Mathematics - 2018-05-23 (page 1 of 3)

Get out of my swamp

I'm still waiting for the Bee Movie / Shrek crossover movie

@KasmirKhaan Good, i was at one point on academic probation my grades were so bad now i got a 8.75 average for the last 3 semesters sometimes it takes time to get into the right stride with mathematics

hell i still spout gibberish most of the time but i blame that on the learning disabilities :P

in the US the scale is 0 to4.0 so americans dont k now what 8.75 means I bet

i dont at least

haha =p

12:05 AM

it means my average is basically above 90%

well it is very hard and alot of things to keep in order

using a 4.0 for A i would have a gpa of 4.0 over the last 3 semesters or 14 math classes

nice!

but at one point i had 1.667/9 and was on academic probation

not sure what that is out of 4 but im gunna go with not good

2.0 is passing in most colleges iirc

just barely passing

12:08 AM

@BalarkaSen did u turn into secret?

its a secret

@Daminark but why?

It's excellent

12:10 AM

your eating all those planets

iv seen this once before , it means dami controlls the world

daminark is just a member of the cult of Thonk

Thonk will dominate the world

i only report to dami

i dont acknowledge thonk

better ask for a promotion

no need

just working for dami is an honor for me

12:11 AM

but how can daminark be real if your thonks aren't real

I am not obligated to answer that

nor continue this discussion

dami is light

I think i have been mindcontrolled for a second there -.-'

@MatheinBoulomenos i didnt know u were undergrad?

12:30 AM

@Faust *you're

i have given up on english like so long ago

language is nonsence

nonsense

i don't know how anyone can understand it.

Daminark

@Kasmir I give you my blessing

@Daminark don't tell me you finally completed that mind control device?

@Faust I don't find that surprising. If you can't figure out how to produce language, it is reasonable to assume that you would have difficulty understanding how to make yourself understood via language. :P

you have no idea i can only explain things to really smart people

but honestly my brain is broken in so many ways thats not all that surprising

12:41 AM

Faust your school probably has counselors?

most people that have whats wrong with my brain never learn how to speak

1 hour ago, by mercio

everyone, I think Secret has killed Balarka and stolen his computer

lol wut?

That's not even remotely weird to be me

O.o

1 hour ago, by Balarka Sen

when $f$ rumplestiltskins at $S$ we get the triumphant penguins with a mirror dancing

wut?

i guess in some sense i am very smart stupid person ^^

@Secret i noticed you didn't deny it...

12:45 AM

everyone is weird in some way, there's no such thing as stupid unless you are a help vampire (but you are not)

A stupid person is willing to learn, help vampires don't

and over time, will become smart

@secret lots of those at my uni

lol

I start out stupid in my grade 3 for example

i dont belive intelligence is so easily swayed by knowledge

intelligence is not about how much knowledge you can cram into our brain, but how you pick the useful bits

It's a craft

i think intelligence is a biological setting thats possible but difficult to change the setting on

but being autistic perhaps that's a jaded point of view

12:49 AM

you sure like talking about yourself Faust

I am aspergic, yet I go all the way from not very good at social to able to understand a bit of the social cues

associating values to what we enjoy is the worst but we still have to do it

@GFauxPas sorry am i bothering you?

nope

ok ^^

12:51 AM

biological limits are like limit points, you might not be able to reach it, but you can get arbitrarily close

intresting proposition

but do we live in a complete space

well, considering that we don't even know if reality is a simulation, I don't know if it has a notion of topology at the smallest scale, let alone its cardinality (which controls whether we have craziness like countably compact but not compact for example)

but coarse grain, we are like $\Bbb{R}^3$

as my topology professor once said

the universe can be factored as the product of two donuts

mercio

oO

12:56 AM

lol

($\mathbb R^3 \cup \{\infty\} \cong T^2 \times T^2$)

That's uh incorrect

why must it be projective?

mercio

my brain is rioting at that statement

What he probably meant was that $S^3$ decomposes as a union of two solid torii.

12:57 AM

I have to go back to my notes thats whaty I recomember

$S^3 = S^1 \times D^2 \cup_\partial S^1 \times D^2$, where the solid torii are glued along the boundaries by the diffeomorphism of $S^1 \times S^1$ switching the two factors.

It's one of my favorite facts of all time

$S^1 \times S^1$ is the duocylinder, it is a very cool shape

Er, it's just a torus :P

o oops, conflated $S^1$ with disks...

anywhere, heading to chemistry seminar, they are talking about polymerisation with small molecule catalysts today

don't let duocylinders haunt you

1:01 AM

okay I found it, sorry for being sloppy, it's

$\mathbb R^3 \cup \{\infty\} \cong D^2 \times \mathbb S^1 \cup_{\mathbb S^1 \times \mathbb S^1} D^2 \times \mathbb S^1$

Indeed.

