Mathematics - 2021-01-19 (page 1 of 3)

Ted Shifrin

12:11 AM

@AlessandroCodenotti Will I do?

@Thorgott if you can define a meromorphic 1-form, just take the dual notion.

White Force

Hello

@TedShifrin you are here from a week ago :)

Fargle

@TedShifrin This answer reads like a joke, outside of the fact that it's also a perfectly sensible answer.

Thorgott

I only know an insensible way of defining meromorphic 1-form, no good ones, that's why I need to learn the bundle perspective

Ted Shifrin

2:10 AM

@WhiteForce I am?

@Thorgott You can talk about a 1-dimensional complex manifold and a local holomorphic chart/coordinate $z$.

robjohn

2:45 AM

@WhiteForce He was not here last night, so I disagree

Physics is my life

5:48 AM

Guys , when yesterday I posted a question of modulus function

then , when question says mod of -5

why is it always 5. In the examples I gave , I also found negative value of x

Like mod of 2x-3 = 9 has values of x = 6 and -3

Ted Shifrin

@Physicsismylife I've never heard of saying mod a negative. The negative makes no difference. Mod $5$, $2x-3=9$ means $x=6$ which is the same as $x=1$ mod $5$. Where did you get $-3$? That isn't right.

robjohn

@Physicsismylife $|x|=\left\{\begin{array}{rl}x&\text{if }x\ge0\\-x&\text{if }x\lt0\end{array}\right.$

so $|x|\ge0$

the case when $|x|=-x$ does not mean that $|x|\lt0$.

Ted Shifrin

Where did absolute value come from, @robjohn?

Oh, I was doing mod 5 arithmetic.

robjohn

Yeah, they mean $|x|$ when they say "mod"

Ted Shifrin

Yeah, I think you're right.

But what is "the mod of $2x-3=9$"?

Physics is my life

6:01 AM

2x-3 = 9 and 2x-9 = -9

robjohn

$|2x-3|=9$

Ted Shifrin

Oh, $|2x-3|=9$ is the question?

Physics is my life

I saw this on YouTube that we write It like this

robjohn

@Physicsismylife or

Physics is my life

@TedShifrin @robjohn

Rithaniel

6:02 AM

I'd assume "the modulus of 2x-3 equals 9"

Ted Shifrin

This is not usual English use for mathematics.

Physics is my life

@robjohn yes. Or we have to write

@TedShifrin did you get how I got -₹

Ted Shifrin

So, for the second possibility, you get $x=0$, not $x=-3$.

Physics is my life

-3

Ted Shifrin

No.

How did you get -3?

Physics is my life

6:04 AM

2x-3 = -9 , 2x = -6 , x = -3

Ted Shifrin

Oh, you wrote the wrong equation the second time.

I read what you typed instead of the original question.

Right, $x=-3$ is correct as the other solution.

Physics is my life

Yes

Ted Shifrin

There are two solutions.

Physics is my life

then why do we say that mod of -5 is always 5

Ted Shifrin

Because mod means distance from the origin (0) on the number line.

The distance from -5 to 0 is 5.

Physics is my life

6:06 AM

But in 2x-3 question , I got x = -3 as well. Which is negative

Ted Shifrin

That's fine. No one is saying you have to take the mod of the answer.

Physics is my life

@TedShifrin ok . Got it thanks

Ted Shifrin

Cool.

Physics is my life

One more thing

|x|/ x = x /|x| right

Ted Shifrin

Yes (assuming x isn't 0).

Physics is my life

6:09 AM

Ok.Great

robjohn

and $x$ is real ;-)

Physics is my life

Ok.@robjohn

Why is the interval of greatest integer function only from (-infinity , x)

Ted Shifrin

Huh?

Physics is my life

Like why is x not till + infinity

robjohn

@Physicsismylife not sure what you mean.

Physics is my life

6:11 AM

[x] = greatest integer in the interval (-infinity,x]

Exact line pasted now

robjohn

[x] is in (x-1,x]

Ted Shifrin

Yes, that's correct, but I would never say it that way.

Just think of it as the largest integer to the left of (less than or equal to) $x$.

