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4:04 PM
@Adam What if Ted, me, Semiclassical, Mathphile, Ultradark, Erico, Alessandro are the same person and thus this chat is just you and avatars ;P
 
Well it's none the bother to me, where else would I go for social interaction, facebook? of course I prefer stack exchange if you think therefore you are far more worth while than most social media streams
 
@Secret lol
 
$0$
$n$
$1+1+\cdots=\omega$
$\omega n$
$(\omega^2 + 1)n$
$(\omega^{\omega} + \cdots + 1) n$
 
4:26 PM
well see here we get to the belly of the beast, because if we let mathematics have it's say on physical matters then half of the pseudo science threads they pump to keep public interest alive in physics gets brought into question and eventually exposed as complete rubbish
 
$(\epsilon_0+\cdots + 1)n$
$\phi(n,0)$
$\phi \binom{1}{n}$
$f(n)$
$f(n)+1$
$f(f(n)+1)$
$f(\alpha)=\alpha$
$f(\alpha+1) > \alpha$
 
 
1 hour later…
5:57 PM
an observation that I don't know what to do with
-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6
3, 6, 4, -2, 5, -5, -1, -3, -6, 2, 0, 1, -4
3, -1, 0, 5, -3, 6, 1, 2, 4, -5, -4, -6, -2
Each column of that array sums to zero, and each row is a permutation of the numbers -6 through 6
I think that this phenomenon should be generic, i.e. for any positive integer n there should exist a 3-row array such that each row is a permutation of -n through n and every column sums to zero
But I don't really see a pattern in the above. (It's a solution that Mathematica generated via the FindInstance command, and therefore isn't expected to be unique.)
 
hmm...
How likely is given a set of integers [-n,n] and a permutation $\pi \in S_{2n+1}$ that:
For all $a,b,c$: $\pi_i(a)+\pi_j(b)+\pi_k(c)=0$
 
think I've found a construction that works:
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
6 4 2 0 -2 -4 -6 5 3 1 -1 -3 -5
0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1
 
sorry typo
$\pi_i(a)+\pi_j(a)+\pi_k(a)=0$
 
so first line is start from -6 and go up one at a time
second is to start from 6 and go down two at a time, wrapping around -8 = 5 mod 13
last is to start at zero and go up 1 at a time, wrapping around 7 = -6 mod 13
those each hit all integers from -6 to 6
and each column sum is shifted by 1-2+1 = 0 (with the 'wrapping' cases being taken mod 13)
 
hmm...
0,1,2,3,4,5,6,7,...,n
$\pi ([0...7])$
 
6:12 PM
and minus that, with zero only showing up once
 
$\pi_a(x)+\pi_b(x)+\pi_c(x)=0$ where $\pi_a(x) = x+a \mod (n+1)$
Thus:
$x+\pi_b(x)+\pi_c(x)=0$ gives:
$x+ (x+b) + (x+c) = 0 \mod (n+1)$
 
wow you guys make quwhite a team
 
$3x + b + c = 0 \mod (n+1)$
Since it has to hold for all $x \in [0,...,n]$ it means:
 
switches to semi classical monitor
damn it
 
Okay, yeah, I'm convinced of this construction
I don't have a good name for this, but it works
 
6:18 PM
$3x$ has to be some additive inverse of $(b+c)$
What is $3x \mod 13$ for $x \in [0,...,6]$?
 
0,3,6,9,12,2,5
 
o wait, my formulation does not work, because it forgot the permutations that are not shifts
Still, it is clear that this property has to satisfy this equation somehow:
$x + \pi_a(x)+\pi_b(x) = 0 \mod (n+1)$
 
my construction for -n...0...n :

-n, -n+1, -n+2, ... , 0, 1, 2, ... , n-1, n
n, n-2, n-4, ... , -n, n-1, n-3, ... , -n+3, -n+1
0, 1, 2, ... , n, -n, -n+1,..., -2, -1
hmm
 
somehow, one gets the additive inverse of $x$ by combining two permutations
 
This stuff is very reminiscent of magic squares and latin squares.
Or at least, latin rectangles.
 
