Mathematics

2:02 PM

I have to solve this integral $$ \int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+\pi/3\right) +4 \sin^2 \left(x+\pi/3\right)}$$ I could see that the denominator is can be written as $ \left(1+ 2\sin \left(x+\pi/3)\right \right) ^2$

I can see that the denominator can be written as $ \left(1+2\sin\left(x+\pi/3 \right)\right)^2$

writing $2\sin \left(x+\pi/3)\right)$ as $\sin x + \sqrt 3 \cos x$

Abhigyan Chattopadhyay

@adeshmishra Are you sure of the numerator?

adesh mishra

yeah

Martin Sleziak

$\cos x + \sqrt 3\sin x%$ in the numerator would make it easier, I think.

adesh mishra

Can we write it that way?

Abhigyan Chattopadhyay

@MartinSleziak You mean $\cos x - \sqrt{3} \sin x$ right?

Martin Sleziak

2:13 PM

Well, Abhigyan Chattopadhyay already asked whether the integral is copied correctly.

adesh mishra

Yes

Martin Sleziak

Yes, I'd have to put minus there to get $\cos(x+\frac\pi3)$.

adesh mishra

@MartinSleziak How did you get $\cos \left(x+\pi/3 \right)$

?

Abhigyan Chattopadhyay

Set the denominator to $t^2$, and the numerator to $At'+Bt+C$

Martin Sleziak

This is what WolframAlpha says

adesh mishra

2:20 PM

Well, finally our integral reduces to $$ \int \frac{\cos x+ \sqrt 3}{\left( 1+\sin x+ \sqrt 3 \cos x \right)^2}$$

Thank you for pointing that out

Abhigyan Chattopadhyay

Right, now set $(1+\sin x + \sqrt{3} \cos x)$ as $t$

And, assume the numerator as $At'+Bt+C$

It'll work out

adesh mishra

It's not coming, $$ u = 1+ \sin x + \sqrt 3 \cos x \\ du/dx = \cos x - \sqrt 3 \sin x$$

Martin Sleziak

I am not really sure whether some of those four options is a correct result.

Anyway, we have $\int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+\pi/3\right) +4 \sin^2 \left(x+\pi/3\right)}>0$ for $x\in(0,\pi/2)$.

adesh mishra

Do you also make mistakes?

Martin Sleziak

So whatever the result is, it should be increasing on that interval.

I should probably go, have a nice day!

adesh mishra

2:33 PM

Have a nice day Sir

The Terrible Puddle

3:18 PM

Greetings

The Terrible Puddle

3:45 PM

Would someone be so kind as to ask how this guy has shown $f_N$ to be extreme points of $K$? https://math.stackexchange.com/questions/3505781/finding-extreme-points-of-closure-of-convex-hull
I don't yet have the reputation to make a comment.

The Terrible Puddle

4:06 PM

Oh hey, can do it myself now. Thanks for the rep

MadSpaceMemer

4:50 PM

Good evening! I keep forgetting what the notation of the polynom $f(X)$ should mean. Is it saying that $f(X) := 0.x+1.x+0.x^2.....$ or is it just a symbol for a random polynom and it is unknown at which $x$ the highest power exists?

COTO

5:13 PM

Figured I'd ask this simple question in here rather than plugging up the board: Is there a standard infix operator notation for the diagonal concatenation of two square matrices? For example, $A \oplus B$? I've been told that $diag[A,B]$ is standard in some cases, but I'm hoping there's some standard that doesn't take up more space than A (+) B?

I've seen operators like [+] and [] in LaTeX. Any of them stand for diagonal concatenation?

Alessandro Codenotti

5:24 PM

What do you mean with diagonal concatenation?

ÍgjøgnumMeg

@COTO do you mean something like $\begin{pmatrix} A & 0 \\ 0 & B\end{pmatrix}$ ?

(block matrix)

Hi @ÍgjøgnumMeg

Hey @Alessandro

Exactly, Alessando.

woah

$A \oplus B$ is standard I think

Matrix addition

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum. == Entrywise sum == Two matrices must have an equal number of rows and columns to be added. The sum of two matrices A and B will be a matrix which has the same number of rows and columns as do A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B:...

Alessandro Codenotti

5:26 PM

That's exactly the matrix representing $A\oplus B$ on $V\oplus W$ so looks good to me

COTO

Ah. It is there on Wiki.

ÍgjøgnumMeg

Aye

COTO

OK. I'll have to convince my supervisor that I'm not just making crap up. :D

MadSpaceMemer

"@AlessandroCodenotti Hi allesandro, could you provide some assistance with a notation problem i am having_

Alessandro Codenotti

I don't know

You should just ask the question and maybe me or someone else will be able to help

(also you don't have to answer if you don't want to share personal informations, but are you German?)

MadSpaceMemer

5:30 PM

No but i do study in germany

@AlessandroCodenotti i have a function that is doing the following $g(f(X)) = f(X + 1)$ i am not sure what is happening here. I was googling what the notation $f(X)$ is (i know its a polynomial but i wasnt exactly sure how it looked like ) turns out that f(X) is just $a_o+a_1x+a_2x^2....$

COTO

Thanks all. Cheers.

