Mathematics - 2015-07-08 (page 4 of 5)

Brave New World is a novel written in 1931 by Aldous Huxley and published in 1932. Set in London of AD 2540 (632 A.F.—"After Ford"—in the book), the novel anticipates developments in reproductive technology, sleep-learning, psychological manipulation, and classical conditioning that combine profoundly to change society. Huxley answered this book with a reassessment in an essay, Brave New World Revisited (1958), and with Island (1962), his final novel. In 1999, the Modern Library ranked Brave New World fifth on its list of the 100 best English-language novels of the 20th century. In 2003, Robert...

Balarka Sen

it was a very quick survey, with many handwaves and whatnots. I hope I have been able to explain the punchline well.

Soham Chowdhury

yeah. this is so interesting, man. thanks.

Balarka Sen

but I'd appreciate if both of you (especially @Remember) would forget I ever talked about this

Soham Chowdhury

2:10 PM

why?

Remember me

why?

Soham Chowdhury

I shall never forget how you looked handwaving -- shamelessly -- in public. :P

Balarka Sen

it was very non-rigorous, and filled with enough interesting stuff to chew youngster's brains.

Soham Chowdhury

@BalarkaSen heh.

Remember me

true

Soham Chowdhury

2:11 PM

@Balarka:

researchblogs.cs.bham.ac.uk/thelablunch/2015/04/…

Balarka Sen

I am not opening this.

Soham Chowdhury

Please do.

It's interesting, what that guy does with kids.

Remember me

i really appreciate your effort to make an idiot like me understand all of this crazy stuuf@Balarka

Balarka Sen

@SohamChowdhury I have been excited about this for -- let's see -- two years.

Soham Chowdhury

He teaches kids to compose knots and whatnot. Minus Khovanov homology. :P

@BalarkaSen well, wow.

Balarka Sen

2:13 PM

Only this month I have found something which I can actually use to see the analogy more clearly.

Soham Chowdhury

what?

Balarka Sen

not bothering to explain.

Soham Chowdhury

ah, okay.

thanks anyhoo.

I have EVS to study now.

Balarka Sen

I have to go.

Remember me

Thanks @Bal

Balarka Sen

2:14 PM

have fun

Remember me

I have to go

Study chem

Semiclassical

@Chris'ssistheartist i'd be curious to see what's generating this sequence, aside from $e^{2x}/\sqrt{1-2x}$ of course. (btw: given the powers of 2 in the denominator, my taste would be to view it in terms of $e^{x}/\sqrt{1-x}$)

oh, and good morning chat

Chris's sis the artist

@Semiclassical Hello

@Semiclassical It's related to this closed form family

$$\frac{\sqrt{\pi }}{2 e},\frac{3 \sqrt{\pi }}{4 e},\frac{11 \sqrt{\pi }}{8 e},\frac{53 \sqrt{\pi }}{16 e},\frac{345 \sqrt{\pi }}{32 e},\frac{2947 \sqrt{\pi }}{64 e},\frac{31411 \sqrt{\pi }}{128 e},\frac{400437 \sqrt{\pi }}{256 e},\frac{5927921 \sqrt{\pi }}{512 e},\frac{99816515 \sqrt{\pi }}{1024 e}$$

@Semiclassical see the numerator

dREaM

-4

Q: How many minimum weights do you need to measure all weights from $1$kg to $1000$kg

You can place weights on both side of weighing balance and you need to measure all weights between $1$ and $1000$. For example if you have weights $1$ and $3$, now you can measure $1,3$ and $4$ like earlier case, and also you can measure $2$, by placing $3$ on one side and $1$ on the side which c...

recreational-mathematics

Semiclassical

right. and i also see that the denominator is powers of 2, hence my comment above

dREaM

2:25 PM

Why did this get downvoted so much?

