Mathematics - 2015-10-08 (page 1 of 2)

Ted Shifrin

00:21

@anon: I don't know that stuff, but I think you're going to have to do more solid reading than just Wiki pages. You might start with MathOverflow and Robert Bryant (my good friend who's an expert on all things exceptional and holonomy). For example, there's this as a starting point.

Bryant's Park City lectures on Lie groups might have stuff for you, for example.

3

MGA

Is it true that if v_1,...,v_m are linearly independent and w is not equal to any of the v's, then v_1 - w, ... , v_m - w are also linearly independent?

Ted Shifrin

@AlecTeal: Yes, I've done it by hand several times. The shape generated by a rotating cube appears as an exercise in three of my books.

anon

@Ted the MO thread seems to be about exceptional groups, not exceptional isomorphisms, but the answer there looks fun.

Ted Shifrin

Right, @anon, it's not exactly what you want, but Bryant is a fabulous source for all such questions.

anon

@MGA in 3D, pick three noncollinear points from an affine plane not containing 0, then translate them until they're in a 2D subspace

MGA

00:29

@anon Awesome, thanks!

Karim Mansour

hey actually @anon

@anon

anon

one @anon will suffice

Ted Shifrin

For someone who wishes to remain anon, even one anon is one too many.

UserX

Our prof said something weird today. That the integers are not a subset of rationals because in rationals you can represent the same number with different ways(equivalence classes)

Ted Shifrin

Hi @UserX !! Sounds like a lie to me.

Karim Mansour

00:32

I actually found that there is at least a sequence of successor of length 2

for example

suppose X is finite

consider P(X)

UserX

It made a little sense, but then he said that N isn't a subset of Z either which sounded a lot more weird

skill patrol

Hello Professor emeritus @TedShifrin :-)

Karim Mansour

consider all of the topologies of X that is less than than P(X) and it is bigger than any other topologies

UserX

Maybe I'm misunderstanding but this goes exactly opposite of what I've been studying about sets and their constructions

Ted Shifrin

Well, perhaps he's being 6000% pedantic, @UserX. There are isomorphic copies that are subsets. I don't like it.

anon

00:33

one could equally well say natural numbers are not a subset of integers because you can represent the same integer different ways (we use equivalence classes to construct negative integers too)

Ted Shifrin

ha ha @skull

Karim Mansour

take $T_0$ in this topology to be one of them

UserX

That's exactly what he said next @anon

Karim Mansour

and $T_1$ to be P(X)

Ted Shifrin

So, @UserX, even with equivalence classes, the equivalence class of $(n,1)$ is a well-defined rational number, and it happens to be an integer.

Karim Mansour

00:34

If X is finite

UserX

Hey @TedShifrin

Karim Mansour

what is the topologies that are less than P(X) in the lattice of topologies ?

@anon?

anon

the fact that two different ordered pairs of integers can represent the same rational number is not really a reason for saying ordered pairs of integers are technically not the same kind of thing as integers.

Karim Mansour

maybe also @TedShifrin you would like to discuss such thing

anon

@KarimMansour all topologies are finer than P(X) (other than P(X) itself)

Ted Shifrin

00:35

Not I, @Karim.

Karim Mansour

you mean coarser

UserX

@anon is it not a good enough reason if you're nitpicking enough?

anon

@KarimMansour no, I mean finer

Ted Shifrin

How can you be finer than the discrete topology?

anon

finer means has more open sets, Karim

Karim Mansour

00:37

well, in my book $\tau_1$ is finer than $\tau_2$ iff $\tau_2 \subset \tau_1$

Ted Shifrin

I'm totally confuzled.

UserX

@TedShifrin wait what? If a set is A is isomorphic to a set B and set A is a subset of a set X, doesn't that imply that set B is also a subset of set X?

anon

@UserX the fact that equivalence classes have different representatives is irrelevant.

Ted Shifrin

That's backwards, @Karim. But I don't understand @anon, either.

anon

@UserX do you think Q is a subset of Z?

Karim Mansour

00:38

I mean the other way

Ted Shifrin

No, @UserX, but in my mind I identify $B$ with $A$ and I don't make a huge deal out of it.

