Mathematics - 2023-06-17 (page 1 of 2)

Ted Shifrin

03:25

@leslie How’s munchkin doing?

leslie townes

napped most of the day, still feverish, but high spirits.

Ted Shifrin

Is she still attacking Olivia?

leslie townes

they did mix it up a little earlier, yeah. it stopped when olivia hit her in the head.

Ted Shifrin

LOL

They’re both more or less as usual.

Sourav Ghosh

@Jakobian Nice :)

Mason

03:53

I haven't written any good math in awhile and that is sad for me.

I have this feeling that I've atrophied and an art that I enjoyed isn't coming as easily as it once did. Writer's block. The thing to do is to exercise. And it just seems hard. Or I have been telling myself that it will be hard but I need to be fun. But it can be both. In fact an art will be like that. It will be hard to improve and one stagnates but one can come back to it. Revisit first principles. Revisit math education. Revisit old books. Read a new book. Get a mathy magazine.

copper.hat

04:13

find a question that you like and answer it.

user4539917

04:41

Or, you could choose the much harder alternative and try to think of the right question to ask.

Mason

05:04

I have this feeling that I've atrophied at an art/exercise that I previously enjoyed and it isn't coming as easily as it once did. *

user4539917

Then, as you mentioned, return to the easy stuff.

Mason

I'm was correcting my jarbled language above. I appreciate your response. I just wrote a question that wasn't very impressive a week or two ago. I think I should be reading more not writing more atm.

Shaun

05:44

Hi :) I just thought I'd advertise a bounty:

1

Q: What is the connection between algebraic groups and topoi?

I have a longstanding interest in topos theory. (See this MSE search of my questions about topos theory.) I am studying for a postgraduate research degree in linear algebraic group theory. Naturally, I wonder what connections there are between the two. A quick Google search produces this page, in...

冥王 Hades

07:12

@Mason If you find it fun it shouldn’t hard to get back into it.

Only reason I do math is because I like solving annoyingly difficult problems, like a puzzle. It’s fun

Silly Goose

07:25

how does one go about showing that a set of invariants is a complete set of invariants?

Thomas Finley

08:01

0

Q: Challenge: Can you point out the Mistake in Here?

Find the $p$-discriminant of the differential equation $F(x,y,p)\equiv 4xp^2-(3x-1)^2=0.$(Note: $p=\frac{dy}{dx}$) We proceed to calculate the $p$-discriminant of $F(x,y,p)=0,$ where $F$ is the function representating the differential equation in the question. So, $F(x,y,p)=4xp^2-(3x-1)^2=0\impli...

calculus

Still unanswered(!)

PNDas

08:14

Let $2<p<q$ and $u_n\to0$ in $L^q(\Bbb R^n)$. Let $\int_{\Bbb R^n}f(u_n)\,dt=1$ where $f(t)=t^p-t^2$. Then Prove that $u_n\in L^p(\Bbb R^n)$

I know that $||u_n||_{L^p}\leq ||u_n||_{L^q}^{\theta}||u_n||_{L^2}^{1-\theta}$

But I can't figure out what to conclude from $\int u_n^p-u_n^2\,dt=1$.

I forgot to mention that $\theta$ is such that $\frac1p=\frac{\theta}q+\frac{1-\theta}2$.

Shaun

08:34

The following was downvoted, so please would someone give some feedback on it?

-1

A: If $A,B$ are square matrices of the same order, given, $A^4=I,A^2=B^2,AB=BA^3$, where $I$ is the identity matrix, show $A^3B=BA$ and $A^2B=BA^2$.

According to GAP, the group given by $$P:=\langle A,B\mid A^4=I, A^2=B^2, AB=BA^3\rangle$$ is the quaternion group $Q_8$: gap> F:=FreeGroup(2); <free group on the generators [ f1, f2 ]> gap> rels:=[ (F.1)^4, (F.1)^2*(F.2)^(-2), (F.1)*(F.2)*(F.1)^(-3)*(F.2)^(-1) ]; [ f1^4, f1^2*f2^-2, f1*f2*f1^...

leslie townes

08:47

it's anybody's guess. if i had to guess, i'd guess that someone feels that the OP's framing of the question suggests that they might not be conversant with GAP syntax (or know what GAP is, or be familiar with using software to explore problems like these), or perhaps more generally that anybody asking or looking up a question like that on MSE might not have that knowledge, so that the answer is not very helpful.

i don't think that's a good reason to downvote, but people don't need a good reason to downvote.

Shaun

Thank you, @leslietownes. That would make sense.