Mike Miller

yeah balarka beat you #rekt

lol

pew pew

oh damn

1:03 AM

oh i just saw it

is there a name for the fact, it's cool

baby example of Heegaard decomposition

Any closed 3-manifold can be decomposed into two handlebodies (solid higher genus surfaces) glued by their boundary surfaces by some diffeomorphism

is "frontier" another word for "boundarz" of a set?

boundary*

yes

old topology term

faaaair, the book I'm using has this

lol

skull

1:05 AM

old textbook?

its pretty much the wierdess word for boundary i can imagine but if someone can 1 up it id be impressed

It's Armstrong's basic topology

1978 I think

skull

yup, old school

it probally has some great pictures

doesn't really give any notation either, do you use $\partial S$ or smth for this?

1:07 AM

thats one of the common ones

$\bar S $

Fair, I think I shall use this

hahaha

sometimes too

This book uses $\bar S$ as the closure of $S$

i prefer the first one

thanks!

1:09 AM

np

I have two exams in under a week and instead I'm starting some self-study

lol

well im exhausted im outie gnight everyone

gn

night man

@BalarkaSen why does the gluing map switch factors?

Alek Murt

1:18 AM

Hey guys, could someone tell me a reference with proof, of the Kondrachov embedding en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem

how is sine usually defined in a formal context?

depends on context

luckily, they're all equivalent!

i know spivaks defines it as some integral

i think of like a circle, idr

its weird though because in my school they start with the triangle stuff and then extrapolate that for larger angles

if you accept the equality of geometry on a plane and $\mathbb R^2$ then the circle definition is formal

if you're not willing to make that assumption you have to use something else

in the context of complex analysis I usually see the power series is the definition, not a consequence

but you can also use it as the solution to

$f'' = -f, f(0) = 0$

really? that uniquely defines it?

Q: Defining sine and cosine via ODE's

1:27 AM

actually you might need $f'(0) = 1$ also?

let me look it up

yes you need that

$f'' = -f, f(0) = 0, f'(0) = 1$ uniquely defines sine

and $f(0) = 1, f'(0) = 0$ gives cosine from the same equation

0

So I read in Simmons book on Differential Equations that via the equation y''+y=0 One can define s(x), c(x) as their solutions with some given initial conditions, that is s(0)=0 s'(0)=1 ; c(0)=1, c'(0)=0 These are of course sine and cosine, but you don't actually need to know it beforehand. ...

but yeah Meow if you accept that $\mathbb R^2 = \text{ cartesian plane }$ with the usual metric corresponding to the distance on a theoretical ruler, the high school definition is rigorous

with reflecting triangles inside the unit circle about the $y$ and $x$ axis

yeah i know, its just that it feels unnatural

1:42 AM

I hear that

remember that the idea of a coordinate plane came about before people thought about pairs of real numbers

and so trigonometry predates analysis

How can I prove that $\displaystyle\int_0^\pi \sin(\cos(x)) dx=0$ ? Any hint?

well you'd like to show that $\int_0^{\pi/2} = - \int_{\pi/2}^\pi$

then you're done because the positive and negative parts cancel out

I get the feeling that every word in this book is old fashioned now lol

maybe change coordinates to be centered at $0$ and show the changed integrand is odd

Change to polar coordinates?

1:48 AM

no no i'm not saying that

instead of $x \in [0..\pi]$ do $u\in [-\pi/2..\pi/2]$

and then the integrand should be odd

Hm and how could I find this new integrand? Sorry I'm a bit rusty, just doing this for fun

$u$-substitution

$x - \pi/2 = u$

$\mathrm dx = \mathrm du$

Oh

Then it becomes $\int_{-\pi/2}^{\pi/2}\sin(\cos(u+\pi/2)du$

Correct? Now not quite sure how to prove it's odd

2:04 AM

now, check your table of trig identities!

or draw triangles

consider $\cos(\theta + 90^\circ)$ on a right triangle if you dont know the identity i'm alluding to

actually i just tried drawing a tirangle and thats not obvious

$\cos(u+\pi/2)=-\sin(u)$ right?

yes

so... the integrand is...

$\sin(-\sin(u))$

is that odd?

$\sin(-\sin(-u)) = \cdots$

wait

yeah that's right

you can simplify $\sin(-\sin(u)) = -\sin(\sin(u))$

then it's obvious its odd

:D

@GFauxPas Boo! >:(

Imagine triangles in your head!

