Mathematics - 2023-06-10 (page 1 of 2)

CowperKettle

00:42

> Eating too much cake is the sin of gluttony. However, eating too much pi is okay, because the sin of pi is always zero.

3

Xander Henderson

00:57

@mick Okay, but that's not the point. The point is that it looks like a no context homework question.

Ultimately, the goal of Math SE is to create a repository of questions and answers which is useful to a large number of people. Textbook questions, or other kinds of questions which might be assigned as homework, are generally not a good fit for that format. Personally, I would like to have a policy which simply says "no homework questions", but the consensus of the community is that this is not an acceptable policy.

Hence we have policies which are designed to "weed out" homework questions. But every question on the site must conform to those policies.

In the example of the problem you have presented, why should anyone believe that the stated result is true (if it were a homework problem, the relevant context would be "[book / article / other reference] claims that this is true." If it isn't a homework problem, then, presumably, you have some other reason to believe it is true.

It is also helpful to know the context in which the question arises. A question about integration is going to have very different answers depending on if the asker is familiar with only the basics of calculus, or knows some real analysis, or is comfortable with measure theory on abstract Banach spaces.

Thomas Finley

01:27

I was recently trying to solve the differential equation eqn by the method of undeterminedcoefficients: $y"+4y=x^2\sin(2x).$ I want to know what should be the assumed (particular) integral (or solution) of this equation. I felt that it should be (Ax^2+Bx+C)sin2x+(A'x^2+B'x+C')cos2x, but then the calculation becomes huge. Am I correct, in my guess ?

I am just enervated to do all the huge calculation. My point is, I just want to verify, that whether the way I am applying to guess the particular integral is justified or not?

Xander Henderson

@ThomasFinley I mean, in order to obtain something which differentiates to a polynomial times a trig function, you are going to have to start with something which looks like a bunch of products of polynomials and trig terms.

Tedious computations often ensue.

Though my first instinct is to apply the double angle formula to get rid of that $sin(2x)$. I don't know that it will make things any easier (it probably won't), but it might.

Thomas Finley

01:43

@XanderHenderson sure, but the guess I did, is educated in the sense, that I came across a suggestion, that "For products of polynomials and trig functions you first write down the guess for just the polynomial and multiply that by the appropriate cosine. Then add on a new gues for the polynomial with different coefficients and multiply that by the appropriate sine". These tactics vary, but the validity of this tip/suggestion seems to be worth verifying considering the scenario I presented.

Thomas Finley

02:20

Finally, got the probable conclusion, that it's an incorrect guess after doing a few pages of calculations. Now, I want to know what will be a correct guess ?

I found out that the correct guess is precisely : Ax^2sin2x+Bx^3cos2x+Cxcos2x. But now, my question changes, that was there any general way or rule to predict the particular integral this way ?

Sourav Ghosh

02:55

Fundamental basis: {1,x,x^2}×{sin2x ,cos 2x}
y_p(x) =(A+Bx+Cx^2) sin2x +(A+Bx+Cx^2) cos2x

PM 2Ring

04:22

@mick That looks similar to the series in a recent article by Simon Plouffe, Pi and the primes (PDF) from plouffe.fr/articles I don't quite "get" that article, but there might be something of interest in it for you.

PM 2Ring

04:37

@TedShifrin :) When a user is removed, it only affects the points from the last 60 days. And in some cases, the staff will "freeze" the points to reduce the collateral damage of the deletion of a very prolific account.

OTOH, a few years ago a very busy member on Physics.SE made a snap decision to delete his account, to force himself to study for an upcoming exam. Quite a few members lost a lot of points that day, several of them lost >100.

PM 2Ring

04:47

@PNDas See the FAQ:

843

Q: What does "user was removed" mean and why did my reputation change because of it?

Message appearing when a user was removed. It seems I lost 1765 reputation points on Stack Overflow yesterday. The reason given in my history is "removed", with the description of User was removed (learn more) I don't understand what that means. What caused this? Return to FAQ index

robjohn

05:04

@PM2Ring he is talking about a question being removed.

I tried to answer their question earlier.

Ted Shifrin

@PM2Ring Oh, I didn’t know that. Over time I’ve lost a few hundred, I guess.

robjohn

when people who have upvoted you a lot get removed, it can hit your reputation a lot.

but, as PM 2Ring points out, that can be mediated by the 60 day limit.

Ted Shifrin

Better to write crummy answers and be unpopular :)

Koro

05:51

I understand now why real part of z^{1/2} is positive.

$z^{1/2}:= exp (1/2 \log z)$, log is the principal branch of log, i.e., $\log: C-\{z\in C: z\ge 0\}\to C$ such that $\log (z)= \ln|z|+i\theta$ where $\theta\in (-\pi, \pi)$.

$\theta/2 \in (-\pi/2,\pi/2)$ so $\cos \theta/2\gt 0$.

one potato two potato

07:21

I still wonder why hecke algebra suddenly appears during the last lecture of Algebra class although that lecture is really a very crash course in representation theory of finite groups.