Silly Goose

It also seems that your answer emphasizes delegating proving OP's property of interest to a computer. However, if it instead emphasized identifying OP's starting data as a presentation of the Quaternions and then explained properties of the quaternions/said to look into properties of the quaternions maybe it would have been more well received by whoever downvoted.

leslie townes

i personally think that there should be room for answers like that. a lot of people go pretty far in learning about software (even specifically math software!) while never realizing that it can shed light on stuff like this. like they cabin their mind into "numerical stuff" where maybe you use computers and "abstract stuff" where maybe you don't. it's a kind of ignorance that will only get more embarrassing in the future.

at the same time, i basically have the view i described up above (minus the downvoting being appropriate), i.e., i think odds of the OP understanding your answer are basically 0

user 85795

@leslietownes do you happen to know a short proof of GCF(a,b) * LCM(a,b) = a*b

Silly Goose

if they knew group theory perhaps they'd understand :D since to me the computational results you cite are just like citing theorems. though i think they were looking for an answer that worked through getting to the result. but the computational result that the group is a presentation of the quaternions i feel would be helpful if they knew what that meant

i would like to get to know some group theory specific software...

leslie townes

08:55

i had a paper rejected once because a corollary stated that something happened for a particular number, and for concreteness' sake i computed it to a few digits in a remark. the referee said something like, i shouldn't include 'experimental' corollaries and should stick to what i could "actually prove." this was astonishingly ignorant (in the given setting, the existence of the number followed from the IVT, but the argument gave no idea where it was on the number line)

if my reviewer was the downvoter, they could have thought of anything involving software as inherently unreliable or experimental or not 'real.' there are still these people out there.

maybe the downvoter was some kind of fanatic who either didn't know that GAP is free software, or who has some very specific quibble with the GNU GPL and doesn't think people should be promoting sacrilegious software licenses

Sourav Ghosh

09:29

I was not selected for the NBHM Scholarship. I thought my interview was good enough to qualify for the final. But again, I didn't manage to qualify. I am crying because I tried really hard.Maybe this is the end of my career.

user 85795

09:50

Let it all out.

Is Physics or Engineering an option?

user 85795

10:36

I gave you immediate treatment: "let it all out," it is okay to cry.

After you calm down, think of other options, pal.

You tried your best and nobody can do any better than their best effort.

冥王 Hades

@SouravGhosh Suicide/self-harm is not a solution to any problem. Please control yourself. I’ve lost a great friend because of it who himself was brilliant when it came to mathematics

He did it because he also couldn’t get a scholarship. It was just bad luck, that’s all. It didn’t mean he wasn’t good enough, no. He was, like I said, extremely capable

user 85795

Don't forget about all the college admission scandals that have been in the news. Rich people buying their kids into the top colleges.

Koro

@user85795 the situation is lot different here.

user 85795

10:52

That is what they want you to think.

Thomas Finley

It's a long time now, I learned elementary calculus in one variable and some basic real analysis(excluding Riemann integration). I always had a question, but never got the time to think out. The thing is, calculus teachers at elementary level always told me NOT to say "$\frac{dy}{dx}$ as "dy over dx" because, $\frac{dy}{dx}$ is not a fraction, it's an operation, loosely speaking just like addition, subtraction blah,blah, blah"

but while integration, I always used the notion of "differentials". To be honest, my notion of "differentials" is only limited to, the fact, that say, $dx$ is nothing but an infinitesimally small quantitiy. I don't know, any operations of differentials, like d(xy), d(y/x), etc. Never, have I got to know any details about the so called term "differentials" in any real analysis book. I found them admittedly in differential equations textbooks.

The problem they never introduced it anywhere. They just started with, "Now the differential dx..." Does anyone know where can I get to know about the topic?

Not casually, but rigorously!

Koro

But sometimes in an exam, it's luck that gets one through, I guess.

user 85795

Luck never made a man wise.

Koro

I know some people who don't know for example what a complete metric space is, but they have in some exams involving analysis scored 'high' marks.

you can guess how.

Thomas Finley

@Koro how?

Koro

10:56

so @SouravGhosh it may be tough, I understand. But suicide is not an option.

write the exam again and try to crack it next time.

user 85795

Think about the people you are leaving behind.

Koro

@ThomasFinley cheating or by shooting arrows in the dark.

in one of our exams, some m no. of people wrote the same answer but not all of them got the same marks.

the one out of these m who got the highest probably just got lucky.

there is no other rational explanation for it.

so sometimes luck does play some role in an exam.

Peter

In Germany , millions of people (including mine) have a dozen reasons to commite suizide , but this should be no reason to actually do it !

Thomas Finley

@Peter how is it in Germany? I am askin this coz many say, it's the "Land Of Knowledge"...

