that's not what your question conveys.

Your question conveys something like "I'm studying. This topic makes no sense. Should I accept it does?"

But the point is it makes perfect sense.

And you're simply misunderstanding things.

Tessa Danger Bamkin

2:51 PM

greetings

Kasper

Hm.. what I meant was, many things in mathematics doesn't really make sense to me (not yet at least). Should I accept that ? Or study the foundation of mathematics better ?

Pedro Tamaroff

I guess what you mean is "I don't fully grasp them."

But it would be strange for you to be studying holomorphic maps and whatnot if complex numbers "make no sense" to you.

@TessaDangerBamkin hey there

There is no need to go to foundations, really.

Kasper

complex numbers make sense to me, as x+iy and i^2=-1, but how complex numbers are constructed doesn't make sense to me. I don't fully grasp that construction if you will. My question is, should I spend time in understanding that construction better ? Or should I just stick with the more intuitive approach.

Same question I could ask for real numbers and studying that construction etc.

Pedro Tamaroff

@Kasper What is it about the construction of complex numbers that you don't understand?

eXtremiity

Difference between black square and white square at the end of proofs?

Pedro Tamaroff

2:57 PM

@eXtremiity Style.

N3buchadnezzar

@eXtremiity color

Pedro Tamaroff

Some people use squares, some people use three dashes, some others use triangles.

Others use QED.

And so on.

eXtremiity

I see.

Tessa Danger Bamkin

can be important in logic (the difference between black and white)

eXtremiity

Thanks guyzzz ^_^

Tessa Danger Bamkin

2:58 PM

but otherwise it doesnt matter

I'm trying to show that the confluent hypergeometric function is the solution to kummers differential equation but not sure how to do it. Never differentiated factorials and sigma notation things.

Kasper

Well, I understand it now a lot better. But what still confuses me, as that you know, from the definition, the real numbers are the scalars, in the construction of the complex vector space. But in some sense, the real numbers are also the complex numbers $(x,0)$. In any text I read, they say $x=(x,0)$, I've asked a question about it, and Bill says it is just some abuse of notation. But I don't like that I think. Allthough I understand they act the same way. Saying that $x=(x,0)$ makes my brain hurt.

Kasper

3:14 PM

@PedroTamaroff That

eXtremiity

3:30 PM

What are some good math drawing programs that one can implement into Latex, preferably LyX.

I'm trying to draw a nice step function.

Sawarnik

4:16 PM

@robjohn You here? Can you confirm the answer here, math.stackexchange.com/questions/703060/… , so I could give it the bounty.

Jasper Loy

4:56 PM

@eXtremiity Use the package pgf and pgfplots.

Mike

Hi @PedroTamaroff

Pedro Tamaroff

@Mike Hey there.

What's up?

Mike

Not much.

Jasper Loy

Mike and Pedro only say hi to each other, not to me, lol.

Mike

Actualy I just realized I need to go

I'll talk later.

Pedro Tamaroff

5:00 PM

@Mike Damn it.

@JasperLoy Yes, feel bad about yourself now.

2

Jasper Loy

Why is Boyce and DiPrima so popular for DEs?

I looked through it, not too many proofs

skullpatrol

@pedro don't tell my pal @jasper to feel bad about himself.

robjohn

@Sawarnik I will look...

Pedro Tamaroff

@skullpatrol You're taking me seriously.

That's interesting.

Jasper Loy

@robjohn Congrats on 100k, I think I played a huge role in that, lol.

Sawarnik

5:07 PM

@JasperLoy Not more than me.

skullpatrol

No, I'm taking you srsly 😒 @pedro

And that my friend is fascinating

Sawarnik

@robjohn So is it good?

Balarka Sen

5:24 PM

I am reading my way through the proof of Hilbert's 17th. Sneaky!

skullpatrol

Do it while listening to Beethoven's 5th ;-)

Sawarnik

Hmm..Balarka finds something spooky in his own field.

robjohn

@Sawarnik There is a lot of undefined terminology introduced. I am trying to figure out what is what. The proof is sparse even if I knew the terminology. So until I work the problem so that I can see where his goes, I won't know.

