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00:28
@SimplyBeautifulArt aq
01:16
hi @TedShifrin
@Maks: Here's the point. The $n$th degree Taylor polynomial $P(x)$ of $f(x)$ (say centered at $a=0$) is the unique polynomial of degree $n$ with the property that the error $P(x)-f(x)$ goes to 0 faster than $x^n$ goes to $0$ (as $x\to 0$). So if $f(x)$ is itself a polynomial of degree $\le n$, then using $P(x)=f(x)$ gives error $0$ and this holds.
hi Karim
A professor liked a proof I did for assignment he asked me to present it in class :D
I am working hard this semester
@Maks By the way, you're allowed to have an infinite series with only a finite number of nonzero terms.
Rehi @Semiclassic
Did you present it well, Karim?
yeah
01:18
Good experience.
yeah
It was about if you have f is a unit in a polynomial ring then a_0, a_1,...,a_n are units
@TedShifrin So that means that the taylor polynomial has the exact same value as the function, with no degree of error
In that case
Right.
And how can I write $(x-1)^2 $ as terms of an infinite sum ?
@TedShifrin can we think of smash products as just cartesian product with deleting points which are common to both X and Y
I mean we are removing a disjoint union right ?
01:20
It won't be infinite, @Maks.
Most of the terms will be 0, but that's allowed.
Following up on what we were talking about earlier: If we start with a Riemann surface with some number of double points, then for each double point we smooth out will raise the geometric genus by 1, whereas normalizing a double point won't change the geometric genus.
@Adeek You're not deleting, you're collapsing. It's sort of like a tensor product on "pointed spaces".
You're not saying that right, Karim. Write down the definition.
@MikeMiller oh ok
Ohhh I get it, but will it still be a series ?
01:21
Sounds right, @Semiclassic.
I see that makes sense I guess with the torus if we apply that smash product to the torus then we get a circle
What about the arithmetic genus?
Can it go from lets say n=0 to 2 ?
I want to think more about this though
A finite sum is still an infinite series where the remaining terms are 0, @Maks.
01:21
And still be a power series ?
Yes. It's $\sum a_nx^n$ where $a_k=0$ for $k\ge 3$.
@TedShifrin remember that question that if n is odd then identity map is homotopic to the antipodal map. First I proved it by just acting on $S^n$ by $C^n$ that is $(x,t) \mapsto e^{i \pit} * x$
You're saying nonsense again, Karim.
sorry @TedShifrin when I write sometimes my brain write stuff I want to write later hah
I adjusted it
I don't understand how that is an action by $\Bbb C^n$.
01:24
because we can think of $S^n$ as living in $C^n$
Since n is odd
@Semiclassic: For a smooth curve, geometric genus agrees with arithmetic genus. Does that answer your question?
Hmm. So this is only a question prior to desingularization.
First, you're acting by $S^1$, Karim.
When you resolve singularities, arithmetic genus becomes the new geometric genus. When you stay in a family, arithmetic genus doesn't change. @Semiclassic
Second, Karim, $S^{2k+1}$ lives in $\Bbb C^k$.
What would a family correspond to here?
Still super sloppy.
01:26
@TedShifrin A follow up question was if we have a non-vanishing tangential vector field, then we get the same thing whether n is odd or even. But yeah we get that homotopy through $(x,t) \mapsto cos(\pi * t)id(x) + sin(\pi * t)\alpha(x)$ where $\alpha$ is the normalized tangential vector field. I guess this conceptually what it does is that since normally when we don't have a fixed way to rotate vectors in $S^n$ for every point, but non-vanishing tangential vector field gives us that.
@Semiclassic: When you refer to "smoothing out," you're doing that smoothing by moving in a family of plane curves.
So I was wondering if we have non-vanishing tangential vector field then n must be odd ?
is that true ?
brb quick I will go grab the train
brb 2 min exactly
You need to know how to decide when the antipodal map can be homotopic to the identity map, Karim. You'll learn about degree.
Now I'm a bit confused. Something doesn't sound consistent there.
If I'm smoothing by moving in a family of plane curves, I'm still in that family. So it would sound like the arithmetic genus doesn't change.
Right. I said that, didn't I? Geometric genus changes, though, because once you're smooth your geometric genus is the same as your arithmetic.
