Mathematics - 2013-02-23 (page 3 of 5)

Khromonkey

04:00

@PeterTamaroff that book is really hard

Peter Tamaroff

I read about compactness in $\Bbb R^n$ in Shilovs (which is kinda old but the proofs are dope), but not read about general compactness in Mendelson.

Khromonkey

I am reading top without tears

Peter Tamaroff

@Khromonkey Why?

Khromonkey

The writer had childhood issues, idk

the point is it is hard

Peter Tamaroff

You do understand the word play in "Topology without TEARS" right?

Because we want to tears in the bijections and stuff?

Alexander Gruber

04:01

@PeterTamaroff oh by the way: yes it is really Grüber

Peter Tamaroff

@AlexanderGruber I thought so!

Khromonkey

I thought it meant it was easy

anon

Qiaochu's diagrams are the first thing to ever make me wish I had an ipad

Peter Tamaroff

@Khromonkey Yes, it is a wordplay.

Khromonkey

where are they @anon

oh

Peter Tamaroff

04:02

@anon Link, braw!

Alexander Gruber

anybody ever try any cryptography?

anon

this + links contained therein

@AlexanderGruber teensy bit

Alexander Gruber

what kind?

anon

I used pwdump and rainbow tables once

actually that's not crypto

but I have vaguely skimmed stuff on public key crypto

Alexander Gruber

i'm working with a guy right now who is a big shot in multivariate public key crypto

anon

04:04

things like PK crypto and rubberhose crypto are beautiful and surprising when you first hear about them

Tim

I am interested in cryptography, but I haven't seriously tried it. I heard it needs abstract algebra basics

anon

ooh, what's multivariate PK crypto?

Alexander Gruber

but i need to brush up on my theory of quadratic forms i think.

anon

well, more elementary number theory than full-on abstract algebra

Alexander Gruber

@anon do you know what the MI system is?

anon

04:05

nope

Peter Tamaroff

@anon You can always make them in Ps, or COREL

anon

multivariate polynomials over finite fields, eh

Peter Tamaroff

Though I don't think you'll get the same "hand written" feel.

Alexander Gruber

yeah!

so if you're the encrypter

you pick two random affine transformations over an $n$-dimensional vector space over $\mathbb{F}_p$ (usually binary)

so letting $k=\mathbb{F}_p$ and $K$ an extension of dimension $n$ of $k$, you let $\phi:k^n\rightarrow K$ be defined by sending the vector to the polynomial in $K$ with those coefficients

like, $\phi((a_1,a_2,a_3))=a_1+a_2x+a_3x^2$

anon

so $\phi:k^n\to K[x]$?

or is $x\in K$ some chosen thingamajig?

Alexander Gruber

04:10

$k^n\rightarrow k[x] \rightarrow k[x]/\langle \text{primitive polynomial} \rangle=k(x)=K$

i.e. i'm representing elements in the extension $K$ as polynomials in $x$, the primitve element

anon

okay, so $K=k(x)$

Alexander Gruber

yeah.

Charlie

@skullpatrol Hi!

Alexander Gruber

choose a $\theta$ so that $(q^\theta+1,q^n-1)=1$, and define a power map $F:X\mapsto X^{q^\theta+1}$.

apply that to the polynomial you sent your message to, then take the resulting polynomial back to $k^n$, and do your other affine transformation to it

so

anon

well, I will read more about it later

Alexander Gruber

04:14

overall, you've got $L_2\phi^{-1}F\phi L_1$ as your encryption.

to decrypt, you need both $L_1$ and $L_2$, but you give the total map $L_2\phi^{-1}F\phi L_1$ out to people as a whole (so they don't know which parts are which).

and that's it.

right - so anyhow it's a lot simpler than it sounds, it's just difficult to type.

skullpatrol

@Charlie How are you?

Charlie

@skullpatrol not very good, and you?

skullpatrol

@Charlie What's wrong?

... I'm ok...thanks.

Charlie

@skullpatrol I'm a bit irritated, can't sleep..., furthermore, it's hot in here

skullpatrol

@Charlie Don't you access to air conditioning?

Peter Tamaroff

04:23

@anon

anon

?

Charlie

@skullpatrol no, but air conditioning usually makes me sneeze, I hate summer

Peter Tamaroff

I am doing that exercise I told you

user19161

The famous Pedro ping is back.

skullpatrol

@?

