Mathematics - 2012-01-24 (page 1 of 3)

Srivatsan

00:01

This much drama is unneeded.

Tim

Does anyone know how and where to run the SE API?

I have never been able to figure out Martin's comments of using the API for my question. Thanks!

Kannappan Sampath

@Srivatsan Are you against the question?

Srivatsan

@KannappanSampath No, did you see Eric's comment?

Kannappan Sampath

@Srivatsan No

I am heading right there!

fabianhjr

00:18

Sup?

Quick geometry question: Are all squares rectangles and rhombuses?

anon

yes

Kannappan Sampath

@Srivatsan I see no evidence supporting his claim about the style of writing!

fabianhjr

:/ Is there any counter argument to it?

Srivatsan

@KannappanSampath Evidence, no. But I myself cannot help but notice similarities... =)

fabianhjr

I would guess the square is more specific in the rations/angles than any of them. Yet, it keeps their properties.

Kannappan Sampath

00:22

@Srivatsan Can you point a few? I see none at all.

anon

The reference answer was very similar of Bill's style. Now if Math Gem had used \rm in an equation...

Srivatsan

@KannappanSampath well, let's drop this. :P

Kannappan Sampath

@Srivatsan ok! if you don't intend to go into the details!

Srivatsan

The post is slowly closing anyway. There are three votes now, and it's only a matter of time. I did what I could..

fabianhjr

00:44

Is wikiversity a good place to learn maths?

en.wikiversity.org/wiki/School:Mathematics

robjohn

I hope I am not inflaming anything with my comment.

I believe that this was part of what Bill was trying to say.

Srivatsan

@robjohn I suppose the reason is that 2 is prime. But then if we say this then we should also mention that integers form a UFD. :=)

robjohn

@Srivatsan well, 2 being prime needs to be mentioned. Furthermore, the basic axiom is that if $p$ is prime and $p|ab$, then $p|a$ or $p|b$, nothing about $p^k$ shows up. That is where the induction is needed.

Srivatsan

@robjohn That is not really an axiom, but a lemma (Euclid's lemma?)

robjohn

@Srivatsan Okay, understood premise. I don't think anyone would want them to prove that, too.

Srivatsan

00:54

@robjohn Well, you and me understand, but I have a feeling you might get a "not a proof" comment+downvote. :-)

robjohn

@Srivatsan for calling it an axiom vs lemma?

or not providing a proof?

Srivatsan

@robjohn Yes. For not distinguishing the definition vs. lemma. (Proof itself may not be needed because it's standard.)

Look: I got what you meant; I am just playing the game putting myself in someone else's shoes. I hope you got that... :)

Ok, I got to go.

(If the last comment wasn't clear, we can chat later.)

robjohn

I am just trying to explain why I don't like Paul's answer. I think, from Bill's comment about $2\to c$, that my example is what he had in mind.

Srivatsan

Precisely.

Dan Brumleve

I answered the drama-inducing question too math.stackexchange.com/a/101553/1284 one thing i'm still stumped by is whether my argument implicitly uses induction and whether induction is necessary to prove this (i.e. is it a theorem of Q?)

Henning Makholm

01:02

With integers, there's always induction implicitly hiding somewhere. The only question is whether you can rope off the conclusion of the induction as a "well-known fact" or not.

Kannappan Sampath

@DanBrumleve On a friendly note, to indicate product use $\cdot$ instead of $\ast$

Dan Brumleve

ahh yes * is ugly

Kannappan Sampath

\cdot is better

robjohn

Henning Makholm

@DanBrumleve A pedant might think it deserves proving that a can be written as 2^x·b with b odd in the first place.

Dan Brumleve

01:05

i guess the induction might be necessary to say a=b 2^x with b odd?

robjohn

@DanBrumleve I really think so in a question this elementary. I also don't like the answers using the Fundamental Theorem of Arithmetic. It is too advanced for this.

