you have to try harder mercio

k from now on I will use an even morepetty and condescending tone

That sounds productive.

@Shreevats: I would have introduced the notation $b_n = a_{n+1}$ and used the limit definition (as the answer with 6 votes did) to find an $N$ so that $n\ge N \implies |b_n-l\ell|<\varepsilon$.

hello, please why $f_n(x)=n\int_x^{x+\frac1n}f(t)dt$ where $f$ is $k-$Lipschitz that is $|f(x)-f(y)|\leq k |x-y|$ is derivable on $[a,b]$

@Poline: So is $f$ continuous? If so, can you use the Fundamental Theorem of Calculus?

yes f is continuous

So, is $\int_x^{x+1/n} f(t)\,dt$ a differentiable function?

What's the derivative?

$n[f(x+\frac1n)-f(x)]$

$-f(x)$

yes i edite it

OK. So that does it.

Weird, so $f_n'(x)$ converges to $f'(x)$.

If $f'(x)$ exists?

Does the Lipschitz condition come in somewhere?

Ya

Not sure.

Hi @Ted

hi, demonic @Alessandro

And @Balarka

Hi @Alessandro

@TedShifrin Maybe it comes in to show that $f_n'$ uniformly converges to $f'$?

(If $f'$ exists)

@TedShifrin it is differentiable because f is continuous right ?

Note that we're only doing the right-hand derivative here.

Um, no, @Poline. That's backwards.

??

@PolineSandra What. Lots of continuous functions which aren't differentiable

$|x|$ is a classic

If $f$ is differentiable, then $f$ is continuous, but not vice-versa.

but i just have f continuous

As Balarka said, $f(x)=|x|$ is a good example to think about. And it's Lipschitz with $k=1$.

yes i understand the example

So what happens with your $f_n$ in that case?

@Balarka: The interesting thing is to see is if $f$ stays Lipschitz but fails to be right-differentiable, does $f_n'$ converge pointwise?

@TedShifrin it is the primitive of a continuous function

Isn't the point of your question to determine whether $\lim f_n'(x)$ exists, @Poline? Perhaps not. I don't see why they gave you the Lipschitz condition rather than just continuity of $f$.

Hello

Hi Demonark.

@TedShifrin Lipschitz i need it after

For what?

imgur.com/a/FUYjlWN my contour can't decide which way to go :(

i don't finish the exercise

I want the diagonal one to be negatively oriented but my code is unhappy, but for some reason it's okay with the circular arc being negative oriented...

rip

Hello chat

Hey chat

Hi @lattice

I'm having some trouble of numerical nature^^

@TedShifrin ressources.unisciel.fr/ramses/517-3-calcul-integral/co/…

it is this the fundamental theorem of calculus ?

rip spelling

yes it is one of the two parts of the FToC

?

yes it is the fundamental theorem of calculus

one half of the theorem

loichot.ch/maths_d/2-Fondamental.pdf

this is complete?

I was writing some function in Matlab and part of it was basically to find a root (approximately) of a continuous function $f$ on some interval $(a,b)$ given that $f(a)\cdot f(b) < 0$.

(the idea behind is to find intersections of an implicit curve or surface with a regular grid.)

What are totally geodesic plane?

I tried using some well known numerical root finding methods, but none of them worked well.

@TedShifrin or @GFauxPas the primitive of a continuous function is diffentiable that is ?

yes that's both parts Poline

yes that's part of the ftoc

well that's the definition of the primitive I should say

So the question is: Which method can be used (efficiently and safely, i.e. it should always converge) to find such a root?

lattice is $f$ differentiable

that's what i need, to say that $f_n(x)=n\int_{x}^{x+\frac1n} f(t)dt$ is differentiable it is sufficient to say that it is a primitive of a continuous function

@GFauxPas

you want to analyze $f_n$ or $f_n \to f$?

