Mathematics - 2016-01-29

Evinda

00:10

Guten Abend @TedShifrin
Could I ask you something?
Why is the norm of a matrix smaller than the root of the sum of the square of the elements ?

Ted Shifrin

Guten Abend, @evinda. Well, maybe equal in a special situation.

What have you tried?

Evinda

This matrix is given: $A=\begin{bmatrix}
a & b\\
0 & c
\end{bmatrix}$ where $a^2+b^2+c^2<1, b \neq 0$ and it says that $||A||<1$ since $||A|| \leq \sqrt{a^2+b^2+c^2}<1$. @TedShifrin

Ted Shifrin

That didn't answer my question. :)

Evinda

Do we use the definition of ||A|| ? @TedShifrin

Ted Shifrin

In this case, thinking about a general $m\times n$ matrix is fine. Yes, you want to think about the length of $Ax$ for various unit vectors $x$. The question is how to think about $Ax$.

mr @pedro !!!!

Pedro Tamaroff

00:14

@BalarkaSen OK. I was trying to sort a proof out, luckily I could do so.

Hello @TedShifrin. =)

Ted Shifrin

@Pedro: When I was a graduate student, Spanier, Vick, and Greenberg were the "only" beginning standard texts on alg top. I owned all three.

Hope you're doing ok these days, mr @Pedro.

Pedro Tamaroff

@TedShifrin Spanier is quite nice. Bummer, do you still have your copy of Spanier? I should fly to San Diego and take it from you.

@TedShifrin Yes, I'm doing quite fine.

Ted Shifrin

No, I kept only a small fraction of the books. Grad students and faculty walked off with most. (Not counting the ones you and Mike got.)

Spanier is not my style. But that won't surprise you.

Evinda

@TedShifrin It holds that $||Ax||_2=\sqrt{(ax_1+bx_2)^2+(c x_2^2)}$. We xould use the fact that ||x||_2=1 if we cound the coefficients by the largest one. Shall I do so?

Ted Shifrin

But he was a decent lecturer and very friendly guy. I knew him pretty well in grad school.

@Evinda: As I said, you probably will see things better if you do the case of a general matrix and don't try to write things so explicitly.

Pedro Tamaroff

00:18

@TedShifrin Well, Spanier is sometimes a bit dry and seems to write hastily, but I was able to sort out his writings so far.

It is a good exercise, given he doesn't leave much exercises to the reader. =)

Ted Shifrin

I don't know about hasty, @Pedro. I don't remember anyone's having found errors in there when we were in grad school. ... I don't like his exercises. Most are formal.

@Evinda: There are two ways to interpret $Ax$. What are they?

Pedro Tamaroff

@TedShifrin Not errors, but he sometimes writes in a way that makes the text unclear.

Ted Shifrin

Well, @Pedro, that's what people who try to use my texts say, too. And they're wrong :D

Spanier gives almost no geometric intuition. So that's why I don't like him.

Evinda

@TedShifrin Ax=\left(\sum_{j=1}^n a_{ij} x_j \right)_{i=1}^n

Which is the other one?

Ted Shifrin

@Evinda: That's one way. But what does it actually mean? What are those entries?

The other way, very, very important, is to take a linear combination of the columns of $A$ (rather than working with rows). But for your question, the rows will do fine.

P.S. @Pedro: I'm very glad you're "doing quite fine." :)

Evinda

00:24

@TedShifrin The dot product of the coefficients with a vector x... What could it mean? :/

Ted Shifrin

The $i$th entry is the dot product of the $i$th row vector of the matrix with $x$, yes.

And what fundamental thing do you know about $|a\cdot x|$?

Evinda

@TedShifrin The Cauchy-Schwarz inequality :)

Ted Shifrin

Yes!! So use it!

Mike Miller

Hi @Ted.

Ted Shifrin

Good night @MikeM. :)

@MikeM: I spent 3+ hours yesterday with about 5 kids doing nothing but ratios, proportions, and slopes of lines.

Mike Miller

00:27

I'd like that, but mostly because I'd get paid an absurd amount.

Ted Shifrin

LOL, I'm paid only in love and admiration :P

Actually, I was having to teach the other helpers at the library how to explain some of the stuff.

The woman in charge hugged me and thanked me for having come back from holiday :)

It's not surprising that even such easy, basic stuff is hard for so many educated adults.