冥王 Hades

Here in Japan suicide due to work and academics are among the highest in the world

Thomas Finley

11:03

But again we mustn't go by people's words, I admit

@冥王Hades Umm, u sure? I thought it was South Korea, idk...

冥王 Hades

South Korea also ranks very high. But Japan has a specific word in its language for death by overwork

Thomas Finley

@冥王Hades Oh, I know this isn't funny but c'mon it's funny....I am sorry

Koro

@user85795 and no one really cared about wise. they want marks/good percentage/high cpi.

user 85795

Then they are missing the the point of an education.

Thomas Finley

What d ya think of an 8.5 SGPA ? Is it good?

(I have a friend who got it, and he's dissapointed because he didn't get a 9)

So, what are ur opinions?

Does the std vary from country to country?

user 85795

11:07

Learning and understanding is not about grades.

Peter

Germany : medical supplies awful , nursing care awful , inflation very high , everything extremely expensive , gendering nonsense and orthography reform , just to mention the main issues.

Thomas Finley

@Peter I once considered moving there for education. What d ya suggest?

People say, it's a nice place for study

I recently read, it's the largest economy in all of EU

Koro

Germany is amazing. I'll visit Germany at least once for sure.

Peter

Do not go to Germany , not even to study ! It is a totally broken country without any humanity left.

Koro

(I like the language too but don't know it at the moment.)

Thomas Finley

11:14

@Peter Germany is a popular country.

But Germany significantly made great contributions in Medicine, Mathematics, etc

But maybe u have a diffe exp altogether...

But nvm, I must be leavin now...

冥王 Hades

@Peter And some people actually think Germany is a good country to stay/work at 💀

You couldn’t pay me to go for a vacation in Germany

Sure that sounds rich coming from someone in Japan but Japan is far more orderly and nicer. I like that

Thomas Finley

@冥王Hades me included

Oh, just to mention, need a help a bit with this:

43 mins ago, by Thomas Finley

It's a long time now, I learned elementary calculus in one variable and some basic real analysis(excluding Riemann integration). I always had a question, but never got the time to think out. The thing is, calculus teachers at elementary level always told me NOT to say "$\frac{dy}{dx}$ as "dy over dx" because, $\frac{dy}{dx}$ is not a fraction, it's an operation, loosely speaking just like addition, subtraction blah,blah, blah"

43 mins ago, by Thomas Finley

but while integration, I always used the notion of "differentials". To be honest, my notion of "differentials" is only limited to, the fact, that say, $dx$ is nothing but an infinitesimally small quantitiy. I don't know, any operations of differentials, like d(xy), d(y/x), etc. Never, have I got to know any details about the so called term "differentials" in any real analysis book. I found them admittedly in differential equations textbooks.

44 mins ago, by Thomas Finley

The problem they never introduced it anywhere. They just started with, "Now the differential dx..." Does anyone know where can I get to know about the topic?

44 mins ago, by Thomas Finley

Not casually, but rigorously!

This ends it. Bye for now...

Koro

11:47

X=[0,1)\cup (1,2) is homeomorphic to (-\infty, -3)\cup [0,\infty)?

apparently, it is but I don't see how.

nvm

it's pasting lemma.

dictionary topology on X\times Y; X,Y are ordered spaces is also called television topology. I don't know why.

on R^2 with dictionary order, is [0,1]\times [0,1] compact?

Yai0Phah

@ThomasFinley There is a notion called differential forms. You can find many materials via Googling.

Lexicographic order topology on the unit square

In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a topology on the unit square S, i.e. on the set of points (x,y) in the plane such that 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. == Construction == The lexicographical ordering gives a total ordering ≺ {\displaystyle \prec } on the points in the unit square: if (x,y) and (u,v) are two points in the square, (x,y) ≺ {\displaystyle \scriptstyle \prec } (u,v) if and only if either x < u or...

Koro

12:04

thanks I didn't know that [0,1]\times [0,1] is compact in that topology.

I don't know though why.

do you have any proof for that? I'm not comfortable with order topology. @Yai0Phah

Stephie

@Peter I must respectfully disagree. It’s not as bleak as you paint it.

Yai0Phah

3

Q: Compact set in the order topology

Let $(X, \leq)$ be a linearly ordered set and let $\mathcal{T}$ denote the order topology on $X$. Prove that $(X, \mathcal{T})$ is compact if and only if every nonempty set of $X$ has a greatest lower bound and a least upper bound. I don't know how to apply the definition of compactness in this t...

general-topology compactness order-theory

Koro

Deutschland :)

PM 2Ring

@Shaun BTW, you can run GAP code in SageMathCell, eg sagecell.sagemath.org/…

Koro

@Yai0Phah thanks. What does $\leftarrow$ mean?

