Mathematics

Shobhit

10:18

I might have to later in this chapter, just started topology :) @AlessandroCodenotti

Tuki

What kind of foreknowledge you should have before learning about topology ?

Shobhit

metric spaces i think

Tuki

is this synonym for euclidean space ?

Shobhit

no

Alessandro Codenotti

That's helpful but not necessary, in my general topology course we just treated metric spaces as a particular kind of topological spaces

Shobhit

10:22

:O

Vrouvrou

Please someone know how we get this

$(w_1^{1/p}|x|)w_1^{1/q}+(w_2^{1/p}|y|)w_2^{1/q} \leq (w_1|x|^p+w_2|y|^p)^{1/p}(w_1+w_2)^{1/q}$

$w_1,w_2>0$

Alessandro Codenotti

@Shobhit ah, I see, it's weird to have Baire's theorem at the beginning of a first course in topology. But it's a cool theorem with quite a few useful applications

Slereah

Depends what kind of topology

you just need set theory for point set topology

Shobhit

no, baire's theorom was not at the start of topology, but at the end of metric spaces @@AlessandroCodenotti

Alessandro Codenotti

Ah, I see, it makes sense to see the "metric spaces are Baire" version then. There's another version of Baire's category theorem that says "compact Hausdorff spaces are Baire"

(The first one is more useful to be fair)

Shobhit

10:26

from what i have been reading, we are only talking on hausdroff spaces, are non-hausdroff spaces not interesting?

Alessandro Codenotti

Most naturally arising spaces are Hausdorff

Shobhit

oh

@AlessandroCodenotti in my book, they talk of continuity of a function by defining oscillation of it around a point x. Let me write that...

Slereah

@Shobhit Non-Hausdorff spaces are great

Alessandro Codenotti

Ah, sure, they want to prove that the set of points of continuity is $G_\delta$?

Slereah

Although they are not easy to deal with

A lot of theorems break down for non-Hausdorff spaces

Shobhit

10:33

let $N_{x}$ denote the collection of all nbd's of a point $x$. Define $w_{f}(x) = inf_{V \in N_{x}} \{sup_{y,z \in V}|f(z) - f(y)|\}$. How does this equating to zero shows continuity. From what i can see they are taking the infimum of the diameters of all nbd's of $x$

@Slereah any particular example of a non hausdroff space thats interesting? or any theorom related to it? From what i read, in non-hausdroff spaces sequences can converge to different limits. I was like...what??? no, not cool. XD

Slereah

How do you define continuity

@Shobhit Leaf spaces of foliations

Shobhit

@AlessandroCodenotti yes they talk about $G_{\delta}$ right after this

Slereah

Or extensions of manifolds

or boundary construction of manifolds

Alessandro Codenotti

Think about this in $\Bbb R$ using the oscillation calculated in $(x-\delta,x+\delta)$ with $\delta\to 0$, convince yourself that this limit being $0$ is equivalent to continuity at $x$ and think about the general case later

Shobhit

i'll write them down, might deal with them later on

Slereah

10:36

Odds are low :p

except the leaf space thing

Shobhit

ohk, thinkinh

Slereah

Leaf spaces are basically the only common occurence of non-Hausdorff manifolds

Shobhit

oh why?@Slereah research thing?

Slereah

Just an interest in 'em

I searched high and low for theorems on non-Hausdorff manifolds

they do have some neat properties

You can build non-orientable manifolds in one dimension

Or two homeomorphic manifolds that aren't diffeomorphic in one dimension, too

Alessandro Codenotti

What's a non Hausdorff manifold? Locally Euclidean, second countable space?