Peter Tamaroff

04:24

$x_j^2,(x_jx_k)^3,(x_ix_jx_ix_k)^2$

I checked that all $x_k=(1k)$ satisfy this.

My question is.

anon

showing the elements satisfy the given relations should be trivial. showing these relations generate all possible relations I don't know how to do.

user19161

@khr You said you wanted to do weights and asked about protein?

Peter Tamaroff

@anon OK, that was the thing. I was going to ask "How do I know those are all the relations I need?"

skullpatrol

@JacobBlack Someone in ELU told me to stop acting like you.

user19161

@skullpatrol In what manner?

Sanchez

04:26

@Peter, what is the exercise?

skullpatrol

@JacobBlack General comments.

Peter Tamaroff

@Sanchez I ought to show that those relations define $S_n$ over $FG^{(n-1)}$

Charlie

@skullpatrol haha

user19161

@skullpatrol I mean how do you resemble me?

skullpatrol

@Charlie ;-D

Sanchez

04:27

@Peter, what is $FG^{n-1}$?

Peter Tamaroff

@Sanchez The free group generated by $n-1$ elements.

Mind the parentheses.

Charlie

@skullpatrol ;D I'll try to sleep, good night, skull :)

user19161

@skullpatrol Do you email Charlie?

Peter Tamaroff

@JacobBlack Nosy, nosy.

user19161

@PeterTamaroff Yes, I am.

Charlie

04:33

No, he does not

user19161

@skull I don't get it, what did they say was common between us?

anon

heh

user19161

@anon It's taking forever to load.

anon

loads immediately for me

awllower

@Sanchez Do you remember you once discussed with me a question on representation theory?

Peter Tamaroff

04:36

@anon I see how you keep it up. You rejoice on people's sadness.

Good work.

awllower

It was about representations over non-algebraically closed fields?

Peter Tamaroff

I'm serious.

@anon I'm working on the exercise. It doesn't feel too crazy.

anon

http://en.wikipedia.org/wiki/Schadenfreude

Charlie

@jacob anon's potato is baking too

awllower

iS Sanchez not here?

user19161

04:38

@Charlie Geezis, I now have no idea what that means. Yesterday I thought of 9000 meanings for it.

Sanchez

@awllower, then?

awllower

In that case, I think I should just ask for some help.
Is the theorem of Wedderburn that states the decompsition of semi-aimple algebras into simple ones valid over any fields?

I am thinking maybe we can still use the Wedderburn theorem to decompose representations, even if the field is not closed.

user19161

@anon Real men don't use iPad.

Sanchez

@awllower, it seems so, according to wikipedia

Charlie

@JacobBlack you did ? I thought that the things I said were stupid for people to think about, not that it's not stupid, it is , but you thought about it

awllower

04:41

OK
Hence there should only be linear irreducible representations!

Because the theorem decomposes the rep. into matrix algebras.

user19161

@Charlie Yes, you must have become a banana.

Sanchez

I don't understand what you mean though. My objection last night was not on the decomposition into irrep, I complained that the equality $kG = \oplus_{irrep} V^* \otimes V$ does not hold for nonalgebraically closed $k$.

Why should there only be linear things?

Charlie

@JacobBlack I am just like you o.o

user19161

@Charlie You know, now I have no idea what potato means. Then again, my banana has 9000 meanings as well.

awllower

I mean there the equality $\sum n²=\midG\mid$

maybe still holds?

Wait, the orthogonality relations should still hold, right?

Sanchez

04:45

No

So, orthogonality is right.

Charlie

@JacobBlack can you say "Good night" so I can go to sleep ? It's almost 2a.m and I'm starting to have headaches

Sanchez

But the problem is that, the entry of your matrix ring may not be over $\mathbb{R}$, they can be any division ring over $\mathbb{R}$, eg $\mathbb{C}$

So one thing that can appear, is $M_1(\mathbb{C})$, which has 2 dimension over $\mathbb{R}$ and is what's happening here.

awllower

I see.

Sanchez

So actually, using Wedderburn gives you a proof that the irrep over $\mathbb{R}$ is either 1 or 2 dim

awllower

I thought that orthogonality implies that the sum of squares is the order of G?

