Mathematics - 2014-09-21 (page 2 of 4)

Mats Granvik

8:06 AM

It should have implications for this sequence: oeis.org/A100995

Analysis

I don't understand Big $\mathcal{O}$ notation myself, could anyone put it in layman words?

Huy

@Analysis: What is your background?

Analysis

@Huy Weak optimization and calculus

@Huy I have seen it, and it seems to be used for terms that are sufficiently irrelevant

Huy

@Analysis: You should be able to understand the given wikipedia entry if you've seen a bit of calculus.

Analysis

Is it essentially a weak variation of another function?

E.g. the function is essentially convergent to another function?

Huy

8:08 AM

I don't understand your question.

Mats Granvik

@Analysis Big O means that there exists a constant "c" such that the absolute value of your function is bounded by "c" times the expression within in the Big O parentheses.

Or at least that is what I believe it means, correct me if I am wrong, someone.

Huy

@MatsGranvik: $\frac{2}{x} \in \mathcal{O}(1)$ as $x \to \infty$, can you find $c \in \mathbb{R}$ such that $|\frac{2}{x}| < c$ for all $x \in \mathbb{R}$?

Mats Granvik

@Huy I am not competent to answer.

Huy

Maybe you could ask Mathematica. ;-)

Analysis

If $x\to \infty$ then any $c$ will do

Oh you said for all $x$ there nevermind

Then no

Huy

8:18 AM

@Analysis: My point is that it doesn't make sense using Landau symbols without saying something along the lines of "as $x \to x_0$" (where $x_0 \in \mathbb{R} \cup \{\pm \infty\}$). Unfortunately @MatsGranvik did forget about that necessary part.

Morning, @Khallil.

Khallil

Good morning, @Huy!

Huy

@Khallil: How are you?

Analysis

So with $e^x = 1 + x + \frac{x^2}{2} + \mathcal{O}(x^3)$ for $x \to 0$, was $\mathcal{O}(x^3)$ the chosen stopping point because the difference in actual value and approximation is less than $x^3$ contributes as $x\to0$?

Khallil

I'm ok. I just woke up. The time is 9:23 in the morning and the weather is nice and sunny. How are you, @Huy?

Huy

@Analysis: Probably. Always depends on context.

@Khallil: I've been up for some hours and it's sunny over here as well.

@Khallil: Maybe you can help me here? matheducators.stackexchange.com/questions/4438/…

Khallil

8:25 AM

I won't be able to help out now as I'm quite busy, @Huy.
However, I will get to answering or leaving a comment on your question when I get the chance!

Huy

Okeydokey.

Kainui

Why don't they teach complex numbers earlier anyway

I don't get why they avoid such a simple and useful concept so long

Huy

@Kainui: Because it is not at all useful before.

Kainui

Rotation in a plane?

usukidoll

omg

kai what theeee

Kainui

8:30 AM

hey

usukidoll

yeah wanna look at my question? ^^

Kainui

Alright send it my way

Huy

@Kainui: Can you explain to me the application of a rotation in a plane for a kid in high school at age 15/16?

usukidoll

math.stackexchange.com/questions/939955/… @kainui

I can't believe you're in STACKS too what the... no wonder you didn't talk that much on skype -_-

Kainui

@Huy can you explain the application of mathematics for ANY kid at the age of 15/16?

Who cares if it's useful?

usukidoll

8:32 AM

I have my characteristic lines right... change of variables.. taking partial derivatives and now I'm stuck... super

Kainui

It's just fun and straightforward, but it is useful

Huy

@Kainui: It is useful to know how much you're gonna have to pay - roughly - at the checkout, e.g.

usukidoll

also... $u_x-u_y-u = z$ like do you use change of variables on the z or what

Huy

@Kainui: I think it's the wrong approach to teach something where kids will completely fail to see literally ANY usefulness in their life.

Kainui

This looks separable @usu

usukidoll

8:34 AM

which one

because my book made these problems easy but what I'm assigned is coo coo

Kainui

@Huy I thought you were talking about teaching Quantum Mechanics though?

Huy

@Kainui: Um, not really, no.

usukidoll

I don't get it... the examples made everything easy..... now I'm stuck?!

