« first day (2735 days earlier)      last day (2582 days later) » 

18:00
@TedShifrin He told me that good job with the solutions, but some of my solutions can be shorter.
Well, cool, Karim
and are you comfortable with the idea that the same notion that you use for the $\liminf$ of a set of real numbers can be applied more arbitrary sets?
he seems like a nice guy
I think a college precalculus book would be fine, @Antonios. I personally am very fond of a book from 1908 or so called College Algebra, by Henry Burchard Fine. (I can't believe I remember that.) I bought it years ago in Dover.
$liminfA_{n} = sup_{m->\infty}inf_{n \ge m} A_{n}$, correct me if i am wrong
18:01
@EricSilva “asymptotics are awesome” is my slogan
It is weird because some of the research mathematicians I met were ***holes @TedShifrin
@Antonios-AlexandrosRobotis Just throw them in the deep end and make them read Felix Klein's "Geometry from an Advanced Standpoint" or whatever it is called
not all
For college calculus, the typical weakness that screws people up is basic algebra, @Antonios, as I'm sure you know, plus the ability to set up word problems.
lol @XanderHenderson
yeah exactly, @TedShifrin.
I'll look into it, thanks for the advice!
18:02
Karim: I don't want to burst your bubble, but (a) we're not all assholes and (b) Vakil probably didn't read your stuff thoroughly. :)
@XanderHenderson yes, it can be applied to them.
Both in the sense of “asymptotics are really cool” and of simple results pointing to deep math
Hi Semiclassic.
I screwed up basic algebra before I had a reason to care for it
Interesting how far you've come/gone, Balarka.
18:02
Hi @ted
You never learn what you are supposed to in the math class that you are taking; you are always learning the material from the last class.
given the definition of $B$, if some $x$ belongs in it, then $x \in A_{n}$ for some $n \ge n_{0}$
Well, people say that, @Xander, but I don't buy it.
@Ted True, but it makes me think most can go down my path in the right environment
Please have a look at this!
4 hours ago, by Silent
@AkivaWeinberger Yes! That's what we have to show that if $\gcd(b-a,n)=1$, then any transposition can be made simple by taking conjugations with $(1,2,\dots, n)$. The question actually arose from here
18:03
Not most, Balarka.
In a calculus class, the actual calculus is all easy examples, so you learn (or perfect, if you prefer) your algebra in calculus (because the algebra gets harder)
It’s the difference between learning and mastering a subject, I suppose
We've watered down college calculus immensely since the 60s and 70s, @Xander.
You don't really learn calculus until your first course in differential equations (for example), where you only see "nice" DEs, but have to work through some more difficult calc exercises.
@Antonios-AlexandrosRobotis I liked "Algebra The Easy Way" and its sequels on trigonometry and calculus back when I was young enough to like that sort of thing
18:04
@TedShifrin I have no objection to that statement.
They surrounded the math by this fantasy narrative for whatever reason
I don't agree with you, though. Depends on the definition of "learn calculus." Don't learn much in ODE.
Ohhh @Antonios-AlexandrosRobotis I remembered a book
@TedShifrin Before coming to Canada I barely knew any basic algebra it took me 6 month to learn algebra calculus and mechanics, and chemistry
One benefit of physics education though is that you have to get good at reading problems
18:05
Yakov Perelman, "Mathematics Can Be Fun"
But even then, I don't think that one ever really learns a topic until they see how it is applied somewhere else.
I think many people don't actually try
I swear that's the book that got me into math
help please
Perelman is an amazing writer
18:05
Calc for the sake of calc is unmotivated; students can learn to go through the motions, but they don't really internalize it until they have to work through problems in a broader context.
@XanderHenderson you don’t actually master it until then, perhaps
He also has some volumes of "cool physics" books
@Semiclassical Perhaps that would be a better word.
However, @Xander, I will say that I found increasingly that students in my senior-level diff geo course truly didn't know how to use the single-variable chain rule, basic multivariable calculus, or linear algebra. So students who earn C's (as opposed to A's) in standard calculus courses typically aren't prepared for much.