@Evinda: Do you see it now?

Mike Miller

I believe that. I have not yet learned patience.

Evinda

@TedShifrin I am making the operations... I will finish in a few!

Ted Shifrin

OK, @Evinda. For extra credit, get an lower bound on $\|A\|$ using the column interpretation :)

@MikeM: So I've noted a time or two :)

Mike Miller

@Ted: I've been trying exercises but having a lot of trouble with them. ;)

Ted Shifrin

00:31

Of course, I quickly lose patience on MSE with people who give away answers when I've been trying to "teach" the OP with hints.

Evinda

@TedShifrin I got $\|A\| \leq \left( \sum_{i=1}^n \sum_{j=1}^n a_{ij}^2 \right)^{\frac{1}{2}}$

Ted Shifrin

Aha, @MikeM: I suspect I'd have trouble, too, after all these years.

Yes, @Evinda. That's exactly what you want!

Mike Miller

@Ted: I mean on patience.

Evinda

Yes!!! @TedShifrin

Ted Shifrin

Oh, you have exercises on patience, @MikeM?

PVAL

00:33

Patients are for doctors. I have more trouble finding students.

Ted Shifrin

Ausgezeichnet, @Evinda. As I said, you can use columns to get a lower bound :P

Evinda

@TedShifrin Danke!!! :D
Could you maybe also give me a hint how to find an example of a $n \times n$ matrix for which it holds that $||A^2||< ||A||^2$ ?

Ted Shifrin

@PVAL: I thought you had 150 of 'em in one class. What are you teaching this semester?

Think about what matrix multiplication means, @Evinda.

PVAL

@TedShifrin I thought a student was one who studies.

Mike Miller

@Ted: We're going slowly on G&H but that's because we all have other things to be doing.

Ted Shifrin

00:35

Aha, you got me on a technicality, @PVAL. :P Most of mine actually did, or they dropped :)

But a number did not and they're part of why I quit, so I understand completely.

@MikeM: I understand that entirely. There's a lot of deep content.

Mike Miller

Also, MSE is an exercise in patience. A long one with many parts.

Ted Shifrin

Well, I'm mostly invisible these days, @MikeM.

@Huy: It's way past your bedtime!

Evinda

@TedShifrin I think that we could pick for example the matrix we were talking about previously.
Then $||A^2||= \sup \{ \sqrt{(a^2 x_1+ b (a+c)x_2)^2+ c^4 x_2^2}: ||x||_2=1 \}$ but $||A||^2=\sup \{\sqrt{(ax_1+bx_2)^2+ (c x_2)^2} : ||x||_2=1\}$

Right?

Ted Shifrin

Ah, I wonder if this is another instance of Huy's computer logging him in without his being there.

Evinda

Does it maybe hold for all matrices that are not diagonal or symmetric?

Ted Shifrin

00:42

I doubt it, @Evinda, but maybe your example works for certain values. You have a typo in the second line.

But you could approach it more conceptually. Why must it be that the $x$ that makes $Ax$ biggest should make $A(Ax)$ biggest? In particular, what could happen with $A(Ax)$?

Evinda

@TedShifrin You mean that I forgot to square it, right?

Ted Shifrin

Yes.

So you could actually work out the Lagrange multipliers problems there, but I'm suggesting an easier conceptual approach.

Evinda

@TedShifrin So we have a matrix A and we look for the vector x so that Ax gets the maximum value. Then we consider again the matrix A and want to multiply it by the biggest possible value of Ax, and we have found the x for which this happens before... Right?

Ted Shifrin

Well, then you'll have equality! But if that doesn't happen, you'll have inequality.

But do you know matrices where $A(Ax)$ gets very, very small for certain $x$?

Evinda

@TedShifrin If we have for example this matrix : $\begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix}$ its square will be the zero matrix...

Ted Shifrin

00:54

Aha.

Evinda

So in this case if we define the matrix as A we have $||A||^2=1$ and $||A^2||=0$, right? @TedShifrin

Ted Shifrin

You need to edit that!

Evinda

@TedShifrin Sorry!!! I edited it...

Ted Shifrin

Yes, now right!!

Evinda

@TedShifrin Great!!! Thanks a lot!!! :)

PVAL

00:57

@TedShifrin The worst is when someone writes a completely unreadable question and then someone answers in detail the question they probably meant.