Kainui

lol ok

Huy

@Kainui: I was just telling you that I don't think any kid would appreciate complex numbers if they were taught earlier.

Kainui

8:35 AM

What? I don't see why not.

Huy

(because you were saying you wish they did teach it earlier)

usukidoll

complex numbers... college algebra learned it in my community college

wasn't too bad...

Kainui

It's incredibly useful because all movement is N/S/E/W

Huy

@Kainui: you failed to give me any actual example of an application.

Kainui

ok

later I gotta go

Huy

8:36 AM

(for that age group)

great argument ;-)

usukidoll

<_<

pft ignoring my question huh

Analysis

@Huy Teach them propositional logic,modulo arithmetic and proofs?

Huy

@Analysis: While I do realise they would probably be able to understand it, I doubt they will care about it.

Analysis

Elementary set theory - Elementary graph theory - Relations & functions - Induction & recursive definitions. I found them all very enjoyable

usukidoll

I prefer number theory over this pos pde book

Analysis

8:42 AM

If this is an optional course, I think they will care regardless of what you give. Or are you trying to sell the idea to them?

Huy

@Analysis: There are other optional courses and I doubt they will take mine if it doesn't appeal to them.

Analysis

So you are trying to sell it. How are you selling it? Will it be written in English on a page, or will there be a poster with graphics?

If poster - Graph theory

Huy

@Analysis: I'm not sure. I think just a short description.

No pictures.

Chris's sis

Here is a question to upvote :-)

0

Q: Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get such a form because of $\cos(2x)$. What other options do I have?

lol, who downvoted me? :-)

usukidoll

omg

math.stackexchange.com/questions/939955/…... anyone??? >__<

or at least tell me how to input a pde in mathematica

mirgee

9:02 AM

Hi guys! Any ideas on how to find $\int_0^{\pi} \frac{\sin((n+1/2)x)}{\sin(x/2)} dx$, please? :)

Khallil

Have a go at using the addition formula for $\sin(a+b)$ in the numerator, @mirgee.

mirgee

@Khallil I have tried that to no avail, but It seems to be $\pi$ $\forall n \in \mathbb N$ so maybe induction?

But it doesn't work for me neither :/

Khallil

@Mirgee, did you manage to reduce it down to the following form as well? $$ \int_{0}^{\pi} \dfrac{\sin \left( n+\frac{1}{2} \right)x}{\sin \left( \frac{x}{2} \right)} \text{ d}x = \int_{0}^{\pi} \dfrac{\sin(nx)}{\tan \left( \frac{x}{2} \right)} \text{ d}x + \underbrace{\int_{0}^{\pi} \cos (nx) \text{ d}x}_{= \ 0} $$

mirgee

@Khallil Yes, precisely

This integral is not easier then the first one

Chris's sis

9:18 AM

This one is an elementary integral. What happens for $n=1$?

And if we consider $\displaystyle I_n=\int_{0}^{\pi} \dfrac{\sin \left( n+\frac{1}{2} \right)x}{\sin \left( \frac{x}{2} \right)} \text{ d}x $ then what is $$I_n-I_{n-1}$$?

Some easy calculations show that $I_n-I_{n-1} =0 \Rightarrow I_n=I_{n-1} $ and since $I_1=\pi$, we have that $$I_1=I_2= ...=I_n=\pi$$

mirgee

@Khallil Hm, I thought induction might work on this one, probably not

Khallil

I was thinking of using complex numbers, @Mirgee.

mirgee

@Chris'ssis There are some recursive formulas for $\sin(nx)$, maybe I should look at those?

Analysis

Is there an easy way to $\frac{1}{\pi}\left[\int_{-\pi}^\pi e^x \cos(nx) dx\right]$?

Khallil

That one can be done with complex numbers, @Analysis.

Consider the real part of $\displaystyle \int_{-\pi}^{\pi} e^{x} e^{inx} \text{ d}x$.

Chris's sis

9:29 AM

@mirgee There you need nothing special, but only to proceed as I did. For these integrals it's good to check the behaviour of the first few terms. Then you have an idea about the way you should tackle it.

Analysis

Must it be done using complex numbers?

Khallil

It can done by integration by parts as well, @Analysis. You'll need to do it twice if I recall correctly.

