Hi @AlexW.

Hi @MikeMiller. How goes it?

Going alright. Putting off work I need to do for a little bit.

Yourself?

Not bad. Just counting down the days now, eager to be done.

How long?

I'm not super pleased with the latest weather change. It's back in the low 40s and rainy. I love to go for long walks in the sun, but it's just so unpleasant out again.

Two weeks from Thursday. :)

Ah... I like my weather to be just around 70 degrees. Luckily, that's what it is lately around here.

Yeah, that's a great spot. Looking forward to that next year. :)

That's not far out at all! What are your plans between the interim months (between job and the move)?

Haha, mainly math, math, and more math. Gotta get serious about prep, start moving through these books more quickly. I'll be home for a little, and probably go visit some family in various places as well. Try to get prepared and detox a bit at the same time.

I would have liked to maybe get a summer internship of some kind, but I thought that might be too exhausting. Plus, it would almost certainly interfere with preparing for the qualifying exams.

@Chris'ssis I will look at it. I see that one of my answers is pretty close to one of your answers posted a couple of years ago. I didn't see it until the question I answered was marked a duplicate.

I see. Yes, that's definitely true. I was lucky that I had a job I could work on now and then over the summer without too much interference with studying... and, as a bonus, it was math.

@Chris'ssis Have a good night.

Oh really? What were you doing, if you don't mind my asking?

Just research with one of the faculty members. It didn't pan out, unfortunately, but I learned some fun stuff.

Oh, that's nice. Were you switched over to topology at that point, or still doing number theoretic stuff?

I guess I was more interested in topology, but the faculty member who did topology was busy over the summer; I studied complex analysis stuff (harmonic measure, which is related to solving the Dirichlet problem over bounded complex domains, and vaguely related to that whole 'can you hear the shape of a drum?' business)

Ah, I see. So pretty far afield from either area. Sounds cool though!

Btw, dumb question if you don't mind: should I have heard something from the department since accepting?

I don't remwmber. Let me check.

@robjohn I slept a while and woke up again.

Oops, I got distracted.

Thanks! Lol, no worries.

@AlexWertheim The vice-chair (then Balmer) emailed me a generic welcome to UCLA letter; some emails about travel reimbursement; in May, they sent me my housing offer.

Stipends are disbursed in August.

Thanks @MikeMiller. Hmm. Do you think I should be concerned if I haven't received anything? I formally accepted the offer and emailed Maida, so I don't think I forgot to do anything...

No, I don't think so. If you're worried you should feel free to email either Maida or Martha.

I think I had similar concerns. I emailed Maida; I got an email that amounts to "I indeed see that you officially accepted UCLA offer on-line - so there is nothing else you need to do at this time. Once we have the bootcamp information available, we will e-mail to all incoming class."

Ok, that makes sense. Thanks for the info!

Oh, that's good to hear. I'll probably just confirm with her then, just to be sure.

I sent you an email.

Got it, thanks!

Hello @MikeMiller. Are you terribly busy or are you ignoring me? =)

@AlexWertheim Congrats for getting accepted.

Thanks, @WillHunting. I'm guessing you're Jasper?

@AlexWertheim Yes, I am.

I tend to try to talk about math when I'm in here, @WillHunting, and avoid not doing so.

As you see, I'm not doing that right now. :P

Sorry! sheepish glances

@MikeMiller I see. I will try not to talk too much to you then.

@WillHunting I am not offended, I just probably will not respond. Thanks, though.

Hopefully, I will get accepted there one day too...

Ah, almost everyone has left chat.

Today is a bad day for me. My anxiety is giving me stomach cramps now.

@JasperLoy I just finished class

@ᴇʏᴇs Shall we talk in the other room?

@WillHunting I do that a lot. I am off to get dinner. We have company tonight, so I won't be back for a while

@robjohn OK, I will see you in my dreams later.

Hey guys. I'm looking for an old post I saw once (I'm pretty sure it was on this site). The question was about why the definition of differentiability in higher dimensions isn't just the derivative = lim ... exists -- like it is for single variable derivatives -- and (s)he provided a suggestion for such a limit definition. An answer then described why that definition wouldn't work (I believe it had to do with it not implying continuity) and an explanation of the actual definition.

