I'have just watched a video on youtube about turning sphere inside out (actually this is my the second time). I just wonder what kind of math i need to know to understand this mathematically ? Is this a difficult field ?

homotopy theory. Also known as algebraic topology.

I'd like to learn those stuffs. It looks really fun.

you might want to major in maths. Because the rabbit hole is deep.

it is not something that can be done with a hobbyist's time

I spy with my little eye a @Mathein

I spy with my little eye a @Daminark

:0

I've been spotted

;)

Gotcha

How's it going?

Goodish

Trying to do this assignment :P

What's it on?

math.soimeme.org/~arunram/Teaching/2018AlgebraicGeometry/…

Problem 1 looks horrible ffs

It's somewhat tedious hahaha

Also wow

Algebraic stack, perfectoid space and fine moduli space are the real bad ones

The second question is very vague to me

webusers.imj-prg.fr/~laurent.fargues/SaltLake.pdf

The length requirement that they have strikes me as asking you to be strangely brief about it

Hi all, just very curious, are you seriously expected to finish Problem 1? :o

Oh the paper is much shorter than I thought

@user31415 Apparently

@user31415 We had only two weeks to do both questions lmao

if you don't know what a manifold is, how are you supposed to figure out what a scheme is and then what a coarse moduli space is..

and there's also perfectoid space.. interesting

Fortunately I already knew what a scheme was, but definitely that's a serious problem for most of the students

I still don't know what a fine moduli space or a perfectoid is yet

This is so strange

Is this for a class?

is it actually possible to "understand" what a scheme is in like a week?

I feel that's quite a stretch if it's your first time

in the same vein doesn't seem like getting a sense of algebraic stack/perfectoid is possible either

@Daminark This is for one of 4 classes I am taking atm

@user31415 Yeah that was my concern. One could certainly read the definition, and write it down. Understanding it is an entirely different story

So they expect you to learn what a perfectoid space is but also feel it's necessary for you to carefully define a metric space and give an example just in case you forgot...

What are the other 3 btw?

I think it might be an exercise in research skills. I suspect the lecturer will give us a rant about how one can read any research paper, since all you need to do is find the definition of each word, and then find the definition of each word that appears in each definition

that sounds more plausible lol

and induct downwards

that reminds me of the course that defined a vector space but expected to know people what a scheme is

Don't mean to intrude...buuuuut...that sounds horrible.

@MatheinBoulomenos Lmao, I am in such a class atm

sounds like "inquiry based learning"?

I guess in principle sure but the issue is that in principle you could just write down what you see without really getting it, and not only that, but the pset feels like it forces you to do so since it's only a 2 week thing

@Daminark string theory, category theory + hom alg, and this one: sites.google.com/site/yapingyanghomepage/teaching/…

Sounds fun!

@Daminark I suspect most people will hand it in largely incomplete, and he'll mark really nicely

Terrifyingly it is worth 20% of the course

20 people dropped it in the first week (of 30)

-__-

HW in general or this pset?

not surprising i guess lol

Just this pset

Shit

If I get a bad mark, I'll drop it and follow along unofficially I guess :P

Good plan

^

But yeah wow I've never seen that kinda weighting

from the classes you are taking it sounds like you are grad student though, so grades probably don't matter anyway

@user31415 Yep, first sem masters

Like, most of my classes have 10-20% of the grade coming from homework total

cool

@Daminark Right, yeah, this is very weird. It's 60% homework grades, 40% 3 hour exam

0-0

@AlexClark wow, hw counts 60% and you have quite unusual problems

Tbh while I can see a case that HW is too easy to copy, so exams need to be worth enough that you're screwed if you don't know what you're doing, I sorta feel like your ability to do math under that deep of a time constraint isn't that useful in the long run, so I'd rather have HW count for a larger proportion

@MatheinBoulomenos You should see what the proof machine actually is too :P

That's my attempt at writing a finite type scheme in proof machine

Though only combo (45% HW) and AG with a particular prof (50% HW) do so

Fuck's sake

What exactly does proof machine mean?