Don't actually draw them

2:13 AM

lol

That wastes paper and kills trees

I hate trees

Oh, well, carry on then.

Oh, viXra...

lol wtf

viXra is the greatest website EVAR!

It has all the BEST maths. It is TREMENDOUS. Totally BIGLY.

2:17 AM

Yes, by symmetry it's odd. how to get to your first statement though? your first equation to integrals. So we have $\int_{-\pi/2}^{\pi/2}-\sin(\sin(u)) du$

In this paper, we define very small numbers and very very small numbers and use them to construct derivatives as ratios of real numbers. We then use that result to rigorously prove that the chain rule treats derivatives as fractions being multiplied.

$-\sin(\sin(u))$ is odd, check it

$\Huge\color{green}{\checkmark}$

zactly

$$\begin{align}\sin(\sin(-u)) &= \sin(-\sin(u)) && \text{$\sin$ is odd} \\ &= -\sin(\sin(u)) && \text{ditto}\end{align}$$

HOLY SH$*$TB$*$LLS! IT'S TRUE!

quod erat DAYUM

2:24 AM

Oh, right. So $\sin(\sin(u) +\sin(-\sin(u)) =0$ and then I'm done?

@GFauxPas That's an interesting question. I guess we can use those functions abstractly by adding ODEs as long they are linear

I don't recall asking a question, interesting or not

No, that question is not asked by you, I am just referring to the question you linked

the theorem that a continuous odd function on a closed interval symmetric around 0 is 0 is well known Christopher. You can just invoke it

Proof. As we know, a derivative is just a ratio of very very small real numbers

Thanks, GFauxPas :) I think I got it. A bit hard to understand a bit for me, but less complicated than I thought

2:27 AM

math takes practice!

proofwiki.org/wiki/Definite_Integral_of_Odd_Function I was referring to this

primitive is a synonym for indefinite integral

Yeah. I wanted to try this because sin(cos(x)) doesn't have a primitive in terms of elementary functions so I got curious. Thanks for the link!

np

I'll note that stuff like cos(cos x) typically shows up in relation to Bessel functions

That's interesting

e.g. one integral representation of the zeroth bessel function of the first kind is $J_0(z)=\frac{1}{\pi}\int_0^\pi \cos(z \sin x)\,dx=\frac{1}{\pi}\int_0^\pi \cos(z \cos x)\,dx$

2:33 AM

Now checking...:

$$\int \cos \circ^n (x) dx$$

It seems Bessel functions mainly have applications in physics

yeah, pretty much

wolframalpha.com/input/?i=integrate+cos(cos(cos+x)))

FAIL

Very interesting. Not yet at that level though, heh :)

where $f \circ^n = f \circ f \circ \cdots \circ f$

2:37 AM

GFauxPas, are you a math student? I used to visit this chat and iirc Semiclassical is a physics major?

yes I am a math student and a wizard. one of those things might be false

Cool haha

defended my physics phd thesis last week :)

5

noice!

Congratulations :)

Secret, does that have a closed form?

It looks hard

2:41 AM

doubt it

Congrats!

thanks

GFauxPas, what courses are you currently taking?

Actually, I am wondering, is there a measure on how hard an integral is, cause what I found integrals of the form:

i just began summer vacation but I'm trying to go through an analysis textbook over the summer

2:46 AM

$$\int f^n(x)dx$$

I'm taking elliptic diffeomorphic hyper congruent modular analysis

quickly ran out of closed forms even for special functions when n increases

I just finished analysis ii, complex analysis ii, top ii, lin alg ii

In contrast, integrals like:

Nice. Which textbook?

2:47 AM

$$\int [f(x)]^n dx$$

does not get hard that quickly, for example f = sin has a closed form

analysis: something lousy, the professor only used it for exercises

complex: mixture of churchill/brown and Alfohrs

geocalc, is that even a thing? lol

lin alg: no textbook

top : no textbook

There is modular forms, but that long name thing, I never heard of it

GFauxPas. Oh I see. And will you aim for a PhD after your degree?

2:50 AM

hope so

@GFauxPas what field you interested in pursuing?

operations research

Churchill and Brown is not terrible, but Alfohrs is da man.

fair

What is the lousy real anal book that you are using?

2:51 AM

Churchill and Brown is much easier though

but its shallower

umm let me see

Q: How do you determine the complexity class of a problem like solving an integral?