^_^

Analysis

And notice that we have the original integral within it, and then bring that over to get $2I$ and divide etc?

Huy

^_^

Analysis

I did that but got it slightly wrong, and didn't want to do it again xD

Khallil

9:31 AM

Something like that, @Analysis!

mirgee

@Chris'ssis I see, nice. I wouldn't think of the $I_n - I_{n-1}$ trick, I haven't seen it before. Thanks :)

Khallil

Let me have a go at it as well.

Analysis

I got a page of garbage now :(

Khallil

I'll let you know if I have any more success, @Analysis.

Analysis

Okay thank you @Kha

Chris's sis

9:33 AM

@mirgee It's just a matter of practice.

Khallil

Imagine if I shortened your username to it's first four letters, @Analysis!

mirgee

@Chris'ssis When they taught us integration at school, they said there are about seven ways to approach any integral, but that is false :)

Chris's sis

@mirgee :-)

Ice Boy

^_^
^_^

Huy

@Khallil: That would be ANAL!

Ice Boy

9:37 AM

:-O

O-:

Analysis

@Khallil Yes, I realised that earlier today xD. I typed A N A LYS I S, as individual posts, with the LYS joined intentionally to avoid it.

Also, what is that gif from, I must know

Khallil

Apparently studying Analysis causes anal lysis because you break your butt trying to learn it!

Huy

@Khallil: You laugh now but you'll soon find out it's true.

Analysis

What is the gif from nooo

Khallil

I've heard a lot of that, @Huy. Don't scare me!

That gif is from an interview that I can't remember. It's Michael Jordan though, @Analysis.

Huy

9:47 AM

@Khallil: It actually won't. Only if you want it to. :> :> :>

@Khallil: Did you already find out which courses you'll be taking?

Khallil

I also managed to finish off that integral, @Analysis. $$ \dfrac{1}{\pi} \int_{-\pi}^{\pi} e^{x} \cos (nx) \text{ d}x = \dfrac{\cos (n\pi)}{\pi (1+n^2)} \left( e^{\pi} - \frac{1}{e^{\pi}} \right) $$

I know of the core modules for my first semester, but I still haven't looked at the options, @Huy.

Huy

@Khallil: What are your core modules?

Analysis

@Kha That is almost exactly what I got hmmm. Mine had a $\cos(-n\pi)$

Khallil

Differential Equations, Analysis (continues into Term 2), Foundations and Abstract Algebra, @Huy.

Huy

@Analysis: $\cos x = cos -x$.

Khallil

9:50 AM

Remember that $\cos (-\theta) = \cos(\theta)$, @Analysis.

Analysis

Oh my god

...

Dear lord.

Cya

Hahahaha!

@Khallil: Weird stuff. :P

Khallil

9:51 AM

Noooo!

Huy

@Analysis: Bye.

Khallil

Good stuff, @Huy!

Huy

@Khallil: I beg to differ. ^w^

I find it interesting though how differently different universities teach maths.

Khallil

Me too. Everywhere's different and that makes things more interesting.

There wouldn't be any fun if all universities studied A, B and C in that exact order!

Huy

@Khallil: It seems in the US there is a rough standard curriculum though.

Khallil

9:54 AM

Really? I'm guessing Analysis comes first then.

Huy

@Khallil: I think they have something like Calc 1-4 or 1-3, no idea what they learn there.

And precalc or so.

Very weird system.

Khallil

At least you'll be able to know what you'll be doing after your final year of high school and will be able to prepare accordingly, unlike the rest of the world.

Huy

@Khallil: Well, we can just check online what we'll be doing. Isn't that the case for most universities? xD

And who on earth prepares for university. .______.'

Khallil

Yep, but you won't know where you'll be going until you get your results about a month before getting to university.

Huy

@Khallil: Huh?

Khallil

9:58 AM

I'm certainly not. I've been watching The O.C. this whole time.

Huy

@Khallil: And why would that be the case?

Khallil

In the UK, you apply to university in the October of your final year of high school with your penultimate year results and your predicted grades. You then get conditional offers based on your final exams that you'll do. Then you do your final exams in the summer, get your results in August and go off to the university you make it into in September/October.

It's even weirder over here in the UK, @Huy.

