5:00 AM
If we devised a psychology experiment where a number of big dots appeared on a screen for a split second and we then had to report how many we saw, anything bigger than 5 or 6 would have us guessing, but 4 is a nice solid number that we can instantly recognize. Thus we don't need to develop intuition for 4; it's already built in. But ask someone unfamiliar ordinals to draw what w^w looks like...

but that is not an intution

Then again, ««intuition about [insert any topic]» seems so misguided» is something you seem to say invariably about anything :-)

people use the word intuition for the most strange things
and then people suffer people they do not "intuit" stuff

the ability to picture something in your head (which I would place under the umbrella term of "intuition"), or to have at least a metaphorical picture, makes concepts aesthetically pleasing and can make it easier to keep track of information as well

but you cannot intuit in that sense properties of the number $e$
or $w^w$.

5:03 AM
you don't tell me what to do >:(
you're not my mother!

study the object in questin
then you develop a sense of what might be true
and what migh be false
just as one does not intuit walls
one learns not to walk into them as a kid
by experiment
the fetish of intuition is dangerous
specially for students, who think they are missing something
and then get discouraged
you can ask an expert on subject X if he expects Proposition A to hold
and he does not consult the fairies of intuition bbut his expertise on the subject,
his familiarity with related matters,

@MarianoSuárez-Alvarez, are you thinking of publishing your poetry? because this is really something "one does not intuit walls"

as I said, I find that using pictures and metaphors and such to keep track of what's going on in a situation, even if I can't articulate them as they exist in my head, helps me keep track of what I'm doing, can inspire new questions, and makes the landscape more pleasant and accessible. I don't despair when I don't have intuition on a topic, nor do I expect as a rule everything to have an intuition, so none of your negatives seem to apply to me.

if only I managed to keep the meter right...

5:07 AM
Give me a minute

@anon, but asking for a picture of $w^w$ is a completely different matter, and a question which can be answered productively

for instance, you talk about ordinal addition in terms of "concatenation." I would definitely call that an intuitive understanding of it.

well, that is the definition, really

Can someone explain me what a limit point is intuitively ? is is like a center of a circle ?

it is like the limit of a convergent sequence

5:09 AM
@MarianoSuárez-Alvarez Can't the students probe their heads and discover if they've learned the thing intuitively?

hahaha

@hyg17 what circle?

@GustavoBandeira, what they can probe is if their educated guesses get better with time

I just find analysis so counter intuitive.

5:10 AM
well, it is

@MarianoSuárez-Alvarez How to avoid intuition then?

I'm self teaching it, so sometimes it's nice to have someone tell me the easier way to understand it.

@hyg17 Same here. =)

I assume most people self teach here

5:12 AM
@MarianoSuárez-Alvarez
Study the object in question,
That you develop a sense of what might be true,
What might be false.
One does not intuit walls,
One learns not to walk into them,
By experiment.
The fetish of intuition is dangerous.
Students, who think they are missing something
And then get discouraged,
You can ask an expert on subject X
If he expects Proposition A to hold;
He does not consult the fairies of intuition
But his expertise on the subject,
his familiarity with related matters.
3

there you go :-)
@GustavoBandeira what do you mean, avoid it?

@MarianoSuárez-Alvarez
Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world,
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.

I am not saying one should avoid it

@WillJagy Read it with this music playing.

@WillJagy :-)

5:13 AM
@MarianoSuárez-Alvarez So you're saying one should control it?

Surely some revelation is at hand;
Surely the Second Coming is at hand.
The Second Coming! Hardly are those words out
When a vast image out of Spiritus Mundi
Troubles my sight: a waste of desert sand;
A shape with lion body and the head of a man,
A gaze blank and pitiless as the sun,
Is moving its slow thighs, while all about it
Wind shadows of the indignant desert birds.
The darkness drops again but now I know
That twenty centuries of stony sleep
Were vexed to nightmare by a rocking cradle,
And what rough beast, its hour come round at last,

So WTH are you saying?

no, I am saying that asking for intution for something is not ging to work in most cases

I am to lazy to read that

5:13 AM
intution for the properties of $e$

Now this bugs me because I can't imagine what it's saying. "A set E is closed if every limit point of E is a point of E"

How can a set have multiple limit points if it's something converging to it ?
How come a limit point cannot be in the set itself if it's converging to it ?
Is it analogous to continuity ?

can't believe it chopped off the last line, that's the best bit.

