Mathematics - 2013-11-22 (page 1 of 2)

Jebediah Kerman

00:00

can anyone answer, wolfram alpha freaked out when I asked it this

what would my next step be?

Mike

$\text{du} = 2\text{dm}$

Jebediah Kerman

okay, yeah I see that

so that means

$\int \frac{du}{u^3}$

i know the integral of $u^3$ is going to be $u^4/4$

what do I do with the dm though?

Pedro Tamaroff

@Charlie HI.

Jebediah Kerman

what nobody knows?

Mike

the $\text{dm}$ is gone... you substitued $\text{du}=2\text{dm}$

also, that's not the integral of $u^3$

it's the integral of $u^{-3}$

Pedro Tamaroff

00:12

@JebediahKerman You have $u^{-3}$, not $u^3$.

Mike

I win!

Jebediah Kerman

okay

why does the dm go away? that the part that confuses me

all the dm and dx just seem to magically go away

Pedro Tamaroff

@JebediahKerman What do you mean they go away?

Jebediah Kerman

@Mike @PedroTamaroff here

Pedro Tamaroff

@JebediahKerman Well, it goes away, but is replaced by another differential!

leo

00:17

holle

Jebediah Kerman

@PedroTamaroff which one?

I don't get that

leo

have realized that seies is a palindrome?

Mike

jebediah do you agree that we have EQUALITY between $du$ and $2dm$? They're the same thing

Jebediah Kerman

..yes?

hold on I am gonna read up on this before I bother you more with my ignorance

Mike

Sorry if that sounded like you were bothering me, that wasn't intended!

We have $\textrm{du} = 2\textrm{dm}$

So where we used to have $2\textrm{dm}$, we just put $\textrm{du}$ instead, as they're the same thing

Jebediah Kerman

00:23

yes

I follow

Mike

So $\textrm{dm}$ didn't go away

We just rewrote it

Jebediah Kerman

okay

I am still gonna read up on it now though

becuase I only vaguely follow

I need to get a more solid intergration foundation

no rhyme intended

Enjoys Math

lulz

leo

if $F\subseteq E$ are fields, $B_1\subseteq E$ is algebraically independent set and $E$ is algebraic over $F(B_2)$, then $\#(B_1)\leq \#(B_2)$?

it holds if $B_1$ and $B_2$ are both finite

Don Larynx

02:27

I really don't wanna study for my discrete math test you guys

i have not even gone to that class for over a week

Eric

03:03

how would I find the sum of the residues of \frac{1}{1+z^3}

residue calculator

agent154

@Eric Need to have dollar signs ($) around your latex code

@DonLarynx I got you beat. I have an exam coming up in like 3 weeks for a course that I haven't gone to in over a month... Hell, since it started in September, I've gone maybe 10 times at most.

The class is taught by a professor that can barely speak English, so most of the class doesn't bother going at all. We get nothing from him.

I've got an assignment due in that class on Monday, of which I know nothing about... so I have to learn the material over the weekend to do so, taking time away from a group project I'm working on.

leo

03:32

@anon hi

anon

hello

leo

haven't the thing I've wrote above already be asked on main?

it seems pretty standard

I mean, this one:

3 hours ago, by leo

if $F\subseteq E$ are fields, $B_1\subseteq E$ is algebraically independent set and $E$ is algebraic over $F(B_2)$, then $\#(B_1)\leq \#(B_2)$?

anon

I don't know if the question has been asked in that specific form

but it probably has to do with transcendence degree

leo

indeed. It's true when the $B_i$ are both finite. When dealing with finite transcendence basis, it's a lemma which allows conclude that any two finite transcendence basis have the same number of elements. In the infinite case, there's another proof, but I wonder if we can proceed by using this again

I'll ask

but it's a bad time to ask

Don Larynx

03:57

I find it really odd and erratic that I'll go on here one day, and one question I frequented will be discussed by my professor the next day.

J. W. Perry

04:27

Can someone please tell me what my commenter is trying to tell me in my answer to this integration? I am not seeing what he is saying.

I would gladly incorporate additional rigor for the benefit of the OP, but I am not seeing it.

anon

do you agree that the square root of cos(t)^2 is not cos(t), but |cos(t)|?