UserX

@anon Is it a trick question? :P I've heard a lot of counter intuitive stuff today

anon

oops, I did mean coarser originally, then mentally switched to thinking about finer somewhere along the way

Ted Shifrin

reprimands @MikeM for his appearance

anon

sorry @Karim

skill patrol

00:39

Hi @MikeMiller

anon

@UserX it's a question designed to make you wonder if you really do believe what you said

Mike Miller

@anon Generically I know precisely about $n \leq 4$. Robert Bryant knows everything about interesting/exceptional Lie groups so you might hunt down MO answers he's posted, or otherwise I'm sure this has been covered on MSE. PS, since I said Robert Bryant, Spin(7) is apparently interesting for reasons I don't understand. (It's one of the rather few possible holonomy groups.)

@TedShifrin: I was requested, approximately.

Ted Shifrin

LOL

I already wrote paragraphs about Robert :)

Mike Miller

Same suggestion? Cool. You would know better than I, I've just seen his MO answers and know he's originally famous for G_2.

Karim Mansour

its okay. I want to know something because I want to use it in my proof is there always topologies in the lattice of topologies that is less than $P(X)$ but bigger than any other topologies ?

bigger in the sense of finer

Ted Shifrin

00:41

I also linked to his Park City lectures on Lie groups, etc.

No, @Karim, the discrete topology is the maximal element.

Mike Miller

Haven't seen those, might be worth putting in the bookmark graveyard.

Karim Mansour

but bigger than any other topologies other than P(X) @TedShifrin

that is

Ted Shifrin

I had a huge folder of a ton of Robert's preprints and reprints, which, of course, I had to throw out in the retirement festivities.

yes, @Karim.

Mike Miller

As an ignoramus I don't really know what Spin(5) or Spin(6) are good for.

Karim Mansour

hahhahaha ted I know how to do it

hhahaha so cool

UserX

00:43

@Anon what set is Q isomorphic to that is a subset of N?

Ted Shifrin

presumably they show up in physics somehow, @MikeM.

Karim Mansour

@anon you will like this too

anon

@UserX any infinite subset of N

skill patrol

How's the air refresher working? @TedShifrin

Karim Mansour

@TedShifrin did you see my question that I posted before

????

Ted Shifrin

00:44

I still smell smoke, @skill, but it's helping, I think. I got a cold — dammit — and so I feel crummy.

Mike Miller

@TedShifrin: Well, Spin(3) and Spin(4) do, but since when has something physical been 5 or 6 dimensional? :)

UserX

That was meant to be Z but still yea

Ted Shifrin

State space, @MikeM.

Mike Miller

OK, I stand by 5.

Ted Shifrin

LOL ... some sort of contact reduction.

Mike Miller

00:45

Yeah, was about to say. Boo.

Ted Shifrin

LOL ... struts for the moment

Mike Miller

I'm not supposed to be out of the salt mines yet. See ya.

Ted Shifrin

Bye

skill patrol

@TedShifrin your system is going to have to build up a whole new defence against California viruses :-)

UserX

@anon But Z is cyclic. It's not isomorphic to Q

Ted Shifrin

00:47

well, I had to take my car to the shop and the one day I walked 2 miles home, it was raining .... I think that got it started, @skull.

We don't mean isomorphic as groups, @UserX. Only as sets.

anon

10 mins ago, by UserX

@TedShifrin wait what? If a set is A is isomorphic to a set B and set A is a subset of a set X, doesn't that imply that set B is also a subset of set X?

you said sets, not groups

Karim Mansour

@anon my claim is the following that there is always atleast a sequence of length 2 of unique successors and @TedShifrin here is how I construct it

suppose X is any set

it could be countable or uncountable

Ted Shifrin

or finite

Karim Mansour

yeah

UserX

Ohhhhh

Ted Shifrin

00:48

although I suppose countable could include finite

Karim Mansour

yeha

Now let ${x_\alpha}_{\alpha \in I}$ be singletons elements that enumerate X. consider $T_0 = \{\phi,X\}$, fix a specific $\lamda \in I$ , let $T_1$ be the topology generated by the subbasis $T_0 U {x_\lamda}$ and we keep doing this process repeatadly until we hit the P(X)

now

The topology generated before the powerset

Ted Shifrin

What does that mean?