Shobhit

10:38

i dont even know what a manifold is

Slereah

@AlessandroCodenotti just locally euclidian

You can throw in second countable, too

(I think they're called $Y$-manifolds if they're second countable)

(but that was only in a single paper)

@Shobhit generalization of a surface, roughly

Alessandro Codenotti

So something like the line with two origins

Slereah

that is the most famous one, yes

Shobhit

@AlessandroCodenotti i am not getting it

Slereah

Basically they're fairly annoying because there's no partition of unity on non-Hausdorff manifolds

So it's hard to build a lot of things

Secret

10:47

I am wondering about the topology of the line with $\omega$- origins

Slereah

not terribly different from the like with $n$ origins

lots of charts, certainly

the weird Hausdorff manifolds are more like

the complete feather

Secret

Is a manifold basically a set of atlas, or is there something more? I knew for topological manifolds, you have a topology and the open sets are homeomorphic to euclidien space

Slereah

You can define a manifold from the charts and transition maps

Secret

right, make sense

charts for putting in a coordinate system and transition maps for gluing

Slereah

pretty much

Akiva Weinberger

12:40

@Slereah Cool

@Slereah The what?

I imagine the nonorientable thing would be like $[0,3]$ where $[0,1)$ and $(2,3]$ are quotiented together by $x\sim3-x$

Like, a circle connected to a line segment by the "double origin" device

Slereah

yeah something like that

It doesn't seem to have a canonical name but I just call it "the noose"

The complete feather is obtained using this procedure :

Take $\Bbb R$ copies of the real line, indexed by $x \in \Bbb R$

Glue each of them along $(-\infty, x]$

That is the feather

Repeat the process $n$ times to obtain the $n$-feather

the complete feather is the limit of this manifold

Every point of the manifold branches

there's variants where you also only glue them at point where $x$ is rational and such

Akiva Weinberger

I don't quite understand

Slereah

Do you know the branching real line

For manifolds

Akiva Weinberger

No

Is this like two copies of the real line with the negatives identified?

Slereah

Yes

Two charts $(\Bbb R, \operatorname{Id}_{\Bbb R})$, $(\Bbb R, \operatorname{Id}_{\Bbb R})$ such that the overlap is on $\Bbb R^-$, with transition $\operatorname{Id}_{\Bbb R^-}$

Akiva Weinberger

12:52

Mhm

Slereah

The feather is the same except for every point

hence the name

Complete feather is the process repeated so that every point of the manifold presents such a branch

Semiclassical

Hi chat

Tuki

Hi @Semiclassical

Slereah

hey

Semiclassical

It’s -11 F / -24 C outside. Why do I live in MN

Akiva Weinberger

13:01

@Slereah Ohh. I think I get it. I have to think about this…

Tuki

MN you mean Minnesota ?

Semiclassical

Yeah

Tuki

We have 2+ Celcius outside

Slereah

Could be Monaco

Or Montenegro

Semiclassical

There are temperature ranges I can cope with

Balarka Sen

13:04

@Akiva It's $\mathfrak{c} \times \Bbb R \sqcup p \times \Bbb R$ quotiented by the equivalence relation $(x_0, y) \sim (p, y)$ whenever $y < 0$.

Tuki

I wonder what would be peak negative temperature in celcius for current year

Balarka Sen

By $\mathfrak{c}$ I mean $\Bbb R$ with the discrete topology.

Semiclassical

But while 0 C may be the freezing point of water, 0 F is my own personal freezing point

Below that I go a bit crazy

Slereah

you can build weird versions of the complete feather where the only homeomorphism is the identity

Tuki

$0\pm 10$ celicus would be what the temperature is usually around here

Balarka Sen

13:07

I meant $y < x_0$, actually

You're shifting the basepoint of the tripod as you're gluing various objects

@Slereah Right, it can be totally nonhomogeneous

Semiclassical

@Tuki fun fact: in kelvin, one usually thinks about all temps as being positive (ie above absolute zwro). But if you use the definition of temperature coming from statistical mechanics then you actually can get negative temps

And they’re actually ‘hotter’ in some sense that positive temperatures

Tuki

negative temps in kelvin ?

Balarka Sen

Oh wait, what I wrote is not a complete feather is it? It's $\Bbb R$ with a copy of $\Bbb R$ glued along $(-\infty, x_0)$ for every $x_0$

So those glued $\Bbb R$'s locally do not look like tripods themselves

Semiclassical

Yeah. This won’t happen for most systems, though

Balarka Sen

The complete feather should be a very non-first countable space

Tuki

13:09

yes i can see why

Slereah

@BalarkaSen very much so

Semiclassical

For it to happen, you need the entropy of the system to go down when you increase the internal energy

Slereah

are there distinctions for manifolds with different cardinalities?