J. W. Perry

sure or certainly the absolute value of a is greater than or equal to zero

anon

I am talking about cos(t), not a

J. W. Perry

Yes I agree absolutely

Karl Kronenfeld

The integral only makes sense if the value of $t$ is within an interval for which the sign of cos(t) is constant though.

anon

04:37

this is a good point

J. W. Perry

I am definitely on my feet, and unexpectedly.

anon

so the manipulation is not valid in general, but is valid in context. if you want, you can explain why in your answer, JW

J. W. Perry

did you read my comments? how would I explain this to my dutiful commenter in any other way than claiming $+c$?

I have some thinking to do

damn

err, rats, strike damn I am not upset, just well...

anon

setting $a\ge0$ by fiat is a rather mundane point

not a serious issue

J. W. Perry

a constant

of this I am sure

anon

04:41

huh?

J. W. Perry

Well ok

anon

(note we need a>0 for the formula to work because arcsin is odd)

J. W. Perry

greater than or equal

anon

well, equal if you want to divide by zero

J. W. Perry

maybe I hould put that up front. I am just trying to think about where to put it now. Up front or in the end

Ah bleh!

anon

04:42

@KarlKronenfeld I am unable to see how the bound $(4/\pi)^sn^n/n!\le\sqrt{|\Delta|}$ yields Hermite-Minkowski (i.e. there are finitely many number fields of given discriminant). ideas?

J. W. Perry

yeah don't want to get kicked out of the math union!

Mike

does anybody know if there's someone I can email if a Math Reviews link to an article is broken?

anon

the few sources on google search results insinuate ramification is involved, but (a) I don't want to think about ramification and (b) stewart/tall doesn't even seem to mention ramification anywhere, so they can't expect us to use it in an exercise

user109886

Let X = {0,1}. List all the elements in ℘(℘(X))

J. W. Perry

@anon thanks, I think I will put the explicit assumption up front.

anon

04:50

@user109886 where are you having trouble?

user109886

I think ℘(X) = {Ø,{0},{1},{0,1}}. But I'm not sure about ℘(℘(X))

anon

well, can you compute ℘(Y) where Y={a,b,c,d}? do this, then set a=Ø,b={0},c={1},d={0,1}

user109886

Is my answer for ℘(X) wrong?

Mike

It's correct

Karl Kronenfeld

@anon Yeah, I don't really have any ideas. Of course only finitely many $n$ satisfy that relation, but (given my limited knowledge of any of this stuff) going from there to finitely many extensions seems nontrivial.

anon

04:56

one can't get [finitely many extensions] from [finitely many n], e.g. there are infinitely many quadratic fields.

Karl Kronenfeld

Yeah, I should have made it clear I know it's necessary to take into account the hypothesis that $\Delta$ is fixed in the latter step.

And I would not know how to use that, esp. without ramification.

J. W. Perry

05:30

@anon I totally get it.

Now I am just wondering what else I have ignored in my seemingly flawless education.

I think probably to not make assumptions and be more critical of "assumed" reasoning in the future.

Don Larynx

05:46

Time to learn boolean algebra in less than 3 hours @agent154

B.S.

08:56

3

:D

Alyosha

How does one go about proving $\Pi(x)= \sum_{r \ge 1}\frac{\pi(x^{\frac{1}{r}})}{r} \Leftrightarrow \pi(x)= \sum_{r \ge 1} \frac{\mu(r)}{r} \Pi(x^{\frac{1}{r}})$? It's obviously a Mobius inversion but I'm clueless as to how to execute it.

badass

@B.S. :D

Pablo Rotondo

10:27

@Alyosha Well, $ \sum_{r\geq 1} \frac{\mu(r)}{r} \Pi\left(x^{1/r}\right) = \sum_{r\geq 1} \frac{\mu(r)}{r} \sum_{j\geq 1} \frac{\pi\left(x^{\frac{1}{r\cdot j}}\right)}{j}$ and the latter is $$\sum_{r,j\geq 1} \frac{\mu(r)}{r\cdot j} \pi\left(x^{\frac{1}{r\cdot j}}\right) = \sum_{r\geq 1} \mu(r) \sum_{r | n} \frac{\pi\left(x^{\frac{1}{n}}\right)}{n} $$

(the second sum is on $n$ of course, not $r$)

(you sum over the multiples of $r$)

(that are not $0$)

try reversing that sum, then, sum first on $n$, and then on $r$, to get a sum over the divisors.

and you should be done

$$ \sum_{r,j\geq 1} \frac{\mu(r)}{r\cdot j} \pi\left(x^{\frac{1}{r\cdot j}}\right) = \sum_{r\geq 1} \mu(r) \sum_{n : r | n} \frac{\pi\left(x^{\frac{1}{n}}\right)}{n} $$

agent154

12:16

@B.S. groan...