Sanchez

04:48

(in the case of your question where $G$ is abelian)

what is orr?

awllower

I have to depart now. Sorry. Per chance the discussion could continue afterwards. :)

Sanchez

Sure.

Ethan

Can someone please evaluate $\lim_{s\to1} \frac{d^2}{ds^2}\frac{1}{\zeta(s)}$ for me in terms of known constants, wolframalpha wont load..

user19161

@Charlie OK good night. I will continue to ponder on the mysterious potato.

Charlie

@JacobBlack :D I tell you later what that means

user19161

04:53

@Charlie Well, do you think I have guessed it correctly?

Charlie

@JacobBlack nope

Peter Tamaroff

@Ethan It returns that it has no computational time

Ethan

ye i know

I can get a numeric result, but I would much rather have it in terms of known constants

anon

$-2\gamma$

Ethan

thats look about right

how do u know?

anon

04:56

mathematica

also you can use stieltjes (sp?) constants expansion and evaluate it analytically

Ethan

its the reciprocal of the zeta function tho

anon

indeed it is

Frank Science

@anon Hi

Ethan

how?

nvm

anon

expand $\frac{1}{(s-1)\zeta(s)}$ partially as a geometric series (using a partial laurent expansion of $\zeta$ around $s=1$), multiply by $s-1$, then look at the resulting coefficients

Sanchez

04:59

@anon, you sure it's not $2\gamma$?

Ethan

no anon is right

Sanchez

geez, my calculations are terrible then.

I really hate it when I get things up to a sign..

Orange Harvester

05:23

A ode to the trolls:
I got your post,
so I posted back
to make sure there was no doubt
if you spam me again
I'll spam you back,
and your servers shall cease to route.
Bikam ka pikkk chik chaa chaaa
Bikam ka pikkk chik chaa chaaa .....

skullpatrol

@OrangeHarvester nice

Orange Harvester

@skullpatrol copied fom userfriendly comics. ;-)

Another one:
Save me from the trolls.
Away from the farce.
Let those know how their comparison is apt to a certain star
when they try to speak, shut them through...
to do so, use a lot of glue.....

This one is completely mine and extempore, and hence somewhat insipid.

Mariano Suárez-Alvarez

It is weird that people upvote questions which, until they are made precise after having had someone ask for claritications in comments, made no sense originally.

Orange Harvester

Because those people have similar difficulties formulating question properly, and they are more apt to relate more to some one who has formulated question in a way similar to them (if poor) rather in a more polished way and hence, more alien?

Sanchez

05:39

@Mariano, what do you think about this question? math.stackexchange.com/questions/311654/…

Mariano Suárez-Alvarez

not much, really :-)

Sanchez

hm okay; I was just wondering if this is a "right" question, I can't judge it myself.

Mariano Suárez-Alvarez

It looks like a fishing expedition

but then, it is the ort of subject where such weird connections do occur, so

Sanchez

Hmm

NullOverNull

06:16

hello

skullpatrol

hi

NullOverNull

Do you happen to know much about simplifying summations?

I want to change a doublesum into a singlesum but am having difficulty

I want to get rid of the inner sum and make it a single sum: $\displaystyle\sum_{k=1}^{N}\sum_{b=2}^{\lfloor\frac{N}{k}\rfloor} \varphi(b)(b^{2}2^{k}k^2)$

wj32

06:41

@NullOverNull: is $\varphi$ Euler's function?

NullOverNull

yes

can it be done?

awllower

I do not believe it is true.

But it turned out to be true...

A little too long maybe.

Chris's sister and pals

Hi

awllower

hi

Ethan

07:05

@NullOverNull

$$\sum_{k=1}^n k2^k=\sum_{k=1}^n\phi(k)k^22^k\sum_{j=1}^{[n/k]}2^jj^2$$

no wait I have it

Orange Harvester

@Chris'ssisterandpals hi

Chris's sister and pals

@OrangeHarvester: hi! :-) Have you seen my last question?

Orange Harvester

wassup? :-)

no

wil see now

Chris's sister and pals

All night I thought of this math.stackexchange.com/questions/311801/…

Ethan

@NullOverNull $$\sum_{b=1}^n\phi(b)b^2=\frac{1}{4}\sum_{k=1}^n\mu(k)[\frac{n}{k}]^2([\frac{n}{‌k}]+1)^2$$

$$=\frac{45}{2\pi^4}n^4+O(n^3)$$

Orange Harvester

07:15

i would guess any $f(n)$ would do such that
\begin{align}
\lim_{n \to \infty} \frac{f(n)}{3^n} = 0
\end{align}

Chris's sister and pals

@OrangeHarvester: hmmm, I'm not sure.