ÍgjøgnumMeg

@MadSpaceMemer something important: a polynomial is a necessarily finite sum, and secondly, if your polynomial is called $f(X)$ then your variables should match, i.e. $f(X) = a_nX^n + \dots + a_1 X + a_0$

rather than $f(X) = a_nx^n + \dots + a_1 x + a_0 $

unless $X := X(x)$ but idk

MadSpaceMemer

No you are right. I wrote it badly.

So what would then $f(X+1)$ mean?

ÍgjøgnumMeg

$a_n(X+1)^n + \dots + a_1(X+1) + a_0 $

MyWrathAcademia

Who can answer this: Given a range of numbers in the range a to b all inclusive. Find the count of numbers divisible by n in this range? e.g. 1: 7 14 7 the answer is 1, e.g.2: 6 14 6 the answer is 2

MadSpaceMemer

5:38 PM

Ohh!!

MyWrathAcademia

I know that if n > b then no number in that range is divisible

but I'm trying to obtain a general formula that works for all cases

Semiclassical

That itself is a polynomial of the form $\sum_{k=0}^n b_k X^k$, of course.

with some linear transformation between the $a_k$'s and the $b_k$'s

ÍgjøgnumMeg

Aye

MadSpaceMemer

@ÍgjøgnumMeg Thank you! now I can solve the question <3

ÍgjøgnumMeg

Nooo worries

The Terrible Puddle

6:07 PM

Operator $M\in\mathcal{L}(l_2,l_2)=\{T:l_2\to l_2 \text{ continous linear functions}\}$ defined by $M(x)=(n^{-1/5}x_n)_{n\geq 1}$ for $x=(x_n)_{n\geq 1}\in l_2$. How would you justify $M$ being compact but not Hilbert-Schmidt?

Thorgott

This is very much pedantic, so feel free to ignore the following. If you have a ring $R$, you can construct the corresponding polynomial ring $R[X]$. Take a polynomial $P\in R[X]$; this can be written in the form $\sum_{k=0}^na_kX^k$ with $a_1,...,a_n\in R$.
The polynomial ring is characterized by a universal property, which allows you to evaluate a polynomial $P\in R[X]$ in an element $x\in S$ ($S$ being another ring) via a ring homomorphism $\varphi\colon R\rightarrow S$ (ommitting a small technical detail). This is given by applying $\varphi$ to the coefficients of $P$ and substituting $…

MyWrathAcademia

6:25 PM

hey @Thorgott do you have any idea how to approach the following problem:

Given a range of numbers in the range a to b all inclusive. Find the count of numbers divisible by n in this range? e.g. 1: 7 14 7 the answer is 1, e.g.2: 6 14 6 the answer is 2

The Terrible Puddle

Nvm, think I've an argument for it being compact. Just need to show it is not H-S now

Alessandro Codenotti

It's compact because it is a limit of finite rank operators

To show that it is not HS compute its norm

The Terrible Puddle

alrighty

Thorgott

@MyWrath what is $n$?

Alessandro Codenotti

The index

He's identifying $\ell^2$ with sequences $(x_n)_{n\in\Bbb N}$

MyWrathAcademia

6:34 PM

@Thorgott n is an integer number

Thorgott

your examples don't make sense without specifying what $n$ is in these examples

Alessandro Codenotti

It's a sequence @Thorgott. $M$ maps the sequence $(x_n)_{n\in\Bbb N}$ to the sequence $(n^{-1/5}x_n)_{n\in\Bbb N}$

The Terrible Puddle

I don't think you two are talking about the same problem

Thorgott

Yeah, I'm talking about Wraths question

Alessandro Codenotti

Oh lol my bad

I thought you were also answering to @TheTerriblePuddle

Sorry

MyWrathAcademia

6:37 PM

Oh I see, apologies. Let me fix it: e.g. 1: 7 14 7 where a is 7, b is 14 and n is 7 the answer is 1. Likewise for e.g.2: 6 14 6 where a is 6, b is 14 and n is 6 the answer is 2

respectively

Thorgott

I think there are two numbers divisible by 7 in the range from 7 to 14, namely 7 and 14

MyWrathAcademia

Oops, your'e right. Thanks.

Oh and by the way I keep seeing dollar signs. I'm assuming code or mathematical equations are not being displayed correctly. How do I get math to display correctly here

Thorgott

See tinyurl.com/cfqcvpc

MyWrathAcademia

Thanks again @Thorgott

Thorgott

As for your question, the answer should be $\max\left(0,\lfloor\frac{b}{n}\rfloor-\lceil\frac{a}{n}\rceil+1\right)$

MyWrathAcademia

6:44 PM

Thanks @Thorgott but I'm trying to install the MathJax renderer you linked because your code is difficult to parse

The Terrible Puddle

@AlessandroCodenotti You mean compute its Hilbert-Schmidt norm right? And if it is not finite it is not H-S

Alessandro Codenotti

yes

MyWrathAcademia

@Thorgott I can't get MathJax to install. After you add it as a bookmark how do you get it to run?

The Terrible Puddle

Maybe I should use that $\|M\|\leq \|M\|_{HS}$?

Thorgott

You click on the bookmark

MyWrathAcademia

6:51 PM

just to comfirm it's only start ChatJax that I need to install right, not render MathJax?

Thorgott

Yes

MyWrathAcademia

@Thorgott I had an add on that was blocking JavaScript, had to unblock it. Its working great and the equations look nice now

Alessandro Codenotti

@TheTerriblePuddle Just write down the series

Thorgott

Nice

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$Mathematics$ Mathematics