Artemisia

12:49

Greetings

Sorry, I am new here

I need some urgent help, just to go over certain things with me... is there anyone who would be willing?

I would sincerely appreciate it :)

badass

askaway

Artemisia

I have asked a question regarding continuity, differentiability and existence of partial derivatives.

and I have typed in my steps

I am new to multivariable calculus, so I find this rather difficult

oops, sorry about that

0

Q: Continuity, differentiability and existence of partial derivatives

Here are a few functions whose continuity, differentiability and existence of partial derivatives are to be checked at the origin. I have given the answers, but I would really appreciate it if someone could check it for me :) $$1. f(x,y)=\sin x\sin(x+y)\sin(x-y)$$ Continuous, differentiable, part...

multivariable-calculus

This is the question

I have shown my steps and my answers... I wanted someone to check them for me so that I know where I am going wrong

Hello...?

N3buchadnezzar

Yo

Artemisia

Haha sorry :) Anyone who could check the link for me?

N3buchadnezzar

@Artemisia Your argument for the first one seems wrong. Can you try to rephrase it here?

Artemisia

13:03

@N3buchadnezzar : Sure :) It is a composition of trigonometric functions and hence is continuous... partial derivatives exist and are equal

N3buchadnezzar

Yes the function is continous, and differentiable. But you do not justify that in your post.

Artemisia

The partial derivatives are lim(h -> 0)1/h(sinh sinh sin(-h) on both sides

Oh I just assumed they are :) because of continuity of trigonometric functions

N3buchadnezzar

Well also you only find that the derivative at $f(0,0)$ is zero, this has unfortunately nothing to do with if the function is continous or not.

Artemisia

@N3buchadnezzar oh... In the first one, you mean?

N3buchadnezzar

Yes

Artemisia

13:09

Ah

@N3buchadnezzar but since it is a product of trigonometric functions... shouldn't it be differentiable as well?

N3buchadnezzar

Indeed! And it is, but your argument does not say that.

Artemisia

@N3buchadnezzar Haha oops :) sorry :)

N3buchadnezzar

A sum of continous functions is continous, and a product of continous functions is continious.

Artemisia

@N3buchadnezzar :) that's what I used as my reasoning.

N3buchadnezzar

Also to prove that a multivariable function is continous one has to show that its value is independant of the path you aproach the point. Luckilly as long as your function does not contain any singularities, you are fine.

Artemisia

13:13

@N3buchadnezzar To be honest... part 1 was a bit of a hand-wave. I am not sure of the rest of them.

Ah :)

Singularity implies a point where function value is 0?

N3buchadnezzar

Something like $$ f(x,y) = \frac{x^2 y}{x^4 + y^2} $$

Artemisia

Oh ok :) I understand

N3buchadnezzar

You would have a problem where $x=0,y=0$, or $x^4 = - y^2$ (is the last case really a problem?).

Artemisia

Sorry... I didn't understand what you mean by the last case

N3buchadnezzar

Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiable. Here you would end up dividing by zero, which never is a good thing.

Artemisia

13:17

OH! haha ok I understand :)

@N3buchadnezzar That means 2. and 4. have points of singularity

N3buchadnezzar

Indeed.

Artemisia

Great :)

Are the rest of them fine? I know my reasoning is skimpy at places :)

N3buchadnezzar

youtube.com/watch?v=_ges8Z6DZkc&list=PLF83D74BA4DE75897 Watch this on, then come back ;)

Artemisia

@N3buchadnezzar Oh I have watched this :)

N3buchadnezzar

Well you are not using it!

Artemisia

13:23

Oh... I thought that's what I did...

I used that to understand why some functions are discontinuous

N3buchadnezzar

Limits are all about the journey towards a point or place. Limits do not care the slightest about what actually happens at the point, only how the function is behaving towards the point.

Artemisia

Yes indeed :)

N3buchadnezzar

So to prove that a function is continous you have to prove that the limit, is the same as the value at the point.

Artemisia

Yes. Absolutely.

N3buchadnezzar

To do this you have to show that no matter which path you choose to get to the poibnt, you always obtain the same value as at the point.

I can not see any argument like this at (4) ?

Artemisia

13:26

Oh someone has already put that in the comments because I forgot to put it on the original post.