I worry about circular reasoning when it comes to proving the FTA

Dan Brumleve

i thought these sorts of problems were easy when i was learning number theory in high school, but it's a lot trickier now without the context of a textbook

ymar

Hi. I have just read this discussion and the discussion in question. I agree with both Bill and robjohn. It not pedagogical go give such answers to a person struggling at this level.

robjohn

@DanBrumleve You need to pay close attention to what you've proven since so much of it is second nature.

Henning Makholm

I think what the OP of that question needs most is some words about being careful and explicit about which assumptions (known facts, axioms, or whatever) he seeks to prove his property from. All of the answers so far just dive directly into proof details without being clear what stage they are playing on.

robjohn

01:09

@HenningMakholm True. I was assuming the level of sophistication from the level of the question. My bad :-)

Kannappan Sampath

@ymar Welcome! How are Prof. Dummit and Foote serving you?

ymar

I haven't had a chance to lay my hands on a copy since yesterday!

Henning Makholm

@robjohn Your bad? I wasn't even aware I was criticizing you there.

robjohn

@HenningMakholm I was criticizing myself (half jokingly). I didn't mean to imply you were :-)

Kannappan Sampath

@ymar Do you know you can get books from air?

Henning Makholm

01:13

In completely different news, could somebody please tell me I'm wrong about part 3 of my answer here, and what I should have said instead?

Dan Brumleve

i agree with bill and yall that Paul's answer is missing some steps, but bill's answer although rigorous was tricky to follow. i was looking for a middle ground in my answer. my answer makes the most sense to myself. :)

ymar

@KannappanSampath I can't! I use a wire. But I don't read books from the computer. It's tiring.

Kannappan Sampath

Even if you haven't been able to check it out from a library or buy your own copy, I suggest you get the book from the `wire' and have a preliminary look!

ymar

@KannappanSampath I did that. The book looks good to me but I can't say anything deeper until I read a chapter or too!

robjohn

@HenningMakholm I don't see anything wrong with it.

Kannappan Sampath

01:17

@ymar or two! OK. Good luck! Looking forward to your telling me that you fell in love with that book!

ymar

I didn't know a comment on an answer could be downvoted. I assume it's only possible when you have a certain amount of reputation?

Kannappan Sampath

You cannot downvote comments, I guess!

ymar

@KannappanSampath OK, it's possible I will!

@KannappanSampath I certainly can't but someona can because one of my comments was just downvoted. :)

robjohn

@HenningMakholm I might have said that $\pi$ commutes with the homomorphism between $B$ and $U\times F$.

Kannappan Sampath

@ymar A link would be appreciated!

Please!

ymar

01:20

Hmm. Maybe not.

Do I lose reputation when I downvote an answer?

Kannappan Sampath

@ymar Yes -1!

ymar

Oh, that explains it. So my comment didn't get downvoted. :)

Kannappan Sampath

@ymar, if I may ask, where do you study?

ymar

I study at the University of Warsaw, Poland. I live here too.

Henning Makholm

@robjohn I don't think that would be accurate -- since the domains and codomains are different, the notion of "commute with" I know of does not apply om this situation.

ymar

01:24

@KannappanSampath Where do you study?

Kannappan Sampath

That's good. I have a friend who is a graduate student at Univ of Lancaster who is also from Poland!

robjohn

@HenningMakholm Well, yes, there is a map $\tilde\pi$... My first attempt may not be accurate, but it might get the idea across.

Kannappan Sampath

@ymar I study at Indian Statistical Institute

ymar

@KannappanSampath Oh. Statistical?

are you interested in statistics?

Henning Makholm

@robjohn Yeah, so. My main fear was that I'd missed a conventional well-defined meaning of "carry over". But then again, it was a quote from Wikipedia. Not always a paragon of clear exposition.

Kannappan Sampath

01:27

@ymar Except for its name and origin, my institute and the courses it offers has nothing to do with Statistics, except possibly on a minor scale, like any other good Math dept!

ymar

@KannappanSampath Aha, OK.

