Mathematics - 2011-11-19 (page 2 of 3)

QED

6:00 PM

Serge Lang. haha

@Matt, so think about how that implies 2541 is divisible by 7 - the same reason holds for the abstract 'a'

Matt

ok

It implies 2541 is divisible by 7 because a 7 was factored from it?

with 7^2

QED

no

Matt

Ok what am I missing?

Srivatsan

@JonasTeuwen may be we should in fact try assigning it as a homework question. =)

QED

@Matt, well the prime factorization of a^2 contains a 7 because of 2541^2 = 7*922383.

Jonas Teuwen

6:07 PM

@Srivatsan Exam question!

QED

and you want to show that the prime factorization of a contains a 7: so you need to relate the prime factorization of a to that of a^2 now.

Matt

Where are these numbers coming from? 2541^2 = 7*922383

QED

I created them

Srivatsan

@JonasTeuwen Hmm, one of the profs in my undergrad gave an option is a course he taught: either a conventional exam or a challenging (take-home) project. But you have no idea what the take-home is going to look like; and if you fail on it, you have no guarantees on passing. How about assigning this as the second option?

QED

I think it's helpful to use actual numbers sometimes

Jonas Teuwen

6:09 PM

@Srivatsan I'd go for the take-home project.

Srivatsan

Read the last sentence now. =) // You'll most likely fail that course if you pick that option. =)

Matt

Ok, but if they're arbitrary how are they telling us the prime factorization of a^2 contains a 7

QED

I guess I sort of slipped back into using 'a' there :/

Matt

Since a can be anything

QED

that's not very good

sorry what I wrote was confusing because I switched from the concrete example back to the symbols when I shouldn't have

J. M.

6:11 PM

@JonasTeuwen One of Knuth's books was asking for the solution of the Diophantine equation x^n+y^n=z^n for n > 2 as an exercise...

Jonas Teuwen

@JM Is it older than Lang's?

QED

(since it's quicker to type 'a' than 2541)

J. M.

Don't know; I'm far away from my personal library...

Asaf Karagila

I hope my cooking works.

I'm playing god with yams and chicken.

Matt

Sounds good.

Srivatsan

6:15 PM

@AsafKaragila yammy!

Asaf Karagila

Yes.

I put the yams for about 15 minutes prior, and I'm gonna check on everything in 10 minutes.

I hope that everything works, and in 20-25 minutes I'll have a wonderful dinner.

I wish we had some wine...

t.b.

@AsafKaragila No chance, Jonas drank all of it yesterday...

Asaf Karagila

I thought that he drank WHISKY...

Matt

He drank both.

t.b.

@AsafKaragila He left a bit whisky just for you, as far as I understood. This despite the fact you closed two questions on him. That's how nice guys those people in the Netherlands are.

Srivatsan

6:23 PM

I was the other non-nice guy.

Asaf Karagila

Only one.

J. M.

Just why did you guys deprive Jonas of t.b.'s gifts? :P

Asaf Karagila

I'm a jerk. Also, Koenig claimed taxes were issued. :-P

Srivatsan

I will make it up to him some time.

May be even post a bad question just so that he can close it. =)

t.b.

@AsafKaragila I still take this Koenig thing as an usurper's claim to fame...

Asaf Karagila

6:27 PM

I just flipped the chicken, it is going to be legen-wait for it-day!

robjohn

@Srivatsan: Or a duplicate.

Srivatsan

@AsafKaragila Legenday, indeed. =)

@robjohn Not much fun IMO. But Jonas can decide what kind of closure... =)

Asaf Karagila

@tb My iPhone just notified me on Dictionary.com's word of the day. Fitting to this discussion, it is "knavery".

@Srivatsan Oh yeah, I prepared a vicious sauce for the chicken:

t.b.

Nice comment by Brian.

J. M.

I think I have the hierarchy down: the first guy is the "judge"; the next three are the "jury", and the last one is the "executioner"...

3

Asaf Karagila

6:30 PM

Garlic, olive oil, cummin, bahart, salt and pepper, soy sauce, date syrup and mustard.

t.b.

Don't forget to mention the Swiss chocolates for dessert...

Asaf Karagila

Have you sent me any?

Jonas Teuwen

I have Swiss chocolate!

Asaf Karagila

I have a lot of dirty dishes!!

Jonas Teuwen

And the book about optimal transportation by Cédric Villani.

t.b.

