Mathematics - 2015-10-22 (page 1 of 3)

Balarka Sen

00:03

@TedShifrin $f(x, y) = x^2y^3/(x^3 + y^9)$, $f(0, 0) = 0$ works.

Oh. I've been stupid. A way more easier and systematic solution is $f(x, y) = 1$ when $y = x^2$ and $0$ otherwise. Very not continuous, and any line $y = mx$ through the origin will have a small nbhd on the line where $f$ is $0$, thus the derivative will also be $0$. Darn.

So much for nothing.

Ted Shifrin

00:25

@BalarkaSen You ignored my warning. Note that this is undefined on the curve $x=-y^3$. Not cool. Yes, your ultimate example is good, but you can fix the other one to make it work correctly :) ... On to unbounded after sleep. :P

Balarka Sen

I have solved unbounded.

Ted Shifrin

You're a very naughty boy, @Balarka. :D

Balarka Sen

$f(x, y) = 1/x$ for $y = x^2$, $x\neq 0$. $f(x, y) = 0$ otherwise.

Same argument shows dir. dervs all 0

But clearly unbounded on any nbhd of $0$

Ted Shifrin

Ah, what if I want discontinuous only at $0$? :D

Balarka Sen

@TedShifrin Yeah, good point. I guess I'd just need to get my exponents to be even. I solved an algebraic equation to get at that, I'll plug in some other value tomorrow. I don't care about that ugly example anyway - can't be seen geometrically.

Ted Shifrin

00:30

The intent of the problem, although I admit I didn't state it, is that I wanted the function good everywhere except at $0$.

Go to bed, now that it's morning, @Balarka :)

Balarka Sen

@TedShifrin yeah, don't think I can keep awake anymore. will think about the follow-up tomorrow.

Ted Shifrin

G'night, @Balarka. :)

Balarka Sen

Byes. And you mean G'morning.

Ted Shifrin

smacks Balarka

Julian Rachman

01:17

@Ted I have sent you an email.

0celo7

01:36

Hey guys, does category theory have any real world applications? Has it ever been used in an industrial process, etc.?

Karim Mansour

Hey @TedShifrin

next semester they are offering applied algebra or lie algebra

which one do you think I should take ?

This is a course in applications of algebra to a selection of topics concerning enumeration, coding, finite state machines and cryptography. *** Prerequisite: MATH 223 or 323 with a grade of at least 60% ***

this is the course description of applied algebra

skull petrol

01:52

your grades given as percents?

Karim Mansour

yh

I think most canada is like that

skull petrol

I see.

0celo7

02:13

@skullpetrol We get letter grades which correspond to percents.

Tien-Cheng Huang

02:28

Could someone list the topics of analysis beyond in R and beyond of functions of single real variable that a new math graduate student should be familiar with? Also could someone list the recommended/standard study materials for these topics? Thanks in advance! (redd.it/3ppxn9)

skull petrol

02:39

@0celo7 you mean you get letter grades that correspond to percent intervals, right?

0celo7

@skullpetrol On tests? No, there we get a percentage. Report cards are letter grades.

No one says "I got a B on that exam."

And "at least 60%" is "at least a D."

Because the lowest D you could get is a 60%.

skull petrol

I meant for the course @0celo7

According to Karim, in Canada they give percent values for the course.

Sanic Hodgeheg

Hello friends.

skull petrol

Hi pal

0celo7

@skullpetrol Well, I can tell you my grade in differential equations is a 100%.

And in physics 97%.

Sanic Hodgeheg

02:46

@0celo7 psh... humblebrag!

0celo7

@SanicHodgeheg my linear algebra grade is like 101% :D

skull petrol

Some places give out GPAs of 4.5

Karim Mansour

lol your lucky that your grades in term of gpa

for example the highest grade I have is 99 that doesn't round to 100

0celo7

@skullpetrol My high school GPA was like 4.4

out of 5

well, really out of 4

if you took nothing but AP classes all 4 years you could get a 5 I guess

but we didn't have AP PE, so dunno how that would work ;)

Sanic Hodgeheg

@0celo7 Have you not already learned pretty much all of ODE/undergrad physics/linear? Why are you retaking it?