Like $\aleph_1$ basis versus $\aleph_2$ basis

Balarka Sen

Semiclassical

In a typical gas, though, increasing the internal energy means the particles move faster and have higher entropy

Slereah

13:11

Is that the Reeb foliation

Balarka Sen

Consider the quotient of this space with the equivalence relation $x \sim y$ iff $x$ and $y$ belongs to the same "leaf"

That's a 1 dimensional non-Hausdorff manifold

@Slereah Yeah

Slereah

I think Reeb's the guy who really started to study non-Hausdorff manifold

He wrote the "big" paper on non-Hausdorff $1$-manifolds with Haefliger

Balarka Sen

mhm

Tuki

I think i've seen this image before but cant remember the context ?

Balarka Sen

I linked it in this chat before

Slereah

13:13

it's the standard image used for foliations I think

“Non-Hausdorff spaces, often regarded as a technical nuisance, sometimes produce a global disaster.”
– Geometry of non-Hausdorff spaces and its significance for physics by M.Heller et al

“Kelley’s book set the cat among the pigeons in 1955 by daring to omit the Hausdorff condition from many of its definitions.”
– On non-Hausdorff spaces by Ivan L. Reilly

Wise words

Akiva Weinberger

13:31

@BalarkaSen I feel like that should be homeomorphic to the first version, though

Quotienting based on $y<x_0$ versus $y<0$

(For the not-complete version)

Oh, wait. It's not

Balarka Sen

I mean the $y < x_0$ thing is homogeneous along the real axis

You can translate it along the real axis (and translate the attached feathers thereof) and that'd be a homeomorphism

No such thing in $y < 0$ where everything is toppled one after another at the single origin

There is no homeomorphism of the line with $n$ positive real axes which takes $0$ to $1$, eg.

Secret

13:58

[Random]

1. $$S =\bigcup_{i\in I} U_i$$

2. $\tau_1 = \{i \in I : U_i\}$

3. $A = \{x,y : xRy\}$

4. $A^{\ell} = \prod_{\ell \in L}A$

Vrouvrou

Hello, i want to calculate $\lim_{t\to0} \frac{|t|^p}{t},p>1$if $t\geq0 $ there is no ptoblem but if $t\leq0$ can i write $\frac{|t|^p}{t}=(-1)^p\frac{t^p}{t}$ ?

Secret

5. $\forall j \in J [\phi_j : U_i \mapsto B \subset A^{\ell}]$

6. $T_{a,b} : \phi_a(U_a \cap U_b) \mapsto \phi_b(U_a \cap U_b)$

Akiva Weinberger

@Vrouvrou The problem is that $t^p$ might be complex or undefined

Consider $t=-1$, $p=3/2$

Vrouvrou

then how to calculat the limit ?

Akiva Weinberger

So I think a better way to do this is to write it as $\frac{|t|^p}{-|t|}$, and use $|t|\to0$

Vrouvrou

14:09

it $t\leq0$ right

Akiva Weinberger

Yeah

Slereah

@AkivaWeinberger the feather without any non-identity homeomorphism is weirder

Vrouvrou

thank you @AkivaWeinberger

Slereah

It's like the branching only occurs at rational points and every branching has a different number of branches

Akiva Weinberger

@BalarkaSen @Slereah I think this is homeomorphic to the set of functions of the form $f:(-\infty,a]\to\Bbb R$, where a basis is given by sets of the form $\{f|_{(-\infty,t]}|t\in(a,b)\}$ for arbitrary $f$

Slereah

14:11

it's not very second-countable

Akiva Weinberger

(The complete feather one)

Slereah

books.google.fr/…

Akiva Weinberger

'Cause of the branching

@Slereah What different number?

Like, the denominator of the rational, or something like that?