And then I edited it to put in my major doubts and queries

N3buchadnezzar

Do just like they did in the video

watch it again if you can not =)

Artemisia

Should I type it in? I calculated it that way for all the functions except 5.

N3buchadnezzar

Like the comments say post a new question, where you show your steps to prove conti

Artemisia

@N3buchadnezzar Ok :) I will do that for 2. and 4. What about 5. and 6.?

N3buchadnezzar

dnno

Artemisia

13:33

Sorry? I don't understand what you mean.

N3buchadnezzar

Well again you have no arguments for either 5 or 6.

Artemisia

@N3buchadnezzar for 5. I just looked at the plot and drew conclusions

For 6. I have given my reasoning...

N3buchadnezzar

Again you only look at $x=0,y=0$. You focus on a single point

To show that a function is differentiable, it must be differentiable for all $x$ and $y$.

Artemisia

Oh... I thought the question asked for differentiability at the origin

N3buchadnezzar

We are very strict on the difference between being differentiable in a point, and being differentiable on a domain or region. Ah, the question states to check origo explicitly.

Artemisia

13:41

Oh. But I think the question asks for just at the origin

N3buchadnezzar

Even so your argument only finds the directional derivate at the origin.

Artemisia

Yes.

N3buchadnezzar

Which is not the same as checking if it is differentiable at $(0,0)$.

Artemisia

But if a function is continuous and has continuous partial derivatives at a point, then isn't it differentiable at the point?

N3buchadnezzar

In that case yes. But from your post it seems that your argument goes like this: Hey I found the directional derivative at origo, hence my function is differentiable here. That is wrong, eg computing the explicit expression at origion is somewhat pointless.

Artemisia

13:45

No no haha sorry about that.

True

N3buchadnezzar

It is enough like you say to say that the function is continous, and so forth.

Artemisia

Ah :) ok :)

That's the reasoning I used :)

N3buchadnezzar

But remember that you have to check the two cases.

Artemisia

Cases? Oh directional derivatives

Kasper

test [ a+b ]

test \[ a+b \]

$ a+b $

Artemisia

13:49

@N3buchadnezzar What about 6.? It is continuous and has continuous partial derivatives and therefore it is differentiable :)

Part 5 I don't really understand, just that partial derivatives are defined

N3buchadnezzar

Being differentiable and having continous partial derivatives are not the same thing.

Artemisia

But if the function is continuous, then it should be differentiable provided partial derivatives are continuous... ?

N3buchadnezzar

I mean the opposite, sorry. Continous partial derivatives implies differentiability.

However differentiability does not imply continous partial derivatives.

An example is this mathinsight.org/…

Artemisia

Yes that's how I understood it :)

@N3buchadnezzar Any other mistakes you spot in the answers that I have given?

If you could point out my mistakes, I can check the concepts once again :)

N3buchadnezzar

14:06

I do not see many "mistakes", since you mostly point out facts without stating why they hold.

Whys is this continous, why is it differentiable eg.

Thats the only "mistake" I see

Artemisia

Oh... but are the facts right? Sometimes I have some very weird intuitions haha. Like 5. was just guesswork in some sense :)

N3buchadnezzar

Yeah, it is correct. You just have to justify it =)

Artemisia

Haha ok :) Thank you for all your help :)

And all the useful pointers :)

N3buchadnezzar

Read the articles on mathinsights, they are really helpfull =)

Artemisia

I surely will :)

I actually have an exam on groups and symmetries round the corner... any advice ? :)

@N3buchadnezzar Apparently the answers are wrong. I was trying this online assignment and that says that my answers are incorrect... I don't know which one is wrong, though.

nullgeppetto

14:22

Dear @DanielFischer, please, save my soul and my mind!!! math.stackexchange.com/questions/556977/…

N3buchadnezzar

Norwegian Magnus Carlsen is the new world chess champion!

Daniel Fischer

@nullgeppetto $S^n_{++}$ means symmetric positive definite?

Artemisia

@N3buchadnezzar I tried it again... I don't see why I am getting it wrong :(

Charlie

@N3buchadnezzar \0/

N3buchadnezzar

14:39

I saw this inequality a while ago, stated as an obvious fact
$$ \frac{x - y}{\log x - \log y} \leq \sqrt{xy} $$
Is there any obvious argument as to why this holds?