He defines a grammar system

math.soimeme.org/~arunram/Notes/grammarContent.html

It's kind of elegant in the end, but goddamn it takes ages

and a one small paragraph definition blows up to that

Oh I cut out above it "Let $Q=(X,\mathcal{T}_X,\mathcal{O}_X)$ and $P=(Y,\mathcal{T}_Y,\mathcal{O}_Y)$"

Where $\mathcal{T}_X$ denotes the topology on $X$ etc

well, have fun defining perfectoid spaces like that

Yeah that ain't happening :P

Why is he that strict with the writing style? I would expect that in papers you just have to be clear instead of having this rigid style to stick to

He says he can't understand math if it's not written like that

:P

Sounds a bit sketch

But... I guess these are the cards you've been dealt

For now I guess

It's just assignment 1 apparently

Lol, I've had a class before where the psets were completely unreasonable but at least they were in principle just 10% of the grade (and actually nothing, as we found out after the final)

Yeah I think it might end up being something like that

He hasn't said how the marks are distributed

So it might be 0% value on most of these insane ones

Lmao, tru

I wish I knew though, given how many hours I've put in so far

If you were to leave this class, what would you take in its place? Or would you just leave it at 3?

Most students take 3 and extend there degree

Hahahaha

W... what's Shutka?

They shut down power generators

@Daminark I have no idea

Wait wait what?

Drinfeld?

The terrorists were using drinfeld modules to shut down the reactors

@AlexClark lmao

Dammit Drinfeld please guard ur modules better

In Episode 11 of Season 4 of Fox's "24", we find Drinfeld modules, the central construction in the solution of the Langlands program over function fields (by Drinfeld for GL2 and Lafforgue for GLn), in use by the Counter Terrorist Unit, spearheaded by Kiefer Sutherland's character Jack Bauer.

That's what I would give for my answer in pset 2

I showed that to my advisor and he thanked my for it and said he showed it to a guest professor who visited for a talk in his seminar

Hahaha

Every talk on homological algebra has the snake lemma proved from it's my turn

And whenever Edward Frankel comes up, we whip out the trailer to his (nude) movie

I'm kinda inclined to send this to Drinfeld himself lmao

He probably gets it sent to him often :P

Lmao, tru

my advisor met Drinfeld before and was like "why terrorists and Drinfeld modules? Drinfeld is such a nice guy"

$$\prod_{p} \frac 1 {1-p^{-s}} = \sum_{n} \frac 1 {n^s}$$

The single formula that makes everything work

my book calls it "the golden key"

Talking about terrorists:

:-D

$$2^{2n} = (1+1)^{2n} > \binom {2n} n \ge \prod_{n \le p \le 2n} p$$

$$\prod_{2 \le p \le n} p \le 2^n \cdot 2^{n/2} \cdots = 2^{2n}$$

$$\sum_{2 \le p \le n} \log p \le 2n \log 2$$

$$\sum_{2 \le p \le n} \log p = \sum_{k = 2}^n (\pi(k) - \pi(k-1))\log k = \sum_{k = 2}^n \pi(k) \log k - \sum_{k = 2}^n \pi(k-1) \log k \\ = \sum_{k = 2}^n \pi(k) \log k - \sum_{k = 1}^{n-1} \pi(k) \log (k+1) = \pi(n) \log n - \sum_{k=2}^{n-1} \pi(k) (\log(k+1) - \log(k))$$

shivers

$$\sum_{k=2}^{n-1} \pi(k) (\log(k+1) - \log(k)) = \sum_{k=2}^{n-1} \pi(k) \int_k^{k+1} \frac1t \ \mathrm dt = \int_2^n \frac {\pi(t)} t \ \mathrm dt$$

hi @Daminark

How's it going?

@Daminark trying to understand prime number theorem

$$\pi(n) \log(n) \le 2n \log 2 + \int_2^n \frac{\pi(t)}t \ \mathrm dt$$

Lmao, I can see that. We did stuff along these lines in complex though my sleep schedule got in the way of my absorbing much

and why are you typing out all the Latex?

maybe i'm crazy

take care then. Students think my math department building looks like an asylum.

$$\prod_{p \le N} \frac 1 {1 - p^{-1}} = \prod_{p \le N} (1 + p^{-1} + p^{-2} + \cdots) \ge \sum_{n \le N} \frac1n$$