PabloZ392

Hello :) I need an example a function : 1. has two minimums, no maximum. 2.has two maximums no minimum.. :)

Slereah

I'm guessing any bijection $\Bbb Q \to \Bbb N$ provides a scheme to get it

Akiva Weinberger

14:13

Hrm

Slereah

The basic trick is that any homeomorphism would map a point near $n$ branches to a point with $n'$ branches, I think

So it fails to work

Akiva Weinberger

$x^4-x^2$: desmos.com/calculator/0l0bgdsq08

@PabloZ392

It has a local maximum but no global maximum

PabloZ392

@AkivaWeinberger Thank you :)

Akiva Weinberger

@Slereah Oh, and so it has no non-identity automorphism

Weird

Slereah

yeah

that's why people ask for at least second countability for manifolds

Akiva Weinberger

14:18

That's not second countable?

Slereah

Hm, i guess that one is, actually

Since it's just $\Bbb N$ copies of $\Bbb R$

That + Hausdorff condition

Secret

no idea

Akiva Weinberger

Strange millipede @Secret

Slereah

otherwise you get the whole bloody manifold zoo

look at that madness

those are just the 2D ones

Secret

Akiva: I think I need to read more closely on the construction again to get the correct picture of all those feather manifolds

Slereah

14:28

oh god those are only the Hausdorff ones, too

I have no idea what's the full set of 2D manifolds is like

I like how $S \times \mathbb L^+$ is called the long drink

Secret

what is $\Bbb{L}^+$, the long line?

Balarka Sen

long ray

Secret

ok

Category of manifolds

In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp. One is often interested only in Cp-manifolds modelled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modelled on a fixed space E is denoted Manp(E). One may also speak of the category of smooth manifolds, Man∞, or the category of...

hmm... so that's how to map between different manifolds of the same differentiabiliy classes...

Category theory sure really satisfy my tendencies to work with general things

Slereah

Gauld's diagrams are quite harrowing to look at

Trey

what trig substitution can be used for $\int \frac{dx}{(a^2 + x^2)^2}$ ?

Slereah

14:36

That looks like one of the inverse trig ones

Try something like $y = \cos(x)$ or $\sin(x)$

Secret

Slereah: What are the arrows? (i.e. what is the directional info that relate the various classes in the diagram)

A->B, A supset of B?

Slereah

Implication.

Secret

ok

Slereah

Some kind of butt foliation

I get a headache just looking at it

Balarka Sen

seems fine

Slereah

14:45

is the long line useful for anything outside of counterexamples, anyway

Balarka Sen

not to me

Secret

well, according to the past discussion in the logic chatroom, $\omega_1$ does not really have a use other than counterexamples and stuff within set theory

Xander Henderson

The long line is entirely useful for [censored].

Slereah

Top secret informations

Secret

I have been asking around for many months on where $\omega_1$ will arise naturally, but it seems uncountable well orderings are kinda a byproduct of the axiom of choice (and ZF in general)

Xander Henderson

14:48

This is why we need to overthrow the Axiom of Choice, and replace it with the Axiom of Monarchal Fiat

Slereah

Just use Peano's axioms

Xander Henderson

YOU GET NO CHOICES, SIR!

Slereah

ZF minus axiom of infinity is just Peano

where everything is nice and orderly

Secret

$\omega_1$ is impredicative and not very constructive. As far we knew, throwing all uncountable ordinals away will not affect most of mathematics

Xander Henderson

but but but... I needs my BANACH-TARSKI!

Slereah

14:50

@XanderHenderson has a sphere business and he just gets free spheres from Banach Tarski

Xander Henderson

ssh!

those are protected trade secrets!

how did you manage to avoid signing the NDA?

Slereah

ZF$\neg$C is worse than ZFC in some ways, tho

Secret

There are many useful results from the axiom of choice, such as nonmeasurable sets, infinite vector space having a basis, baire cateogry theorem and other things that makes real analysis straightforward to formulate

so I won't necessary want to throw it away

Silent

Does $a_n=1+1/2+\dots+1/n$ fit for requirements here?