Semiclassical

but what's generating that closed-form family?

interesting

Chris's sis the artist

@Semiclassical Yeah, it is. :-)

Semiclassical

my thinking, then, would be to resum them as $\sum_{n=0}^\infty \frac{t^n}{n!}$ and hope that the cosecants become simple

Chris's sis the artist

@Semiclassical I wanna add to my book the version with $n=1$.

Semiclassical

ahh

Chris's sis the artist

2:28 PM

@Semiclassical The best way I think is to reduce all to the gamma function. :D

Well, it actually is a gamma function form, the way it is, but a disguised one.

Ramanewbie

@hippa irc python

Hippalectryon

@Ramanewbie ? I'm already on irc ...

Semiclassical

hmm. if i'm looking at this right, resumming as i suggested gives you a factor in the integrand of the form $e^{-t \csc^2x}$

Ramanewbie

@hippa Ok I thought you were afk but marked as online

Semiclassical

which would amount to a modification of the $n=0$ case

Chris's sis the artist

2:32 PM

@Semiclassical Well, I'd also like to see other ways. Pls approach it as you like, I didn't try other ways.

(not yet)

Semiclassical

well, i have the advantage of having that generating function interpretation

Chris's sis the artist

@Semiclassical :D

Semiclassical

without that, i imagine it's a lot harder to spot

not impossible, perhaps, but not so nice

Chris's sis the artist

OK, let me try the generalization in a different way

Semiclassical

it looks to be a bit subtler in that direction than i expected, actually. i'll have to think about it some more

Chris's sis the artist

2:39 PM

I'm done.

Just to check my work one more time.

Semiclassical

neat.

which naturally explains the $\sqrt{pi}$ as well

Chris's sis the artist

@Semiclassical It's so awesome! :-)))

@Semiclassical Yes.

@Semiclassical Maybe I should propose it in some magazine. It's a cute one.

Semiclassical

i can't advise you on that, i'm afraid :)

Chris's sis the artist

@Semiclassical Why? Is it something wrong with it? :-)

Semiclassical

didn't mean it like that

dREaM

2:46 PM

coz he's a noob

Semiclassical

...yeah, pretty much. publishing isn't something i know anything about.

so i'm not in a position to advise what's notable or not.

Chris's sis the artist

@Semiclassical ah, I see.

dREaM

don't worry, you will someday fulfill your dREaMs

Chris's sis the artist

@Semiclassical It's about proposing it for the problems section for students in some magazine, it's not a hard question.

Semiclassical

gotcha.

Chris's sis the artist

2:47 PM

Anyway.

Semiclassical

a proposed problem rather than an article

that makes sense

Chris's sis the artist

@Semiclassical Yes, not an article at all. :-)

Hippalectryon

@dREaM Hoi TheEmperorofIceCream :D

dREaM

That was supposed to be a secret, what blew my cover?

I still haven't solved your problem on stuttering sequences.

Hippalectryon

Aha it's a hard one

A very hard one iirc

dREaM

2:50 PM

do you like the drawing in my profile picture?

Semiclassical

modified bessel?

Hippalectryon

@dREaM Yeah it's cool !

Chris's sis the artist

@Semiclassical and you also meet Bessel function for other odd values. :-)

Semiclassical

@Chris'ssistheartist: oh, and the reason my approach isn't as simple as I thought: it'd work if the summation converged. but $\csc x$ diverges at zero, so that creates problems

dREaM

I also like the picture of a bear in your profile picture

Chris's sis the artist

2:51 PM

@Semiclassical Yes.

