Bye @orlp :-)

@orlp I mean if I have this equation: $\sum_{k=1}^{i} \binom{n}{k}$ then Is there any formula that equals to this term

since we have a formula that says "\sum_{k=1}^{n} \binom{n}{k} = 2^k - 1" but how about this

@YOUSEFY I don't believe so I happened to look into that one yesterday

@orlp So there no formula for this $\sum_{k=1}^{i} \binom{n}{k}$

I can't say for certain

but I believe not

as in, a closed form

Understand you! Thanks!

but if you write $f(n, i) = \sum_{k=1}^i \binom{n}{k}$ I'm sure there's relations you can write about $f$

@Wildcard are you thinking about the problem? any progress?

@orlp "relation" you mean "inequalities"!

@YOUSEFY no I don't

@orlp Can you be more specific about what do you mean of "relation"!

I mean recurrence relations

now I have to read hours of transcript to see what transpired in my chatroom

@anon For the full context, you should also read the chatroom that was created in order to move the drama way from here

Oh, and some messages were deleted

I can and have read the deleted messages.

@anon Hi anon :D

hi

I got a question on that question we discussed yesterday =p

@SteamyRoot There was drama and I missed it? Now where will I get my entertainment?

if i can show that GL_2(Z/2) and S_3 are generated by 2 elements

and on those elements , isomorphism crtieras are obyed

You could always start more drama! :^)

(not a serious suggestion, by the way, so please nobody get offended)

can I conclude that it is obeyed for all the 36 "products" ?

:^)

Instead of doing f(xy) = f(x) f(y) for all elements

I do that for the geneatators only

@SteamyRoot this is a serious site streamy root , let us focus -.-

I'm offended by your suggestion to not get offended @SteamyRoot

@SteamyRoot plz no kthxbai

Btw semi

yesterday anon showed me an awesome idea of how to find the 1-1 correspondance between S_3 and GL_2(Z/2)

using vectors

yeah, I saw that

:D

that was like magic :D

But how did he come up with that -.-

I mean I dont think of such things

amusingly enough, I came across that too..._after_ I'd tediously worked out how to get it just by trial and error

grrrr :D

nice

But he said that was a group action

group actions

@KasmirKhaan To show that some group is isomorphic to a symmetric group, it is very natural to try to find a nice set on which that group acts

But I still dont know what that is so ><

Well, I think the hint is this. It's a matrix representation, so the hope is

In this case, the group is defined in terms of an action on a set

We did not do group actions yet @TobiasKildetoft

But after knowning that

hah, the best way I know how to say this is using bra-ket notation :)

how can we proceed?

let me post my idea

$M(g)|v\rangle = |gv\rangle $

@Semiclassical That notation never makes anything clearer. It just tricks physicists into thinking they know what is going on :)

pffffft

it's mostly handy in this case because it's easier to read.

@anon is what I wrote correct?

@Semiclassical I am not actually sure what the notation does there.

@KasmirKhaan It is enough to show it maps generators to generators which satisfy the same relations, but knowing why that works requires an understanding of group presentations that you don't have.

@anon hmm so that idea is out >< thanks =p

although it's possible you could figure out that understanding

@orlp Thank you orlp :) I also want to make sure about something: we cannot take the expected value for "event" so we should define a random variable to have the expected value. Is this right or do you have any idea about this!

@anon did you mean advanced formulas? we only did 4 lectures so far, we went thru a little bit about groups and just started cyclic ( did not finish it yet )

we introduced isomorphism and hom, but only definitions

@YOUSEFY no idea, I have trouble understanding exactly what you're asking

@KasmirKhaan never said anything about formulas

if you wrote out 6x6 multiplication tables for GL(2,Z/2Z) and S_3 you would be done by now

@anon the two generators I found was (123) and (23) , I know what matrices those maps to, it should be reasanble to assume that knowing how the generators act, we can conclude that all the group act the same way, because the generators are the group itself

I did that

but is that concedered to be a proof?

if you notice that your relabelling of the elements turns one multiplication table into the other, then you've just verified f(xy)=f(x)f(y) for every x and y

I thought about that, but did not concidered it a good way of proving, i mean if we had a bigger case , like group of order 100 it would not be easy to do that

@TobiasKildetoft eh, it's perhaps clearer if I make it concrete: If $M[(12)]$ is a matrix representation of $(12)\in S_3$, then it should act as $M[(12)]|1\rangle = |2\rangle$, $M[(12)]|2\rangle = |1\rangle $, and $M[(12)]|3\rangle = |3\rangle$

it should be other way to show that they are the same

I'm probably writing it in a bad way, but oh well

@KasmirKhaan "if we had bigger cases" you would use ideas and methods you haven't yet had time to develop. there's a reason you're given an exercise with small numbers.

it's a bit silly here, though. I could just as well write my vectors as $v_1,v_2,v_3$ and then have $M(g)v_k = v_{g(k)}$ where $g\in S_3$.