01:30
Hmm. So the arithmetic genus was already bigger than the geometric genus.
For your example, normalizing made geometric genus 0 but moving in a family made it genus 1. No contradiction.
Right.
Let me take a precise example. I start with a pair of Riemann spheres touching at two double points.
If I normalize one of the double points but smooth the other by moving in a family, I should end up with a single geometric genus 0.
I guess my confusion is: Does normalizing change the arithmetic genus?
Yes, evidently, because in the end the arithmetic genus equals the geometric genus ('cuz it's smooth). I said that already.
Hm. I'm probably getting myself mixed up somewhere.
cool @TedShifrin I will think about it
01:35
One process maintains geometric genus. The other maintains arithmetic genus. You end up with different Riemann surfaces at the end and different genera.
Let me ask a dumb question, then. What's the arithmetic and geometric genus of a single Riemann sphere with one double point?
My guess would be arithmetic genus is one, geometric genus is zero.
Yes, that's the example you started with!
@Semiclassical what are you doing ?
The pinched torus you started with is a Riemann sphere with a double point :)
Mmkay. Somehow I got confused on that.
Well, yes :P
01:36
Are you doing Riemann surfaces ?
Hence why I wanted to be sure!
But, to sum up: the arithmetic genus is what the genus would be if I normalized all singularities, whereas the geometric genus is what the genus would be if I instead smoothed all singularities.
No, backwards.
crawls in a whole hole
Yeah, it'd better be backwards if I'm going to have arithmetic genus > geometric genus.
So arithmetic vs. geometric genus of the initial singular curve serve as upper/lower bounds on the genus of any desingularized curve.
01:39
OK.
Okay, enough genus discussion for a Friday night.
Back later.
Night!
How've you been?
Was that addressed to I? I'm in bunches of different places ;) ... I've discovered that assisting in science lessons from 8 AM to 2:30 PM (arriving at school at 7:30 AM) with 4 4th grade classes is extremely exhausting at my age.
Yikes.
01:57
back sooner rather than later, it seems.
02:34
I expect to reach n=10,000,000,000 in 24 hours...
and I don't expect to find that I want
02:45
@Socrates haha, ok then
@DHMO ?
@SimplyBeautifulArt n!=a!b!...
O_O
Hm...
We have $1!=1$
$2!=2$
$\vdots$
$5!=120$
from here, take the last non-zero number. That is, take $5!\to2$
$6!\to2$
$7!\to4$
I wonder if theres a pattern
so, factorials mod 10?
@Semiclassical not really
last non-zero number
02:58
@Semiclassical the mod increases with the factorial
Yeah, that....seems even less likely to have a pattern.
1, 1, 2, 6, 4, 2, 2, 4, 2, 8, 8, 8, 6, 8, 2, 8, 8, 6, 8, 2, 4, 4, 8, 4, 6, 4, 4, 8, 4, 6, 8, 8, 6, 8, 2, 2, 2, 4, 2, 8, 2, 2, 4, 2, 8, 6, 6, 2, 6, 4, 2, 2, 4, 2, 8, 4, 4, 8, 4, 6, 6, 6, 2, 6, 4, 6, 6, 2, 6, 4, 8, 8, 6, 8, 2, 4, 4, 8, 4, 6, 8, 8, 6, 8, 2, 2, 2, 4, 2, 8, 2, 2, 4, 2, 8, 6, 6, 2, 6
@Semiclassical yes it's always even
because in the prime factorization, the number of 2s must > than the number of 5s
hm, okay.
> The decimal number 0.1126422428... formed from these digits is a transcendental number; see the article by G. Dresden.
yay transcendence
03:00
Interesting that it's probably transcendent.
lmao, OEIS!!! wooo
hm, ok then
@Semiclassical I thought it is transcendent
Woops, typo. Should've been provably.
Ok. Well, I'll be back later
(Python)  # replace triple dots by spaces

def a(n):

...if n<=1: return 1

...return 6*[1, 1, 2, 6, 4, 4, 4, 8, 4, 6][n%10]*3**(n/5%4)*a(n/5)%10
Can we prove this code?
basically, [1, 1, 2, 6, 4, 4, 4, 8, 4, 6][n%10] returns the n%10-th item in the list
3**(n/5%4) is 3^((n/5)%4)
(% is mod)
03:12
that's some inefficient code there
you're probably right
I just copied it from OEIS
Enciclopedia online de numeros enteros ?
si
replace triple dots by spaces ?
can you give an example ?