@Iza_lazet why, are the walls padded? :P

nah. Probably because everything looks white and sterile

$$\sum_{p \le N} \ln (1-p^{-1})^{-1} = \sum_{p \le N} \sum_{k=1}^\infty \frac{p^{-k}}k = \sum_{p \le N} \frac1p + \sum_{p \le N} \sum_{k=2}^\infty \frac{p^{-k}}k \le \sum_{p \le N} \frac1p + \sum_{p \le N} \sum_{k=2}^\infty \frac{p^{-k}}2 \\ = \sum_{p \le N} \frac1p + \sum_{p \le N} \frac{p^{-2}}{2(1-p^{-1})} = \sum_{p \le N} \frac1p + \sum_{p \le N} \frac{1}{2p(p-1)} \le \sum_{p \le N} \frac1p + \frac12$$

$$\sum_{p \le N} \frac1p \ge \ln \prod_{p \le N} \frac1{1-p^{-1}} - \frac12 \ge \ln \sum_{n \le N} \frac1n - \frac12 \ge \ln \ln n - \frac12$$

Now for any $q$ and any $x$

$$\pi(x) \le q + \frac {\varphi(q)} q x$$

$$\frac{\pi(x)}x \le \frac q x + \frac {\varphi(q)} q$$

take $q = \displaystyle \prod_{p \le N} p$ then $\displaystyle \frac{\varphi(q)}q = \prod_{p \le N} (1 - p^{-1}) \le \left(\sum_{n\le N} \frac1n \right)^{-1} \le (\ln(N+1)+\ln2)^{-1}$

$$\sum_{2 \le p \le n} \log p = \pi(n) \log n - \int_2^n \frac{\pi(t)} t \ \mathrm dt$$

@LeakyNun What is this ? i don't understand anything :(

@Sou $\pi(x)$ is the number of primes less than or equal to $x$

$$\sum_{2 \le p \le n} \log p = \pi(n) \log n - \int_2^n \frac{\pi(t)} t \ \mathrm dt$$

$$\theta(n) = \pi(n) \log n - \psi_1(n)$$

@Leaky @Mathei if you're still interested in infinite products of measures there's a very nice exposition of Kolmogorov's extension theorem in Tao's measure theory book, starting at page 235 (it is the very last topic in the book)

thanks

The idea is that we have a family of measure spaces $(X_\alpha,\Sigma_\alpha,\mu_\alpha)_{\alpha\in A}$, it's easy to build a measurable space $(X_A,\mathcal{B}_A)$ where $X_A=\prod X_\alpha$ and $\mathcal{B}(A)$ is the coarsest $\sigma$-algebra making all of the projections continuous

The problem is defining a measure on that space. For all $B\subseteq A$ we have a projection $\pi_B:X_A\to X_B$ (with $X_B=\prod_{\alpha\in B} X_\alpha$), if we had a measure on $X_A$ we'd also have a measure on $X_B$ by pushing it forward with $X_B$

But we do have measures already on some $X_B$, all of those corresponding to finite $B\subset A$. So when can we reconstruct an unique measure on $X_A$ by knowing its pushforward measure on all the finite products? That's the question answered by the Kolmogorv extension theorem

@AlessandroCodenotti of course the projections here are meant to measurable rather than continuous, sorry

Hi. I have a condition $a_n = O(Q_n(\tau))$ as $n\to\infty$ for each $\tau\in(0,\epsilon)$ and some $\epsilon>0$, where $Q_n$ is the quantile function of some nonnegative discrete distribution $F_n$. First, I think I can write just $a_n = O(Q_n(\tau))$ as $n\to\infty$ for each $\tau\in(0,1)$. But it feels somehow redundant since I care only about being able to take as small $\tau$ as possible. Is there some better way to write the condition?

$$\psi(n) = \sum_{2 \le p \le n} \log p \left\lfloor \frac {\log n} {\log p} \right\rfloor = \sum_{2 \le p \le \sqrt n} \log p \left\lfloor \frac {\log n} {\log p} \right\rfloor + \sum_{\sqrt n < p \le n} \log p \le \pi(\sqrt n) \log n + \theta(n) - \theta(\sqrt n)$$

$\psi(n) \ge \theta(n) - \theta(\sqrt n)$

$$\psi(n) = \sum_{2 \le p \le n} \log p \left\lfloor \frac {\log n} {\log p} \right\rfloor \ge \sum_{2 \le p \le n} \log p = \theta(n)$$

$$\pi(\sqrt n) \log n - \theta(\sqrt n) = \pi(\sqrt n) \log n - \pi(\sqrt n) \log \sqrt n + \displaystyle \int_2^{\sqrt n} \frac{\pi(t)}t \ \mathrm dt = \pi(\sqrt n) \log \sqrt n + \displaystyle \int_2^{\sqrt n} \frac{\pi(t)}t \ \mathrm dt$$

