Mathematics - 2016-01-14 (page 2 of 4)

L33ter

15:00

if you have self-correcting it is better

Balarka Sen

What's $Z^k_f$?

L33ter

let us write it as $F_q^k = F_q (+) .... (+) F_q$ k-times

the book uses the notation I've written before

Balarka Sen

Ok, so $Z_f^k$ is the same as $\Bbb F_f^k$? That's weird notation, plus unusual to denote a number by $f$, so just making sure.

L33ter

yeah

that is correct @BalarkaSen

Balarka Sen

Fine. So encoding is just a injective morphism of direct sum of finite fields, and decoding is a surjective morphism in the other way.

What're the mathematical structures arising from this?

(by that I mean, "what next?" :D)

L33ter

15:08

I don't know yet I am still learning @BalarkaSen

haha

give me like 1 week and I will tell all about algebraic coding theory xD

Balarka Sen

Ah, alright.

L33ter

I will do my project on either gappa code or elliptic cryptography in that class

Balarka Sen

Can you define the noise thing you mentioned above?

L33ter

yes, so for example suppose we send as above a orginal message which composed of bits

suppose we send (1,0,1)

but in the end after decoding we receive (1,1,1)

then we know that there was noise affecting our message, which changed our original message

Balarka Sen

Sure, that can happen. So how to deal with this noise?

L33ter

15:11

with coding theory by implementing error correcting codes

for example you know ascii ?

they use something called even parity

the characters we use in the computer is 128 characters

but the ascii character has 1 extra bit they use for error detection

so what they do is that they change bits before sending such that the number of bits is even before sending

if you receive odd number of bits then something is wrong

you can do the even parity thing mathematically as follows

the encoding function is given by $E(x_7,...,x_1) = (x_8,...,x_1)$, where $x_8 = x_7 + ... + x_1$ where addition is done in $Z_2$

I have a good book for you if your interested to know mor

more it is the one I am reading from

Abstract Algebra
Theory and Applications
Thomas W. Judson
Stephen F. Austin State University

It is actually a more recent book done in 2009

Hippalectryon

@L33ter 1993 ed ?

Oh

Balarka Sen

Interesting, but not sure if I digest that. What's guaranteeing that if I send a message with even number of bits, and if I encode and then decode it, then if it's wrong it'll consist of odd number of bits?

Thanks for that recommendation, but I was merely curious about it. Not planning to study anything, I assure you :D

L33ter

let us for example take an example

A in ascii is 65

65 in binary is 1000001

After encoding it is still 1000001, since it has even number of bits

now suppose we send this and receive 1000011, then we know something is wrong

Hippalectryon

@L33ter What if we receive 100001 or 10000011?

Balarka Sen

^

L33ter

15:20

it will not detect error in this case

it is not a perfect encoding scheme the ascii

it has problems such as above you mentioned

but, using algebraic coding theory we get better methods

Balarka Sen

Certainly interesting, but I'd be more excited when you can tell me efficient methods to do this :) Let me know when you learn some of this stuff.

Semiclassical

i imagine there are tradeoffs to be had, i.e. the more error protection you want the longer the encoding must be

i vaguely remember stuff like hamming codes but not in any detail

L33ter

alright

yeah that reminds me

we can put a metric on the information

called hamming distance

Balarka Sen

"information"?

L33ter

let us define something called a code word

a code word is just subset of the alphabet so in the case above it is subset of our field above in the decoded information.

Balarka Sen

15:24

I am confused. We were working with $\Bbb F_q^n$'s.

L33ter

yeah

Balarka Sen

What are the codes here?

L33ter

so a code word is subset of it

Balarka Sen

Subset? Or an element?

L33ter

code word is the alphabet we use it doesn't have to be the whole thing

and a code is an element

we can define hamming distances on codes as follows

Balarka Sen

15:26

Oh. So a code is an element, and a codeword is an element of some factor in the direct sum?

L33ter

yeah

Balarka Sen

Makes sense.

Go on.

L33ter

now we can define a metric on the the codeword as follows d(x,y) = number of bits that are different from x and y

Balarka Sen

Yeah. Sure.