03:16
oh, python
def a(n):
   if n<=1: return 1
   return 6*[1, 1, 2, 6, 4, 4, 4, 8, 4, 6][n%10]*3**(n/5%4)*a(n/5)%10
I can probably write some python
or not. :D
@SimplyBeautifulArt I have a different code in mind if I were to generate a list from the beginning
mhm
or did you just copy it from OEIS?
Xam
Xam
Hello everyone :)
03:18
@SimplyBeautifulArt I said I copied the above code from OEIS
but there's an algorithm here: oeis.org/w/images/4/48/AlgLastFinal1.txt
Xam
Xam
What are you talking about? Seems like programming
@Xam hola
@Xam So we have the factorial
@Xam estamos hablando de esto
03:19
and we're trying to find the last non-zero digit in a factorial
:-/ I don't speak that language
Xam
Xam
@DHMO Hola, ya que hablas español xd, pero es permitido hablarlo en este chat?
Wo hui shuo yi diar zhong wen
@Xam por que no? :p
pero alguien no entiende espanol
@SimplyBeautifulArt ni hao
Xam
Xam
@DHMO porque el resto probablemente no entenderá xd
03:20
@Xam probablamente
Xam
Xam
@SimplyBeautifulArt seems interesting
@SimplyBeautifulArt wo hen hao xie xie
The most random things always seem interesting
lol
and OEIS has everything
Xam
Xam
03:21
OEIS is life hehe
Whenever you want to look at an integer sequence you can find it on OEIS.
I didnt get what you wanted to do xD
pretty much
@Simple Ni hao
@SimplyBeautifulArt hello
@Maks Take the factorial
write it out and take the last digit that is not a zero
$\int_1^x f(x)dx$
03:24
@Topologicalife Yes?
Hi.
I'm trying to turn on my mathjax.
Xam
Xam
I wonder if any of you can help me :/
@Xam Hopefully
@Xam que esta el problema?
> Just ask; don't ask to ask.
Xam
Xam
03:26
This seems to be a silly question, but I can't prove that in a DIP the expression of an ideal as a product of prime (maximal) ideals is unique up to order.
PID*
Pero el ultimo termino de un factorial no es siempre 1 ? O.o
@Maks no
6! = 720
Please translatee for me lol
@Xam what is pid
Yeah, I tried it @Simply
Xam
Xam
03:27
A PID is a principal ideal domain
O en español dominio de ideales principales (DIP)
Ahh osea, del resuleto
resuelto*
y queres hacerlo sin resolverlo ?
@Topologicalife So do you have a bookmark with the stuff?
Porque sino, convertilo en lista
y busca el ultimo numero que no sea 0
@Maks esto es muy lento
Yeah.
Dunno why is not working
Xam
Xam
03:29
I tried to write it as p_1\cdots p_n=q_1\cdots q_m and I proved that p_1=q_j, let's say q_1, but I can't cancel ideals like it they were elements.
No tenes un truco matematico para hacerlo ?
@Maks en cada paso solo necesitamos tener un digito
si hacemos la secuencia desde 1
@Topologicalife Did you click the bookmark to activate it yet?
@Simple how would you say Happy Chinese New year?
03:32
I copied that code and pasted it into my direction's bar.
$\ddot\frown$
Does it work now?
@SimplyBeautifulArt xin nian kuai le
Not on this chat.
@SimplyBeautifulArt Xin Nian Kuai Le
03:33
Oh right, I'm stupid
(don't bother me, I'm still learning XD )
@Simple ni shi zhongguo ren?
yes
@Topologicalife Did you hit the bookmark while you had this pageo open?
@DHMO Dui
03:34
Gonna reload it all.
@DHMO Y si necesitas rapidez entonces porque python ??