Ted Shifrin

Moral of this discussion, @Evinda: Sometimes it helps to think a bit more conceptually :)

Evinda

Yes :) @TedShifrin

Ted Shifrin

Glad I could help you.

Well, probing to get the person to say what he actually meant takes work, @PVAL.

OK, back to reviewing a geometry book.

Learner

01:12

please help stats.stackexchange.com/questions/193015/…

Evinda

@TedShifrin And something else... If we have a linear function $T: \mathbb{R}^2 \to \mathbb{R}^2$ with $T(x_1, x_2)=(ax_1+bx_2, cx_1+dx_2)$ how can we apply the Caucchy-Schwarz inequality to bound this $||T(x_1, x_2)||_2$ ?

Josué Molina

Is a singleton closed in a general metric space?

Mike Miller

Yes. Try to prove it.

Evinda

@TedShifrin A ok... I found it by myself... :)

I'm an artist

01:32

Very late here but again I got a mind-blowing result ...

L33ter

01:45

Suppose X is a metric space consider a arbitrarily point $x \in X$, Let $U = X - {x}$, let u be a point in our set U. What can you do after that @JosuéMolina ?

I'm an artist

I challenge you all to calculate $$\int_0^1 \frac{\text{Li}_3\left(\frac{x}{2}\right) \log ^2(1-x)}{x} \, dx$$

and

$$\int_0^1 \frac{\text{Li}_3\left(\frac{x}{2}\right) \log ^3(1-x)}{x} \, dx$$They are very nice, and that challenge it's an actually an invitation to Let's have fun!

L33ter

You need to construct a open nbhd of u that is contained in U

I'm an artist

@Huy your turn!

Wait! It's very late here.

I have to sleep a bit.

@r9m see above ^^^

Josué Molina

02:15

@L33ter I suppose I need to show $U$ is open to conclude $\left\{x\right\}$ is closed. Since $X$ is open, there is $\varepsilon_1>0$ such that $B_{\varepsilon_1}\left(u\right)\subseteq X$, and since $u\neq x$, $0<d\left(u,x\right)=\varepsilon_2$. Setting $\varepsilon=\min\left\{\varepsilon_1,\varepsilon_2\right\}$, then $B_\varepsilon\left(u\right)\subseteq U$. Does this sound alright?

Semiclassical

@TedShifrin what kind've geometry book, out of curiosity?

Josué Molina

@L33ter Wait, I don't think I even need the $\varepsilon_1$ I posted above.

Balarka Sen

07:21

@PedroTamaroff Sure, OK.

What are you studying?

Tobias Kildetoft

09:04

Anyone familiar with the Chevie package from GAP 3 (unfortunately not ported to GAP 4)? I can't get it to tell me the cells of a Coxeter group when it is not a Weyl group, even though the documentation does not seem to require this.

Samuel Yusim

09:17

just finished the last of four assignments due yesterday/today. what a relief.

Tobias Kildetoft

@SamuelYusim What subject?

I'm an artist

@robjohn hey. It seems no one is able to calculate

and I wonder why. Maybe the exponential integral scares them a huge lot.

@TobiasKildetoft do you think, counting all students and professors from your uni, is any able to calculate this integral?

Oh, I also posted this night

$$\int_0^1 \frac{\text{Li}_3\left(\frac{x}{2}\right) \log ^2(1-x)}{x} \, dx$$

Tobias Kildetoft

@I'manartist No idea. I think nobody would be very interested in trying unless it had a connection to something related to their research

I'm an artist

$$\int_0^1 \frac{\text{Li}_3\left(\frac{x}{2}\right) \log ^3(1-x)}{x} \, dx$$

Tobias Kildetoft

But even then, no idea if any of them would be able to

I'm an artist

09:24

@TobiasKildetoft that seems fair.

Balarka Sen

Hi @TobiasKildetoft.

Tobias Kildetoft

@BalarkaSen Hi

pingwindyktator

hey, quick answer neede ;D if we've got polynomial w over complex numbers and w(c) = 0, does it implies that w(conjugation(c)) = 0? I know that if polynomial is over real numbers, that is true

Tobias Kildetoft

@pingwindyktator No, if the coefficients are not real then it need not hold

just consider a linear polynomial

pingwindyktator

@TobiasKildetoft right. Thanks!

