Mathematics

12:39 AM

@TedShifrin I'm having trouble with a question on parametrization (already asked it, but I still don't see how it's possible)

Ted Shifrin

12:54 AM

@psa What's your question?

psa

Let C be the curve in the xy-plane that consists of the half-line y=−x, x≤0, and the half-line y=x, x≥0. Give an example of a vector-valued finction r:R→R^2 which parametrizes C and such that r(t) is differentiable for all t.

how is this even possible?

Ted Shifrin

Ah, note that it just says $r(t)$ is differentiable. Why do you think that's impossible?

psa

because that just looks like y=|x| !

how can you ever make that differentiable at 0?

Ted Shifrin

Yes, and there's a corner at the origin.

That function $f(x)=|x|$ is not differentiable, of course. But if you were driving a point-size car, couldn't you drive on that path?

psa

I'd have to stop for a moment

Ted Shifrin

12:57 AM

Aha.

So if you have zero velocity at the corner, there's no longer a contradiction, right?

psa

That's it? You just define $|v| = 0$ at the corner and you're done??

Ted Shifrin

Well, you still have to write down a function $r(t)$ that works.

psa

right then the issue does go away...

Ted Shifrin

In fact, you can have an infinitely differentiable function that parametrizes that path.

But they just asked for one derivative, right?

psa

I think that's implied yeah

so I should find a function that models those half lines but for which I'm stopped at the origin

so the obvious choice $t\mathbf{i} + |t|\mathbf{j}$ doesn't work as expected

Ted Shifrin

1:01 AM

Of course that doesn't work :P

psa

@TedShifrin still getting stuck. I have $\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} \to \mathbf{v}(t) = \frac{dx}{dt}\mathbf{i} + \frac{dy}{dt}\mathbf{j} = \mathbf{0} \to \frac{dx}{dt}\mathbf{i} = -\frac{dy}{dt}\mathbf{j}$ with $x(0) = y(0) = 0.$

that tells me nothing

but I know that I need $\mathbf{v} = \mathbf{r} = \mathbf{0}$ there

Ted Shifrin

You need y=|x|, so what can you do to your original parametrization to get zero derivative at the origin?

Don't be abstract. Be specific.

Bob

@TedShifrin thanks for your comment

Ted Shifrin

Mostly worthless, @Bob, but I answered :P

Bob

I find the price of some text books shockingly high

Ted Shifrin

1:15 AM

Oh, you get no argument from me. I fought on my second book with the editor to keep the price low, and after one year, it went through the roof. I complained, and he said his boss gave him no choice. ... At any rate, that's why my fourth book is free .pdf, despite the fact that 5 or 6 publishers have asked to publish it.

Bob

what is your free text book on?

Ted Shifrin

At the AMS Open Notes site you can find mine and lots of other books people have posted there for free.

It's differential geometry (undergraduate).

Bob

that subject is over my head

Ted Shifrin

LOL, maybe. The prereqs are sophomore multivariable calculus and linear algebra, not super fancy stuff.

Bob

maybe not then

no analysis as a prereq

Ted Shifrin

1:18 AM

And my 112 multivariable math lectures (for which the book is disappointingly expensive) are free on YouTube.

Nope, no analysis.

There is analysis stuff in the 112 lectures, though :P

psa

It just gives an impossibility. If $\mathbf{r}(t) = t\mathbf{i} + |t|\mathbf{j}$ then $\mathbf{v} = \mathbf{i} + \frac{t}{|t|}\mathbf{j} \to |\mathbf{v}| = \sqrt{2} \not= 0.$ I'd need another $t$ term for the $\mathbf{i}$ term in $\mathbf{v}$ and I'd need that term to be -1 when squared to cancel out the $(t/|t|)^2 = 1$ term.

Bob

I downloeaded the book

psa

Am I thinking TOO specifically now?

Ted Shifrin

@psa ... You have to use different functions, of course. What do you have to put inside the absolute value signs to get a differentiable function at $0$?

Cool, @Bob.

It may or may not be of interest to you. My feelings aren't hurt if it is not.

Bob

@TedShifrin I could not find anything related to finance math. I could not select that pull down. I am wondering if there is an issue with the site.

psa

1:22 AM

Do you mean the $t/|t|$ term?

Or just the $|t|$ term in $\mathbf{r}$?

Ted Shifrin

@psa. We know that $|t|$ isn't differentiable. I'm asking what function of $t$ you have to put in there to get something differentiable at $0$.

You mean the AMS site? I have no idea, @Bob.

psa

I'll have to think about that, but apparently it's obvious to you :P

Bob

@TedShifrin maybe the AMS site does not have any papers there on finance mathematics

anakhro

Hi.

Ted Shifrin

@Bob, that would surprise me, because it's a hot topic, but it's certainly possible, if every other drop-down works.

@psa, we already know we want the velocity vector to be $\vec 0$. What functions of $t$ have zero derivative when $t=0$?

Howdy, @anakhro.

psa

1:31 AM

there's tons

sin(t), tan(t), t^n (for any n), etc.

Bob

@TedShifrin thanks and good night

Ted Shifrin

Sure thing, @Bob.

@psa, well, some of your tons aren't right.

Care to be a smidgeon more careful?

psa

which? sin(0)=0, tan(0)=0, 0^n = 0 (except when n = 0)

Ted Shifrin

You didn't think about what we're talking about or read what I typed.

Jacksoja

2:26 AM

@LeakyNun Heya sorry did not see the message

@TedShifrin Hi Ted, how are you sir?

Leaky Nun

its ok

Jacksoja

How are you leaky ?

it has been a long time !