Huy

@Khallil: I see. I don't think that applies for all of Europe though. In Switzerland, graduating from high school will suffice in order to study at any arbitrary university, independant of your final grades.

Khallil

Yep, that whole paragraph only applies to the UK as far as I can remember, @Huy.

Huy

@Khallil: I know in Germany it also depends on the final grades whether you will be accepted or not.

Khallil

10:01 AM

I like the system in Switzerland. It's much nicer than having to worry about your final grades right before you get into university.

Huy

I prefer our system.

Khallil

Me too. ^_^

Huy

@Khallil: It can only be realised by trying to have a certain "skill standard" after high school graduation.

Apparently, in other countries such as Germany, skill levels vary extremly from high school to high school. :<

Khallil

Which is easier said than done in a country like the UK that makes the exams easier to make it seem as if everybody's doing better.

Huy

@Khallil: That doesn't sound like a very good educational system. :(

Khallil

10:03 AM

It really isn't. University is completely different though. ^_^

Huy

Yeah, I hope so.

Analysis

$=2\left[\left[\frac{-2x^2 \cos(2n\pi x)}{2n\pi}\right]_{-1}^0 - \int_{-1}^0 \frac{-2x\cos(2n\pi x)}{n\pi} \, \mathrm{d} x\right]$

How do I make the $\int$ bigger?

Khallil

\displaystyle

Huy

$\bigint$ ?

Analysis

Huy that didn't work?

Huy

10:06 AM

I think it requires a package.

Analysis

Does display style need a package?

Khallil

I think you mean \Big\int, @Huy. Anywho, \displaystyle is better!

Nope.

Analysis

$\big\int_{-1}^0$

Huy

There's a bigints package, iirc.

Analysis

How to use display style?

Khallil

10:07 AM

Just type it in before your code, @Analysis.

Like this: \displaystyle \int.

Analysis

Thx

Khallil

^_^

Analysis

:) : $=2\left[\left[\frac{-2x^2 \cos(2n\pi x)}{2n\pi}\right]_{-1}^0 - \displaystyle \int_{-1}^0 \frac{-2x\cos(2n\pi x)}{n\pi} \, \mathrm{d} x\right]$

Very pretty

Huy

@Khallil: I just checked online and it seems in Germany, almost 50% graduate from high school. Do you know the numbers for the UK?

Khallil

Not a clue. At which age do the students graduate from high school, @Huy?

Huy

10:11 AM

@Khallil: Typically 17/18, I think.

Khallil

Ok, I'll check out the figures for A-Levels.

Huy

What do you mean by figures for A-levels?

Khallil

Do you know what A-Levels are, @Huy?

Huy

I'm not sure I do.

Khallil

When you're 16, you get to choose 3-5 subjects to continue with for the next 2 years until you're 18. Universities use the grades you get in A-Levels as a condition for getting in to their undergraduate programs.

Huy

10:14 AM

Okay. And what about A-levels?

Those are A-Levels!

Those 3-5 subjects?

Yep!

I see.

I can't seem to find a figure for the no. or percentage of students that finish their A-Levels, @Huy.

Huy

10:16 AM

=_=

In the United Kingdom, the average household net-adjusted disposable income per capita is 25 828 USD a year

:o

Khallil

Is that low?

It is, isn't it.

Huy

Apparently it's over average, but it seems ridiculously low. :<

@Khallil: Isn't rent in e.g. London incredibly expensive?

Khallil

It is.

Huy

Like 600£ per month for a 1-room apartment?

Khallil

Elsewhere it's a fair bit cheaper.

I don't know the figures, but that seems like a reasonable guess.

Huy

10:21 AM

And groceries etc. are kind of expensive too, there.

Khallil

Almost everything is more expensive in London.

Huy

Yeah.

Analysis

Do I need a package to strike through things?

Huy

@Analysis: I think ulem or soul.

Khallil

\usepackage{cancel}

Huy

10:23 AM

Oh yeah, that would work too.

Analysis

Do you use TeXworks?

Huy

I don't, if you asked me.

Analysis

Asked you both :)

Huy

I prefer TeXnicCenter.

Khallil

I use TeXShop.

