Mathematics - 2015-04-30 (page 4 of 5)

Chris's sis

19:06

Pour tout réel $x>0$, $\displaystyle \psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$. Montrer que pour tout réel $x>0$
$$\psi(x+1)-\psi(x)=\frac{1}{x}.$$

@LeGrandDODOM many can be done in one line I think like the one above. One only needs to wisely put all the essence of the answer in a terse way.

Samuel Yusim

hi, peeps. I'm having trouble understanding the difference between limit points of a set A and points that are the limit of a sequence whose terms are all in A

apparently if my space is not first countable these aren't the same thing

LeGrandDODOM

@Chris'ssis I concur. I took this one as well, which was way harder marocprepa.com/temp/2015_conc/xens/2015_xens_b.pdf What's your opinon ?

Samuel Yusim

math in french is always weird

Karl Kronenfeld

@SamuelYusim That's correct. :)

Mike Miller

what's your definition of a limit point?

Karl Kronenfeld

19:18

Kind of having trouble figuring out what precisely is your difficulty.

Mike Miller

i partly ask because the definition varies, and also because i don't remember what the hell a limit point is

Samuel Yusim

my difficulty is that I've only inferred this from the text's exposition but I don't know why it's true, @KarlKronenfeld

"we say a point $p \in X$ is a limit point of $A \subseteq X$ if every neighbourhood of $p$ contains a point of $A$ other than $p$"

Karl Kronenfeld

Ah, I am way too lazy to type out $U_{\alpha}$'s and stuff, so lemme see if there are any MSE answers that do a good job explaining it.

Mike Miller

@SamuelYusim: One example to think about is the cocountable topology on an uncountable set.

I don't think that it's false for every space that's not first countable; I think I have an example where it's true and the space isn't first countable

Samuel Yusim

you know, I feel like the problem with topology as a subject isn't that it's poorly behaved, but that we happen to define stuff as if we expect it to only be used in the nice cases

Karl Kronenfeld

19:24

"Basically", you cannot "exhaust" the set of neighborhoods of certain points in a non-first-countable space with a countable collection of neighborhoods. Now, consider the precise definition of lim a_n and note that some limit points might not be limits of sequences.

@MikeMiller You're correct.

Mike Miller

you've thought too much about this, @Karl

@Samuel: we define stuff motivated by the nice cases, and see how it behaves. that's all

Ted Shifrin

hi @LeGrand, @Karl, goodnight @MikeM

Karl Kronenfeld

Hiya @TedShifrin

Mike Miller

morning

LeGrandDODOM

Bonsoir @Ted On a eu une magnifique épreuve sur Gamma.Beta.Digamma aujourd'hui :)

Ted Shifrin

19:28

Si ça te plaît bien, alors, @LeGrand :P

Samuel Yusim

I feel like in that case the stuff shouldn't be defined for the aspiring student outside of the nice cases on their first time seeing the material then, @MikeMiller

Karl Kronenfeld

@SamuelYusim Compare: Set Theory :D

Mike Miller

I can't talk about my pedagogical opinions on topology anymore, @Samuel, Ted will yell at me

Ted Shifrin

@MikeM: Not to worry. Ted is leaving.

Mike Miller

I'll wait til I see it. I don't want my feelings hurt is all, @Ted ;)

Chris's sis

19:30

@LeGrandDODOM The questions are pretty interesting. Given the nature of the questions flow and taking into accoung that it's a concours d'admission I suspect these questions are already found in some books, paper and the students need to solve the whole set of them from which some will be selected for the concours. At least this is the way the admission takes place here.

Samuel Yusim

I would have been far less confused if the definition of a limit point was given after that of first countability, even if there was a warning beside it that it would still be useful, if funny, outside of that context

Ted Shifrin

I learned/taught limit point way before I learned/taught about countability issues.

Karl Kronenfeld

@SamuelYusim But, the point of these counterexamples to elucidate the difference between the two concepts...

Ted Shifrin

But one certainly understands that sequences are insufficient in many situations, and some people are fond of nets for that reason.