Semiclassical

so one needs to do something a bit more complicated, say by introducing a regularization

Chris's sis the artist

@Semiclassical Exactly.

dREaM

sorry, now that I look at it it's clear it is a hamster

Semiclassical

not sure what the smartest one is, but the most obvious is to introduce a cutoff $\epsilon>0$ on the lower bound to make the summation well-behaved

Chris's sis the artist

That might work.

dREaM

2:54 PM

I have a problem for Chris's sis

prove $x^8+1$ has no irrational roots in $\mathbb R\cup \mathbb Z$

Semiclassical

isn't $\mathbb{Z}$ superfluous there? if it's not rational, it's certainly not an integer

Chris's sis the artist

And it's also nice the evaluating $$\int_0^{\pi/2} \frac{e^{\csc ^2(x)} \csc ^2(x)}{e^{\csc ^2(x)}-1} \, dx$$

dREaM

yes, that is a good first step

Semiclassical

...come to think of it, $x^8+1$ doesn't have any real roots period

dREaM

that is a good second step

Semiclassical

2:57 PM

since $x^8=(x^4)^2$ is necessarily positive. so it's really pretty obvious

Khallil Benyattou

Would it be $2|P|+1$ where $P$ is a partition, @Huy?
(Sorry I took so long to reply!)

Huy

@KhallilBenyattou: Yes, but that's still not the final answer to your question.

Khallil Benyattou

Is it not the maximal cardinality or are you referring to parts $(ii)$ and $(iii)$, @Huy?

Chris's sis the artist

Oh, there was a mistake above.

Huy

@KhallilBenyattou: Can you send me the questions again?

dREaM

3:00 PM

Ok, now a harder question. Let $k\in \mathbb Z$ be such that $f(k)>0$ for every polynomial with real positive coefficients, prove $k$ is necessarily of the form $a+b\sqrt2$ with $a,b\in \mathbb Q$

Khallil Benyattou

Of course, @Huy! :-)

yesterday, by Khallil Benyattou

Huy

@KhallilBenyattou: I am refering to part i).

@KhallilBenyattou: If it was a test, I'm sure your answer would not give full points.

Chris's sis the artist

So, we get the beautiful result $$\int_0^{\pi/2} \frac{\csc ^2(x)}{e^{\csc ^2(x)}-1} \, dx=\frac{1}{2} \sqrt{\pi } \text{Li}_{1/2}\left(\frac{1}{e}\right)$$

@Hippalectryon ^^^

Hippalectryon

;-)

Semiclassical

@dream the only intersection between $k\in \mathbb{Z}$ and the set you suggest is $k=0$.

Khallil Benyattou

3:04 PM

Really, @Huy?

Huy

Yes, really @KhallilBenyattou.

Khallil Benyattou

(It's an assignment from a few years back, @Huy)
Hmm, I don't see what else I could include ...

Semiclassical

wait, derp. the only intersection is the integers themselves, since one can take $b=0$.

Khallil Benyattou

It's constant on the open intervals, of which there are $|P|$. Between the partitions, there are $|P|-1$ values that $\psi$ could take and finally there are 2 endpoints. After summation, there are $2|P|+1$.

Semiclassical

in which case the claim is trivially true: If $k$ an integer which satisfies some condition, then $k$ is an integer.

Huy

3:07 PM

@KhallilBenyattou: Still not enough. :D

dREaM

Actually that's not necessary, if $k$ is in $\mathbb Z$ then it is just $k+0\sqrt{2}$ where $k,0$ are rational numbers.

Semiclassical

yes, hence my "$b=0$" comment

dREaM

Here is a harder problem. Two distinct ellipses share one focus, prove they intersect two times at most.

Chris's sis the artist

BBL (I need to write up some solutions)

Semiclassical

from roots to geometry, i see

Khallil Benyattou

3:13 PM

Isn't it just a matter of solving simultaneous equations, @dREaM?

Something to do with a discriminant being positive ...

Semiclassical

one also has to ensure that they've got the same focus, and i forget what that means at the level of a quadratic equation

probably the simplest way, though, is to take the common focus to be at 0, pick two arbitrary foci $c,c'$ and write down the most generic equations satisfying this

(there's probably some cute projective geometry way of saying it, but i don't know that well enough)

Khallil Benyattou

It probably has to be more general than that, @Semiclassical.

Picking the common focus to be at the origin restricts things pretty heavily.