@hyg17, look at the definitions of the terms you are mentioning

Principles of Mathematical Analysis by Walter Rudin

5:15 AM

what edition?

@WillJagy The "see full text" will reveal everything. :)

Third

x is a limit point of A if there are elements of A arbitrarily close to it (metric space definition). there is nothing about this definition that precludes A from having more than one limit point. there is nothing that precludes A containing its limit points, nor is there anything that forces it to contain its limit points. For instance, if A=[0,1), then both 1/3 and 2/3 are lim pnts in A, and 1 is a lim pnt outside of A.

I didn't know there was a third lol, I have the 1st one I think its 40 some years old

5:16 AM
@MarianoSuárez-Alvarez What would you recommend?
@Ethan That's a MILF book.

just learn stuff, Gustavo

@J.M., I know, it just interrupts the flow.

@MarianoSuárez-Alvarez Yep. I'm doing.

well, then

I've read a lot of math books, but this one is really killing me. I don't see any motivation behind the calculation...

5:17 AM
"intuition" will come
don't look for it
Allez en avant, et la foi vous viendra, as D'Alambert used to say to his students

@MarianoSuárez-Alvarez all in due time, too.

got to be D'Alembert. Like the cheese.

Eventually, you get the hang of it, and even forget that you once thought it was too unintuitive.

@WillJagy haha

@hyg17 Reading something without motivation is a motivation. It's a challenge!

5:19 AM
your motivation should be to learn it

Never mind, I actually "saw" it myself intuitively. I'm wishing to become a professor one day, so I would like to be able to teach my students so that they "get" it, rather than shoving it up their head.

http://en.wikipedia.org/wiki/Camembert
http://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alembert

as a student, your motivation for the subjects you have to study is that if they are part of your required stidies they probably are useful
after you finnish stidying, you provide your own motivation

@WillJagy Yes, a nice cheese with the white wine...

Thanks everyone, I actually feel better now that most of you actually think that having no motivation is a motivation for me to work harder on it.

5:20 AM
personally, my motivation to study when I was a student was that I enjoyed it
I could not care less for what the use of what I studied was
I remember talking with a class mate about measure theory: we converged to the argument that if something ends up taking up several walls in a library, then the subject surely is worth being studied

@J.M., they were feeling their way with Silence of the Lambs, when Anthony Hopkins pronounces Chianti the way he does. The last book has him as Lithuanian nobility. Or, he was mocking Clarice's supposed original regional accent.

I would be less frustrated if I were not studying for the GRE now, on other days I actually take my time studying until I get it.

"we converged to the argument"
lol

@WillJagy By last book, it's the prequel, no? I didn't realize that Hannibal was just pulling Clarice's leg there...

5:23 AM
lol

hm, Chianti

I removed the comment, because I didn't think I was in a position to be handing out advice on life

Can someone help me out ?

I'm studying for the GRE subject test and I would like to practice a bit on Laurent Series expansion and the Residue Theorem.

Is there a page that I can look at, or can anyone give me a basic example ?

every textbook on the subject as exercises

@Ethan Then you can give advice on death

5:25 AM
and there are uncountably many textbooks on that subject

@GustavoBandeira my life is about 20% over

@Ethan Why?

Because I don't think I will live much past 85, do you?

"uncountable" is shorthand for "speaker was too lazy to count", yes? :D

I don't have a book. All I have is a GRE practice book and I
am out of example problems.

5:27 AM
whats gre?

@J.M. have you been to the complex analysis section of a math library?

there are uncountably many complex analysis textbooks!

@MarianoSuárez-Alvarez there are math librarys?

