Mathematics - 2011-10-25 (page 2 of 2)

mafutrct

15:03

@Gortaur going to contemplate a little further, if I get around to it I'll post it on MSE. Thanks so far! See you around

Gortaur

@mafutrct I was of small help I think, but you're welcome

robjohn

@JM out of manna to rain on people?

Gortaur

@robjohn From Dust?

robjohn

?

Gortaur

they changed the title apparently

From Dust

|genre = God game, sandbox |modes = Single-player |platforms = Microsoft Windows, Xbox Live Arcade, Playstation Network |ratings = |media = Download, cloud computing }} From Dust is a video game, designed by Eric Chahi and developed by Ubisoft Montpellier. Ubisoft described it as a 'spiritual heir' to Populous, a British game developed by Peter Molyneux and Bullfrog Productions in 1989. The PC version was originally scheduled to be released at the same time but was delayed to August 17 due to undisclosed reasons. From Dust is distributed online via Steam, OnLive, GamersGate...

J. M.

15:05

But, but... I answer because its fun! That they're actually helpful is a happy coincidence...

(Okay, not entirely true. But I'm missing the sort of questions I like and are yet unanswered...)

Gortaur

how do you find time, I'm curious

J. M.

Who, me?

robjohn

My son probably knows a lot about From Dust, but I don't game online much.

Gortaur

I've spent almost all the day reading the book and don't have just a 30 min to focus on a small problem

@robjohn is it online??

@JM yes

robjohn

@Gortaur I don't know. It sounded as if it were, but I didn't read more than the part of the page that showed on chat.

Gortaur

15:09

@robjohn the only online game I played was World of Tanks, there I've got Koenig Tiger, played a bit with this lovely monster and stopped playing: to go further I either have to spent too much of the time or to start wasting my real money on this game

it was for free, but if you pay you have +50% of all the stuff. no way.

robjohn

That is why I don't start playing games on the computer. Board games with people don't tend to be such a time-sink.

Gortaur

@robjohn sometimes I play in NFS: hot pursuit just to relax my mind

because grand piano is not always available unfortunately (

J. M.

@Gortaur: Hmm, find time for math? Just the same way I find time for my other hobbies...

Gortaur

@JM: no, I mean that sometimes it seems that you're doing it without any break ) though maybe it's because it's me who slow doing this stuff

J. M.

What do you mean? I have an actual job, too... :)

Gortaur

15:17

@JM I don't have any doubts about it. I wonder how do you manage to work so much on MSE the same time working on your job

J. M.

Also, I do take breaks from math.SE . Sometimes as long as a few months.

@Gortaur: believe it or not, math.SE is one way I unwind.

Jack Schmidt

jm: your picture for the pole is good

Gortaur

unwind=relax?

J. M.

@Jack: I had the picture prepared, but Sasha already wrote an answer... :D

@Gortaur: yes indeed.

Gortaur

@JM ok, then it's my lack of knowledge

J. M.

15:20

After a grueling job, it's fun to either a. make quips or b. solve a few puzzles...

math.SE fills that particular void for me

However, when I'm too baked to even think about math, then I watch TV...

Holy everloving... where are all these people coming from? Look at the number of views!

Jack Schmidt

15:37

if only they came with questions about the structure of finite groups

2

Q: Involution centralizer does not determine the group

Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution? The Brauer–Fowler results show that if a finite group has no non-identity normal subgroups of odd order (so is core-free), then its order is bounded by a function...

group-theory

one thing I like about symmetry and graph theory is that if I have question there, I just ask my 150 students it, and one or two at least find me an answer by the end of class

J. M.

Heh. It requires a some straining of the imagination, though. :)

Well, two grateful customers today! I guess this day isn't so bad after all...

robjohn

15:57

@JM On MSE or IRL?

J. M.

Well, MSE. :)

robjohn

Got your priorities straight at least :-)

Gortaur

)

sometimes I think that my posts like ')' took too much space vertical compare to their horizontal size

J. M.

@rob: In RL, no customers; just technicians. :) I don't have the necessary charm to face customers... :D

Gortaur

1

Q: complete metric space

Prove or disprove: $(A_i)_{i=1}^\infty$ are closed subsets in a complete metric space. Assume that there is an open ball in the $\bigcup\limits_{i=1}^\infty A_i$ , so exists $k$ s.t $A_k$ contains an open ball as well. Thank you!

homework metric-spaces

I wonder why the [general-topology] tag was deleted. quite topological I think

J. M.

16:05

You could put it back, you know... along with a comment explaining why it should be there.