Ajay

12:14

@PM2Ring I managed to catch you!

I wanted to ask if you knew anything about image compression and how it can be achieved through the use of the banach fixed point theorem?

PM 2Ring

@SouravGhosh Please see a doctor or counsellor! We understand that you feel bad right now, and you will probably still feel bad tomorrow. You are undoubtedly stressed, which makes the emotional impact of your situation feel even worse. But a professional can help you.

@Ajay I know a little about compression algorithms, including image compression, but I haven't thought much about it in the last decade or so. But yes, there are compression techniques related to fixed point stuff. See en.wikipedia.org/wiki/Fractal_compression & Iterated function systems

shintuku

germany social democrat dream

PM 2Ring

Of course, no compression scheme can always compress its input, due to the pigeon-hole principle. OTOH, when compressing images, we're mostly interested in images that contain a fair amount of pattern. The trick is to extract that pattern efficiently.

Ajay

12:30

I'm doing a research paper on this topic

as part of it, I was wondering how I could analyse the actual steps which occur during the image process

Koro

the set of all positive definite matrices in the space of n by n real matrices is connected?

PM 2Ring

In a sense, every fractal generator is a special-purpose image compressor with an extremely high compression ratio. But people want to compress cat photos, not just Mandelbrot sets. ;) Barnsley realised that you could apply fractal generation techniques to a wide class of structures.

Barnsley fern

The Barnsley fern is a fractal named after the British mathematician Michael Barnsley who first described it in his book Fractals Everywhere. He made it to resemble the black spleenwort, Asplenium adiantum-nigrum. == History == The fern is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Like the Sierpinski triangle, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers. Barnsley's 1988 book Fractals Everywhere...

Koro

by our one and only

8

A: Intuition for connectedness of positive definite matrices

The most straightforward intuition comes from the fact that the set $P$ of positive definite matrices is convex and convex sets are path connected by straight lines. If $\langle x, A_k x \rangle > 0$ for all $x \neq 0$, then for $\lambda \in [0,1]$, we have $\langle x, (\lambda A_1 +(1-\lambda) ...

The set of all invertible real matrices is not compact: pf: it is not closed.

Shaun

12:46

@PM2Ring Yep :)

Sourav Ghosh

Four interview questions can decide one's selection; it doesn't matter how you perform in the exam.

Jakobian

I'm tired

user 858770

@SouravGhosh then it is a popularity contest

shintuku

@SouravGhosh

not the same situation, but somewhat related

user 858770

Give a popular well liked answer and you win.

Koro

12:52

@Jakobian try hot chocolate :}

ChrisW

Maybe this is the end of my career. :63805739 I did Maths at Cambridge. The system is England is based on doing well at exams and I did well on Maths exams. But I knew I wouldn't work as an astrophysicist or a quantum mechanic after I graduated, so what was the point? My first employer let me learn telecom software engineering, which I found fascinating (and much easier than Mathematics). I've been doing that for last 40 years or so. In other words maybe your career hasn't even started yet.

Koro

@SouravGhosh I hear the result was declared today. Do they also tell marks obtained in interview?

Sourav Ghosh

No

Koro

This year they combined MSc+ Ph.D entrances both but the cut off was different for both.

Jakobian

@Koro physically and mentally

Koro

12:55

@SouravGhosh I understand that through the exam, one can go only to certain colleges. Do you want to go to only those colleges for Ph.D?

Jakobian

I'm already drinking coffee. Idk how good is it

shintuku

@Jakobian me too, been awake an hour and a half and brain still not activated

user 858770

@ChrisW could you please remove the flagged comment?

Jakobian

I'm just stressed a little

shintuku

how much time since you've taken a full two days off

Koro

12:58

I tried to sleep but couldn't as my classmates were debating over something really loud 😃

user 858770

🙉

Jakobian

@shintuku no it's not that, just things happened today that stress me out, and that I can't really talk about

shintuku

gl

Koro

I forgot to go outside to get mangoes straight from the trees. It was raining...

now they may have picked up all the mangoes.

Jakobian

oh, I thought the plural was mangos

Koro

13:00

issues here are different haha

shintuku

both correct spellings

user 858770

Does spelling affect the taste :P

Jakobian

they say even the visuals can affect the taste, so why not spelling

user 858770

How about pronunciations ;-)

Soumik Mukherjee

Mangoes to pick up mangos

user 858770

13:04

🙈🙉🙊

PM 2Ring

A female mango is a manga. ;)

user 858770

Do magos grow in Australia?

shintuku

there are absolutely hispanic sorcerers in australia

PM 2Ring

13:25

Yes, we grow wonderful mangoes in Australia, mostly in the more northern regions.