Physics is my life

For example if it is 4 , then interval is from 3,4 right

robjohn

(3,4]

which means 4

Ted Shifrin

Just think of rounding down to the nearest integer.

robjohn

6:13 AM

because it is an integer

Physics is my life

Why did my teacher write - infinity then ?

robjohn

no idea

Physics is my life

@TedShifrin and @robjohn got that part

Thanks still.What you wrote makes sense as well

Ted Shifrin

The teacher is saying look to the left of x (including x) on the number line (so that goes down to $-\infty$) and take the largest integer. I wouldn't say it that way, but it's OK.

robjohn

what they wrote is not technically wrong, but it is not the easiest way to understand it, imo

Physics is my life

6:15 AM

Hmm yeah

Also , I can’t say that this interval means range of x right

Ted Shifrin

Nope.

Don't get sloppy with your language.

Physics is my life

Ok.

Semiclassical

this escalated quickly: math.stackexchange.com/questions/3990810/…

robjohn

@Physicsismylife: if you want to get the most from this room, you might look into MathJax and install ChatJax (the installation page is listed at the top of the sidebar to the right)

Ted Shifrin

Good grief. Well, damn, there sure looks to be something missing. Did you click to see if there were unviewable characters?

Semiclassical

6:19 AM

doesn't seem to be

Ted Shifrin

I have gotten people mad at me when I've complained that things don't make sense, but this seems a clear case. What is that person's problem?

Semiclassical

shrug

i mean, I think the "OP looks defensive" remark probably didn't help but

Ted Shifrin

Somebody called me sarcastic and mean or something a day or two ago in here, and I wasn't even at that point — perhaps a day earlier I had been, though.

Semiclassical

overreact much

yeah

Ted Shifrin

I mean, seriously, if someone doesn't know what "at least" means, we have a long way to go with probability problems.

Semiclassical

6:21 AM

i do wish I'd said my comment a bit more clearly, e.g., "You say you have a calculus-based solution, but you haven't actually said what the answer is supposed to be."

Ted Shifrin

Yes, that would have helped.

Semiclassical

so it goes

Karl Kroningfeld

$P_{\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\},\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}\}}$ ain't nothing.

Ted Shifrin

It ain't much to look at.

Karl Kroningfeld

lol

Physics is my life

6:28 AM

@robjohn ok.

guys , if you see this . This is how my sir drew the [x] on the number line

robjohn

@Physicsismylife the latter one is $-4$

Physics is my life

He took it to - infinity range

Ted Shifrin

All you have to do is start walking to the left at $x$ and stop at the first integer you come to. You are never going to have to go all the way to $-\infty$, of course.

If you understand what's going on, stop insisting on his definition over and over.

Physics is my life

@TedShifrin Yes.Got that from your point.But why would he draw this diagram then? It’s just that I can’t be easy to say that my sir is entirely wrong here. You getting it right

Ted Shifrin

We've already told you he is NOT wrong.

We also told you none of us would explain it that way. We can't keep discussing it. Ask him why he chooses to use that instead of something else.

Semiclassical

6:33 AM

your sir used a description that made sense to him and thought it would make sense to you

Physics is my life

Ok.@TedShifrin

Semiclassical

if it didn't, then asking "why" isn't going to help much

Ted Shifrin

I'm sure plenty of times my students wondered why I used a different explanation or definition. Of course, when they asked me, I tried to convince them why I was right :)

Physics is my life

Ok@TedShifrin and @Semiclassical

Karl Kroningfeld

Ok, this is pretty ironic. A misprint in a textbook I am reading has a paragraph beginning with Step ${}$.

Ted Shifrin

6:38 AM

LOL ... the missing step.

Or ... step lightly?

Karl Kroningfeld

Suppose $X$ is irreducible of dimension $0$. Doesn't sound like a killer step. Maybe intentional.

Ted Shifrin

Well, Step 0 is allowed.

robjohn

@KarlKroningfeld what kind of an object is $X$?

Ted Shifrin

an irreconciliable object?

Karl Kroningfeld

@robjohn It is the hazy middle between W and Y.

Or, to quote my stupid bio, What does it mean to find 'x', anyway?