6:32 PM

@AsafKaragila Yes, they were hidden in the package of empty bottles you refused to accept :)

Asaf Karagila

@tb Drats :-)

J. M.

As a matter of pure opinion: when I think of Swiss chocolates, I don't count the ones by Nestlé...

t.b.

Who would?

That would be, like, calling Coke a delicious beverage...

J. M.

Good point. :)

Asaf Karagila

Or calling beer an alcoholic beverage...

Jonas Teuwen

6:34 PM

I just have "Swiss Extra Noir". Well only 78%, I wouldn't call that Extra noir, but oh well.

t.b.

Ah, Lindt, I presume

Jonas Teuwen

Yes.

Just a blend.

J. M.

@JonasTeuwen Depends on what they merely call "Noir"...

Jonas Teuwen

I have Lindt 99%.

Asaf Karagila

My friend was in Germany a while ago. He brought the most awesome pickled cucumber. Do you have those in Switzerland too, @tb?

Jonas Teuwen

6:35 PM

I used to have some 100% "single malt" chocolate, but the shop doesn't sell that anymore :(.

J. M.

Now that is noir...

t.b.

@AsafKaragila Yes, we do have them. But they get more and more replaced by the smooth water-balloons the Tulip-factories produce.

Asaf Karagila

@tb Say wha?

QED

@Matt, any luck?

Jonas Teuwen

Do the Swiss have any good drinks?

Belgium has awesome beers, France awesome wine, Schotland awesome whisky.

So...?

Asaf Karagila

6:37 PM

germanpickles.com/spreewaldhof-get-one-salt-dill-pickle.htm

t.b.

@JonasTeuwen We have awesome milk :)

Matt

@QED, ehhh, reading my textbook, will let you know

Srivatsan

I googled for single malt chocolate, and aptly enough for this conversation, this was the first hit: theochocolate.com/store/products/specialty-chocolate/….

t.b.

@AsafKaragila Well, in the super markets we have the perfectly round tomatoes from the Netherlands and also the Cucumbers. They essentially consist of water and color.

Asaf Karagila

I wonder if this is the same Theo... :-P

Jonas Teuwen

6:37 PM

@Srivatsan I mean single sourced cacao of course ;-).

QED

@Matt, What I was trying to say was if a^2 = 3^2*7^2*11^4 then a = 3^1*7^1*11^2

Jonas Teuwen

@tb True. But they cost almost nothing!

t.b.

@JonasTeuwen But they don't deserve the name!

Jonas Teuwen

Yeah, there is an inflation of names going on. The "Excellent" tomatoes are actually normal ones.

Matt

@QED right

t.b.

6:39 PM

Right, qed.

Matt

@QED im not sure I covered prime factorization yet

QED

yes

I outlined how to prove it

Matt

@So im seeing if there is another way to prove without it

QED

you would be better off learning about prime factorization

t.b.

When QED's right, QED's right.

Matt

6:41 PM

Haha, ok

Jonas Teuwen

I wonder if the Monge-Kantovorich problem has any connections to harmonic analysis.

Matt

Is there a prime factorization theorem?

QED

yes

Theorem Unique prime factorization: let n be a natural number, then n can be written uniquely as a product of prime powers

Asaf Karagila

Wow. The chicken is AWESOME.

The yams are not yet ready.

Srivatsan

@JonasTeuwen It seems to me that I am hearing about the Monge-Kantorovich problem (spelling, Jonas =)) more and more often. I wonder why that is...

Jonas Teuwen

6:46 PM

=).

@Srivatsan Because it has many connections to many things. :P.

t.b.

@Srivatsan Be easy on him. He almost wrote his Master on the Ornstein-Uhlbeck operator

3

Srivatsan

Can you explain one or two connections? The high level story (and where I can read more about it)...

Jonas Teuwen

@tb ... :D.

@Srivatsan Well one are the "gradient flows" by Ambrosio and guys like that.

Villani's book seems to have a lot of information about that.

@tb Eventually I wrote it the right way.

Srivatsan

Ok. Thanks.

Asaf Karagila

Right. Dinner time. I will see you folks later.

Matt

6:58 PM

@AsafKaragila: Enjoy! We had roast chicken too, with roasted parsnips and steamed sprouts. It was quite good. : )

J. M.

...and bedtime for me. Later, y'all.