Orange Harvester

@Chris'ssisterandpals any specific reason why it may not work? (I think we can plug in the analysis in your previous question directly here.)

skullpatrol

@Charlie Good night.

Orange Harvester

@skullpatrol she will get up in an hour or so! :P

skullpatrol

@OrangeHarvester Orly ;-)

Orange Harvester

07:27

yeah.

user19161

07:39

Wow guys, Firefox 19 comes with a built-in PDF viewer!

Mariano Suárez-Alvarez

07:51

sigh

a new guy posting his homework and askign for detailed solutions

it must be urgent, too...

grr

Orange Harvester

@JacobBlack yeah.

Mariano Suárez-Alvarez

a sign that I am getting old is how much people saying «solve this integral», «solve this expectation» and so on annoy me :-/

compute stuff, solve problems

3

at most, solve the problem of computing the integral

2

Orange Harvester

there there mariano :-) However annoyed you get, we still like you. :-)

NullOverNull

hm how to render tex/latex/whatever it's called on Android?

Mariano Suárez-Alvarez

:-)

user19161

08:00

@MarianoSuárez-Alvarez Yeah I noticed how common it is here, I wonder why.

Orange Harvester

@NullOverNull if you are talking about mathjax in chat on android, I am not sure it is possible. If you are talking about main site, you need javascript enabled browser.

NullOverNull

yeah the chat

works on main site

user19161

If one can read the comments and some of the revisions on the now deleted posts with regard to the recent hooha on meta, one may become very disturbed. I think someone owes the community an apology.

NullOverNull

Jacob what happened

user19161

Never mind if you don't know.

user19161

08:02

I am being deliberately cryptic here.

Mariano Suárez-Alvarez

usng mathjax in the browser in android should be exactly the same as in any other browser

NullOverNull

doesn't render here

Orange Harvester

@NullOverNull Do you know how to enable Mathjax in chat on PC?

Jonas Teuwen

It needs some optimizations if you use it on a mobile device.

It can be quite possible different config files are served to mobile devices...

Mariano Suárez-Alvarez

@JacobBlack ?

Jonas Teuwen

08:06

I am in terrible pain.

I need a new spine!

Dominic Michaelis

a lil topology question

user19161

@MarianoSuárez-Alvarez Well, never mind. I don't want to hurt more feelings by being too obvious. =) But at the same time, I just could not help expressing myself, because I am quite disturbed by some of the lying and insult there.

Jonas Teuwen

@DominicMichaelis Go to math.stackexchange.com.

Dominic Michaelis

@jonas it's just a yes or no and it's totally basic :)

Jonas Teuwen

If it is so basic, why can't you answer it yourself? 8-).

Mariano Suárez-Alvarez

08:07

@JacobBlack, if there is insult going on, please raise a flag

Jonas Teuwen

@MarianoSuárez-Alvarez Are you... threatening me?

Dominic Michaelis

the closure of A in an arbitrary topological space is defined as the intersection of all closed subset V with the property A \subset V

Mariano Suárez-Alvarez

I am always threating everyone

2

no news there

Jonas Teuwen

Ah! Good boy.

Dominic Michaelis

@jonas someone commented that $\overline{U} \subset U$ is not true in general so i was unsure

Jonas Teuwen

08:09

It is almost never true.

user19161

@MarianoSuárez-Alvarez Well, nah, it's just the lying about what has been done in the comments on a deleted post, and also saying someone has an inflated ego for pointing out the act when pointing it out is the right thing to do.

Jonas Teuwen

(except when closed).

@JacobBlack I suggest to not give a shit about the small things and be awesome in the big things.

4

Comments are little things.

Mariano Suárez-Alvarez

Can you please raise a flag on the thread?

Dominic Michaelis

@jonas sry i wrote it wrong i mean $U \subset \overline{U}$ is not true in general

user19161

@MarianoSuárez-Alvarez It's all deleted already and the community already knows about the threads, and 10k users can see. So I think no need to flag.