Balarka Sen

07:32

its atypical

one potato two potato

I guess it's the professor's characteristic. There are a lot of likes and dislikes.

Balarka Sen

I prefer characteristic 0 professors.

one potato two potato

he couldn't finish his goal: linear representation of $GL_2(\Bbb F_q)$. So he uploaded lecture slides about that after the class.

Balarka Sen

haha both my representation theory courses were like this

the instructors get bored when doing these computations. GL2(Fq) was delegated to an exercise for us

one potato two potato

He's a number theorist. He's probably characteristic $p$.

Silly Goose

07:40

does anyone have or know of an exposition of Lie groups which outlines what each piece of structure buys us? e.g. Lie groups are things with group structure and differentiable manifold structure. the diffe m structure buys us...

Balarka Sen

i think its got to do with the subject, to be entirely honest. rep theory seems like a big mess of ideas and computations which is impossible to put together into one coherent theme

i never understood it

and i have taken effectively 3 courses on it

Silly Goose

naively, the diffe m structure allows us to perform calculus on the group structure

one potato two potato

we don't have a representation theory class but probably he would count harmonic analysis class as a representation class.

Silly Goose

also why do people seem to use the word continuous to conjure up the image of differentiability when continuous does not imply differentiable

PNDas

08:29

In Axler's exercise, $T$ is given to be compact and $(e_n)_{n\in\mathbb N}$ is an ONB of $\text{cl}(\text{Range}T)$. And $P_n$ is an orthogonal projection onto span of $e_1,\ldots,e_n$. Then he asked us to prove $||T-P_nT||\to0$.

Here I don't see where we used compactness.

one potato two potato

one useful characterization of compactness of (separable) hilbert space $\mathcal{H}$ is that if $\{e_i\}$ is an orthonormal basis of $\mathcal{H}$ then a subset $S$ is compact if and only if it's closed, bounded and for given $\epsilon>0$, there exists $N$ such that $\sum_{n>N}|\langle s,e_i\rangle|<\epsilon$ for all $s\in S$ (This is nontrivial btw).

I don't like that precompactness definition

PNDas

08:56

@onepotatotwopotato, we in general have $\sum_{k\geq N}|\langle s,e_k\rangle|^2<\varepsilon$. Right?

one potato two potato

yes by bessel's inequality

PNDas

@onepotatotwopotato Do you have any simple example where it doesn't hold.

one potato two potato

nonseparable space? Idk

PNDas

@onepotatotwopotato Oh you are saying one epsilon works for all elements of $S$. But if $s\in H$, we don't generally have $\sum_{i>N} |\langle s,e_i\rangle|<\epsilon$. e.g. In $\ell^2$, take $(\frac{e_i}i)$ .

one potato two potato

yes epsilon is uniformly fixed

PNDas

09:04

I am just verifying if my understanding is correct.

Thanks

math.mit.edu/~rbm/18-102-S14/Chapter3.pdf found this. Your statement is on 77 page

@onepotatotwopotato, can you please give any suggestions to my question chat.stackexchange.com/transcript/message/63768620#63768620?

Nevermind, I actually found the solution in the notes I gave. Lemma 31 says: K is compact iff $u_n\rightharpoonup u$implies $Ku_n\to Ku$.

Jakobian

0

A: Constructing continuous functions with prescribed preimage

Note that a cozero set is a complement of zero set i.e. $Z$ is a zero set if there exists continuous $f:X\to \mathbb{R}$ with $f^{-1}[\{0\}] = Z$. Theorem. If $U\subseteq X$ is a co-zero set and $V\subseteq Y$ is non-empty proper open set, $Y$ is arc-connected, then there is continuous $g:X\to Y$...

Hopeful Whitepiller

09:19

Another day, another close vote war

PNDas

Nah lemma 31 doesn't help. I was writing without thinking.

Okay we have $f_n\rightharpoonup f\implies I_K (f_n)\rightharpoonup I_K(f)\implies \langle I_K (f_n)-I_K (f), e_i\rangle\to0$. To prove $I_K (f_n)\to I_K (f)$: We know $||I_K (f_n)-I_K (f)||^2=\sum |\langle I_K (f_n)-I_K (f), e_i\rangle|^2$. So taking limit we are done. I just need to think about why taking limit is possible.

Hmm got it.

Thomas Finley

09:58

Recently, I was, going through a definition of envelope. I know that the definitions can be written and the exposition of it, might vary, but the following picture, which I am hereby attaching has the definition of an envelope given with the reference to a differential equations of 1st order, and probably 2nd degree, and I am unable to make up /decipher the meaning. Here it is:

First of all, I dont understand the fact that it says, " The curve corresponding to the two consective values of c...". I feel this is erroneous , for c is an arbitrary real constant and there is nothing as, consecutive real numbers.

