Mathematics - 2018-01-30 (page 4 of 5)

Akiva Weinberger

19:00

@MatheinBoulomenos What is this about?

Counting primes…?

PVAL-inactive

I'm sure theres lots of tactics in mental arithmetic that I don't know that would improve my arithmetic substantially, and it'd be a complete waste of time to learn them (really at any stage of my life)

MatheinBoulomenos

Analytical and algebraic number teory

Shobhit

when i was small i was boggled when a neighbour's kid started counting his fingers in opposite 10,9,8... For first five fingers in first hand he said 10,9,8,7,6 and 5 fingers in the other hand. 6+5 =11 and therefore we have 11 fingers :O

MatheinBoulomenos

it deals with qudratic forms over Q, mostly

Ted Shifrin

I'm not talking crazy idiot savant things, PVAL. I'm not that good at mental arithmetic.

PVAL-inactive

19:01

I don't know even for 72-39 I don't naturally have any tricks

Akiva Weinberger

I once stunned an auditorium by knowing sqrt(5776) off the top of my head

but that was just a lucky thing since I already knew that 25^2=625 and 76^2=5776 were the only two-digit numbers whose squares ended in themselves

MatheinBoulomenos

I'd say there is no point at all in a speed competition on matrix calculations. You should be able to them in reasonable time, yes. It's better to test some understanding of the concepts. And if you care about applications, it's better to learn how to implement this stuff on a computer

Ted Shifrin

One of my AoPS students knows all sorts of esoteric things I don't know. I don't say we should all be good at competition math.

PVAL-inactive

ah mental linear algebra is probably more important than mental arithmetic to me.

Alessandro Jacopson

Hello, Donald Knuth wrote in his The Art of Computer Programing, Vol 1, about his notation for Fibonacci numbers "The notations we are using in this section are a little undignified. In much
of the sophisticated mathematical literature, $F_n$ is called $u_n$ instead, and $\phi$ is
called $\tau$." Why does he define is notation "undignified"?

Ted Shifrin

19:03

@Mathein: I made sure my test questions came out simply in very few steps. Occasionally, I would give the echelon form and they only needed to do one step to get reduced echelon form to read off the kernel. Again, I don't object to using technology for any number of things once you have the basic skills. But I insist that you acquire those first.

He's being funny, @AlessandroJacopson.

Akiva Weinberger

I don't get it

Ted Shifrin

I guess I've always used the unsophisticated notations.

PVAL-inactive

He's being $F_un$ny

Tuki

but i don't see how computing inverse matrix of 5x5 is necessary to do by hand ?

i terms of learning

MatheinBoulomenos

@TedShifrin I said that you should be able to them in reasonable time, yeah

Ted Shifrin

19:05

I would never ask students (or me) to do that, Tuki.

Alessandro Jacopson

@TedShifrin Thank you, I am Italian and maybe I am missing something in his English.

@PVAL-inactive :-)

Ted Shifrin

Wow, we have two @Alessandros coming to chat now :P

MatheinBoulomenos

We had to compute the Smith normal form of a 7x7 matrix with coefficients in $\Bbb F_{11}[x]$ or something like that. We had no time pressure, this was on a homework sheet

Tuki

We have had tests that had this type of question

PVAL-inactive

@MatheinBoulomenos This sounds like a programing problem.

Ted Shifrin

19:06

Well, I think I give fairer (and more interesting) tests than that.

Tuki

This is partly the reason i don't like exams very much

Akiva Weinberger

The most efficient way to do that is to write $[X|I]$ and reduce I think

Like that's how computers do it

Ted Shifrin

Although my students often whined at the time, in hindsight they all said my tests were fair and not that hard.

Yeah, DogAteMy, that is.

In general.

PVAL-inactive

if its big there are other tricks probably

row reduction is pretty cost inefficient.

MatheinBoulomenos

In our exam, we were given the Smith normal form of $XI-A$ with base-of-change matrices and we were asked to compute the Jordan normal form of $A$ from that

PVAL-inactive

19:08

especially if you only need an approximate answer.

MatheinBoulomenos

I thought that was reasonable

Tuki

someone told me that LU decompositon is also good for this use ?