Semiclassical

not really. it just amounts to a translation

Chris's sis the artist

Oooooooooooooooo, I just got another amazing result!!!!!!!!!!!!!!!!!!!!!1

This made my whole weeeekkkkkkkkkkkkk!

dREaM

hole*

you also spelled week wrong..

it was a joke

Hippalectryon

3:18 PM

@dREaM You spelled joke wrong :3

dREaM

did you understand my joke Hippa?

Hippalectryon

I'm not sure xD depends on how many levels it can be understood

Khallil Benyattou

Hippalectryon

Wow, cats are dangerous

neko neko

user183297

Hi, everybody. Can anyone explain me this: math.stackexchange.com/q/1353923/248291

As is see in this hours of the day the activity is low, so i invite you personally :)

hope you dont mind

dREaM

3:22 PM

It made my whole week ----> It made my hole weak

hope I don't get banned for too long

Khallil Benyattou

See you in an hour, @dREaM

anon

@user183297 what don't you understand specifically?

Khallil Benyattou

:-P

anon

@user183297 If $a$ is any number and $(s_n)$ is any convergent sequence, then $(as_n)$ is also a convergent sequence. in particular, if $s_n$ is the $n$th partial sum of $\sum c_n$, this tells us if $\sum c_n$ converges then $\sum ac_n$ converges. thus, if $\sum a_nx^n$ converges we can multiply by $x^r$ to get that $\sum a_n x^{n+r}$ converges, for any integer $r$ (assuming $x\ne0$ at least). and conversely.

you should not accept an answer you don't understand BTW

7

user183297

Yes, I got it. Thank you very much as well )

Have a good day

Khallil Benyattou

3:28 PM

@anon's explanation is clearer in my opinion

dREaM

I am a master of procrastination

user183297

Its excellent!

Chris's sis the artist

@Semiclassical you should also see this one ... (wait a bit)

$$\int_0^{\pi/2} e^{-\csc ^2(x)} \csc ^2(x) \left(\text{Chi}\left(\cot ^2(x)\right)+\text{Shi}\left(\cot ^2(x)\right)\right) \, dx$$

@Semiclassical ^^^

Semiclassical

...not going to lie, that's kind've terrifying

Chris's sis the artist

@Semiclassical It's $0$. :-)

Semiclassical

3:32 PM

hah

Hippalectryon

@Semiclassical I had the same reaction :P

Graphically it's nowhere near obvious btw

Khallil Benyattou

What are Shi and Chi, @Chris'ssis?

r9m

@Chris'ssistheartist :-) Sweeet!!

Chris's sis the artist

@KhallilBenyattou hyperbolic sine and cos integrals

@r9m :D

@Semiclassical It seems hard, but it's not that hard. It depends on the approach you have.

Semiclassical

i can believe that, but i'd need to remember more about those two integrals than i do

Chris's sis the artist

3:43 PM

OK

Remember me

Hey@r9m

Khallil Benyattou

Shouldn't it just be $3$ for the maximum cardinality, @Huy?

If for each partition of $[a,b]$ a squelch function $\psi : [a,b] \to \mathbb{R}$ is constant on each of the open intervals of the partition, surely the overlap of all the possible partitions would be the set $[a,b]$ itself, and so $\psi$ would be constant on $(a,b)$ and can only take on different values at $a$ and $b$.

Huy

Yeah.

anon

4:01 PM

let ${\sf Set}_n$ be the category of sets of cardinality $n$ with bijections for morphisms. denote $[n]=\{1,2,\cdots,n\}$. any functor $F:{\sf Set}_n\to{\sf Set}$ turns $F[n]$ into an $S_n$-structure (since any permutation $\sigma:[n]\to[n]$ gets carried to a permutation $F(\sigma):F[n]\to F[n]$). I suspect one can find inequivalent functors $F,G:{\sf Set}_n\to{\sf Set}$ such that $F[n]\cong G[n]$ as $S_n$-sets (but $F\not\cong G$ as functors). But I haven't been able to find an example yet.