Analysis

10:25 AM

I'll look into both, thanks

Khallil

TeXShop is on Mac as far as I can recall, @Analysis. ^_^

Analysis

Are you two around here every night?

everyday at this time* (It's 8:30PM here - Australia)

Huy

Not really. Especially not during week.

Analysis

I'll miss you two then $:'$(

Huy

Don't worry. I'll be back.

I'm currently preparing my first exercise class in linear algebra for freshmen. :3

Analysis

10:32 AM

Was that the optional one?

Also where do freshman sit? What age?

Freshman -> sophmore -> senior?

Huy

@Analysis: I usually call first year undergrads freshmen.

Isn't that the common notion?

@Analysis: No it's not optional, it's a compulsory course for all first year undergrads in maths/physics.

Analysis

In Australia, high school is Grade 8,9,10,11,12 and university, first year, second year, third year, Honours year

Huy

@Khallil: The weather is phenomenal today. We went to play football yesterday because we thought it would be raining today. And now this. i.imgur.com/maZFMR2.jpg

Analysis

@Huy that is beautiful, did you take that?

Huy

@Analysis: In Switzerland, you have 6 years of high school and afterwards 3 or 4.5 years of university, 3 for your BSc and another 1.5 for your MSc, ideally.

Khallil

10:36 AM

If only I had such a fantastic view to take a photo of, @Huy!

Huy

@Analysis: yes, that's from my windows next to my PC. :P

@Khallil: I just want to play football!

Khallil

I've got football tonight. It's 7 a side and I don't really want to go because I feel tired. T_T

Huy

I'll go instead!

Khallil

I'm not usually on at this time during the week either, @Analysis.

I just derived a really strange result. Can you guys confirm it's true?

Huy

I think it's false, upon observations so far.

Khallil

10:40 AM

Huy

Again?

Khallil

Yep, I lost my paper from last time.

For part b, I got $\alpha^2 + \beta^2 + \gamma^2 = -12$.

$\begin{aligned} x^3 + 2x^2 + 8x - 5 & = (x-\alpha)(x-\beta)(x-\gamma) \\ & = x^3 - (\alpha + \beta + \gamma)x^2 + (\alpha\beta + \alpha\gamma + \beta\gamma)x - \alpha\beta\gamma \end{aligned}$

Comparing the coefficients of the LHS and RHS, we get the following
$(1) \qquad -2 = \alpha + \beta + \gamma$
$(2) \qquad 8 = \alpha\beta + \alpha\gamma + \beta\gamma$
$(3) \qquad 5 = \alpha\beta\gamma$

Huy

Yeah, should be correct, then.

Why is that weird?

Khallil

I think I assumed that all of the roots are real.

Huy

@Khallil: Why would you ever assume that? :D

Khallil

10:47 AM

Not a clue! =P

Huy

Does anyone know what to call symbols such as $\implies, \iff, \neg$? I know $\forall$ and $\exists$ are called quantors, but what about the others?

Khallil

Logical symbols?

Huy

That would be a way, thanks.

Khallil

List of logic symbols

In logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, the following table lists many common symbols together with their name, pronunciation, and the related field of mathematics. Additionally, the third column contains an informal definition, and the fourth column gives a short example. Be aware that, outside of logic, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings. == Basic logic symbols... ==

^_^

I thought $\forall$ and $\exists$ were called quantifiers, @Huy.

(At least, that's what they are referred to in Liebeck's A Concise Intro to Pure Math.)

Huy

@Khallil: Yeah, in German we call them "Quantor" and I thought it'd be the same translated.

but apparently the correct translation is quantifier.

Analysis

11:12 AM

¬ is negation, $\Longrightarrow$ conditional, and yes we call them quantifiers

$\Longleftrightarrow$ Biconditional

Khallil

11:26 AM

Cool!

I got $x^3 - 4x^2 - 40x - 281$ as my final cubic equation. Is there any way I can check it, @Huy and @Analysis?

Huy

@Khallil: I can check later if you want, right now I'm a bit busy.

Khallil

Ok. Whenever you get time!

Analysis

Sorry Kha, I am just finishing off some Fourier thing that is taking forever :\

Mainly typing it

Khallil

Ah, that integral did look very familiar!

Parth Kohli

11:49 AM

Pardon my ignorance in this matter, but what does OP mean by |...| here?