It looks similar to the Lipshitz condition, or perhaps Jensens Inequality. But it is not quite the same.

nullgeppetto

@DanielFischer, exactly!

Pablo Rotondo

@N3buchadnezzar fix $y$ and let $x\to \infty$, does it make sense?

N3buchadnezzar

@PabloRotondo No?

Pablo Rotondo

@N3buchadnezzar Exactly. That couldn't have been the inequality, unless the source was wrong.

Artemisia

Hi :) How can I start a private chat with someone?

N3buchadnezzar

14:52

@PabloRotondo Silly me

$$ \frac{x - y}{\log x - \log y} > \sqrt{xy} \ , \qquad x>y$$

I posted it as a question instead =)

theUg

15:41

Anyone knows what would be the purpose of calculation of the cube of the adjacency matrix of the complete graph?

robjohn

@theUg finding trips of length 3?

theUg

So, there is the practical purpose, besides purely computational masturbation?

I need to find A^3 via matrix tree theorem without actually computing it. And I don’t know where to start. :/

Chris's sis

16:06

Greetings people!

Ron Gordon

@Chris'ssis Hi ChrisSis!

Chris's sis

@RonGordon Hello! How are you? Long time I haven't seen you around! :-) (referring to chatroom)

Ron Gordon

Oh, yeah...I don;t think to come here often

I guess I should be more social

Chris's sis

@RonGordon Yeah, that could be a good idea :D

N3buchadnezzar

@RonGordon We miss you long time!

Ron Gordon

16:10

Really? You get plenty of me on the main site...

I certainly know what most of you are up to

N3buchadnezzar

I always look at your answers and think "Hah! The physicist strikes again". In regard to your silly placement of symbols..

Ron Gordon

@N3buchadnezzar :)

N3buchadnezzar

Because of the people who frequent the chat, the topic often sway towards integration, sums and problem solving. So being absent from chat has a few downsides, including missing out on cake and soda.

Ron Gordon

Mmmmmm....cake...

I feel like I have been acting a little arrogant the past week or so. I apologize to anyone who has been offended by anything I have said or done.

N3buchadnezzar

^^

$$ \int_{0}^{\infty} \frac{x^{29}}{\left( 5x^2 + 49 \right)^{17}} \,\mathrm{d}x = \frac{14!}{2 \cdot 49^2 \cdot 5^{15} \cdot 16!} $$

Old present! (No beta, or gamma allowed)

Ron Gordon

16:21

Residue theorem...involving a pair of 16th derivatives

Oh fun!

Because it is one-sided, you need to introduce a log term or something like it

N3buchadnezzar

Easy to solve with a few elementary subs

theUg

Anyone has any hint on that? chat.stackexchange.com/transcript/message/12303929#12303929

Chris's sis

I was wondering if there is some hope for computing this by real methods $$\int_0^{\infty} e^{-nx} (\cot(x)+\coth(x))\sin(nx) \ dx =\frac{\pi}{2}\left(\frac{1+e^{-n\pi}}{1-e^{-n\pi}}\right)^{(-1)^{n}}$$

theUg

@robjohn Why would it be so? I mean, I am trying to understand the sense of it, so I know where to look for the solution. Let’s say I have a complete graph K_5, then there would be no trips of length 3, would there? They all would be longer, because all vertices are connected.

Chris's sis

I feel like I'd prefer to attend a series this beautiful evening. Maybe this one $$\sum_{n=1}^{\infty}\frac{1}{3^n}\left(\frac{\pi }{\pi^2-3\pi+3}\right)^{1/3^n}$$

robjohn

16:29

@theUg Not round trips. The cube of the adjacency matrix would show where you could go in three hops.

theUg

@robjohn So, how exactly would I interpret this: wolframalpha.com/input/…

This crap is totally over my head. Graph theory was easy up until now.

@robjohn Pardon: wolframalpha.com/input/…

robjohn

@theUg You have three nodes, each connected to the other two. There are two paths to get back to the same node (e.g. 1->2->1 and 1->3->1 are the ways to get from 1 back to 1 in two hops) and only one way to get to another node (e.g. 1->3->2 is the only way to get from 1 to 2 in two hops)

theUg

And by 1, 2 and 3 in those → schemes you mean v_1, v_2 etc?

robjohn

@theUg for that one, you can get from 1 to 1 in three hops in two ways: 1->2->3->1 and 1->3->2->1. To get from 1 to 2, there are 3 ways in three hops to get from 1 to 2: 1->2->3->2, 1->2->1->2, and 1->3->1->2.

theUg

@robjohn Okay, that makes sense now, but does not help me relate it to matrix tree theorem. I just do not understand relation fundamentally. And just looking at two pages of dense text of proof of this theorem just scares me.