Jakobian

I created a new tag realcompact-spaces, and I'm wondering about adding this tag to old questions. Should I do it?

does anyone know how it works

PM 2Ring

@Jakobian Sure! But don't retag a bunch of old questions at once. Do maybe 5 to 10 a day. People get annoyed if a bunch of old questions flood the front page.

Jakobian

ah, thanks!

PM 2Ring

If you like, you can ask about it on meta. Or search there for old questions on retagging.

Also, don't bother retagging old questions unless they're valuable in some way. If the question might be useful to future readers, then fine, retag it. But if it's an obscure old question that's of little interest don't retag it merely because your new tag applies to it.

Jakobian

hmm... I see

冥王 Hades

13:39

Find $x$ as a function of $a$ and $b$

PM 2Ring

BTW, one of the 10k privileges is that you can Edit tags inline math.stackexchange.com/help/privileges/moderator-tools But editing tags through the normal editor is pretty quick anyway. Of course, when editing a question it's a Good Idea to fix any other problems (like typos) at the same time.

PM 2Ring

13:54

@冥王Hades Cute. sagecell.sagemath.org/?z=eJyrsE3UTgIABSwBpA==&lang=html

冥王 Hades

@PM2Ring I mean, it’s correct, but how?

Koro

A chart in an atlas of a smooth manifold is C^\infty.

PM 2Ring

@冥王Hades Ok. I first did it analytically, using the equation for $\tan(2\theta)$. But it's even easier by drawing a rotated copy of the b triangle adjacent to the a triangle, and seeing that together they make an isosceles triangle.

Koro

Why is it true?

I think every chart should be C \infty following definitions.

But I have an example where chart is not C \infty.

Let $(U, \phi)$ be a chart in a manifold M. $\phi: U\to \phi(U)\subset R^n$ is C^\infty at $p$ if the composition $\phi\phi^{-1}: \phi(U)\to \phi(U)$ is C^\infty at $\phi(p)$.

but the identity is always C \infty.

so \phi is C^\infty.

冥王 Hades

@PM2Ring Yeah the second method was what I did as well

Koro

14:08

but this looks wrong. What is wrong in this?

Thorgott

nothing's wrong, Koro

冥王 Hades

@PM2Ring here’s a harder problem

2

Q: Compute the area of Quadrilateral $ABCD$

As title suggests, the question is to solve for the area of the given convex quadrilateral, with two equal sides, a side length of 2 units and some angles: I have solved the problem with a synthetic geometric approach involving some angle chasing. However, I believe my solution (which I will pos...

geometry trigonometry euclidean-geometry

I don’t need to say this but do not look at the answers

monoidaltransform

Is the $Z_p$ actions on $S^3$ to get lens space an isometric action?

冥王 Hades

No calculators by the way

Thomas Finley

Hey guys! I hit myself with a strange question. I was given a high-school differential eqn, y'=\sqrt y, with the condition that y(0)=0. Is this question at all legit?

Jakobian

14:17

@PM2Ring maybe I'll reach 10k rep in the next 5 years

Thomas Finley

I mean isn't the only solution is y=(x/2)^2

Wolframalpha affirms me, but I am feeling weird.

Koro

@Thorgott thanks.this is a starred exercise in Tu's.

So I wonder why it was starred.

16 mins ago, by Koro

A chart in an atlas of a smooth manifold is C^\infty.

Thomas Finley

Koro

@Thorgott but then $(R, x\mapsto x^{\frac 13})$ is not C infty.

@ThomasFinley isn't y=0 also a solution?

s.harp

@Koro depends on the atlas you choose on R

Thomas Finley

14:23

This scrnshot is the proof that WolframAlpha and I , both are on the same page...

@Koro That's covered in y=(x/2)^2 , isn't it? I mean if x=0

Koro

@s.harp let's take Atlas as the singleton set containing the chart $(R, x\mapsto x^{1/3})$.

@ThomasFinley careful there! take your time and think about why not.

冥王 Hades

I really like the puzzle aspect of that problem

s.harp

@Koro with this atlas the map is then smooth

冥王 Hades

Just one big puzzle. The calculations are the easy part

s.harp

provided the target (Which is also R) has the standard atlas

Koro

14:27

I'm afraid I don't understand. I thought that since the map $x\mapsto x^{1/3}$ is not differentiable twice, hence not smooth.

s.harp

as a map R --> R, where both R have the standard smooth structure, it is not smooth. But if you give the first (or the second) R a different atlas then it can be smooth

Thomas Finley

@Koro is it because, I obtained, $y=(x/2)^2$ from the general solution $y=(x+c/2)^2, by plugging $c=0$( using the relation y(0)=0) and y=0 is not a particular solution, i.e not obtainable by assigning real values to $c$ in the general solution, and so, y=0 is not a part of general solution ?