differentiability of each member of a sequence of functions doesn't imply the limit $f_n \to f$ is differentiable unless the convergence is uniform

just $f_n$ is it differentiable

i don't need convergence

just $f_n$

yes if it is the primitive of a function $f$ then it is differentiable with derivative $f$

wait no

i'm getting confused

wait i was right

if it's the antiderivative of something, you differentiate it to get that something

@GFauxPas Hmmm this is a tricky question^^ In most of the applications of this, $f$ will be differential, even smooth. But in some situations where interesting things happen it might be too restrictive to only work with differential functions.

my favorite algorithm for roots of a non-diffable function is the method of false position. it's usually faster than the bisection method, which is simpler

both the bisection method and the false position method always converge for continuous $f$

if there's a solution

the idea is this

Yeah, I tried false position.

do people here know about communication complexity?

oh, it didnt work?

But... I don't know if I did something wrong perhaps, but I tested my program, and it works well for many points on the grid, until at the 123rd point it suddenly does not converge, but jump back and forth^^

what language

I didn't try bisection yet, cause I want to reach a tolerance of $h^4$ where $h$ is the step width of the grid, and that would take many iterations I think.

Matlab

and the 123rd point isn't a good root?

what was the function you tested it on

F4=@(X,Y,Z) (X.^2+(9/4)*(Y.^2)+Z.^2-1).^3-(X.^2).*(Z.^3)-(9/80)*(Y.^2).*(Z.^3);

yay I fixed my graph

(This is some heart-shaped surface, I wanted to impress my girlfriend xD)

I gave up trying to parameterize the graph backwards and instead parameterized it forwards and GRAPHED it backwards, I'm so smart lol

awwwww

that's like the sweetest math thing ever

what's . in matlab

pointwise multiplication

(Cause I evaluate this function at the beginning on all grid points simultaneously)

Let me check what exactly happens at those two ever-repeating iterations.

if it's a math question, the chat here can help you. if it turns out it's a question specific to matlab, there's a code stackexchange called stackoverflow. but they're jerks who love downvoting

Haha okay^^

My question was originally an (applied) maths question I would say.

But it could be that regula falsi actually should work and it just does not work because of my implementation (or even because of Matlab, but I doubt it).

So regula falsi converges for any continuous function, right?

yes and wikipedia states problems where it converges slowly, but it will always converge

en.wikipedia.org/wiki/False_position_method check out the "improvements" section

please how we prove in a metric space that a set $B\subset A$ is dense in $A$ ?

closure of B = A

with sequences what it means ?

it means every point in $A$ is a limit point of a sequence in $B$

classic example, every real number is a limit of a sequence of rational numbers

so $\mathbb Q$ is dense in $\mathbb R$

@GFauxPas Then perhaps I really just have one of those situations where it converges very slowly, maybe so slowly that Matlab does not even see the improvement over the iterations (cause of rounding errors) and runs infinitively long^^

@Blue Yes.

Thank you for your help!

try the improvements in the wikipedia article then

Yes, I will check on that.

good luck! you're an awesome boyfriend/girlfriend!

Well, last day at my job today. Returning to education on Wednesday to do institutional research for a community college :)

awesome, what are you researching

As a first year math student, this website and this chat are a fascinating source of information

IDK at this point, but the impression I get is I'll be doing data analysis to help with general operations and assist with recommending resources to help underperforming students

Hello, people. Is it really necessary to master all of real analysis to start learning topology?

*point set topology

@ParthaSarker certainly not "all of it". It will be helpful for motivation, especially if you know metric spcaes

you don't strictly need real analysis, but some things might be very unmotivated if you don't know real analysis

Also the later stuff like measure theory is probably unnecessary

Btw hello @Mathein!

Hello @Daminark

@ParthaSarker My linear algebra professor, who specialized in topology, told me I could handle topology after taking the class, despite that on paper, Real Analysis was the prereq. The impression I get is that it's one of those "mathematical maturity" things. Unfortunately, I haven't learned topology yet.

yeah you definitely don't need measure theory to start learning topology

continuity, convergence and metric spaces are useful to have seen before

(any maybe some things like compactness, dense subsets etc.)