@J.M., I guess that's the question, I can't be sure. That seems one possibility, although they may not have been sure about what was in the prequel; assuming Thomas Harris was somewhat involved in consulting on the script for the movie, it depends on what he told them, and whether he had made up his mind at that stage. Certainly in the movie, Lector says "Good nutrition has given you some length of bone, but I bet you're not one generation removed from poor white trash."

5:27 AM
@MarianoSuárez-Alvarez I have, I got lazy counting the books on analysis, much less complex analysis. :)

Unfortunately I don't live close to a library... but that sounds like a good idea.

if you count calculus textbooks, you get way past large cardinals

@Jay Hey!

@pen Hi!

5:29 AM
@Ethan or math sections at universsity libraries

@MarianoSuárez-Alvarez Notice that I said "analysis", not "calculus". If I count calculus as well, it'd be more productive to wait for the universe's heat death.

Part of my friend Dmitry's dissertation turned out to be a theorem by Lindelof. i found it in half an hour in Evans Hall library. I still don't quite understand how Dmitry did not find out it was due to Lindelof.

@JayeshBadwaik Surely gonna be selected! :-D

but yes: my math department has a library, and there is nothing in it but math books

@pen Good. :-)

5:30 AM
@WillJagy Maybe somebody took it out when he was searching...

@WillJagy well, that's better than having part of one's dissertation be a false claim. At least Lindelof checked the details and agreed :-)

@JayeshBadwaik Yeah. Can you help me with my question?

Wasn't there an MO thread on stories like that?

Could be. Dmitry and i talk by phone about once a week, he said recently that he was proud that he had found something Lindlof found. I think that is the right attitude.

Is there a Ph.D in here ?

5:32 AM
I agree with him, in fact!

Several Ph.D.s I am also sober

@hyg17 why

when I was a student I discovered Lefschetz's fix point theorem en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem
i was so proud!

whats the theorem?

Lefschetz, that's very good.

5:33 AM
I have a question about what it is like to research in Math.

nvm, I can't understand it lol

@pen Will see.

Many times before, I reckon out a nice algorithm, and then find out that I'm at least ten years too late after a literature search...

Homological algebra

@Ethan it gives a nice criterion for a function $f:X\to X$ to have a fixed point

5:35 AM
@MarianoSuárez-Alvarez by fixed point do you mean

@hyg17
It's like rain on your wedding day
It's a free ride when you've already paid
It's the good advice that you just didn't take
Who would've thought... it figures

a point x such that x = f(x)

@pen Heed the comment.

@MarianoSuárez-Alvarez that sounds interesting lol, especially atractive fixed points

5:37 AM
I want to prove something historically awesome, like showing that $pi + e$ is transcendental. Which by the way, I still am not sure why e or pi is transcendental.

@Jay $(c - b)^3 - (a + b)^3 = \left(c - a - 2b\right)\left(c^2 - 2cb + ac - ba + bc + a^2 + 2ab + b^2 \right)$ Is that correct?

@Ethan Lefschetz' is not as shallow as you are, and he finds all fixpoints, attractive or not!
:-)

@MarianoSuárez-Alvarez the dottie number is one my favorites :)
or more like the only one I know lol
@pen lol $$(a+b+c)^3+(a-b-c)^3-(a+b-c)^3-(a-b+c)^3=24abc$$
Is bountins identity

@Ethan It's not taught at school. -_-

@pen Identities of this sort are actually very nice, I found that one on some guy on heres blog
pizzaz? or somthing

5:41 AM
@Ethan It's nice, but I can't really use it to show my work.

lol

What is the class of study that shows numbers being transcendental ?

@pen where did that come up?

@Ethan ?

where did you encounter the problem of proving the equivalence of those two expressions

5:42 AM
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways. Transcendence The fundamental theorem of algebra tells us that if we have a non-zero polynomial with integer coefficients then that polynomial will have a root in the complex numbers. That is, for any polynomial P with integer coefficients there will be a complex number α such that P(α) = 0. Transcendence theory is concerned with the converse question, given a complex number α, is there a polynomial P with integer coefficients such that P(α) = 0? If no...