Sawarnik

@robjohn Ok, but it seems the answerer is online now, so you can ask him for the details.

skullpatrol

Defining terms is 80% of the work imo

Chris's sis

5:40 PM

Greetings

I just skimmed a calculus book of about 500 pages, nothing interesting :-(

Jasper Loy

@Chris'ssis Spivak?

Chris's sis

@JasperLoy No, by some authors from my country.

Jasper Loy

@Chris'ssis I like Vrabie's Differential Equations.

Chris's sis

@JasperLoy Yeah, I know the guy. :-)

Jasper Loy

@Chris'ssis Is he your student, lol

Chris's sis

5:49 PM

@JasperLoy Sorry? I didn't catch your joke. :-)

Jasper Loy

@Chris'ssis My jokes are all stupid!

Chris's sis

@JasperLoy :-)

Tessa Danger Bamkin

please anyone want to help show the solutions of kummers differential equation. im on page 3 of algebraic manipulation and i feel that is just too much and im not getting any closer to 0

Jasper Loy

@TessaDangerBamkin Have you asked the question on the main site?

Tessa Danger Bamkin

not yet, there would be a lot of equations to write out to show what ive done already

Jasper Loy

5:53 PM

I seriously doubt anyone in here would help you.

Tessa Danger Bamkin

blunt much?

r9m

@Sawarnik Precalculus ? :)

Sawarnik

@r9m You like that! ....trig?....algebra?....geomtery?....elementary calculus?

r9m

@Sawarnik all of that :P

Sawarnik

@r9m Good. Rarely seen anyone like you. So here is a question, $f(f(x))=-x$, then show $f$ is discontinuous.

Typo fixed.

r9m

6:01 PM

@Sawarnik how to do that ?

Sawarnik

@r9m hee:) Start by seeing some properties of $f$.

r9m

@Sawarnik properties like ?

Sawarnik

@r9m Jus wait...coming in 1min.

Chris's sis

Let $(a_n)_{n\ge 1}$ be a sequence of real numbers in $(0,1)$ and
$$(1-a_n)a_{n+1}>1/4, \space n \ge 1$$
Compute
$$\lim_{n\to\infty} a_n $$

Sawarnik

@r9m Prove it is invertible...Do you want the answer>?

r9m

6:11 PM

@Sawarnik ya sure

Sawarnik

@r9m math.stackexchange.com/questions/312385/… .

Chris's sis

I find the problem above really nice.

Sawarnik

@r9m Howazit. Want more?

r9m

@Sawarnik NIce :) sure ..

Sawarnik

@r9m Ok. I will get back to you in some time. Balarka trying to teach me something :/?

r9m

6:17 PM

@Sawarnik okay :)

Complex analysis

@Chris'ssis nice problem , but don't know how to proceed =P

Chris's sis

@Complexanalysis Neither do I.

Complex analysis

@Chris'ssis what about considering $\lim_{n\to \infty} = \ell$ , would that make sense ?

Chris's sis

@Complexanalysis It's not a bad idea.

@Complexanalysis It's true the limit is $1/2$.

Complex analysis

@Chris'ssis So we have one inequality i.e. $\ell > 1/2$

Sawarnik

6:39 PM

@r9m Since Balarka has disappeared, you want some functional equations?

r9m

@Chris'ssis can we argue like $1 > a_n + \frac{1}{4a_{n+1}}\ge 2 \sqrt{ \frac{a_n}{4a_{n+1}}} \implies a_{n+1} > a_n$, and the rest follows from monotone convergence ?

@Sawarnik sure :)

Sawarnik

@r9m $$f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$$

$f'(0)=1$

$f:\mathbb{R}\to\mathbb{R} \text{and is differentiable everywhere}$

Find $f$, you might have seen it.

r9m

@Sawarnik is everywhere differentiability necessary ?

Sawarnik

Yes.

@r9m Sorry. Remove the $f'(0)=1$ condition. Messed it up with a related q :(

Chris's sis