Congrats @Semiclassical

All I can say about topology is that I tried self-studying with Munkres and utterly failed

So, what should I study before I start a chapter on Euclidean topology?

@ParthaSarker the things I mentioned, continuity, convergence and metric spaces will be helpful. But you don't need any of that

@ParthaSarker The impression I get - again, I don't know much topology - is that if you know how to write a proof, you could just get started.

@Clarinetist that's true, but when you don't know the important special case from analysis, some things will be unmotivated

@Clarinetist topologywithouttears.net You cam check this out.

Yeah, like I said, I don't know much about topology

@ParthaSarker I read that book about a year or two ago. Wasn't a fan. I don't recall why. This summer, I plan on reading the text by Adams and Franzosa

amazon.com/Introduction-Topology-Applied-Colin-Adams/dp/…

@Clarinetist Allen Hatcher has a free pdf textbook on topology but I've only looked things up in it, I never "read" it

That's an algebraic topology text.

But he also has a freely available note on point-set on his website

oh I glossed over the words "point-set" sorry

Hmm how do i find out about topics that are for my level ?

@Tuki Well, how much math do you know, and do you have any idea what you're interested in?

i've studied some basic linear algebra and (calculus 1 and calculus 2)

recently

Any idea what you want to do in the long term?

Well i would like to study more rigorous / in more depth something

no not really

We didn't cover any proofs or anything in detail

My recommendation would probably be to study linear algebra in further depth

Do you know what eigenvalues and vector spaces are?

yes

Yeah, I'd say there are a few directions you could go

You could, for instance, learn multivariable calc with linear algebra (I recommend Ted Shifrin's book - he happens to be active on here occasionally)

You could learn linear algebra in greater depth or start going into numeric linear algebra. Insel's text comes to mind; I don't know of resources for numeric linear algebra

Hmm i could actually go and buy this

amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/… ?

You could even learn about probability (my favorite subject). I recommend Wackerly's text

Yeah, that's the one

Yes i've done basic course on statistics / probability

Was quite interesting topic indeed

Yeah, Wackerly would be prob and stats with calculus

Or, if you want to continue down the traditional math curriculum, you could start learning about real analysis. You could try Understanding Analysis by Abbott

Personally, I learned from Bartle and Sherbert's Introduction to Real Analysis

There are a lot of options with what you know now

Yes i can see that

If you want something that's REALLY out there, you could get into much more applied mathematics, e.g., convex optimization (which is available online for free, btw! web.stanford.edu/~boyd/cvxbook )

or you could study matrix computations (amazon.com/Computations-Hopkins-Studies-Mathematical-Sciences/…)

I'll quit listing everything :P

So you have read the Shifrin's book ?

@Tuki Yes, I have, but not in as much depth as I would like. I know enough that I can tell you I like his writing style and explanations

Do these books have practise problems in them or ?

Once i've bought so far have included some

All of the books I've listed have plenty of examples

If you're looking for a solutions manual, you will have to get used to not having one

Well how do i know if my result is correct or not ?

You mean by solution manual by a separate book which contains step by step solutions to problems or ?

@Tuki You'll develop an understanding over time where you should be able to judge whether or not your solutions are correct, but you could always ask questions on Math StackExchange. I do it all of the time when I'm in study mode

@Tuki Yes, that's what I mean by "solutions manual"

Wooooo I leave this job permanently in less than an hour

Can you reliably tell if your own answers are correct or not ? When you have some more experience ?

@Tuki Yes, that's what I mean

Well the thing is that i don't think i have yet this ability ^^

@Tuki Ask on MSE if you have questions.