And it kinda sucks too, because I hardly ever did matrices before (I have trouble with basic properties), and now this is on exam all of a sudden.

Oh, by the way, it is not for K_m, it is for K_4, so, at least, it is finite.

robjohn

16:41

@theUg using matrices for graphs does not really require that you've done a lot of matrices in the past. As long as you know how to multiply matrices, that is usually good enough to get started.

theUg

Things I suppose I could use are adjacency, degree, incidence, what else?

robjohn

@theUg I am not sure what you are asking. You mean what kind of matrices are used with graphs?

theUg

@robjohn Well, I get how to multiply them, but it is hard to keep track of operations (all the dot-products), since I had no practise with it.

@robjohn I am asking what can I use to solve my problem. What notions out of matrix tree theorem I should look at.

robjohn

@theUg I am not sure what you mean by the matrix tree theorem, so I don't know what to say.

theUg

@robjohn Like I said, there is literally two pages of proof, and it boggles me. And all this just for a 1/4 point problem. So it should be something obvious, but I do not see it.

Kirghoff's theorem

Kirchhoff's theorem

In the mathematical field of graph theory Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time as the determinant of a matrix derived from the graph. It is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph. Kirchhoff's theorem Kirchhoff's theorem relies on the notion of the Laplacian matrix of a graph that is equal to the difference between the graph's degree matrix (a diagonal matrix with ...

Ron Gordon

16:46

@Chris'ssis where do you get these sums from?

Chris's sis

@RonGordon These ones I received from some kids that have fun challenging me. I usually teach them some things.

Ron Gordon

@Chris'ssis Nice.

theUg

@robjohn Degree matrix for K_4 is just:

robjohn

@theUg Perhaps someone who is more acquainted with this may be of more help. I would need to do a bit of research to help more. Sorry.

theUg

3 0 0 0
0 3 0 0
0 0 3 0
0 0 0 3

robjohn

16:48

@Chris'ssis I assume you know that this has a nice answer?

Chris's sis

@robjohn Yes.

robjohn

@theUg That makes sense.

theUg

@robjohn So I wonder if there is something from making Laplacian matrix out of those [D - A]

Though, I suppose it would be just:

3 1 1 1
1 3 1 1
1 1 3 1
1 1 1 3

Rather, -1s

3 -1 -1 -1
-1 3 -1 -1
-1 -1 3 -1
-1 -1 -1 3

@robjohn Then there's this. Is this where I should look?

0

A: How to calculate square matrix to power n?

One method is induction. Another way to calculate $A^{n}$ for a $2 \times2$ matrix generally is Hamilton-Cayley Theorem: $A^{2}-Tr(A)\cdot A +\det{A} \cdot I_{2}=0$. This is a very useful theorem which can be applyed for any $n \times n$ matrix. for example if you have a $2 \times 2$ matrix wit...

Sushma

hi

Old John

@Sushma Hi there

I am running the 35 version on pari/gp right now - but it might be slow

and it has occurred to me that there might not be any solution at all - depends on quadratic residues, I recall

Sushma

16:59

Sir, I tried the same code in pari/gp 2.5.5 but for n=35, it didn't give

Old John

@Sushma That might mean that a solution does not exist :(

Sushma

ok, quite possible, but that prime is cong. to 1 mod 35, so I thought, a soln might exist

Old John

I think it might depend on whether 35 is a square mod p

Sushma

ok I will check

Old John

35 seems to be a square mod p

Sushma

17:03

yes

Old John

and 21 is also -

Sushma

yes, for that I got

as 442263555, 97162880914

I meant for 21

Old John

ah yes - just seen that

I think 23 is going to fail - it is not a square mod p :(

Sushma

ok, thanks

Old John

and it looks like 35 is taking some time on my machine here - but I can leave it running for a long time if necessary

Sushma

17:06

i was using nzmath, for python developed in tokyo metropolin university, for these type of problem,

yes sir, don't take trouble please

Old John

@Sushma It sounds as if GAP might be a better method from one of the other comments on the question

@Sushma Oh - it is no trouble :)

Sushma

ok, so should I code it

in Gap

or a command exists

ok, I can install Gap and check

Old John

@Sushma Just re-read the GAP comment - unfortunately, gap only seems to give $x^2+y^2$ representation

Sushma

yes I think so

Old John

pity

Sushma

17:09

why pity?