It depends what one is doing
18:07
@Silent If we have that $\gcd(b-a,n)=1$, then we have that $(b-a,2(b-a),3(b-a),\dots)$ is a full cycle. If our subgroup contains both that and $(b-a,2(b-a))$, the we can use the same proof that shows that $\langle(12),(12\dotsb n)\rangle=S_n$ to show that it's the entire group
@Shobhit: You're trying to define liminf of a sequence of numbers or of a sequence of sets?
It's the same thing but with relabeled elements
sequence of sets
Shouldn't you be using unions and intersections, then?
And as I explained before, the group generated by $(a,b)$ and $(1,2,\dots,n)$ contains both those permutations
18:08
If you’re doing plain vector calc for an electromagnetism , then the foundational aspects matter less
@TedShifrin Yeah, there is clearly an issue of preparation; but I think that I would phrase it in terms of a class ensuring that you are on the right track to mastering the material in the future.
Too many teachers teach/test just memorization of algorithms, and students retain/understand very little, @Xander.
for diff-geo, though, those foundational points become crucial
Not so much, Semiclassic. I'm not talking graduate manifolds. I'm talking curves and surfaces.
18:10
My notes don't expect a theoretical proof-based multivariable background.
More explicitly, if $h$ is the permutation that sends $x$ to $(b-a)x\pmod n$ (and note that this is a permutation iff $\gcd(b-a,n)=1$), then $(b-a,2(b-a),\dots)=h(1,2,\dots)h^{-1}$ and $(b-a,2(b-a))=h(1,2)h^{-1}$.
@Silent
I mean I mention the derivative as a linear map, but I use just standard science/engineering stuff.
This was my question @TedShifrin. The liminf is defined as union and intersections. let $(A_{n})$ be sequence of subsets of $X$. Define $B = \cup_{m \in N} (\cap_{n=m}^{\infty}A_{n})$. Then B is called the limit inferior of the sets $(A_{n})$. I know the definition of liminf, but i am not able to nderstand how is this the liminf.
I fall under the category you’re talking about unfortunately
It's a definition. So what do you mean you're not able to understand how it's the liminf. It's not the same as for sequences of real numbers.
18:11
Something's in $B$ if it's eventually in $A_n$ for large enough $n$, I guess
@TedShifrin Indeed. It is one of my biggest frustrations with the standard precalc and calc curricula that exist right now.
All problems are sorted into "types" that have a memorized algorithm for answering.
It's basically because most university faculty are lazy bums, Xander.
So for sets, this is the "definition" for liminf? @TedShifrin
Well, that, and there is a great deal of institutional pressure to (a) include the kitchen sink in the curriculum and (b) increase pass rates
Yes, @Shobhit. Now you should sit down and think about what it means.
How is liminf different from limsup?
18:12
@Shobhit Have you heard of the Iverson bracket? Essentially, it returns $1$ if something's true and $0$ if something's false. So $[4<5]=1$ but $[\pi\in\Bbb Q]=0$.
@XanderHenderson yeah, that’s pretty universal
$[x\in B]=\liminf_{n\to\infty}[x\in A_n]$ for all $x$.
@TedShifrin you should tell me if you come to Canada one day
You could take the special case, @Shobhit, of a sequence of real numbers and let $A_n = \{a_k: k\ge n\}$. Then is this the usual notion?
@Semiclassical Yarp.
Okay, I really need to get some work done. Laters.
18:13
That's the conection to the idea of "lim inf" in the real numbers.
DogAteMy: liminf and limsup of sets are defined in total generality in any set, so what you wrote doesn't make sense.
There’s pressure on the intro physics course aimed at premed students to both lower the credit load AND include more stuff
@TedShifrin Why not?
How is your RHS defined?
Which would be funny if it weren’t so frustrating
18:14
I mentioned I was using the Iverson bracket
I have no idea what that is.
1 min ago, by Akiva Weinberger
@Shobhit Have you heard of the Iverson bracket? Essentially, it returns $1$ if something's true and $0$ if something's false. So $[4<5]=1$ but $[\pi\in\Bbb Q]=0$.
Oh.
Ah, that's interesting
The main thing I associate Iverson with is exchanging the order of (finite) summations
@Semiclassic: I am capable of covering a lot of material and having a good sense for what's important and what isn't, but even in the AoPS classes I've been finding they cram in way more than I could possible cover in the allotted time (if I expect the students to actually work out stuff themselves).