L33ter

for example suppose x = 1100, y = 0011, so d(x,y) = 4

Balarka Sen

15:27

Right.

L33ter

we can let $d_{min}$ define that minimum distance in all elements of the a specific code

Balarka Sen

@L33ter I think you mean code, not codeword here.

L33ter

yeah

Mike Miller

morning

Balarka Sen

Morning, @MikeMiller.

L33ter

15:28

morning @MikeMiller

Semiclassical

morning

L33ter

we can prove the following theorem

C be a code with $d_{min} = 2n + 1.$ Then C can correct any n or fewer errors. Furthermore any 2n or fewer errors can be detected in C

the proof is actually easy

Balarka Sen

Interesting. Thanks, I think I get the idea. Exciting.

Mike Miller

@SemiC: I seem to have lost the review you sent me. any chance you can resend it? I have time now

Semiclassical

i don't, but i think i remember the title of the book being reviewed: "Fukaya categories and Picard-Lefshetz theory" (no guarantee on spelling)

Mike Miller

15:34

found it, thanks

clickhole.com/article/…

L33ter

yeah it is very interesting stuff @BalarkaSen

Semiclassical

@MikeMiller trying to convince the internet to stop employing a certain meme seems a pretty sisyphean task

PVAL

o that book

Brennan.Tobias

hi

Semiclassical

i'm curious what you get out of that. it's quite beyond me @MikeMiller

Mike Miller

15:47

alan rickman died

Semiclassical

ugh. that sucks

Mary Star

16:19

Could someone of you take a look at my question:

0

Q: Taylor expansion - How can we deduce that?

We have that $$\tilde{E}=\frac{u^2}{r^2}+\frac{Gv^2}{r^4}, \ \ \tilde{F}=\left (1-\frac{G}{r^2}\right )\frac{uv}{r^2}, \ \ \tilde{G}=\frac{v^2}{r^2}+\frac{Gu^2}{r^4}$$ and that $$u^2(\tilde{E}-1)=v^2(\tilde{G}-1)$$ By considering the Taylor expansions of $\tilde{E}$ and $\tilde{G}$ about $u = v...

differential-geometry

?

Balarka Sen

@MikeMiller uh-oh

lol that emoticon buisness

Hippalectryon

@MaryStar you write By the Taylor expansion of E~E~ we get $\tilde{E}=\frac{u^2}{r^2}+\frac{v^2}{r^4}G+\text{ remainder }$... but $\tilde{E}=\frac{u^2}{r^2}+\frac{v^2}{r^4}G$ ! That's no 'expansion', you've just rewritten the same thing twice

Superstar Monica

@BalarkaSen you should be suspended for a very long time, and if the mods don't do it I send what you told me to MSE team.

Besides that I delete my account, and this is not cool for many users.

Balarka Sen

@SuperstarMonica as you wish. you were the one who attacked and called names, not me, so I'm not really afraid. I doubt action would be taken against me.

Superstar Monica

'I generally assume @SuperstarMonica is simply suffering from some sort of psychological trouble. I can't find any other explanations for the stuff she talks about. '

^^^

Balarka Sen

16:33

if you're rude to people, don't expect they would be nice to you.

Superstar Monica

@BalarkaSen Hope you're out for at least a year. Not for me because I don't come here anymore.

Balarka Sen

whatever

Superstar Monica

@BalarkaSen Do you realize that you would talk like that with mathematicians like Euler, Gauss, Ramanujan? Because you simply don't care the person you talk about.

Mike Miller

@Semiclassical: Really enjoyable summary.

Superstar Monica

To be clear once and for all

@BalarkaSen I'M AT A DIFFERENT LEVEL!!! (my math is not kindergarten)

Anyway.

Hippalectryon

16:37

What the hell is going on here ._.

Krijn

Hey @BalarkaSen

Ropstah

Hi everyone

Krijn

I did quite a lot on my project on cohomology

Mike Miller

@SemiC: See in particular the discussion on the bottom of page 6. Very cool.

Ropstah

I read about the "principal ideal domain" on Wikipedia (https://en.wikipedia.org/wiki/Principal_ideal_domain):

Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm).