Karim Mansour

?

Ted Shifrin

The power set most likely does not have an immediate successor, let alone a unique one (even in the finite case).

Karim Mansour

I will give you an example

take X = {a,b}

$\tau_{1} = \{X,\phi,\{a\}\}$

add b to this $\tau_{1}$

and consider the topology generated by that set

it will be P(X)

Ted Shifrin

00:54

So you're assuming some sort of listing (enumeration) of all of $X$, which is not necessarily possible.

Karim Mansour

yeah

I am adding singletons to topologies and move on

Ted Shifrin

you will need some sort of transfinite induction, and I stand by my comments.

Karim Mansour

yeah that is why I used

I to be some index set

Ted Shifrin

I is just another name for X.

morphic

Hi @TedShifrin

Ted Shifrin

00:58

heya mr eyeglasses; how be you?

Karim Mansour

yeah

morphic

@TedShifrin Kind of terrible because I'm doing so poorly in my classes

How about you

Ted Shifrin

well, mr eyeglasses, I'm sorry for that, but maybe it's time you realized that grad school in math is not for everyone.

I have a cold, so I feel like ****.

morphic

Awww

Ted Shifrin

if I can help on a question or two, let me know, mr eyeglasses

skill patrol

01:01

Why are there so few proofs from geometry on standardized tests @TedShifrin

Ted Shifrin

how do you expect them to test proofs on multiple choice, @skill?

besides, they've all but 100% disappeared from the US curriculum

UserX

A semantics question; We can follow these paths in undergrad; Analysis-Pure math-Geometry-applied math which is vague by itself but anyway where does number theory fall under?

Ted Shifrin

that makes no sense to me, @UserX

UserX

Me neither. All of these are somewhat connected

Karim Mansour

hey @TedShifrin or @anon I want to prove that for finite case that there is a sequence of length two for successor as I am seeing a list of topologies I am noticing something quite picular

Ted Shifrin

01:06

number theory falls mostly under algebra at the undergrad level ... so pure math, I guess, although cryptography/computational number theory is applied.

Karim Mansour

that topologies even the ones that aren't comptaible

always have the same size

is that true ?

UserX

What does number theory have to do with algebraic structures other than working in some of them?

Ted Shifrin

I doubt that seriously, @Karim.

You use ring theory, group theory, etc., in number theory, @UserX.

Karim Mansour

that is that can you a topology of cardanility n aside from the P(X) that is unique for the topologies on a set X

?

that is

there doesn't exist a topology that have same cardinal

Ted Shifrin

Rewrite that sentence, @Karim.

Karim Mansour

01:08

ok I will

UserX

I'm definitely not that advanced in NT

Ted Shifrin

some schools teach a totally elementary number theory course, but it's better if you know some algebra, @UserX.

UserX

I thought NT was about finding relationships between primes

Ted Shifrin

It's about all sorts of things.

Karim Mansour

Suppose you have a set X and now suppose $\tau_1$ is a topology that is neither the discrete or the indiscrete topology, then does there always exist a topology $\tau_2$ such that $|\tau_1| = |\tau_2|$ ?

@TedShifrin ?

Ted Shifrin

01:09

Are you assuming finiteness, @Karim?

Karim Mansour

yeah

Ted Shifrin

Well, then take the action of the permutation group on the set of topologies :P

Karim Mansour

coz I have a idea of using group action of $S_n$

yeah exactly

Ted Shifrin

good idea

Karim Mansour

I was gonna prove that there is only a sequence of length two on finite topology that by using action of $S_n$ on the topology

yeah for finite case there should be only sequence of successor of length 2

Ted Shifrin

01:12

ok, dinnertime for me ...

g'night all

skill patrol

Later

Karim Mansour

g`night @TedShifrin

I will be thinking of this problem

I need to take a shower though

I am stinky

Pies

02:09

Howdy.