This is the first time I've ever seen Iverson. Weird.
18:15
[Trump is the president of united states] = [We are doomed]
I'm not sure that's IFF, Balarka.
no no no no
liminf is the supremum of all infimums of $A_{n}$ as n increases, and limsup is the infimum of all supremums of $A_{n}$ as n increases. I have written this very badly, but i think i know what they are. @TedShifrin
It also comes up in $[x\in A]+[x\in B]=[x\in(A\Delta B)]\pmod2$, where $\Delta$ is the symmetric difference. But I've never seen that written anywhere
18:16
@Ted i suppose you are right
The only context I've ever actually seen it used was what Semi mentioned
[Half the US voted for Trump]
But figure out what it is for sets, @Shobhit. What you're writing still only makes sense in $\Bbb R$, basically.
That’s the correct LHS
(and I think that's the reason it was invented) (re: summations)
18:17
our algebraic topology lecturer wrote down the exact sentence “$\pi_1$ is a functor from the category of pointed topological spaces to the category of groups” along with the quotation marks
Hi, demonic @Alessandro.
@BalarkaSen
What is free probability @Semi
@Leaky It is true
Just had my functional analysis exam today and I went to book some driving lessons afterward, beware :P
18:17
There's a typo in that sentence, Leaky, Balarka.
ROFL, Alessandro. I'm very skeered.
fixed
I guess you didn't fail the functional analysis. :)
@TedShifrin or->of?
@EricSilva tbh I dunno but it has to do with noncommutative stuff
theres still a typo but who cares
18:18
No, there was originally an extra "category of" or something, DogAteMy.
@BalarkaSen the fact that he placed it in quotation
Lol I see
Heya Eric.
Nope, it went very well (today I had the oral part of the exam, I did the written one last week)
@TedShifrin hints please
18:18
@Leaky It is true
@TedShifrin I like oral exams
Hey @Ted
@Shobhit: Try examples!
@EricSilva Terry Tao has a section of his notes devoted to it: terrytao.wordpress.com/2010/02/10/245a-notes-5-free-probability
@TedShifrin I think the best way to learn is to have 1 final which is composed of 1 oral exam and 1 written exam
18:19
@TedShifrin going by the defintion it means if an element belongs in $B$, then it belongs in all but finitely many $A_{n}$
Try examples where the sets are nested and example where they're not necessarily contained in anyone ...
then you can just stay at home and learn things by yourself
I learned that most of my students don't learn so much by themselves ... or understand far less than they think they do.
It depends for me I learn much better by myself than going to classes
OK, whereas limsup would be what, @Shobhit?
Karim: That's only you judging what you've truly understood. :)
18:21
@Adeek that's how most courses work here. I prefer the oral exams usually
Ohhh I should check it out then, I love Taos writing
Ya
It gets mentioned a lot in the context of random matrix theory
@TedShifrin I think maybe a professor you should discuss with. I took higher level mechanics class as reading course, but ofcourse I had to write final and midterm like other students. I got actually 95 % the professor told me it was the 2nd highest grade he gave for the past 20 years.
@TedShifrin "you know less than you think" are words to live by I think
@TedShifrin limsup is also given in the exercise, with the union and intersection symbols exchanged with one another.
18:23
I think probably the best way to learn is something like apparentice type teaching
where you discuss with professor and read and solve books by yourself
Right, @Shobhit. So what is the limsup?
An element will be in the limsup set, if it belongs to finitely many $A_{n}$. @TedShifrin
sorry
Karim: If you truly learned better just by yourself, then why would you waste the 112 hours watching my lectures?
infinitely many*
OK, cool, @Shobhit. I think you understand now :)
18:25
@TedShifrin I never took multivariable analysis. Plus, your specail case you explain things very conceptually :D
@Ted What's the extra geometric structure provided by flowing along a Hamiltonian vector field over just flowing along a symplectic one?
ty :)
Say $A_i$ is $[0,2]$ for $i$ even and $[1,3]$ for $i$ odd
112? Is that a year long course?
Yes, year-long, @Alessandro.