Matt

Good night!

Srivatsan

@QED @Matt This is called the fundamental theorem of arithmetic.

Matt

That rings a bell

Matt

@tb: Don't forget the tomatoes from Spain (Almería) which I think are even worse than the ones from the Netherlands.

Srivatsan

7:10 PM

@Matt How did you make your gravatar, Matt?

Matt

@Srivatsan: PS and AI

Srivatsan

PS?

Matt

Photoshop

Matt

@QED Ok so I found the theorem in my book

But I am not seeing where you got this: a^2 = 3^2*7^2*11^4 then a = 3^1*7^1*11^2

Matt

AI = Adobe Illustrator

Matt

7:12 PM

Here the theorem is: imgur.com/AJnJb

QED

dividing by two

Srivatsan

@Matt Thanks. I figured that one after a bit of googling.

QED

each of the exponents

Matt

: )

Matt

So because a is divisible by 7, from the unique prime factorization theorem we know a^2 = 3^2*7^2*11^4?

QED

7:13 PM

no

Matt

*becasue a^2 is div by 7

Srivatsan

I like the graininess of that gravatar.

QED

let's forget about the concrete example and go back to the abstract

Matt

ok

QED

You know that the prime factorization of a^2 contains p

Matt

7:14 PM

@Srivatsan: Me too. It's a logo of something, I didn't design for this site.

QED

and you want to prove that the prime factorization of a contains p

Matt

ok

Srivatsan

@Matt Oh really. Can you tell us for what it is a logo of?

Matt

@Srivatsan: I'm afraid that is classified as of now. Why?

Srivatsan

@Matt That's cool. Curiosity, that's all.

Matt

7:16 PM

And it's not particularly interesting either, trust me.

QED

so by unique prime factorization, if a contains p^i (we aim to show that i is nonzero) then a^2 contains p^{2i}

Srivatsan

@Matt Ok. =)

Matt

@Srivatsan: I'll tell you once it's finished : )

Srivatsan

Sure. Thanks.

Matt

@QED ok kind of get that

QED

7:17 PM

that proves the theorem

Matt

that prove: If a^2 is divisible by 7 then a is divisible by 7?

QED

set p = 7 to prove that

Matt

hmm

This seems backwards: if a contains p^i (we aim to show that i is nonzero) then a^2 contains p^{2i}

QED

so the thing we want to prove is p|a^2 -> p|a

Matt

yes

QED

7:22 PM

so we want to prove p|a given the hypothesis p|a^2

the hypothesis tells us something about the prime factorization of a^2

by the theorem, we can state that 'a' has a prime factorization and then we can relate that to the prime factorization of a^2 to incorporate the information we have

Matt

ok

Asaf Karagila

That was good.

Matt

Im not putting it all together :/

QED

that is it all

Matt

hmmm

by unique prime factorization, if a contains p^i then a^2 contains p^{2i}

a^2 does contain p^{2i} (p=7)

thus, a must contain p^i

QED

7:32 PM

by "contains" in this context I mean that, say, 6456681 contains 11^4 - and does not contain 11^3 - since it equals 3^2*7^2*11^4.

sometimes people double bars for that like 11^4||6456681

Matt

and as long as it contains 7^i, it is divisible by 7

QED

actually that isn't note isn't needed, what you wrote is perfect

Matt

ok let me try to wrtie this out formally

QED

make sure you know exactly when unique prime factorization is used too

Matt

@QED how is this: imgur.com/SJHwX

robjohn

7:46 PM

@Matt Photoshop and Illustrator to edit a small bitmap? That seems a bit overkill. Even Photoshop is a sledgehammer, but if it is the only graphics editor available...

QED

it's not perfect but the reader will understand

Matt

Is it valid?

Matt

@robjohn: How would you have done it? I am using these two to make all the graphics so it seemed like the least work.

QED

you wrote e.g. p^{e_1} p^{e_2} but it should be p_1^{e_1} p_2^{e_2}

robjohn

@Matt Well, I don't know all that you did for it, but what did you use Illustrator for?

QED

7:49 PM

but also if a = 7^{e_1} p_1^{e_2} p_2^{e_3} ... then a^2 = 7^{2 e_1} p_1^{2 e_2} p_2^{2 e_3} ... is just algebra, it doesn't need theorem 4.3.5

Matt

@robjohn: Inking for example.