But then again, I am having a hard time convincing the situation, when y=(x/2)^2 is taken as the particular solution, then y=0 is attained if x=0 .

冥王 Hades

We had a thermodynamics class and someone said “I carnot understand anything”

Koro

@s.harp I think I'm starting to understand what you are suggesting:

Le me write out the definition here first: A continuous map F: M\to N;M ,N are smooth manifolds of dim m and n resp. is said to be C^\infty at p if there are charts around p and F(p) such that composition from R^m to R^n is C^\infty at some image of p.

Thomas Finley

@Koro What did you mean if you consider sharing ?

Koro

14:34

@ThomasFinley hint: what is a solution to an ODE?

so as a map from R to R where both R are given the standard smooth structure (I think this means the atlast is singleton consisting of (R, id: R-->R).), then the composition id f id^{-1} is not C^\infty at 0.

but if I give domain the atlas consisting of singleton set $(R, f)$, where f is the cube root function, then the composition id f f^{-1}: R--> R is just the identity hence C^\infty.

Thomas Finley

@Koro This confused me more. I know what's a general solution of an ODE, a particular solution, a singular solution and all three of them are solutions of a random ODE. Or what are you trying to imply?

Koro

@s.harp and @Thorgott I suppose this is correct.

s.harp

@Koro yes, thats fine

Koro

thank you so much :)

now, I want to understand why the wording of the exercise is as below:

35 mins ago, by Koro

A chart in an atlas of a smooth manifold is C^\infty.

Thomas Finley

@Koro I gave those notions, in terms of single variable calculus

Better, if you dispel the suspense!

s.harp

14:41

@ThomasFinley some ODEs have more than one solution for a given initial conditon, the ODE y' = y^{1/2} with initial condition y(0)=0, is one such example.

Koro

@ThomasFinley solutions are y= g(x), right?

Thomas Finley

@Koro yes ...

Koro

@ThomasFinley if x=0, then you just get a point (x,y)=(0,0).

Thomas Finley

@Koro yes, umm so?

I might be missing something huge

@s.harp The thing is, what else solutions are there apart from y=x^2/4? Wolfram also gave that particular solution

Koro

y= 0 (x- axis) and (x,y)=(0,0) are two different things, no?

Thomas Finley

14:44

@Koro That came below the belt. I am reeling

The thing, is, the solutions are y=0 and y=x^2/4, right?

s.harp

@ThomasFinley There is one other solution that you can find if you spend some time thinking about it. I think its worth the time spent for you to find it on your own. (As a hint: As soon as you see it you will hit yourself for not finding it sooner)

Koro

@ThomasFinley uniqueness could be tricky:

Picard–Lindelöf theorem

In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. == Theorem == Let D ⊆ R × R n...

(how do you know that if you start with some method (say variables separable) to solve an ODE, then you have got all the solutions?)

Thomas Finley

@s.harp if you were talking about finding solutions like y=0, then, I bet I would not hit it soon, becoz till now, I was just gazing over points not a whole axes. @Koro The point is, is there a way to come up with "these" sorts of solutions ?

I mean you instantly came up with it, @Koro. So, did you just came by it, by chance or is there some trick you're using. If that's the later case, I genuinely want to know it!

@Koro I read this before, but the thing is, I am interested to find solutions like these. That's the part bothering me now.

Thomas Finley

15:08

@s.harp I ask the same question to you: Is there any standard /known way to come up with these peculiar solutions ?

s.harp

@ThomasFinley Not as far as I'm aware. If your ODE has bad regularity the standard uniqueness theorems just dont apply, but there are techniques tha t show uniqueness of solutions belonging to a certain class.

Constructing additional solutions outside of that class is something ive only ever done ad-hoc

that said the example you give is a very common one

and there are even more solutions than the two youve found

Thomas Finley

@s.harp So it's purely experience, right ? The more exprerience you have the more number of preculiar type of solutions you can come up with ?

s.harp

@ThomasFinley I don't know, I'm not an expert on ODEs. But I would guess yes, there is no pretty theory for making extra solutions

Thomas Finley

@s.harp Ok, that's fine. Thanks!

monoidaltransform

15:27

Is there a natural SO(2) action on S^1?

Thomas Finley

@s.harp I will just poke you once more, to ask, if by any chance do you know how to find "singular solutions" of a diff eqn of 1st order and 1st degree? Or atleast know some resources that you can suggest ?