Henning Makholm

Anyway, I should be going. Just came here to find out what was going on after I seemingly moved up a slot in the all-time rankings without passing anyone. 'Night.

robjohn

@HenningMakholm I think of "carry over" as somehow "commuting" with a change of coordinates, or a homomorphism, or somesuch

ymar

@KannappanSampath I should thank you for your kind note under my question the other day.

It was reassuring.

Kannappan Sampath

@ymar when and where? (You exposed my poor memory!) =)

ymar

01:30

Here math.stackexchange.com/questions/101487/…

How do I post an elegant link here, like yours a to the Indian Statistical Institute?

Kannappan Sampath

This way [Description](link with http://)

Notice there is no space between the brackets!

ymar

@KannappanSampath Thanks. Let's try: Here

OK, works! :)

Kannappan Sampath

Brilliant. That's it!

@ymar I am writing blogposts on group theory, I'll let you know once I post substantial material there!

ymar

@KannappanSampath Where do I look for them?

Kannappan Sampath

@ymar Do you want a link straight away? But Disclaimer: It has only one post on a problem in group theory!

ymar

01:35

@KannappanSampath Sure I do. Why not?

Kannappan Sampath

@ymar this is my blog with one post and only one post =)

@ymar Is the problem interesting?

Dan Brumleve

Kannappan if K is a subgroup of G and also a strict subgroup of every subgroup of G then mustn't it strictly contain itself? i'm not very familiar with group theory so i'm probably missing something

ymar

@DanBrumleve I believe $\subset$ simply means inclusion there, not strict inclusion.

@KannappanSampath Yes, I'm thinking about it.

Kannappan Sampath

@DanBrumleve Exactly, as @ymar points out, I don't intend a strict inclusion!

Dan Brumleve

nod got it

$\subset$ does mean subgroup though right?

anon

01:41

ymar is correct. In group theory \subset generally means not-necessarily-strict

Dan Brumleve

so K is a subgroup of every H which is a subgroup of G?

Kannappan Sampath

@DanBrumleve "every nontrivial" H!

Dan Brumleve

nodnod

Kannappan Sampath

I might face a day, when I am accused of advertising for my blog and getting hits from here!

ymar

OK, so isn't it a direct corollary of the fundamental theorem of finite abelian groups?

Kannappan Sampath

01:45

@ymar The whole point of the post is to push through the fact that using such a big theorem is not good in this context!

In fact, I know of a proof that uses this as a Lemma in its approach, but Navarro wrote it a long back before I got to it myself!

And I linked his paper there!

ymar

Ah, OK. I'll try to think about it.

Oh, there's a proof there!

Kannappan Sampath

@ymar Please do leave your comments on the style/clarity or usefulness/unworthiness of the post.

I'd love to see that!

ymar

OK, but not now. The post is long and it's 3 am here.

Kannappan Sampath

@ymar Sure, take your time!

ymar

Eh, 11 upvotes, a bounty of 50. What does a question need to get an answer... :/

Kannappan Sampath

01:53

@ymar Aren't you satisfied with the answers you already have received?

ymar

@KannappanSampath On the question about the difference between groups and semigroups? I am. But I'm talking about another one now. I didn't get any answers. Just two comments -- helpful, but only comments.

Kannappan Sampath

@ymar I cannot be of much help in both these cases!

I am sad, I have not nuch knowledge in these things!

There comes @BrianMScott

Brian M. Scott

Yep; now that I have groceries, I can eat for a while.

ymar

@KannappanSampath It's OK. I just think it can't be a very difficult question...

Hi, Brian.

Brian M. Scott

@ymar Hullo. Which question was that?

ymar

02:00

This one

Kannappan Sampath

@ymar That's better! You've learnt it man!

ymar

:-)

You taught me!

Brian M. Scott

@ymar Not really up my alley, but it looks like a good question. It may be too open-ended to get a good answer unless someone really expert happens upon it.

ymar

@BrianMScott Well, some parts of it are open-ended. But a reference request isn't I think...

Kannappan Sampath

@BrianMScott Then may be MathOverflow is a better place? Not sure, it can have drastic effects!

ymar

02:06

@KannappanSampath Oh, man. I've read some answers there. I don't understand half of what these people say... :)

Brian M. Scott

@KannappanSampath From the little that I’ve seen, I’d be very hesitant to ask anything on MO; it’s not a very welcoming place, though some folks there are perfectly reasonable.