6:25 PM
\begin{pmatrix}
-n &-n+1 & -n+2 & \cdots &0 & 1 & 2 & \cdots & n-1 & n\\
n & n-2 & n-4 & \cdots & -n & n-1 & n-3 & \ldots & -n+3 & -n+1\\
0& 1& 2& \cdots & n & -n & -n+1 & \ldots & -2 & -1
\end{pmatrix}
 
$[-n,..,n]$ is a set that forms an additive group
 
bah, still wrong
 
$\pi_a(x)+\pi_b(x)$ forms the additive inverse of $x$
 
okay, that shouold work
@PM2Ring it has that flavor, yeah
 
Let $y+x=0$, it means that we have $y = \pi_a(x)+\pi_b(x)$, that is, for each $x$, there exists a unique decomposition of $y$ into pairs of numbers
 
6:28 PM
@Secret yeah. typically one would write the elements as 0 through 2n+1 but it's the same group
 
What's mysterious is why there always a $\pi_a,\pi_b$ such that $\pi_a(x)+\pi_b(x) = -x$ for all $x$. Its as if $\pi_a,\pi_b$ quotient the additive group somehow
(each column can be thought of as cosets of this additive group with the property they sum to the identity)
 
If I were to guess, I'd suspect that one should view this in terms of $[-n,n]\times [-n,n]\times [-n,n]$
which is itself a group
and now we're considering the orbit generated by repeatedly adding $(1,-2,1)$
Since 1 and -2 both generate $[-n,n]$, each element shows up exactly once for each component
and by construction one has that the sum of the three components remains zero
 
yeah
 
Only issue I could see is that we're only strictly guaranteed that the sum remains zero mod 2n+1
 
hmm...
 
6:35 PM
yeah, there's something a bit delicate going on.
 
0,1,2,3 will fail
 
yeah, it's definitely possible to have the sum of the three components be 13 if I pick a generic seed and orbit it
I don't think that the seed my construction above was using runs into this, tho
 
0,1,2,3
1,3,0,2
3,0,2,1
fail
 
well, for my construction with n=3, it'd be
-3 -2 -1 0 1 2 3
3 1 -1 -3 2 0 -2
0 1 2 3 -3 -2 -1
which looks legit
On the other hand, if I take the left-most column to be 0 0 0 then I get
0 1 2 3 -3 -2 -1
0 -2 3 1 -1 -3 2
0 1 2 3 -3 -2 -1
in which case the column sums are 0, 0, 7, 7, -7, 0
so zero mod 7 but not strictly zero
 
Hi guys, I started a StackExchange page about Graphing Calculator 3D here:
https://area51.stackexchange.com/proposals/120787/graphing-calculator-3d
It already has 316 members, it needs 75 more members with 200+ reputation before it can go public. If it doesn't reach that number this week it will shut down.
It'd be great if you support it by joining it.
3
 
6:46 PM
So it does seems that it boils down to whether given m rows, the underlying set forms an additive group and can be decomposed into a product of m groups and that the m possible shifts has a solution such that it sums to zero mod 2n+1
 
Sounds right? I'm not touching the case of m rows tho
 
hmm...
0,1,2,3
1,2,3,0
3,0,1,2
that still fails despite it is made by (0,1,3) which all the orbit actions should cancel
I think we need the additive group to have unique additive inverses except zero
which here it clearly fails because 2+2=0 mod 4
0,1,2,3,4,5
3 will fail
0,1,2,3,4,5,6,7
so at least for 3 rows, this will fail for all even n
since there always exists an element (the median) such that it is its own additive inverse
It will be useful though if our investigations above can be generalised to semigroups because then we have a way to tackle the sunset sum problem
In computer science, the subset sum problem is an important decision problem in complexity theory and cryptography. There are several equivalent formulations of the problem. One of them is: given a set (or multiset) of integers, is there a non-empty subset whose sum is zero? For example, given the set { − 7 , − 3 , − 2 , 5 , 8 } {\displaystyle \{-7,-3,-2,5,8\}} , the answer is yes because the subset { − 3...
 
