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00:58
Hello guys!!
If $\gcd(a,b)=1$ can we ensure that at least one of both $a$ or $b$ is a prime number? Justify why
What is $\text{gcd}(15,14)$?
@Rithaniel oh, either $15$ and $14$ are not prime numbers hehe. Thank you!
No problem.
$\ddot\smile$
01:52
Hey there!
Is an affine parametrization only defined for geodesics? or also for any arbitrary curve in spacetime?
02:18
I've got a question about the name of a particular optimization problem.
I have coins of about 20 different denominations. I have a large number of every kind of coin. I want to buy an item from a merchant who does not give change. So, I have to give the merchant coins which at least total the price of the item, but subject to that constraint, I want to give the merchant as little money as possible.
And the question is, how many of each coin should I give the merchant in order to accomplish this?
I'm just wondering if anyone knows a name for this problem offhand.
Going by Wikipedia, it looks like it's an integer linear programming problem. I don't know if it has a more specific name.
 
3 hours later…
05:06
Heyo chat!
any thoughts on this one-
2
Q: Obtaining positive eigenvalues of the matrix $A$?

BAYMAXLet us consider the matrix $A$ which has three parameters $R,C1,C3$. This is from the Ikeda map in real form. It is defined as $$x \rightarrow R+(x \cos(\tau)-y \sin(\tau))$$ $$y \rightarrow x\sin(\tau)+y\cos(\tau)$$ The Jacobain matrix is given by: \begin{equation*} A = \begin{bmatrix} \cos...

05:41
I'm back from bannation
:'D
@TannerSwett it starts with an F
the name of that problem
I saw it in a book actually today
I looked for 10mins, can't find the book
but it was an integer programming book
The coin problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that asks for the largest monetary amount that cannot be obtained using only coins of specified denominations. For example, the largest amount that cannot be obtained using only coins of 3 and 5 units is 7 units. The solution to this problem for a given set of coin denominations is called the Frobenius number of the set. The Frobenius number exists as long as the set of coin denominations has no common divisor greater than 1. There is an...
Frobenius
Discuss
*the news article
I can't believe there's no one here. Where are people in China
India
IIT
Bomb bay!
effin all nighters man
one's coming on to me as I type this
Effin cigarettes and caffeine and so on...
And of course the math
Math can be deadly that way
Gettin tweaked on math, faces of math, simplices
:D
 
4 hours later…
cis
cis
10:31
Let A,B,C points of the plane. Where are points X with AX + BX = CX ?

X need to lie on the arc under AB of the circumscribed circle (of triangle ABC) .

Spontanious idea how to proof?
 