4

The P and NP classes relate to decision problems, but what about calculus problems, specifically computing an integral? How does one figure out if a certain class of integrals is in P or NP? Can something like this be rephrased in terms of a decision problem? Or is there another, indirect method?

integration computational-complexity

C&B is a reasonable undergrad text, but, as you say, it doesn't get very far

I also find it a bit pedantic

I have a really difficult integral

We used Gordon? I don't really remember it much

Lebesgue Integration on Euclidean Space by Frank Jones

he told everyone not to buy it

2:53 AM

Operations research.. looks like it has to do with optimization and statistics (Stochastic models, Markov..) looks interesting

Lebesgue integration in an undergrad real anal class? Interesting...

im grad

I would have expected something more like Rudin or Spivak

OH!

well, okay, then

I made an assumption after seeing Churchill and Brown ;)

Liouvillian function

In mathematics, a Liouvillian function is an elementary function or (recursively) the integral of a Liouvillian function. More explicitly, it is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, constants, solutions of algebraic equations (a generalization of nth roots), and antiderivatives. The logarithm function does not need to be explicitly included since it is the integral of 1 / x {\displaystyle 1/x} . It follows directly from the definition...

we need to expand this set to the currently defined special functions

the prof called it the "baby book" and Alfohrs the "adult book"

2:55 AM

c.f. Integral Project talked about and then deferred n years ago

yes

Daminark

S c h l a g

Best complex book tbh

I have encountered a lot of complex anal books that I don't really like

Alfohrs is the big exception; I love that book

Definition: A closed form of an operation is an outcome that can be uniquely defined by finite number of symbols in a formal language which does not contain implicit or explicit symbols that denotes infinity

(Most general one I ever came up of)

My institution likes to use Conway, which is okay-ish

2:58 AM

What about $f(n+1)(x) = f(n)(x), f(0) = \cos x$ Secret

If I have a family of sets $\mathcal{F}$ and I write $\bigcup \mathcal{F}$, how would you interpret this? The context in the book seems to suggest the union runs over the members of $\mathcal{F}$..

Sorry to spam with questions like this lol

@ÍgjøgnumMeg I've seen notation like $\cup \mathcal{F}$ to indicate the union over the entire family

i.e. $$ \cup \mathcal{F} := \bigcup_{F\in\mathcal{F}} F$$

@GFauxPas Well, that technically uniquely defines the cos function nested up to n times so it counts

a "better" notation would be $\bigcup_{F \in \mathcal F} F$

e;f,b

I'm going to claim that I beat you to it, @GFauxPas :P

3:00 AM

fingers too slow

@GFauxPas @Xander Right, that's what I thought

lol

you did, I lost ;___;

However something like $\lim_{n\to \infty} f(n) (x)$ will be forbidden

so do infinite sums

I'll limit your $f$ of $x$ to $n$!

@Semiclassical This is one of the greatest recent horror games I have seen a playthrough of.

3:05 AM

glad it is not a jumpscare

psychological horror is more fun

Random fact of the day:

@Secret It's nuclear holocaust horror. Very fresh stuff

The gameplay looks beyond entertaining

If you had one shot, one opportunity... would you integrate f(x)? or just let it slip away...

we live in a society

$$\int f(x) dx = F(x) + C$$

QED

3:14 AM

:O

The more serious answer: We need an Extended Galois theory and a better way to categorise what is a special function and what isn't

@Secret differential Galois theory is a thing

@Secret en.wikipedia.org/wiki/…

yeah, but that only applies for liouvillian functions. Remember what we discussed 2 years ago, we need to extend that to the special functions if possible. User21820 also said we need a more systematic definition of special functions so that it does not sound ad hoc whenever the history add a new one

"functions you never heard about in middle school"

3:18 AM

For a lack of better word: The set we are talking about is the closure of all special functions and their integrals

I am not sure if the literature already had that since I am never free enough to study it in detail, plan to do that after the PhD

I have an integral

There's some literature along the lines of "closed form functions" and "closed form constants", yes

right, those are the info we need to develop a framework that unify them (or so that it is impossible)

Tbh ,I don't know if that set will suffer the same non closure problems like e.g. algebraic numbers vs transcendental numbers

(i.e. we all knew that transcendentals don't even form a ring)

google "ring of periods"

How do you integrate (a)^(1/log(x)) from x=0 to x=1?

3:25 AM

> Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods".

Interesting, that seemed to be a concrete direction that can be worked on, hmm...

oops misread

Geocalc has been thinking about complex powers for days :)

mathematica gives the answer as 2 BesselK[1, 2 Sqrt[Log[a]]] Sqrt[Log[a]] for a>=1

I am guessing it will probably a reduction formula involving the logarithmic integral

deadsniped