Mathematics - 2011-12-02 (page 2 of 4)

Srivatsan

4:00 AM

Well, another option is to occupy a different room. I am not sure this discussion is appetising to all members here.

J. M.

@Srivatsan You'd be surprised...

Skullpatrol

@JM Thank you

Srivatsan

@JM I first thought that your comment is a reply to my previous comment just above yours. And I presume Skullpatrol did so too... =)

J. M.

@Sri Sorry, couldn't tell; I can't "see" ;) what he's saying...

Skullpatrol

Do you agree that separating 0/0 into 0*1/0 is not allowed Algebraically according to the Algebraic definition of division
because 1/0 is not permitted

QED

4:04 AM

no

Skullpatrol

@QED why?

QED

/ : R x R\{0} --> R

Skullpatrol

This is the heart of the whole arguement

t.b.

:2593826 Use %29 instead of the closing parenthesis

Srivatsan

@tb Yes, I figured something like that. I must learn the art of writing valid URLs. =)

J. M.

4:07 AM

@Srivatsan Ideally, we shouldn't have to, but as Jeff says, that wiki's URLs can be screwy...

Skullpatrol

@QED The proof that 0 times any number is 0 not 1 is the heart of the whole argument

t.b.

@Srivatsan You only need a few of those listed here on a regular basis.

Srivatsan

Bookmarked. Thanks.

I am not committing it to memory yet; hopefully it'll stick with use.

J. M.

"it'll stick with use." - that's how I remembered them. :)

Srivatsan

Btw, what did you guys think when you saw this question?

Honestly, I thought it was another cranky question, which is obviously true or false. =)

Skullpatrol

4:13 AM

@QED I agree that this formal rigorous proof is not normally given to students and teachers do a lot of hand-waving to convince them of it.

Srivatsan

@JM Let's see. The trouble is I keep avoiding the use.

QED

I don't care what teachers do though

J. M.

@Srivatsan It looked hard. (I upvoted Brian's nice answer.)

Skullpatrol

@QED me neither

QED

@Srivatsan, I thought I solved it negatively :S

Srivatsan

4:14 AM

@JM That was only my first instinct. I went nowhere with that instinct though. =)

QED

I realize now my comment was totally wrong..

Skullpatrol

@QED but if you start with that proof and build it up to the definition of division it does make sense, pal

QED

@Skullpatrol, I disagree

J. M.

(i.e. if I were smart, I could've come up with Brian's answer, because I actually understand his answer.)

t.b.

@Srivatsan I thought, well, glad to have Brian around, so I don't have to think about it. He pulled that nice transfinite construction trick out of his hat a couple of times already. The first time I heard of him (and he made his nice entry here) was a somewhat related question.

Skullpatrol

4:18 AM

@QED What part do you disagree with?

QED

with what you said

J. M.

@tb Awesome. :D

t.b.

@JM yep, that's what I meant :)

J. M.

(I wonder if there are other examples of authors correcting posts invoking their papers.)

t.b.

SW?

Maybe?

J. M.

4:20 AM

"That's not what I said in my paper..."

t.b.

Here's another instance of such a construction.

Skullpatrol

@QED separating 0/0 into 0*1/0 is not allowed Algebraically according to the Algebraic definition of division because 1/0 is not permitted... is a tough pill to swallow

Srivatsan

Guys, that some scary red dot that just appeared next to my name -- is that a "perk" for joining the 10k+ league?

t.b.

@Srivatsan That's the suggested edit thingie.

Click on it and approve/reject, but you have to be quick...

Srivatsan

Yes, just approved one suggestion.

@tb Why should I be quick?

J. M.

4:23 AM

Of course, do a quick runthrough before clicking the button.

Dylan Moreland

There's no time!

Srivatsan

Yes. This is what I am confused about: I could approve edits even before this. But why the notification now and not earlier?

@DylanMoreland Ok, you're approved, Dylan. =)

t.b.

@Srivatsan They are so annoying that people usually don't let them stand for long. Often edits are approved too quickly as far as I'm concerned.

@Srivatsan I don't know why it is the way it is, but if I remember correctly, you could approve suggested edits only if you chanced upon them when seeing edit (0).

Srivatsan

Hmm, ok. Thanks for the info. Let's see how this goes.

Skullpatrol

The result of dividing zero by zero, according to the definition, can be any number since c*0 = 0 in all cases.