I recall it being quite good, but in my naivete I didn't favorite it and can't seem to find it now. Any help would be appreciated.

Can i switch two entries in a matrix ?

what do you mean @Rememberme?

I mean if i have a matrix like [-3 1 -2] can i write it as [-3 -2 1]

no

two matrices are equal if each of the corresponding elements are equal

okay

now

no

what else can i do to this

row reduced echelon form all numbers below the leading ones have zero below them

i know that but i can do anything more............

delete the 12 entry

easily

which one do you mean ?@KarimMansour

$R_2$ = $R_1 * 3$ + $R_2$

@Rememberme You finished Apostol but you can't rref this?

i am doing linear algebra from hoffman kunze @ᴇʏᴇs

Apostol i felt was out of sorts on Linear algebra

@Rememberme The LA problems in Apostol are much harder than rref this, how did you manage to do them

who i said i did them.....i am doing linear algebra from Hoffman kunze@ᴇʏᴇs

@Rememberme You said you finished all the problems in Apostol

i said in calculus not linear algebra

Linear algebra is part of calculus

But Linear algebra is better in hoffman kunze @ᴇʏᴇs

How did you do all the vector calculus problems without linear algebra

i dosent introduce linear algebra before vector calculus it introduces vector algebra first then linear algebra

@ᴇʏᴇs and sometimes you make mistakes in most of the easiest problems like this one :p

@ᴇʏᴇs!!!!!!

Hello! Sorry. My bad!

hi @JulianRachman

how are you?

@Rememberme

great

(P.s., You are sayan right?

)*

yup

Hey @ᴇʏᴇs, I am going to sleep soon...

@Remember Ok! Cool!

@ᴇʏᴇs you have done linear algebra right?? i have a doubt

What you been working on?

@Rememberme You did not finish Apostol, then asked about groups, then went on to Hoffman, so I won't give you any more advice from now. Not that you need any from someone like me.

if we have $3^{x - 1}$ = 1 (mod x) does it imply that x is prime ?

@Remember ^

Have you tried any examples? @KarimMansour

@WillHunting i have finished apostol calculus......i am doing linear algebra from hoffman kunze and those little facts about groups everyone here knows.......

no I am trying to prove Pépin's test @DiscipleofBarney

I am at the last step

@Rememberme You have not finished Apostol because you did not do the linear algebra in Apostol.

Prove that if $3^{{F_n - 1 }/2} \equiv -1$(mod $F_n$) then $F_n$ is prime.

Here is my proof

I didnt like that linear algebra.....i did linear algebra from other book @WillHunting how does it matter......hoffman kunze also does introduce it like apostol

squaring both sides we get $3^{F_n - 1}$ $\equiv$ 1 (mod $F_n)

@Rememberme Never mind, you can do things your own way, I won't say anything from now.

why i havent done something wrong....have I @WillHunting

@Rememberme My "advice" is clearly rubbish to you, so I won't say anything anymore.

You know @WillHunting if i had wished i would have just ran to topology but i didnt

i am doing linear algebra.....and i am not even rushing at it

so ord(3,$F_n$) | ($F_n$ - 1) = $2^{2^n}$ since ord(3,Fn) doesn't divide (Fn - 1)/ 2 since ord(3,$F_n$) | $2^{2^n}$ the possible values of the ord(3,$F_n$) must be the divisors of $2^{2^n}$ so it must be one of the following 1,....,$2^{2^n}$ / $2^2$,$2^{2^n}$/2,$2^{2^n}$ but ord(3,$F_n$) doesn't divide Fn - 1 / 2 = $2^{2^n}$/2, so we must have ord(3,$F_n$) = $2^{2^n}$ = Fn

@DiscipleofBarney

now now we know ord(3,$F_n$) = Fn - 1 I don't know how to proceed further to justify that number is prime any hints @DiscipleofBarney ?

So $\phi (F_n) = F_n -1$?

yes

Okay can $ \phi (n) = n-1$ if $n$ is composite?

I don't understand why is $\phi(F_n)$ = $F_n$ - 1 ?