Does anyone know any examples of domains in which it isn't possible to use the Euclidean algorithm to find a GCD?

Semiclassical

16:44

@MikeMiller glad you found it interesting, heh

Superstar Monica

@BalarkaSen and if one day it happens to go crazy completely, well it was for the total dedication to the math, and that must be appreciated. You understand?

Learn what real dedication means. No one to ever talk about my math.

Now I'm definitely out.

Mary Star

@Hippalectryon But do we not get that when we apply the above formula?

Mike Miller

@Ropstah: Any example? How would you take the gcd of 8554739202 and 57849208433?

Hippalectryon

@MaryStar No. Show me how you're applying it to, for instance, $\tilde{E}$

Ropstah

@MikeMiller: From reading Wikipedia I'd say using Euclid's algorithm, or using LCD

Mike Miller

16:55

Oh! completely misread your question. To be exactly opposite what you asked. (I thought you were looking for an example where you can use Euclid's.)

Ropstah

Or is the answer simply a matter of computational feasability

:)

I guess using complex numbers is a case?

Mike Miller

There isn't bexessarily a Euclidean algorithm in an arbitrary PID.

Ropstah

No exactly, that's what I'm also getting out of the statement, however I'm looking for an example where this is explicitly the case (so where it is not possible to solve using Euclidean algorithm)

Mike Miller

Take $\Bbb Z[-\sqrt{p}]$, $p$ prime. There exists some $p$ such that this is a PID but has no Euclidean algorithm. (Challenge: Write down a Euclidean algorithm for $p=2,3$.)

@Ropstah: The thing is that when I hand you an arbitrary ring "Euclidean algorithm" doesn't make sense until you write down an algorithm. Look at the Wikipedia page for Euclidean domain.

Semiclassical

i guess 'which $p$' is the obvious question

Ropstah

16:59

Ok I think I get it

Mike Miller

even in cases where one exists, you still have to find it. You can't just say "Use the Euclidean algorithm"

@SemiC: I don't remember, hence the existence statement.

Semiclassical

ah

Ropstah

163 is a case

Just a wild guess haha, thanks for the explanation!

Mike Miller

Occasionally I should be adjoining 1/2 there, I forget at which primes.

Ropstah

I'm going to look at $p = 2, 3$ ;)

Mike Miller

17:04

Try $p=2$. I wouldnt do $p=3$ because I mucked something up.

Ropstah

ok

Semiclassical

Does Euclidean domain = rign possessing a Euclidean algorithm?

Balarka Sen

@Krijn Hi.

@SuperstarMonica Look, I never said anything about your math (if I do, I do apologize), I do respect what you do. It'd be silly to say computing integral & series is not really math, even thought it's a rather isolated area of math (hell, technically any branch of math is - mathematics is big). I have issues about one and only one thing - calling good mathematicians like Ted and Tobias names. If you quit that, then I have no complaints.

Mike Miller

@Semiclassical Yes.

Semiclassical

from Wikipedia's page on PIDs, one has the statement "An example of a principal ideal domain that is not a Euclidean domain is the ring $\mathbb{Z}[(1+\sqrt{-19})/2]$."

Mike Miller

17:12

I knew I was messing up the 2

w/e

Semiclassical

i wonder if 19 is the smallest one?

Mike Miller

probably else they would have had the smaller example

Semiclassical

yeah

Balarka Sen

@Krijin What did you study in cohomology? :)

Semiclassical

so far that's the only counterexample i've seen cited while googling

i'm sure it's not the only one, but the lack of other examples is a bit annouying

Balarka Sen

17:20

what about Z[\sqrt{-5}]?

Clarinetist

For positive integers $p, n$, can the expression $$\sum_{k=0}^{p-1}\binom{p}{k}\left(1^{k}+2^{k}+\cdots + n^{k}\right)$$
be simplified?

Mike Miller

@BalarkaSen Not a PID

Balarka Sen

oops, right.

Mike Miller

done anything interesting?

Balarka Sen

I can prove the immersion and submersion theorem.

Mike Miller

17:30

Dunno what that means.