Daminark

Rehi everyone

MatheinBoulomenos

Rehi @Daminark

PVAL-inactive

Re: Rehi everyone - Hi

Eric Silva

19:10

@Ted i cant think of an example for the sectional thing but im brain dead and have lab today so i will try to think of one later

Akiva Weinberger

@Dam, said Amsterdam

Ted Shifrin

LOL, ok, @EricSilva.

Daminark

@Akiva kek

How's it going @PVAL?

Ted Shifrin

Oh, maybe a torus in $S^3$?

Not the Clifford tori, of course.

Daminark

@EricSilva "have lab today" I am sorry. I will say goodbye to the sun for you while it's still up

Eric Silva

19:12

@Daminark sobs uncontrollably

Ted Shifrin

Too many drama queens.

PVAL-inactive

What about the boundary of a tubular neighborhood of the whitehead double of the (3,-5,7) pretzel knot? @Ted

Ted Shifrin

Huh?

Daminark

Though if you're in E&M at least the labs are far more interesting than those in mechanics so long as that oscilloscope or whatever it's called isn't going haywire

PVAL-inactive

That's a torus in S^3

Eric Silva

19:13

the oscilloscope was going crazy last time

lol

Ted Shifrin

With everywhere positive curvature, PVAL?

Daminark

F

PVAL-inactive

Does the torus even admit an every positive curv. metric?

Ted Shifrin

Oh, duh, that was dumb.

Akiva Weinberger

If you have a torus in a very hyperbolic space can you get it to be positive everywhere?

Ted Shifrin

19:14

No, no ... Gauss Bonnet.

PVAL, did you see my original question?

PVAL-inactive

You want a product manifold with everywhere pos. sectional curvature?

was that the question?

Eric Silva

pls no spoilr

Ted Shifrin

Right. The Hopf conjecture is that there is one on $S^2\times S^2$.

MatheinBoulomenos

I think the Hopf conjecture is that there is none on $S^2\times S^2$

Eric Silva

well the conjecture is that there isnt one

yeah

Ted Shifrin

19:16

Oh, sorry.

Eric Silva

cause no one believes that there is one

Akiva Weinberger

Elon Musk has sold 15,000 flamethrowers online at $500 each to raise money for a company that has nothing to do with flamethrowers

I'm still having trouble believing this is real

Ted Shifrin

Right. I don't see any product example with nowhere vanishing sectional curvature.

Shobhit

atlast, a joke i can understand, lol @AkivaWeinberger

Eric Silva

at least it would be weird to me if there were considering $\mathbb{R}P^{2} \times \mathbb{R}P^{2}$ doesnt have one

Ted Shifrin

19:18

I'm trying to think about inversive geometry and writing some exercises for my class for the weekend. So I'm distracted.

Eric Silva

oh inversive geometry that's coooool

Ted Shifrin

They're making me do Möbius transformations, so I want to give them something geometric ...

Akiva Weinberger

Peaucellier–Lipkin linkage

The Peaucellier–Lipkin linkage (or Peaucellier–Lipkin cell, or Peaucellier–Lipkin inversor), invented in 1864, was the first true planar straight line mechanism – the first planar linkage capable of transforming rotary motion into perfect straight-line motion, and vice versa. It is named after Charles-Nicolas Peaucellier (1832–1913), a French army officer, and Yom Tov Lipman Lipkin (1846–1876), a Lithuanian Mathematician and son of the famed Rabbi Israel Salanter. Until this invention, no planar method existed of producing exact straight-line motion without reference guideways, making the linkage...

^That thing has to do with inversive geometry apparently

PVAL-inactive

@Ted Do you want the metric on the product to split?

MatheinBoulomenos

Aren't Möbius transformations geometric?

Akiva Weinberger

19:20

Ted Shifrin

Sure, DogAteMy ... Inversion will in general turn lines into circles :)

Akiva Weinberger

Though ^that's the simplest straight-line mechanism by far

Ted Shifrin

@PVAL: IF it splits, then of course you get some sectional curvatures 0.