Xander Henderson

As an analyst, I cannot imagine a world without AoC

Secret

14:52

On the other hand, in ZF, there are these infinite dedekind finite sets and amorphous sets which are very interesting and it can actually support a lot of structure

Xander Henderson

well, that's not entirely true; I just don't want to live in such a world

Slereah

Let's also get rid of the axiom of the excluded middle

Xander Henderson

HERETIC!

Secret

♦ $Mathematics$ Logic

This room is meant for discussion about logic, including found...

1

Slereah: You will like this chat room

user21820 is working on a type theory system that does not necessary have law of excluded middle (LEM)

Slereah

@XanderHenderson Let's switch to en.wikipedia.org/wiki/Paraconsistent_logic

Xander Henderson

14:54

GAH!!!!!!!!!!!

NONONONONONONONONONONONO!

Secret

Why choosing, just pick them all:

Metalogic

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and...

Xander Henderson

also, time to take the small person to school

laters

Slereah

later

Secret

in Logic, yesterday, by user21820

@Taroccoesbrocco: And I've in fact build my own alternative foundational system, which is most similar to a kind of type theory, but is based on Kleene's 3-valued logic instead of intuitionistic logic. I wrote a bit here to briefly describe the motivation. I understand the intuitionistic viewpoint, but I prefer to treat true and false on equal footing. =)

Slereah

I think Robert Anton Wilson had a weird 7-valued logic

though not a formal one

"Robert Anton Wilson in The New Inquisition developed a non-Aristotelian system of classification in which propositions can be assigned one of 7 values: true, false, indeterminate, meaningless, self-referential, game rule, or strange loop. Wilson did not devise a formal system for manipulating propositions once classified, but suggested that we can clarify our thinking by not restricting ourselves to simplistic true/false binaries."

that's the one

Alessandro Codenotti

15:13

@Secret if you want to construct the Borel sets iteratively you can't stop at any countable ordinal but need $\omega_1$ iterations. Dunno if this counts aa natural

mickep

15:23

Hi, has anyone seen the improper Riemann integral $\int_0^{+\infty} \frac{\sin(\pi x)}{\ln x}\,dx$ calculated somewhere?

Slereah

15:34

Did you try Gradshteyn and Ryzhik

Vrouvrou

Hello, i have this function $\int_0^{|t|}\phi(s)ds$ where $\phi(t)=0$ if and only if $t=0$ , if i suppose that $\int_0^{|t|}\phi(s)ds=0$ how i can prove that t=0 ?

Slereah

Is it continuous?

Vrouvrou

yes

$\phi$ is continuous

Silent

Please reply:

53 mins ago, by Silent

Does $a_n=1+1/2+\dots+1/n$ fit for requirements here?

I mean, is this sequence a good counterexample?

Vrouvrou

15:51

@Slereah

i just say by continuity of $\phi$ ?

i have also that $\phi(t)>0, t>0$ and $\phi$ is invreasing

Alessandro Codenotti

@Silent isn't that the "standard example" from the body of the question?

Silent

omg! so sorry!

Tuki

Is there any analytical method which could tell me the largest decreasing direction if i have function $f(x,y)=2x^2+3y^2+4\sin(x)\sin(y)$

gradient would give the largest increasing direction

$-\nabla f(x,y)$ is also increasing direction in this ?

at point 0,0 i mean

Slereah

16:12

@Vrouvrou Then either $\phi > 0$ or $\phi < 0$ on $(0,t)$

Vrouvrou

?

Slereah

Because $\phi \neq 0$ on that interval

and it is continuous

To switch from positive to negative it would have to cross $0$

Vrouvrou

$\phi(t)>0$ if $t>0$

then on (0,t) $\phi(t)>0$

@Slereah

Slereah

Since $\phi(t)$ is $>0$ on $(0,t)$, pick any point $t' \in (0,t)$, then $$\int_0^t \phi(s) ds = \int_0^{t'} \phi(s) ds + \int_{t'}^s \phi(s) ds \geq \int_0^{t'} \phi(s) ds + \phi(t') [t - t'] > 0$$

(That's assuming $t \neq 0$, of course)

Semiclassical

warning, use of $s$ as both a dependent and an independent variable detected :P

Slereah

16:26

Hence contradiction

Oops

it was supposed to be $t$

Semiclassical

ah, i get you

nitsua60

@Slereah do you want it edited?