It is usually preferable in algebra (in order not to violate the uniqueness of division) to consider division by zero to be impossible for ALL cases.

In mathematics the art of asking questions is more valuable than solving problems.

t.b.

4:28 AM

@Skullpatrol Does your LP have a scratch or something?

Really, it's getting boring.

Dylan Moreland

@tb The edit seems innocuous. Am I missing something?

t.b.

@DylanMoreland A crucial minus sign...

Dylan Moreland

Oh, minus sign.

J. M.

@tb Careful, he might not know what an LP is, if my guess of age is correct...

Skullpatrol

@tb Do you find the quotation "In mathematics the art of asking questions is more valuable than solving problems." boring?

t.b.

4:31 AM

Yes, when repeated for the tenth time. The other one is von Neumann's quote.

Dylan Moreland

It's kind of an idle question.

Skullpatrol

@tb I only counted 8 times ;-)

sorry about that miscount

t.b.

@DylanMoreland Nice job on the closed forms (i.e. locally exact) question, by the way. I wouldn't have guessed that definition.

Dylan Moreland

@tb Oh, thanks. I just wanted to see it through to the bitter end. I had to ask for his definition; I guess it makes sense.

If I had thought that the answer wouldn't just be "d(f dz) = df ^ dz" then I probably would have skipped it.

Skullpatrol

@tb What is your favorite math quote?

t.b.

4:37 AM

@DylanMoreland It surely makes sense. It also is a good definition if you only have to deal with 1-forms (as is the case for most complex analysis), so you can avoid mentioning the wedge product which is a major stumbling block for many students.

Skullpatrol

@tb or do you have one of your own

Rajesh D

@Skull : IMHO asking same question repeatedly and wasting time of the experts (i do not count myself in this list) may not be a good idea, perhaps thinking about the problem at hand on your own for a considerable time would do a lot good than hanging around here............may be you may come back to this question (if you still feel you have something) a couple of days later..........this is only my humble opinion

Dylan Moreland

@Skullpatrol It's interesting to note that it's far more common to have technical chops than an actual feel for what might be true; it's good to remind yourself that proofs aren't everything. But people have fine careers either way. A lot of it is up to personality.

I don't know how much time you really want to spend on that question, though. Matt was talking about this the other day, but he has tenure and he can do whatever he wants.

Srivatsan

Which Matt has tenure?

Dylan Moreland

Matt E. I don't remember why it came up.

Srivatsan

4:41 AM

Oh, a third Matt. Lucky for him he doesn't appear in chat. =)

J. M.

@Srivatsan He's too busy answering tough questions... :D

Srivatsan

@JM I was only saying he won't be pinged by our comments to Matt_1 and Matt_2. ;)

Dylan Moreland

This was in person, not here.

J. M.

Heh, "theorist" is now an insult.

Dylan Moreland

Is analitic a word

his spelling is otherwise very good

Srivatsan

4:44 AM

@JM Micro lost the "string" attached to "theorist". I don't think of it as an insult anymore. =)

J. M.

Point taken. :)

Srivatsan

[just kidding; I don't have anything against or for string theory.]

Dylan Moreland

Despite the excellent spelling I have no idea of what he is saying

t.b.

@Skullpatrol JvN has a couple of good ones: There's no sense in being precise when you don't even know what you're talking about. or If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

4

Anyway, I couldn't decide for a single one.

Srivatsan

Thanks a bunch!, but no upvote or accept? :-/

J. M.

4:47 AM

Just the fact that math "works" for things as complicated as "real life" is a surprise in itself.

Dylan Moreland

@Srivatsan He's new. You could give him a nudge.

J. M.

@Srivatsan He can't upvote, since he's unregistered. But you can nudge him into accepting.

t.b.

@JM The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or more generally, to any science which interprets experience on a higher than purely descriptive level.

JvN again.

J. M.

That guy's a fountain of neat quotes...

Srivatsan

@DylanMoreland Voting is a touchy issue here. :)

Dylan Moreland

4:50 AM

The von Neumann quote that always sticks with me isn't really about math, but it's when he says to Feynman (they're working on the bomb), "You don't have to be responsible for the world that you're in."

I still don't know what to think of that.

J. M.

@DylanMoreland Oh, that one in "Surely You're Joking"...

Dylan Moreland

Grothendieck is also full of good lines.