Jonas Teuwen

08:10

Otherwise 1) You are frustrated about the comments 2) You are frustrated that you are frustrated about it. Double negative is not positive here.

@DominicMichaelis But it is. It is defined as the smallest closed set enclosing ...

Dominic Michaelis

ok so i was right thanks :)

Jonas Teuwen

You're welcome.

Anyway, I'm trying to be more awesome at the bigger things, so I'm off crawling to the library.

Mariano Suárez-Alvarez

I have absolutely no idea what you are talking about, really

It is impossible for me to judge

but oh well

user19161

@MarianoSuárez-Alvarez Hehe, I don't think you need to do anything. I am just voicing my own frustration, please ignore me. =)

Orange Harvester

@JacobBlack Mail.

BenjaLim

08:15

@MarianoSuárez-Alvarez Hey

quick question

@MarianoSuárez-Alvarez Do you have a super fast proof as to why all finite subgroups of $\Bbb{C}^\times$ are cyclic?

@MarianoSuárez-Alvarez You always don't reply everytime I ping you :(

user19161

@BenjaLim Mariano is a busy man... Just ask on the main site...

Orange Harvester

Is $\Bbb{C}^\times$ the multiplicative group of complex numbers?

user19161

@OrangeHarvester Replied.

user19161

@JonasTeuwen Well, as long as you don't use "asshole" you are safe bro!

Mariano Suárez-Alvarez

Fiite subgroups are entired composed of roots of unity

user19161

08:30

@MarianoSuárez-Alvarez You must be tired, you mean entirely.

Mariano Suárez-Alvarez

if the exponent if $n$, then it is contained in the set of $n$throots of unity

and you know that that group is cyclic, generated by \exp(2pi i/n)

since everysubgroup of a cyclic group is cyclic, you are happy

(in general, every finite subgroup of the multiplicative group of any field is cyclic)

user19161

I notice that typing an answer on the site is different from writing one on paper.

user19161

It is so much easier to make mistakes when typing.

Orange Harvester

Depends on your typing vs writing speed.

user19161

Maybe because one tries to think and type at the same time, and so fail in both.

Orange Harvester

08:38

for me typing is better than writing.

user19161

That is why I type things like 1+1=3 quite often.

Orange Harvester

I can think and type better than think and write.

@JacobBlack replied.

user19161

Yesterday I had to edit 3 times to get a simple answer correct, sigh.

user19161

I try to get it right the first time, but fail miserably in doing so.

robjohn

09:21

@JacobBlack I often have to type things several times. :-)

darn, he was not here, so he didn't get the pings. >8(

robjohn

09:33

@skullpatrol: how are things?

skullpatrol

@robjohn Fine thanks and you?

robjohn

I've been dealing with difficulty on meta. Lots of busy time.

skullpatrol

Nice way to spend your time :-)

user19161

I have been dealing with difficult people my whole life.

user19161

I am glad they are all gone now, but the difficulty is not gone.

Ilya

09:46

hi folks

Chris's sister and pals

@JacobBlack: how about me? Am I difficult too? :D

user19161

@Chris'ssisterandpals No, trust me, you cannot imagine how "difficult" these people are...

Chris's sister and pals

@JacobBlack: See the positive part of them .:-)

Orange Harvester

@Chris'ssisterandpals :D :D

user19161

@Chris'ssisterandpals What if they have none?

Chris's sister and pals

09:54

@JacobBlack: we need to rigorously prove that they have none. If it happens to be so, no one is perfect. :-)

user19161

@Chris'ssisterandpals Yes, no one is perfect, but some are much more evil than others. =)

Chris's sister and pals

@JacobBlack: the power of positive thinking is amazing. :D

user19161

@ora So I tried out Ubuntu 12.04 and 12.10 with GNOME shell and also Ubuntu 12.10 GNOME respin. They all froze multiple times, which really surprised me. Therefore, I think Ubuntu is history for me. =)

Orange Harvester

@JacobBlack Okay, I will try to address this slowly. 12.04 froze on you? I am sceptical about that.

user19161

@OrangeHarvester Yes, it did. I am surprised. Somehow Ubuntu doesn't go well with GNOME shell, whichever variant.

user19161

10:01

So I think if you use Ubuntu, just stick to Unity shell.

user19161

Of course, I appreciate everything Ubuntu has offered. But it is not for me.

