Mathematics - 2020-04-12

skullpatrol

12:26 AM

mathematicians are shocked that this paper got published

CaptainAmerica16

@skullpatrol I keep seeing articles about this everywhere. The whole thing is starting to sound kind of shady XD

skullpatrol

yup

slim shady

2

Thorgott

wow

Balarka Sen

kurims kyoto slash tilde mochi

inter galactic teichmuller should just stay in that meme webpage instead of being published

dont even care if its right or wrong

it should be where it belongs

CaptainAmerica16

oof

Masterphile

12:35 AM

what are prerequisites for the study of spectral theory? some real and complex analysis and linear algebra, i guess?

skullpatrol

how long does your online camp last for? @BalarkaSen

Balarka Sen

2 more days

im done with my talks

skullpatrol

coolio

Thorgott

intergalactic lol

Knight

1:46 AM

@Thorgott You there? Can you help me before going to sleep?

I’m studying Fluid Mechanics and in my book it says: $$\mathbf F = - grad~U$$ Where the minus sign is prompted by the relation to the potential energy. The existence of potential function $U$ is not sufficient, $U$ must also be single valued within the space occupied by the liquid

Now, my doubt is: $U$ is a scalar and it is a function then it is very obvious that it will have a single value as an output, then why the book says “ the existence of potential function $U$ is not sufficient, $U$ must also be single valued within the ...”

Balarka Sen

3:33 AM

Ok, take the hyperboloid $t^2 - x^2 - y^2 = 1$ equipped with the metric $ds^2 = -dt^2 + dx^2 + dy^2$. Let's parametrize the hyperboloid as $(t, x, y) = (\cosh(\phi), \sinh(\phi)\cos(\theta), \sinh(\phi)\sin(\theta))$.

The metric should change in a reasonable way, let's see. $dt = \sinh(\phi) d\phi$, $dx = \cosh(\phi) \cos(\theta) d\phi - \sinh(\phi) \sin(\theta) d\theta$, $dy = \cosh(\phi)\sin(\theta) d\phi + \sinh(\phi)\cos(\theta) d\theta$

$ds^2 = -\sinh(\phi)^2 d\phi + \cosh^2(\phi)\cos^2(\theta) d\phi + \sinh^2(\phi)\sin^2(\theta) d\theta +\cosh^2(\phi)\sin^2(\theta) d\phi + \sinh^2(\phi)\cos^2(\theta) d\theta$

$ds^2 = d\phi + \sinh^2(\phi) d\theta$ if I did everything correctly.

Which sounds about right

So $(\Bbb R^2, dx^2 + \sinh^2(x) dy^2)$ is a model for $\Bbb H^2$.

Balarka Sen

7:37 AM

Consider $\Bbb R^2 \to \Bbb R^2$, $F(x, y) = (f(x) \cos(y), f(x) \sin(y))$. $F^*(dx^2 + dy^2) = (f'(x)\cos(y) dx - f(x) \sin(y) dy)^2 + (f'(x) \sin(y) dx + f(x) \cos(y) dy)^2$. Which is equal to $f'(x)^2 dx + f(x)^2 dy^2$

Leaky Nun

@BalarkaSen what are you doing

Balarka Sen

Thinking

Isn't it curious that the spiral map/weird polar coordinate changes the metric like that

Balarka Sen

8:23 AM

$F(x, y) = (f(x)\cos(c y)/c, f(x) \sin(c y)/c)$ gives $f'(x)^2/c^2 dx^2 + f(x)^2 dy^2$, no?

Yeah it does

Alessandro Codenotti

9:13 AM

@TedShifrin By the way I'm pretty sure now that doing this for $C^\infty_c(\Bbb R)$ followed by a density argument works, so there's no danger in assuming continuous

Edward Evans

Mo

Rning

Lol

Alessandro Codenotti

Now I want to show that the map $t\mapsto (f(x)\mapsto f(x+t))$ is not continuous as a map $[0,\infty)\to\mathcal L(L^p(\Bbb R))$ though

Hi @Edward

Edward Evans

Hiya :)

Alessandro Codenotti

You're doing functional analysis this semester, right? :P

Edward Evans

Yeaaaaah

Balarka Sen

9:22 AM

What is $\mathcal{L}(L^p(\Bbb R))$ man

Alessandro Codenotti

Bounded linear operators $L^p(\Bbb R)\to L^p(\Bbb R)$

With the topology induced by the operator norm

Balarka Sen

I know it's true that small translate of $L^p$-functions are $L^p$-close to the original function