I do it

That's what i've done so far

Worked just fine

@Tuki Well, I'd say, keep at it, and of course, if you know anyone in person who knows this material, talking about the material with others is always helpful

I think that's the main thing I miss about full-time schooling: being able to talk to others about what I'm learning. I've learned a lot more through those conversations than I've learned through textbooks

Right now i don't have anyone to discuss math related topics (if you exclude math SE and this chat ) but hopefully this will change soon

This chat is very nice though. It is highly likely that all regulars who visit this chat know more about mathematics than i do so the probability that someone is able to help me is quite high

What kind of prerequisite i need to be able to study topology ?

there are no prerequisites other than some elementary set theory (like intersetions, preimages etc.), but knowing some real analysis can help a lot with motivation @Tuki

ok

What are geodesic planes in the context of hyperbolic surfaces?

Hi is there a quick way of showing that if $x \notin B^\circ$ (interior) and $x \notin \overline B$ (closure) then $x \in \partial B$ (boundary)?

hi @mercio

Do you have an idea on that math.stackexchange.com/questions/2795609/… ?

Oh, I have a mistake above. I meant $x \notin U \setminus \overline B$ (complement of the closure)

Conjecture: given a topology $(X, \tau)$ and a set $S \subseteq X$, $S$ is discrete iff any set-theoretic function $S \to X$ can be extended to a function $X \to X$

@philmcole define the three terms

@LeakyNun counterexample. Consider the indiscrete topology on $X$

what if $X$ is Hausdorff?

@philmcole context is very important

"context" meaning those things in the beginning that start with "let ..."

This is part of a proof that a set $B$ is jordan measurable iff $B$ is bounded and $\partial B$ is a Lebesgue nullset.

@Nûr I don't like "find something equivalent to this property" questions

Please is it right to say $$\sup_{x\in[a,b]}|n\int_x^{x+\frac1n}f(t)dt-f(x)|=n\sup_{x\in[a,b]}|[\int_x^{x+‌​\frac1n}f(t)-f(x)]dt|$$

Excuse me, I apologize if this is too trivial to bring it up here, but I'm learning to add subspaces in linear algebra, and while I feel like I understand the basic idea, I'm having a hard time applying it.

Could someone help me by walking me through the following example? $S = \{(x,y,z) \in \mathbf R^3 : x = z\}$ and $T = \{(x,y,z) \in \mathbf R^3 : z = 0\}$

I thought I could ask the above without showing context since this would be very cluttered then @Leaky

@philmcole for instance, I had no idea you're in a metric topology

what are you trying to show, that $S \oplus T = \mathbb R^3$?

Ok sorry, I thought you know my level is not that advanced :p

@GFauxPas

metric topology then

@JakeS so the golden rule is that $\vec v \in S+T$ iff there is $\vec s \in S$ and $\vec t \in T$ such that $\vec v = \vec s + \vec t$

@PolineSandra

I have to calculate the sum and also determine if it's direct, which I believe it is in this case.

Please is it right to say $$\sup_{x\in[a,b]}|n\int_x^{x+\frac1n}f(t)dt-f(x)|=n\sup_{x\in[a,b]}|[\int_x^{x+‌​‌​\frac1n}f(t)-f(x)]dt|$$@GFauxPas

@PolineSandra yes

@LeakyNun Alright, I understand that idea.

why please why we don't put $\frac1n f(x)$

instead od $f(x)$@GFauxPas

Sometimes $\partial B$ is defined exactly as $\overline{B} - B^o$

Thanks! Can I prove this quickly using my definitions above @loch ?

I've been reading this answer, and I'm trying to see what is an example of an interval contiguous to the Cantor set, could someone lend me a hand?

@PolineSandra are you sure it's correct? I agree it should be $\dfrac 1 n f(x)$

@LeakyNun ?

@philmcole You can try and prove that if $x\not \in Y$ and $x\not \in \partial Y$, then $x\not \in \overline{Y}$.