Gortaur

could I? you mean not rollback but edit one more time?

J. M.

Well, you have the rep for the task...

Gortaur

aha. lame start of the week on MSE for me

J. M.

Also, Chris looks like a reasonable guy; if you have a good reason, he shouldn't have trouble with you restoring the tag.

Gortaur

@JM looks like

J. M.

16:11

I've only seen his activity, but I haven't actually talked to him (unlike, say, with you); hence "looks like".

Gortaur

@JM I just agreed with you |

and put my supersmart answer 1 minute before your comment ) while you were editing I guess

J. M.

Well I wasn't sure! I don't know him as much as I know you.

If I still had votes I'd vote your answer up...

Jonas Teuwen

If I have weak convergence in L^p (1 < p < infty), what are some conditions that would allow me to extract a (sub)sequence that converges pointwise a.e.?

Gortaur

@JM 2 is more then enough for such answer ) but you can upvote it tomorrow ;D

maybe I shouldn't complain: +95 today for doing nothing for 20 minutes overall, not so bad

see you all tomorrow

J. M.

See you!

Jonas Teuwen

18:25

Would it be inappropriate to write on homework assignment you've been correcting "Well done!" if they scored less than 30%?

Jack Schmidt

18:43

@JonasTeuwen For the United States: I recommend very conservative grading. Indicate clearly what they've done wrong and what they've done right. Don't encourage bad behavior by joking about it, and don't encourage complaints by mocking them. Remind them about office hours and any on-campus tutoring services.

Jonas Teuwen

I think it is less formal here (The Netherlands).

But I won't do it, but I was tempted to do that sometimes...

But thanks for the advice @JackSchmidt.

Jack Schmidt

no problem. i figured the direction of the advice would be close, but the magnitude would probably be wrong.

robjohn

19:59

"Well done" and less than 30% might be applied to cooking meat and the fat content thereof; however, I prefer less heat and more meat. As for homework, not so much. :-)

Jonas Teuwen

21:18

It can be slightly frustrating that easier questions get much more votes. But, there are people that spend a lot more time than me on long (and complicated) posts and still don't get many votes.

robjohn

@JonasTeuwen I agree. Look at some of my answers. :-)

I will collect a sampling that display this tendency.

Jonas Teuwen

Yes :).

robjohn

Of course, in so doing, I may get more votes ;-)

Jonas Teuwen

Why do we actually like those votes so much? :).

karmic_mishap

Perhaps this tendency reflects a tendency of users on the site to vote on what they understand.

robjohn

21:21

@karmic_mishap that is true. However, they should all just understand that my answers are correct :-D

Mike Spivey

Of course, all of this means that those of us who do understand a particular long and complicated post should upvote it!

Or we could just automatically upvote all of robjohn's answers. ;)-

Asaf Karagila

Sheesh, that question is open again :|

karmic_mishap

I am beginning to think that I serve as a warning to others

robjohn

@MikeSpivey I commented on one of your fine answers regarding a binomial sum \binom{n}{k} 2^{k^2}

Mike Spivey

@Asaf: What question?

Asaf Karagila

21:23

@Mike: That "Inherently discrete concepts" thing. He even added tags and a bounty.

Jonas Teuwen

I often just do +1 on very long posts that look good before I have even studied them completely...

Mike Spivey

@robjohn: I saw it. Thanks.

robjohn

@MikeSpivey this one

@Mike sorry, I have trouble with the links.

Asaf Karagila

@JonasTeuwen Yeah, like "Good for you for sitting so long..." I get those a lot when I write incredibly long posts about the axiom of choice :D

robjohn

I thought Asaf was asking which question I was talking about

Asaf Karagila

21:27

@karmic_mishap If that was true then some of my posts would not have gone past 3-4 votes. Incidentally, some of those got beyond 20.

robjohn

And to display my craziness, I kept replying to the same message, expecting a different result.

Asaf Karagila

@rob: I did the same thing over and over again and my sound came back. So it might not be insane to do that :D

Jonas Teuwen

@AsafKaragila I don't have such a flagship yet.

karmic_mishap

Hmmm @AsafKaragila, I guess that means that there may be a fair amount of what @JonasTeuwen was referring to.