Akiva Weinberger

Ya gotta think outside of the box (or plane) sometimes

PVAL-inactive

oh i see

Ted Shifrin

19:21

@MatheinBoulomenos: If you do projective geometry, of course. But in this elementary course, we're only doing a few ad hoc things.

Eric Silva

young pups dont need any fanciness

Daminark

@AkivaWeinberger this makes me dizzy

PVAL-inactive

I'm trying to remember any manifolds which are a product but not obviously.

Akiva Weinberger

Too late for me to edit it, sorry

Ted Shifrin

I am enamored of the inverse form of the triangle inequality (Ptolemy's theorem).

Akiva Weinberger

19:22

If it gets starred, will it show up as a gif in the sidebar?

Ted Shifrin

LOL

MatheinBoulomenos

it will be a link

Ted Shifrin

Maybe if we make Demonark dizzy enough he'll stop his relentless ill-punnery.

Actually, he's been sorta dull the last few times I've seen him.

MatheinBoulomenos

I think it's weird how some trivial answers get a lot of upvotes, but answes that took a lot of effort barely ger any upvotes

Daminark

@TedShifrin shrug

IRL I make more puns for the most part but also I'm just too tired for wit, if we will be so generous as to call it that

PVAL-inactive

19:24

I wrote like a 5 character answer that got 3 or 4 upvotes and got accepted.

Ted Shifrin

@Mathein: I've complained about that for years. For me the only one that took serious thought that got lots of upvotes was the answer to Zev's Thurston question. But numerous others get hardly a vote.

Akiva Weinberger

@MatheinBoulomenos Accessibility.

If it's on a topic not a lot of people know about, most people won't read it, let alone answer it. On the other hand, if you correctly state that 4+4=8, that's accessible.

MatheinBoulomenos

math.stackexchange.com/a/2624012 it took me quite some time to come up with that stuff. But I might submit one of the lemmas to the stacks project, so there's that

Akiva Weinberger

My most upvoted answer is this, and it's something anyone here could have written.

Ted Shifrin

The vector bundle PDE question I just answered, finally, took hours and hours and hours. But no one's gonna notice.

Yikes. That answer is so long I'd never read it, Mathein.

Eric Silva

19:28

that answer is a book

Daminark

I think my most upvoted answer was on a question about fake proofs

My answer was the classic fake proof of Cayley-Hamilton and it got over 10 upvotes I think

MatheinBoulomenos

The question was really vague, about "characterizing" certain rings, whatever that means

Hi @Tobias

Tobias Kildetoft

@MatheinBoulomenos Hi

Daminark

Hey @Tobias!

Tobias Kildetoft

hi

Ted Shifrin

19:29

Heya @Tobias

Eric Silva

@TedShifrin at least you know it's a good answer i guess

MatheinBoulomenos

@TedShifrin basically I'm giving criteria for when the Zariski topology on an affine scheme is Noetherian (ignoring the sheaf)

Kasmir Khaan

@MatheinBoulomenos @TedShifrin@TobiasKildetoft hi :D

MatheinBoulomenos

@KasmirKhaan hi

Ted Shifrin

Well, I actually sent Rafe the link cuz he had a murky PDE argument I didn't follow. @EricSilva

Kasmir Khaan

19:30

:)

Ted Shifrin

hi Kasmir

Eric Silva

what was the idea @Ted

Tobias Kildetoft

Tomorrow I will have a meeting with the external evaluator. Then we will see if he agrees with me or if I have been too nice in my grading.

Ted Shifrin

@EricSilva: Here's what he wrote me. "The general theorem that if E is a C^\infty bundle which carries a partial connection \dbar
which satisfies \dbar^2 = 0 then E is holomorphic is a softer version of Newlander-Niremberg —
it is not quite so coupled since E is not the tangent bundle. The proof is a bit of PDE,
and I really think it cannot be reduced to Frobenius. The idea is simply to produce a local
frame of holomorphic sections. If E has rank k, then first write down k independent smooth sections, say

Wow, @Tobias. Is this just 'cuz it's your first time lecturing there?

Eric Silva

hrmmm

Kasmir Khaan

19:34

So what is the idea of algebraic numbers

Tobias Kildetoft

@TedShifrin No, most courses require an external evaluator (or examiner if the exam is oral)

Kasmir Khaan

are numbers divided into algebraic and transcental?