You just said it was....

we know that ord(3,$F_n$) = $F_n$ - 1

What does that notation mean?

it is the smallest number c such that $3^c$ = 1 (mod Fn)

So the order of $3$ is $F_n -1$ in the multiplicative group (unit group)

yes

Well you can go from there

why do we now get $\phi(F_n)$ = n - 1 then ?

@KarimMansour Your latexing in here is terrible, LOL.

why ? @WillHunting care to elaborate ?

@KarimMansour For example, the previous line. Why didn't you just put the $ right at the end? =)

@KarimMansour Anyway, sorry to interrupt. Good luck for your exams, and good night!

I thought it wouldn't make a difference since I didn't use any math notation in that last bit at n - 1

np ! @WillHunting thank you !

IN this www-inst.eecs.berkeley.edu/~cs61a/fa12/hw/hw3.html question 4 says: In passing, Ben also cryptically comments, "By testing the signs of the endpoints of the intervals, it is possible to break mul_interval into nine cases, only one of which requires more than two multiplications." Write a fast multiplication function using Ben's suggestion:

when I observed the pattern, it looks like this..

Well it looks like trash and nobody wants to read trash... @KarimMansour

so am not sure, what does it mean to say nine cases?

You can figure it out, I mean you guys have done stuff with $\phi$ functions and multiplicative groups right? @KarimMansour

alright np will improve my latex typing @DiscipleofBarney in my algebra class yes but not in number theory.

If I get some help on this, I can complete my python rogram

Well algebra, even better

ok will go think about it, since its just 1 step I should figure it out soon :D.

@robjohn Thank you.

@PedroTamaroff Thanks, but there is nothing to congratulate me for! Congratulations to YOU!

@DiscipleofBarney Oke I got it the reason we have all numbers below Fn - 1 must be coprime to is that since if we had an integer such m | Fn - 1 then this would mean $3^m $ = 1

but ord(3,Fn) = $F_n$ - 1 so each number below Fn must be coprime to Fn

@overexchange Are you still there?

@KarimMansour That is not why, and that isn't even true

@overexchange Looking at the examples, there is a pattern depending on the sign of the endpoints... ping me to discuss it further if you haven't figured it out already.

I don't know @DiscipleofBarney

I have no idea how to deduce that last step

When $\gcd ( m,n )>1$, is there an element $m^{-1}$ so that $m m^{-1} = 1 \mod n$? Why? What can we say about our situation $\mod F_n$? @KarimMansour

@DiscipleofBarney do i have to introduce new matrices in order to do a row exchange

no gcd(m,n) > 1 then we can't do this since m wouldn't be invertible

since elements of n are invertible iff gcd(m,n) = 1

@DiscipleofBarney

no @Rememberme

Ok......so i can change them as i want them in order to get 1 as the pivot

@Rememberme I am not really sure what you are talking about. (also you should read the section ;P )

no @DiscipleofBarney it doesnt talk about row exchanges it talks about row operations which are fine for me

order(3,$F_n$) = $F_n$ - 1 means that all elements less than Fn are invertible correct? @DiscipleofBarney

@KarimMansour Does it?

@Rememberme You should read what the row operations are...

Read @DiscipleofBarney

yes I think so @DiscipleofBarney

@KarimMansour That does not instil confidence... You know you should have ideas on how to prove these things

@Rememberme What?

I read row operations they were pretty easy @DiscipleofBarney

yes I am thinking don't tell me it I will try and figure it out

What did it say? @Rememberme (because it doesn't sound like you did based on what you said)

What i said absolutely is not realted to it ...

Anyways row operation is a special type of a function which which associated with each m X n matrix A an m X n matrix e(A)@DiscipleofBarney

@Rememberme "elementary row operations on an $m \times n$ matrix $A$.... (3) interchange two rows of $A$."

I just gave you the definition and the three points were given by you.......

you have hoffmann kunze with you?

They are in the book @Rememberme

But in the example they dont do any row exchange.....

It says the interchanging is a row operation, how is that got nothing to do with row opertations and interchanging two rows?

Example 5

So?

Just because they didn't do it doesn't mean its not an operation

When it clearly says it is an operation

@DiscipleofBarney In that case will i also consider transpose as an row operation?

Does it say it is?

we are exchanging rows with columns so it should be

Does it say that is an operation?

You can definitely define that operation if you want

it doesn't......