Mike Spivey

@Asaf: I see. Yikes, especially that the question would elicit that comment from Arturo (which I will refrain from reposting here).

@JonasTeuwen I do that, too, sometimes - particularly if the answerer is someone whom I have had reason to trust in the past.

Asaf Karagila

21:30

@karmic_mishap Yes, we all do that. When you see a post that you feel is trustworthy, you can vote it. Sometimes just long votes that are not complete junk would be subjected to several upvotes. Not always, I'd guess it depends on the user as well.

Jonas Teuwen

I wonder if someone already wrote a script that enables me to have this channel in irssi.

Asaf Karagila

It goes to the Vote Early, Vote Often ideology.

Jonas Teuwen

What is the maximum number of votes you can give in a day?

Asaf Karagila

30 votes. 40 if you spend on questions more often than on answers.

Mike Spivey

@Jonas: 40

Asaf Karagila

21:31

Oh, now it's just 40? I recall you have to have at least 10 votes on questions there.

Mike Spivey

@Asaf: Maybe I'm wrong.

karmic_mishap

@JonasTeuwen the closest thing I've seen so far is this project that's still being worked on: github.com/ghewgill/soirc

Mike Spivey

@Asaf: It looks like you're right: "A maximum of 40 votes can be cast per user per day, however, to reach the maximum you must vote on at least 10 questions." stackoverflow.com/faq#reputation

Jonas Teuwen

@karmicmishap Looks far from finished.

Mike Spivey

@rob: How was Griffith Observatory?

robjohn

21:37

It was a very nice lecture about the supermassive black hole at the center of the Milky Way.

Asaf Karagila

Awesome. Does it suck hard?

robjohn

The person speaking was very knowledgeable.

@AsafKaragila extremely.

Mike Spivey

@Asaf: I laughed hard enough at that that I started choking.

robjohn

@MikeSpivey The Observatory is usually closed on Mondays, but they have a series of lectures for the members of their support group.

Mike Spivey

@robjohn Cool.

Jack Schmidt

21:41

Anyone like noncomm rings? I think this question may have two "wrong" answers, but my only answer is pretty round-a-bout.

2

Q: every $R$-module is free $\Rightarrow$ $R$ is a division ring

From Grillet's Abstract Algebra, section VIII.5. I'm trying to solve exercise 9. I suspect it should be formulated: Let $R$ be a unital ring. Then $R$ is a division ring $\Leftrightarrow$ every unital left $R$-module is free. Attempt of proof: $(\Rightarrow)$: theorem 5.2. $(\Leftarrow)...

linear-algebra abstract-algebra modules

Asaf Karagila

@MikeSpivey Well, then it is a good thing that I know the Heimlich maneuver!

robjohn

Is that an alien using a wireless headset?

Jack Schmidt

i suspect a ninja dj

robjohn

aha.

Mike Spivey

@AsafKaragila If you can perform the Heimlich maneuver all the way from Israel to the U.S. then I am mightily impressed.

2

Jack Schmidt

21:43

if you cover one eye, it becomes a pirate dj

Jonas Teuwen

I have one vote left for today. What is a topic that deserves my final vote?

robjohn

If no one else is going to speak up, you can look at this.

Jack Schmidt

@anon: hey, I couldn't get those notes to solve my problem. the graphs i've checked have girth 3, and the left hand side is about 4

robjohn

It's good for a laugh

anon

@Jack: I didn't mean to sound like I knew the answer! I have no idea.

robjohn

21:47

graphs with girth?

Jack Schmidt

cool. I think i'm going to make some string and bead models of them tonight

Jonas Teuwen

@robjohn That was a good one.

Mike Spivey

@J.M.: You probably saw this question from a couple of days ago. Did you catch that I used basically the same argument as on that interesting asymptotics question from last month?

This interesting question, I mean.

Asaf Karagila

@MikeSpivey It's a magic trick that I'd learnt.

Mike Spivey

@AsafKaragila I bow to your magic powers.

Asaf Karagila

21:51

@MikeSpivey You mean? I thought only robjohn is mean.

@MikeSpivey No need to bow. I charge 50 bucks :-)

Jonas Teuwen

That's like 2 euros.