JeSuis

@Hippalectryon dsl j'avais plus internet hier... Pour ton exo aucune infos sur la continuité de $f$ ou monotonie ?

Hippalectryon

@JeSuis Aucune info, mais tu peux commencer par le cas $f$ continue/monotone (en gros pour le cas général tu dis que c'est la même démo sauf cas pathologiques)

Evinda

Suppose that we gave $x(t)=c+ \int_{t_0}^t \Phi(x, \xi) d\xi$.

If $x$ is continuous, how do we deduce that $\Phi(x,t)$ is continuous?

Since the left side of the equality is continuous, so will be also the right side, i.e. $c+ \int_{t_0}^t \Phi(x, \xi) d\xi$. So we deduce that $\int_{t_0}^t \Phi(x, \xi) d\xi$ is continuous. Do we deduce from that that $\Phi$ is continuous?

Balarka Sen

$M, N$ be manifolds with $\dim M < \dim N$. Then given a smooth map $f : M \to N$, if for a given $p \in M$, $Df_p$ is injective (i.e., $f$ is a immersion at $p$) then there exists local coordinates in which $f$ is the canonical inclusion $\Bbb R^{\dim M} \hookrightarrow \Bbb R^{\dim N}$ by setting zeroes on $\dim N - \dim M$ coordinates.

JeSuis

@Hippalectryon Ok je vais essayer ça

Balarka Sen

17:33

This is the immersion theorem. Submersion theorem is for $\dim M > \dim N$, $Df_p$ a surjection.

(then $f$ is locally a canonical projection, by deleting $\dim M - \dim N$ coordinates)

Mary Star

@Hippalectryon We have $$\tilde{E}\approx \tilde{E}(0,0)+u\tilde{E}_u(0,0)+v\tilde{E}_v(0,0)+\frac{1}{2}[u^2\tilde{E}_{uu}(0,0)+2uv\tilde{E}_{uv}(0,0)+v^2\tilde{E}_{vv}(0,0)]$$

Since $\tilde{E}=\frac{u^2}{r^2}+\frac{Gv^2}{r^4}$ we have
$\tilde{E}(0,0)=0$

$\tilde{E}_u=\frac{2u}{r^2}+\frac{G_uv^2}{r^4} \rightarrow \tilde{E}_u(0,0)=0$

$\tilde{E}_{uu}=\frac{2}{r^2}+\frac{G_{uu}v^2}{r^4} \rightarrow \tilde{E}_{uu}(0,0)=\frac{2}{r^2}$

$\tilde{E}_{uv}=\frac{G_{uv}v^2+2G_uv}{r^4} \rightarrow \tilde{E}_{uv}(0,0)=0$

Mike Miller

Cool. What's the correct statement for Banach manifolds?

Hippalectryon

@MaryStar Looks correct to me

JeSuis

@Hippalectryon si tout vas bien, quand $x$ est très grand, $f(x+1)-f(x)$ tend vers $0$

Evinda

@Semiclassical Hi!!! Did you see my question?

Mike Miller

17:38

(You may assume that the correct conclusion is "Is locally given by a continuous linear map", which is all you're saying above. I'm interested in the correct conditions.)

Semiclassical

did, but it's not the kind i can help with

Mary Star

@Hippalectryon So is what I have done in my question correct?

Hippalectryon

@MaryStar I don't see any $G_v$ in your question, so I don't think so

Evinda

@Semiclassical A ok... no problem...

Does anyone else have an idea?

Mary Star

@Hippalectryon Oh yes... I will change it and calculate again the taylor expansion of $\tilde{G}$...

Balarka Sen

17:41

I'm trying to think. Inverse function theorem holds just fine for Banach manifolds by previous discussion (I have a complete ball), so that isn't a problem.

So presumably something in the proof.

Mike Miller

Right, well what about implicit, which is what you're using here?

Balarka Sen

But we don't use implicit function theorem in the proof of immersion theorem?

Mike Miller

Ok, whatever, the proofs are literally the same. Feel free to think about the proof instead then.

Balarka Sen

Ah, ok.

