Mathematics - 2016-12-22 (page 5 of 6)

4:00 PM

@GFauxPas I've now completed the proof. Can you tidy it?

Given a real function $f$, if $f(x)$ is differentiable everywhere, then is the following true?
$f'(a)>0 \implies \exists b \in \Bbb R: f(b) > f(a)$

Mary Star

Let E/F an extension and T a subset of E, that spans E algebraically over F.
We have that $T=B\cup \{b_1, \ldots , b_m\}$. Does it holds that B spans also E algebraically over F ?

DHMO

@MaryStar können Sie mir helfen?

SteamyRoot

So formal :O

DHMO

well I don't know which environments are considered formal

I've seen her exercises use "Sie"

@SteamyRoot Kannst du mir helfen?

SteamyRoot

Written text is (nearly) always for formal. For a chatroom conversation, "du" should be fine :)

DHMO

4:10 PM

I see, thanks

Balarka Sen

@DHMO Assume there ain't. Then $a$ is a maximum. That forces $f'(a) = 0$, no?

Mary Star

@DHMO Ich werde ein bisschen später Zeit haben :-)

DHMO

@BalarkaSen sure but I don't know which theorem to use

SteamyRoot

Hmmm

Balarka Sen

Fermat's theorem.

SteamyRoot

4:12 PM

What if the derivative isn't continuous.... hmmm

Balarka Sen

Doesn't really matter, @Steamy

GFauxPas

@DHMO you know I did the hard part

DHMO

@BalarkaSen but I'm thinking of something like mean value theorem

SteamyRoot

Hrmm...

DHMO

@GFauxPas as if my part weren't hard

Balarka Sen

4:13 PM

So you're literally asking for a proof of Fermat's theorem? There are plenty on google.

DHMO

@BalarkaSen no i'm not...

GFauxPas

I'm joking :D

Balarka Sen

Then what are you asking? I don't even understand the question

GFauxPas

if you prove the hypothesis is false you dont need to prove antyhing about the conclusion so you can make the proof a little shorter

SteamyRoot

$$f(x) = \left\{\begin{array}{ll}
x & \text{ if } x \leq 0\\
x-3 & \text{ if } 0 < x < 2\\
... & \text { if } x \geq 2
\end{array} \right.$$

GFauxPas

4:16 PM

Unless you want to get rid of the split between forward and reverse implication and prove iff the whole way through

DHMO

@SteamyRoot it isn't differentiable when x=0

Balarka Sen

It's not even continuous there

DHMO

@GFauxPas I don't understand "the hypothesis is false"

SteamyRoot

Hmmm, yeah, right, I'm confusing things

DHMO

@BalarkaSen I was just expecting mean value theorem

GFauxPas

4:17 PM

$\forall x : P(x). \implies. Q(x)$

if you disprove the LHS it doesnt matter what the RHS is

Balarka Sen

If you're not continuously differentiable what you can say at best is that it might not be strictly increasing in a neighborhood of $a$, @Steamy. But you can still say it's not a extremum.

SteamyRoot

Yeah, that's what I was thinking...

Balarka Sen

Stuff like $x^2\sin(1/x) + x$

DHMO

@GFauxPas so you're asking me to disprove "a=e"? I don't understand.

GFauxPas

"Thus the right hand side is not true." "Thus the left hand side is not true."

Semiclassical

4:19 PM

Hi chat

GFauxPas

either we merge the necessary and sufficient conditions into one big proof, or we dont need to prove LHS is not true

DHMO

@GFauxPas but why merge if you already have the necessary condition?

I used the necessary condition many times in proving the sufficient condition lol

GFauxPas

right, so them, you don't need to show "thus the left hand side is not true"

maybe I'm confused

DHMO

oh ok

(referring to just edited version) is this what you mean?

GFauxPas

I've gotten 6 texts from my bus service telling me changes in the schedule and 1 text from amazon letting me know my order was shipped and so I was disappointed 7 times it wasnt my gf texting me :(

Balarka Sen

4:21 PM

@Semiclassical Been reading a small booklet of sort on Bose-Einstein statistics, which I found gathering dust on my bookshelf. Really neat stuff, although I only superficially understand it.

GFauxPas

right exactly DHMO

DHMO

@GFauxPas I'm saying that the second case contains so many lemmata that you can separate some results

for example I proved that the equality case of e^x >= x+1 only happens when x=0

more importantly I proved that a^x-x-1 is convex whenever a>0

Semiclassical

@BalarkaSen Neat

Balarka Sen

I am forgetting, but you're a stat-mech person, yep?