@robjohn: When you design characters PS is good but you want crisp lines so you ink the lines in AI...

@robjohn: I'm kind of reluctant to learn gimp, because I really have no spare time.

robjohn

@Matt I believe if you expand the bitmap using the correct method, the lines will be crisp. I will try.

@Matt I am not advocating that you learn anything new, just wondering why you used Illustrator, too.

Matt

@robjohn: Sure. I'm interested in better ways to do the things that I do.

Matt

Ah yea true

Matt

@robjohn: I copied it from here vimeo.com/982474

Matt

7:55 PM

@QED improvement?

@QED only thing is that a isnt explicityly stated to be > 1

QED

no

oh then that's fine

you should say it explicitly

Matt

oh wait thats flawed

QED

a = 7^h p_1^{e_1} p_2^{e_2} ... where h is a non-negative integer and e_i are positive integers

since 7|a^2 we know 2h is nonzero

Matt

a=p_1^(e_1 ),p_2^(e_1 ),p_3^(e_2 )… would be better for the first bullet point

then if a=7^(e_1 ),p_2^(e_1 ),p_3^(e_2 )… then a^2=7^(〖2e〗_1 ),p_2^(2e_1 ),p_3^(2e_2 )… for the 2nd like I have

QED

I think doing it that way is better than starting out with 7 involved

but both work

Matt

7:59 PM

Cause basically what I am saying in the 2nd bullet is

QED

you have a lot of freedom when writing proofs

Matt

If the prime factorization of a has a 7 in it

then the prime factorization of a^2 has a 7 in it

the prime factorization of a^2 has a 7 in it

so the prime factorization of a has a 7 in it

Waitt.....

Is that a logical error

if...then

p -> q

q

therefore p

QED

yes may be easier to avoid thinking "if A then B" and instead think "A implies B"

Matt

This is the converse error

uh oh

Or should I not think of it this way

Since we are using actual numbers

QED

converse error isn't a valid inference

your proof doesn't use it though

Matt

8:06 PM

ok

It just is in the same shape sort of

But it doesnt apply because we are using actual algebra

QED

what do you mean

do you think you used it somewhere?

Matt

How is that for a final polished proof

QED

I don't think you should say "if a = 7^{e1} ..."

just delete that line

also "is true" is redundant

you can just say "X" rather than "X is true"

also on the second last line you repeat "since ... then, by algebra ..." you can just write "by algebra ..." (it's easier to read that way)

Matt

ok

Also I ha p1,p2,..

QED

and you can delete the last bullet point too, since "triangle of dots a is divisible by 7" covers it

Matt

8:12 PM

it should be p1*p2*

Since they are factors not a series right

QED

yeah

Skullpatrol

@Matt May I suggest you add "QED" at the end.

Of your polished proof.

Matt

I have the therefore symbol

Skullpatrol

oh ... I didn't see that , sorry

Matt

Thanks all ;)

Skullpatrol

8:17 PM

@QED What do you think about replacing "therefore" with "QED" Sir.

QED

You should do the proof of the unique factorization now

@Skullpatrol, I think 'triangle of dots' should be replaced with "therefore" (but that's just personal) also I use a box symbol to end proofs rather than "QED"

@Matt, I'll help you if you want

Matt

@QED, there is actually a 2nd part to this study problem:

b) Use a) and the quotient remainder theorem and case reasoning to prove that the square root of 7 is irrational.

QED

That's not as interesting

Matt

lol, what as proving the uique factorization theorem?

QED

yes

or (a)

Skullpatrol

8:23 PM

@QED I agree, there are already a lot of symbols in the proof and Latin phrases aren't that popular anymore ... with all respects to your user name Sir.

Matt

Haha, according to my text, the proof for the UFT isnt outlined until chapter 8

this test is on chapter 5 :p

I proved the sqrt(7) was irration using a proof by contradiction before

Not sure how to do it with case reasoning and Quotient-Remainder Theorem this time however

robjohn

@Matt: choose Image > Image Size... then select Nearest Neighbor as the scaling method. That will do the pixelated scaling that it appears you wanted.

Matt

@robjohn: Let me try...

robjohn

@Matt: However, I see you are trying to follow the tutorial, which indicates a way of doing painted line art, it seems.

Matt

@QED why do you say this is not as interesting as 1a

Matt

8:31 PM

@robjohn: Yes, I'm using a tablet. Not for this avatar though.