Ted Shifrin

6:46 AM

I like my answer better.

Karl Kroningfeld

:P

Ted Shifrin

I was out of the room when you made your retort.

Physics is my life

[x+[x+[x]]] , can we also where this like this [x]+ [x]+ [x] .

Ted Shifrin

No. Try lots of examples.

Karl Kroningfeld

Hindsight is 20/20 @Ted

Ted Shifrin

6:52 AM

That was your erased comment?

Physics is my life

Why did we write it the 1st way which is a bit complex than 2nd one

Karl Kroningfeld

@TedShifrin Who knows?

Any historians?

Ted Shifrin

Because it's different. Please follow my suggestions and do examples, @Physicsismylife.

Physics is my life

@TedShifrin ok.

Leaky Nun

actually he's right

because [x+n] = [x]+n for any integer n

Will

7:07 AM

"The Malagasy Ariary is the official currency of Madagascar. The Ariary is one of only two circulating currencies in the world whose division units are not based on a power of ten."

So I dont understand how this means we calculate the fractional part of this currency

I'm working on an app involving currency conversions - I just did a quick search to see whether there were currencies with odd fractional parts and found this edge case.

Semiclassical

do you have an example in mind?

like, of what the dollar value of X ariaries would be (or vice versa)

Will

I dont even know how to *think* about decimal parts of money not based on power of ten :(

Closest example I can give: converting USD to XFA - I know USD uses 2 decimal places (for presentation, although more are used in sensitive calculations) and XFA does not have a fractional component.

If I were to convert 5 USD to XFA at a rate of 1 USD == 568 XFA, I'd end up with 2840 XFA
If I were to convert 5,000 XFA to USD, I would get 8.80282 USD

Semiclassical

hrm

both of those seem unambiguous, though. do you have a case where you're not sure?

oh, but that's XFA not Ar

robjohn

@LeakyNun $ \begin{align} \lfloor x+\color{#C00}{\lfloor x+\lfloor x\rfloor\rfloor}\rfloor &=\lfloor x+\color{#C00}{\lfloor x\rfloor+\lfloor x\rfloor}\rfloor\\ &=\lfloor x\rfloor+\lfloor x\rfloor+\lfloor x\rfloor\\ \end{align} $

copper.hat

when i grew up it was pounds shillings & pence (amusingly denoted lsd).

Will

7:21 AM

I do not - That currency is not even part of what I need to account for at the moment.

From wikipedia: "Today, only two countries have non-decimal currencies: Mauritania, where 1 ouguiya = 5 khoums, and Madagascar, where 1 ariary = 5 iraimbilanja.[1] However, these are only theoretically non-decimal, as in both cases the value of each sub-unit is too small to be of any practical use and coins of sub-unit denominations are no longer used."

Semiclassical

culture is confusing

Will

Just thought it wise to leave a note in the code about how to handle it later

Leaky Nun

@robjohn yep

Will

I suspected that while coins of these sub units aren't used, it might be relevant in future to account for such tiny units when doing sensitive calculations - just how you wont find sub-cent coins in the US, but sensitive calculations in USD use sub-cent values

Semiclassical

i don't have anything useful to say, but you might be amused by this old bit of TV

First CityWide Change Bank 2 - Saturday Night Live

►

Physics is my life

7:25 AM

I am having confusion in how to make notes actually . For example , you sir gave taught you about quadratic functions today . Then , started with trigonometry. Now, let us say there were some examples you had to write before starting trigonometry for quadratic functions . You then write those examples in the chapter of trigonometry. So this way I think things can become a clutter when you are going to revise notes.

So , should I make a notebook or just wrote on separate pages

separate pages mean like not a notebook but just A4 sheets

That way , I can just attach whatever information is needed at right position

Will

@Physicsismylife any chance you can use a binder with hole punched pages?

as opposed to a notebook

Physics is my life

The problem with sheets is that when I use my punching machine and put them in a file

those pages tear up after some @Will

copper.hat

haven't heard a guinea for at least a month...