Kannappan Sampath

@BrianMScott That's why the 'drastic' tag to it!

Brian M. Scott

@KannappanSampath That’s what I figured.

ymar

@BrianMScott Why not welcoming?

Brian M. Scott

@ymar Some of the folks there can get very snippy if they think that a question isn’t up to the standards of the group.

ymar

02:10

@BrianMScott OK. I'm quite sure my questions never will be so I'll give the idea of posting there a rest.

Dan Brumleve

MO has very high standards. i completely switched to math.SE when it launched because it is more welcoming to amateurs like myself. now i'm very hesitant to post anything there. many of the same people are here anyway. unless i'm certain my question is "research-level" i will stick to math.SE.

anon

I went there once and was scolded for double-posting on MSE and MO. Then, naturally, it was answered on MO and therefore closed on MSE.

ymar

I'll going. I've got laundry to hang and four hours of sleep to get. Good night.

Kannappan Sampath

@ymar Catch some good night's sleep Bye!

anon

Arturo is almost 100k... (indeed, 99k + 7^3)

Brian M. Scott

02:20

@ymar Only four hours?

Dylan Moreland

05:07

Whoa, they suspended Bill?

anon

Mmhmm.

Dylan Moreland

30 days seems like a long time.

Dan Brumleve

enough time to crochet a scarf at least

Asaf Karagila

I'm cold.

Dan Brumleve

05:23

all the snow melted in illinois

Dylan Moreland

Yes. I stepped into a lot of puddles today.

Dan Brumleve

some january eh

tstorms last night too

not at all like i remembered it being

Dylan Moreland

That was a surprise. I was heading out to grab a book from my office and saw a strike near my door. The book was not worth getting zapped, I felt.

For Serre things might be different.

Asaf Karagila

A strike?

Dan Brumleve

/\/\/

anon

05:33

What.

Dan Brumleve

$\cup $\cap $\cup $\cap$ ?

Dylan Moreland

@AsafKaragila Of lightning.

Dan Brumleve

lightning dude

Asaf Karagila

@DylanMoreland Ah.

Man... that Vassillis Parasitic is dense. meta.mathoverflow.net/discussion/1287/…

Dan Brumleve

if there were only enough room in the margin for everyone

... we would have x new answers, y new questions, and x^k + y^k = z^k argumentative discussions

Asaf Karagila

05:43

Either way, I have to go.

anon

Your people need you.

Mariano Suárez-Alvarez

ALL CAPS makes it look so tragic

Zeeshan Mahmud

06:02

@DanBrumleve Speaking of crochet you do realize there is this :)

2

Dan Brumleve

a nice cozy for one of these kleinbottle.com

oops i guess you'll need two :)

errr actually, i'm not sure that one won't suffice?

sewing a side to itself, hmm that is tough

shouldn't distract me else i might leave math.SE forever and disappear into a ball of fluff

i hope bill returns at least because that dude is really smart and has helped me so many times

Zeeshan Mahmud

06:19

Speaking of fluff, I have a soft question: If I want to start a math blog, how do I make sure math is right and I am not embarrassing myself?

Meaning how to get peer reviewed?

anon

if it's a blog, your peers are your audience, and the review comes in the form of comments. no?

Dan Brumleve

dunno but i'm curious about that too

i would post worked problems too but i'd be embarrassed by major flaws

Mariano Suárez-Alvarez

You have to both be careful and become resilient to that "embarrassement"

Everyone makes mistakes

And there are many. many mistakes even in papers printed in reputable journals, with good editors, good referees and even better authors.

anon

no pain no gain

Dan Brumleve

i'm comfortable enough with math.SE. i would blog too if i had the confidence but i think it will take me some time before i reach that point. if you are in university maybe you have fellow students who can review before you publish?