@LeakyNun oh yeah that's quite important - the reason why the riemann zeta function has anything to do with primes

$$\le \sqrt n \log \sqrt n + \sqrt n$$

@loch lol you appeared on and off here and finally you speak

uh

i just woke up

$$\theta(n) \le \psi(n) \le \theta(n) + \sqrt n (1 + \log \sqrt n)$$

$$\theta(n) \le \psi(n) \le \theta(n) + n$$

$$\theta(n) = \pi(n) \log n - \int_2^n \frac{\pi(t)}t \ \mathrm dt$$

$$\theta(n) \le \pi(n) \log n \le \theta(n) + n$$

$$\psi(n) - n \le \pi(n) \log n \le \psi(n) + n$$

mean value theoremn't

@loch

what lol

that isn't mean value theorem right

oh yeah no

MVT isn't even valid right

$$-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \Lambda(n) n^{-s}$$

$$\psi(n) = \sum_{k=1}^n \Lambda(k)$$

$$\psi_1(n) = \sum_{k=1}^{\lfloor n \rfloor} (n - k) \Lambda(k)$$

$$\psi_1(n) = -\frac1{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{n^{s+1}}{s(s+1)} \frac{\zeta'(s)}{\zeta(s)} \ \mathrm ds$$

$$\frac{\psi_1(n)}{n^2} = -\frac1{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{n^{s-1}}{s(s+1)} \frac{\zeta'(s)}{\zeta(s)} \ \mathrm ds$$

desmos.com/calculator/37ufjlx6cp

click this button in the desmos link after guessing what it would do :P

@AlessandroCodenotti

Hello, I'm self studying probability and I got to the portion of the mixtures of distributions. The source I'm using is a study manual for actuarial science and I feel there isn't too much why's in it, or at least as much as I'd like.

I want to know more about why we assign weights to the distributions and not simply add them together, and when we're looking for mean, or any moments really we can't simply find it from the mixture but instead we have to go back to the parts that made the mixture in the first place. I'm completely fine with a reference as an addition to self study, something verbose perhaps.

Hello, I'm trying to get the chatroom for Group Theory up and running. You're all invited to contribute :)

o..o

I wonder if a (a,b,c,d) profile will say anything about the nature of those critical points as the attractors abruptly changes

perhaps there's a bifrucation somewhere near them

That's an intersting idea

Although visualizing it is, well, difficult

so like, plot a x,y,a graph, holding b,c,d constant, see what the shape is like

and then repeat for b,c,d to get some slices

I could animate through the variable a, if that is what you mean

yeah, that would help

Okay, I'll make an animation of that, the rendering of the atractor itself doesn't take that long

brb

My code is finally working, making some picture sequences rn

@Shaun I am usually up for discussing some group theory. But so far not much has been going on in that room

hi @TobiasKildetoft

@LeakyNun Hi

@TobiasKildetoft PNT is so cool

Is there a name for a subgraph where A<->B, B<->C, but not A<->C? That is, a triangle minus an edge

nontransitive graph ?

@Julius trianglen't

Thanks, nontransitive sounds good. Now I realized that it's also a path graph $P_3$

transitiven't

a graph whose connected components are not complete ?

good lord, they set up all the lemmas, and still needed 6 pages to show that $\displaystyle -\frac1{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{n^{s-1}}{s(s+1)} \frac{\zeta'(s)}{\zeta(s)} \ \mathrm ds \to \frac12$

(as $n \to \infty$)

That's because I need more contributors, @TobiasKildetoft! Why not start be sharing what your favourite theorem of group theory is or perhaps your favourite group?

@Shaun I prefer others to get the discussion rolling

(I am lazy)

I want to show that if every conjugate class of a group $G$ consists of a single element then $G$ is abelian. So suppose that $G$ is not abelien, then there exists $x, y \in G$ such that $xy \neq yx$ which implies that $xyx^{-1} \neq y$. How can I conclude from this that the conjugacy class of $y$ has cardinality at least $2$?

you just did

o..o'

@G.Ünther hmm. a=2.44 seemed to look like something is splitting up, and as we head towards a=1.04, it seems 4 objects fused into 2 and then disappear, before appearing again. a=3.04 may be a singularity or something. I wonder if when holding a=3.04 and changing b,c,d will collapse the pattern further and give some kind of point attractor in the (a,b,c,d,x,y) space?

@LeakyNun I don't see how $xyx^{-1} \neq y$ implies that the conjugacy class of $y$ has cardinality at least $2$

Any fixed points you can exploit in that system and only plot those, they might show how the bifrucations happen?