-__-

Tobias Kildetoft

in fact, one of the requirements for a bachelor's degree here is a certain number of credits from courses with external evaluation

MatheinBoulomenos

@KasmirKhaan a lot of things get better if you add "algebraic" as an adjective

Ted Shifrin

Interesting, @Tobias. That's way too much work/effort/time for the US universities to do something like that.

Tobias Kildetoft

19:35

@TedShifrin Yeah. It is meant to make sure that the quality of the universities stay consistent (I think)

Kasmir Khaan

@MatheinBoulomenos what does that mean mathein :D nd btw I kinna need your help to understand tensor Product :D

MatheinBoulomenos

I was just joking

Kasmir Khaan

-.-

not good time for that -.-

Ted Shifrin

But the external evaluators may have inconsistent standards, @Tobias. Certainly if I were one, my standards would be far more stringent than those of any number of my former colleagues.

Tobias Kildetoft

@TedShifrin Sure, but at least it is not possible for the university itself to lower standards, since the external evaluators will be from outside.

Ted Shifrin

19:37

Yeah, I get that.

MatheinBoulomenos

One nice thing about algebraic numbers is that if $\alpha$ is algebraic and you take the subvector space of $\Bbb C$ spanned by all the powers of $\alpha$, this vector space is just finite-dimensional

that's equivalent to $\alpha$ being algebraic, in fact

Tobias Kildetoft

@MatheinBoulomenos you mean the subspace as a rational vector space

MatheinBoulomenos

oh yeah, I should've mentioned that

Eric Silva

jesus christ geometric analysis computations are off the wall

skullpatrol

do the students have any say in this evaluation system? @TobiasKildetoft

Tobias Kildetoft

19:38

@TedShifrin I have no idea if it really serves that much of a purpose, but it is one of those things that has been done this way for a very long time

Kasmir Khaan

Hmm you guys kinna lost me there :D

the numbers can be algebraic or transctenal right?

Tobias Kildetoft

@skullpatrol not as such. They can of course complain if they don't feel they got the right grade, but not apart from that

Ted Shifrin

@EricSilva: Was that addressed to me?

skullpatrol

I see.

Eric Silva

no it was just a general exclamation of my suffering

im working on some Simons' equation stuff

Tobias Kildetoft

19:40

@skullpatrol I mean, the system just consists of me mailing the exams to someone else who also grades them, and then we discuss those grades we did not agree on

Ted Shifrin

I adjudicated a number of grade complaints when I was associate department head. Most times, I was able to reassure the student that on balance the grade was actually fair. I might have been more generous on these problems, but less generous on those problems and the grade came out roughly the same.

Kasmir Khaan

@MatheinBoulomenos can I invite you to a room tonight? :D I kinna need some help :)

Tobias Kildetoft

(or rather, I have the department mail them, as I want no responsibility if they get lost)

Kasmir Khaan

@MatheinBoulomenos only if you dont have other things to do ofc :D

MatheinBoulomenos

I don't have that much time, I'm doing homework on the side @Kasmir

Kasmir Khaan

19:41

@MatheinBoulomenos okay good luck :D

Ted Shifrin

@Tobias: The problem — aside from how that's a ton of work for the external person (does he/she get paid?) — is that this delays students' grades by a month.

PVAL-inactive

I don't think I've ever written a long answer on this site.

skullpatrol

Kinda like cross validation @TobiasKildetoft

Ted Shifrin

I've written some involved ones, PVAL, but never that length!

Eric Silva

i think ive only written like 3 answers

Tobias Kildetoft

19:42

@TedShifrin They do get some amount of compensation (no idea if it is reasonable, but since it is voluntary to be part of the pool of evaluators, I assume it is not terrible)

Eric Silva

i hate answering questions on mse

PVAL-inactive

I found this one though math.stackexchange.com/questions/260065/…

which I don't have any recollection of.

Tobias Kildetoft

As for the delay, yes, but not by that much, as we only have a total of 4 weeks before the students must have their grades.

Kasmir Khaan

In what order should one read serres books?