Zeeshan Mahmud

06:26

Point taken. (@MarianoSuárezAlvarez my concern was not so much as the trivial errors rather than coming out as crank or hack since my interest is phil. of maths. which can be receptive to varying degrees of taste...)

Mariano Suárez-Alvarez

the first thing you see when you get copies of a paper you've just had printed in a journal is the typos...

The best plan, Zeeshan, is not to be a crank or a hack. :)

Given that, blog away!

Dan Brumleve

nobody will mistake you for a crank if you are honest about your own shortcomings, although self-indemnifying qualifiers don't help a curious reader, so keep it balanced.

Zeeshan Mahmud

Agreed about the typos...actually SE helps too. Because there is maturity in everything. I think I am asking more to the point questions now than vague ones like before...btw Silver Badge beckoning tomorrow ;) )

I remember when I asked in MO: Should there be concept of God in maths.. :facepalm: yet I am still alive.

Dan Brumleve

se has helped me a lot in the last year, if i continue someday i think i will feel ready to start a math blog too

Mariano Suárez-Alvarez

heh

anon

06:31

Wow.

Zeeshan Mahmud

yes

Dan Brumleve

doh the set of all surreal numbers it is yoda knows, yes.

Mariano Suárez-Alvarez

Lots of people have enourmous qualms about posting on MO for fear of embarrassement.

I don't quite understand that.

Zeeshan Mahmud

@Mariano May be because people of weight are present there and reputation is at stakes? :]

What I mean is MO is handful of sites where superstars tread than other ones...

anon

You take that back, Zeeshan. Mathematicians are not fat!

Dan Brumleve

06:34

peter shor answered one of my questions there and i realized i was in the wrong league. then math.SE started up

Mariano Suárez-Alvarez

haha

Dan Brumleve

somehow i escaped without too much damage aside from mathoverflow.net/questions/27967/…

Mariano Suárez-Alvarez

Peter Shor's the nicest guy

Dan, showing understanding of one's errors (as opposed to stubbornness, also quite frequent...) is a great way to impress people

Zeeshan Mahmud

Okay fellas, here is your chance to fame and immortality! What should be my blog name? ;]

omegapoint is taken... so no dibs

Dan Brumleve

... as a nice guy at least, but not as any sort of mentor :)

niceguymentor.com

(which i strive to be, often unsuccessfully :))

anon

06:41

In early high school I had a blog with the subtitle "From the Riemann sphere to the blogosphere." Wasn't I a clever lad.

Man, looking back at stuff you wrote young makes you really cringe.

Dan Brumleve

"the pursuit of itself"

that will be the title of my blog though so you can't use it.

Zeeshan Mahmud

hilbertshotel is available...but i might go for turingcomplete :)

Dan Brumleve

dan@hq:~$ whois quinequine.com |grep No
No match for "QUINEQUINE.COM".

anon

lol

Dan Brumleve

No match for "no match for 'QUINEQUINE.COM'".QUINEQUINE.COM?

you should choose something based on your philosophical focus.

not even wrong, goedel's lost letter...

Mariano Suárez-Alvarez

06:53

or something silly and marketable

Dan Brumleve

if you are wealthy enough to afford that kind of domain

Zeeshan Mahmud

@Dan yeah i was actually looking at interesting math blogs questionn... but it's chosen now, i will put it on my page to boost profile views and pique curiosity :+]

Mariano Suárez-Alvarez

well, buying nike.com to put a math blog there would be a waste of money :P

Dan Brumleve

x.com would be cool but ebay has it

Mariano Suárez-Alvarez

x.com belongs to ebay?

I would have imagined the xorg guys owned it

Dan Brumleve

06:58

merged with paypal in y2k, bought by ebay

in the early-mid 90s i don't think it was possible to register single-letter domains

Mariano Suárez-Alvarez

x.org has been there forever

Dan Brumleve

Created On:18-Jan-1997 05:00:00 UTC

found this: en.wikipedia.org/wiki/Single-letter_second-level_domain

Mariano Suárez-Alvarez

a long while ago a guy registered a bug on evolution (the gnome mail thingie) because it was rejecting his email because it thought it was malformed

the guy's email was: johndow@ie

Dan Brumleve

dan@hq:~$ whois xyzwv.com |grep No
No match for "XYZWV.COM".

if you like a lot of variables that is

anon

you have those letters in a seemingly arbitrary order

Dan Brumleve

07:08

i start at x and then i run out, but optimism persists

unfortunately xyzvw.com is taken

but those are terrible math blog names, people are attracted to insights not variable names or numbers.