I haven't seen a lot of Weil quotes but I have to imagine that he has some gems. Of meanness.

Srivatsan

Of meanness? Do you have any example in mind?

Dylan Moreland

Nothing immediately comes to mind. But he wrote a lot. And he was the sort of person who would show up to a talk by someone he didn't like, sit in the front row and open up a newspaper in front of his face.

J. M.

Ballsy.

Dylan Moreland

4:56 AM

So he didn't mince words.

The whole Grothendieck philosophy, the "you shouldn't try to prove anything that isn't obvious" and so on. That's an interesting viewpoint. Probably dangerous for the most part.

QED

I like the Erdos way.. comes and sleeps through your lecture and solves an open problem you mention at the end

Dylan Moreland

mathoverflow.net/questions/7155/…

There's a trove.

t.b.

@Srivatsan Read his memoirs... Here are a few characteristic excerpts.

QED

I wish I could follow some of Grothendiecks work

Srivatsan

Thanks, tb.

Have to leave now. See you all.

J. M.

5:00 AM

See you.

t.b.

See you!

Dylan Moreland

Goodbye.

I find it very hard to answer this person's questions: math.stackexchange.com/users/19485/tashi

They're sort of medium-difficulty algebra questions, and I think he/she really needs answers that start from the beginning of algebra.

And that's going to take wayy too long.

QED

it's a bit frustrating my only real question is too soft to ask on the site

Dylan Moreland

@QED We get really soft questions, don't we? How bad could it be?

:)

QED

@Dylan, my problem is trying to find a subject to study

most things seem far to hard

Dylan Moreland

5:05 AM

Okay, that might be too soft.

QED

haha

Dylan Moreland

You've got to start somewhere. If you're in a department, there should be someone you can talk to.

QED

no I'm not

Dylan Moreland

Ah. I don't want to probe too much.

But this doesn't seem like a question you can solve over the internet.

As someone who was out of school for a few years, I empathize.

QED

I think you're right

Dylan Moreland

5:09 AM

Have you ever had contact with research faculty anywhere, though?

QED

no

Dylan Moreland

Those are the people you want to talk to. Someone who knows you and knows what they're talking about.

QED

I see

Dylan Moreland

Well, I'm not saying anything of substance.

J. M.

We asked QED to make a bucket list of things he's interested in a while back...

QED

5:11 AM

and I did!

although it wasn't very good

Dylan Moreland

I don't want to ask more personal questions, but it seems like getting some sort of stepping stool degree is a prerequisite to going further.

I don't know if you've had an undergraduate education in another field, etc.

But you have to get your foot in the door somehow.

Until then it doesn't matter whether you know what you want to do.

J. M.

"Some of them can be downloaded. Not the students, the papers." :D

Dylan Moreland

If you can figure out what you need to do in that respect, then that could be a good question. I feel that a question like, "what are some masters programs in the US that offer financial support" has to be okay.

Because that has specific answers and it's not so easy to figure out. Most programs don't have terminal masters degrees and it's not clear to me that they ever want to subsidize these things. It's tough.

And it's broadly useful. It's not "what should I do in my specific situation that only applies to me"

QED

yes

Dylan Moreland

Anyway, I hope you figure something out. People should be able to pursue their dreams as long as they're willing to work like crazy.

2

QED

5:19 AM

well I don't really have any dreams but I thought I would like to keep studying mathematics so I don't waste spending so much time on it before

yunone

Out of curiosity, is it unusual to see people do a math master's program at one place, and a phd program elsewhere?

Rajesh D

@tb : What is meant by 'derived' series of Fourier series ?

Dylan Moreland

@yunone Depends where you are.

J. M.

Not sure about math, but I do know a lot of guys whose master's and doctorates are from different institutions.

Dylan Moreland

It seems like in the UK people do part III really often and then go off to grad school.

In America it's less common. Maybe if you graduated with an English degree, decided you wanted to do math, and needed to establish a track record for grad school.

yunone

5:22 AM

I see, I'm not really familiar with any of the process. I just remember reading on physicsforums or something that having a math master's made it more difficult to get into phd programs, which I found hard to believe.

Dylan Moreland

I've heard that you might be held to a higher standard

I don't know whether that's true

yunone

I think the explanation was that a master's degree was seen as a copout in one's first try at graduate study, but I guess you shouldn't believe everything written on the internet anyway.