Tobias Kildetoft

09:34

In fact, if that holds for all roots (counting multiplicities), then the coefficients are real

pingwindyktator

@TobiasKildetoft nice, I didnt know about it. One more question. polynomial (z-c)(z-conjugation(c)) would't divide w, right? It's true just for (z-c)

Tobias Kildetoft

@pingwindyktator Right, because $(z-a)$ dividing it means that $a$ is a root

pingwindyktator

;)

Jessy Cat

0

Q: For what $a$ is $t^{a}\sin \frac{1}{t}$, $t > 0$ uniformly continuous?

Without paying a whole lot of attention to what $a \in \mathbb{R}$ is, I was able to show that, since $t^{a}$ and $\sin \frac{1}{t}$ are both continuous on $(0,1]$, $x(t)$ can be extended to $y(t) = \begin{cases} t^{a} \sin \frac{1}{t}, & t> 0 \\ 0, & t = 0\end{cases}$, which is continuous on the...

pingwindyktator

10:20

Can someone calculate for me volume of shape stretched on point (0,0,0), (0,0,1), (0,1,1), (0,1,2), (1,1,0), (1,1,1), (1,2,1), (1,2,2)? I just want to know answer and I really dont know how to calculate it with for example wolframalpha

and google dont want to help me

Evinda

10:57

@DanielFischer I am rereading your answer. We have shown that a $M$ exists for all linear maps $A: \mathbb{R}^n \to \mathbb{R}^n$. How do we deduce that this also holds for all the continuous maps?

Daniel Fischer

11:19

@Evinda Sorry, what's the context?

Evinda

11:36

@DanielFischer We want to show that $A : \mathbb{R}^n \to \mathbb{R}^n$ is continuous iff there is a $M>0$ such that $||Ax||_2 \leq M ||x||_2$ for each $x \in \mathbb{R}^n$. If there is such a M then $A$ is Lipschitz continuous, so also continuous.
I tried to prove the other direction as follows:

$||Ax||_2^2= \sum_{i=1}^n \left( \sum_{j=1}^n a_{ij} x_j \right)^2 \leq \sum_{i=1}^n \left( \sum_{j=1}^n a_{ij}^2 \right) \left( \sum_{j=1}^n x_j^2 \right)= ||x||_2^2 \sum_{i=1}^n \sum_{j=1}^n a_{ij}^2 \Rightarrow ||Ax||_2 \leq ||x||_2 \left( \sum_{i=1}^n \sum_{j=1}^n a_{ij}^2 \right)^{\frac{1}{2}}$

hhh

Is there someone here who can instruct about Gröbner basis computational results?

0

Q: Gröbner basis from ideals, how to verify hand-calculated results computationally?

M2 stands for Macaulay2. The goal is to learn to use M2 and learn to calculate Gröbner basis from ideal with Buchberger's algorithm. The reduced GR basis should not be needed to calculate by hand here. So how can you calculate GR basis from ideal and verify the result computationally? Helper q...

math discrete-mathematics symbolic-math symbolic-computation

I am able to calculate non-reduced GR basis while I have found so far M2 to calculate only reduced GR so how to verify hand-done calculations computationally?

Daniel Fischer

12:02

@Evinda Not really, and yet in some sense we do. The point is that all linear maps $\mathbb{R}^n \to \mathbb{R}^m$ are continuous, which for example this calculation shows. In the calculation, you use the explicit form of the linear map, and without continuity, you couldn't have such a convenient explicit form.

Evinda

You mean that this :$||Ax||_2^2= \sum_{i=1}^n \left( \sum_{j=1}^n a_{ij} x_j \right)^2$ holds only if A is continuous? @DanielFischer

Daniel Fischer

@Evinda You have such a representation only when the spaces involved are finite-dimensional. And so, it's true that it holds only if $A$ is continuous, but that's a pretty vacuous statement, since there are no discontinuous linear maps between such spaces. It becomes non-vacuous when you move to infinite-dimensional spaces. For this specific calculation, you need a Hilbert space, since you're using the Cauchy-Schwarz inequality.