A layman would take those as absolute value bars, and that statement which OP asks to prove is actually true, but they also ask about "norms" and "inner products".

Huy

@ParthKohli: The absolute value is a norm, and some norms are induced by an inner product.

Parth Kohli

So should I undelete this proof?

mathh

Hey everyone. Is it true that the points $A(3;-1;2), B(1;2;-1),C(-1;t;-3)$ are never collinear with any value of $t$? It seems to me so, since $\frac{1-3}{-1-1}=1$, but $\frac{-1-2}{-3+1}=-1.5\neq 1$.

Parth Kohli

I answered it with the perspective of a usual algebra student, not a real-analysis one, so hmm.

Analysis

12:49 PM

$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$ What to do?

Khallil

1:22 PM

What's the question, @Analysis? Is it just to find a simpler form of $f(x)$?

Analysis

Originally that was what I wanted to do, but I realise now that it is in its simplest form. The following question is "Using the fact that $\sinh(x)$ and $\cosh(x)$ are the odd and even components of $e^x$, write down the fourier series of $f(x)=\sinh(x)$ and $f(x)=\cosh(x)$

Should I be noticing some simple way of doing this from the above expression?

Khallil

I have hardly any experience with Fourier series, @Analysis. Sorry!

Analysis

That's alright xD @Kha

Khallil

If you don't mind, I have a very simple differentiation question, @Analysis.

Analysis

Sure

Khallil

1:29 PM

Would this solution be ok?

I didn't want to sketch the graph, so I just tried to describe everything I could.

Should I have said that $f$ is continuous over $\mathbb{R}^2$ instead?

Also, does everything seem water tight, @Analysis?

Analysis

You will lose marks if you don't graph it, and yeah $\mathbb{R}^2$ would be better.

Looks good!

@Kha Lose marks if it is an assignment that is

Khallil

Yep, it's just a pdf I picked up from the net.
Thanks, @Analysis.

^_^

Analysis

Good to hear xD

Chris's sis

Let me share some crazy stuff with the world

Analysis

Okay

Khallil

1:40 PM

Blow. Our. Minds.

$1=0$

Spagett

=P

xD

1:44 PM

@robjohn see above :-)

Khallil

What was that final $F_3$, @Chris'ssis?

It's argument looked complicated.

Chris's sis

@Khallil the hypergeometric function

Khallil

It looked really cool!

Chris's sis

@Khallil Thanks! That one is pretty hard to derive. It's a crazy integral from the same class with this one

2

Q: Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get such a form because of $\cos(2x)$. What other options do I have?

calculus real-analysis integration definite-integrals

Analysis

I must sleep now. Goodnight everyone, I hope to talk another time @Khallil

Chris's sis

2:00 PM

I need to check that with Mathematica, some signs might be missed. Just to be sure.

Chris's sis

2:13 PM

Indeed, there was a small problem that is now fixed.

Khallil

2:46 PM

See ya later, @Analysis. ^_^

Maths Lover

3:32 PM

Hello, Does any one know how to write ( A bar ) using LaTeX ?

Huy

@MathsLover: \bar{A}

Maths Lover

@Huy, Thank you very much :)

Balarka Sen

3:52 PM

Anyone familiar with how to underscore in LaTeX?

Daniel Fischer

Why would you want to do that @BalarkaSen?

Balarka Sen

Oh nevermind $\text{_}$ works well.

Daniel Fischer

And in text mode, or math mode?

Balarka Sen

@DanielFischer $\mathcal{Hom}_A(G, \text{__})$, that's why.

Daniel Fischer

Ah, a place-holder.

Also, $\operatorname{Hom}_A(G,\underline{\hphantom{M}})$.

Balarka Sen

3:57 PM

that'd do too. thanks.

Khallil

4:11 PM

When considering an or statement in math like '... if $m>0$ or $0>M$', do you consider both cases at the same time?

Alizter

@Anastasiya Ignore me I am being an idiot today

Alizter

4:49 PM

@BalarkaSen

4

Q: The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the n th square number - it is n^2 - but we do not have an exact formula for the n th prime number p_n ! God may not play dice with the universe, but something strange is going on with the prime numbers. (Paul Erdos, 191...