Koro

@ThomasFinley take a look at the wiki link I shared.

s.harp

@monoidaltransform yes, S^1 is the unit circle of R^2, and SO(2) preserves the lenght of vectors. That said SO(2) can also be identified with S^1

Koro

@ThomasFinley did you look up Coddington's?

Thomas Finley

@Koro I read it before, but I am not accustomed to some portions.

@Koro no, is that a book ?

Koro

15:30

yes, it's a book.

Thomas Finley

Name ? (Of the book) @Koro

monoidaltransform

So $A\dot x=Ax$ is the natural action @s.harp ?

s.harp

@ThomasFinley For the example you have given: If y' = f(y) and f is singular at some value y*, check if the constant solution is a solution, also look for solutions to y(0)= y* + epsilon and see if any of them cross the line y*

@monoidaltransform yes

Thomas Finley

@s.harp What do you mean by, "singular at some value y" ?

s.harp

@ThomasFinley for example f is not differentiable at y*

for your ODE f(y) = y^{1/2} fails to be differentiable at y=0, the function has bad behaviour here, its "singular" at this value

Thomas Finley

15:43

@s.harp Oh, I see, but I wanted something general for say, an equation of the form, Mdx+Ndy=0 (?)

s.harp

For "reasonable" M, N this kind of an equation will only have bad behaviour around some points. Local uniqueness for solutions starting at some y(x_0)=y_0 is given if the functions M, N are not too shitty near (x_0,y_0). You need M, N to have a derivative "going to infinity" at (x_0,y_0) or something like that to get bad behaviour near this point

That said I don't know how to find other solutions if M, N are bad. In the kind of situations I am in I always try to show uniqueness of solutions (at least for solutions having some kind of regularity)

Peter

16:01

@Stephie You can disagree , but this does not change anything about the reality in Germany. And this reality is a real nightmare. I know it , I live there for more than 50 years.

shintuku

try the third world

冥王 Hades

Being worse off isn’t an achievement.

shintuku

no it is a relative measure of how well-off one is

by all reasonable standards germany is one of the best countries to live in

Thomas Finley

@shintuku absolutely...

s.harp

I live in Germany too, this country has many issues. But I have lived in other countries as well, and I find that gives a good perspective. Overall I think Germany is a place that can provide many opportunities for people from other places.

Thomas Finley

16:05

@s.harp nice, how's the education there ?

I mean undergrad studies, grad studies, blah blah blah...

I heard lots about it being one of the best in the world

s.harp

@ThomasFinley I can't really compare that

shintuku

(it's free)

s.harp

more or less

Thomas Finley

@s.harp That's good.

Thomas Finley

16:22

@Koro I googled that authors name and found a book, but nah, it's not there.

Peter

@ThomasFinley My description was in good faith and realistic , if you want to follow other people apparently seeing this different , I can unfortunately not help you. But remember that I warned you.

s.harp

@Peter where would you prefer to live? Just curious. You complained about orthography reforms and gendering in the same breath as medical care, these things are not comparable at all in their importance

Koro

@ThomasFinley Did you see 'theory of ordinary differential equations'?

or 'an introduction to ordinary differential equations'?

Thomas Finley

16:46

@Koro This was the book I was talking bout. Nope, it's not there.

Koro

@ThomasFinley Coddington has atleast two books. Did you check the other one?

theory of ordinary differential equations

(I didn't find it there.)

Sourav Ghosh

17:03

Thank you @SpencerG for you support. I am ok right now. I believe I will come back strongly.

Koro

@ThomasFinley you may find this helpful, it mentions a reference also.

Ted Shifrin

That looks like the book @Thomas has been trying to read.

shintuku

17:19

suppose I have a grid of 9 points, 0 through 8. what's the less visually cumbersome way to represent a topology on this? say i start with three non intersecting sets

Sourav Ghosh

A $T_1$ topology on a finite set will be a discrete topology.

shintuku

yeah but i'm hoping to represent T_0 topologies

s.harp

Give the open sets or closed sets not containing any other open sets / closed sets

Sourav Ghosh

There are many non homeomorphic non T_1 topologies.