2 hours later…
12:19
@ShineOnYouCrazyDiamond That doesn't quite sound right. I'm looking for the smallest monetary amount (above a given threshold) that can be obtained using only coins of specified denominations.
Example: if I have coins of denomination 69, 110, and 161, what's the smallest amount of money I can make which is at least 371?
Correct me if I'm wrong but $H_0(\Omega S^k) = \mathbb{Z}$ for all $k \geq 1$ right?
@Perturbative I'm not very sure about $k=1$
surely $H_0(\Omega S^k) = \pi_0(\Omega S^k) = \pi_1(S^k)$
Ohh I forgot about how the loop space functor works on homotopy groups, that's a nice way to see it
yeah $H_0(\Omega S^k) = 0$ for $k > 1$ then
that should be right
12:35
Thanks Leaky!
@Perturbative a path between loops is a homotopy right
that should give you a natural correspondence between $H_0(\Omega X)$ and $\pi_1(X)$
well
wait
it should be $H_0(\Omega X) = \Bbb Z^{\pi_1(X)}$ insetad
i'm dumb
13:09
It has a product and a co-product which makes it into the Hopf algebra $\Bbb Z[\pi_1(X)]$
13:55
Hopfffff
So no one's going to discuss the recent publication and news about the sensitivity conjecture for boolean circuits?
It's another AKS-like paper
Gives us amateurs hope
14:35
@LeakyNun $\mathbb{Z}[\pi_1(X)]$ being a group ring right?
Ah okay, I should try and flesh out the details on my own for that a bit later
Let $V\subseteq\mathbb{R}^N$ be open, $g\in C^2(V,\mathbb{R}^m)$, $c\in g(V)$ be a regular value and $M:=g^{-1}(c)$. Let $p\in M$ and $v\in T_pM$. By definition, there is a $C^1$-curve $\gamma\colon(-\varepsilon,\varepsilon)\rightarrow M$ for some $\varepsilon>0$ such that $\gamma(0)=p$, $\gamma^{\prime}(0)=v$. Does there always exist another curve satisfying the same properties that additionally is twice differentiable at $0$? If yes, is $g\in C^2(V,\mathbb{R}^m)$ (rather than $C^1$) necessary?
14:59
Maybe I'm missing something, but why is it in Hatcher's calculation of $H_q(\Omega S^k)$ using the Leray-Serre spectral sequence, that $E^2_{k, 0} = H_0(\Omega S^k) = \mathbb{Z}$?
Sorry, Hatcher uses $n$ instead of $k$ above. so $E^2_{n, 0} = H_0(\Omega S^n) = \mathbb{Z}$
15:18
I am not sure what the confusion is
You're worried about $n>1$?
The loop space is uninteresting for $n=0,1$
$\Omega S^0$ is a single point, and $\Omega S^1$ is homotopy equivalent to the discrete space $\Bbb Z$
15:42
oooh spectral sequence
o0oo wow (drake meme)
@MikeMiller Yep, $n > 1$
I don't understand what your question is
Anyone want to talk earthquake theory?
Okay, so assume $n > 1$, I'm trying to use the Leray-Serre spectral sequence to calculate $H_q(\Omega S^n)$. To do this we use the path-space fibration (I think that's what it's called), $\Omega S^n \to P \to S^n$. Now the $E^2$ page of the spectral sequence is only non-zero on the $0$-th column and $n$-th column (and only in the first quadrant). And in particular $E^2_{n, 0} = H_0(\Omega S^n)$ (I've skipped some steps but we end up here).
Now in Hatcher's text the diagram above seems to assert that $E^2_{n, 0} = H_0(\Omega S^n) = \mathbb{Z}$ but we know that $H_0(\Omega S^n) = 0$ so I must be doing something wrong somewhere
So my question is why does $E^2_{n, 0} = \mathbb{Z}$?
16:21
@Perturbative yeah $H_0$ should be $\Bbb Z$ not $0$
it's $\Bbb Z^{\text{number of path-components}}$
I confused it with just the number
maybe you were confused with reduced homology
Okay I'm a bit more confused now, so I don't mean to jump around, but in Lecture Notes in Algebraic Topology by Davis and Kirk it's stated that $H_0(\Omega S^k) = 0$
On pg 245
But I see that $H_0(\Omega S^n) = \mathbb{Z}$ since essentially $\Omega S^n$ is path connected for $n > 1$
Or possibly pg 247 of Davis and Kirk's book (depending on which version you're using I guess)
Possible typo?
16:47
@Perturbative the errata for Davis & Kirk can be found here: indiana.edu/~lniat/typos.pdf
in particular, the top of page 10 gives the following: *p. 245, line 1. Replace “H0(ΩS^k) = 0” by “H0(ΩS^k) = Z”
So yeah, the typo is on page 245
@Semiclassical Thanks a lot! I googled the errata but couldn't find it
yeah, i had to search a bit
@RyanUnger thanks for the heat equation suggestion. You said the heat equation is instantaneously analytic. Here's a quick (unrelated) question for you: Given $N$ dipoles (with the particles at the boundary of a unit disk), it is generally very difficult to calculate the all the forces between the particles correct?
on jim davis's site he's got his books listed here: indiana.edu/~jfdavis/books
and the errata is nicely right there
@Ultradark uhhhh I guess so but you'd have to ask a physicist
16:59
Okay I will
I don't think computing all the forces is particularly hard. What would be hard is to compute how the system would evolve in response to those forces
i mean, what makes the three-body problem is not "compute all the forces between the three particles"
right
because in real life things aren't that static