But in the infinite-dimensional case, you have to worry about convergence. Does the formula even make sense? You need the "matrix" to satisfy some constraints for it to work. In the finite-dimensional case, you only work with finite sums, so a convergence issue could not arise in any way.

aaadddaaa

12:47

I would love to get help with a problem in linear algebra:
Let P be self adjoint matrix P=P* (Over the Complex field)
T is a linear transformation defiend as T(X)=(P^-1)xP i need to prove that T is also sefl adjoint

Clarinetist

Suppose $t, X \in \mathbb{R}$. Can I find $$\lim_{n \to \infty}\dfrac{(tX)^n}{n+1}$$ (I can impose restrictions on $t$)

I'm not sure how to do this one

Might just want to use L-Hospital

(now that I think about it)

so that gives me $$\lim_{n \to \infty}n(tX)^{n-1}$$

Sigh, I should just look for something on convergence of MGFs

Huy

what do you know about $tX$?

Clarinetist

That's in $\mathbb{R}$, and I can impose restrictions on $t$ if I want

What I know for a fact is that apparently $t$ has to be in a neighborhood of $0$, but IDK why

The general problem is this

Huy

pretty sure it'll grow too fast if $|tX| > 1$ and otherwise it'll go to 0

Semiclassical

Is X a random variable here?

Clarinetist

12:55

@Semiclassical Yes

The general problem is this

Consider \begin{equation*}
e^{tX} = \sum_{n=0}^{\infty}\dfrac{(tX)^n}{n!} = 1 + \dfrac{(tX)^2}{2!} + \dfrac{(tX)^3}{3!} + \dfrac{(tX)^4}{4!} + \cdots\text{.}
\end{equation*}

Where is this going to converge?

I thought maybe, do a ratio test

Oh man

The limit computation is WRONG

It should be $$

$$\lim_{n \to \infty}\dfrac{tX}{n+1}$$

which is significantly easier

So this limit is $0$

Hmm, I feel like there's something missing here

I guess it is absolutely convergent, without restrictions on $t$.

Now the expected value though, that might be a different case

Because you would be taking the integral of an infinite sum...

But this works since the sum is absolutely convergent.

Huy

you wrote something entirely different

Clarinetist

Yeah, that was my bad

Huy

don't know anything about random variables but pretty sure it'll converge everywhere

just like the usual exponential function

if I remember correctly, if $X$ is Gaussian, the exponential $e^{tX}$ will have expectation value $e^{\frac{t^2}{2}}$ or something like that

Clarinetist

@Huy @Semiclassical By the way, I made the realization yesterday that $\mathbb{E}$ is a mapping $\text{Func}(S, \mathbb{R}) \to \mathbb{R}$

and I thought, how strange

Huy

what do you mean by Func(S,R)?

Clarinetist

13:08

@Huy The set of functions from a sample space to $\mathbb{R}$

Huy

which sample space though

Schwartz space?

Clarinetist

Now there, you've got me. I haven't learned the technical details of that yet

Huy

probably Schwartz or something like that because you want distributions

or wait, its dual is a distribution space

@Clarinetist: where are your topology questions though

:P

Clarinetist

@Huy Sigh, been procrastinating on Topology. I have quals to study for xP

Huy

excuses

Clarinetist

13:13

But it seems fascinating

Hopefully I can knock out both quals this year, and then I can study all the Topology I want :)

Huy

cool

hope so too

Balarka Sen

@Huy pot calling kettle

Huy

@BalarkaSen: you're supposed to be dead

Balarka Sen

yes, thanks to you, I am almost. I am sick.

Clarinetist

I'm trying to teach myself the material for one of the quals. My background is substantial enough for me to find it worth the effort to just learn the rest of the material

Partly Putrid Pile of Pus

13:14

@TobiasKildetoft Fairly unspecific question I suppose: When reading a research paper, and coming across some concept or word you haven't come across, do you have some method of working out it's meaning, if you can't glean it from context?

Balarka Sen

fever + shivers + stabbing pain on left ribcage

Tobias Kildetoft

@PartlyPutridPileofPus I usually google it and look for a wikipedia page on it

Partly Putrid Pile of Pus

How many links would you expect to click to find the meaning, if there wasn't a wikipedia page? (If you've made some rough observation)

Balarka Sen

silent cheers, @Huy? :P

Partly Putrid Pile of Pus

Maybe I am lacking tenacity.

Tobias Kildetoft

13:18

@PartlyPutridPileofPus It is quite rare for this to happen, so hard to say

Partly Putrid Pile of Pus

I see, so normally at the level that you are producing research, all research related to your specialization is either in your vocabulary, or developed in the paper that you are reading?