Karl Kronenfeld

You can't say you understand either concept if you cannot differentiate it from something that actually is different.

Ted Shifrin

19:33

Perhaps point set topology, more than any other undergraduate subject, is best learned by testing out new concepts of lots of old examples/counterexamples.

OK, now Ted is leaving.

Samuel Yusim

yeah but since they're so closely related, why not put them together? I mean just because you define first countability first doesn't mean you can't have asides where you discuss pathological cases of limit points in contexts other than first countable spaces

if I were to write a book, that's how I would do it

Karl Kronenfeld

Why do you want them to be interchangeable?

Bye @Ted

Sufyan Naeem

Are you there @jeremyradcliff

jeremy radcliff

Sufyan

yes!

do I have to write @SufyanNaeem?

Chris's sis

@LeGrandDODOM maybe one day you can bring here some of the professors prepared those tests (in case you know some of them well).

jeremy radcliff

19:39

@SufyanNaeem ah ok, I think I see how this works, I can reply directly

Sufyan Naeem

No you have no need. Tell me your problem precisely but don't be concise

LeGrandDODOM

@Chris'ssis Nope, we don't have a list of questions we know are going to be at the exam. It's totally random, but of course our concours have their classics, such as the Gamma function, Jordan normal form, ... These exams, at least those dedicated to Polytechnique and the Ecole Normale Superieures aim at selecting the very best students (ingenuous, smart, quick-witted).

@Chris'ssis This was the hardest exam this year drive.google.com/…

jeremy radcliff

@SufyanNaeem Ok, I was reading the post by Blue that you mentioned made you understand my OP, and I think it's starting to make more sense.
In general I'm confused as to why things are defined certain ways in math. In this case, I just didn't understand why the unit circle and coordinate systems were chosen to expand the Trig functions for obtuse angles. Was it just convenience, does it have to be this way? etc...

Mike Miller

@samuel limit points do not end up being a big deal in topology, mostly because you can't transfer the metric space ideas. you can work with things called nets instead but meh; i'd rather work with open neighorhoods.

Karl Kronenfeld

dissing nets. ugh

Mike Miller

19:42

(there is, in particular, absolutely no way of topologically encoding the notion of a cauchy sequence.)

nets are stupid. point set topology is stupid. ur stupid

Karl Kronenfeld

referring to the stupid-filter :D

Samuel Yusim

okay but then why define limit points at all?

Sufyan Naeem

@jeremyradcliff yes it turned out be convenient and there was a need as well.

Mike Miller

don't ask me

Karl Kronenfeld

closure?

Chris's sis

19:43

@LeGrandDODOM I can understand that, but you tell me you don't have any dedicated stuff from which you prepare for the Polytechnique concours? If so, then for students the entrance admission might be pretty difficult.

Mike Miller

intersection of all closed sets containing your set

aka minimal closed set that contains your set

boom

jeremy radcliff

@SufyanNaeem, so you mean "there was need" in the sense that there was a need to define Trig functions for angles > pi/2, and then the unit circle and coordinate system was the most convenient way to extend the definitions of Trig functions in a coherent way with the already established definitions?

Sufyan Naeem

Yes of course bro.

LeGrandDODOM

@Chris'ssis We do have a syllabus cache.media.education.gouv.fr/file/special_1_MEN_ESR/42/4/… as anywhere else, but the entrance exams to Polytechnique and the ENS go way beyond

jeremy radcliff

@Su

Sufyan Naeem

19:46

Yes?

jeremy radcliff

@SufyanNaeem haha, thank you that makes sense

Sufyan Naeem

Was that all, you want to understand?

jeremy radcliff

@SufyanNaeem, I think before I used to think there was an "absolute" reason for things in math, i that makes sense.

Chris's sis

@LeGrandDODOM Are you given marks from 1 to 10?

LeGrandDODOM

@Chris'ssis 0 to 20 (20 is best)

Chris's sis

19:47

@LeGrandDODOM What is the minimum (accepted) to enter there? It also depends on the students' scores, that's true.

jeremy radcliff

@SufyanNaeem, Yes it's clearer now, I'm going to read Blue's long post carefully. Thanks for your help, I appreciate.