@Perturbative do you know what conjugacy class is?

Yes

what is it?

@Secret The question is what are the fixed points

O ...

hmm...

If $G$ is a group, then the conjugacy class of an element $g \in G$ is the set $\{h \in G \ | \ xgx^{-1} = h \text{ for some } x \in G \}$

I can fiddle around a=3.04 up close and observe what happens if I change another variable. It could be because there is a=b

@Perturbative is $y$ in its own conjugacy class? What about $xyx^{-1}$, is it in the conjugacy class of $y$?

$t \to \infty$, $\zeta(c+it)^{-1} \in \mathcal O((\log t)^7)$

@Secret here is a closeup around a=0.6: youtu.be/ZveuYBnm6Us

Oh derp thanks @AlessandroCodenotti. So $y$ is certainly in its own conjugacy class and since $xyx^{-1} \neq y$ we have that $xyx^{-1} = a$ for some $a \in G$, so then $xyx^{-1}$ is in the conjugacy class of $g$. So the conjugacy class must have cardinality at least 2, contradiction.

a =0.6004 is very strange. Does it simply disappear like that?

Yeah

a = ~0.7 has these two cluster of points that does not seemed move much when you continue to increment. Might want to zoom in there and see the dynamics

This is right before it

And this right after

it just goes snap

Is it possible to increment in even smaller steps between 0.74005 and 0.7411?

say 0.00000001?

I'll give it a shot

I think I saw those two points rapidly expanding as that shape snaps into view

There is something fishy about those two points, because between 0.74 and 0.6, the two points move up and down out of the shape and disappear

There seem to be two cycles or something

these two clusters each have one

and they jump between them really fast

as if which one got chosen is random

sounds like a 2-cycle, I am suspecting there might be doubling going on there, and then the shape appeared as it becomes chaotic or something

I am not sure how you can produce a brifucation profile for this system though to investigate further

but 0.7411 seemed to be within some chaotic regime as there is a strange attractor

What do the clusters look like up close (as in zooming in in x,y not in increments of a), do they look like strange attractors or just circular shaped attractors?

they aren't circular

wobbely and dented

yeah, I am pretty sure there is birfrucation doubling near those two points, if you run that video where you move from 0.6 to 0.7, you see the twinkling decreases in frequency before the strange shape appeared

I'm rendering a closeup of that rn

I don't know, but if I were to drop the x-coord and push all of the points onto the y-axis and then plot variable a unto the x-axis, if that would look like the bifurcation diagram. It obviously isn't, but idk if that'd be close

@Perturbative $g$ should be $y$, but yeah that's right

The blinking is so irregular

It could be highly dependent on the starting point

Then random rounding errors determine where to land

Hmm... there are couple things I want to try, what program you are using to execute the code?

That's just visual basic that make that

to be more exact, Microsoft Visual Studio

That weird shape

how can a linear combination of sins and cos produce something like that... hmmm

the power of iteration I guess

ah right, you are iterating a lot of sins and cos, anything can happen there

too bad that's not a linear system of equations, thus preventing an analytic way to solve for the fixed points (if any)

@G.Ünther the beauty of maths

Yeah that's true, the transformation is complicated

I guess we aren't looking at fixed points, but rather cluster points?

The two points I highlight are very fishy, with those loops highlighted by G. Ünther

but we are only looking along one 3D slice of a 6D object, thus we cannot drew any conclusions yet

what do you mean 3D slice of 6D object?

The whole thing has 4 parameters a,b,c,d and two coordinates x,y

and we are varying the a parameter

I still don't understand how the images are generated

what is the red things?

It just is a map showing how often each pixel is hit during the iteration

More yellow equals more hits

yellow?

and what does it mean if the whole picture is black?

Black means no hits at all

how can no points be hit?

The second to last one posted looks a bit like the Feigenbaum plot, except that is usually the other way around

is that the logistic map?

Yeah for example if you have a fix point and you just hit that fixed point over and over again, then just one pixel would be hit

yeah, that was just me screencaping wikpedia because I initial thought there is a birfrucation going on

but G. Ünther showed the blinking between the two cycles is random

@LeakyNun I guess. It is $ax(1-x)$ plotted with $a$ along the $x$-axis and each time iterated a bunch

and.... it has been a very long time since I use visual studio, I forgot how to run scripts on it

I could make an improved version in processing if you like

tfw the paper uses half a page to prove something when a nice picture would do: $$\sum_{n=1}^{\lfloor X \rfloor} \int_n^X sx^{-s-1} \ \mathrm dx = s \int_1^X \lfloor x \rfloor x^{-s-1} \ \mathrm dx$$

ok

sometimes I don't really like this paper

Gonna sleep soon, so I will follow up on the King Dream fractal later

Perhaps... I might dreamed about it tonight and see something weird...