«The condition [that a permutation is an $n$th power] is equivalent to the statement that the number of $r$-cycles in the permutation is divisible by $d_n(r)$, where $d=d_n(r)$ denotes the largest divisor of $n$ such that $r$ and $\frac{n}{d}$ are relatively prime.» This seems like a typo, but I'm not sure how to fix it.

So it's after studying a lot of number theory, huh.

How fascinating.

@hyg17 I bought a book on this. =)

How was it ?

I can't even read the cover of the book, but I bought it!

5:44 AM
lol

It will accumulate dust for now, but the future...

@anon permutation of what?

permutations of a finite set

@hyg17, that usually falls under analytic number theory. About research, if you are in the U.S. and in college, a very good thing is these Summer R.E.U. programs, groups of students working on a few related projects, writing up individual reports.

@WillJagy R.E.U?

5:47 AM
A kid who asked a question on MO about his reu project turned out to have something publishable. It's just that he did not know enough to interpret it. By the time it all came together, he was in graduate school doing algebraic topology. But I have told a few people. Ethan, Research Experience for Undergraduates. And the occasional unusual high school student. Very well thought out program, federal funding.

@WillJagy how do they define research?
miss ping'

I have graduated school 3 years ago with a very low grade, and ever since then I have been studying alone to overcome my week parts in math. So far, I have been able to have a solid understanding of secondary school math, a good understanding of linear, abstract algebra, non-euclidean geometry and multivariable calculus. Now I am challenging myself to working on analysis because I've never taken it is school. So, I am trying to go back to school by taking the GRE for now.

whats the GRE

@WillJagy Do you know if there's something online?

Meanwhile, a kid i know in India got a plum summer program, but it was one on one with a postdoc who had little time for him. @Ethan, for the R.E.U. it is just something that they work on together. In the one kid's case, he thought he was doing modular forms, but it was a very simple result in quadratic forms.
@GustavoBandeira, yes, let me look

5:52 AM
An exam you take to go to grad school in the us

how does one confuse whether one is working on modular or quadratic forms?

@WillJagy I want to do something like this, but I'm from Recife, Brazil. I guess these kind of oportunity is rare by here.

@GustavoBandeira, my impression is that these are for U.S. residents, even if you could afford to pay your own way. Noone I asked was entirely sure about that. And, Brazil may have something, it is a matter of how to find out.

@WillJagy It has. But here it's rare and for such a thing I guess I would have to travel ~2,5k Km.

See you guys around, it was fun talking.

5:56 AM
nvm, I need to study more lol

@anon, that is my way of describing it. The basis problem is the attempt to write reasonable modular forms as linear combinations of theta functions, of positive integral quadratic forms in this case. He made an understandable error of restriction in his computer program, so he never saw that it amounted to Siegel's local density formula (if he found out what that was). I can tell you that there is an ongoing markey for people who know both modular forms and quadratic forms-there are few.

wow

@WillJagy That seems shocking.
That only a few people know both

ongoing market. i have wound up writing papers with modular forms people, it is amazing what they don't know, that is standard and easy quadratic forms stuff.

@WillJagy sounds interesting, I am planning on learning some combinatorics, and then geting into integer partions and the like
I will see where it takes me lol

6:00 AM
@WillJagy What are you defining as "quadratic forms stuff"? I mean, any number theorist knows a bit, at least enough to talk intelligently about Brauer groups.

@AlexYoucis, I guess I would say that many of the people who know both have bigger fish to fry than I do. Also, there are about four camps of mutually incompatible viewpoints on quadratic forms. One is related to Lie algebras. I solved something, I now need to choose whether to post something on the arXiv, as my co-author is very ill. Oh, my own interest is always "what numbers are integrally represented?" This is very similar to quadratic fields in the binary case, but drifts in more variables
So, for instance, I keep answering Pell equation problems one way, with the word "automorph" which is a bit outdated, and Andre keeps putting these things that are just field element raised to the power n.

@WillJagy have you ever seen sites.google.com/site/tpiezas/Home.