Sufyan Naeem

If it makes sense then feel free to go to my post and mark it as accepted :)

Karl Kronenfeld

@MikeMiller and a point x in the closure is a point that is in the complement of any open set containing the given set A. so there does not exist a neighborhood of x that does not intersect A, etc.

jeremy radcliff

@SufyanNaeem, sounds good. Thank you, take care :)

Mike Miller

i know these things, and yet i find them incredibly dull and only barely worth saying

in any case, my opinion on point-set is (apparently?) nonstandard

Sufyan Naeem

19:50

See ya!

LeGrandDODOM

@Chris'ssis the minimum to be on the list of "admissible" contestants is a mean around 11/20. Only 200/1000 contestants are "admissible" each year and only 100/200 are "admis"

Chris's sis

@LeGrandDODOM I see.

buddhababe

Prove that a subset S of R^n with Jordan content 0 has an empty interior.

hola :)

Karl Kronenfeld

@MikeMiller ignore the points?

it's really just algebra with open sets amirite

Chris's sis

@LeGrandDODOM How many of the students reach 180-200 points (in general)?

LeGrandDODOM

19:54

@Chris'ssis less than 10

Chris's sis

@LeGrandDODOM And the perfect score?

LeGrandDODOM

@Chris'ssis I don't know. I guess no one

Dave

can some one check if my calculation is correct for my equation

hopefully i've understood

Chris's sis

@LeGrandDODOM Well, I entered Polytechnique with a perfect score (200/200), but the questions were not that hard, and besides that we had a book of over 1500 (or much more - I don't remember) problems or so from which they selected some for the exam.

evinda

At the step simplify it should be -17*13 @Dave

Chris's sis

19:58

@LeGrandDODOM let me see if I find a version on internet.

Dave

Which simplify?

first or second one

Karl Kronenfeld

first one

Dave

oh so its 13*(7k/13) - 17*13

Karl Kronenfeld

dat distributive law

Dave

should i be multiplying 13*3t how ever

or should i wait for it to be 13(3t + 17)

buddhababe

20:00

nevermind

Chris's sis

@LeGrandDODOM here victoritadutu.files.wordpress.com/2011/10/…

Karl Kronenfeld

@Dave either way is correct

Dave

so like that?

Karl Kronenfeld

the first three steps are correct. Of course, you have to go through and fix the rest

Dave

so the second simplify onwards is wrong?

evinda

20:07

@Dave Look at the step Add 17

Samuel Yusim

looks like the step where you added 17 was problematic

Dave

how so ?

adding 17 the left side to cancel it out on the right side?

Samuel Yusim

well, you can't expect 17 to cancel out 17*13

remember how order of operations works

Dave

oh so its actually 221

good point

forgot about that -_-

oh thats meant to say 7 not 21 on the last line

Samuel Yusim

yeah, and I assume you meant divide by 7 and not divide by 21

otherwise that looks good

Dave

20:12

yeah

thanks for the help !

a lot easier than i first thought

Chris's sis

@LeGrandDODOM I don't even need to mention that one can be very smart and fail the test though if not prepared for that kind of questions given on exam.

Chris's sis

20:31

This reminds me of my research questions that I might post here and probably no one answered, but this is just normal to be so since I did a lot of research in some particular corner of mathematics and I'm in an advantage from this point of view. I'd also fail like you without that research.

Dave

when you get an answer for an equation how do feel confident if you are right

i got an answer but i just keep feeling i got it wrong

Karl Kronenfeld

you're referring to that one you showed us @Dave?

Dave

no the one i just did after it

Karl Kronenfeld

similar type of problem?

Dave

i can show you what i got if you want. im not confident with my answer though

yeah its same idea

Karl Kronenfeld

20:38

you can try plugging in a value for the indepedent var. of the equation at the beginning, getting an answer for the dependent var., then substituting that into your solution. You should always get the value you started with.