Mariano Suárez-Alvarez

thisdomainistaken.com

for a student of logic, thisdomainisnottaken.com sounds good

or a Matisse fan

Dan Brumleve

i've heard many times something like "math is interesting, but i can't handle all the numbers". imo it is a problem with math education not people (no not that imo)

Mariano Suárez-Alvarez

it is taken, though :(

Zeeshan Mahmud

darn! that was a cool idea.. @Mariano

Dan Brumleve

No match for "THISDOMAINISNTTAKEN.COM".

now that looks like it could be quined....

oh wait, it is already :)

Mariano Suárez-Alvarez

07:20

thisdomainshouldbetaken

Zeeshan Mahmud

08:07

Thanks for the suggestions everyone. I have to retire..

Dan Brumleve

08:19

goodnight zeeshan, everyone

Asaf Karagila

08:33

What about banach-tarskibanach-tarski.com but write everywhere that the domain title is actually Banach-Tarski.com

Matt N.

Morning.

@AsafKaragila Do you have two SE accounts? : )

Asaf Karagila

No, why?

Is my name Banach-Tarski?

Matt N.

Mouse-over the brown avatar to the right of yours.

Asaf Karagila

Ah.

Matt N.

: D

Asaf Karagila

08:46

instant rimshot effect

Blah. I have to finish a homework assignment due last week, and I'm just not in the mood for that.

Matt N.

Commutative algebra still?

Asaf Karagila

Yeah.

Matt N.

I have to do set theory and I'm not in the mood to do anything at all.

Asaf Karagila

I hope that there will be not fourth assignment, then I'll only have to prepare a lecture on the Whitehead problem, and I'm done with the course.

Today is the last lecture in algebraic topology too. I have to solve 7 questions and take an exam on 3 of those (in which I have to solve only one) and then I am done with this course as well.

Matt N.

I wish I knew the exam questions in advance! Jealous.

Asaf Karagila

08:51

Yeah, the exam is only 20% of the grade. He gave us the choice of three questions from the homework assignments and he'll give them priority 100%/66%/33% and we can only solve one question of those. However remembering the proof for three questions is not a big deal.

Matt N.

I'm starting to feel sorry for myself.

Asaf Karagila

Oh, don't be. I've had plenty of hard exams... this exam is only a formality since they are trying to eliminate all the "grade purely on a final assignment" method in elementary grad-level courses.

Gigili

10:07

Anyone available?

Hello, I have a question. I have problems solving limit questions, but I cannot ask all of them on the main site. What should I do?

Matt N.

Ask here, maybe we can help.

Gigili

Hello Matt.

Matt N.

Hello Gigili.

Gigili

What's this sign called in English? [x]

I have problem when it's included in limit questions.

Matt N.

I'm not sure but maybe it's used to denote the floor function?

Andrey Vihrov

10:10

You'd have to provide an example, because [x] can mean several things.

Matt N.

Nearest integer function looks promising.

Gigili

Yes, that. Like $\lim_{x \to 0^+} [x]=0$

$[1^+]=1$

The question is, for example: $$\lim_{x \to 10^-} \frac{[x^3]-x^3}{[x]-x}$$

I don't know what happens to $[(10^-)^3]$

Or $$\lim_{x \to (1/2)^-} [4x][\frac{2}{x}]$$

How many questions I'm allowed to ask per day?

Matt N.

I'm not sure, I've not had to deal with $[ \cdot]$ before. Is $\lim_{x \to 0^+} [x] = 0$ the first question in the book? I'm asking because I'm not sure what one would have to do to show that the equality indeed holds.