Mike Spivey

@AsafKaragila He's mean squared, right? I'm not even in the same league!

@JonasTeuwen Is the exchange rate that bad now?

Jonas Teuwen

No, I actually have no idea.

Asaf Karagila

@MikeSpivey Yeah, I think he is also the root of all mean. So he's the squared root of himself... so he's either 0 or 1

Jonas Teuwen

The Euro isn't doing very well either :).

Asaf Karagila

21:54

@JonasTeuwen Google says it's more like 36 Euros.

Jonas Teuwen

Sounds more like it.

J. M.

Hey Mike. Neat bits! I'll upvote in two hours (and I still owe Hans an upvote).

Mike Spivey

@JM @Thanks!

Asaf Karagila

Hi @JM

Mike Spivey

@J.M.: Do you ever run out of upvotes for a day catching up on all the voting you would have done the day before had you not run out of upvotes then? :)

J. M.

21:57

Guten morgen, König. :)

Jonas Teuwen

Hi @JM.

J. M.

@Mike... exactly the situation I have. (If I didn't know any better, you'd have scared the daylights out of me.)

Hi Jonas!

Mike Spivey

@J.M.: And, on a serious note, Thank You for all the voting you do. I'd probably have half the votes I do in some of the tags like optimization and LP if it weren't for you.

Jonas Teuwen

I have reached my voting limit for the first time today!

Asaf Karagila

Congratulations.

J. M.

22:00

@Mike: Lucky for you, I like those subjects... :)

Mike Spivey

@JM Yep! As I said, thanks! :)

J. M.

(Those guys asking about euro-dollar conversions: xe.com)

Asaf Karagila

@Mike: You said something about writing a paper based on a question here. What is it about? (Did I ask that already? I forget.)

Jonas Teuwen

My 8 most popular tags are all involving analysis.

J. M.

@Asaf: check Mike's blog.

Mike Spivey

22:01

@Asaf: What J.M. said. A couple of posts down, I think.

Jonas Teuwen

This?

Mike Spivey

@Jonas: Yep!

robjohn

@AsafKaragila It is also mentioned here

@AsafKaragila The Lah Numbers link

Asaf Karagila

Yeah, I figured it out :-) Thanks.

Mike Spivey

It's more of an expository paper than a research paper, although I think the fact that the Lah numbers show up as the coefficients in the derivatives of e^(1/x) is actually new.

I keep thinking that there's a slick way to answer this question using complex variables, but I can't seem to make it work. Srivatsan's answers are nice, though.

Asaf Karagila

22:09

Well, I have filled my quota of Sigmas for the week :P

J. M.

22! Well, that product does collapse neatly...

Asaf Karagila

Say, from an ultrafinitistic point of view... the factorial is a bit problematic no?

I mean, after something like 40-50 there is no such number anymore, as 50!

Mike Spivey

@Asaf: I know not that of which you speak.

J. M.

50! is \approx 3e64. Yeah, quite big, but I don't know what finitists consider as "too big".

Jonas Teuwen

Woohoo, the Vox Populi badge!

Asaf Karagila

22:17

@JM Not finitists, but ultrafinitists. I think that they would all agree that googleplex is moot.

Jonas Teuwen

So their fields are not closed under addition? What?

J. M.

Sorry, I tend to lump them... :D anyway, there's around 10^83 particles in the universe, so 3e64 is still a bit short.

Asaf Karagila

I see.

So 50! exists :D

What about 100!?

J. M.

9e157 is sort of high... ;)

Mike Spivey

O.K., so I just spent some time looking up "ultrafinitism," which eventually lead me to this interesting article/rant‌.

J. M.

22:23

Well, it's DZ... why won't it be a rant? :)

Asaf Karagila

Oh that claim that R is not "true mathematics" but rather an approximation of a part of the rationals?

Ramana Venkata

@robjohn Hi

Mike Spivey

The entire perspective, really.

robjohn

@RamanaVenkata how's it going?

Ramana Venkata

Fine

@robjohn I have one question to ask

robjohn

22:26

@RamanaVenkata go ahead :-)

anon

I remember when I was a young teenager arguing with some English-speaking religious (Islamic) scholar on his blog over whether or not the universe was discrete or continuous. Fun times.

Jonas Teuwen

@MikeSpivey Interesting rant...