dREaM

wasn't nineteen years ago a better time to plant a tree than today)

?

r9m

@Rememberme hello

Balarka Sen

4:19 PM

what's equivalence of functors, again, @anon?

anon

@BalarkaSen natural transformation comprised of isomorphisms

Balarka Sen

hm, ok.

anon

(also, fun fact: the only "natural" permutation groups besides alternating and symmetric groups of every degree, is the klein four group)

Balarka Sen

how do you define "natural"?

anon

I suspect $FX=\{$ cyclic subgroups of ${\rm Perm}(X)$ of order $|X|\}$ and $GX=\{$ cycle graphs with vertex set $X\}$ works but I haven't tried proving inequivalence.

@BalarkaSen define a funcor $F:{\rm Set}\to{\rm Grp}$ for which there exists a natural transformation $F\to{\rm Perm}(-)$ comprised of injective maps

if one writes down what it means for there to be a natural transformation, one finds that $FX$, as a subgroup of ${\rm Perm}(X)$, must be normal.

Khallil Benyattou

4:23 PM

Was it, @dREaM? ^_^

Balarka Sen

you just have to classify normal sbgrps of S_n, yeah

anon

so, trivial, alternating, symmetric groups in general, with the one exception of the klein four group (out of ${\sf Set}_4$ instead of $\sf Grp$)

Balarka Sen

it's A_n for n > 4

and there is V_4 for n = 4

yep

hey @Soham

@anon I no longer agree with you that fibers are just barren sets. I think I can build $\pi_1 X$ out of fibers of finite-sheeted covers.

Soham Chowdhury

hi.

Balarka Sen

well, $\widehat{\pi_1X}$

Soham Chowdhury

4:28 PM

@Balarka, I implore you, listen to this.

Johann Sebastian Bach - Chaconne, Partita No. 2 BWV 1004 | Hilary Hahn

►

what are you doing right now?

Balarka Sen

well, that's useless, because I am not going to.

@SohamChowdhury I? Taking a break from doing school-work.

Soham Chowdhury

ah, you're just like Chris'ssis in this. there's all sorts of cool things you should open yourself to. :)

@BalarkaSen ah. I'm studying, uh, EVS. :(

Remember me

I am studying projectile motion @SohamChowdhury

anon

@BalarkaSen what do you mean by that?

Balarka Sen

@anon $\pi_1 X$ acts on fibers of finite sheeted covers $E \to X$ by aut. however, it preserves the symmetries of fibers of finite sheeted covers lying below. so, I think, $\widehat{\pi_1 X}$ should be inverse limit of automorphism groups of the fibers, with bonding maps being pushing automorphisms of fibers down below

Soham Chowdhury

4:31 PM

@Rememberme exercise: derive the equations of motion for a projectile where the acceleration is $g+\alpha v^2$, where $\alpha$ is a constant.

this simulates the drag force.

Balarka Sen

similar thing works with $\mathsf{Gal}(k^{alg}/k)$, at the very least, where fibers over the point $k \hookrightarrow k^{alg}$ are $\mathsf{Hom}_k(L, k^{alg})$

Remember me

We don't have drag though @SohamChowdhury

Chris's sis the artist

@Hippalectryon are you done with my problem? :D

Soham Chowdhury

@Rememberme you don't have path-connectedness either ;)

Hippalectryon

@Chris'ssistheartist O_o which one ? I still have many problems from you :P

Balarka Sen

4:32 PM

@SohamChowdhury must be a pain

Hippalectryon

But as aways I lack time :(

Balarka Sen

we don't have EVS.

Chris's sis the artist

@Hippalectryon $$\int_0^{\pi/2} e^{-\csc ^2(x)} \csc ^2(x) \left(\text{Chi}\left(\cot ^2(x)\right)+\text{Shi}\left(\cot ^2(x)\right)\right) \, dx$$

Soham Chowdhury

or do this: given an inclined plane of height $h$ and angle $\varphi$, what angle should you shoot a projectile at from the base so that you cover the greatest possible distance along the inclined plane?

