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psa
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)
 
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?
 
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?
 
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
 
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??
 
Well, you still have to write down a function $r(t)$ that works.
 
psa
right then the issue does go away...
 
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
 
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
 
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
 
Mostly worthless, @Bob, but I answered :P
 
Bob
I find the price of some text books shockingly high
 
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?
 
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
 
LOL, maybe. The prereqs are sophomore multivariable calculus and linear algebra, not super fancy stuff.
 
Bob
maybe not then
no analysis as a prereq
 
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?
 
@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 $\mathbf{r}$?
 
@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
 
Hi.
 
@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
 
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)
 
You didn't think about what we're talking about or read what I typed.
 
2:26 AM
@LeakyNun Heya sorry did not see the message
@TedShifrin Hi Ted, how are you sir?
 
its ok
 
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 ?
 
no clue
 
I would really like to learn some advanced topics
but feel like I dont have the basis to anything haha
 
get started and you'll know what you need
 
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?
 
 
2 hours later…
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.*
 
 
2 hours later…
6:56 AM
@Jacksoja Silverman/Tate doesn't have much prerequisites
It depends on the level you want to approach the subject at
 
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
 
For algebraic Geometry you want to know some commutative algebra
 
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 ?
 
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
 
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
 
7:11 AM
go back through your school textbooks
 
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 ?
 
7:25 AM
why not continue on from the last textbook that you understood?
after all, there are no royal roads
 
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
 
why do you need to specifically study elliptic curves?
 
 
1 hour later…
8:56 AM
12:20 it goes Bb5, just to physically stop the b7 from its run
 
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.
 
 
4 hours later…
1:35 PM
@Jacksoja sorry, I meant Tate/Silverman Rational Points on Elliptic Curves
 
1:48 PM
@Jacksoja yes, it's me, Mathein
 
2:03 PM
aha! i new it :P
 
2:31 PM
Hi
 
@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).
 
nptel.ac.in/content/storage2/courses/111101002/downloads/… ... im not conviced that the equivalence relation in the first definition is an equivalence relation
 
@topologicalmagician you're right, it my not be that $x \sim y$
when $y \sim x$
unless you have a homeomorphism
 
I’m studying adjunction spaces and im having difficulties with the definition
 
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
 
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
 
That's good, keep it that way :)
 
@ShineOnYouCrazyDiamond any ideas on the issue im having?
 
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:
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...
 
when Lee sais “generated by a\sim f(a)..$ he means by $\sim$, right?
 
Your subset is $R = \{ (a, f(a)) \}$ I think
 
2:40 PM
Hmmmmm
 
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
 
What makes you say that that is my R? I thought the same but im not convinced
Yeah, i will
 
$R$ is a subset of (whatever) not the equivalence relation
 
@topologicalmagician in this case you identify $a$ with $f(a)$ and everything else doesn't get identified with anything
 
You take $R$ and generate the smallest equive relation containing $R$, call it $E$. It's always possible for any subset of "$A \times B$" where $A, B$ are (whatever) in your problem.
Reason it's always possible is because $A \times B$ is itself an equivalence relation containing $R$ so that the collection is non-empty. The intersection of an arbitrary collection of equiv relations is an equiv relation. Prove that
 
2:43 PM
When does you mean when you say identity? Because i thought that term is used while referring to quotient maps, no?
Hold on
I have no idea why i wrote when does
What do...*
indetify*
identify*
 
Dude, what are you typing lol
 
Sorry im on my phone
Lol
What do you mean when you say identify? Because i thought that term is used while referring to quotient maps, no?
 
Sure, you're quotienting by an equivalence relation, so you have a quotient topology and a quotient map
 
Reflexivity may not be there and symmetricity may not be there. Those are all you need to form an equiv relation, so as Alessandro says, you include $(x, y)$ whenever $(y, x)$ is there
Transitivity is not there either unless $f^2 = f$
 
2:47 PM
I would draw a finite picture and work it as an example to build intuition
 
Yeah im gonna do that
 
Take $f : X \to Y$ where $X, Y$ are finite sets given some course topology
 
Also is there a shorter proof you know of for attaching top manifolds along their boundaries? The proof in the book is literally the next page and is really long lol
 
Google probably knows
Yeah, I hate inelegant and long proofs
 
Can you see why if $Y$ is a singleton then this adjunction space is the same as the usual quotient identifying $A$?
What do you get if $Y$ is a singleton and $A=\{0,1\}$ is a closed subspace of $[0,1]$?
 