0celo7

02:52

no, no, no

not retaking what I don't know

knowing some GR does not make me a god, idk why some of you think that

"you" includes physics chat

Karim Mansour

you took GR?

Tien-Cheng Huang

Could someone list the topics of analysis beyond in R and beyond of functions of single real variable that a new math graduate student should be familiar with? Also could someone list the recommended/standard study materials for these topics? Thanks in advance! (math.stackexchange.com/q/1491747/275935)

0celo7

@KarimMansour no

read some books

Karim Mansour

I see

nice

0celo7

I'm a freshman

Currently reading Arnold Mathematical Methods of Classical Mechanics to get some proof writing skills

Karim Mansour

02:57

I am gonna take it next semester although it will be taught by physicists not mathematicians so it will not be covered in general way.

Sanic Hodgeheg

@KarimMansour check his answers on Physics stack.

0celo7

the exercises are murder

@KarimMansour I just found out my school has a GR for mathematicians class.

Karim Mansour

I would rather go through actual proof book maybe munkres or something

that would be awesome to take

0celo7

@KarimMansour eh, doing topology in school in a year

I know some basics, not really motivated to read a book on it (on my own).

Arnold is nice because it's applied math, but still has theorems and whatnot

and lots of exercises

Sanic Hodgeheg

@TedShifrin I found a copy of your book at my school's library, so I'll send that e-mail asking for HW problems. Thank you!

Karim Mansour

03:00

cool

yeah I have heard about that book

I might read it next summer

I have it on my pc

0celo7

my reading list is 100 books and papers long

if I can touch it by the time I die, I'll be content

what is so special about his book?

@KarimMansour wait

Ted's book or Arnold?

Karim Mansour

arnold

0celo7

Ah, yes. I like it.

Some sections are terrible, the two lines on cohomology will confuse you if you've never seen it before.

@KarimMansour This theorem. This theorem makes the book worthwhile.

Karim Mansour

oh I see

PVAL

Fellow followers of Thoth. I am elated.

0celo7

03:09

...welcome

skull petrol

Yo

Galois in the Field

Does $V^{\otimes 0}=\Bbb F$ by definition? Is there a mathematical reason for this choice?

Sorry if that's a dumb question

0celo7

as someone who doesn't know algebra, what do you mean by $V^{\otimes 0}$

Galois in the Field

The $0^{th}$ tensor power(but that's kind of my problem haha)

Haven't really ever dealt with tensor products

0celo7

@GaloisintheField well that I could have told you

how are you defining the $n$-fold tensor product

Galois in the Field

03:20

$V^{\otimes n}=V\otimes V\otimes \cdots \otimes V$ with $n$ factors

0celo7

well, yes

what does the rhs of that mean

Galois in the Field

$\{v^{i_1}\otimes v^{i_2} \otimes \cdots \otimes v^{i_n}: 1\leq i_k \leq n\}$

0celo7

dude, stop answering "what is $\otimes$" with more $\otimes$

because now I want to know what $v^{i_1}\otimes\cdots$ means

Galois in the Field

I don't know, it's a connective symbol for constructing bases??

I don't believe its an actual operation

It's a way of constructing new vectorspaces from a product of old ones

@0celo7 That's the basis of the tensor product of vector spaces

0celo7

I know what it means...I was trying to do a socratic thing

only to discover my idea was wrong :P

ignore me!

Galois in the Field

03:25

Oh haha

That's alright, I thought it was an operation at first but couldn't find anything

Now I have no idea why a 0th tensor power is just the base field, although it seems intuitively right, I have no reason as to why

0celo7

probably a definition

the geometric definition does not make sense with $n=0$ either

Galois in the Field

For some reason I feel it should just be the trivial vectorspace

skull petrol

Take it as an axiom and move on :P

Galois in the Field

@skullpetrol That sort of action makes me sad :P

0celo7

@skullpetrol physicist!

son of a bricklayer!