@anon all right , that reasoning convinces me =p thanks again !

I guess I like using kets here because the letter v in there is pretty superfluous. what matters is how $g$ permutes the labels. so writing vectors as $|k\rangle$ instead of $v_k$ is tasteful to me

@orlp I have this book that says "A basic characteristic of a random variable is its expectation. The expectation of a random variable is a weighted average of the values it assumes" Thus, the "event" is just by definition outcome of experiment. So, expected value of "event" is just its probability and its weighted average. But if it doesn't have, then it has no meaning. For example: define R = event where we have one heads.

So, E[R] = Pr[E]*a. if a is not defined, then it has no meaning. So, I was thinking that "random variable" is the one that should be taken in order to get an "expected value"

@TobiasKildetoft but don't take that too seriously :)

So, when the book says "basic characteristic of a random variable is its expectation" then we understand that by "event" we cannot get the expected value!

Do you have any argument!

@YOUSEFY Presumably the codomain of all random variables in that book is the reals

I'm sorry, this is too many communication issues for too little math for me

@TobiasKildetoft Why you say that Tobias!

@YOUSEFY Because then the statement about expected value makes sense

as tobias said, in most contexts random variables are real-valued'

Yes, in the same book he define the random variable as " A random variable X on a sample space is a real-valued function on the sample space. ..."

Is there another definition of random variable (in discrete probability)?

@YOUSEFY So that answers the question

@TobiasKildetoft sorry Tobias "which question do you mean?"

@YOUSEFY The one you asked about defining expected value

So, Tobias you agree with me that we cannot take expected value of "event", right!

I am not even sure what that would mean

Suppose I defined an "event" that we have 2 heads by flipping two coins. Now, what is the expected value of this event?

Why would it have one?

noticing that heads and tails are not numbers is not an earth-shattering insight

just like your book says, expected value is something we associate to random variables, not to events

You'd first have to associate actual values to the possible events

So, what I'm saying is that "we need to define a random variable" to get "expected value". Do you agree?

Necessary but not sufficient.

@Semiclassical Ammm, why you say that!

Because events aren't values.

Heads and tails are events.

You can choose to associate numbers with those events, e.g. heads is one and tails is zero

@Semiclassical But assigning values is precisely what a random variable is

hmm

Yeah, I'll yield the point

Is there a neat solution to the above?

Bar trial and error

hi, anyone has a recommendation to a Measure Theory book ? I've got Rudin but im not sure i like it.

@IPAddress What do you mean by solution?

@TobiasKildetoft hi Tobias

I have a silly question

sure

why does n/ gcd( n,k) divided n

gcd(n,k) is a divisor of n, meaning n is an integer times gcd(n,k)...

am proving that each subgroup of a cyclic group , its order is divisible by n

@anon its N / ( gcd (n,k) )

if the cyclic group is <g> and a nontrivial subgroup is H, let g^r be the element of H with the smallest positive exponent r... (then do stuff)

I don't know what you're trying to say by typing "its N / ( gcd(n,k) )"

omg ><

I got it =p thanks

Stuff like these drives me crazy sometimes

@TedShifrin how are you? message or email me whenever. Lin alg is really bad, too easy, :( someone told me it picks up with eigenvectors though. Calc is really cool the prof sent us some set theory notation to learn. Physics seems like it'll be fun, the prof is very laid back.

@TobiasKildetoft I mean is there a way to algebraically solve this??? Or another method except "trial and error"?

@IPAddress I don't see anything to solve here

@Dodsy What are you doing in linear algebra?

Also hey everybody!

@Liad what do you not like about Rudin? The answer to that well influence the resulting recommendation

@Daminark so teacher did vectors but was basically like okay boat goes 10 m/s north and then said okay stream goes 5 m/s east now how boat goes? right away I was like square root of 75 m^2 /s^2, but never said it. Then it took about 30 minutes to get there. Then everyone was confused because of direction but never said anything finally he said he could only give us a formula for direction (took about 20 minutes) then he realized that you could get, then gave "homework" which was bs as well.

so it was shitty

like grade 9 math shitty

he asked us if we ever learned trigg

daminark?

Oh God, tab that's no fun. For linear algebra the book I used and loved is Hoffman and Kunze

so far teacher hasn't talked about curriculum or book

but it's poole i believe.