2:52 PM
I dont have paper at this time, and its hard to do in my head. As soon as i get back ill try to see... but right now, i have no idea
 
I think books are BS
Why read a book like you're living 200 years ago
it's no one's fault, it's just inertia of industry
And lack of proper software
I need to link with my brain and download
 
I wanna skip the chapter on adjunction spaces till a later time but alot of material in the following chaoters depend on them :/
 
Evening all
 
Hullo :)
 
3:02 PM
Are adjunction spaces called aomething else?
 
@topologicalmagician You're getting lost in technical details and missing the simple intuition behind this idea. You're literally glueing together two spaces
 
Hi @Alessandro
Hey @Lukas , how'd it go?
 
Hey @ÍgjøgnumMeg it went well I think
 
Niiiiice one
What did you have to answeR?
 
Some weird questions like what makes a good neighbour
 
3:08 PM
Hi @ÍgjøgnumMeg @Lukas
 
I said being quiet hahaha
 
What are you talking about?
 
hahaha nice
 
Ciao @Alessandro
Parliamo del mio esame di Inglese
Ho appena finito
 
That is correcto
 
3:10 PM
Ciao! Per caso sai rispondere a questa domanda? La trovo molto interessante
(I'll switch to English because it seems kinda rude toward ÍgjøgnumMeg otherwise)
 
Yeah wtf
also I feel somewhat pressured to change my name to my actual name
 
Is that actually the kind of stuff one does in an harmonic analysis course? Because I'm definitely taking harmonic analysis next term in that case lol
We already know your actual name anyway :P
 
yeah but now I'm the only one with a non-name
and it annoys everyone
rofl
Alright yo
wait
I changed it so it'll change at some point lol
 
Yeah takes a while to update in the chat
 
@AlessandroCodenotti non so la risposta, ma la domanda è interessante
 
3:14 PM
You might need to leave the room and reenter. Maybe clear the cache or something
You're the tech support guy, don't ask me lol
@LukasHeger is there a nice relationship between the spectrum of a C*-algebra and that of its dual? Because $M(G)$ is the dual of $C_0(G)$
 
I don't know much about harmonic analysis
 
lool oh well
 
I don't know anything about it, I only know a little about operator algebras and Pontragyn duality, but as soon as it becomes harmonic analysis I start guessing wildly
@ÍgjøgnumMeg uhm I guess it'll update eventually
 
Doesn't really matter anyway hehe
@TedShifrin will be overjoyed
 
Now almost everyone has a real name. Leaky is using an anagram, so that's close enough
 
3:28 PM
I refuse to believe Balarka is using his real name until he publishes a paper.
 
hahaha
I have so much pfand to awaybring
 
The money you get on the glass bottles is very low, so me and my flatmates always end up with a mountain of them, which then are too heavy to bring back to the supermarket
So they remain in the kitchen. And the mountain keeps growing
 
And to think I just recycle things for free. :(
 
Yeah I have two Kasten full of empty bottles and then like another Kasten worth of bottles sitting on my shelves rofl
 
At some point we end up bringing them to the glass container close to our place for free
 
3:37 PM
lol we always get someone with a car to drive us to Rewe
 
Makes sense
The plastic ones though are totally worth bringing back
 
Yeah defo
The first time I bought beer here I didn't know the word for "6 pack" so I accidentally bought an entire crate and then had to carry it for 2km on my shoulder
which really hurt hahaha
 
It's quite nice, my flatmates brings them back and gets cleaning stuff or similar that we all use with the money
 
that's a cool idea
 
@ÍgjøgnumMeg there's no word for 6 pack in German, you get the whole crate or nothing
 
3:39 PM
hahahaha
truth
 
I actually usually buy Scottish beer though, I think that makes me an heretic here in Germany
 
yeah definitely
There's actually a map of the German Bundesländer by favourite beer brand
 
What's that fireballer stuff in germany?
 