Physics is my life

Slowly as you keep openers them

Openers = opening them I meant here

Karl Kroningfeld

Take mental notes

Will

7:28 AM

Understandable :) I remember lugging around an ugly binder in high school with pages sticking out

Hanging on for their dear lives by only 1 hole

Physics is my life

Yeah.@Will

any idea

@KarlKroningfeld I am not that great like you yet

Will

nothing that doesnt subject you to loose pages still unfortunately

Physics is my life

Hen you open the binder right. Slowly the pages also start to to torn away easily. That makes it look worse .

Will

@Semiclassical "OPTIONS! OPTIONS OPTIONS!"

Semiclassical

"How do we make money? The answer is simple: Volume."

Will

7:31 AM

haha

Physics is my life

Guys , any idea to help me

save my notes and make them easy for me so that I can revise them

Will

@Physicsismylife the best idea i have is to use onenote. I have long discarded physical notebooks, for the ability to copy paste my notes to shuffle around as needed

Not practical for everyone unfortunately

Physics is my life

Yes

Karl Kroningfeld

Typing notes sounds like a bigger pain in the ass than it is.

Will

I use a surface book - onenote is amazing with a nice screen and pen

copper.hat

7:33 AM

is pen & paper passé?

paper is cheap.

Will

eh

Physics is my life

Use pages and write page numbers on them

What if I use paper clips instead of binder

Will

I used to do this back in my country haha. The amount of notes we got was insane throughout middle school. No one handed out printed sheets of material for you to put in your binder like they do in the US. Teacher wrote everything from their notebook and students copied it all :sobs:

Everynotebook I got, i spent a few hours writing numbers on all pages - sometimes skipping numbers by mistake lol. It was great for writing thing like "see X on page X"

copper.hat

i think there is some learning value to writing stuff down.

Will

Looks like thats your solution

Physics is my life

7:37 AM

What if you wrote sth on a page that you don’t need

will you remove the page

Will

Okay dude are you a troll

This is funny, the exceptions you're trying to consider haha

Physics is my life

@Will why ?

Will

To answer your question - No, i wouldnt remove the page. I leave it, possibly cross out the bad content

Physics is my life

@Will they are actually true and I am in tension of it

copper.hat

@Physicsismylife there are bigger things to be tense about!

Will

7:39 AM

Understandable, are you just now entering a stage where you must take more notes than you;re used to? or you've entered a very competitive academic scene?

Physics is my life

Competitive

JEE

copper.hat

jee?

Physics is my life

Entrance exam to engineering

Out of 1.5million , 10000 get best colleges

I want to study computer science which is given to top 1000 students only

Will

I see :D

For your notes, just take the time to write page numbers when you buy notebooks. Refer to content on specific page numbers as needed.

@Physicsismylife my broda :D

Physics is my life

Ok.

I’ll try this

Will

7:44 AM

@Physicsismylife Good luck on your path to a Computer Science career!

I enjoy it so much although it can be so frustrating at times - ever wanted to beat up your computer? XD

"behave and do what I want, dammit!" - while coding something wrong but not realizing lol

Physics is my life

Hahaha

copper.hat

8:02 AM

33 hours to go...

Karl Kroningfeld

yes :D

Physics is my life

If f(x) or y = $(x-1)^2$ , then let us say g(x) or z = x^2 . Then z-1 or g(x-1)= x^2 - 1 right and not $(x -1)^2?$

copper.hat

could you express that less clearly?

Physics is my life

If f(x) = $(x-1)^2$ , then let us say g(x) = x^2 . Then why is g(x-1) = $(x-1)^2$ ??

copper.hat

i really have no idea what you are asking.

what has the specification of $g$ got to do with the specification of $f$?

Physics is my life

8:07 AM

It is just written g(x)to explain more

copper.hat

ok, if you have a quick actual question i can help, otherwise i am off to bed...

Physics is my life

Did you get it but ?

What I am trying to say

copper.hat

no idea. i need full sentences, actual questions, etc.

Physics is my life

f(x) = x-1) ^2 . Got this

copper.hat

is that a definition of $f$?

Physics is my life

8:09 AM

.It is just written that way

copper.hat

are you asking a question or just typing stuff?

Physics is my life

I am asking question

copper.hat

i don't mind either way, but if it is the latter i am off to sleep.