18:25
last question, in your display picture, what is that figure? @TedShifrin
Ugh, Balarka, I don't remember.
I have met only 3 professors who explains things as conceptually as you @TedShifrin
(Cont'd) I think the lim inf will be $[1,2]$ and the lim sup will be $[0,3]$
That's for you to figure out, @Shobhit.
18:26
@Shobhit That's a puzzle :P
@TedShifrin Shucks. Thanks for having a look, though.
Maybe I should reiterate the definitions?
I have to leave in five minutes, Balarka. I don't use this stuff enough to remember definitions.
ok :O
I'll be back later.
18:27
Cya
i have met no professors who explain as good as @TedShifrin
Personally, I try to always imitate his teaching whenever I am doing something
whenever I am teaching someone something
Thanks for the kind words, folks.
I'm off to see a movie. Bye.
@TedShifrin For example, I took your way of explaining vector subtractions
haha
cya :) have fun
Enjoy @Ted
18:28
bye @TedShifrin
I am also off to work
cya guys
Cya pal
Which movie? And bye! @Ted
Hi all; I'm stuck on the following problem:
I've tried a bunch of tactics, including
@Astyx j'ai un espace $L^{\Phi}(\Omega)$ avec $\mu(\Omega)=\infty$ le but est de montrer que cet espace est un espace de Banach, donc je prends une suite de Cauchy $(f_n)\subset L^{\Phi}(\Omega)$, j'ai une proposition qui dit que si $X$ est borné $L^{\Phi}$ s'injecte de maniére continue dans $L^1$
18:33
First I began by rewriting ${ T }_{ n }$ as ${ T }_{ n }=n(n+1)$, using the formula for the sum of consecutive integers. I also found some symmetry in the fractions, such as:$\frac { 1 }{ 2 } +\frac { 1 }{ 2*3 } +\frac { 1 }{ 3*4 } +\frac { 1 }{ 4*5 } $, but I'm not sure how to use that. I also tried adding the first and last terms, but that doesn't work either.
Using 0,3,4,7,8 we create two numbers.
From these two numbers we create the smallest difference and the greatest sum.

Could you give me a hint?
$\frac{1}{2*3} = \frac{1}{2} - \frac{1}{3}$ @DarkRunner
I thought it didn't work since we're adding an odd # of integers, however, when I tried to see if ${ T }_{ 1 }+{ T }_{ 44 }={ T }_{ 2 }+{ T }_{ 43 }$, it turns out it isn't equivalent
write all like that, they will cancel each other
@Shobhit Wow; How did you notice that?
Thanks for the help
18:35
experience i think haha
np :)
@MaryStar smallest difference, greatest sum? what do you mean?
@Shobhit As far I understood that, we create so the numbers that we have the smallest difference and the greatest sum. Or is this not possible?
that is certainly possible.
for smallest difference - the smallest can be 1 and can be achieved by taking 3 and 4.
did you mean 5 digit numbers?
@Shobhit yes
repetitions allowed?
I think each digits hould be different.

We have two numbers of the form: abcde and ABCDE.
We get the greatest sum if we take twice the numbers 87430 , right?

The smallest difference is when we subtracht the bigest number from the smallest, ie. 03478 - 87430, or not?
@Shobhit
or is the smallest difference meant to be 0, instead of a negative number? @Shobhit
18:53
hey I want to calculate P(X>1 | Y=y) in terms of Q function for a standard bivariate normal. I know that for P(X>1) the answer is Q(1) based on the definition. But I don't know how to incorporate the condition
@Alessandro congrats!
19:11
@MaryStar sorry, had some work. Can you put the exact question (as given in the book) here?
That would be great if someone can spot the flaw in this one
1
Q: Computing error limits with total differential.

Tuki Problem Tower height is measured with angle measurement from two points $A$ and $B$ which are in same same direction (relative to the tower). Measured angles are $50\pm1$ degree,$35\pm 1$ degree and length between points $A$ and $B$ is measured $100\pm1 $ meters. What is tower height and how l...

Thanks @Dami I'll probably stop bugging you with functional analysis questions now :P
Let G be a bipartite connected simple graph. Is it true that if G is Hamiltonian then there exists a perfect matching in G? How does one explain it?