Dave

thats a bit difficult with this answer because i get a weird negative

Karl Kronenfeld

Ah, but -(-k)=k

Dave

on the last line -4k/k = -k right?

what you mean -(-k)

Karl Kronenfeld

the opposite of the opposite of k

Dave

but wouldn't that mean the right hand side numbers all need to be swapped

so -33t^2 + 21tk / -4 ?

Karl Kronenfeld

20:41

exactly

Dave

oh so i did technically get the right answerr

.

Karl Kronenfeld

I didn't check your work carefully, but you do have the problem that you did not isolate the variable

notice that the variable k appears on both sides

Dave

isn't tk not same as k though

Karl Kronenfeld

yeah, you're right, but the expression on the right depends on both t and k, when you need it to depend only on t

Dave

unless you mean basically on the second simplify i should not merge 3t * 7k

keep them seperate until i can remove 3t?

Karl Kronenfeld

20:43

indeed you cannot determine the value of tk or (3t)(7k) knowing only the value of t

so you need to do something different at some point. too lazy to help :P

Dave

lol okay

Samuel Yusim

does first/second countable imply hausdorff?

my guess is no

Ted Shifrin

no way, @Samuel

Samuel Yusim

just as I thought

Ted Shifrin

Do you have a favorite example of a non-Hausdorff space?

Samuel Yusim

20:48

usually I go to {1, 2, 3} given the topology {{} {1, 2}, {1, 2, 3}} but I'm new to this stuff so I can't say I've frequently needed to

Mike Miller

well... does that have a countable basis? a countable local basis at each point?

Samuel Yusim

oh, duh

Ted Shifrin

Maybe you should have a nice infinite example, too

Samuel Yusim

$

oops

Ted Shifrin

hi @Disciple of Wilma

Samuel Yusim

20:50

$\mathbb{R}$ given the trivial topology?

Ted Shifrin

yeah, but that's a boring example, @Samuel ...

Samuel Yusim

so what's a non-boring example in your mind?

Disciple of Barney

Hey @TedShifrin, enjoying retired life?

Ted Shifrin

an example that's Hausdorff everywhere except for two points, @Samuel

not yet, @Disciple of Wilma ... two final exams coming up

Disciple of Barney

Oh, just get some grad student to do that

Ted Shifrin

20:52

@Samuel: Take two copies of $\Bbb R$ and glue them together everywhere except at their respective origins.

Nope, @Disciple of Wilma ... doesn't work that way.

Samuel Yusim

what does it mean to glue them together?

Ted Shifrin

it means that we have one point for each $a\in\Bbb R$ except for $a=0$, and we have two different copies of $0$.

Disciple of Barney

@TedShifrin Well hopefully these exams don't send you into a deep depression. :D

Ted Shifrin

A basis for the topology is usual intervals in $\Bbb R$.

Thanks, @Disciple

Samuel Yusim

uh, do you mind giving a more explicit construction? I'm not sure I get it

Dave

20:54

@KarlKronenfeld think i managed to isolate it

got k = -33t^2/9t

Karl Kronenfeld

check it

also can be simplified further, one is important, another simplification is a matter of choice.

Dave

how would i check for it ?

Karl Kronenfeld

pick something nonzero for t

using your last equation determine what k will be.

plug in t and k into the equation you started with and see if both sides are equal

(you might get false positives of course)

Ted Shifrin

More formally, but no more explicitly, @Samuel: Take $\Bbb R\cup\Bbb R/\sim$ where $x_1\sim x_2$ (by which I mean the same real numbers in the two copies) UNLESS $x_1=x_2=0$.

Karl Kronenfeld

@dave a (much harder) way to be sure you're correct is to plug in the whole expression -33t^2/9t for k everywhere it appears in the given equation, then you try to simplify and manipulate it down to 0=0, but that is an error-prone procedure in general

Samuel Yusim

21:01

isn't $\mathbb{R} \cup \mathbb{R}$ just $\mathbb{R}$?

Karl Kronenfeld

yes!