QED

It doesn't take any new ideas

robjohn

@Matt In general, what they say there might be reasonable; but for making that avatar, it seemed a bit much. However, if you have the tools and you know how to use them to get what you want, it's all good.

Matt

@robjohn: I'm not sure how to get the number of "pixels" to be exactly what I chose for the logo using scaling with nearest neighbour.

@robjohn: I have the tools but I want to use them optimally (which presumably I don't)

@robjohn: Can you reproduce my avatar using scaling?

robjohn

@Matt Lemme try...

Skullpatrol

@QED Could you please take a look at this proof Sir math.stackexchange.com/questions/26445/division-by-0/…

Jonas Teuwen

8:38 PM

@robjohn Stein's Harmonic analysis is written using Emacs :)).

QED

@Skullpatrol, any questions?

robjohn

@Matt How big do you want it?

@JonasTeuwen I am not too surprised. Emacs is very popular.

Skullpatrol

@QED No, just looking for an opinion Sir

QED

I don't like it

Matt

@robjohn: Try avatar size. Looks like approx. 100 x 100 px

the one next to where you enter text

Skullpatrol

8:41 PM

@QED But why? Sir

QED

You don't need to prove a0 = 0 that's immediate from the definition

robjohn

@JonasTeuwen TeX, X Windows, and GNU Emacs. TeX and not LaTeX?

Jonas Teuwen

@robjohn Yes, that surprises me too.

Skullpatrol

@QED Which definition?

robjohn

@JonasTeuwen When I was there, people were using TeX, but that was before LaTeX was around.

QED

8:42 PM

of 0

robjohn

@Matt Just a sec...

QED

well no that's true

Jonas Teuwen

@robjohn Indeed, Timothy Murphy has written the book.

QED

it's proved exactly as you did

Matt

@robjohn: You have all the time you like ; )

QED

8:43 PM

from field axioms

@Skullpatrol, and it's a very long post

I don't think there's much to say on this topic

robjohn

@Matt Did you want the jaggies, or did you want a nice sloped line?

Matt

@robjohn: Let's see jaggies.

Skullpatrol

@QED Thank you for your time and attention Sir

Matt

@robjohn: Do you make graphics for your son's games? Or does he make board games?

robjohn

Try this: i.stack.imgur.com/Rqi4M.png

QED

8:47 PM

The question itself doesn't even make sense

robjohn

That's 128x128 expanded from a 9x9 source.

Matt

@robjohn: Awesome. And how did you draw it?

QED

@Skullpatrol, nobody seems to be starting with the definition of a/b

Matt

@robjohn: Just rough shape with mouse or trackpad?

robjohn

@Matt I copied your avatar to get the right shape, then used the pencil tool to adjust colors

with the trackpad actually

Skullpatrol

8:50 PM

@QED I tried to start with the definition of a/b Sir

Matt

@robjohn: But the challenging bit is making the shape!

: D

Not the scaling...

Matt

@QED minus.com/mbdI3DAuzr#

robjohn

@Matt well, I wanted to get the same shape as you had. I could recreate the shape using the pencil

Matt

That is how I proved this problem previously using proof by contradiction

QED

@Skullpatrol, ah yes this is good but I looked past it because i was focused on the proof

robjohn

8:51 PM

@Matt Gads these proofs can get complicated.

Matt

@robjohn: Using the pencil in PS is a pain in the neck and boils down to what I did in AI : )

QED

@Skullpatrol, The comment "Assuming 0 has no reciprocal is begging the question" is bad

robjohn

@Matt For a 9x9 pixel image, it is not too bad.

QED

but it might have been posted in regard to a previous edit of the answer

Matt

@robjohn: No it's cool. I'm quite pleased, I still learned something: when I scale this I should use nearest neighbour to avoid blurring : )

robjohn

8:54 PM

@Matt I used Intaglio to do my avatar

Skullpatrol

@QED Yes, that is why I had to prove 0a = 0 Sir

robjohn

@QED: were you recently knighted?

Matt

@robjohn: I've not heard of Intaglio before. Did you buy it to make your avatar? Or is digital art a hobby of yours?

robjohn

I got it as a replacement for MacDraw which is what I used to make diagrams for math and astronomy papers.

I save the images as EPSF and they go right into my LaTeX documents.

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$Mathematics$ Mathematics