Semiclassical

Occasionally

GFauxPas

4:24 PM

I alternate between saying "lemmas" because "lemmata" I'm afraid sounds pedantic, and saying "lemmata" because it's fun to say

DHMO

@BalarkaSen somehow I know that "a is a maximum" implies "f'(a) = 0" but I'm not convinced that its contrapositive is true lol

GFauxPas

it's not true

also it's not true either

DHMO

??????

PearlSek

hey guys

Balarka Sen

Contrapositive, not converse

GFauxPas

4:26 PM

$f(x) = -|x|+2$

has a max at $0$

$g(x) = x^3$ for the converse not being true

oh wait

contrapositive

but anyway, $-|x| +2$, so be careful

Balarka Sen

f'(0) does not even make sense for that

PearlSek

when we have derivation rules like $(e^u)' = u'e^u$, what are the limitations ? because if $u$ is 1, then $(e^u)'=u'e^u=0$. u needs to be not constant, right ?

GFauxPas

right Balarka, so I'm saying that you need to be careful when you say "$a$ is a maximum implies $f'(a) = 0$

Balarka Sen

oh, sure. he indicated $f$ is everywhere diff before

GFauxPas

Oh I didn't see that XD

@PearlSek when using the chain rule, it's less confusing if you don't use prime notation

DHMO

4:29 PM

@PearlSek well it is true if u=1 also? u is a constant, so (e^u)' must be 0

Balarka Sen

@Semiclassical What kind of a person you are most of the time? Condensed-matter something something?

Semiclassical

Yeah.

GFauxPas

or if you do, use $f'(x)$ because then you show what you're differentiating with respect to

PearlSek

i am stupid sorry

:facepalm:

GFauxPas

You're not stupid, I've noticed a lot of people think they're stupid if they dont understand something immediately in math

don't know why that is, but the "stupid" people are the people who are afraid to ask and so never learn

so in other words, you're stupid not because you asked, but because you called yourself stupid. ?????

I'll be quiet now

anyway Pearlsek check out the Khan Academy videos on implicit differentiation

they helped me understand the chain rule when I was in calc I

Semiclassical

4:32 PM

i try to resist calling myself stupid when I make a mistake. I prefer "silly"

Seems healthier

DHMO

@GFauxPas are you following lol

Fargle

@Semiclassical This is an impulse I could use.

Mike Miller

My good friend Kevin Carlson posted his first (non-REU) paper today. I have no idea what it says.

Kari

@PearlSek Looks like 'thestudentroom.com' syntax

GFauxPas

Sweet, I got an A in Physics II!!

Kari

4:45 PM

Congrats!

GFauxPas

:)

ty

Balarka Sen

@MikeMiller Nice, but the paper looks scary.

DHMO

@GFauxPas should we extract some paragraphs as lemmas/theorems?

GFauxPas

Maybe DHMO but I don't feel like it righ tnow sorry :(

DHMO

sure

Mike Miller

4:46 PM

@BalarkaSen I'm trying to read it now.

Kari

Anyone seen Herbert Gross' MIT complex analysis lectures?

robjohn

@MartinSleziak Oh! Okay. I thought maybe you were replying to something I said.

GFauxPas

I wonder if I'll ever have to go back and learn non-analytic geometry

I never learned it

Kari

Like intrinsic geometry?

GFauxPas

Euclid

compass and straightedge stuff

Kari

4:52 PM

Oh that stuff

I doubt it

GFauxPas

I want to go into differential geometry, just because I've asked 3 professors what grad math I should go into , and I told them my favorite subjects, and they all said "differential geometry"

Kari

What are your favourite subjects, @GFauxPas?

Balarka Sen

I think most of the Euclidean geometry you'd need can be picked up from working through a bunch of geometric constructions. There's a couple dozen of them you'll ever need in your life.

GFauxPas

I told them I like matrix calculus and smooth curves/surfaces/hypersurfaces

Find the solutions to $\mathbf J = \mathbf 0$ on $w = x^2 + \sin 2xyz$, things like that

well no, that's not really a good exmaple of what I like

DHMO

@GFauxPas in light of the fact that characterisation is used throughout Proofwiki, could you move the page accordingly?

