Mathematics - 2016-11-23 (page 4 of 7)

Hiro

17:03

so can you express $$\sum_{k=3}^n 1/6^k$$ as $\frac{18(1-\frac{1}{6^n}}{5}$?

Danu

@BalarkaSen What do you mean? The curvature tensor definitely doesn't need to vanish at any point... The standard embedding of $S^2$ shows that much.

Balarka Sen

17:15

@Danu I am talking topology, not geometry. Being locally flat means the embedding (need not be smooth) locally looks like the embedding of a linear subspace of $\Bbb R^n$ inside $\Bbb R^n$.

Danu

@BalarkaSen Hmm, okay. Do you mean locally Euclidean (as appears as part of the definition of a manifold)?

Balarka Sen

No.

Danu

Can you give me a definition?

Balarka Sen

It's an embedding $M^k \to N^n$ such that there is a chart $U$ on $N$ around every pt so that $(U, U \cap M)$ is homeom to $(\Bbb R^n, \Bbb R^k)$.

Anyway google is always your friend.

Mike Miller

Any progress?

@BalarkaSen I doubt it.

@Danu Locally flat, in simpler words, means that locally it extends to an embedding of a tubular neighborhood.

Danu

17:22

I see.

Balarka Sen

@MikeMiller Nothing so far. Akiva has probably more useful contributions to make on this question than me though.

Maks

@DHMO are you there ??

Krijn

Oh yay, I did a thing today

Mike Miller

congrats on doing a thing

Krijn

Thank you, I am indeed proud of myself

Cobordism

17:31

I haven't done a thing yet today.

I hope I will do a thing soon.

You are an inspiration, @Krijn

Mike Miller

i mainly want to see if I can get Daria to play

Cobordism

Daria being?

Balarka Sen

maybe today is the day when i'll actually watch a film and not plan on watching one

Mike Miller

internet is barely good enough to download arxiv papers, so I am skeptical

woo, got it

Alessandro

I still suggest the lobster @Balarka

NaCl

17:39

0

Q: Closed form of recursive function

Given the following function $$f(n, x, y)=\left\{\begin{array}{ll}x\cdot y&\mbox{, if }n=0\\x&\mbox{, if } n>0\wedge y=0\\f(n-1, f(n, x, y-1), f(n, x, y-1) + y)&\mbox{, else}\end{array}\right.$$ I managed to find the recursive term $x_y=x_{y-1}\cdot(x_{y-1}+y)$ with $x_0=x$ by testing some valu...

recursion

@Null

Alex

Does anyone know why for tensor products of finite dimensional vector spaces $V$ and $W$ we can't simply define $V \otimes W$ as linear combinations of symbols $v_i \otimes w_j$ rather than formal linear combinations of these symbols. See my post if you have an idea, thanks. Or you could just tell me...

Krijn

@Cobordism Then I did two things today

Amazing progression

Cobordism

The second being becoming an inspiration? Thanks that'd mean I did one thing today: making someone else do a second thing today.

I feel great.

Krijn

We book amazing results this way

Mike Miller

how do people here watch HBO shows

Danu

17:49

@MikeMiller Whatcha mean?

Max

I've been thinking about using integrating factor as a way of solving differential equations, and one thing really bothers me. If you have a general linear differential equation $f(x)y' + g(x)y + h(x) = 0$, you can rearrange it to $y' + \frac{g(x)}{f(x)}y + \frac{h(x)}{f(x)} = 0$. However, in that case you're assuming that $f(x) \neq 0$. Does this mean that solutions where $x$ can be equal to $0$ are excluded?

Mike Miller

like, streaming

Danu

What about it?

arctic tern

go to my neighbors house

I hear GoT is the world's most torrented show

Mike Miller

"what website service does the kind gentleperson use to watch television shows offered by the HBO channel"

I would do that if it was reasonable to torrent on chromebooks, tho I'm actually looking for westworld

arctic tern

17:51

excellent choice

Mike Miller

got all the streaming things except HBO now and I'm not going to burn another whatever/mo for a single show

Danu

@MikeMiller Oh. There are numerous streaming websites... Streamlord seems pretty okay.

THE LONE WOLF.

Should I edit this question for reopening. reply please

arctic tern

from ad-revenue-based websites, streaming video quality sucks, and also lags, it'd be better to use a browser add-on to rip streams and watch later then delete.