PVAL-inactive

From front to back.

3

Ted Shifrin

19:43

@PVAL: Your $d$'s need to be $\partial$'s :)

Kasmir Khaan

Pval I wish someoen could hit you now

Ted Shifrin

I've never read Serre's books, so I cannot comment.

Kasmir Khaan

okay thanks Ted =p

Daminark

Pure gold

PVAL-inactive

I used iff too.

Yuck.

Kasmir Khaan

19:44

Ehmm so dami ?

I guess you know the answer?

Tobias Kildetoft

Same. They are supposed to be good, but I usually find that newer books on representation theory have a huge edge since a lot of the concepts have been developed a lot in the mean time

Which allows for a much more structured approach to many things

Daminark

I've never quite read Serre's books, I am intending to look through his rep theory at some point but rn I'm in the middle of other stuff

Kasmir Khaan

Hmm ._. i meant for general knowlegde Tobias

I want to read about stuff in advance

Ted Shifrin

Kasmir: Probably you should find more readable sources, until you're far more advanced.

MatheinBoulomenos

I liked his book on rep theory, but Tobias knows a lot more than I do, so he's probably right

Kasmir Khaan

19:45

since the time at school aint enuf for me

Good Point Ted ._.

Ted Shifrin

Humphreys and Fulton/Harris have modern style representation theory books. The latter is very chatty and has lots of concrete stuff in it.

PVAL-inactive

I tried reading parts of GAGA at one point. That's all I know about Serre.

Kasmir Khaan

I meant more like hmm, algebra books

but in certain order

Eric Silva

@Ted i just figured out why minimal surfaces suddenly become horrible in dimension 7

it's cause 25>24 apparently

Ted Shifrin

Oh, the cone.

Eric Silva

19:46

yeah the bombieri-de giorgi-giusti guy

Ted Shifrin

I remember hearing about this from Lawson and Frank Morgan independently.

But I don't remember the crucial numbers.

Kasmir Khaan

Ted there is an online version ?

Daminark

Idontbelieveyou

►

Kasmir Khaan

I googled it and found soemthing , could be it ?

Ted Shifrin

I'm not the right person to ask such questions, Kasmir.

Tobias Kildetoft

19:47

@TedShifrin Yeah, Humphreys' book on Lie algebras is still my goto for introductory stuff, despite being old enough to have some things done in somewhat different ways than one would do them now (mainly in terms of notation)

MatheinBoulomenos

@Ted Isn't Humphrey's only about Lie algebras? (Maybe I'm thinking of a different book)

Eric Silva

@Ted ambient has dim > 7

Ted Shifrin

Right, but what happens to the 7 to make 25>24? :P

He has lots of books, Mathein.

Eric Silva

the whole construction boils down to the inequality $(n-2)^{2}/4 \geq n- 1

MatheinBoulomenos

I know he has a book on Lie algebras and one on algebraic groups

Eric Silva

19:49

so false until 7 then you get $25/4 > 24/4$

MatheinBoulomenos

but I think Kasmir is looking for stuff on finite groups

(which are technically algebraic groups, but I don't think that helps much)

PVAL-inactive

What's the smallest order non-interesting finite group?

Ted Shifrin

Oh, I guess you're right that his representation theory is of Lie algebras. I stand by Fulton/Harris. I like their mathematics and their exposition.

Eric Silva

and then the singular sets go fucking crazy and have codimension 7 for higher dimensional dudes and i think knowing more about the singular sets properties is like a black hole where we know nothing beyond the basics

Ted Shifrin

I don't remember at all where that would come from, Eric, but OK :)

Oh right ...

Eric Silva

19:51

it comes from spectral estimates for the Jacobi operator

Ted Shifrin

Nah, there's a GMT viewpoint.