Tobias Kildetoft

@PartlyPutridPileofPus Mostly yes, though sometimes I need to read through how some term is used in the proofs to see what is meant

Huy

@PartlyPutridPileofPus: usually you should find some half-decent definition within 3 links

at the very most

Partly Putrid Pile of Pus

@Huy Sure, but it's hard to tell when they lead you to more words you need to google. I am sure the chain ends somewhere though.

Tobias Kildetoft

@PartlyPutridPileofPus Yeah, the chain should end somewhere, and in the meantime you are learning valuable concepts (hopefully)

Huy

13:26

@PartlyPutridPileofPus: if you have a chain of length 10, maybe that means you need a bit of a better background to seriously tackle the paper :P

Partly Putrid Pile of Pus

13:36

@Huy That's true, but it also helps give me an indication of where I should be looking to learn these things

Tobias Kildetoft

@PartlyPutridPileofPus Possibly one of the main things that can cause it to be hard to find stuff is when the authors just reference some "standard reference" for an entire page of stuff (which is then scattered across several hundred pages in that reference)

Huy

CTRL + F is your friend

Tobias Kildetoft

but often that is a clue that this reference might be a good thing to get acquainted with

Partly Putrid Pile of Pus

@TobiasKildetoft That's why I asked about MM1-6, I wasn't sure if perhaps it is best to read a motivating chain of papers first.

Tobias Kildetoft

@PartlyPutridPileofPus In this case not necessarily. Those papers require a lot of background in 2-category theory, while the new one does not

Partly Putrid Pile of Pus

13:39

@TobiasKildetoft Seems reasonable. Did you fully read all of those papers btw?

Tobias Kildetoft

though certainly much of the motivation might be a bit hard to see if one is neither familiar with categorification nor KL theory

@PartlyPutridPileofPus Yeah, that was more or less the first thing I started doing when I started my current position, since the methods from those were a driving factor in the results in my first paper with Mazorchuk

Mike of SST

Hello mathematicians. I don't want to derail your conversation too much, but please could someone point me in the right direction to get advice on whether I should ask a particular question on Math.SE: is chat the right place, or meta?

Tobias Kildetoft

@MikeofSST chat is probably faster

Mike of SST

OK. Is this the room to do so?

Tobias Kildetoft

yeah

Mike of SST

13:48

Great. I recently posted a question on StackOverflow as it's fundamentally a programming problem, but I have come to believe that, if there is a solution, it will be a mathematical one.

Tobias Kildetoft

Well, if you can phrase it purely as a mathematical problem, then it sounds like a fine question for MSE. Do put the programming aspect as a motivation, though, so people know what sort of answer will be useful

Mike of SST

I'm not a good mathematician and don't have much idea about how to go about phrasing the question in mathematical terms, but I'll have a think about it. Thank you. :-)

DylanSp

15:28

Hi, I have a question on an answering etiquette. If I post an answer, then realize it's wrong due to comments, should I delete the answer or just let it be downvoted?

Mike Miller

If an answer's wrong, unless it's instructivelt wrong (and you add a disclaimer at the top that it's wrong), I would delete it.

aaadddaaa

15:58

Hey, can anyone help me with this question:
http://math.stackexchange.com/questions/1631991/linear-transformation-defined-by-self-adjoint-matrix

Mike Miller

Hi @AndrewT.

L33ter

16:26

@MikeMiller can you discuss with me something in algebraic topology

why in general we can build a genus g surface from a 4g polygon

I can imagine the torus

iwriteonbananas

What's your definition of the orientable surface of genus $g$?

L33ter

I am just seeing hatcher introduction chapter 0

he didn't give a definition of it

I understand genus is the amount of holes we have in an object

iwriteonbananas

Right, well to my knowledge the orientable surface of genus $g$ is usually defined as the surface with surface word $[a_1,b_1]\cdots [a_g,b_g]$

which is, by definition, a quotient of a 4g-gon.

L33ter

oh I see

I wanted to see the picture geometrically

I can imagine first one

iwriteonbananas

Well, try to prove that the orientable surface of genus $g$ is homeomorphic to a connected sum of $g$ tori. The latter description is the geometric picture you're thinking of.

L33ter

16:38

I see

iwriteonbananas

In general, the connected sum of two surfaces given by words is homeomorphic to the surface corresponding to the concatenation of both words.

L33ter

I see

@iwriteonbananas where did you learn algebraic topology ?

hatcher?