@GFauxPas i found it in a book

you dont need to tag me every line, it makes an annoying noise :P

sorry

So given the sets I described, I need to find $v \in S + T$ where $v = s + t$, and in this case $s = (x,y,x) \forall s \in S$ and $t = (x,y,0) \forall t \in T$

@loch Ok thx!

it could be a mistake even though it's in a book, I agree with you that it hsould be $1/n$ in front. I have to go now good luck

@LeakyNun are you here >

Jake and you have to show theyre unique or otherwise show that their intersection is trivial iirc those two things are equivalent

?

@philmcole Let $\varepsilon > 0$. $x \notin Y^\circ$, so $B_\varepsilon(x) \nsubseteq Y$, so there is $z \in B_\varepsilon(x)$ and $z \notin Y$, so $B_\varepsilon(x) \cap (X \setminus Y) \ne \varnothing$; $x \notin X \setminus \overline Y$, i.e. $x \in \overline Y$, i.e. $x \in Y$ or $x \in \partial Y$. In the latter case, we are done, so assume that $x \in Y$. In that end, $B_\varepsilon(x) \cap Y \ne \varnothing$, so we are done.

Thank you for your help. I understand that idea, but I don't have any worked out examples to platform off of, haha...

I think once I figure out how to work out one exercise I should be able to figure out the rest of them.

@PolineSandra $\displaystyle n\int_x^{x+\frac1n} [f(t) - f(x)] \ \mathrm dt = n\left[\int_x^{x+\frac1n} f(t) \ \mathrm dt - \int_x^{x+\frac1n} f(x) \ \mathrm dt \right] \\\displaystyle = n\left[\int_x^{x+\frac1n} f(t) \ \mathrm dt - \frac1n f(x) \right] = n\left[\int_x^{x+\frac1n} f(t) \ \mathrm dt \right] - f(x)$

@JakeS we need to fine all the $\vec v$ that can be expressed in the form of $(x,y,x) + (x',y',0)$

what vectors, do you think, can be expressed that way?

Could someone explain this to me please? It seems like they are saying that partial(f)/partial(t) is the same as dx/dt, and later they are saying it is different. Also, what is dt?

as in, dt by itself

All of the vectors of the form $(x+x', y+y', x)$? Forgive me if I'm missing something obvious.

@JakeS is there a vector that cannot be expressed in that form?

bear in mind that $x$ and $x'$ are different

so are $y$ and $y'$

I'm thinking about your question, but I don't see one if there is

can you prove that?

Every vector in S can be written in the form $s = (x, y, x)$ and every vector in T can be written in the form $t = (x, y, 0)$, thus it follows that every vector in $S + T$ can be written as some combination of these two?

@LeakyNun Nice, thanks very much.

Hey @Ted

LMAO

Have you seen this before? :P

I saw this before

it's great

It's like my spirit jam

How dare you ask me that question

lfmao

that looks off

Did you know that guitars are not very common in Germany? That's why metal sounds like this here:

dundudundundundurunrunrdudnundundun

beautiful

Amazing

I saw them live, it was great

this is like Darude Sandstorm metalified

Pretty cool idea though

Lel, from a classmate's TeX'd notes

tis true

A proof here even says "I don't get it"

(Ever since complex analysis became analytic number theory we've all been a bit lost)

There's a cool motivation that can be turned into a proof if you try hard enough. So dividing by $n$, we get $\frac{\varphi(n)}{n} = \prod_{i=1}^k(1-\frac{1}{p_i})$ in this form, you can just interpret both sides as the probability that if you pick a number uniformly from $\{0, \dots, n-1\}$ will be coprime to $n$ (since that's equivalent to not having a prime factor with $n$ in common)

ye

Ah, I like that

The proof I did was just, it's easy for prime powers, then use multiplicativity

sure, that's easier to write down

but doesn't really provide intuition

It's also easy to just enumerate

inclusion exclusion

boom bom bom schoom zoom zom

Hello world!

@Balarka I wanna see you write that in a paper