it's to compute how those particles will move in time as a result of the forces at that time and all future times
right
probably chaotic and related to the $n$ body problem
to compute the forces don't you need to solve the relevant Poisson problem
17:02
hmm
it does depend on boundary conditions, i guess
Thanks @Semi
i mean, if it's just "N dipoles in free space" then you just add them up
and thank goodness for the linearity of the Poisson equation
right if the dipoles are somehow nailed down
right. say, they're attached to some ring
and their orientation is also 'nailed down'
if, by contrast, you allow them to rotate
then you're not just looking for what the forces are, but what configuration of those orientations is stable
and that's definitely much harder
are you talking 3-dimensional now or still 2d
17:05
hmm
i mentally was thinking 2D
shouldn't matter
i.e. that the orientation was still in the plane of the ring
you just have a different integral kernel
log vs 1/x
yeah I was thinking it wouldn't fundamentally change the problem
if you have more than 2 of these guys and they're allowed to move then it's a well studied but fundamentally impossible problem in general
17:07
by 'allowed to move', do you mean translation or rotation (or both)
i could believe that just allowing rotation would be enough to make it intractable
so if you fix them so they can only rotate along the boundary of the disk this might work
if they're completely free then it's a 3 (or more) body problem
so are they all rotating at the same speed around the disk keeping the same space between them
well, what i had in mind was to specify the locations of the dipoles along the ring (and their dipole moments) but not their orientations
i can see that getting hard by itself, though. suppose I arrange three such dipoles at the corners of a triangle
oh yeah the orientations...impossible problem lol
i'm pretty sure that you run into issues of geometric frustration
i mean, if you've got two dipoles which can freely rotate, they'll orient themselves to point along the same line
but there's clearly no way to do that with a triangle
17:17
so considering the two dipoles. say they are situated on the disk equally apart from one another and fixed at these locations. dipole A spins counterclockwise while dipole B spins clockwise
and so the dipoles are fixed in space and rotate at the same velocity
what happens physically inside the disk?
the field lines don't interact right?
between the two dipoles
17:48
Any one here a CS major and want to publish a paper with me? I think two heads are better than one
I'm independent from any university, so I ask here
Not to shock you or anything, but P=NP is seeming more and more likely to me
Minus that, we have an interesting way of analysing a certain problem that hasn't been published before
with regard to the certain problem
*anything major, as long as you can understand formal math proof
Anyway, you have my email
I have a bit of a debate going on in the comments section of an answer, and I'm wondering if someone with a bit more of a mathematics background can chime in. Here's the answer in question: security.stackexchange.com/a/161509/213456 The debate is surrounding a sentence in the "Interesting note" section at the bottom. It says, "a 1000 possibility combination lock, if guessed randomly, actually has a 1 in 630 chance of getting it correct."
I disagree with that statement in the comments, and further think it's worded in a rather confusing manner.
@ShineOnYouCrazyDiamond were you formely banana cats?
18:11
@Ultradark yep
@Ultradark that project took another turn. It has to be rewritten in C++ / Qt Creator / Qt Quick for mobile + desktop and graphics scene speed.
Is Jack Zimmerman here?
But yes, I am formally creator of the BananaCats app.
:D
Which of course I am the only user of in its primitive state
And my useage is 100% accompanied by a debugger
A person did email me, trying to find them
This is the only place I posted the wanted ad
crickets
@rschwieb actually if you want to learn about category theory, you can read about it on my blog :) I'm currently writing on an introductory series about category theory. Start here: wlou.blog/2019/06/18/a-brief-introduction-to-categories-part-1 you may also like my posts about representation theory of groups (which I develop from a non-commutative algebra point of view)
@MatheinBoulomenos props on category theory perspective
bookmarked
@ShineOnYouCrazyDiamond it's just some basic stuff, but I'm always glad to have readers :) Even if you know the theory, there might be some new examples or perspective for you
I humbly request pictures, lots of pictures. Category theory in particular is very visual
18:21
CD's is what I mean
it's a pain to get commutative diagrams to work on wordpress, so I'm a bit lazy on that (but not too lazy!)
0
A: Is there a way to show Yoneda equality visually instead of compositional algebraic symbols?