@Gigili sorry what were you saying?

@DiscipleofBarney how do i prove that there is no operation in the world which is not an arithmetic operation?

@Rememberme What do you mean by that? What is an arithmetic operation? Do you have a reason to believe your statement?

The four basic operations are +,-,*,/ that is what i have known till now.....

ok @DiscipleofBarney I understood it now here is my argument since we know ord(3,$F_n$) = $F_n$ - 1 then we must have that there exists $Fn$ - 1 then all 1,$3^1$,... are all distinct since there is n elements in $F_n$ then the elements of $3^c$ form multiplicative group of units so that is we get our result

what do you think @DiscipleofBarney?

@Rememberme Normally to be a binary operation it is only required that it is a function $X \times X \to X$, so there are tons of other operations. Think about it, if you have learned any group theory there is a ton of operation that are not like that

@KarimMansour What are the inverses of these elements?

hm

the negative powers for each c

but no

I am tired atm to function tomorrow I will figure it out

good night guys

@overexchange I was trying to help you finish your Python code

Consider 16 cases where P1 to p$ have different signs, but some cases have the same results, e.g. all negative and all positive (in ascending order)

Good night

@DiscipleofBarney For you?

@DiscipleofBarney Oh I see

@KarimMansour Good night

@Incurrence Hello, how was complex test

Your exams are over @Incurrence

how was it

It was great

good

Only made 1 mistake, which was converting $\cosh 4$ to $\frac{e^4 + e^{-4}}2$ at the start instead of leaving it alone

So I got ungodly fractions all over the place

oh.....

So I probably got 95%

Now I have to work until exactly 24 hours from now

why..

Assignment due in 23 hours and 59 minutes

Assingment.....hmmm

@Incurrence after 2 days i have finally understood what subspaces of vector spaces clearly without any doubt....

That's good

They are subsets that are vector spaces in their own right

so not only R but all lines going through the origin are subspaces of R^2

and the trivial one that is [0,0]

And in $\Bbb R^3$

We have all planes and lines through the origin and the trivial

any plane going through the origin...

you just took what i was about to say:p

and in $\Bbb R^{124124543}$

We have all $124124542-$hyperplanes, and all $124124541-$hyperplanes $\dots$ and all lines and the trivial

all planes ,lines,surface(i dont what you call it R^3),.......and going through the origin

Gotta go do assignment, talk later

fine....good luck

Greetings

@robjohn did you manage to do some work on the integral I showed you yesterday?

@robjohn no worry, that is just a natural way of doing things. Perhaps the most natural way.

@Chris'ssis Had to restart my computer just now... No, I haven't yet.

@robjohn OK

hi

@Rememberme ok, enumerate subspaces of R^n.

Anyway, this is too advanced.

hi

are there equations for which there exists a certain (real) solution but we cannot solve without any guessing because we lack the algorithm to solve such?

yes

then my next question would be. Can we prove that such an algorithm for above equations does not exist or does it possibly exist, yet we did not figure it out yet?

yesw

that wasn't a yes/no question

we can prove that some equations aren't analytically solvable

and require brute force

@pZombie depends on the definition of "solving"

by equations, i presume you mean polynomial equations?

I believe he means solving analytically

i was meaning equations in general, but polynomial is fine

then every polynomial equation can be analytically solved

pzombie, polynomials above degree 6 can't be solved

false, @Mew

i find it strange that there can be equations with a solution, yet there is no "mechanical" way to solve them, other than guessing some values

some can, but not all polynomaisl can

Balarka, what is the solution to this x^7 + 25x^6 + 16x^5 + 4x^4 + 2x^2 + 98x + 4 = 0

polynomials of degree 6 cannot be solved in terms of analytic functions of single variable, that doesn't mean there is no analytic solutions.

@Mew there is certainly a solution in $\Bbb C$. You're thinking of Abel-Ruffini theorem, that's why i meant what one means by "solving".

Abel-Ruffini theorem says that polynomials in $\Bbb Q$ of degree 5 or above cannot be solved in "special extensions" of $\Bbb Q$.

So Balarka, are you saying you work out the values x can take in my 7th degree polynomial without using any guess and checK?

yes.

how

Taylor series :)

how can you apply it here

x is just a number