So, in short, I dont what is meant by consecutive curves, neither do I get the meaning of the term, "ultimate point of intersection."

Finally, they conclude the defn stating "The limiting position of these points....". Well, what's a limiting position ?

Silly Goose

10:20

Here it seems like we are assuming to endow the topological superset $R^{2n^2}$ with a metric (we are using Heine-Borel $\implies$ treating $R^{2n^2}$ as a Euclidean (metric) space. From searching online, it doesn't seem like for a general Lie group you can endow the manifold with a topological metric. So, is this definition correct?

^ in a natural way. i.e., endowing a topological metric is an arbitrary choice

robjohn

11:41

@Koro for a $z\in\mathbb{C}$, what does $z\ge0$ mean?

Balarka Sen

@SillyGoose There is no "canonical" metric on a Lie group in general, no.

robjohn

@TedShifrin In my second to last undergrad quarter, I had grad real analysis, grad algebra, upper division differential geometry, but I took an upper division British folklore and mythology course to keep from burning out.

Jakobian

12:12

@leslietownes about the question I had, turns out the answer is yes, you simply take $x\in U$ and $y\in U^c$ and a path $p$ from $x$ to $y$. Then letting $t_0 = \inf\{t : p(t)\in U^c\}$, we obtain that $p(t_0)\in \partial U$ and $U\cup \{p(t_0)\}$ is path-connected.

@AlessandroCodenotti you might be interested in the above post, I analyzed some cases for when given open $U\subseteq X$ and $V\subseteq Y$ there exists continuous $g:X\to Y$ with $g^{-1}[V] = U$.

Koro

13:07

@robjohn it means that $z$ is non-negative real.

fact: clopen connected subset P of a space X is actually a connected component of X.

for if $K\supset P$ is a connected set $K\subset X$, then $K= P\cup (K-P)\implies K-P=\emptyset\implies K=P$.

Jakobian

13:31

you might be interested in checking what quasi-components are

one potato two potato

pseudo-, quasi-

Jakobian

what's a pseudo-component?

Secret

13:57

Thinking about functional equations lately
The solution of $f'(x)=1+f(x)$ is straightforward if we knew about trigonometric identities, but supposed we knew nothing about them, what will be a first approach to investigate the possible nature of the solution?

Sha Vuklia

I typed out a question in Overleaf, since I wanted to include commutative diagrams. Here it is:

Sourav Ghosh

Intersection of a sequence of dense $G_{\delta}$ sets is a dense $G_{\delta}$

sunny

Basic question incoming. In the definition of a limit of a two-variable function, why can we write $0<|x-x_0|<\delta$ and $0<|y-y_0|<\delta$ instead of $0<|\sqrt{(x-x_0)^2+(y-y_0)^2}|<\delta$, as suggested here? How are these equivalent?

The reason for the question is that in my lecture notes, for a uniformly continuous function, they write $0<|x-x_0|+|y-y_0|<\delta$. As suggested here, continuity depends on the distance function. This confuses me since I don't know which distance function makes a given function continuous or not, i.e. how did the authors of my lecture notes figure out that the distance function that reads $0<|x-x_0|+|y-y_0|<\delta$ works?

Sourav Ghosh

Those two metrics are topologically equivalent.

sunny

@SouravGhosh thanks, you mean $|\sqrt{(x-x_0)^2+(y-y_0)^2}|$ and $|x-x_0|+|y-y_0|$?

anak

14:08

Yes.

You can show that the identity function is bicontinuous between the metric spaces with those corresponding metrics ("l^2" and "l^1").

sunny

@anak Ok. How is then $|y-y_0|<\delta \wedge |x-x_0|<\delta$ equivalent to $|\sqrt{(x-x_0)^2+(y-y_0)^2}|$? Which metric is $|y-y_0|<\delta \wedge |x-x_0|<\delta$?

anak

That's just controlling the size of a sum by the size of its summands.

sunny

Ok.

Koro

@sunny max metric.

sunny

Ok.

anak

14:13

I say "just", but really I am unjustified in saying this. Think about the proof that the limit of a sum is the sum of the limits.

(assuming you have seen that one).

Koro

$d((x,y), (x',y'))= \max( |x-x'|, |y-y'|)$

hi anak!

anak

I am guessing from the topic of your question that you likely have not seen equivalence of norms.

@Koro! Hi! How are you?

sunny

No :), but helpful anyway

Koro

I'm good, thanks. It's summer holidays time :). How are you?

anak

@sunny you can take a look at this if you are curious: math.mit.edu/~stevenj/18.335/norm-equivalence.pdf

@Koro I am better this week than last week, so let's hope next week is the same.

Still studying for quals?

Koro

14:15

my 1st year of Masters is complete.

1 more year left :).

anak

Congrats!

Koro

thnx

anak

Are you doing a masters thesis?

Koro

no, I have only course work.

anak

Are you hoping to go on to a Ph.D. after, or are you entering reality after? :P

Koro

14:18

I don't want to do Ph.D now. I want to return to industry after my masters.

anak

Any particular industry?