Recent exposure to politics seemed to make my dreams more capable of problem solving...

$$\sum_{n\le X} n^{-s} = \sum_{n\le X} \int_n^\infty sx^{-s+1} \ \mathrm dx = s \int_1^{\lfloor X \rfloor + 1} \lfloor x \rfloor x^{-s+1} \ \mathrm dx + \lfloor X \rfloor \int_{\lfloor X \rfloor + 1}^{\infty} x^{-s+1} \ \mathrm dx$$

$$\sum_{n\le X} n^{-s} = \sum_{n\le X} \int_n^\infty sx^{-s+1} \ \mathrm dx = s \int_1^{\lfloor X \rfloor} \lfloor x \rfloor x^{-s+1} \ \mathrm dx + \lfloor X \rfloor \int_{\lfloor X \rfloor}^{\infty} x^{-s+1} \ \mathrm dx$$

$$\sum_{n\le X} n^{-s} = \sum_{n\le X} \int_n^\infty sx^{-s+1} \ \mathrm dx = s \int_1^X \lfloor x \rfloor x^{-s+1} \ \mathrm dx + \lfloor X \rfloor \int_X^{\infty} x^{-s+1} \ \mathrm dx$$

yes, all three cuts are valid

but the last one is more useful

and I forgot $s$ in all three lines

$$\sum_{n\le X} n^{-s} = \sum_{n\le X} \int_n^\infty sx^{-s+1} \ \mathrm dx = s \int_1^X \lfloor x \rfloor x^{-s+1} \ \mathrm dx + \lfloor X \rfloor \int_X^{\infty} s x^{-s+1} \ \mathrm dx$$

$$= s \int_1^X \lfloor x \rfloor x^{-s+1} \ \mathrm dx + \lfloor X \rfloor X^{-s} = s \int_1^X x^{-s} \ \mathrm dx - s \int_1^X \{x\} x^{-s+1} \ \mathrm dx + \lfloor X \rfloor X^{-s}$$

$$= \frac s {s-1} (1 - X^{-s+1}) + \lfloor X \rfloor X^{-s} - s \int_1^X \{x\} x^{-s+1} \ \mathrm dx$$

$$\zeta(s) = \lim_{X \to \infty} \sum_{n \le X} n^{-s} = \frac s {s-1} - s \int_1^\infty \{x\} x^{-s+1} \ \mathrm dx$$

$$= \frac s {s-1} - s \sum_{n=1}^\infty \int_n^{n+1} \{x\} x^{-s+1} \ \mathrm dx = \frac s {s-1} - s \sum_{n=1}^\infty \int_0^1 t (t+n)^{-s+1} \ \mathrm dt$$

eh... replace all my $-s+1$ with $-s-1$ lol

$$\zeta(s) = \frac s {s-1} - s \sum_{n=1}^\infty \int_0^1 t (t+n)^{-s-1} \ \mathrm dt$$

$$\left| \int_0^1 t (t+n)^{-s-1} \ \mathrm dt \right| \le |n^{-s-1}| = n^{-\Re(s)-1}$$

so the infinite sum converges absolutely whenever $\Re(s) > 0$

and this is the extension of $\zeta$ to the plane with positive real-part

Why is the integral of 3/x constant

3ln(x) is variable, no?

they meant +

turn the first = to +

Thanks

interesting stuff going on here.

happy 5th of aug

It's the 4th where I'm at.

^

for me as well

how the student reversed the role on his teacher

Hi chat

from a set of data that contains some errors

how to find out which value that if any data is above it , it should be removed?

@Jacksoja I don't think there is a way to do that in all cases

@Jacksoja There may very well be exceptional data that arises

the example worked is accually from coordinates in real life

ofc we dont get perfect data from the gps system

and the precess is to calculate each distance / angle multiple times to collect data

then to somehow make it adjusted

@AlexClark it is a problem in statistics , and its not my area at all :/

hopefully i find someone here who can give me some hints or idea

So the scenario is that some gps-tracked-device moved along the same path numerous times, and you are only given inaccurate sample-path data?