@ÍgjøgnumMeg urgh I don't want to see that
Pretty sure it'll be some kind of Kölsch for NRW which is awful
 
3:42 PM
It's Eichbaum here
which is great
 
It's actually Krombacher. That might even be worse
 
rofl
Oettinger in Bayern
which is Pißwasser
 
What did you expect?
 
In Austria it was Mohrenbräu which is amaaaaazing
 
Also what's wrong with people in Bremen
 
3:45 PM
Becks?
Rofl
 
Yeah, that's really atrocious
 
truly
@Leaky @Akiva @Lukas Half of all students of English transcribe "marriage" with a final [tʃ], so I guess it's true that 50% of marriages end in devoice
 
lol
 
laughs in autism
 
Now get someone with a boston accent to tell that joke.
 
3:54 PM
hahaha
divwoaaas
I only know the Boston accent from Good Will Hunting
 
4:24 PM
A+ pun
 
Hey Balarka.
What have you been up to?
 
Yo @Balarka
 
Oh real name
What is up with you all
 
Oh look my real name hath appeared
@Alessandro
 
4:39 PM
Now you also need to use your real name @Balarka
 
I felt peer-pressured
 
@Alessandro Can you index a collection of sets over an empty set
 
Now pinging you is actually harder @Edward!
 
hahaha
 
Very important question
 
4:40 PM
@BalarkaSen Your collection better be empty I guess
 
Say I look at the collection of subsets $\{A_\alpha\}_{\alpha \in I}$ of a set $X$ indexed over some arbitrary set $I$
I know for a fact that $A_\alpha = X$ for all $\alpha$
 
You can most definitely index over $\emptyset$.
 
Suppose I tried to compute $\bigcap_{\alpha \in I} A_\alpha$ as $\bigcap_{\alpha \in I : A_\alpha \neq X} A_\alpha \cap \bigcap_{\alpha \in I : A_\alpha = X} A_\alpha$
 
That's why we have empty intersection and union conventions.
 
Then I get $\bigcap_{\alpha \in I : A_\alpha \neq X} A_\alpha \cap X$
Which will be $X$, but how does $\bigcap_{\alpha \in I : A_\alpha \neq X} A_{\alpha}$ make sense?
 
4:44 PM
the empty intersection is $X$ by convention
 
Yeah so there isn't any precise way to do this
Foundational issues!!
 
@EdwardEvans what happened?
@BalarkaSen why are you dealing with set theory
 
Just picking on Alessandro
 
I suppose if you work in NBG set theory, you could say every empty intersection is the class of all sets, you can intersect a class and a set just fine
 
Ah I like that
 
4:47 PM
how about don't
 
@LukasHeger that's not a convention, it's a computation. $x\in\bigcap_{\alpha\in\varnothing}A_\alpha\iff \forall\alpha(\alpha\in\varnothing\implies x\in A_\alpha)$
 
1. SF 11 is out!
2. the SF 11 eval of the ongoing Anand - Carlsen game is currently +0.6 at depth 34
 
(which is problematic in ZFC because the intersection over the empty set becomes $V$ so we don't talk about it)
 
just work in the lattice P(S)
then the empty infimum is S
 
Why is it problematic in ZFC, @Alessandro
 
4:49 PM
We'd like intersections of a family of sets to be a set
 
@LeakyNun yeah I guess that works for all practical purposes
 
If there weren't problems in ZFC, then we wouldn't have to pay set theorists anything. It follows that every mathematician should work in set theory and clear out all the problems, once and for all.
 
@Alessandro So did I find a genuine foundational problem just in the process of writing a troll question
 
It's not really a problem, we just say that $\bigcap A$ is defined for $A\neq\varnothing$. You're fine in this case because everything lives inside $X$
The problem is when you try to define $\bigcap$ as a unary operation on sets, like $\bigcup$
 
Admit that set theory is bollocks
Admit it
 
4:51 PM
lol
@BalarkaSen I literally just posted a set theory question. It better not be
 
set theory doesn't know if set theory is bollocks
 
Quite the opposite, if ZFC is consistent, so is ZFC+not Con(ZFC)
 
that isn't the opposite
if ZFC is consistent then ZFC+Cons(ZFC) and ZFC+¬Cons(ZFC) are both consistent
so ZFC doesn't know if ZFC is bollocks
 
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