Koro

although I planned to Ph.D when I came to this college for masters but I realized that Ph.D is not for me.

@anak related to my engineering stream

anak

Ph.D. is not for everyone, so that makes sense. There are different commitments for different people.

Engineering is cool though---any particular brand you are heading towards? Like computer/software,etc.

Koro

I'm a civil engineer, don't wanna go to an IT company.

:)

oneofvalts

In a unitary and commutative ring, I'm trying to show that for all $r\in R$, $(-1)r = -r$. Is the following valid? We have $(-1)r = (-1 + 1 + (-1))r = (-1)r + r + (-1)r$, therefore $(-1)r + r = 0$, thus $(-1)r = -r$.

Balarka Sen

14:28

Hi @anak

You're at ISI Kol now, aren't you @Koro

Koro

yes

Balarka Sen

I saw some of your opinions about teaching there, found it interesting

robjohn

@Koro Since $\ge$ is not defined on $\mathbb{C}$ maybe $\mathbb{C}-\{z\in\mathbb{R}:z\ge0\}$?

Koro

I have stopped talking about them and have now moved on :).

robjohn

sorry

Balarka Sen

14:32

I personally know some very good mathematicians there. But I have also heard many complaints about overall course quality from some of my friends who studied there.

It's a bit strange to hear, but not entirely unbelievable.

For example, I can buy that they're very unforgiving with the course material and there's a couple courses which can be rather uninspired, both of which I have heard from multiple accounts by now.

Koro

@robjohn Conway writes C- {z: z$\ge 0$} (proposition 2.19 in their complex analysis text) so I interpreted that as- what do we mean by $z\in C$ such $z>0$? it should mean that $z$ is real else > or < don't make sense.

Balarka Sen

Yeah I think Conway's convention is, whenever he writes "z >= 0" he means "z is real and z >= 0"

It's an old book with funny notation choices

anak

Hi @BalarkaSen!

Koro

ohh :). I found that in complex analysis section in library. Among others, there was one by Gamelin and Greene- it had a whole chapter/section on Mandelbrot too.

anak

Have you had the chance to teach any courses yet?

Balarka Sen

14:37

@anak No, thankfully.

anak

Heh. Not a fan of teaching, or just happy to focus on research?

Balarka Sen

I was a TA for a bit, but that just means I graded homeworks.

I like to teach, but teaching a course seems to require a lot of work and planning. Seems hard work if you do it responsibly.

Koro

that's right.

anak

It is indeed time consuming.

Koro

if one prepares 'what to say? and 'the order of delivery of ideas' before coming to class for teaching, then the lecture remains more successful.

Balarka Sen

14:40

I prefer cornering unsuspecting grad students at random places and times and starting an elaborate spiel instead

Teaching but less responsibility, more fun. More wiggle room to be incorrect.

@Koro I have been taught courses by people who used to do this and still mess it up :P

Koro

not like: 'writing the proof of Tychonoff theorem in class and then facing objection by students -"Sir, you have not yet 'taught' product topology." and then 'teaching' product topology in like 10 mins so that the Tychonoff theorem's proof can be written on board.

Balarka Sen

Lol, Tychonoff before product topology is horrifying.

Koro

I'll say no more on this in view of the following:

11 mins ago, by Koro

I have stopped talking about them and have now moved on :).

Balarka Sen

I think Samik is good at teaching, or not? I have heard he sets nuts exams and is unnecessarily harsh with grades.

Were you taught by Kingshook? I think he left or is leaving ISIK

Koro

@BalarkaSen I think that if one solves the problems that he keeps assigning from time to time, then one shouldn't have getting a good score in his exams.

@BalarkaSen I don't know him.

Balarka Sen

14:48

oh ok cool

What about Swagato? He seems like a good teacher

Secret

Ok figured out
$f'(x) = 1+ f(x)^2$
The left hand side gives how much the right hand side increments so we can roughly have something like
$1+f(x)^2, 1+f(x+1+f(x)^2)^2, 1+f(x+1+f(x)^2+1+f(x+1+f(x)^2)^2)^2, ...$
Then working through the cases $f(x) > 0$, $f(x) < 0$ and $f(x) = 0$, the graph traced out by the increments will rise and fall without limit, while locally linear at $f(x)=0$ thus producing the familiar tan x curve

Koro

@BalarkaSen I got not so good marks in midsem in algebraic topology so he asked me (kindly): you attend every class what was the problem? I said- I was trying to study from Rotmann because I was having lot of difficulty with Hatcher's. So he said-you have other subjects too, studying a book is not possible within the time that we have.

So he opened Hatcher's ch. 2 in front of me and said - see how lengthy this chapter is!

so if you just follow my notes and solve the practice problems I give then there is no problem. Think about it: if you know what I teach in class, then will you have any difficulty in understanding a portion of Hatcher's that I didn't cover in class?