19:29
@AlessandroCodenotti lmao, so you're not taking functional analysis 2: electric boogaloo?
not the bogaloo meme again
it's a classic
well-aged meme
It's one of those memes that kinda dies but maintains a moderate showing until the fact that it keeps being used becomes the meme
And then just induct
i guess only 90s kids appreciate the meme
3old5me
19:42
Is Stein, Shakarchi 's Real analysis book a grad level text?
@Daminark maybe next year
This category theory questions isn't is actually quite easy once you understand all of the components that your trying to prove just bunch of tricks @MatheinBoulomenos
I didn't understand the diagram correctly with this Yoneda's lemma, but after understanding it. It is just basics
@PVAL-inactive advanced undergrad
I might have a legit undergrad to mentor
He claims to have worked through Artin and parts of that and middle Rudin.
Claims?
19:48
I don't know I didn't see the work.
@PVAL-inactive yeah it's still good work if an undergrad goes through that
It's easier then their Complex book right?
Or should I be intimidated?
20:04
I never did their complex book. I think it’s an easier measure theory books than most others. I used it when I was an undergrad
i have a quick math question is it ok if i ask here?
In solving the nonhomogenous heat equation with source term f(x,t) and homogenous Dirichlet boundary conditions. How come many books just say that a basis is the Fourier sine series. They say that this becomes clear if you fix t this becomes clear. But it's not.
Separation of variables becomes really tricky if not impossible when theres a source term
Hello )
:)
20:17
@Adeek you're right, once you understand what is going on, it's easy
Let $\mu=\frac{1}{2} (L^1_{[-1,0]}+\delta_2)$ is probablity measure. What is the cumulative distribution function?
@hola I thought the idea there was to write the source term in a Fourier basis as well?
@MikeMiller gchat's not working for some reason. Emb(D^2, Cyl) is just homotopy eq to Z/2 by disc theorem, isn't it
Well it gives an isomorphism in pi_0 idk about other things
20:32
@Astyx vous etes là ?
 
1 hour later…
21:50
So this guy has edited his comment on one of my posts multiple times trying to get me to select his answer as the answer.
I’m choosing to avoid a response.
I find it interesting and weird that he keeps editing it after so long
I'm struggling with a few problems;
I tried breaking it down to find some sort of symmetry, like:
So you essentially want to find $\sum n(n-1)$
$\frac { 2 }{ 2018 } +\frac { 6 }{ 2018 } +...+\frac { n(n+1) }{ 2018 } $, where n is the term #
@AkivaWeinberger Yeah
But I basically can't find anything worthwhile; I know the property that $\frac { 1 }{ ab } =\frac { 1 }{ a } -\frac { 1 }{ b } $, but that doesn't help me here
I believe $\sum(n+1)n=\frac13(n+2)(n+1)n$, but I don't know a good way to motivate that
Numerically it works though
@AkivaWeinberger I don't understand how you came up with that
As in, how would you derive that/know it's true?
22:01
I vaguely remembered it from somewhere
and just tried a numerical example to make sure I got the details right
Although-- I think you can do something with Pascal's triangle here.
The bad thing is I don't even have a solution/answer key
See my answer here:
6
Q: Calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$

I.GandakovHow can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?

Note that that question is asking for $\sum\frac12(n+1)n$
In the end I get (D) 2690
Sorry, but how did you get that? If you could clarify please
Assuming for the moment that $\sum(n+1)n=\frac13(n+2)(n+1)n$ is in fact true
then we have $\dfrac{\frac13(2019)(2018)(2017)}{2018}$
or $\frac13\cdot2019\cdot2017$
and 2019/3=673 apparently
so $673\cdot2017$
and then the question wants us to write it as the sum of the two factors
so $673+2017=\boxed{2690}$
So you just break it down into a prime and a factor
22:08
@DarkRunner Googling reveals that both 2017 and 673 are prime numbers so that's the only way you can factor the number
Ah ok Thanks @AkivaWeinberger
I get it now
22:26
For $G$ any group and $g\in G$, and for $m,n\in\Bbb Z$, does $g^{mn}$ only depend on $g^m$ and $g^n$?