Ya know, math is pretty awesome. You just gotta look at it from the right angle.

morphic

@Pies omg

ಠ_ಠ

Pies

:D

Well then again, it can be a triggy subject...but no one can deny its absolute value in life

morphic

ಠ_ಠ

Pies

Anyway...

"Hello everyone, my interests are mathematics and"

Erm...ya finish that sentence or...?

morphic

Yes

Pies

02:23

So, ya in high school?

morphic

No, college

I should be in high school though

Pies

Freshman?

morphic

Senior

Pies

.-.

So then how old are ya?

morphic

I'm too dumb to be a college student

Pies

02:24

Well what're ya majoring in?

morphic

26

Math

I'm failing pretty miserably though

Pies

That sounds fun though...

You get to learn about the Universe

morphic

It's fun when you can understand the material

Pies

At least you get to learn. It's my junior year in high school and my calc teacher just left a week ago

Now we don't know if we'll learn it

morphic

I'm not even learning anything

Pies

02:25

?

morphic

I'm trying to learn but I'm not smart enough to learn anything :(

Pies

Ah come on. Ya just gotta have confidence

Maybe a tutor?

morphic

I'm too poor to afford a tutor

Pies

Ya got a job?

morphic

No

Pies

02:28

Try to fit one into your shedule

morphic

Last time I tried to work while studying I failed all my classes

Pies

Hmm...ya tried working in a study group?

morphic

No but I'm trying to make friends

I haven't spoken to any classmates yet and I promised myself I would before I graduate

Pies

Work with someone in your class. You'll help each other out

That's what I did and my grades went from Cs to As

morphic

Well I don't care about my grades, I just want to know the material properly

I got all As in my classes but the classes here are bad so I didn't really learn anything

Pies

02:31

Better grades will help ya out a bit later. But groups will definitely help with the latter

Anyway, it's almost 2:00am. Gotta go, 'night

morphic

Night

I wish I was born in Europe instead

skill patrol

Why?

morphic

Grass is greener

skill patrol

On the other side of the ...

morphic

world

skill patrol

03:05

and it's a big world out there.

Karim Mansour

03:15

@anon in few hrs I will present a proof atleast for the finite case that its only possible to have a list of sequence of two successors only

I like my proof alot so far

Huy

04:44

@morphic: You can compensate by asking many questions here!

Karim Mansour

05:28

@Huy

how r u doing

Huy

hungry, haven't had breakfast yet

u?

Karim Mansour

I am so happy btw I proved that for finite set X we can only have a sequence of of length 2 for the successors

and their is only one way to obtain it

Huy

05:50

cool

what are you studying atm, @Karim. still munkres?

Karim Mansour

yeah

Huy

I borrowed it when I was studying topology because I didn't want to pay at that time. Now I want to have the book myself but now it's 20 bucks more expensive.

As soon as I'm working 100% I'll add so many great textbooks to the few I have already.

Karim Mansour

06:09

why point set topology

I can't wait until I finish point set topology to study algebraic topology

Huy

to improve my foundations

Karim Mansour

yeah your right

Huy

I forget stuff I don't use on a regular basis very quickly

and I get sad when I realize that so I want to revise

Karim Mansour

yeah

I wanted to ask a question

Huy

what is that?

Karim Mansour

06:12

if we take maybe the standard topology

and union that with a singeleton

do we get a topology that is finer than standard topology?

Huy

so is the singleton open or?

sorry I just woke up maybe I'm being slow

Karim Mansour

that is consider B1 to the the basis of the standard topology

take $B_1 U {2}$

will it be finer than $B_1$?

Huy

intuitively I'd say no

you mean strictly, right?

Karim Mansour

yeah

Agawa001

08:08

@AlecTeal that reality is really annoying, do you think im satisfied of the fact that a 15 min answer in reverse engineering brought me more points than a 1 week long answer ????? folks just like visual

Remember me

$\sum_{n=1}^{\infty}(n^3)$ diverges right?

Tobias Kildetoft

@Rememberme Yes, of course

Agawa001

of course

to infinity

Remember me

@Tobias So is it true that $\sum_{n=1}^\infty(n^3) \neq (\sum_{n=1}^\infty (n))^2$

Tobias Kildetoft

@Rememberme Neither side is defined

Remember me

08:17

So it is not true.