I find hatcher difficult to read sometimes

Partly Putrid Pile of Pus

Is the transcendence base of $E/F$ just the subset of $E$, consisting of all transcendentals of $E$ over $F$?

iwriteonbananas

@L33ter Hatcher was the book I most frequently used. You'll get used to the difficulty eventually. There's no algebraic topology book out there which is more hands-on and easy to read than Hatcher (to my knowledge).

L33ter

I see

PVAL

17:27

@MikeMiller So uh this questions still not dealt with math.stackexchange.com/questions/1630512/…

robjohn

17:50

@I'manartist: Did you see $\displaystyle\sum_{n=1}^\infty \left(H_n-\log(n)-\gamma-\frac1{2n}\right)$?

I'm an artist

18:15

@robjohn Hi. Back. I think I did something like that in the past.

@robjohn simple summation by parts, the rest is an easy story.

@robjohn (+1) for the proof.

@robjohn I wanted to use this one $$1+\sum_{k=2}^n\left( \frac1k -\log\left(\frac{ k}{k-1} \right) \right)=H_n-\log(n)$$ for a generalization (I mean another problem in a generalized form), but someone took my opportunity.

Mike Miller

@PVAL: Isn't there a version of Alexander duality for taking complements in arbitrary closed oriented manifolds?

I'm an artist

@robjohn I try to finish now some amazing stuff for a journal, I'll show it to you at some point. Something you don't see neither on MSE, nor on MO.

Evinda

18:35

@DanielFischer A ok... Thank you!!!! :)

Samuel Yusim

@TobiasKildetoft they were for algebraic geometry, lebesgue integration and fourier analysis, category theory and homological algebra, and complex analysis

I'm an artist

18:54

@Huy is it normal for a notebook to have sometimes a weird behaviour of the computer after plugging the stick, more exactly slow motion. After unplugging it all is fine.

Mouse pointer is moving in slow motion till the moment I unplug the stick.

@RandomVariable have you ever tried these integrals?

$$\int_0^1 \frac{\text{Li}_3\left(\frac{x}{2}\right) \log ^2(1-x)}{x} \, dx$$

$$\int_0^1 \frac{\text{Li}_3\left(\frac{x}{2}\right) \log ^3(1-x)}{x} \, dx$$

I love this stuff.

user276387

Is there a quick way to decide whether $2016x+4031y = 2014201520162017$ has integer solutions?

Mike Miller

Yes.

I'm an artist

Yes.

Random Variable

@I'manartist I don't think I have.

I'm an artist

@RandomVariable OK

user276387

19:07

@MikeMiller by computing $\frac{2014201520162017}{\gcd(2016,4031)}$?

Mike Miller

Try computing the gcd first and see if it's clearer what I mean.

user276387

Ah, I see. The $\gcd(2016, 4031) = 1$. Thanks.

Huy

@I'manartist: define "after plugging the stick"?

Mike Miller

@PVAL: There is, in the form $H^q(X) \cong H_{n-q}(M,M \setminus X)$. That's probably useful. I don't really want to think about this though.

I'm an artist

@Huy You attach the stick to the computer through an USB port, that's the meaning of plugging the stick.

Huy

19:20

@I'manartist: does this slow motion only last for a few seconds or does it last for a long time?

I'm an artist

@Huy Sometimes disappear immediately but sometimes it lasts longer, so long I have to unplug the stick.

Huy

how long?

I'm an artist

@Huy 15 min? I didn't let it more like that since I couldn't use my computer.

robjohn

@user276387 Since $\gcd(2016,4031)=1$ it has a solution.

Huy

ok, no, that's definitely not normal.

is the USB stick new too?

robjohn

19:22

@user276387 Ah, I see

I'm an artist

@Huy No, it's not.

@Huy One more thing, the stick becomes hot after a while during the period it is plugged.

Huy

@I'manartist: does this behaviour occur with any other machine that you plug the stick into?

I'm an artist

@Huy No, it doesn't.

Huy

what version of Windows are you using?

robjohn

@I'manartist bad stick

I'm an artist

19:24

@robjohn it might be!

robjohn

@I'manartist that is odd. Perhaps a bad dll

I'm an artist

@Huy windows 8.1

Huy

the first thing I'd do is to see if you have the correct and up-to-date drivers installed

I'm an artist

For instance, I have the device from a wireless mouse and all is perfectly fine (which is in the other USB port).