J. M.

@anon: You gave him stuff on quanta, no? How'd he take it?

anon

I can't remember much, I was like 13 or 14.

Ramana Venkata

Can arbitrary intersection of infinite sets be infinite? I think its false. Am I right?

Jonas Teuwen

22:28

What is a "Naked Brain" human? According to Zeilberg I am one.

robjohn

@RamanaVenkata It depends on what you mean by arbitrary intersections

anon

@J.M. My contention was that continuous models make more sense but theoretically the universe could be discrete, and his contention was that logic dictates the universe is discrete definitively. (And I don't mean "quantized.")

robjohn

You can have an uncountable intersection of sets still have full measure.

A_x=R^2\{0,x} for x in R

The intersection of all the A_x is just R^2(x-axis)

J. M.

@anon: Okay. On the other hand, I don't understand how you're drawing the line between "discrete" and "quantized" though...

Ramana Venkata

@robjohn I am using arbitrary in the sense that I can use any possible combination of Infinite sets for intersection.

J. M.

22:31

(but I do agree continuous models work and are appropriate most of the time)

robjohn

@RamanaVenkata did you see the A_x I wrote above? Does that violate your setup in some way?

anon

@J.M. quantum mechanics doesn't say everything exists on a grid with finite differences, it just says certain global quantities of a system are discrete, if I understand correctly.

J. M.

Ah, alright then. :)

Ramana Venkata

@robjohn Yeah

J. M.

(On the other hand, note the flurry of attempts to quantize everything in the recent years... ;) )

Put another way: I've seen the theory, but I remain unsure about there being a graviton, for instance...

Asaf Karagila

22:39

Wait until they start stringin'

J. M.

That's another kettle of fish. :)

Asaf Karagila

Are you out of barrels?

J. M.

Actually, I'm fresh out of cans of worms...

...so I settled for a kettle of fish.

@Jack: That Möbius question is nice! Any reason why you didn't add a topology tag?

Asaf Karagila

You settle for a kettle?

anon

that reminds me. Yau's the shape of inner space is online for free: stringworld.ru/files/…. Not sure if it's legal or not. It's one of the top search results for the book's name (so you find it without trying) but it happens to be on a .ru server, and if places like library.nu or bib.tiera.ru are any indication it's pirated...

J. M.

22:44

@Asaf: Why not? I have the mettle...

@anon: I'll take a(n objective) look and try to forget that Yau shabbily treated Perelman for the time being...

anon

I was not aware of any scuffle between them. I shall have a look and see...

Asaf Karagila

@JM Sounds like a troublesome fettle.

"A New Kind of Hammer" - does he speaks of Mjolnir?

J. M.

Mjolnir ain't new... it's older than dirt!

Asaf Karagila

Older than time, too.

J. M.

Now that I think about it, I don't think I've ever seen Bill Thurston comment on his conjecture being dispatched anywhere...

Jonas Teuwen

22:53

Good night, guys.

J. M.

Good night, Jonas!

Asaf Karagila

Bill Thor-stone??

That's just Bill Mjolnir.

J. M.

My, you're feeling punny today.

What have you been drinking?

Asaf Karagila

I'm all puckered out from lack of sleep.

I am not tired, though.

Mostly hungry.

J. M.

I see at least two tags here that aren't quite comfortable where they are...

Asaf Karagila

22:59

@JM: I see four. And the bounty is ridiculous. It requires a complete answer.

J. M.

Given the tendency to generalize everything... I'm not quite sure there'll be a better answer than Srivatsan's comment.

Asaf Karagila

Me neither.

robjohn

Bunchawimps :-)

They were complaining about how complicated the numbers got. The hardest thing to do was square 8.5.

J. M.

I got what you got (I was running through the calculations when your answer popped up.) I don't know what's their definition of "simple"...

robjohn

I like DJC's comment

he had answered that linear scaling was lost. I commented on his answer that that was for norms, not metrics.

The OP accepted his answer.

J. M.

23:10

Well, Zev nuked DJC's answer per DJC's request...

robjohn

Yes, I didn't see his comment until it was gone.

J. M.

DJC's reply to your comment: "You're absolutely correct. I'll delete the post if the OP designates another answer as correct."

robjohn

Did Zev nuke it or did DJC delete it?