@BalarkaSen very much so

dREaM

@SohamChowdhury is the answer $4$?

anon

4:33 PM

@BalarkaSen what do you mean by covers lying below?

Remember me

Think of this question :
A projectile is thrown from a height of 10 m to an inclined plane inclined at an angle 45 degree . It hits the inclined plane at 90 degree. Find the initial velocity of the projectile

Soham Chowdhury

@Chris'ssistheartist did you ping the wrong person? :P

okay, this is not Physics.SE. let's stop.

Chris's sis the artist

@SohamChowdhury Yeah, sorry.

Soham Chowdhury

@dREaM what?!

dREaM

Is the answer $4$?

Balarka Sen

4:34 PM

@anon covers lying below $E \to X$ are covers $Y \to X$ such that $E \to Y$ is a cover.

Remember me

@Soham the answer 45 degree

no.

oh

@Rememberme no

@BalarkaSen so you're defining $\pi_1(X)$ using more than just the set $p^{-1}(x)$ no?

Soham Chowdhury

4:35 PM

did you even understand the question, @Rem?

Balarka Sen

while I say cover, I always mean galois cover.

Oh wait, is it $7$?

At a height okhay...

I think it's $7$

you want to cover the greatest distance "along the inclined plane", not along the ground.

Balarka Sen

4:35 PM

@anon well, yes, kind of. I am considering the set of all the fibers of all the finite covers.

dREaM

yeah, $7$ then. My bad.

Balarka Sen

looking at a single set of fibers is worthless, I agree.

Soham Chowdhury

if it were the ground, then $\frac\pi4-\varphi$ is correct, so that the total angle is $\frac\pi4$.

@dREaM no. no offense -- are you on something?

Remember me

Wait.. I got it

Chris's sis the artist

@SohamChowdhury I'm opened to the whole stuff, but there is something that I've realized a long time ago, in the area I'm in at the moment there is so incredibly much work to do, not much time left for other things now. I wanna let behind me an epic amount of work, not just to walk through all areas without having any significant result anywhere.

dREaM

4:36 PM

@SohamChowdhury no, it is $7$ because $7,8,9$

Soham Chowdhury

@Chris'ssistheartist don't miss the forest for the trees, is all. :)

Balarka Sen

@Chris'ssistheartist there are persons who have walked through several areas of math with significant result everywhere (Paul Erdos). just sayin'

Remember me

@Soham can you give me the link for latex... I am on my phone

Chris's sis the artist

@BalarkaSen I'm not Erdos, I'm a simple person.

Soham Chowdhury

@Chris'ssistheartist John von Neumann.

@Chris'ssistheartist so was Erdos ;)

Chris's sis the artist

4:38 PM

@SohamChowdhury :-)))

Remember me

@Soham the answer is pi/4-\alpha/2 . Where alpha is the inclination of the plane.....

Balarka Sen

@Chris'ssistheartist Well, if you're as hard-working as you claim to be, you have the potential to.

Remember me

That's the answer ... I got it by maxima and minima @Soham

Well only maxima

Chris's sis the artist

@BalarkaSen There is much work to do in my area, believe me. Well, believe me might not mean much to you, but I cannot convince you ... (for example, at the moment I'm very focused on some connections between some integrals and series, I might find something cool in a few hours or not - working on that - nothing works in the blink of an eye)

Remember me

@Soham my answer is right ?

dREaM

4:42 PM

what is purple and commutes?

Balarka Sen

good luck, then, @Chris'ssistheartist. :)

Chris's sis the artist

@SohamChowdhury btw, I'm listening to your song that is very nice.

@BalarkaSen Thanks ;)

dREaM

Listen to this song plz, it is sehr gut youtube.com/watch?v=qGKrc3A6HHM