Galois in the Field

03:28

I'll ask it on main, but I bet I will be downvoted

0celo7

ask a prof or grad student

or wait until someone smart gets here

cheerio folks

gotta sleep

skull petrol

Later pal

Galois in the Field

03:48

@0celo7 Cya later. Also I found my own logic for my problem

The $n^{th}$ tensor power of $V$ has dimension $(\dim V)^n$

So the $0^{th}$ tensor power should have dimension $1$, i.e. the vectorspace being the underlying field

skull petrol

What does your logic say about $0^0$ in the set of real numbers? @GaloisintheField btw nice question :-)

Galois in the Field

@skullpetrol Thanks xD. My logic says that we define $a^0=1$(which is also how we have defined $0^0$) but I am not fully happy with it, since $0^a=0$ for all $a$ except $a=0$ :P.

Alex Youcis

@GaloisintheField There is a very natural reason if you are working with, say, algebras over a field.

Galois in the Field

1

Q: $0^{th}$ tensor power, $V^{\otimes 0} = \Bbb F$, definition, or mathematical construction?

95% sure I will be told it's just a definition, move on etc: Is there mathematical reason why the $0^{th}$ tensor power is defined as: $$V^{\otimes 0} = \Bbb F$$ Where $\Bbb F$ is the field that vectorspace $V$ lies over. We define the $n^{th}$ tensor power of $V$ as: $$ V^{\otimes...

Alex Youcis

@GaloisintheField Does that mean you want me to answer it here.

there*?

Galois in the Field

03:56

@AlexYoucis Nono, I just posted it for the record

(obviously you can if you want haha)

Just maybe your answer is similar to Qiaochu Yuan's

Alex Youcis

@GaloisintheField It might be. I commented on Qiaochu's answer.

skull petrol

Proof by definition :P

Move on, don't waste time pal.

Alex Youcis

@skullpetrol I don't think that's a very enlightened approach. It certainly isn't something worth devoting copious amounts of time to, but it's still a reasonable question to ask.

Alias User

I read this quote in an introduction to probability paper: "Pair-wise independence does not generally imply mutual indepen- dence. For example, suppose that A1,A2,A3 are pair-wise independent, p(Ai) > 0 for all i and A1,A2,A3 =∅, then p(A1,A2,A3)=0 and the factorization rule does not apply" - if A1,A2,A3 =∅, wouldn't they not be pairwise independent?

Ohh - never mind, they're saying that the intersection of all three A1 is the empty set, not that they all equal the empty set.

skull petrol

Agreed @AlexYoucis :-)

Haven't seen you for awhile pal?

Anthony

04:11

Anyone here not horrible at probability?

@AlexYoucis You're at Berkeley right? I wonder if I've ever seen you...

Alias User

Anthony - I'm pretty miserable, but I'd take a look :)

Anthony

lol for some reason I have no intuition

if you pick two random numbers between 0 and 1, what's the probability their difference will be less than a half?

I realize there's some picture I can draw on $(0,1) \times (0,1)$, but I feel like there should be intuition that carries over from discrete probability.

In particular, I'd be interested in hearing the solution for the case of drawing numbers randomly from $0$ to $100$, and asking what the probability is that there difference is smaller than 50.

Alias User

That's actually a good problem - and my intuition is pretty shaky as well ;). If it's absolute difference, I would probably parameterize over the first random number x and find the probability given x what the value of y would be.

For instance, if x = 0, P(E) = 0.5, whereas if x = 0.25, P(E) = 0.75. If you plot P(E|x=k) over k, you'll get a distribution that has two lines meeting at a sharp peak at 0.5. Then multiply P(E|x) * P(x) to get P(E) - it doesn't have much more intuition, but I'm picturing one number varying along a line, and the odds at different points that the other number is close to where it is. Hope that helps!

Anthony