Shine On You Crazy Diamond Now here shows some features that my app should have including commutation color highlighting. What your not seeing is the diagram of category $C$ on the left with $\text{Hom}(A, \cdot)$ and $F(\cdot)$ arrows going to this category diagram (essentially representing functors), the fact that mos...

Turn them into images
yeah that's what I do
You won't get near expressive enough power in LaTeX / TiKzCD as you would with another tool or image creator
Anyway, see my answer at the bottom of that post (link above)
@ShineOnYouCrazyDiamond it seems to me that you're knowledgable about cats. Feedback is always welcome
18:23
I've never seen that proof of Yoneda ANYWHERE! It's a visual one-diagram thing
I invented that proof
you're welcome to spread it
*use it
It's more of a mock-up for an app I'm creating
but that's on the back burner right now
I will look into your app, but not today. I just got home from a conference, so there are a lot of emails to write and papers to read etc.
The app is not ready for you
but when it is, you'll know about it because I frequent MSE
I'm looking forward to seeing that
18:24
Or just keep in touch [email protected]
It's called BananaCats
Apple computer, fruit => banana, category = cats!
Do you also like Fun Cat Pics?
Yes when they come around naturally, I don't seek them out or anything
for that splash screen I found it on google image search
that's my favourite fun cat pic
18:26
That's a beautiful definition
Indeed. It's marvellous
I would however use D and take out the C'
or some letter
yeah
I also like \mathcal for the name of a category
BananaCats supports auto-indexing using a simple method it's nice
@ÍgjøgnumMeg what's that?
18:28
a seminar I registered for
@ÍgjøgnumMeg sounds perfect for you! I don't think you'll get credit for a proseminar as a masters student, but if you want to learn some topology (which you should), this looks like a great opportunity
Oh now I see your name
@Mathein I'm not sure if you do or not (maybe just one or smth?)
and you're even doing covering spaces, that's a great topic
but yeah this will be nice lol
yeah I figure I can maybe talk about the connection to Galois theory too
that would be great
18:30
yeeee
:)
Good jerb! Send us videos
That's unlikely to happen lol
@ÍgjøgnumMeg Banagl will give you enough leeway to talk about the analogy
@ShineOnYouCrazyDiamond the seminar is in German
I sprache Sie eine a bitte Deutsch
@Mathein ah nice! I'm wondering about the personalities of the lecturers
most of mine during undergrad were nice
18:31
@ÍgjøgnumMeg I can brief you on that, but not in a public chatroom
hahaha fair
Whatever you do, invite me to where you talk about it! I want to hear
lol
@ÍgjøgnumMeg btw when you arrive here, you should go to Venjakob's office hours right away
just tell him you want to do Iwasawa theory and he will find a way
@Mathein how come?
Oh woah
yeah
you can go to Michael Fütterer as well
18:33
hahaha I figure he's an authority on Iwasawa theory
he's a post-doc of Venkajob who wrote a Spanish book on Iwasawa theory
I want to do Iwasawa theory too, can you talk to Elon Musk, get him to rocket me over to Germany, I'll be there in an hour or less. Thx
and he's a cool dude
he offered me to give me a short intro to Iwasawa theory before
That's amazing, isn't CFT standard in Iwasawa theory tho?
yeah kinda
but you can blackbox that
you'll learn it in ANT2
18:34
Sure
I can't believe I'm in a chat room full of geniuses!
Well that's exciting!
lol
Iwasawa theory is neat
Could you describe it?
do you know some algebraic number theory?
18:35
Rudiments
mostly abstract algebra
gets popcorn
I have yet to attempt ANT
@MatheinBoulomenos I'll definitely take a look at your blog. Actually I have learned the absolute rudiments of category theory, the problem is that none of it imparted any fluency or intuition for the useful basics that everyone seems to know (like which functors commute with which, and how all the exact sequence lemmas work.)
@rschwieb to really appreciate category theory you have to get engrossed in a field that uses it regularly
I mean, i worked through four chapters of Mac Lane's book twice, I think, and while it's very clear, I didn't get any intutition.
18:37
@rschwieb I try to give some intuition as well, not just dry definition-lemma-proof
I've found it to be a skill that atrophies very quickly
@MatheinBoulomenos Yeah, exactly. I never had any chance to exercise it a lot, beyond idenitfying what i had encountered before.
1st world problems :)
@ShineOnYouCrazyDiamond @ÍgjøgnumMeg I'll talk some Iwasawa theory, just give me a moment
"god, I pray to you, please let me exercise category theory more often than not"
18:38
cool
K, gonna have a smoke
brb
@MatheinBoulomenos Actually the next book I want to read on category theory is that book by Brendan Fong and David Spivak that Baez promoted