Dylan Moreland

I don't really understand that remark. Some people need to build up the grades. Most graduate students just do masters-level work during their first year. No real difference.

Maybe not at Harvard.

yunone

Yeah, I didn't see much sense in that remark either.

Rajesh D

I'll be glad if anyone could get me this paper ?

J. M.

5:29 AM

Your library doesn't have a subscription?

QED

that's odd, I thought JSTOR opened up the old stuff

Rajesh D

no

J. M.

@QED Depends on the journal.

t.b.

@RajeshD I would guess it is the Fourier series of the derivative of f. More precisely, it's the series with the coefficients ik a_k or something like that.

J. M.

Funnily enough: old AMS stuff is easily accessible from their site, but JStor still requires a fee.

Rajesh D

5:32 AM

@tb : do you mean the derivative of the fourier series ?

there is a subtle difference between derivative of Fourier series and Fourier series of the derivative function

anyway i got the conclusion after some references that derived series means derivative of the Fourier series

especially when $f$ is not differentiable it makes a difference

t.b.

Well, the formal derivative of the fourier series of f (interchanging summation and differentiation). Just the series you get when replacing a_k by ik a_k.

Rajesh D

I need a reference (available on net) or a book for the cesaro summability of derived series of conjugate Fourier series

yes thanks @tb

Skullpatrol

@tb What do you think of the quote "In mathematics the art of asking questions is more valuable than solving problems."

Rajesh D

@Skull : it should be asking questions to onself and not to bug others !

Dylan Moreland

@Skullpatrol Now you're just trying to drive him insane.

J. M.

5:43 AM

It's no use, you guys.

yunone

Time to use that "hide posts" option.

2

Skullpatrol

@DylanMoreland The person who wrote it did die in an insane asylum ... didn't he?

t.b.

@Skullpatrol It is mainly a mistranslation...

J. M.

"All triangles differ from each other. They are not the same and they are not similar." - *facepalm*

Dylan Moreland

@JM That guy is great.

Skullpatrol

5:49 AM

@tb Did Godel's work have anything to do with Cantor's?

t.b.

@Skullpatrol ?

Skullpatrol

@tb Was that a dumb question?

I apologize.

I recently found out that Godel died of starvation ... what a way to go for such a brilliant mind

Skullpatrol

6:09 AM

@tb In mathematics the art of proposing a question must be held of higher value than solving it. en.wikiquote.org/wiki/Georg_Cantor

I agree that this quotation is out of place in such a practical and application oriented web site, but it still has some value here.

or anywhere where mathematics is discussed ... "the essence of mathematics is its freedom"

another Cantor quotation

Matt

7:16 AM

What a breathtaking sunrise...

Literally, because of the cow stink.

Skullpatrol

@Matt All cows stink, some cows stink, or there exists at least one cow and it stinks at least one time ;-)

Rajesh D

@Skull : what are you doing..student/professor/engineer ?

Matt

7:37 AM

I got a nice answer here.

Ilya

7:51 AM

seems like @JM now cannot stop thinking about curves with holes )

J. M.

Hey Ilya.

Ilya

8:04 AM

@JM hey-hey

I recently realized that I have a taxonomist tag markov-chains

I'm forcing Russian in tags apparently )

J. M.

Speaking of which... I wonder why nobody's thought about making martingales?

Ilya

well, our stochastic-process tag is mostly about them )

Rajesh D

pardon me if it is not relevant here...........How can i download code from this site...it shows all code in aspx pages........there must be some way for me to download..

Ilya

the whole advanced probability theory is not so popular - so I would even merge brownian-motion with stochastic-processes

J. M.

@Ilya That actually sounds like a good idea. Why not propose it on the long thread?

Ilya

8:08 AM

@JM you mean, on meta?

J. M.

Yep.

Ilya

I don't have much evidences since I haven't thought seriously about it :)

Matt

@RajeshD I don't understand your question. Here: corp.koders.com/downloads/demo

Rajesh D

@Matt : I am only able to search and view code on the browser, but i am not able to download the source code file for example quaternion.m ( a mtlab code) is displayed in the browser but i am not able to save the .m file to my computer....how to do that is my question

Matt

8:24 AM

Which download are we talking about? IDE Plugins? Where do you browse source files? Can you post the url?