7:16 PM
Evening chat
 
welcome back
 
7:49 PM
@user170039 Ah no but I have pictures of the lecture notes that I can share.
 
Hello, if $a_n$ are real numbers and $I_n$ are half open intervals of type $[a,b)$ and $\sum_{n=0}^\infty |a_n|\mu(I_n)<\infty$, then is it true that $\sum_{n=0}^\infty a_n\chi_{I_n}$ converges almost everywhere? ($\mu$ is the Lebesgue measure so that if $I_n=[a,b)$ where $b>a$ then $\mu(I_n)=b-a$, $\chi_{I_n}$ is the characteristic function of $I_n$)
 
This recent SMBC was uncomfortably on point: smbc-comics.com/comic/stress
 
Yikes
 
8:18 PM
@BalarkaSen If $f:M\to N$ is an $A$-module hom and $\mathfrak a$ is an ideal, do we always get an induced hom $M/\mathfrak aM\to N/\mathfrak aN$ or is there some condition
seems like it should be fine.
 
looks like a commutative square
 
8:35 PM
@RyanUnger This is true, yes.
$f(\mathfrak{a}M) \subset \mathfrak{a}N$, so it passes to the quotients
 
9:06 PM
hi chat
 
9:36 PM
Hi guys
One question:
is there any nice book to read about triangles and its properties, or maybe one covering all geometry, but from a very low level
?
 
what do you mean by low level ?
 
Well, showing where sin, cos, and so one come from
and the different types of triangles
and so on
not one starting with complicated theorems
 
There are plenty of resources on the subject on diverse websites
If you want a book specifically I don't know
But any basic introduction to trigonometry should cover these things
 
yes you're right, I'm diving on the web to see if I found something good
 
For instance, googling "introduction to trigonometry" gives you mathsisfun.com/algebra/trigonometry.html
Which might cover what you're asking about
 
9:50 PM
Hey guys, anybody know if there's a "standard" $\sigma$-algebra for $\mathbb{N}_0 \cup \{\infty\}$?
 
@RyanUnger yeah, that was on point too. Reminded me of an old onion video: youtu.be/NYSxkqL9l_8
 
I'm reading some probability theory stuff and the book claims that a certain class of $\mathbb{N}_0 \cup \{\infty\}$-valued functions $N \to \mathbb{N}_0 \cup \{\infty\}$ are measurable for a certain $\sigma$-algebra $\mathcal{N}$ on $N$. But I'm wondering what the $\sigma$-algebra on the $\mathbb{N}_0 \cup \{\infty\}$ the measurability is referring to.
 
@Astyx that's cool, thanks
 
No reference to it in the book. So I assume that it's somehow a standard one.
 
I'm trying to learn mathematics not through many formulas but by ideas and visual thinking, but I can't find very good books
Some from walter warwick sawyer are nice
 
9:59 PM
If $k$ is a perfect field and $P\in k[X]$ then the Galois extension $k[X]/(P)$ has the symmetric group of order $\deg P$ for its Galois group iff $P$ is irreducible right ?
 
is $N$ the natural numbers?
 
Depends on context @happyEddie
 
sorry, it was meant for zxmkn. what does he mean by $N$?
 
ah ok, no problem
btw what I wrote above about Galois groups is wrong and I know why (in case someone wants to answer it)
 
10:15 PM
It's not even necessarily Galois
 
Is it not ?
 
e.g. $\Bbb Q[X]/(X^3 - 2)$
 
Oh right, I meant $k[x_i]$ where the $x_i$ are the roots of a polynomial $P$
Didn't realise that wasn't the same thing
 
You mean a splitting field of $P$ I think
 
Yeah, probably
 
10:19 PM
@zxmkn: if $N$ is a topological space then there's the Borel sigma-algebra which is generated by all the open sets
 
Are there any non trivial cases where the Galois group of a finite Galois extension of a perfect field is the symmetric group ?
 