Balarka Sen

There's probably some merit to what he's saying. Should take less effort to make notes from his class.

Koro

yeah

Balarka Sen

Also Rotman has a completely different approach.

Koro

14:55

many people don't like him in my class. But I don't have any problem with him.

he seems reasonable to me.

Balarka Sen

Nice.

Koro

I have already expressed my views on the 'teaching' part here and that stands for every teacher here (at least those who have 'taught' my class).

Balarka Sen

There's not enough inspired teachers out there, for sure.

Koro

lot of people failed in algebraic topology. Though, I passed but not with very high score. He asked again- what was the problem this time? I said- you said that there will be stuff from pre-midsem as well so I spent too much time on that but the end sem was mostly based on post midsem syllabus. He said-I understand. We didn't have enough time to discuss many things for last couple of days of our classes.

(he asks me because I attend almost all his classes. 😅)

Balarka Sen

It's good that he's friendly. That has generally been my experience with him.

(not in terms of courses he took, but other interactions)

Sourav Ghosh

15:03

1) An example of a non Borel set $B$ such that $B\cap K$ is Borel for all compact subsets $K$.

Koro

Balarka: I still don't understand cellular homology though. I even asked a question here math.stackexchange.com/questions/4694526/…

The question is as you would expect from a beginner.

But it was not well received.

Balarka Sen

Question seems great. I can chime in if you have further questions.

Koro

I think 'understanding topology of CW complexes' will help me with that. There is a book that does CW complexes in a DIY way via exercises.

It'll take me some time though to reach that chapter in the book.

Balarka Sen

Personally I found that much of my confusions regarding CW complexes vanished after working through the entirety of Hatcher Ch 0

Sourav Ghosh

[Contd.] $B_n=B\cap [-n, n]$ Borel for all $n\in\Bbb{N}$ implies $B$ is Borel.

Koro

15:07

I think confusion to me arose from looking at 'what is an n-cell?' from different sources.

Balarka Sen

An $n$-cell is just a closed $n$-disk, $D^n$.

Sometimes "an $n$-cell of a CW-complex $X$" indicates an open $n$-cell, which is the interior of $D^n$.

But this is a confusion that rarely matters, and many people are (including myself) flimsy with this.

Koro

sometimes homeomorphic image of D^n too, right?

Thomas Finley

0

Q: Absurdity while finding the singular solution of a Differential Equation

Find the singular solution of the differential equation $$4xp^2=(3x-1)^2,$$ where $p=\frac{dy}{dx}.$ As we know the singular solution, of a first order differential equation, is represented by the envelope of the family of curves, represented by the general solution of the differential equation. ...

calculus ordinary-differential-equations

Jakobian

Borel sets with respect to some topological space?

Balarka Sen

@Koro Yes, we generally don't distinguish homeomorphic spaces in topology. If $X$ is homeomorphic to $Y$, we call it "the same space"

Koro

15:09

@BalarkaSen I understand this having solved some of the exercise problems in ch 0 through ch 2.

Thomas Finley

Guys! Need a help with this....

Sourav Ghosh

@Jakobian In ($\Bbb{R}, \tau_{std}) such non Borel set doesn't exists.

oneofvalts

chat.stackexchange.com/transcript/message/63771860#63771860 I can't find this online or on proofwiki.

Jakobian

As you noticed, any $\sigma$-compact space has no such set

Koro

I did feel that AT should be studied differently than the way I would study real analysis for example.

Sourav Ghosh

15:10

@Jakobian Any topological space.

Jakobian

First thing that comes to mind, maybe $X = \omega_1$ serves as a counter-example

Koro

because otherwise would mean getting stuck at something like 'definition of n-cell' for hours, sometimes days (for me).

Balarka Sen

@Koro Very different, yes.

Koro

like a cylinder is a sphere.

(but don't worry too much about maps)

Balarka Sen

Cylinder is not a sphere, do you mean sphere with the north and south pole removed?

Koro

15:12

I mean cylinder with both its ends fit.

Balarka Sen

Ah

Yes

Koro

see

it's confusing

😅

Balarka Sen

People generally don't call that a cylinder.

A cylinder is $S^1 \times I$

Koro

I think I now have had enough 'getting used to with the subject'.

Sourav Ghosh

@Jakobian Thank you. Lower limit topology is not sigma-compact.

Compact sets are countable

Koro

15:14

most likely, I'll take topology 3 next sem.

(this is ch. 3 and 4 of Hatcher's)

cohomology onwards

Balarka Sen

That should be fun. Essentially you need to know Ch 0 and Ch 2 really well for Ch 3.

Ch 0 is paramount

Many flimsy arguments on the face of it can be made completely rigorous using Ch 0 materials

Sourav Ghosh

@Jakobian Done!

Koro

I didn't like Hatcher at first. But Pierre Albin's videos help a lot.