Krijn

4:57 PM

Eyooo

GFauxPas

basically I love Calculus and Linear Algebra

Both are used I think DHMO, we never had a consensus as to whether to use British or American English

one of my professors says that if you dig deep enough into any topic that's based on Calculus you'll find linear algebra, do you guys agree

Kari

They go pretty well together

I'd try reading some texts on functional analysis

SteamyRoot

Well, as soon as you come across a Jacobian you're in linear algebra, no?

GFauxPas

ya agreed

DHMO

@GFauxPas ok

GFauxPas

5:03 PM

@DHMO it's that primemover is the biggest contributor, that's why

I think

DHMO

@GFauxPas Is there anything I can do now?

GFauxPas

you mean, for that proof?

DHMO

for anything on ProofWiki

GFauxPas

I think PW likes $\vdash\ \ \dashv$ \dashv \vdash instead of $\therefore \ \ \because$

um you can help me with my stuff leading up to laplace trasnforms :D

DHMO

link?

GFauxPas

5:08 PM

basically i'm not sure what the most elementary way is to deal with the origin in these

proofwiki.org/wiki/Real_Power_is_of_Exponential_Order_Epsilon

proofwiki.org/wiki/…

for a function to be of exponential order, it needs to be piecewise continuous at the origin from the right, and I know how to prove that these things are, but I don't know what the exposition should be

hichris123

@TheGreatDuck That was a joke...

Sorry for the confusion I guess.

GFauxPas

I really need to learn more LaTeX than mathjax

I'm going to have to write papers evnetually

DHMO

@GFauxPas isn't it continuous at the origin also?

GFauxPas

do I need to prove it, or is it sufficiently obvious?

Is $0^e = 0$ obvious?

it's not $\exp \left({e \ln 0}\right)$.

DHMO

look, rational powers aren't defined using exp and ln

GFauxPas

5:19 PM

so that takes care of that

DHMO

yes

GFauxPas

so what about $t^e \vert_{t = 0}$

DHMO

rational or real?

GFauxPas

$t$ is a positive real

$e = \exp 1$

well

non-negative real

you see the problem?

the thing is, in the context of integration, it's not important. but I need to show it's not important.

DHMO

so you're talking about the first link

well I'm confused

what the hell are the domains

Kari

5:24 PM

@GFauxPas There's really not much of a jump

All you need to get by is Google and a bit of editing

GFauxPas

proofwiki.org/wiki/Real_Power_is_of_Exponential_Order_Epsilon how about now

DHMO

@GFauxPas I suspect a circular proof here proofwiki.org/wiki/…

@GFauxPas nice

GFauxPas

Induction proofs often "feel" circular, I'm not good and identifying circular induction proofs

DHMO

I have raised the objection in the talk page

please participate

TheGreatDuck

@hichris123 as i said many times over. I was not referring to your comment. I was referring to an earlier comment by a moderator telling me to stay out of chat and that if I caused any more trouble I would be banned. Sorry for the confusion. I figured you were joking.

LeGrandDODOM

5:47 PM

@TheGreatDuck you vote to close your own questions, why is that ?

GFauxPas

$\Large \mathfrak {u \ U} \ \mathcal U \ \mathscr U \ u \ U \operatorname{U \ u} \ \texttt {U u} $ favorite?

for the heaviside step function

probably best to go with default $u$

TheGreatDuck

@LeGrandDODOM I really dont know why. I think I was being too harsh towards myself about what is and is not allowed on the site. I legitimately believed I would be banned for asking those questions.

Perturbative

Hey everyone, quick question, given a set $X$ and a basis $\mathcal{B}$ for a topology on $X$, then the result that a collection $\mathcal{T}$ generated by $\mathcal{B}$, is a topology on $X$ should be classified as a Theorem or merely just checking a definition?

Null

$\mathfrak {u \ U}$

^those two

(but completly unrelated to your question, i just find them nice)

GFauxPas

thanks

the quality of being bounded is "boundedness"?

DHMO

6:05 PM

> Why did the chicken cross the Möbius strip?

To get to the other ... -- no, wait ...

GFauxPas

proofwiki.org/wiki/… that was easy

DHMO

@GFauxPas I think you should use $K$ instead of $M$

GFauxPas

done. you can do that to my proofs without asking me if you think its a good idea to change things

DHMO

@GFauxPas can an upper bound be smaller than a lower bound?