Dave

okay - ill try my best see what i get

Karl Kronenfeld

@SamuelYusim he means disjoint union

Samuel Yusim

I get the feeling you're trying to make this somehow a "disjoint" union where elements of the first $\mathbb{R}$ aren't in the second $\mathbb{R}$

yeah

Karl Kronenfeld

where you take two distinct "copies" of $\mathbb R$

Ted Shifrin

21:03

yup I meant two separate sets where you identify corresponding elements

Samuel Yusim

so okay I guess I can take it for granted that this is doable

by which I mean I could probably do it myself given two minutes

Karl Kronenfeld

just tag the elements of the first with $1$s and the elements of the second with $2$s (using ordered pairs)

Ted Shifrin

Now understand why it has all the properties of $\Bbb R$ you can think of ... except Hausdorff.

Samuel Yusim

oh, sure I get it

Ted Shifrin

it's a nice example to keep in your bag of tricks

Samuel Yusim

21:05

yeah, definitely

topology is pretty messed up, frankly

Ted Shifrin

Well, life isn't just $\Bbb R^n$ and metric spaces ...

Disciple of Barney

I sort of like the messiness of (general) topology

Its fun

Samuel Yusim

well it's something, anyway

Ted Shifrin

It's good for developing critical thinking skills ... playing with examples/counterexamples.

Samuel Yusim

21:19

what's a metric space that isn't second countable?

Mike Miller

discrete on uncountably many elements

Samuel Yusim

derp

ᴇʏᴇs

21:33

Just walked out of an exam 30 seconds ago and realized I made an error

Karl Kronenfeld

you must have channeled me

sorry

i'll try to be less error-prone when you channel me in the future

ᴇʏᴇs

Thanks, I would really appreciate that @KarlKronenfeld

robjohn

@evinda I am not sure what you are looking for in your question, so I posted a computation of what I believe the problem is asking. I may have missed the point.

Victor

21:52

@robjohn - May you help on this question? math.stackexchange.com/questions/1258191/…

robjohn

@Victor sorry. I don't have any idea about that.

Victor

@robjohn - Is that hard to understand or hard to answer or for other reason?

edited: Is that question hard to understand or hard to answer or for other reason?

robjohn

@Victor How do we classify ODEs and can we even determine if one doesn't have a closed form? For two equations of the same type, there may exist a closed solution for one and not for another when they only differ in a constant. I think it may be very hard to answer, but since I am not an expert, I may not know some classification result.

Ted Shifrin

hi @robjohn

Victor

@robjohn How should i improve my question in order to ask it on math overflow?

robjohn

22:00

@TedShifrin Hey, Ted. How are things?

Ted Shifrin

Well, my house is on the market. I'm starting to get used to drinking no water :P Finals tomorrow and Tuesday. You doing ok?

Victor

@robjohn - Thanks in advance.

robjohn

@Victor I don't know. Did it get rejected from MO in its current form?

ᴇʏᴇs

Hi @TedShifrin, are you feeling anything as your day of departure is closing in

Ted Shifrin

@Victor @robjohn: That's really a differential algebra question.

mr eyeglasses: I will miss most of my students. Some of them will miss me. But it had to end sometime.

robjohn

22:02

@Victor Try adding the differential-algebra tag.

Victor

i wonder if it is okay for the strict member of math overflow to able to interpret the nontechnical question of mine...

Ted Shifrin

Don't put it on mathoverflow, @Victor.

Victor

@TedShifrin - Why?

Ted Shifrin

It's not a research-level question, IMO.

I just don't want to think hard enough to answer it.

robjohn

@Victor heh... so it is too hard for me, and too easy for others :-)

Victor

22:07

@robjohn @TedShifrin - Also i am wonder that why there are no chat room for general question in math overflow.

Ted Shifrin

because serious research mathematicians don't have time to spend in chatrooms

only us lessers spend time here

ᴇʏᴇs

Damn, I walked out of that exam so happy thinking I killed it and then as I was walking down the hallway I realized an error and it wiped the stupid smug smile off my face

Ted Shifrin

smug is usually not good in math, mr eyeglasses

ᴇʏᴇs

@TedShifrin Qiaochu Yuan seems to say things often in the homotopy theory chat

Ted Shifrin

Qiaochu, brilliant as he is, is a grad student still ...

is there chat on math overflow? or are you talking about chat here?