@Ethan, new one on me. It's late. The best book for this is Cassels, Rational Quadratic Forms. I keep seeing confused questions by people who are plowing through Lam or other books and don't know how to do down-to-earth things. Anyway, Cassels does arbitrary dimension, but does not go wild on all possible fields of coefficients. So it's a bit easier for a beginner.

1 hour later…
7:14 AM
Nothing like the feeling of your cherries on a cold chair.

7:31 AM
I have different types of chairs.
The cushion is often too instable to sit and then it hurts like a mofo.

8:27 AM
14

From my experience, it seems that although unstable is more commonly used, instable is often preferred in engineering and scientific contexts, e.g. "aircraft instability", "instable algorithm". Are there any differences in the implied meaning of the two terms? Should unstable be preferred?

8:55 AM
Anyone have ideas on this? "There are n participants in a swimming competition. Each of them takes k seconds to cover the pool where 12 <= k <= 17, and k is an integer. How many times will any 5 participants get the same time? (5 <= n <= 10)"
Sounds like a case for the bookkeeper rule, but I don't know how to use it properly.

9:19 AM

hi guys :)

@libjup Hello!

2 hours later…
11:22 AM
I post it because it seems like someone is actually listening. I bet I was just telling some nonsense and they think it is funny 8-).
(nobody with a Dutch nationality can be seen in this picture!)
I look like a giant on these pictures!

@JonasTeuwen You have more than 2 meters high.

With shoes, yes, without not.

You should have taken a photo like this:

Well, I did, but then he could not make any pictures.
Even more obvious here.
(I am the chairman of the thingie that organizes these things)

I took this pic some time ago:
This is the son of my cat. Look how he's sleeping.

11:39 AM
this cat does look hungry

Why?

every other cat i know looks a bit fat but i hardly can see this one :D

@DominicMichaelis He's very young.

oh maybe thats why

In another angle.

11:41 AM
oh thats how i know cute cats :D

Hello everybody

@skullpatrol hey)

@Nimza What's the meaning of $\smile$ in $f^\ast (a \smile b) = f^\ast(a) \smile f^\ast(b)$?

@robjohn hi, do you know some upper estimates on grow of $\pi$ function $\pi(z) = \frac{1}{\Gamma(z+1)}$ along vertical lines?

11:44 AM
@Nimza There is a question about that... just let me look

@GustavoBandeira it is cup product, do you know what is it for $a \in H^{p}$ and $b \in H^q$?

@Nimza No.

@GustavoBandeira okay, let $x$ be a singular ($p+q$)-simplex in topological space $X$, i.e. continuous map $x \colon \Delta^{p+q} \to X$, where $\Delta^{p+q}$ is a standard $(p+q)$-simplex

@Nimza Take a look at this answer

@GustavoBandeira then we can define operation of taking front $p$-face and rear $q$-face of it
@robjohn thank you!
@GustavoBandeira Then $(a \smile b) (x) = a(front\;face\;of\;x)b(rear\;face\;of\;x)$, but our homology groups need to be computed over the ring to be available to multiply here
so $a \smile b \in C^{p+q}$ if $a \in C^p$ and $b \in C^q$
@GustavoBandeira sorry, I told you about cup product in cochains, but it descends on cohomology groups obviously

11:52 AM
@Nimza But I'm not understanding what you're saying.
I'm still a noob.

@GustavoBandeira on which stage? I'm noob too but this thing seems very easy

@Nimza Starting analysis.

aa
@GustavoBandeira we can define it on usual simplexes, if $a$ if a function that takes a $p$-simplex as input and returns for example a real number and $b$ is the similar function that takes $q$ simplex as input then $a \smile b$ is a function that takes $(p+q)$-simplex as input as returns a real number that is $a$(front $p$-face)$\cdot b$(rear $q$ face), is it clear?)

11:58 AM
@GustavoBandeira for $(p+q)$-simplex $[a_0,\ldots,a_p,\ldots,a_{p+q}]$ front $p$-face is simplex $[a_0,\ldots,a_p]$ and rear $q$-face is a simplex $[a_p,\ldots,a_{p+q}]$ if we speak about geometric simplexes
@Chris'ssisterandpals hi

@Nimza I'll have to go out urgently now. Can I talk about that later?
I bookmarked the chat, later we can proceed.