Jakobian

Ah, yeah. Just take any non-Borel measurable set (like Vitali set) and $B\cap K$ is countable so Borel for all compact $K$

Koro

and with the exercises of notorious chapter 0, ch 0 felt complete.

Sourav Ghosh

15:16

Yes. B= Bernstein set

Koro

(complete in the sense of understanding it.)

Sourav Ghosh

@Jakobian Lower limit topology isn't locally compact.

Jakobian

because the Borel sets Sorgenfrey line and standard topology on R agree

Sourav Ghosh

@Jakobian Yes.

Does there exists any locally compact space with such a non Borel set?

Balarka Sen

@Koro Currently you have summer vacations, I think? When does next semester start usually?

Koro

15:20

in august

Balarka Sen

Cool. Do you have plans regarding what to read during the vacations, or not really going to do that much math in the holidays?

Koro

some people are doing reading projects with some professors here. I decided not to do that because I thought I should revise sem 1 and sem 2 stuff.

so I have been revising topology, complex/real analysis etc.

I also do numerical analysis.

Balarka Sen

Cool. I saw you were asking some questions about point-set topology. Why point-set, out of everything else?

Jakobian

@SouravGhosh in $\omega_1$, if $K\subseteq\omega_1$ is compact, then $K\subseteq [0, \alpha]$ for $\alpha < \omega_1$. Thus for any $B\subseteq \omega_1$, if $K$ is compact then $B\cap K$ countable hence is Borel

so it's enough to show there are non-Borel subsets of $\omega_1$

Koro

because you see I felt uncomfortable with: topologist sine curve is NOT path connected. I mean I'm able to follow its proof but don't really understand why it was done etc. And I was not comfortable with 'local connectedness, local p.c., components, path components, proof of Tychonoff, and in thinking examples on my own'.

So I'm trying to build all that.

Balarka Sen

15:27

Fair enough. Are you doing this from a book, or?

Jakobian

I mean, it's "visible" that topologist's sine curve is not path-connected, all left is to prove it

and it doesn't really matter that it's a sine curve, since any zig-zag would be equally good

Koro

yes, from some books (that I studied from and some that are in my to be studied list).

Jakobian

I like that example because it shows us that connectedness is not really what we would think it is

Koro

@Jakobian yeah but visibility alone isn't enough in the world these days.

Balarka Sen

I have a suggestion.

Jakobian

15:30

@Koro you were asking for why it was done, I provided insight. All left is tedious details

Koro

@Jakobian left as an exercise.

Balarka Sen

Ditch point-set topology. Hatcher has a 50 page notes (pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf) on point-set topology; for most students of mathematics, that is all that is ever required. Just read that much.

Jakobian

you can really come up with the proof yourself once you observe it shouldn't be path connected

Balarka Sen

Instead, use Chapter 1 of this book maths.ed.ac.uk/~v1ranick/papers/cartert.pdf as bed-time read.

Jakobian

@BalarkaSen yeah, that's true

Koro

15:31

@BalarkaSen thanks. :)

one of my classmates also said that there is no more research in point set topology.

(like nobody works on it these days)

Jakobian

there is research, just not a lot of it, since it's very done field

Balarka Sen

Not true, Jakobian will protest immediately.

Soumik Mukherjee

Is there any youtube playlist to follow alongside with Raymond Wells' Differential analysis on complex manifold book?

Koro

@BalarkaSen I expected that too from Jakobian but even he seems to agree with this.

Jakobian

I'm not delusional, what I'm doing, I'm doing for fun

Koro

15:33

@BalarkaSen noted, thanks.

Jakobian

there are open problems and I'll try solving them eventually, but right now I'm just learning

2

Sourav Ghosh

One way to prove the topologist's sine curve is not path-connected is to use the Darboux property of the coordinate functions.

Koro

@SouravGhosh too difficult (to me)

I have one direct way.

I'll share:

Sourav Ghosh

Very easy :)

porridgemathematics

honestly i would say for most students of mathematics, lees intro to topological manifolds is better than hatcher

i like how he gives a pretty complete proof of classification of surfaces

something that is missing from hatcher as far as i remember

Balarka Sen

15:39

Hatcher isn't a manifolds text for sure. Almost nothing is done upto homeomorphism.

porridgemathematics

yeah, of course its a super well known AT book

Koro

@Jakobian I think that's fair enough. We want to learn, that's what matters. Who cares if there is no more research in it!

Soumik Mukherjee

Currently following Prof. Borcherds' lecture series for Hartshorne, but I feel to have some video lecture guidance for Wells' book too.

porridgemathematics

but for a graduate sequence if someone has to take a course from one or the other I would recommend lee , without any more info on their interests

Balarka Sen

@Koro Some cool point-set topology problems for you: Suppose $X, Y$ are compact metric spaces.
(0) If $X$ is homeomorphic to $Y$, then $X \times \Bbb R$ is homeomorphic to $Y \times \Bbb R$. The converse is not true, find an example.
(1) If $X \times S^1$ is homeomorphic to $Y \times S^1$, prove that $X \times \Bbb R$ is homeomorphic to $Y \times \Bbb R$. (Easy)
(2) Prove that the converse of (1) is true. (Hard; @Jakobian you may enjoy this.)