Also, the second limit looks as if it evaluated to the same value as the expression without the $[ \cdot ]$ but again I'm not sure how to rigorously show it.

Gigili

I'm typing something which may help @MattN ...

Wait a minute please

Matt N.

Sure, take your time. : )

@Gigili You have to read $x \to 10^-$ as "$x$ approaches $10$ from below". $10^-$ on its own is just $10$ and $[(10^-)^3] = [(10)^3] = 10^3 $.

My second sentence here doesn't really make sense as $10^-$ needs to be inside an "$x \to $" clause but you get the idea.

Gigili

10:26

Yes, right.

$$\begin{eqnarray*}
[x] =
\begin{cases}
x, &\text{if }x \in Z,
\\ \text{the greatest integer number smaller than x} , &\text{if }x \notin Z.
\end{cases}
\end{eqnarray*}$$

If that makes any sense at all ! I have to translate everything in my mind to English.

for example $[2.5]=2$

Matt N.

Ok. Now I'm trying to think of values where this differs from the floor function.

Gigili

I guess you're right, maybe it's the floor function you're saying.

$[0^-]=-1$ but $[0^+]=0$

Matt N.

This looks like the floor function. The nearest integer function looks like this.

Because it always rounds down. The nearest integer function might round up.

For example $1.75$ would become $2$.

Gigili

Right, yes.

Matt N.

Ok. Now let's see...

Here's an idea: do you know about sandwiching limits?

Let me think about whether this works here.

Gigili

10:37

I know it, but how it works here?

Matt N.

The sandwiching might help. After reading this I think you can use $x-1 < [x] \leq x$ to get $-1 < [x] - x \leq 0$

Although this doesn't help yet for the first limit since we can't replace the denominator with zero.

Gigili

Yes, I cannot apply it ..

What should I do when I'm not good enough at solving limit questions @MattN? like the ones I said or limits of trigonometric functions. My book has bunch of limit questions but without solution

It's not nice to ask many limit questions, is it?

Matt N.

@Gigili Just post them here or on SE and get the answers there!

@Gigili I wouldn't worry about this for now.

I can post one for you if you like, now I'm also interested in seeing the solution to the first one. : )

Would you want me to do that?

Gigili

Sure, so kind of you.

Great, thank you for your time Matt. You're awesome. I'll check your responses later.

Have a nice day.

Matt N.

Ok, then I'll post the first one and will link it here in chat. You too, be seeing you : )

Matt N.

11:04

Now I think it's not actually the floor function. For $x \in Z$ you'd have $floor(x) = x$ whereas in your book you get $[x] = x - 1$.

Andrey Vihrov

@MattN But the definition stated $[x] = x,\text{ if }x \in Z$, where does -1 come from?

Matt N.

@AndreyVihrov Can you add dollar signs for the mathjax?

@AndreyVihrov True!

Andrey Vihrov

@MattN Sure. I didn't think it worked in the chat, since it actually doesn't here :)

Matt N.

@AndreyVihrov It does : ).

Andrey Vihrov

@MattN Very tricky :)

Matt N.

11:13

But then I think we have it. What do you think of this: Use $x^3 - 1 \leq [x^3] \leq x^3$ to get $-1 \leq [x^3] - x^3 \leq 0$ then $$ \lim_{x \to 10^-} \frac{1}{x - [x]} \leq \lim_{x \to 10^-} \frac{[x^3] - x^3}{[x] - x} \leq 0$$

No. That doesn't lead anywhere.

@AndreyVihrov It's really a simple matter of copy pasting the js code into a bookmark of your browser. Takes 30 seconds. : )

But I don't want to convince you, of course.

@Gigili It's here.

Gigili

12:17

@MattN Thanks a lot.

Asaf Karagila

13:05

@Willie!

Willie Wong

Yo

Gigili

Hello all

What can be said about real roots of $x^7+2x+C=0$ where C is a constant?

I just know it has at most 7 real roots but it's not in the answers.

Asaf Karagila

@WillieWong How be you, Guillaume?