GFauxPas

in absolute value, why not

DHMO

6:11 PM

right

GFauxPas

though I had to ask myself that question twice when writing the proof :)

DHMO

@GFauxPas don't use $L$, just use $\max (U,0)$

GFauxPas

you do it

DHMO

so you don't need a lower bound

but you would need to change the name of the theorem as well

bounded above function etc

GFauxPas

but if it's unbounded below I don't know if its of exponential order

is it?

DHMO

6:12 PM

why?

SoumyoB

can someone explain to me why is the usage of axiom of choice in any kind of proof not particularly liked?

I mean people always try proving things without using it

GFauxPas

$-e^{x^2}$?

SoumyoB

what's with the hatred for that poor axiom?

GFauxPas

Soumy, some people don't like the AOC because it's equivalent to some counterintuitive things

DHMO

@GFauxPas why isn't it order 0?

SoumyoB

6:13 PM

@GFauxPas for example?

GFauxPas

Banach-Tarski Paradox

Because $\vert e^{x^2} \vert > e^x$?

DHMO

@GFauxPas oh, sorry, my fault

GFauxPas

Except AOC is also equivalent to plenty of things that are very intuitive

Like that every vector space has a basis

DHMO

@GFauxPas would it be a good idea to replace "K > ..." with "K = ... + 1"?

GFauxPas

and that the cartesian product of nonempty sets is nonempty

DHMO

6:15 PM

@GFauxPas it is not intuitive that R/Q has a basis

GFauxPas

hm

has a basis in $\mathbb Q$ you mean?

Semiclassical

An old joke: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

3

GFauxPas

has a basis over what?

SoumyoB

@Semiclassical why would the well ordering principle be false?

DHMO

i've no idea what it means, i just repeat it lol @GFauxPas

GFauxPas

6:17 PM

Soumy those things are all equivalent to the axiom of choice

:)

SoumyoB

@GFauxPas damn is there absolutely no proof of basis of vector space without AOC? :O

Semiclassical

It's a joke. The axiom of choice, the well-ordering principle, and Zorn's lemma are all equivalent.

SoumyoB

oh

DHMO

@SoumyoB it just means that the principle is counter-intuitive

SoumyoB

the well ordering principle?

GFauxPas

6:18 PM

AFAIK Soumy in ZF it's equivalent

DHMO

@SoumyoB yes

GFauxPas

I don't even understand all the forms of the Axiom of Choice

trilolil

Hello everyone
I just learned about iterated function systems, fractals and the chaos game.

Is the following statement correct: The chaos game uses iterated function systms to render fractals. ?

Semiclassical

I should note that the above joke is credited to Jerry Bona.

mercio

the chaos game ?

SteamyRoot

6:21 PM

Still one of the best jokes if you've dealt with all 3 a bit :D

trilolil

never mind actually I found the answer to my question just on wiki

Thanks :)

DHMO

@GFauxPas so you're studying order now?

trilolil

Chaos game

In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration. Repeating this iterative process a large number of times, selecting the vertex at random on each iteration, and throwing out the first few points...

GFauxPas

i'm studying laplace transforms

every function of exponential order has a laplace transform

sufficient, not necessary

and probably other integral transforms as well

trilolil

@mercio Howver I don't see the link with chaos theory...

DHMO

6:24 PM

@GFauxPas magic: proofwiki.org/w/…

GFauxPas

-__-

DHMO

do you like it?

GFauxPas

maybe cosine and sine are important enough to have their own proofs

I mean the proofs are so short already

I dunno

DHMO

I don't see the difference

GFauxPas

hmm you're right

DHMO

6:25 PM

If you like it then I'll perform the same magic here

proofwiki.org/wiki/…

GFauxPas

No keep that one

you can change the sine one though

DHMO

done

GFauxPas

thanks

constant is so much "more" than bounded

so i think it should have its own

DHMO

lol

@GFauxPas can I change the constant though

I found a nicer proof

GFauxPas

what is it?

DHMO

6:28 PM

well, |C| < (|C|+1)e^(0t)

no need that case splitting

GFauxPas

hmm I think pm is changing your edits back

good call

DHMO

nice! another meaningless bureaucratic arm-wrestling

i'll split proofs then

SteamyRoot

actually, @Semiclassical, I just noticed your joke is wrong :O

Adeek

hey just want to check if my notation is correct.