Eric Silva

the dude that comes from the second variation formula

ah maybe,i dont know it

Ted Shifrin

Too bad I burnt all my files and notes. :(

Daminark

Press F to pay respects

skullpatrol

:(

Kasmir Khaan

19:52

Hmm

Ted that book has stuff on lie groups

hulton

Ted Shifrin

fulton/harris

Alessandro Codenotti

Hi chat

Hi @Ted

Kasmir Khaan

fulton/harris yes

Ted Shifrin

hi @Alessandro: You had a namesake here earlier :)

skullpatrol

Hi

Kasmir Khaan

19:53

second part is lie Groups onward

but first part was rep theory :)

Tobias Kildetoft

Yeah, Fulton/Harris is a good one. Lots of examples, and makes you work through them to get to the base ideas

Alessandro Codenotti

@TedShifrin what do you mean?

Ted Shifrin

Another Alessandro was here

Kasmir Khaan

@TobiasKildetoft can you please give ISBN ?:D i Think i found wrong book

Eric Silva

I've been doing more the pde side of minimal guys so far this quarter I guess

Ted Shifrin

19:53

There's only one book by those two, @Kasmir.

PVAL-inactive

@Ted Now I know why California has been having so many issues with fires.

Kasmir Khaan

@TedShifrin What did I find then >< let me look again

Ted Shifrin

Why, PVAL? Other than droughts ...

PVAL-inactive

That was supposed to link to your comment about burning your notes.

Alessandro Codenotti

Oh, nice, the Alessandros will conquer the chat

Ted Shifrin

19:54

Ohh, no, that burning happened in my office in GA.

Kasmir Khaan

link.springer.com/book/10.1007%2F978-1-4612-0979-9

Look Ted

MatheinBoulomenos

Hi @AlessandroCodenotti

Kasmir Khaan

that link , second part is lie groups

should I read just first part?

Tobias Kildetoft

@KasmirKhaan Sure, Lie groups and their representations

Ted Shifrin

It's representation theory of Lie groups.

Kasmir Khaan

19:55

We dóing finite Groups only

MatheinBoulomenos

finite groups are zero-dimensional compact Lie groups

Ted Shifrin

Oh.

For finite groups, you could read a little bit in Artin's Algebra, too.

Kasmir Khaan

I too can make up new Words mathein :D

Alessandro Codenotti

Hi @Mathei

Kasmir Khaan

okay :D because serres book so far

very dense

Ted Shifrin

19:56

@AlessandroCodenotti: I didn't ask the other Alessandro if he was going to run me over with his car.

skullpatrol

\o @AkivaWeinberger

Akiva Weinberger

lol

MatheinBoulomenos

@KasmirKhaan did you look at the book I recommended you?

Akiva Weinberger

lo/

Kasmir Khaan

@MatheinBoulomenos Yes, it has fourir analysis approch

skullpatrol

19:58

\o/

Kasmir Khaan

@MatheinBoulomenos did not take that course yet

Akiva Weinberger

ö <- face

Kasmir Khaan

I want to follow one book from start to end, without having to deal with stuff I did not take yet ><

Ted Shifrin

@Kasmir: As you get farther in mathematics, it gets harder and harder to separate everything perfectly like that.

skullpatrol

:O

Kasmir Khaan

20:00

@TedShifrin hmm ._. each time I feel am missing something, i feel like in a loop

like never took a course i was ready to take from strat

allways extra work

-.-

the Thing is , they assumed only knowledege of linear algebra and group theory

Alessandro Codenotti

@TedShifrin is he Italian? In that case most likely

Kasmir Khaan

then one sees modules there

-___-

skullpatrol

math builds upon itself

Ted Shifrin

@AlessandroCodenotti: Yes, Italian also.

OK, I'm off to make lunch. Bye.

MatheinBoulomenos

modules are just linear algebra over rings

skullpatrol

20:03

Cya

MatheinBoulomenos

Bye @Ted

Daminark

See you @Ted!

Tuki

Hmm what eigenvectors in hessian matrix represent ?

The vectors that stay on their span during linear transformation but what does this mean in terms of second derivatives ?

ÍgjøgnumMeg

When we have a short exact sequence of groups $1 \longrightarrow N \longrightarrow G \longrightarrow Q \longrightarrow 1$ with $N \trianglelefteq G$ and $Q = G\ N$ and say that "$G$ is an extension of $Q$ by $N$" is this kind of like reversing the act of quotienting? If I didn't know what $G$ was but knew $N$ and $Q$, is it possible to determine the structure of $G$?