Willie Wong

13:25

@Asaf: well, you saw the bruhaha over at meta, right?

@Gigili: do you know Descartes' rule of signs?

Asaf Karagila

@WillieWong I stuck the last nail in the coffin, as you can see.

Willie Wong

Descartes' rule of signs

In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for determining the or negative real roots of a polynomial. The rule gives us an upper bound number of positive or negative roots of a polynomial. It is not a complete criterion, i.e. it does not tell the exact number of positive or negative roots. Descartes' rule of signs Positive roots The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial ...

@AsafKaragila Well, I have you to thank. I was going to write an answer, which in hindsight may be ill-advised, but you closed the question before I finished composing.

Asaf Karagila

You're welcomed :-P

I take my gratitude in scotch and in arak.

Willie Wong

Heh. But now I am re-learning lattice theory on one hand, and on the other writing entries for MathReviews.

Asaf Karagila

Which hand holds the scotch/beer/etc.?

Also, I assume that you have another hand for coffee...

Willie Wong

13:32

Obviously the hand that learns lattice theory is for scotch, and the head that writes entries for MathReviews is for coffee.

Asaf Karagila

Ah, I see.

I was starting to think that you're an Indian deity or something like that.

Willie Wong

(I can't learn lattice theory at the same time as I write a MathRev, and lattice theory requires me being awake, and most math papers go down much better with a bit of alcohol...)

Asaf Karagila

Can you make anything out of this?

All I got is that the tag is wrong.

Willie Wong

So you monitor (set-theory) just to right those wrongs?

Asaf Karagila

Also for axiom-of-choice stuff.

:-P

Willie Wong

13:37

@Gigili Also, is this in the context of a calculus course or a pre-calculus course?

Asaf Karagila

And you know, the occasional answer.

Willie Wong

Funny story about Zorn's Lemma:

I was doing some reading the other day, and saw a statement asserted without proof.

The author refers to two references: one is a book in Russian, the other is a paper in German.

Asaf Karagila

I love when they do it.

Willie Wong

Russian I could not read, and German only a little bit. So I decide to try my hand first at this seemingly harmless statement.

I spent the better part of the afternoon getting nowhere. So I hit the library and looked for the German paper.

The title said something about transfinite induction.

The section in which the theorem I wanted was proven began with "Let us recall Zorn's Lemma".

So I figure, ah! I didn't use transfinite induction nor Zorn's Lemma, that must be the missing ingredients.

So I went back and spent the better part of the evening and still got nowhere.

The next day I checked the Handbook of Analysis and found a proof of a slightly weaker statement, but if you do the right substitutions of definitions could be copied word for word to prove the statement I want.

And it emphatically did not use either transfinite induction or Zorn's Lemma.

Asaf Karagila

So?

Willie Wong

13:44

So I've been trying to figure out using my limited German, why the heck the author of that article mentioned either of those two things when the main result of his paper required neither.

Asaf Karagila

I used tb's assistance to translate German stuff before, but he's gone AWOL and was last seen quite some time ago.

Willie Wong

I should probably add that this German paper was published before the independence of AC was shown, in the era when people were fairly allergic to using AC when they don't have to...

Asaf Karagila

If you tell me what is the result then perhaps I can help you.

Willie Wong

Okay, the theorem goes something like this: Let $(E,\leq)$ be a complete lattice. Let $\mathcal{C}$ be a subset of $E$ closed under infimum. Let $h:E\to E$ be the hull operator sending elements $x\in E$ to $h(x) = \inf \{ c\in \mathcal{C} | x \leq c\}$.

Then if $(E,\leq)$ is an algebraic lattice (every element is the sup of all preceding compact elements) and if $h\circ sup = \sup\circ h$ applied to sets of compact elements, then $\mathcal{C}$ is inductive (that it is closed under chain supremum).

Asaf Karagila

MathJax died on me, the bastards.

Willie Wong

13:57

The interesting thing is that the partial converse, that $\mathcal{C}$ being inductive implies that $h$ commutes with $sup$ on compact elements, holds without the algebraicity assumption on the lattice.

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