Rajesh D

In this URL there is a directory structure of a project...root/make3d click on the '+' mark on side of 'Third party' and in that click on qtfm and then in that on @quaternion,and click over @quaternion to display the files in that folder on the right side, to see a display of .m files. Upon clicking on the m files they open in the browser...but i need to download them directly to my computer as .m files

@Matt

Matt

@RajeshD: Here: make3d.cs.cornell.edu/code.html

I don't want to d/l so I didn't check but I think this is what you're looking for.

Rajesh D

8:45 AM

@Matt : I wonder how you figured it out

Matt

@RajeshD Experience ; ) There was a link at the top of the page, right before the text.

Rajesh D

i was searching for make3d on stanford site..thats where i did wrong

Matt

If you google for "make 3d" it's the first hit : )

Rajesh D

ok...i had attached the word stanford to make3d while searching on google............thanks a lot for the help @Matt

Matt

No problem : )

Matt

9:01 AM

@Ilya: What's your argument with the prime numbers here: math.stackexchange.com/questions/87677/sets-and-countablity

Ilya

@JM I was also thinking to retitle probability into elementary-probability as it is with set theory

@Matt Hi. What is your question?

Matt

I think you meant to write integers there.

If you want to show that the countable union of countable sets is countable you don't need prime numbers:

Ilya

@Matt that's how we prove it in Russian forests :)

Matt

: D

Assume you have f_n : N -> A_n then you can count A using F: N x N -> A via F(i, n) := f_n(i)

Ilya

oh yeah, then I would say, two primes are enough )

@Matt could you remind me: when do we need infinitely many prime numbers to make the bijection? never?

Matt

9:07 AM

I don't know because I don't know where the primes come in...

Ilya

@Matt first sentence carried a secret code. after 100 years it will be triggered to you, hahaha (maybe, quite a long time - but my algorithms are not focused on the computational/time complexity)

Matt

I thought primes were only used in number theory.

@Ilya : D I won't be around anymore by then.

Ilya

@Matt If they manage to produce the pills for the infinite life, you'll regret )

Matt

True : D

@Ilya: Have you been reading Gandalf de Grey?

Martin Sleziak

@Ilya This meta discussion is related: Sophistication level tags?

Ilya

9:12 AM

@MartinSleziak my pleasure

@Matt nope, or you've forgotten that Gortaur have some troubles with Gandalf?

Matt

@Ilya I didn't know he had any.

Asaf Karagila

@Matt: Rasmus' answer is not that great.

Ilya

@Matt well, latter Mr. G. picked up 4 hobbits to finally destroy the former Mr. G.

Matt

@AsafKaragila I don't want to judge its greatness but at least he answered c).

Asaf Karagila

But he uses extra, and unneeded assumptions :-)

Ilya

9:15 AM

@Matt Asaf is very accurate with people who use greed proofs in set theory ;)

Matt

@Ilya I read LOTR about 10 years ago. Funnily I don't remember Gorthaur even though I read it in English.

Ilya

@Matt I don't remember if it appears somewhere in the text of LOTR, maybe only in Silmarillion

I read it about 6 years ago the last time

Matt

@AsafKaragila: Plus one for you. Anything wrong with my b) part? You wrote that only my a) part gives him a good idea...

@Ilya I didn't enjoy it that much so I didn't read any of his other books. I mean, it's alright and nothing wrong with it but it's highly overrated imho.

Ilya

@Matt even if you say that it is wrong, I wouldn't argue, I'm not that fan of this stuff

Asaf Karagila

@Matt Nope, this is what I wrote in my answer to (b) that inductively it's fine to do that.

Rajesh D

9:29 AM

I have a problem in understanding this answer, hope someone could help me with this

Ilya

I should read meta more often

Matt

9:50 AM

Can you tell me if the following is correct: Assume that I have a function that enumerates Z^{n-1}, let's call it f: N -> Z^{n-1}. Then I want to construct F that enumerates Z^n. I define it as follows: F: N x N -> Z^n defined as (i,j) -> (f(i), g(j)) where g(j) = k + 1 for k even and g(j) = -k otherwise

Assume that I showed N x N is countable then this tells me that Z^n is as well.

Ilya

@Matt I've already replied him

Matt

I know. But I still want to know.

Ilya

then no. you don't count 0 any some others

Matt

What does it mean?

Ilya

because g(2k) = k+1 and g(2m+1) = -2m-1

0 is not in a range of g

Matt

9:56 AM

Shoot.