Indiscrete, particular point topologies

Jakobian

@shintuku pretty sure they correspond to partial orders on $9$ elements

using the specialization preorder

Sourav Ghosh

17:27

@Jakobian Correct✅

shintuku

oh i can give a complete representation with arrows

Charlie

Am I correct that for a tensor in $V\otimes V$, while it isn't true that $v_1\otimes v_2(\omega_1,\omega_2)=v_2\otimes v_1(\omega_1,\omega_2)$ it is true that $v_1\otimes v_2(\omega_1,\omega_2)=v_2\otimes v_1(\omega_2,\omega_1)$ because $v_1\otimes v_2(\omega_1,\omega_2)=v_1(\omega_1)\cdot v_2(\omega_2)=v_2(\omega_2)\cdot v_1(\omega_1)=v_2\otimes v_1(\omega_2,\omega_1)$? Where $v_i$ are vectors in $V$ and $\omega_j$ are elements of $V*$

Jakobian

Hasse diagram you mean?

shintuku

probably that

Sourav Ghosh

See mathoverflow.net/q/447280/483536

Jakobian

17:30

all the types of covering axioms and metrization theorem related things, the amount of these things gives me a feeling of "staring into the abyss"

shintuku

"topological spaces on finites sets are in one-to-one correspondence with preorders" @SouravGhosh thanks this is what I'm looking for

ty to Jakobian too

Thomas Finley

@Koro My goodness! I am shocked. Just a minute.

s.harp

what does the T in T0 T1 etc stand for? Topology?

Koro

Trennungsaxiom?

s.harp

right

makes sense

Koro

17:33

(German for separation axiom )

Deutschland :)

Thomas Finley

1

Q: Challenge: Can You Point out the Mistake in Here?

Find the $p$-discriminant of the differential equation $F(x,y,p)\equiv 4xp^2-(3x-1)^2=0.$(Note: $p=\frac{dy}{dx}$) We proceed to calculate the $p$-discriminant of $F(x,y,p)=0,$ where $F$ is the function representating the differential equation in the question. So, $F(x,y,p)=4xp^2-(3x-1)^2=0\impli...

calculus

This is where my doubt is, @Koro !

Unfortunately, in all other cases p-discriminant, is calculated using partial derivatives and in this case specifically, it's calculated using this different way. That's really strange. Is this something special ?

Again, I got a hold of gold, but just when I started to take it in my possession, it turned out to be a replica built just like the same awry way.

What a mysterious puzzle this is.

@Koro Not to forget, Thanks! Atleast I found something accurate to my issues, though it didn't quite solve things, but it's exciting.

Jakobian

17:52

Q: How to learn a topic without growing crazy

s.harp

Most secure way: lots and lots of time

Second way: By doing it poorly and shoring up gaps later

Koro

@ThomasFinley it doesn't look complicated. Check out the footnote at page no. 1 as well.

Jakobian

How do I learn things like developable spaces, quasi-developable spaces, stratifiable spaces etc. it's a massive amount of different spaces and they just keep on pilling up with next to no intuition of mine

Koro

Q is: why are you finding p/c discriminants?

to find singular solutions. (3x-1)^2 doesn't do the job.

@Jakobian I feel the same about finding face maps in cellular homology.

absolutely no idea what goes on there in the background.

shintuku

18:15

nvm

Jakobian

@Koro hmm... I think we have different experiences

shintuku

18:35

$S$ not closed in $X$ if and only if there is no choice $C_1, C_2$ of open sets that separate $S$ from $X - S$ right? (more formally: no choice of collections $C_1, C_2$ s.t. $S \subseteq C_1, X - S \subseteq C_2$, and $C_1 \cap C_2 = \emptyset$)

Jakobian

the statement on the right means: $S$ is not clopen

shintuku

:/ need more thinking, ty

this instead: $S$ not closed in $X$ if and only if there is no collection $C$ of open sets such that $X-S \subseteq C$ and $C \cap S = \emptyset$

oops i want $\bigcup C$, not just $C$

Alessandro Codenotti

18:58

@s.harp Tseparation

northerner

19:21

I'm helping someone do the coding portion for analyzing data for a science experiment. Can a science experiment have 1 dependent variable and multiple independent variables?

s.harp

@AlessandroCodenotti Tsheparashion sounds like how Sean Connery would pronounce seperation

shintuku

@northerner yes, the speed of a car at a point in time is dependent on air friction, motor acceleration, car weight, etc.

Ted Shifrin

@northerner Think about temperature as a function of time and location.

northerner

If the experimenter intentionally changes a variable, does it effect if it's dependent or independent? For example if the experimenter reruns the test with a different car weight, the car weight would still be an independent variable and the speed would be the dependent variable? @shintuku

shintuku

@northerner ted's example is simpler: you can redo a measurement of temperature by changing the time and location, and you obtain a different temperature

usually the experimenter affects the independent variable to see how it changes the dependent variable

Sine of the Time

19:58

Consider a closed set $C$ say in $\Bbb R^2$ and a point P, I want to calculate the distance of P from $C$. The distance in $\inf_{Q\in C}|P-Q|$. In general, $C$ is not bounded, and my professor claimed that "inf" is actually "min" since the distance is a continuous map on a compact. Is he correct? My intuition says that the "inf" is a "min" since $C$ is closed

Transcript for

$Mathematics$ Mathematics