One starts by choosing local parameterizations $U \to M$ and $V \to N$ and looks at the open set level map $g : U \to V$. $Dg_0$ is injective, and one changes basis so that the $n \times m$ matrix has identity in the $m \times m$ block and zero on the rest. I think this is where things start to fail, because this argument cannot be generalized in infinite dimensions.

($n = \dim N$, $m = \dim M$)

I'll have to study now, so I'll get back to you later. Sorry!

Mike Miller

The xhange of basis is inconsequential. Especially if I only want the final result to be of the form "...is locally linear".

Krijn

18:20

@BalarkaSen Oh just regular cohomology in Hatcher up to the Universal Coefficient Theorem

We have to present that in an hour, so that should be enough

Joshua Lamusga

When I take an upper or lower sum for a Riemann sum (I'm in Calc II), is there any special way to account for the fact that the sample points of each rectangle can be at different positions? (It's the "finding the area under the curve" precursor to integration)

As I understand it, upper sums use left and right endpoints for rectangle approximations based on whichever produces the largest area (and similarly for smallest area with lower sums). How can I express this mathematically using sum notation?

Joshua Lamusga

18:39

I figured it out now. The sample point positions don't matter.

user17753

19:33

Hi guys

I had a quick question on Taylor Series for e^x as shown on wolframalpha.com/input/?i=e%5Ex+taylor+series wondering why they put O(x^6) as a stand-in for the remaining terms if they are much higher order. How is O(x^6) an upper bound on those?

TheGreatDuck

It's wolframs way of trying to prevent over calculation/designate infinite series.

At least, that's what I'm thinking

Does anyone know of a summation from 0 to x, that yields x^2 as the result?

the closest I have is the summation of odd numbers, but then at x = 0 I get -1

Martin Sleziak

@user17753 $O(x^6)$ means that the absolute value of the remainder can be estimated by $Cx^6$ for some constant $C$. Wikipedia: Big O notation, Taylor's theorem.

@TheGreatDuck Is $x$ an integer? What about $\sum_{k=1}^n (2k-1)=n^2$?

TheGreatDuck

A sum of 2k-1 does not work. K = 0 results in -1. Unless I am misinterpreting summation that should not work.

Huy

$n=0$ results in $0$ as desired

TheGreatDuck

So, does the summation not add anything when n = 0, similar to how a for loop would not execute if the starting value is already at it's limit?

Huy

19:46

for (int i=1; i <= n; i++) {}

n = 0 < 1

Hippalectryon

@user17753 It's for $x\to0$. Hence $x^{j>6}<<x^6$

@Huy What about $\sum_{n=0}^x\delta_{n,0}x^2$ ? :P

Huy

don't know what you're summing over

user17753

So do I always assume x to 0 if they don't specify it?

Hippalectryon

@user17753 Look closer

Huy

@Hippa: where did you learn to tex

Hippalectryon

19:56

@Huy I'm too lazy to add \displaystyle

TheGreatDuck

Huy

Hippalectryon

TheGreatDuck

i said from 0 to n

Hippalectryon

@user17753 ^

Huy

why would you add display style

@TheGreatDuck then use Hippas cheat solution

Hippalectryon

19:56

So that the indices look better on the sum

Huy

@Hippa I was refering to the $\sum_0^x \to \sum_0^{n=x} \to \sum_n=0^x \to \sum_{n=0}^x$

Hippalectryon

:P

user17753

Is there something I have to do to get the mathjax to show up in chat? Or do you guys just read it as code?

TheGreatDuck

im not entirely sure what you just said

Huy

does this sequence split?

Hippalectryon

19:57

@user17753 math.ucla.edu/~robjohn/math/mathjax.html

@TheGreatDuck $x^2=\sum_{i=0}^n\delta_{0,i}x^2$

TheGreatDuck

i simply want a series with the lower limit of n=0, an upper limit of x, and a result of x^2

Huy

@TheGreatDuck he gave u1

TheGreatDuck

@Hippalectryon: Sadly, I don't know how to read that math jax. Are you trying to do a composite function.

Huy: he slipped in before me. :p

Hippalectryon

@TheGreatDuck math.ucla.edu/~robjohn/math/mathjax.html to render mathjax

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$Mathematics$ Mathematics