(we're talking about the upper limit on the third integral, yes?)

Slereah

It wil do

Silent

@AkivaWeinberger, $1/n$ converges, and difference of each successive term is $\frac 1{n(n+1)}$, while $1+1/2+\dots+1/n$ diverges and difference between each successive term is $1/n$. Does 'more' difference in second sequence has to do something with divergence?

Akiva Weinberger

I guess yeah, in the sense of the difference test, if $a_n$ and $b_n$ are positive but $\sum a_n$ diverges and $\sum b_n$ converges then $a_n>b_n$ for large enough $n$

(Here $a_n=\frac1n$ and $b_n=\frac1{n(n+1)}$)

I don't know if you want something deeper than that, though

Silent

16:43

Thank you!

Xander Henderson

ugh! The proof I wrote two years ago is not bullshit; there is a very clever multiplication by 1 that I did not make explicit, and which is non-obvious in retrospect

YAY!

WHY DID IT TAKE ME TWO DAYS TO FIGURE THIS OUT?!

I've made it explicit now. $\delta^{Q-Q} = 1$. Don't forget that.

Nitz

16:58

Hey! Can anyone help me with an integral problem.How can I find the volume below a paraboloid and above the triangle on xy plane?

Slereah

Substraction?

Trey

how is $\frac{sin 2u}{2} = sin(u)cos(u)$ ?

Xander Henderson

@Trey Magick.

$$\sin(a+b) = \cos(a)\sin(b) + \sin(a)\cos(b) \implies \sin(2u) = \sin(u+u) = \cos(u)\sin(u) + \sin(u)\cos(u) = 2\cos(u)\sin(u). $$

Semiclassical

there's a bunch of proofs of the double angle identity

so it really depends on what you take as your starting point

Xander Henderson

it is really just an application of the angle addition formula for sine, which follows directly from the angle addition formula for cosine, which is DEEP MAGICK

or, as @Semiclassical you can find another proof that you like better ;)

Trey

17:10

@XanderHenderson now I see it. thank you!

Semiclassical

okay, this is a funny failed mathematica animation

my eyes!

Abcd

@Semiclassical is there any proof through euler's formula.

Semiclassical

sure

that's a sensible one

Xander Henderson

@Abcd YES! And it is lovely!

Abcd

@XanderHenderson please show.

Xander Henderson

17:13

urg... let me see if I can remember how it goes.

Semiclassical

start from the right-hand side. then write down the euler's formula expressions for sine and cosine, multiply it out, and then express in terms of sine again

Abcd

$\sin2\theta= e^{i\theta^2}$, as a starting point?

Semiclassical

no.

Leaky Nun

@Abcd no

Abcd

Then $e^{i\theta}= cos\theta + i\sin\theta$

Semiclassical

17:16

Yes.

Xander Henderson

$$\sin(2u) = \frac{\mathrm{e}^{i2u} - \mathrm{e}^{-i2u}}{2i}$$

Abcd

I think we'll have to use De MOivre's theorem too.

@XanderHenderson Oh, I wanted to try too...

Let me continue without seeing yours.

Xander Henderson

factor the top as a difference of squares, then simplify, using the same idea for cosine

heh

Abcd

$\sin 2\theta = (\cos 2\theta + i\sin 2\theta)$

Xander Henderson

that doesn't look right... or am I being dense?

$\mathrm{e}^{i\theta} = \cos(\theta) + i\sin(\theta)$; that's what you are trying to use, no?