Ilya

wait. g(j) = k+1 - what is k?

did you mean g(k) = k+1 for k even and g(k)=-k for odd?

Matt

I wanted to write g(j) := k + 1 for j = 2k

Daniil

Hello

Matt

and g(j) := -k for j = 2k + 1

so g(1) = -0 = 0

Sorry for the mess up.

Skullpatrol

@Daniil Hi

Ilya

9:59 AM

@Matt I think, it is ok

Matt

Gotta go, bbl!

Daniil

crave.cnet.co.uk/gadgets/… haha

Skullpatrol

@Matt See ya later ... nice chatting with you

Rajesh D

10:19 AM

@Skull : are you a student ?

Skullpatrol

@RajeshD If by student you mean someone who has an interest in learning, then yes. How about you?

@RajeshD I simply have a genuine interest in learning, and mathematics is, in my opinion, the best subject to exercise the mind.

Rajesh D

@Skullpatrol What do you do on daily basis, i mean occupation ?

Skullpatrol

@RajeshD Why do you need to know that?

Rajesh D

ok. never mind

Ilya

10:35 AM

@Asaf: are you here?

Skullpatrol

@RajeshD Do you agree Rajesh that if you have a genuine interest in learning then mathematics is the best subject to exercise the mind???

2

Rajesh D

@Skullpatrol it depends.........in my opinion, it differs from person to person

Why do you need to know that?

Skullpatrol

@RajeshD ok. never mind

Rajesh D

good

Skullpatrol

;-)

Rajesh D

10:43 AM

now you are on track

Skullpatrol

@RajeshD Thank you for putting me "on track."

I agree that I do get derailed sometimes

due to all the confusion

and/or frustration...

Rajesh D

google.co.in/…

Skullpatrol

@AsafKaragila Do you agree Asaf that if you have a genuine interest in learning, then mathematics is the best subject to exercise the mind???

2

Skullpatrol

11:08 AM

@AsafKaragila We had a few harsh words earlier and I hope your not holding a grudge, I didn't mean to offend you.

as I told Rajesh I do get derailed sometimes

but if you're not passionate about what you're doing then its not worth doing it that is why, in my opinion, math is the best subject to exercise the mind ... you can be passionate about it but at the same time don't get derailed

robjohn

11:34 AM

@Srivatsan only one of us was pinged. Both of us got highlighted pseudonyms, but only one seemed to get the ping

@Ilya: Good afternoon! I haven't seen you for a while.

no mirrors about?

Ilya

@robjohn hi, I can say the same :)

robjohn

I was out all day yesterday. I had a dentist appt in the AM and then went Xmas shopping with my wife for most of the rest of the day.

I checked in on iPhone once in a while (while waiting at the dentist and while my wife was in a dressing room)

It gets hectic around the holidays.

Ilya

@robjohn I rather meant that I was not checking out MSE for about a week and hence haven't seen you

but happy to hear that you're serious about your teeth ;) after the last time I was to dentist my teeth became so sensitive that I don't know if I want to visit one of these guys ever

robjohn

@Ilya I've noticed :-)

My dentist is very good. Find a good dentist. It's better than losing teeth.

Ilya

@robjohn you've straightened his nose ))

@robjohn you shink sho?

J. M.

11:43 AM

@Ilya But you're supposed to see them annually. They're like taxes; a bit inconvenient, but needed.

robjohn

@Ilya He had some problem, but it was easily fixed ;-)

Ilya

@JM well, in NL I've never been to one. I should ask Jonas how does it work here )

robjohn

@JM I see mine twice a year; I guess that is what my insurance covers.

J. M.

@robjohn Teeth are serious business. :D

robjohn

@JM After grad school, it a long time until I saw a dentist, and the metal fillings I'd gotten as a child had shrunk (oxidation, whatever) and left my teeth with hollow spaces. It was only a matter of time until eating a tortilla chip that I noticed something hard in with it. I thought to myself that it looked as if someone had lost part of a tooth in my chips, then I realized it was mine :-)

So I got a cap and then have been going to the dentist regularly. I had all the fillings replaced with ceramic ones.

J. M.

11:50 AM

Yeesh.

robjohn

I take good care of my teeth, though. I did not get a cavity over all the years I didn't see the dentist after grad school.

But the incident with the filling got me started again with the dentist.

Ilya

yeah, motivating story )

robjohn

Heh. I think this was a drive-by questioning. :-)

Ilya