Physics is my life

but trying to make you understand my question first

Then , g(x) = x^2

copper.hat

my comprehension is fairly good, but you need to communicate clearly.

Physics is my life

8:11 AM

@copper.hat I am trying now.

@copper.hat got this

copper.hat

what do you mean when you say 'then $g(x)=x^2$?

what is the then part?

Physics is my life

Ok.It is just another function g(x)

copper.hat

so the then is superflous.

Physics is my life

Then is just English. Ofc it doesn’t mean anything in maths

right now

copper.hat

all we have is english here, i am a monoglot.

and this has taken 5 mins to get nowhere!

Physics is my life

8:13 AM

Thrn I write g(x-1) = (x-1)^2

go ahead. please.

Got this part

excruciatingly

So , I didn’t get this part actually

why is it not x^2 - 1^2

copper.hat

what part? so far there are just definitions.

sorry, until you actually pose a question i cannot comment.

Physics is my life

8:14 AM

g(x-1) the way we converted it from g(x)

copper.hat

huh???

Physics is my life

I didn’t get the conversion

copper.hat

who converted what>

Karl Kroningfeld

What is $g(y)$? @Physics

Physics is my life

One second . I’m thinking @KarlKroningfeld

g(y) = x-1^2

with brackets ofc

copper.hat

8:16 AM

huh?

Physics is my life

if you mean f(x) = y

Karl Kroningfeld

No, $y$ is just some new variable.

Sorry for the confusion, if you're thinking about the xy-plane. Not that y.

Physics is my life

I am not getting what to write it equal to then

Karl Kroningfeld

g(x)=x^2 is telling you what to do if you give it an input x. Namely, square the input x. But I can call x something else. I am suggesting to call it y, because why not?

Physics is my life

Yes@KarlKroningfeld right . Do you mean x^2 = y right

g(y) = y

I suppose

Karl Kroningfeld

8:19 AM

No, no.

I'm just calling the variable by a new name, 'y' instead of 'x'.

So, the way in which g operates isn't changing.

Physics is my life

ok. g(y) is new variable ok

Got this

Karl Kroningfeld

Nope :) 'y' is the new variable. You can tell what g(y) is, actually.

Here's something more concrete. What is g(2)?

Physics is my life

4

Karl Kroningfeld

What process did you use to get 4.

Physics is my life

g(2) = 2^2 I used

Karl Kroningfeld

8:22 AM

Ah, so you replaced x with 2 in the expression g(x)=x^2

Physics is my life

Yes

replaced 2 with x . I got 2^2

Karl Kroningfeld

I am suggesting the same, except instead of 2, you have y.

So, it should be g(y) = ...

Physics is my life

ok then it y^2

Karl Kroningfeld

Yes. :)

Physics is my life

Thanks got it so far then

Karl Kroningfeld

8:23 AM

Now, here's the wizardry: you have this equation g(y) = y^2.

You can swap y out with something else. How about x-1?

So if you replace y with x-1 on the left, you get g(x-1).

How about the right?

Physics is my life

then y = x-1

yes

it is x-1 to whole square

Karl Kroningfeld

What's that in symbols?

Physics is my life

Where @KarlKroningfeld

user 85795

or g(*) = *^2

Physics is my life

g(x-1) =( x-1)^2

Karl Kroningfeld

8:25 AM

yeah

and that is the same thing as f(x)

2

Physics is my life

@user85795 yes.Got it l thanks a lot @KarlKroningfeld

user 85795

::applause::

copper.hat

i completely missed what the issue was there.

user 85795

notation

copper.hat

$\lambda$ substitution.

good night folks.

user 85795

8:29 AM

cya

The substitution principle is simple only in principle.

Will

Haaaaaah ahaha

Karl Kroningfeld

8:45 AM

hm

user 85795

whatever

Karl Kroningfeld

5 minutes is picayune in the scheme of things

Will

@KarlKroningfeld why do you know this very particular word lol

user 85795

GRE vocabulary

Will

Uncommonly used words which have simpler, more common equivalents seem like very weak/worthless components of a vocab exam :/

Karl Kroningfeld

8:52 AM

I like to spice up my expressions.

With words like scheme.