@Huy Good point.

robjohn

@I'manartist Does this persist if you restart the computer?

I'm an artist

19:26

@robjohn Yes.

@robjohn It might happen that if I unplug the stick and plug it again, the bad behaviour doesn't repeat.

robjohn

@I'manartist Then it might be a corrupt dll. Reinstall drivers.

I'm an artist

@robjohn OK

I'm looking for an update.

I'm an artist

19:40

@robjohn @Huy all is fine now.

Thanks.

More tests, no delay, all is perfect, that was after reinstalling some drivers (downloading the last ones).

robjohn

20:09

@I'manartist Great! It would be bad if your stick blew up :-|

I'm an artist

@robjohn :-)))))

Vader

Hello

I don't understand this step

I don't see how 5^x can be factored out of 125 when there is no 5^x in 125

robjohn

@Vader Distributive property of multiplication over addition

Noting that $5^x=1\cdot5^x$

Vader

Yes

robjohn

$$\begin{align}&=5^x+125\cdot5^x\\ &=5^x\cdot1+5^x\cdot125\\ &=5^x(1+125)\end{align}$$

Vader

20:22

why the 1+, isn't that increasing value?

nvm

robjohn

@GBeau Congratulations!!

guest

You are right @Vader there is no 5^x in 125. Replace 5^x with the letter "a" and then just factor it out :-)

Vader

@guest so you say I should factor? a + 125 * a?

guest

Yes.

Vader

a(125+1)

but then using the distributive property I get 125a + 1a

so basically 126a or 126*5^x

which I know can't be right

guest

20:33

Why?

Vader

why what?

guest

Why is it not right?

Vader

because that's not what I started with

125 used to be a constant now it's got 5^x with it

guest

You just simplified it.

Vader

I'm going bald over this

guest

20:36

:D

Ilya_Gazman

Hey guys, can you please help me with linear algebra. I don't understand something very trivial.

guest

You started with two terms, now you have one. @Vader.

Vader

I'm going to watch some khan videos

guest

Have fun.

He does a lot of good work.

user174558

@I'manartist I am now Superman.

guest

20:41

:O

Vader

@guest it all makes sense now, thanks

Vrouvrou

please if i an integral over $A\cap B$ can i devid it into two integrals onve over $A$ and the other over B ?

guest

thanks for asking @Vader :-)

Vader

thanks for answering @guest

Vrouvrou

@robjohn have you an idea please

Ilya_Gazman

20:54

0

Q: Solving the matrix in Quadratic sieve

I am trying to implement the quadratic sieve and I don't understand how to solve the matrix at the end. I will show you what I did and where I got stuck. So I am trying to factor $149 * 103 = 15347$ I picked B smooth $11$, so my vector primes are $2,3,5,7,11$(I know that I was supposed to actual...

calculus number-theory

guest

Thanks for trying to understand @Vader

Josué Molina

Good appropriate positioning of the sun relative to the horizon, everyone.

guest

Good day/night.

Vader

bye

guest

Later

I'm an artist

21:07

@Jasper I'm not Superman in the last hours. Glad for you.

I'm in a terrible fight with a devilish integral, I'm in the middle of a real battle now.

Vrouvrou

21:22

hi someone here ?

Vader

21:36

How do I multiply

guest

Use the rule of exponents for a power of a quotient first.

(5/2)^x = (5^x)/(2^x)

Vader

so (5^x * 125) / (2^x * 8)?

625^x / 16^8

guest

Nope. You can't just multiply 5 * 125

Change 125 to 5^3

Do the same for 8 to 2^3

Vader

so 5^3 / 8?

I see

ahhhAAHAHAHAhhhaaahHHHa

guest

21:52

Once you have the same base you can add the exponents, right?

Vader

yes

it makes sense now

thanks

It's just been so long since I did some math

guest

np

Idle001

22:13

look at those joky proofs in yahoo!

Forever Mozart

look a

hello?

user174558

23:11

Should Lebesgue integration be taught in an undergrad analysis course? Should differential forms be taught in an undergrad analysis course? What are your opinions?

Thomas Andrews

23:31

I recall differential forms in Real Analysis being extremely mystifying. I hadn't take a several variables class, so I didn't know the various low-dimensional examples, and Rudin's approach felt like abstract nonsense.

Transcript for

$Mathematics$ Mathematics