2

Koro

15:41

@SouravGhosh I meant to say the following: we prove that any path starting at $0$ must be a constant path.

(in {(0,0)}\cup {(x,\sin 1/x): 0<x<=1})

this uses only the fact that [0,1] is connected.

@BalarkaSen I'll try these. Thanks.

$\ddot\smile$

Sourav Ghosh

I will learn mathoverflow.net/q/427536/483536 in future.

Jakobian

@SouravGhosh math.stackexchange.com/q/268794/476484 according to this post, we have a description of Borel measurable subsets of $\omega_1$, from which it follows that there exist non-Borel subset of $\omega_1$. Since $\omega_1$ is locally compact, we have what you wanted.

Sourav Ghosh

Thanks.I will verify all the details:)

So local Compactness isn't sufficient.

Soumik Mukherjee

Can anyone help in this matter?chat.stackexchange.com/transcript/message/63772208#63772208

Jakobian

Solovay theorem I read proof of before, I guess the characterization of Borel subsets of $\omega_1$ is by transfinite induction

Koro

15:49

for (0): my initial guess is if $f:X\to Y$ is a homeo., then $f\times id_R$ should be a homeom. from $X\times \mathbb R$ to $Y\times \mathbb R$.

Let $f\times id_R=:g$. $g$ is continuous because its projections are continuous.

Balarka Sen

Yup

Koro

$g$ is invertible and $g^{-1}= f^{-1}\times id_R$ so we are done if we prove cont. of $g^{-1}$.

this notation means that: $g^{-1}(y,r)= (f^{-1}(y),r)$.

but this is continuous because the projections are continuous.

I didn't use compactness of X and Y anywhere.

Balarka Sen

And that's because you don't need them.

Not for this part of the problem.

Koro

I think that's because of multiplication by R.

I recall one theorem: $X\cong Y\implies X\times Z\cong Y\times Z$

(if Z is locally compact Hausdorff)

I'm not sure if that was the statement.

Jakobian

No, this holds for any Z

Balarka Sen

15:57

You don't need any hypothesis on $Z$ for that to be true

Koro

ohh, that was for quotient maps then.

Balarka Sen

I actually think (1) is wrong. It is true if $X, Y$ are both simply connected.

But I don't know a counterexample off the top of my head. Take that as exercise (difficulty: not sure)

Thomas Finley

@SouravGhosh Do you mind helping me with finding the singular solution of the differential equation , I posted here? I don't understand the absurdity

Xander Henderson

@ThomasFinley Honestly, I don't think that this is a good guess. The fact that many of the coefficients end up being zero is not something which can be known a priori. As @SouravGhosh points out, the space of possible solutions is generated by a larger collection of terms.

@Koro Mechanical engineers build weapons systems. Electrical engineers build weapons guidance systems. Civil engineers build targets. :P

3

Koro

@XanderHenderson touché :)

Thomas Finley

16:09

@XanderHenderson the irony is:

15 hours ago, by Thomas Finley

I was recently trying to solve the differential equation eqn by the method of undeterminedcoefficients: $y"+4y=x^2\sin(2x).$ I want to know what should be the assumed (particular) integral (or solution) of this equation. I felt that it should be (Ax^2+Bx+C)sin2x+(A'x^2+B'x+C')cos2x, but then the calculation becomes huge. Am I correct, in my guess ?

Xander Henderson

@Koro Are we playing old school epee rules, or do I need to win another 14 touches?

Thomas Finley

That was my prior guess, but that didn't work out well...

Xander Henderson

I don't see the irony. Computation is often tedious, and you will make mistakes.

But it seems to me that you are thinking about this far too procedurally, and that you aren't understanding the deeper theoretical idea that problems like this are supposed to help get across, namely that the space of possible solutions is going to consist of linear combinations of the "kinds" of terms which show up in the DE.

The idea of undetermined coefficients is that you choose an arbitrary element from this space of all possible solutions (i.e. you start with a linear combination of basis elements), then work out the values of the coefficients by comparing the arbitrary thing to what you want.

You cannot a priori assume (in general) that you can drop certain terms.

Thomas Finley

@XanderHenderson and that is where the word "luck" comes in. Yes, calculations are tedious, but I have no issues with it, but when asked to do in a limited time, that makes thing horrible if not bad. The thing is, I am trying to look up for some strategies that might reduce the work. So far as the idea is concerned, I mean yeah, that's how I made my first guess.

Suddenly when calculations went down the road, things got messy and a fruitful desparation let me drop a few terms and the answer came out magically smooth.

Xander Henderson

@ThomasFinley The problem is that you can run into more general situations where there may be many solutions. Finding one solution does not guarantee that you have found all of the solutions. Getting into the habit of dropping terms early on is going to cause problems for you in the long run.