Tobias Kildetoft

they both diverge to infinity

but not at the same speed

Remember me

So they cannot be said to be equal?

Tobias Kildetoft

woops, actually at the same speed

both are polynomials of degree 4 in $n$

well, with proper interpretations to make things actually make sense

Agawa001

Cube (algebra)

In arithmetic and algebra, the cube of a number n is its third power: the result of the number multiplied by itself twice: n3 = n × n × n. It is also the number multiplied by its square: n3 = n × n2. This is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power. Both cube and cube root are odd functions: (−n)3 = −(n3). The cube of a number or any other mathematical...

Tobias Kildetoft

@Agawa001 ??

Agawa001

08:19

yes same speed i was late :p

Remember me

I know it is true when taken sum to a finite value is it true when sum is taken to infinity ?

Tobias Kildetoft

@Rememberme that question does not make sense

Remember me

Well I mean to say that $\sum_{n=1}^N{n^3}=(\sum_{n=1}^N{n})^2$ but is this true when the sum goes to infinity@Tobias

Tobias Kildetoft

@Rememberme The sum does not "go to infinity"

Remember me

I have to go for lunch that is why I am using such weird terms . Is this true
$\sum_{n=1}^\infty(n^3) \neq (\sum_{n=1}^\infty (n))^2$

Tobias Kildetoft

08:23

Unless you mean as formal power series (I am fairly sure you are not familiar with those). And for formal power series I think it might be true, but I would have to do the calculation

@Rememberme they are either trivially equal, or you need these to be formal power series to be able to compare them

Agawa001

08:35

^ i doubt really if this pile of lines have already been posted

thats same as saying, $lim_{x->\infty} (x+1)^2 \neq lim_{x->\infty} (x^2+2x+1)$

Tobias Kildetoft

08:57

@Agawa001 Well, I think the question is better asked about formal power series, but I also think that is beyond the level of @Rememberme at the moment

Kaj Hansen

I've posted a question: math.stackexchange.com/questions/1470083/…

Agawa001

@TobiasKildetoft i think not, he had the craving of posting something here and go. he could elaborate in less sophisticate examples, asking if $(\infty)^2 == \infty$

Tobias Kildetoft

@Agawa001 Right, in the stated form, the question is either meaningless or trivial. But as a question about formal power series I think it is a fine question (which I haven't checked yet)

Agawa001

@TobiasKildetoft iv checked it before, that sum of powers found by bernoulli numbers.

$\sum(n^3)$ is just a case

Tobias Kildetoft

@Agawa001 There are no bernoulli numbers in that power series (just $1$s)

Agawa001

09:17

i was reading this starred post by skillpatrol, lol, tbh, the lady has right to distrust the vaccination

Tobias Kildetoft

@Agawa001 What? Please don't start spouting anti-vax stuff here

Kaj Hansen

More importantly @Agawa001, we should take pause before making anything mandatory in such an extreme regard. Strongly recommended? Sure. Ostracizing people in the form of restricting access to public property? Sure. But actually giving people a very real choice between getting vaccinated or getting executed is ridiculous.

Balarka Sen

hi @KajHansen, @TobiasKildetoft

Tobias Kildetoft

@BalarkaSen Hi

Kaj Hansen

Pretty much every problem that has ever arisen in policymaking has found its inception when short-term good, excitement, and zealousy have been allowed to overshadow, obfuscate, and make harder to predict long-term disaster.

Agawa001

09:30

@KajHansen i gave the lady the right to abstain, nothing more to tell here, its her own blood, and her own children. i know people who do so, like my neighbor doctor he never let her daughter take that school-vaccination. im not intimidating people, but any such medical procedure souldnt be obligatory, thats all, people have right to choose independently of their level of knowledge or "not being allowed science" or "allowed citizen" or anything like this, back to my maths now.

Kaj Hansen

That's what I'm saying @Agawa001

Refusing someone participation in as intrinsically human of an activity as democracy, science, or mathematics is a very extreme and cruel move.