Danu

@arctictern Ain't so bad in many places.

arctic tern

17:53

interesting

Mike Miller

usually terrible in the US

Danu

Streamlord, for instance, is pretty high quality.

Mike Miller

audio doesn't sync with video, 240p is common

really I think I need to find someone with HBO Go

I'll check that site out when I land

THE LONE WOLF.

@MikeMiller,dan, arctic please help me. You can talk about your HBO shows later, please

Danu

^screw that

3

Max

17:55

with all due respect, they can talk about what they want

arctic tern

lolwut

Mike Miller

I tend to be strictly less likely to help someone if they interrupt conversation to ask me to.

Balarka Sen

helping you can wait too

THE LONE WOLF.

@BalarkaSen okay. I m waiting.

Astyx

Hi everyone

Krijn

18:03

Hi @Astyx

THE LONE WOLF.

Hi @Astyx

Astyx

How are you ?

THE LONE WOLF.

Well, you say

An old man in the sea.

Hello, may I interest you in this question?

0

Q: Product rule of multilinear mappings for directional derivatives

How does one calculate $DT(f_1,\cdots,f_k)(a)(h_1,\cdots,h_k)$? I guess we'll use the theorem above, but I don't see how exactly, in the sense how each $h_i$ will be assigned... Any help would be appreciated.

multivariable-calculus multilinear-algebra

Astyx

I'm personnaly fine, thank you

Balarka Sen

18:08

@Anoldmaninthesea $T(f_1(a), \cdots, Df_i(a), \cdots, f_k(a))$ is still a multilinear map, not a number. You just feed $(h_1, \cdots, h_k)$ to it.

And then sum over them like in the theorem.

Silent

@BalarkaSen, What is meant by reflection here : The set of rotations together with the set of reflections in the lines through $0$. The latter are given analytically by $(x, y)\rightarrow(x', y')$ where
$x' = x \cos θ + y \sin θ, y' = x \sin θ — y \cos θ$

meow-mix

good afternoon

Balarka Sen

Just rotate the coordinate plane fixing the origin by angle $\theta$ (counterclockwise I guess)

Astyx

Hi @meow-mix

Silent

@BalarkaSen, you mean that if $\theta=90$ then $(1,0)\rightarrow (0,-1)$ ?

Danu

18:11

It should go to $(0,1)$

Astyx

Why ?

Balarka Sen

^ you're working counterclockwise

Danu

Oh, you're working clockwise?

Silent

@Danu, then what is differenv\ce between reflection and rotation?

Danu

Lol

@Silent ???

Silent

18:13

@Danu so it will take to (0,-1) right?

Mike Miller

wifi is too crappy for spotify

:/

Danu

@MikeMiller This is in uni? Or at home?

Mike Miller

on plane zoom zoom

Danu

@BalarkaSen Who is? :P

@MikeMiller You're on a plane.

Balarka Sen

everyone does

Danu

18:15

And you're complaining about the wifi?!?!

Mike Miller

yes; zoom zoom

Danu

@BalarkaSen Good.

$1^1$-st world problems.

Tristen

Woah, I just got access to the chat, and this looks awesome! Hi everybody!

Danu

o/

arctic tern

hello stranger

Balarka Sen

18:16

Hi @Tristen

Welcome.

Mike Miller

youtuve seems better as long as the songs aren't music videos

and they've got this new feature that just lets you play albums

Balarka Sen

that's neat; I didn't know about that

Tristen

Can you post latex equationsin here? $\sqrt{2}$

Astyx

Hi @Tristen

Danu

@Tristen Yes---use the ChatJax stuff (look at the top right corner of the chat for info on how to do that)

Mike Miller

18:18

luckily all but the first song of "love this giant" is video-free

An old man in the sea.

@BalarkaSen Yes, you're right. but my doubt is precisely how to feed the function. Since, it's each of the Df_i(a) who will receive one of the h_i... but which one?

Mike Miller

now to decide if I'm liable to get any work done today

Tristen

So, I was wondering, is Wolfram alpha making a mistake when I plug in things like $\sqrt{5}^X+\sqrt{5}^{-X}$ because it gives me an equation with a Root of $\frac{i(2\pi n+\pi)}{\log(5)}$ Can that be correct?

Astyx

@Tristen What function did you use ?