Mike Spivey

DJC asked how to delete an accepted answer on meta, which I appreciated. I had an old (> 1 year) incorrect accepted answer that I've been wanting to get rid of for a long time. I didn't realize that a mod could delete it for me.

J. M.

It was Zev. OP hasn't come back to undo his check.

robjohn

23:12

Oh, Zev was not the OP. Okay.

Asaf Karagila

Oh man. I just realized.

I am taking two courses this semester...

Mike Spivey

@robjohn: You can't delete your own answer if it has been accepted.

Asaf Karagila

Commutative and homological algebra; and Basic Concepts in Topology and Geometry.

For the love of chaos, I wish there were more set theory courses here...

robjohn

@MikeSpivey Thanks. I didn't know that. I guess that is why all the talk about the mod deleting things :-)

Did Zev unaccept the answer as well, or is it still accepted?

Asaf Karagila

Still accepted.

J. M.

23:15

I don't think mods can undo accepts of users.

robjohn

Sorry I have to ask these things; I am still below 10K :-)

I can't look at deleted answers.

Mike Spivey

@robjohn I found out the hard way. I posted a wrong answer, the OP accepted it, and then somebody else pointed out what I had done wrong. I tried to delete the answer but was unable to. The OP, meanwhile, never came back to the site.

robjohn

So can the OP unaccept a deleted answer? If the mod can't unaccept it, that acceptance is in limbo.

Mike Spivey

@robjohn At the rate you're going, though, it shouldn't be much longer to 10K.

J. M.

Agreed. Like I said, if you only joined earlier, you'd be 10k at this point... :)

Mike Spivey

23:17

@rob: See DJC's question on meta.

Asaf Karagila

I think I'll go to sleep early today... 1:30 :-)

robjohn

That is where the "chaos is fun" happened :-)

@AsafKaragila for shame!

Asaf Karagila

Chaos is fun!

@robjohn I couldn't function today with only 4 hours of sleep for three nights straight :\

robjohn

@AsafKaragila Piker! :-p

J. M.

@rob: Eh let him sleep. It'll be a challenge for us to deal with the resulting silence... ;)

Mike Spivey

23:22

@J.M.: How are the hypergeometric edits coming? I pretty much rewrote all of the Faa di Bruno section, but I think I'm converging on being done with my part.

Asaf Karagila

And I had a visitor from TAU today, and we didn't do any actual work. Just came to the conclusion that my idea from the last time we met was not good in its naive form.

So it's back to the drawing board :\

robjohn

@AsafKaragila sleep on it. It will come to you in a dream.

that you will forget upon waking :-p

J. M.

@Mike: Since the ref was begging for some historical background (if I read what he said correctly), I'm still digging up stuff on Kummer. I am also trying to see if I could squeeze in the Vandermonde approach to proving Kummer's transformation...

Asaf Karagila

@robjohn If only my dreams were that simple :\

J. M.

(Remember when we thought it might make the paper too long?)

Asaf Karagila

23:24

Well, so long folks.

J. M.

See you, Asaf.

Mike Spivey

Catch you later, Asaf.

@J.M.: Yes, I do. That's been part of my problem with several of the edits. How to provide more background info on the techniques without detracting from the main line of the argument and making the paper too long? That's a large part of the reason why I rewrote the Faa di Bruno section. It doesn't use Bell polynomials anymore, but the argument is much more (IMO) transparent.

J. M.

But you at least left a mention that the business can be done with Bell polynomials, right?

Mike Spivey

@JM Only a mention. I'm actually using a different version of Faa di Bruno's formula now. Hopefully I can finish up my part of the edits tomorrow. If I do, I'll send you what I've got by email.

With the new version I actually prove Faa di Bruno's formula.

J. M.

Sounds great!

Mike Spivey

23:32

Hopefully that will make the ref happy that we aren't just pulling a new technique out of a magic hat.

And I'm checking out, too. Time (once again) to go pick up the kids. Catch you folks later...

J. M.

See you, Mike!

robjohn

Missed Mike while reading some answers.

I should probably go, too. I have some errands to run before walking the dog.

J. M.

I still have an unread answer backlog and a bunch of IOUs. Two for you, if I've kept track right.

anon

23:49

maybe SE should develop a voting cache :)

J. M.

I think so, too. There have been days where I didn't quite get to use up my votes...

Off for now. Later.

Transcript for

$Mathematics$ Mathematics