And, again, computations often get messy. That is the nature of the beast. You should not be afraid of tedious computations (particularly when studying DEs, where things can easily get quite out of control very quickly).

It might help to review some linear algebra (particularly Gauss-Jordan elimination). That should make some of the computation easier.

Thomas Finley

16:20

@XanderHenderson ha ha...not afraid, just looking the web to make life less tiring...

@XanderHenderson I was looking for something like this

Thanks!

@XanderHenderson If you do not mind, can I ask you to share some insights to solve for singular solutions of a 1st order differential equation ?

I meant problems like this:

0

Q: Absurdity while finding the singular solution of a Differential Equation

Find the singular solution of the differential equation $$4xp^2=(3x-1)^2,$$ where $p=\frac{dy}{dx}.$ As we know the singular solution, of a first order differential equation, is represented by the envelope of the family of curves, represented by the general solution of the differential equation. ...

calculus ordinary-differential-equations

Xander Henderson

@ThomasFinley I have no insights. I took a year of PDE during my first year of grad school, and have not thought very much about differential equations in the 10 years since then.

Thomas Finley

@XanderHenderson Oh! Ok, thanks!

Best wishes !

Thomas Finley

16:46

I need some help regarding the method to partially differentiate the equation, $4xp^2=(3x-1)^2$ wrt $p$ where $p=\frac{dy}{dx}$

I felt the result should be, $8xp$,

But somehow I have a premonition, that it's not so

Can it be really differentiated partially ?

Till now, I am only aware of the algorith to partially differentiate i.e" treat all the terms except the variable of differentiation as constants". But is there anything more to it ?

That's what I wanna know precisely

Sourav Ghosh

Partial derivative of f(x, y) w.r.t. $x$ means ordinary derivative of $g(x)=f(x, c), where $c$ is constant.

Koro

Charlie 777 is such an amazing movie.

One of the best movies I have ever watched.

So is Hachiko and Togo as well.

Sourav Ghosh

@ThomasFinley Yes. It's a polynomial in $p$.

Thomas Finley

@SouravGhosh Any idea where this is going wrong?

0

Q: Absurdity while finding the singular solution of a Differential Equation

Find the singular solution of the differential equation $$4xp^2=(3x-1)^2,$$ where $p=\frac{dy}{dx}.$ As we know the singular solution, of a first order differential equation, is represented by the envelope of the family of curves, represented by the general solution of the differential equation. ...

calculus ordinary-differential-equations

The partial derivative I am talking about appears here

Koro

you seem familiar, like someone that I talked to here like months ago.

your writing style matches too much with them...

Thomas Finley

16:54

@SouravGhosh All in all, what would be the correct partial differential I mentioned, will it be 8xp ?

@Koro You know what, I am a professor at ur univ hahahahahahahahahha

Koro

similarly, I suspect that one of my favorite users krm is a new user who joined the site around the same time as krm was last seen here.

Thomas Finley

@Koro and I know what youre doin now! I can see you! Who knows, I might be lurking behind you?

Look back

Sourav Ghosh

17:15

@ThomasFinley I don't know. $4p^2x=(3x-1) ^2$ . Then $8px=0\implies 8 \times \pm (3x-1)\times \sqrt{x} =0 \implies x(3x-1) ^2=0$

Thomas Finley

17:30

@SouravGhosh You have just written a paradox! What a strange absurdity!

I feel both of us, are correct and wrong at the same time!

Can I add your comment in my post @SouravGhosh ?

It will be really interesting to add on a new absurdity.

Koro

18:57

I take $X=[0,1], Y= \{e^x: x\in \mathbb R\}$. Then $X\times \mathbb R\cong Y\times \mathbb R$ but $X$ is not homeo. to $Y$ because Y is not compact.

$Y\times R$ is just the region between $\ln x$ and $y-$ axis.

But you want Y also to be compact.

@BalarkaSen

Ted Shifrin

19:13

I don’t see how a manifold with boundary is homeo to one with none.

Yai0Phah

20:46

It is impossible. The local homology around a boundary point seems to be different from that around an interior point.

Ted Shifrin

21:22

Invariance of domain. Same thing.

s.harp

I recall from a lecture that there is a classical "hard" (but solved) problem related to homeomorphisms of (open sets of) R^n. Do you know what I mean?
Is it invariant domain? (injective continuous images of an open set between same dimensional R^n are open)

Ted Shifrin

21:50

Right. Proving that dimension of a topological manifold is well-defined (trivial in the smooth case).

Astyx

Let $B = CAC^{-1}$ for integer $2\times2$ matrices $A$, $B$ and $C$, with $C \in GL_2(Z)$. Is there an algorithm for recover $C$ from $A$ and $B$?

leslie townes

off the top of my head, i note there are problems with indeterminacy e.g. if A = 0 or A = I where any C will satisfy the equation with B = A, and maybe generally this is a problem with "repeated" eigenvalues.

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