@Simple danshi ni de profile shuo ni shi meiguo ren
Wo shi mei guo ren he zhong guo ren
@Maks buena cuestion :p
and Vietnamnese, but I don't know how to say that one
03:35
@SimplyBeautifulArt ni bu shi shuo ni zai xue zhongwen ma
@SimplyBeautifulArt yuenan ren
Xam
Xam
@DHMO cuantos idiomas hablas? xd
yuenan ren?
oh, lol
It doesn't work, meh.
Nvm.
@SimplyBeautifulArt yuenan = vietnam
@DHMO I study at America
03:36
mhm, I figured
@Xam tan mucho como ves
I was born in America
Xam
Xam
@DHMO nice
I was wondering if someone can help me to figure out the steps to complete this proof: math.stackexchange.com/questions/814500/…
It is the last answer on the page.
03:37
@SimplyBeautifulArt I see
It seems obvious but I don't know how to proceed.
Yeah. And I suck at chinese
@SimplyBeautifulArt why is everybody learning chinese?
Well, my mom is chinese...
:| I have an obvious reason
I see
03:40
@Topologicalife what do you need help with?
Oh, I fixed my latex.
How to prove the claim with that hint.
I don't know how to complete the steps in order to prove the claim.
i.e: what should I do from here? ${(r-\epsilon)}^{1-{m/n}}<\frac{{x_n}^{1/n}}{{x_M}^{1/n}}<{(r+\epsilon)}^{1-{m/n‌​}}$
That is equal to $\dfrac{{x_n}^{1/n}}{{x_M}^{1/n}}<|{(r+\epsilon)}|^{1-{m/n}}$
Sum all sides for when that inequality is valid.
lol
I can practice my chinese here
haha, the things I find on a math site
what do you mean with 'sum all sides? which one's?
@DHMO Say, did you derive an extension to the factorial?
@SimplyBeautifulArt what extension
03:44
@Topologicalife The double inequality one
sum it all up
I don't know what you mean
do you mean this ineq ${(r-\epsilon)}^{1-{m/n}}<\frac{{x_n}^{1/n}}{{x_M}^{1/n}}<{(r+\epsilon)}^{1-{m/n‌​‌​}}$?
@DHMO tengo un codigo
te lo paso aca ?
@Maks podes darmelo por pastebin.com tambien
Xam
Xam
Good night/day everyone
@Topologicalife you may need to look at the proof en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem
03:52
does anyone here know a lot about things like nilpotent groups, normal series
@Topologicalife Petro's answer somehow answer your question, if you need more detail, this math.stackexchange.com/a/1618234/127116, may help you
@Simple I'm trying to complete Swapnil Tripathi's answer.
I proved it in my own ways, I'm interested just in that way.
@DHMO def last_fact_n (n):
    start_time = time.time()
    "Return the last non-zero digit of N!"
    f2 = f5 = lznd_trailing_zeros(n)
    r = 1
    for i in range(1,n+1):
        while f2 > 0 and i%2 == 0:
            f2 -= 1
            i /= 2
        while f5 > 0 and i%5 == 0:
            f5 -= 1
            i /= 5
        r = (i*r) % 10
    print("It took {} ms".format((time.time() - start_time)*1000.0))
    return r
o tambien tarda mucho ??
@Maks haz clic ctrl+K
gracias
que tiempo esperas mas o meno ?
04:03
tentalo y buscalo :p
pero tu algoritmo es O(n)
el algoritmo dado es O(log(n))
Es el factorial
El ultimo digito de n!
pero no necesitas saber el entero factorial para saber el ultimo digito
Sorry, internet was down
@DHMO I was talking about this: $$1=\lim_{n\to\infty}\frac{n!(n+1)^z}{(n+z)!}$$
@SimplyBeautifulArt ranhou?
@SimplyBeautifulArt oh it was easy
Just let $n \to \infty$ in the last inequality and we are done.
If I'm right ofc...
04:18
Tu codigo no me funciona en python3
@DHMO
@Maks reemplaza "/" por "//"
Sisi, hice esa porque no te toma el float, pero el resultado no es el correcto
a(6) tiene que ser 2 o no ?
si
@DHMO ranhou?(?)
@SimplyBeautifulArt "then"
04:20
sorry, internet is currently horrible
然后
@DHMO me da 6.784354...