Abcd

17:19

@XanderHenderson I am trying to use $e^{i\theta^2}$

Xander Henderson

so whence $\sin(2\theta)$ on the left?

also, do you mean $(\mathrm{e}^{i\theta})^2$, or do you mean $\mathrm{e}^{(i\theta^2)}$?

your notation is ambiguous

and exponentiation is not generally associateive

Abcd

@XanderHenderson 1st one

Slereah

Oh wait

My integral proof was wrong

You should use the inf of $\phi$, not $\phi(t')$

Abcd

$\implies \sin2 \theta (1-i)= \cos 2\theta$

Slereah

It exists because continuous on an interval

Abcd

17:22

$\implies \sin 2\theta = \dfrac{\cos2\theta}{1-i} $

Now we should write both numerator and denominator in euler's form, I think.

$\implies \sin 2\theta = \dfrac{\cos 2\theta}{(e)^{-1}}$

How to write $\cos 2\theta$ in euler form?

Lol, is this even true: $\sin 2\theta = e \cos 2\theta$

18

A: Reasoning that $ \sin2x=2 \sin x \cos x$

$$\sin(2x)=\mathrm{Im}(e^{2ix})=\mathrm{Im}(e^{ix}e^{ix})=\mathrm{Im}((\cos x+i\sin x)(\cos x+i\sin x))=2\sin x\cos x$$

Best^!

17 mins ago, by Abcd

$\sin2\theta= e^{i\theta^2}$, as a starting point?

My this assumption was wrong^

Trey

is it possible to simplify $sin(arctan(x))$ ?

Abcd

@Trey You might like that proof^

Peter Sheldrick

17:39

wow this chat is now buried even more

i'm looking for more info on the Transform $T$ that appears at the top of this question

did any of you see that before?

Xander Henderson

Poor Bouligand. He introduced two notions of dimension in a single 1928 paper: the Minkowski dimension, and the Assouad dimension.

He gets his name on neither. :(

Abcd

Is it possible to find the function whose graph is like this^?

All horizontal lengths= all vertical lengths = 1 unit.

Xander Henderson

That isn't the graph of a function; it fails the vertical line test.

But if you assume that the vertical segments are "jumps", then that looks like the graph of the greatest integer function, i.e. $x \mapsto \lfloor x \rfloor$

Peter Sheldrick

floor/ceil?

Xander Henderson

Does anyone here read French and want to translate a couple of papers for me, gratis?

Anyone?

Peter Sheldrick

17:46

try google translate

Xander Henderson

Bueller?

Abcd

@XanderHenderson How would you represent the greatest integer function? I mean what is its value?

Xander Henderson

@PeterSheldrick That would be neat, if the papers were in a machine readable format

all I have a low quality scans :(

Peter Sheldrick

ohhh

wasnt thinking of that

Abcd

like $floor(x) = ....$

Xander Henderson

17:47

@Abcd USE TEH GOOGLE! google.com/…

Abcd

Hmm, okay. I had a parabola and the greatest integer function. I was trying to find their intersection. Seems like it is not possible.

Xander Henderson

@Abcd Finding the intersection of a parabola and the GIF is a problem in Diophantine equations.

Those are, typically, quite hard.

"Ensembles impropres et nombre dimensionnel..." I'm just going to assume that those are all cognates, and that French grammars is exactly the same as English grammars

Abcd

Oh, interesting. Anyway, it was a physics problem. I should rather discard this approach and think of other methods to solve my problem.

Xander Henderson

ooo... it looks like Bouligand uses mother f'n' mathfrak for content... maybe I'll do that for the Minkowski content from now on... how does $\mathfrak{M}_{\ast}^s(E)$ look?

hrm... no, I don't like that

nevermind

"Une limite supérieure de l'approximation sera $$\frac{\sqrt{3}}{N\partial^2} \mathfrak{M} $$ en appelant $\mathfrak{M}$ la masse totale." == "An upper limit from the approximation is given by $$\frac{\sqrt{3}}{N\partial^2} \mathfrak{M} $$ where $\mathfrak{M}$ is the total mass." (Again, I'm just going to assume that mathematical French is really just English with a funny accent.)

♦ $Mathematics$ Logic

Transcript for

$Mathematics$ Mathematics

♦ Logic

Transcript for

♦ $Mathematics$ Logic

$Mathematics$ Mathematics