@Perturbative Step back and ignore errata. This is something very basic you should understand on an intuitive level. For any space $X$, there is a canonical isomorphism $H_0(X;A) \cong A^{\pi_0 X}$, with the map from the first group go the second group given by counting the number of points in each component, and the map in the other direction given by inclusion of a point in each component.
@MatheinBoulomenos Get a chance to check that one out?
@Mathein looks like they cancelled the seminar on quadratic forms
braces why are quadratic forms so interesting to need a seminar
I mean... what are those
does something crazy happen over general fields/rings
Yes, math happens
it's crazy
@Ryan I think the main interest is in representing integers by quadratic forms
rofl
18:42
i mean, quadratic forms are interesting enough but they don't seem altogether novel
underwhelmed
Those pesky whole numbers
@RyanUnger why are you so opinionated on other people's interests
Oh, there are infinite twin primes, it's just going to take 200 pages of analysis and 20 years of study
I'm not
that was a legitimate question
18:42
@RyanUnger oh
I'm sorry
hence the "does something crazy happen over general fields/rings"
what interesting questions remain in quadratic forms, yeah
I can talk about quadratic forms, but I can't do that while also talking Iwasawa theory
Please, proceed
Whatever you want to talk about
@ÍgjøgnumMeg
@ShineOnYouCrazyDiamond
Okay, so fix a number field $K$ and a Galois extension $L_{\infty}/K$ such that $\mathrm{Gal}(L_{\infty}/K)=\Bbb Z_p$
18:45
oooh... I need to study p-adic stuff first
:D
no wait
I can describe it in elementary terms
is that p-adic or Z mod p
number theorists should invent a notation for that
very confusing to be the same as Z mod p
we have tho
18:46
$L_{\infty} = ?$
$\mathrm{Adic}_p$
just all other people need to stop using $\Bbb Z_p$ incorrectly for $\Bbb Z/(p)$ :)
^^^^^^^^
yeah but it's like number theory vs. everyone else
I agree, or $\Bbb{Z}/p \Bbb{Z}$
but that's usually for group thoery
18:47
@RyanUnger idc. If you want to read some NT, get used to the convetions. That's not unique to NT
for a cyclic group I just write $C_p$
that clarifies that I'm not thinking of a ring
A number field = some algebraic extension of $\Bbb{Q}$?
finite extension
I mean "okay" when I said K
lol notations...
18:49
I know, I was joking
So we're dealing with a field containing a subfield, and the field $K$ can also be viewed as a vector space over the smaller field of dimension $n \in \Bbb{N}$.
So anyway, the datum of $L_\infty$ is by infinite Galois theory equivalent to giving a tower of field extensions $K \subset L_1 \subset L_2 \subset \dots$ such that all $L_i$ are Galois over $K$ with $\mathrm{Gal}(L_i/K)=\Bbb Z/p^i\Bbb Z$
And you took an inverse limit?
we then have $L_{\infty}=\bigcup_{i=1}^\infty L_i$
@ShineOnYouCrazyDiamond a colimit (=union) of field extensions corresponds to an inverse limit of Galois groups (due to contravariance)
this tower of fields thing should explain the notation $L_{\infty}$
I was going to talk about it anyway
What's the first neat thing that happens in the theory?
I'm listenin'!
please try to practise some patience
Please be patient with my learning to practice patience
:D
18:53
So for every $n$, you can look at the $p$-component of the class group of $L_n$, (i.e. the unique $p$-Sylow subgroup of the class group)
(the $p$-Sylow is unique as the class group is abelian)
Okay, I give up. I see what you're implying I need to study ANT first
yeah
@ÍgjøgnumMeg you're with me?
Book recommends?
@ShineOnYouCrazyDiamond kinda depends. Do you prefer conciseness or comprehensiveness? How much background on commutative algebra do you have?
18:55
I know a little Module Theory up to Nakayama's lemma there abouts
and I know basic homological algebra concepts
homological algebra will be important for later, but irrelevant for the basics
I want a modern treatment written after 2000
just look at Neukirch - Algebraic Number Theory
I think I have that in my books folder lel
Dude, I do have that!
@MatheinBoulomenos what is the class group?
18:57
I should read it soon
if you can work through chapter I, that would be sufficient to get the gist of Iwasawa theory
@Ryan group of fractional ideals of a number field modulo the principal fractional ideals
Cool, thank you for teaching!
@RyanUnger geometrically, line bundles over $\mathrm{Spec}(\mathcal O_K)$ which form a group under tensor products where the inverse is the dual line bundle (this works for manifolds as well, you do that mostly for complex manifolds, I think)
18:58
I know it just keeps on going forever (the nested definitions)
@ShineOnYouCrazyDiamond yeah makes trying to learn some number theory on the side impossible
it's a full time job
NT is tough
I'm not gonna lie
but in my subjective opinion it's worth it :) that depends on your taste of course
Yes, I like prime number problems
:D
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