Should be yes, actually. $m$ and $n$ are uniquely determined modulo the order of $g$.
@AkivaWeinberger could you make that statement more precise thanks
If $g^m=g^a$ and $g^n=g^b$ then $g^{mn}=g^{ab}$
for $a,b,m,n$ integers and $g\in G$
Fixed
So $g^{nm}$ only depends on $g^n$, $g^m$, and $g$
which is in fact trivially true
@LeakyNun
I see
The Diffie–Hellman problem (DHP) is a mathematical problem first proposed by Whitfield Diffie and Martin Hellman in the context of cryptography. The motivation for this problem is that many security systems use mathematical operations that are fast to compute, but hard to reverse. For example, they enable encrypting a message, but reversing the encryption is difficult. If solving the DHP were easy, these systems would be easily broken. == Problem description == The Diffie–Hellman problem is stated informally as follows: Given an element g and the values of gx and gy, what is the value of ...
^The reason I was thinking about that
$g\in\Bbb Z_p^\times$ usually
for a very large prime
actually, $g^{mn-ab} = g^{(m-a)b+m(n-b)} = (g^{m-a})^b (g^{n-b})^m = e^2 = e$
22:39
or actually it could be other things as well nowadays but in the original formulation it was just $\Bbb Z_p^\times$ I think
@LeakyNun Mhm
Yeah it was the original solution to the key distribution problem in cryptography, before RSA was discovered
You and I agree on a large prime $p$ and a primitive root mod $p$ called $g$
@MatheinBoulomenos haha yeah I guess this is true for everything lol. Once you understand what is happening then it is easy lol.
You choose a number $a$ and keep it secret; I choose a number $b$ and keep it secret.
At this point, $p$ and $g$ are not secret since we have to tell them to each other over nonsecure communication lines
You tell me $g^a$ mod $p$ and I tell you $g^b$ mod $p$
so those are no longer secret either
but it's really hard to find $a$ from $g^a$ so $a$ is still secret, similarly for $b$
You calculate $(g^b)^a$ and I calculate $(g^a)^b$
and we both end up with $g^{ab}$, which we can then use as a key for some cypher
Diffie–Hellman key exchange (DH) is a method of securely exchanging cryptographic keys over a public channel and was one of the first public-key protocols as originally conceptualized by Ralph Merkle and named after Whitfield Diffie and Martin Hellman. DH is one of the earliest practical examples of public key exchange implemented within the field of cryptography. Traditionally, secure encrypted communication between two parties required that they first exchange keys by some secure physical channel, such as paper key lists transported by a trusted courier. The Diffie–Hellman key exchange method...
^Explained graphically as a way to share a secret color of paint
23:23
hey akiva
maybe I am blind
but what fact are they is the exercise refering to ?
@AkivaWeinberger
@Adeek what is (a)?
a is yonneda lemma
"corresponding fact" means something modified from something mentioned before
i.e bijection between natural transformations and morphisms
sure I will upload it
c is yoneda lemma again?
23:25
normally people call a Yonneda's lemma because that's where most of the work lies
but what is (b)
I proved (a)
I don't understand (b)'s statement ?
(b) Suppose $A$ and $B$ are objects in a category $\scr C$. Given a bijection between the natural transformations $h_A \to h_B$ of contravariant functors $\mathscr C^{op} \to {\bf Sets}$ and the morphisms $A \to B$
Well, a bijection between those and something, at least
and the morphisms $A \to B$
isn't that a?
I guess arrows are switched
@Adeek that's what "corresponding" means
it means "slightly modified from what is previously mentioned"
23:32
oh
okay
i.e. (b) is the version of (a) with covariant functors replaced with contravariant functors
sorry I am blind sometime
I see
yeah I see now sorry
no problem
23:53
Yoneda's embedding "contravariant it is"
hahaha
What sort of things do you learn in an undergraduate course on combinatorics?
@AkivaWeinberger I never liked combinatorial thinking so I never took a course in it.
I guess you can say basic abstract algebra is combinatorics
Yeah, it's not my favorite branch either
I like structural thinking or visual thinking or analytic
I don't like counting
I hated group theory

« first day (2735 days earlier)      last day (2582 days later) »