C:\Users\XXX>python
Python 3.5.1 (v3.5.1:37a07cee5969, Dec  6 2015, 01:38:48) [MSC v.1900 32 bit (In
tel)] on win32
Type "help", "copyright", "credits" or "license" for more information.
>>> def a(n):
...     if n<=1: return 1
...     return 6*[1, 1, 2, 6, 4, 4, 4, 8, 4, 6][n%10]*3**(n//5%4)*a(n//5)%10
...
>>> a(6)
2
>>>
@Maks
Ahh me falto el otro divisor
04:27
$$z!=\lim_{n\to\infty}\frac{n!z!(n+1)^z}{(n+z)!}$$
$$=\lim_{n\to\infty}\frac{n!(n+1)^z}{(1+z)(2+z)(3+z)\dots(n+z)}$$which is defined for all $z\in\mathbb C$ excluding negative integers.
what's everyone up to
@SAWblade A lot of randomly knitted together things
@SimplyBeautifulArt where do you learn that equality
i love the theory of braids
yuk yuk
@Simple It comes rather elementarily
Notice the following:
04:32
@DHMO copado, es rapidisimo
$$\lim_{n\to\infty}\frac{n!(n+1)^z}{(n+z)!}=\lim_{n\to\infty}\frac{(n+1)^z}{(n+1‌​)(n+2)\dots(n+z)}=1$$
@Maks porque es O(log(n))
By comparing powers and coefficients @Simple , then multiply both sides by $z!$.
@DHMO lo rompi jajaja, le di un numero muy grande
@Simple As per where you "learn" about such things, for me it was Wikipedia and WolframAlpha on the "Gamma function"
04:35
ok
@DHMO tenes ganas de ayudarme con algo ?? Ya que la tenes mas clara en matematica
@Maks con que?
Tenia ganas de hacer un programa que cree y resuelva sudokus
A standard Sudoku puzzle contains 81 cells, in a 9 by 9 grid, and has 9 zones, each zone being the intersection of the first, middle, or last 3 rows, and the first, middle, or last 3 columns. Each cell may contain a number from one to nine; each number can only occur once in each zone, row, and column of the grid. At the beginning of the game, many cells begin with numbers in them, and the goal is to fill in the remaining cells. Players may use a wide range of strategies to solve Sudoku puzzles, and this article goes over a number of methods for doing so. == Techniques == === Backtracki...
04:39
El tema es que el algoritmo que use para crearlos, (lo hice yo) es muy basico, osea, seguro que tiene muchos pasos innecesarios
mmm haber
Y como lo creo ?
2
Q: Algorithm for solving Sudoku

Rag SagarI want to write a code in python to solve a sudoku puzzle. Do you guys have any idea about a good algorithm for this purpose. I read somewhere in net about a algorithm which solves it by filling the whole box with all possible numbers, then inserts known values into the corresponding boxes.From t...

en cual lenguaje?
Python, C o Java el que venga
hasta haskell y go
buscar "sodoku solver program java" y lo encontraras
Quick question, has anyone in here have any knowledge over googology, the study of large numbers?
@SimplyBeautifulArt what is your question?
04:42
Whether or not anyone has knowledge over said field
is that a subfield of number theory?
Pero, otra vez, yo le tengo que dar el tablero con los numeros
yo quiero que me genere un tablero al azar
what is googology? I can't find a wiki about it
3
Q: Java Sudoku Generator(easiest solution)

HunderingThoovesIn my last question seen here: Sudoku - Region testing I asked how to check the 3x3 regions and someone was able to give me a satisfactory answer (although it involved a LOT of tinkering to get it working how I wanted to, since they didn't mention what the class table_t was.) I finished the pr...

google es tu amigo
(⊙0⊙)
04:51
I've heard a quote that I can't find on the internet
(paraphrasing) at some point a number gets large enough we have to use the strength of the axiom that defines the number to compare it
I don't know why I can not connect Google. I use Yahoo instead, but Yahoo is bad
@DHMO hi
@KasmirKhaan hi
that was a wise thing to say
indeed it is
@KasmirKhaan god morgon?
@DHMO I just wanted to say hi in person lol I dont have questions I need help with atm ><
I just wanted to say good morning lol
Morning sir!
Where are you from ?
hong kong
what time is it there?
12:59

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