Orange Harvester

See, if you do not want stuff to freeze on you, you should not try the latest cutting edge release. Ubuntu has their LTS releases for a reason and cutting edge releases for a reason. Hence, to say Ubuntu is bad because of their instability of a 12.10 release (which is not LTS) is not fair.
Now, coming to Ubuntu 12.04 LTS freezing, does it still freeze on the latest updates? Have you updated it completely?

user19161

@OrangeHarvester It might be because I used Chrome on it. Possibly. Anyway, I am currently on Debian and Iceweasel.

Ethereal

Meta looks very messy today.

user19161

@robjohn I saw something about you there. Don't worry, I think you are a great moderator.

awllower

10:10

A shocking claim:

user19161

@Ethereal Yes, I just saw some more new stuff which is shocking me hahahaha!

Ethereal

@JacobBlack I saw that and it is true that RobJohn the Great is the greatest!

user19161

@Ethereal I also did not expect nonregular chat users to read the transcript.

user19161

So be careful what you say in chat! The world is reading!

robjohn

@JacobBlack I probably shouldn't have discussed the possible outcome on chat, but other than that, I don't think anything was improper.

user19161

10:17

However, nonregulars often misinterpret or misunderstand the context.

user19161

All this stuff shocks me because I do know some people more than just the random user of MSE.

Ethereal

As I saw with the other answers, actions were taken.

@MarianoSuárez-Alvarez Slow machines like me cannot compute the awesome stuff you do here. :-)

user19161

@gnometorule I just want you to know that robjohn is a great moderator and great man to me, even though I don't know him in real life. We get very informal and humorous in chat, and this in no way reflects how flags or real problems are handled. — Jacob Black 36 secs ago

user19161

@robjohn For you. ^

robjohn

@JacobBlack Is that in another chat?

user19161

10:25

Because I think gnometorule's comment was too strong there, so.

robjohn

I see it. nvm

user19161

To say this is "power abuse" is not correct. I have to speak up.

robjohn

I would not make a humorous comment if I thought that plagiarism had been involved. That :-D comment was a few weeks before this incident, and I had pretty much put it out of mind.

user19161

One can also make a humorous comment and still investigate thoroughly, no contradiction.

user19161

Of course, I can understand that gnometorule must have been feeling a bit emotional after all this.

Ethereal

10:33

How do you get the permalink for comments?

user19161

Copy link location of the comment time stamp.

robjohn

@JacobBlack I hope that everyone involved reads my responses. With the possible exception of the comment I made about the possible outcome, I don't see any reason to do anything differently.

Ethereal

@JacobBlack Oh, works. Thank you Jasper

user19161

@robjohn Your comments are harmless, QED.

user19161

One cannot just take a comment or two and interpret 9000 things out of it.

4

user19161

10:36

One must take several comments, see the entire context, think a few days, and then interpret a few things out of it.

user19161

One must also know that with new info, old info is reevaluated.

Ethereal

@JacobBlack Have a star for that great thought. :'-)

user19161

@Ethereal My thoughts often put me in conflict with this world.

user19161

There are various people in real life who have judged me to be immoral based on certain things I have done, but they know shit about me.

Orange Harvester

The wheels just keep on turning.
the drummers begin to drum
I don't know which way I am going
I don't know which way I've come.

robjohn

10:44

user19161

@robjohn I think he is referring to site comments.

Ethereal

@robjohn I meant comments.

Yes.

robjohn

@Ethereal Ah. I think you can right click on the timestamp and copy the link location

Which is just what Jacob said :-)

Ethereal

@robjohn Thanks again!

user19161

@robjohn Jacob took his carrots, as advised!

Ethereal

10:50

@ora Are you passionate about your work? Just asking...

user19161

@Ethereal You need at least 3 letters.

robjohn

@Ethereal You need at least three characters to ping :-)

user19161

Jinx.

Orange Harvester

@Ethereal Yes. Very.

Ethereal

@OrangeHarvester OK, that is nice.

Orange Harvester

10:53

@Ethereal However, I have grown from experience to temper that reckless passion with some measured caution which I have found has been important in my case. (I still have my moments of recklessness/instinct though.)

Ethereal

Why do people write their email addresses like abc <at> xyz <dot> com instead of [email protected]?

Transcript for

$Mathematics$ Mathematics