Tobias Kildetoft

@ÍgjøgnumMeg No, certainly not knowing only the groups

quallenjäger

20:21

@TedShifrin Thanks Ted

Daminark

$1\to \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 1$ and $1\to \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 1$ give non-isomorphic groups

Yet they're both extensions of $\mathbb{Z}/2\mathbb{Z}$ by $\mathbb{Z}/2\mathbb{Z}$

Akiva Weinberger

(Semi)direct products

Tobias Kildetoft

@AkivaWeinberger assuming the sequence splits

Akiva Weinberger

Oh

quallenjäger

Suppose $f$ is continuous, and a<b, under which circumstance can I say $sup_{u\in[\frac{j-1}{k},\frac{j}{k}]} |f(u)-b|<sup_{u\in[\frac{j-1}{k},\frac{j}{k}]} |f(u)-a|$?

Apart from $f(u)>b$

Dattier

20:40

@Secret : about the empirical reasonning, yes, it's better if even the rules are discover, the only fixed affirmation is the définition of, the EC affirmations (the rules)

Akiva Weinberger

@quallenjäger I don't have an answer, but draw pictures

The Great Duck

0

Q: Are two functions $f$ and $g$ with linear independent derivatives neccessarily linearly independent and vice versa?

Suppose that there exists functions $f$ and $g$ defined on the real numbers and differentiable everywhere. If their derivatives $f'$ and $g'$ are linearly independent on some nonzero interval then are $f$ and $g$ linearly independent on the same interval? Similarly if $f$ and $g$ are linearly ind...

does anyone know if that is true?

quallenjäger

@AkivaWeinberger Hi Akiva

Well, let me formulate the original problem

Tuki

Now this is a good question

quallenjäger

I have a two dimensional $C^1$-path with $|x'(t)|+|y'(t)|=L$ at every point $t\in[0,1]$, where $L$ is the length of the path under $l^1-norm$, which is defined as $|(x,y)|=|x|+|y|$.

The Great Duck

20:50

@Tuki who?

Tuki

@TheGreatDuck your question is quite good

quallenjäger

I want to estimate $|\frac{x'(t)}{L}-\frac{|\Delta_i x|}{|\Delta_i \gamma|}|$

Akiva Weinberger

Say $\alpha f'+\beta g'=0\implies\alpha,\beta=0$. Suppose $\alpha f+\beta g=0$. By differentiating, we see that $\alpha,\beta=0$.

Thus, if $f'$ and $g'$ are linearly independent, so are $f$ and $g$.

@quallenjäger What are $\Delta_i x$ and $\Delta_i\gamma$?

quallenjäger

where $\Delta_i x=x(\frac{j}{k})-x(\frac{j-1}{k})$ and $|\Delta_i \gamma|=|x(\frac{j}{k})-x(\frac{j-1}{k})|+|y(\frac{j}{k})-y(\frac{j-1}{k})|$

$j/k$ is just the partition of [0,1] into k pieces.

Akiva Weinberger

I imagine the Mean Value Theorem would be relevant

quallenjäger

20:55

I have rewritten everything in integral, especially $|\Delta_i \gamma|=|\int x'(t)dt|+|\int y'(t) dt|$

The thing is, I cannot really pull the absolute sign under the integral, as the derivative might change the sign

The Great Duck

@Tuki thanks

Akiva Weinberger

If $x'\ne0$, then there exists a small neighborhood around $t$ in which $x'$ does not change sign

quallenjäger

otherwise I would have $|\Delta_i \gamma|=\int |x'(t)|+|y'(t)|dt=\frac{L}{k}$

Akiva Weinberger

due to $x'$ being continuous

The Great Duck

@AkivaWeinberger oh. duh. Would the equality be preserved by differentiation, though?

Akiva Weinberger

20:58

@TheGreatDuck If $\alpha f+\beta g=0$ is true on the whole interval, yes

The Great Duck

@AkivaWeinberger ah

ok

quallenjäger

If $x'$ doesn't change the sign, the $y'$ would still be able to change the sign.

The Great Duck

proof by contradiction shall yield my third law of piecewise trivial exclusion then.

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