SteamyRoot

It's the "well-ordering theorem", not the principle (which is obviously true)

DHMO

6:31 PM

@GFauxPas what's happening between link1 and link2?

SteamyRoot

That might explain @SoumyoB 's confusion about why it would be false.

GFauxPas

Because $\mathcal L \{ \sqrt t \}$ is simpler than other rational powers

and I'll use it later

DHMO

@GFauxPas also, I'm not seeing $\forall t \ge M$ in any of your proofs yet it is stipulated clearly in the definition

GFauxPas

also you can define square roots before defining general rational powers

DHMO

@GFauxPas is there any Laplace transform that links to those results?

GFauxPas

6:33 PM

I haven't gotten to them yet

Sometimes I say "for $t$ sufficiently large", sometimes I forget

Like proofwiki.org/wiki/… I say $t > 1$

DHMO

then fix them

GFauxPas

ugh DAD

later, I've been dealing with proofwiki for long enough today

DHMO

proofwiki.org/wiki/…

Very handwavey

but I need to go now

GFauxPas

how do you want me to fix it

DHMO

I'll fix it when I come back

GFauxPas

6:36 PM

k

Alessandro Codenotti

I think Zorn's lemma actually feels intuitively true if one spends 10 minute understanding the definition of the objects involved and what the lemma is saying

SteamyRoot

Hmmm... I don't know.

GFauxPas

what counter intuitive results besides Banach Tarski are equivalent?

I know that's the big one

SteamyRoot

It feels natural for sets with certain structure order than being a poset

The Well-Ordering Theorem is pretty counter-intuitive (hence the quote)

Alessandro Codenotti

I mean it's not like it's obvious, but it's reasonable at least

Semiclassical

6:45 PM

@SteamyRoot that may be true, but it's not the quote

SoumyoB

@SteamyRoot I have to agree that seems more reasonable

GFauxPas

I guess it is pretty weird that every set can be "put in order"

Balarka Sen

@Alessandro I think of it in terms of some of it's most obvious algebraic applications.

SteamyRoot

@Alessandro True, but in my opinion it's reasonable as well that for certain things it would not work.

Alessandro Codenotti

Maximal ideals? @balarka

Balarka Sen

6:47 PM

Also that f.g. implies Noetherian.

SteamyRoot

@GFauxPas Exactly. For rather easy sets it can already be incredibly difficult to come up with a well-ordering.

Let alone any set.

Balarka Sen

I should reparse. What I mean is the fact that Noetherian (ascending chain condition) implies every nonempty collection of submodules has a maximal element.

Alessandro Codenotti

Hm, but is it true that f.g. implies Noetherian?

mercio

@trilolil I don't think there is a link

Balarka Sen

f.g. is equivalent to being Noetherian, in fact

SteamyRoot

6:50 PM

@Semiclassical True, but it ruins the joke. Annoyingly, the joke is first cited in a 1997 handbook referring to the source as "private communication in 1977". So there's no telling if Bona misspoke, or if the first person writing down the joke wrote down the wrong thing. :P

Alessandro Codenotti

Every ideal being f.g. is equivalent to Noetherian? Or is it enough for the ring?

Balarka Sen

The former.

I am being sloppy today, sorry.

Alessandro Codenotti

No problem, but that looked suspicious :P

Ted Shifrin

Hi @Alessandro, sloppy @Balarka

Balarka Sen

Morning, @Ted

Alessandro Codenotti

6:53 PM

Hi @Ted

Balarka Sen

(I guess it's about midday there)

Ted Shifrin

A bit before ...

Kari

Am watching a video on complex analysis and supposedly the Cauchy Riemann equations lead us to those differentials being exact. I'm not sure how it follows :-/

Ted Shifrin

Not true. They're closed, hence locally exact.

Or exact on a star-shaped domain.

Balarka Sen

The point being locally a holomorphic function has an antiderivative. Globally that's false for many reasons (can you come up with an example of one?)

Mike Miller

6:58 PM

Many reasons? ...One reason.

GFauxPas

Ted are you familiar with the CUNY system

Ted Shifrin

not hugely

Kari

I just read about star regions. Seems ok. You just gotta be able to find a point from which every other point is reachable by a line segment entirely contained within the shape.

GFauxPas

I need to take some courses to shore up my math education to get ready for a pure math phd program

Kari

No idea what closed differentials are :-/

GFauxPas

6:59 PM

my major is in applied math

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