Old John

it is a pity that it does not seem to have a quick way to do representations in the form $x^2+dy^2$

Sushma

no, when I was without any answer I got your answer, so will wait

last year, I had run the command in nzmath to get a representation of a prime no. with125 digits, but this is not a huge number I do not know why this time it is taking such a long time

Old John

There might be another reason why a solution is impossible ... did you know a guy called Cox wrote a whole book on "Primes of the form $x^2+dy^2$"?

I have a copy - and it is a hard subject!

Sushma

yes I read that book, this is the side effect of that);

Old John

@Sushma I haven't finished it - don't know enough algebra, as I am basically an analyst

Sushma

17:18

I didn't finish the full book; the last chapter is still remaining, of complex multiplication, rest I finished, but initially I didn't understand, now slowly knowing that subject

N3buchadnezzar

@OldJohn Hey Old man =)

Old John

@N3buchadnezzar Hi there young fella

Sushma

I also read a paper by Lenstra and stevenhagen on"Mersenne primes and artin reciprocity", which is nice,

Old John

@Sushma sounds hard

Sushma

no, it is a joy, gothrough once

Old John

17:20

@Sushma excellent

Sushma

@OldJohn yes really

N3buchadnezzar

@OldJohn Do you remember how to add spoilers for answers? I tried <! but alas.

Old John

@Sushma I will have to go shortly - will be back later, and I can leave my program running

Sushma

@OldJohn, have a good day(night), thanks

Old John

@N3buchadnezzar not sure off-hand, but I am sure Makoto Kato did something similar recently, so it might be worth looking at his answers to his own questions

@Sushma Thanks - and if you want to contact me, my email is on my profile page

robjohn

17:23

@Chris'ssis I am trying something that looks promising, but just to check, I ran this through Mathematica and it gets an answer of $0.4886437056940325001$ and the ISC finds nothing for that. So the nice answer is not nice enough for the ISC :-)

Sushma

@OldJohn, thank you nice of you, Bye will in next chat

Old John

@Sushma Bye for now - speak again sometime

robjohn

@theUg I am sorry. I just am not familiar enough with this to make a suggestion.

Chris's sis

@robjohn Oh, the answer looks nice $$\frac{1}{\log(\pi-2)}+\frac{1}{3-\pi}$$ :-)

robjohn

@Chris'ssis Hmmm... That doesn't match what I got on Mma

Enjoys Math

17:33

$S \otimes_R R \approx S$, where $f : R \to S$ is a given hom of rings

robjohn

@Chris'ssis Do you have a proof that this is the sum, or is this what you were told is the sum?

Chris's sis

@robjohn I was told this is the sum. I'm sure this must be the sum. (well, not 100%)

robjohn

@Chris'ssis The series converges fairly quickly, so I believe Mma's approximation, but it only matches the answer you give above to 3 places.

Chris's sis

@robjohn Yeah, I noticed that. I don't know what to say at the moment. I'll ask for the answer again.

MickLH

Am I on a wild goose chase trying to separate $\frac{1}{x+x^2}$ into something "regular"

robjohn

17:46

@MickLH Do you have ChatJax installed?

MickLH

not on this computer, let me grab the bookmarklet

robjohn

@MickLH $\frac1{x+x^2}=\frac1{x}-\frac1{x+1}$ Is that what you are looking for?

MickLH

I believe so, I am trying to encode polynomials for a computer program, let me get back to you in a moment

derp I don't know how I didn't notice that, thanks!

robjohn

@Chris'ssis If we let $\pi$ be a variable and let $\pi\to3$, then that function limits to $\frac12$ which agrees with the sum if we substitute $3$ for $\pi$...

@MickLH anytime :-)

Chris's sis

@robjohn I see.

robjohn

17:58

@Chris'ssis: I got an upvote today for my answer to one of your questions. It is 9 months old, so it was a bit surprising.

Chris's sis

@robjohn Let me see. I missed that.

robjohn

@Chris'ssis The high-school level answer, not the EMS one :-)

Chris's sis

@robjohn Oh, this is an amazing answer! math.stackexchange.com/questions/307117/… I like that very much! Indeed! It's an elementary approach.

Transcript for

$Mathematics$ Mathematics