@GustavoBandeira of course)

Thanks. Cya.

later

bye
@robjohn hm, in this answer you speak about upper estimates on $\Gamma(z)$ but in my case it is $\frac{1}{\Gamma(z)}$

12:02 PM
This morning I woke up with a question: Is it possible to prove that $\sum_{k=1}^{\infty} \frac{1}{k^4}=\frac{\pi^4}{90}$ by using squeeze theorem?

@Chris'ssisterandpals squeeze theorem? I like when it is called two policemans theorem)

@Nimza hehe. I never heard of this name. It sounds nice.:D

@Chris'ssisterandpals it is official name of this theorem in Russia :D in any book it is called like this

@Nimza really???

@Chris'ssisterandpals aha! ru.wikipedia.org/wiki/…

12:07 PM
A funny name! :-)

@Chris'ssisterandpals for second paragraph google translate gives "The name of the theorem comes from the fact that if the two policemen held together with the offender and go to the camera, then the offender is also forced to go there."

hehe, I see.

))))

12:26 PM
These days I only met problems that seem to come from other worlds.

@robjohn ah, the upper estimate on $\frac{1}{\Gamma(z)}$ can be obtained from yours by $\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin{\pi z}}$ or via complex Stirling, right?

@Nimza notice that $\sin(iy)=i\sinh(y)$

@robjohn you missed a - there

@DominicMichaelis I don't think so... I will check

@robjohn aha

12:32 PM
$$i\cdot \sin(iz)=\frac{i}{2i} \left( \exp(i^2 z) -\exp(-i^2 z)\right)$$ $$\frac{1}{2} \left( \exp(-z)-\exp(z)\right) = -\frac{1}{2} \left(\exp(z)-\exp(-z)\right) = - \sinh(z)$$

Hey.

@DominicMichaelis divide by $i$ and you get what I said

hey

@robjohn oh right

12:34 PM
@DominicMichaelis You can move the $i$ in and out of $\sin$ and $\cos$ as long as you add or remove the 'h'

Somewhere here I met this question but I cannot find it now: $\lim_{n\to\infty} \frac{p_1}{p_1-1}\cdot\frac{p_2}{p_2-1}\cdots\frac{p_n}{p_n-1}, \space p_n$ is the $n$th prime number

$\sin(ix)=i\sinh(x)$ and $\sinh(ix)=i\sin(x)$

lol, maybe it's a question asked by my bro. I need to check that.

it looks like eulers produkt
@chris doesn't it diverge? its the harmonic series isn't it ?

@DominicMichaelis I only had to mention that it's required an elementary proof.
@DominicMichaelis yeah, it diverges.

12:37 PM
as i said the harmonic series

Right.

Am I unjustified in finding this person's questions/comments to be pretty annoying? math.stackexchange.com/users/61948/jane-ke

i have somewhere the proof but only in german

Oh, it's clear now. I have the proof. (just sent by Chris)

@Chris'ssisterandpals I would guess that it would grow like $\log(p_n)$

12:40 PM
@robjohn that's correct.

@Chris'ssisterandpals I got that just by assuming the density of primes is $1/\log(n)$
I think I can work up something a bit more rigorous if needed

@robjohn nice

@Chris'ssisterandpals I think a slight modification of this answer, changing a 2 to a 1, will work

@robjohn oh, that looks great
Let $f:\mathbb{R^{}_{+}}->\mathbb{R}$ be continuous in $1$ and we know that $f(x^2)=f(x), \space \forall x\in \mathbb{R^{}_{+}}$. I have to prove that $f$ is constant.
I don't know why my text above looks so horrible.
Let $f:(0, \infty)->\mathbb{R}$ be continuous in $1$ and we know that $f(x^2)=f(x), \space \forall x\in \mathbb{R^{*}_{+}}$. I have to prove that $f$ is constant.