Physics is my life

How to solve this

Hello guys..

Hello there

I am solving exercises from book.not getting these questions

a unique number because it is neither a prime number nor a composite number. It has only one factor, i.e, the number itself

Is this what they mean by in 2nd question

Anyone,any idea

user71207

9:25 AM

would anyone know of a place/website/forum where I can get good, difficult high school math questions? Or where people trade maths problems?

For example, I am looking for rates of change problems similar in difficulty to this:

of course, I appreciate even harder questions ;)

Perhaps something liek JEE or Korean CSAT type questions

My countries math syllabus isnt as strong as those countries

sorry if thats a bit of a wierd question haha

Physics is my life

9:45 AM

Solve JEE advanced previous years

Most Toughest exam

quesrions

you can get them anywhere online

@user71207

user71207

thank you

do they usually have worked solutions?

love_sodam

10:37 AM

any example of non-analytic function that satisfies cauchy-riemann equation?

Leaky Nun

no

love_sodam

why is that?

Leaky Nun

because satisfying the Cauchy--Riemann equations makes a function analytic

feynhat

@AlessandroCodenotti Hi.

Alessandro Codenotti

Hi

satan 29

10:50 AM

hello everyone

Any ideas on solving the equation r^2(d/dr* (r^2 dv/dr)))=cv, for v=v(r), where c is a constant?

using elementary methods...

feynhat

Is CET a commonly used abbreviation? Like if I say 15:00 CET, would an average European know what I mean?

Alessandro Codenotti

@Balarka by any chance do you know if there is some kind of results along the lines of "if $G\curvearrowright X$ with a $G$-invariant measure $\mu$ and $G$ is topologically transitive, then the action is ergodic"?

Astyx

@feynhat Central european time?

feynhat

Yeah

love_sodam

@LeakyNun Well I know if the function satisfies the Cauchy-Riemann equation and each real and imaginary part functions have continuous partial derivatives then the given function is analytic

Alessandro Codenotti

10:53 AM

@feynhat I know that the abbreviation refers to the central european time zone, but I need to look up when that is exactly because of DST (sometimes it is the same as my time, sometimes there is an hour of difference, I think)

But yeah if you're giving the time for a seminar or something it will be recognized

Leaky Nun

oh

feynhat

Okay cool.

Leaky Nun

@love_sodam according to this article, the strongest statement seems to be:

love_sodam

so anyway just satisfying Cauchy-Riemann equations does not ensure the given function is analytic

Leaky Nun

here is a counter-example if you only talk about satisfying CR at one point:

2

Q: Existence of partial derivatives & Cauchy-Riemann does not imply differentiability example

I learned about the Cauchy-Riemann equations today, and my instructor used the following example to show that differentiability is not guaranteed if the partial derivatives are not continuous. Let $$ f(z)=f(x+iy)= \begin{cases} \frac{xy(x+iy)}{x^2+y^2},&\quad\mbox{$z\neq 0$}\\ 0+0i,&\quad\mbox{$...

love_sodam

11:14 AM

strange.. the example satisfies Cauchy-Riemann on (0,0) but not in general.

It seems hard to find the function satisfying Cauchy-Riemann on the whole domain but not differentiable

Leaky Nun

I mean you can just look at the theorem I quoted

or the article

love_sodam

yes then what if the function is not bounded on D?

Leaky Nun

then maybe the article has a counter-example ^^

user2103480

11:51 AM

@Astyx @Semiclassical our prof just defines $f(T)(x) = \sum_n f(\lambda_n)\langle x,e_k \rangle e_k $ for an ONB of eigenvectors to nonnegative eigenvalues

Astyx

that ought to work

Huh wait actually I'm a bit surprised

Usually you'd define the square root of $-\Delta$ as multiplication by k, and this gives negative eigenvalues whereas your prof's def only yields positive eigenvalues

I'd have to give this some more thought but I'm a bit short on time rn

user2103480

Weird

Mike Miller

This is normal, everyone disagrees on whether $\Delta$ is pos def or neg def

I think it's a positive semidefinite operator

And if you want to do heat flow you look at e^{-t Delta}

Transcript for

$Mathematics$ Mathematics