@LeakyNun do you know what are the prerq to study elliptic curves?

and projective geometry ?

Leaky Nun

no clue

Jacksoja

I would really like to learn some advanced topics

but feel like I dont have the basis to anything haha

Leaky Nun

get started and you'll know what you need

Jacksoja

2:29 AM

alright

but for algebraic

geometry

what I need is some knowledge of algebra right?

that course i have done, but what else? any idea?

@LeakyNun what are you doing ? playing chess?

psa

4:56 AM

@TedShifrin I did think about it, and I did read what you typed. You just asked for functions of $t$ that have zero derivative when $t$ is zero... There are lots. Any function that maps to $\mathbf{0}$ at the origin fits that category. I don't doubt there's something there that you're giving me (that I don't quite see yet), but you shouldn't assume that I'm not listening.

Any function that's constant when $t=0$ fits that category.*

Lukas Heger

6:56 AM

@Jacksoja Silverman/Tate doesn't have much prerequisites

It depends on the level you want to approach the subject at

Jacksoja

Hello thanks

I took some courses in analysis

and undergrad algebra so I did not have anything serious yet

But if I want to prepare for an intro level self study for these topics

1) Elliptic curves , 2) projective geo and 3)algebraic geo

Lukas Heger

For algebraic Geometry you want to know some commutative algebra

Jacksoja

What would you advice me to do ? k

I have a book for undergraduate

but it was too hard

by miles reed i think the name of the author

I want to learn some serious math on my own, and so far, i find it super hard to tackle many books

Any useful tip or comment are much appriciated !

@LukasHeger your logo looks like matheins, are you him ?

skullpatrol

in my opinion, if you want to be a completely independent self-studier; your only choice is to find a book at a level that you completely feel comfortable in trying to understand it and then read it and master it from cover to cover before moving up to the next level

Jacksoja

Yes i agree, am asking for such books

that sorta prepare me from 0 to that point of understanding the topics at hand

am intrested in Elliptic curves the most

skullpatrol

7:11 AM

go back through your school textbooks

Jacksoja

we do not cover this topic at universty

we did small part in crypto course

But It might be a good idea to ask on main, for recomendation of books that require so little background as possible

@LukasHeger that book you gave me is way too advanced

arithmatic of eliptic curves? H.silverman ?

skullpatrol

7:25 AM

why not continue on from the last textbook that you understood?

after all, there are no royal roads

Jacksoja

thanks that is a good advice

but i think i did not explain myself very well haha

I mean , am doing mathematics at the moment, but what I need is a certain topic

if i study different books , then i get better at those topics covered there

but there should be a road map to study particular topics that connect to EC

skullpatrol

why do you need to specifically study elliptic curves?

Adam

8:44 AM

Archer goes to Russia real Father episode

►

Leaky Nun

8:56 AM

Peter Svidler analyses Anand vs. Ding Liren, Norway Chess 2019

►

12:20 it goes Bb5, just to physically stop the b7 from its run

skullpatrol

are you a member of chess.se yet?

::checks leaky's profile::

:P

psa

9:19 AM

@TedShifrin I got it. I was overcomplicating it! Thanks.

Lukas Heger

1:35 PM

@Jacksoja sorry, I meant Tate/Silverman Rational Points on Elliptic Curves

Lukas Heger

1:48 PM

@Jacksoja yes, it's me, Mathein

skullpatrol

2:03 PM

aha! i new it :P

topologicalmagician

2:31 PM

Hi

Shine On You Crazy Diamond

@TedShifrin what name do you suggest for my math software (diagramming) . I'm coming up with a new design so far called Proof Adventurer y Math Explorer (PAyME).

topologicalmagician

nptel.ac.in/content/storage2/courses/111101002/downloads/… ... im not conviced that the equivalence relation in the first definition is an equivalence relation

Shine On You Crazy Diamond

@topologicalmagician you're right, it my not be that $x \sim y$

when $y \sim x$

unless you have a homeomorphism

topologicalmagician

I’m studying adjunction spaces and im having difficulties with the definition

Shine On You Crazy Diamond

Smoke an adjoint, that might help you think about it in a diff way

j/k

Smoking is bad for you. You should eat the cannabinoids instead

topologicalmagician

2:36 PM

Suppose X,Y are top spaces and A is a closed subspace of Y and $f:A \rightarrow Y$ continuous... let $\sim$ be the equivalence felation on the disjoint union $X \cup Y$ generated by $a \sim f(a)$ for all $a\in A$

Im not sure what is meant by “generated by....”

Lol, smoking is bad, i dont smoke, unfortunately :P

Shine On You Crazy Diamond

That's good, keep it that way :)

topologicalmagician

@ShineOnYouCrazyDiamond any ideas on the issue im having?

Shine On You Crazy Diamond

Yes

Given any subset of $A \times B$ you can generate an equivalence relation I think

it's a closure of the subset

that includes all required symmetries and etc

Via intersection of all equiv relations containing the subset

see here:

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c: a = a (reflexive property), if a = b then b = a (symmetric property), and if a = b and b = c then a = c (transitive property).As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they...

topologicalmagician

when Lee sais “generated by a\sim f(a)..$ he means by $\sim$, right?

Shine On You Crazy Diamond

Your subset is $R = \{ (a, f(a)) \}$ I think

topologicalmagician

2:40 PM

Hmmmmm

Shine On You Crazy Diamond

Then you take every equivalence relation containing $R$, the collection is nonempty, show that

And intersect all of them

That's too formal though, so read that article snippet for more useful ideas

topologicalmagician