@Astyx sure, to take my previous example, the splitting field of $X^3 - 2$ has $S_3$ as its Galois group
 
@zxmkn: but maybe try to find if $\mathcal{N}$ is defined somewhere else in the book
 
Hmm true
 
@happyEddie No, N here is actually a particular set of $\mathbb{N}_0 \cup \{\infty\}$-valued measures. I was thinking about the Borel sigma-algebra for $\mathbb{N}_0 \cup \{\infty\}$, but I don't know if that has some obvious topology.
 
10:24 PM
That's because $3$ is prime though
It can't happen with $S_n$ where $n$ isn't prime, can it ?
 
@Astyx You can construct a splitting field for a polynomial by successively factoring your polynomial in each extension into irreducibles, picking one, and then quotienting
so that the degree of the splitting field is bounded above by $\deg f!$
 
@happyEddie The definition for the measure space $(N, \mathcal{N})$ is clear. But not the sigma-algebra on $\mathbb{N}_0 \cup \{\infty\}$ that the $\mathcal{N}/?$-measurability refers to. Thanks for the response, though!
 
@zxmkn: sorry, i misread, you were looking for the sigma algebra on the codomain, not the domain
 
Right
I think I'm asking whether that bound can always be reached
 
@zxmkn: would it make sense if the sigma-algebra were the whole power set of $\mathbb{N}_0 \cup \{\infty\}$
 
10:29 PM
In fact according to some other post, most polynomials will have $S_{\deg f}$ as their Galois group
 
@happyEddie Oh, right. I should check if that works out. If so, then it doesn't matter which sigma-algebra was actually meant, since measurability would then work for any sigma-alg. on the codomain. Thanks!
 
Hmmm that seems a bit counterintuitive to me for some reason
Thank you for the link
 
No worries :)
 
@zxmkn: yeah then it would actually make sense that the sigma-algebra on the codomain was not specified
 
10:45 PM
What a generator of the splitting field of $X^3-2$ over $\Bbb Q$ ?
 
Meh I'll give this some sleep
Thank you again for your help
Bye chat
 
no dont leave
anyways, i have a question
regarding math, of course
 
@zxmkn: or another way to interpret that: measurable with respect to whichever sigma-algebra on the codomain, hopefully it works out
 
@Astyx given that its Galois group is $S_3$ it doesn't have just one generator
 
10:50 PM
hello?
Is there anyone active in this chat?
 
@elipson_-1 don't ask to ask, just ask :)
 
Aren't all finite Galois extensions over perfect fields monogeneous ? I was thinking of $i\sqrt{3}\over{2^{1/3}}$
 
@Astyx I see sorry, I misread your question
This is called the "Primitive element theorem"
 
Right
So if I find any element of order $6$ (as for instance the one I gave above) I know it generates the extension right ?
Anyway II'm gone for good now
Good day/night
 
how come my silkroad room got deleted that's hate speech
 
11:03 PM
@Astyx if that were the case then you could write down a $\Bbb Q$-linear map $T : \Bbb Q(\sqrt[3]{2}, j) \to \Bbb Q\left( \frac{i\sqrt{3}}{\sqrt[3]{2}}\right)$
where $j^3 = 1$ and $j \notin \Bbb R$
 
@Adam try asking in the math mod office
 
@Astyx actually that's not quite enough because they can be isomorphic as $\Bbb Q$-vector spaces
in fact I think you're right so ignore me
lol
 
11:35 PM
is it possible to find the solution to:
$x^3+y^3=1$ and $x^2+y^2=1$?
for $x,y \in \Bbb R(0,1)$
 
Uh..., that's just circles in l2,l3 missing out the axes intersections?
 
@skullpetrol well it's a website I don't know if it will ever get THAT liberal
 
@Secret yes i want find the point where they intersect
 
That's just (0,1), (1,0)
set x=0 or y=0 and the value of the remaining one becomes obvious
 
$x,y \in (0,1)$
the interval is open
 
11:42 PM
In that case, you have no intersections
 
oh I made an error
I meant $(1-x)^3+y^3=1$
intersecting $x^2+y^2=1$
 
It's some cubic, so the two intersections are algebraic, any other details I am not sure
 
@Secret do you know what a groupoid is
plotted some points^
the natural density is $\approx .457$
 
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