Tobias Kildetoft

@Agawa001 That is a terrible argument. People refusing vaccines are not only hurting their own children but also potentially immuno-compromised children whose parents would have liked nothing better than for their children to be vaccinated.

(also, since when is it ok to do something that hurts your own children, just because they are yours?)

Kaj Hansen

Anyone with the proper command of the language they speak can be arbitrarily powerfully persuasive if only they worked hard enough. We should strive for the betterment of society via good rhetorical skills and peaceful persuasion rather than the force of law, in the sense that @Agawa speaks (forcing procedures upon people)

Joshua A

Just a warning @TobiasKildetoft:

in The 2nd Monitor, Oct 2 at 13:14, by rolfl

He is a known troll, a pretty good one, and if you engage with him you get all the crap and frustration you deserve

Kaj Hansen

Another thing that would go a LONG ways is for the government to stop doing shit that makes people distrust it. The CIA and related organizations have a long history of experimenting on people, and this is definitely not just relegated to the conspiracy theories.

Tobias Kildetoft

09:37

@JoshuaA which of them?

Joshua A

@TobiasKildetoft That message is in regard to Agawa

Kaj Hansen

Cry wolf too many times, and people stop trusting you. In the same way, it's now harder for the government to convince people vaccines are safe, and that is not something these skeptics should be blamed for. That's an entirely rational response.

Tobias Kildetoft

@JoshuaA I see

@KajHansen There is nothing rational about ignoring all scientific studies

Kaj Hansen

I know this @TobiasKildetoft. However, when a largely distrustful and dishonest organization is the one doing a lot of the salesmanship for something that is as good as vaccines, therein lies the problem.

Tobias Kildetoft

@KajHansen which organization is this?

Kaj Hansen

09:41

The government @Tobias. The US government has a long history of ghastly, unethical human experimentation that has been done as recently as the 80's, and probably since then as we could find out when more recent documents become declassified in the future.

Tobias Kildetoft

@KajHansen But the recommendation is coming from multiple sides, most of them unrelated to the government (and please, let's not make "the government" into a single uniform entity here).

but anyway, this is getting way off topic

Kaj Hansen

Consider, for example @TobiasKildetoft, that the US Public Health Service sponsored infecting minorities with syphilis at Tuskegee university as recently as 1970 under the guise of vaccinations.

Anyways, I have a serious amount of cryptography to get done over the next 4 hours.

Mary Star

We have that t, n, b consist the orthonormal basis of R^3. We know that the vector a is perpendicular to n, and the angle between a and t is theta. How can we write a using the orthonormal basis? Which are the coefficients?

Kaj Hansen

As a final note, my goal here was not necessarily to defend anti-vaxxers, but rather I think it is immeasurably valuable to understand where misguided opinions are coming from. Understanding why your opponent feels a certain way is the first step towards actually effecting change within people.

Balarka Sen

@KajHansen Can you tell me what a Berkovich space is?

Tobias Kildetoft

09:48

@MaryStar So $a$ is in the plane spanned by $t$ and $b$ and you know the angle to $t$. This does not tell you $a$ completely

it only gives you the direction of $a$, not the length

Balarka Sen

I have been planning to have a discussion with someone about that.

Kaj Hansen

@MaryStar, I don't have time to explain in detail, but take dot products to yield a system of two equations that your vector coordinates of $a$ must satisfy, recalling the relationship between the dot product of two vectors and their lengths / angle between.

@BalarkaSen, absolutely, but not when I have crypto due so soon. I'm trying to not be distracted as I am with chat right now :P

Balarka Sen

ah, sure. let me know when you get time.

I have exams ahead of me too, I am just procrastinating.

Kaj Hansen

@BalarkaSen, one consideration of many I've thought about recently: math.stackexchange.com/questions/1470083/…

Balarka Sen

The reason I am interested in Berkovich spaces is two fold : (1) I have heard that one can do ergodic theory with them in the utmost arithmetic sense (2) I have heard they are some kind of p-adic analytic manifolds or a generalization of them, and I have been wondering about whether one can have actions of Gal(\bar Q_p/Q_p) over those

Agawa001

09:52

oh please im off for a couple of days

Balarka Sen

@KajHansen I'm having a look.