Tristen

None, just $\sqrt{5}^X+\sqrt{5}^{-X}$

Well, its a root, so it would be $\sqrt{5}^X+\sqrt{5}^{-X} = 0$

arctic tern

18:30

what makes you think it's a mistake?

Astyx

wolframalpha.com/input/…

Balarka Sen

@Anoldmaninthesea. I am sorry, I have to go now. I recommend explicitly looking at an example. Maybe someone else would be able to help you though.

Astyx

A link could have been useful

Tristen

Oh, Sorry! Will do!

wolframalpha.com/input/…

That part

Astyx

@Tristen This seems likely true. You can rewrite it as $5^x + 1 = 0$, ie $e^{x \log 5} = -1$, and therefore $x = {i(2\pi n + \pi)\over \log(5)}$

Tristen

18:33

x^2 = -.5 then

arctic tern

huh?

Tristen

wolframalpha.com/input/…

arctic tern

it says n = - 1/2 is a solution to i(2pi n + pi)/log(5) = 0

which is both true and completely irrelevant

Tristen

wolframalpha.com/input/…

Astyx

Wolframe is powerful, that doesn't mean it's clever

arctic tern

18:34

also don't see how zeta is relevant...

Astyx

(for instance take the future president of the USA)

Sylent Nyte

hey guys, need some help with functions

Astyx

Gotta go eat i'll be back

Tristen

Ok, so give me a second to figure out exactly what is bugging me to post a better more concise question

Sylent Nyte

i have a function $f(x) = kx^3 - 6x^2 + 12kx$ and I need to "discuss the number and stationary points as k varies"

easiest method, desmos

arctic tern

18:36

fixed points, f(x)=x?

so, the number of real solutions to f(x)-x

since you can factor x out and use discriminant to determine number of real solutions to quadratic, that seems doable

Sylent Nyte

yea I can do it

its just I dont know how to display this mathematically

for lack of a better word

for example, I know that when $-1<k<1$ there are no solutions

no stationary points*

arctic tern

x=0 will always be a fixed point no?

Sylent Nyte

yes

oooh

wait

arctic tern

actually, wiki says stationary point is when the derivative is 0

so you're looking at the discriminant of f'(x) as a function of k

Sylent Nyte

derivative shows stationary points

An old man in the sea.

18:40

@BalarkaSen ok, no problem... ;)

Sylent Nyte

okay

Tristen

Ok, so a better way to phrase this, is why does the solution to $\sqrt{5}^X+\sqrt{5}^{-X}=0$ have a zeta plot like this wolframalpha.com/input/…

wait sorry, like this: wolframalpha.com/input/?i=plot3D+zeta(i*((2+%CF%80+n+%2B+%CF%80))%2F‌(log(5))) Copy and past link, for some reason it isn't working correclty

Sylent Nyte

ive got my discriminant of $f'(x)$ as $\frac{12+\sqrt(144-144k^2)}{6}$

@arctictern wait, why am i looking for the discriminant? I want to find stationary points as k varies

arctic tern

@Tristen first of all, the solutions to that equation involve only integer n, whilst that graph you have uses real values for n. second, what in the world does your equation have to do with the zeta function? third, the (correct) graph looks like it does because you plug in those values and get what you get. it is what it is, not sure what else you're looking for. there is no obvious geometric description of the zeta function's graph.

@SylentNyte the number of real solutions to real quadratic=0 depends on the discriminant

(that is in fact why it is called the discriminant)

Sylent Nyte

but how does that help me exactly? I cant see the link

Tristen

18:45

Hmm. Ok!

arctic tern

@SylentNyte you're trying to find the solutions x to f'(x)=0 right? and f'(x) is a real quadratic function of x.

Sylent Nyte

im not looking for solutions

arctic tern

the number of solutions

stop being silly

2

Sylent Nyte

stationary points

solutions is where y = 0

am looking for local maximum and local minimum

arctic tern

a stationary point is where f'(x)=0 right?

Sylent Nyte

18:47

oh

yes

sorry

being silly

Danu

@arctictern Hah, I didn't make that connection before.

arctic tern

:D'

Tristen

A little bit of a different question, so I like geometry, and I was wondering why I can't find a formula anywhere that doesn't involve trig functions to solve the sides of an isosceles triangle, if one side is known and the angle is 45 degrees where the 2 equal sides meet?

Sylent Nyte

@arctictern can you help me? i think ive done bad things

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$Mathematics$ Mathematics