Mike Miller

22:10

mathoverflow

Ted Shifrin

oh, so overflow has chat too? I inferred from Victor's question that they didn't.

Mike Miller

nobody uses the general chat

the only used one is a homotopy theory chatroom

Ted Shifrin

oh

Victor

@TedShifrin Why does mathematician in overflow only want to discuss algebraic topology in the chat room?

Ted Shifrin

most people in overflow don't spend idle time in chatrooms

not very many do here, either

Mike Miller

22:12

only us idiots here

Ted Shifrin

some of us should do way less looks at @Mike @robjohn @tedshifrin

robjohn

@TedShifrin this seems to be a MathOverflow room

Samuel Yusim

maybe all those guys are just too boring to have anything to talk about

Karl Kronenfeld

hah

Ted Shifrin

yeah, @robjohn, with days lapsing between lines :P

serious research math takes time to ponder

unlike high school or undergraduate homework problems here

Victor

22:13

@TedShifrin i mean why algebraic topology is more important than the other subject?

Alessandro

I can understand a few words in their posts, sometime. (Also, they seem to use that chat once every 3 days)

Ted Shifrin

maybe there's a few people in alg top willing to go in there, @Victor

heya @Alessandro

I only came in here years ago because @Pedro forced me

Alessandro

good evening @TedShifrin

(@Ted you live in America, right? I recently realized that I always greet you by saying good evening, but it should be afternoon for you)

Karl Kronenfeld

a little after six

Ted Shifrin

no, @Alessandro, it's 6:15 PM, evening

Mike Miller

22:16

lol

Ted Shifrin

of course, for @MikeM, it's still morning

wait, @MikeM, isn't your talk around now?

Mike Miller

two hours

grad student seminar is in the evening

Ted Shifrin

what? seminar at 5 PM?

yikes

well, grad students go out for beer after :P

Mike Miller

the colloquium is thursdays... we grab the interested audience for what comes afteter

correct

Ted Shifrin

I don't like the model of having to sit through 2 hours of hard lecture straight

maybe I'll just meet you after it all sometime :D

Mike Miller

22:18

colloquium is at 3

Ted Shifrin

oh, so an hour break

well, give a great lecture, @MikeM

robjohn

@TedShifrin are you in the area now?

Dave

damn my answer was miles off the actual answer

robjohn

@TedShifrin Ah, perhaps I was not getting the entire context.

Mike Miller

@TedShifrin I'll give it a shot

Clarinetist

22:28

Hello Math SErs

I have a very difficult decision to make and would like some input

Dave

always the second decision

Clarinetist

A healthcare tech company has given me an offer. I was originally going to opt for them until a position that I'm much more interested in (a nationally-known institution for health research and education) decided to expedite my application after I told them of this offer and because accepting that offer would prohibit me from working for this institution, due to the non-compete

I am interviewing with this institution tomorrow

Via Phone, so it is a 1st interview

I don't want the tech company to wait longer than a week, plus I need to figure out moving plans

So I would like the institution to get back to me within the week after the phone interview if they want me to do an on-site and/or maybe with an offer

I'm not sure how to word my requests for the tech company to wait and for the research institution to speed things up

Suggestions are appreciated

I'm a newbie at this

ᴇʏᴇs

Hi @Clarinetist

Karl Kronenfeld

You chose math.se?

Disciple of Barney

Obviously we are not very good at making decisions we are in the math chatroom...

Clarinetist

22:34

Well, I've tried Reddit, and it seems like Workplace has a negative attitude toward the what-would-you-do questions. And people know me here :P

Karl Kronenfeld

"Let $f$ the function which returns the amount of time you would like the given company to wait..."

ᴇʏᴇs

$f(x) = \frac{1}{0}$

Transcript for

$Mathematics$ Mathematics