Interesting question, @KajHansen. Maybe there's some intrinsic property of the conics that make them appear frequently.

UserX

11:38

@Chris'ssistheartist are you here? I summon you

Huy

12:26

@Hippalectryon: did you ever start looking at the book on functional analysis I sent you yet ?

Chris's sis the artist

13:01

@UserX Yes, now.

skill patrol

13:54

J.M.? Is that you?? You're back!!! :-D @J.M.isback.

Welcome.

J. M. is back.

Yes.

skill patrol

Welcome Back Kotter Theme Song HQ (Opening)

►

HQ for an HQ user.

J. M. is back.

I'm way better looking than Gabe Kaplan. :P

skill patrol

:P

UserX

@Chris'ssistheartist any faster ways for $\int_0^1 \frac{x^2}{1+x^4}dx$ other than the classic computations?

Chris's sis the artist

14:07

@UserX Is it from $0$ to $1$? If it's from $0$ to $\infty$ you can apply Cauchy-Schlomilch transformation.

That is

$$\int_0^{\infty} \frac{x^2}{1+x^4}dx=\int_0^{\infty} \frac{1}{x^2+2}dx=\frac{\pi}{2\sqrt{2}}$$

@UserX Did you think of using geometric series?

Using geometric series all reduces to calculating $$\frac{1}{4}\sum _{k=0}^{\infty } \frac{(-1)^k}{k+3/4}$$

Not sure if you recognize the series but it's a classical one.

$$\sum_{n=0}^{\infty} (-1)^{n} \frac{1}{n+1+x}=\log(2)-\psi(x)+\psi(x/2)+1/x$$

On this page you also have the needed special values

Digamma function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. == Relation to harmonic numbers == The digamma function, often denoted also as ψ0(x), ψ0(x) or (after the shape of the archaic Greek letter Ϝ digamma), is related to the harmonic numbers in that where Hn is the n-th harmonic number, and γ is the Euler-Mascheroni constant. For half-integer values, it may be expressed as == Integral representations == If the real part of x is positive then the digamma function has the following integral representation...

that can be derived with the formulae presented on the page (if you wanna do that).

Gauss's digamma theorem

That's all.

Now let me address to myself the following question: can I get the answer in a more brilliant way? Maybe, but I need some more time.

Of course, one does not need to use special functions since you have a nice primitive there. For a complete elementary way I might suggest you using a system of equations in integrals.

That is use $\displaystyle I=\int_0^{\infty} \frac{1}{1+x^4}dx$ and $\displaystyle J=\int_0^{\infty} \frac{x^2}{1+x^4}dx$ and add them and subtract them. All flows beautifully with a bit of work.

I mean the indefinite ones too

$\displaystyle I=\int \frac{1}{1+x^4}dx$ and $\displaystyle J=\int \frac{x^2}{1+x^4}dx$ (that is you use the classical substitutions in Cauchy-Schlomilch transformation)

Addem

14:34

Is there a name for the shape that's a sphere contained within a larger sphere? Like an annulus rotated about an axis through its center?

J. M. is back.

@Addem "spherical shell"?

Chris's sis the artist

For instance, we can write that $$I+J =\int \frac{1 + 1/x^2}{(x - 1/x)^2 + 2} \ dx$$ whence all is clear. And we proceed similarly for $\displaystyle I-J$.

OK, let me see another way, a 3rd way. Well, beta function can be used too.

UserX

@Chris'ssistheartistc it's 0 to 1. 0 to inf would hint me to complex analysis

How do you come up with all that lol

Any brilliant ways for 0 to 1?

And no I've never heard that transformation before

Chris's sis the artist

@UserX You can use the way with system of equations or beta function.

Chris's sis the artist

14:57

(beta function doesn't seem to work directly)

But rather what is known as incomplete beta function according to the definition here

129.81.170.14/~vhm/formula_html/final11.pdf

@UserX ^^^

$$\frac{1}{4}\int_0^1\frac{x^{1/4}}{1+x} \ dx$$ where you use formula $(1.1)$ - but this is more or less what I was suggesting at the beginning.

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