@TobiasKildetoft sorry I'm just mixing my terminology up... apologies... is there a way to find mathematically the square numbers between a certain equality?

Hi @Dodsy, @Daminark

but the teacher is just really bad

he said that $\sqrt {5^2 + 5^2}$ $=>10$

@IPAddress Well, once you have two of them, you can find the rest by using that the distance to the next always grows by 2

(the two smallest one I mean)

I never corrected him though

ofc

too old and too shy

he said that it's ~11

@TobiasKildetoft May I have an example please

very bad teacher

just no math brain

@IPAddress well, let's say we found the consecutive squares 16 and 25, so the difference is 9 and the next one will be 25 + 9 + 2 = 36

then repeat

Okay @Liad detailed meaning, it abuses the word "clear"?

@BalarkaSen hey duude

what's up

nm

just had first day of class

most of it was great

coolio

first teacher was very smart

learning set theory

8)

or set notation sorry

noice

but like I said my lin alg teacher thought $\sqrt{5^2+5^2}=>10$

ugh

@Daminark that or just writing things (some i familiar with and i think should be explained) without explaining. but i can't say it's not a good book so if there isn't something better i will stick with it.

yeah so he disagreed with a students answer

it was tough

I kept wanting to say something

you should have

before someone said "we use pythagoreans theorem" I wanted to say "it's the square root of 75"

Hmm, I've heard Stein-Shakarchi explains details more

yeah I should have.

That and Royden

I'm just very shy

@Daminark I'm only familiar with S-S's Complex Analysis

I haven't been in class in a long time Balarka

Which is like the vaguest, most handwavy CA textbook I've ever touched :P

true

But those books start by doing things on R/R^n and then repeat in general which irks me a bit

@SteamyRoot I like Stein-Shakarchi

I think it approaches things in a lot intuitive way that it should be I guess

I was so nervous going to class

@TobiasKildetoft I've never seen this occurrence before. THANKS for point it out.... much appreciated!!!

@IPAddress It is pretty easy to show

The only instance of handwaviness I saw in Stein complex was the "toy contour"

I much prefer Greene & Krantz :P

better than Rudin @SteamyRoot ?

@TobiasKildetoft is there a name to this occurrence?

@BalarkaSen I was thinking I would send you some uwo content though

Never read Rudin

@IPAddress I don't think so

I know you're probably far ahead of first year math

what's the name of the book?

but I think you could probably share it with others

just content from professors

Function Theory of One Complex Variable

lol I remember being in a reading course with someone on Stein-Shakarchi and whenever the professor would see some not very good paragraph which is handwavy enough to irritate him, he'd blurt out "This... this bit is written by Shakarchi"

kek

@Dodsy what's the content

some set theory/lin alg

so far

as in class notes

I am a big believer that education should be free, but I don' really know many people I could share the content with

sure

The reason I liked Stein for complex is that it an Freitag seemed to be the two books I found which did complex with an eye toward number theory

and assignments and readings

from physics as well

I could upload a massive document at the end of the semester or upload content as I go

don't they put it up on their webpage

I'm saving everything though

this is all on a student login page called OWL

Ah

just a though

pm me if you want my email and I'll send you some stuff

@Daminark Well, if you like the NT aspect, then sure

or if you don't want stuff, I'll still give you my email

But, for the sake of teaching a course focused purely on CA, I find it, well, crappy.

@Dodsy I'll shoot you a message if I need them, thanks

Is it true that $(A \cup B) - (C \cup D) = (A-C) \cup (A-D) \cup (B-C) \cup (B-D)$?

@BalarkaSen if you know anyone as well

I think education should be public access information.

I mean what other instances of handwaviness are there?

gotcha

And lol Greene/Krantz does seem p good

@Daminark I found the order of growth of functions chapter hard to follow

also you're one of my frans :3

Does complex analysis means measure theory?

No, it means analysis of functions $\Bbb C\to\Bbb C$

Nope, Steamy was just saying that given how he doesn't like Stein complex so he's a bit suspicious of that series

I don't have the book anywhere near me at the moment, but I recall a lot of the proofs being somewhat vague or skipping over a lot of details

Measure theory is basically trying to formulate various notions of size

I don't think it's a bad book by itself, but I think it's far from ideal to teach from.

so your book is about measure theory or CA @SteamyRoot ?

CA

huh, im loooking for a boook about MT :P

It's perfectly plausible that other books by S-S don't have that same feeling, though :P

@Dodsy yeesh

@Semiclassical whats wrong dude?

my storyies?

CA is pretty "inviting" to vagueness and handwaviness