Oof, that's confusing

Alessandro Codenotti

Ok let's fix some reasonable notation. Let $T(t)$ denote the map $L^p(\Bbb R)\to L^p(\Bbb R)$ that maps $f(x)$ to $f(x+t)$

What you're saying is that for every $f\in L^p(\Bbb R)$ the map $t\mapsto T(t)f$ is continuous as a map $[0,\infty)\to L^p(\Bbb R)$, which is true

However the map $t\mapsto T(t)$ itself is not continuous

Well I want to show that

Balarka Sen

Got it

Scary

Edward Evans

9:51 AM

$\sum_{k=1}^n \tau_\ell(\alpha_k)\tau_k(\alpha_j)$ is still just gonna be the trace of $\alpha_i\alpha_j$ i guess, where the $\tau$ are embeddings of a number field into C. The embedding is fixed in the first factor of the Summand and the Alphas vary with k, so all conjugates get hit, while in the second factor the alpha is fixed but the embedding varies, so again all conjugates get hit

Or maybe thats wrong af

Mike Miller

@AlessandroCodenotti What does this mean in practice? I guess that there is a continuous function $c(t)$ vanishing at 0 so that $|T(t)f - f| \leq c(t) |f|$, uniformly in $f$, using $L^p$ norms on both sides. Why should I get a contradiction here?

Well in particular if $|f_n| = 1$ for all $n$ but $f$ does not converge --- eg, take the $f_n$ to be the indicator function of the "even length $1/2^n$ intervals in the unit interval", so 1 on [0,1/2^n], [2/2^n, 3/2^n] and so on. Then we should have $T(1/2^{n+1})f_n - f_n \to 0$ in $L^p$. But this translation is precisely the indicator function of…

That gives a contradiction but no intuition for me

Hey @ShaVuklia!

elo Balarka:D

Long time

ikr

10:06 AM

@MikeMiller Hmm I'm misunderstanding your $f_n$ I think, I don't see why $|f_n|=1$. Maybe it should be that indicator function times $2^n$?

But then it doesn't converge wait

Mike Miller

@AlessandroCodenotti Sorry, $|f_n| = 1/2^{1/p}$. We have that $|f_n|^p$ is constant on a set of measure roughly $1/2$ (and zero elsewhere) at the value $1^p = 1$, so that the integral is $1/2$, and thus the $p$-norm is $|f_n| = 1/2^{1/p}$

Alessandro Codenotti

Ok

Mike Miller

Maybe it's still unclear with $f_n$ is? $f_n(x) = 1$ if the $n$'th binary digit of $x$ is 0

AKA $f_n(x) = 1$ if $\lfloor 2^n x \rfloor$ is even

Balarka Sen

Horrible

Alessandro Codenotti

Got it, I drew some pictures

Ok I see the rest of your argument too now

I'm not sure if this counts as intuition but I can tell you why I already knew that this map had to be discontinuous

Mike Miller

10:19 AM

Sure

Alessandro Codenotti

$t\mapsto T(t)$ is obviously a semigroup representation $[0,\infty)\to\mathcal L(X)$ (where $X=L^p(\Bbb R)$), and it is also strongly continuous because of the thing me and Balarka mentioned earlier

The infinitesimal generator of this semigroup is the derivative operator for the functions where it makes sense, I mentioned this in the garbo room a while ago, but I can elaborate if this is not clear

Mike Miller

It seems clear

Alessandro Codenotti

Ok perfect, do you also agree that this operator is not bounded?

Mike Miller

Sure

Alessandro Codenotti

There is a general result that says that for a $C_0$-semigroup of operators on $X$ having bounded generator is equivalent to the map $t\to T(t)$ being continuous as a map $[0,\infty)\to\mathcal L(X)$

The semigroups for which that map is continuous are called uniformly continuous semigroups, so the thing is that a semigroup $T(t)$ is uniformly continuous iff its generator $A$ is bounded iff $T(t)=e^{At}$ for all $t\geq 0$

While the generator of the translation semigroup on $L^p$ is unbounded, so the semigroup cannot be uniformly continuous

This is also the reason why interesting semigroups usually have unbounded operators (but it can be shown that the generator of a $C_0$-semigroup is always closed and densely defined, so they're about as nice as possible for an unbounded operator)

Mike Miller

10:30 AM

Yeah that's good

Edward Evans

11:43 AM

@Alessandro how does one usually go about calculating the Lebesgue volume of a set? I have a specific set that I need to calculate the Lebesgue volume of but I have no idea how to go about it lol

Alessandro Codenotti

volume means measure or is it something else?

Edward Evans

I assume it just means measure

I think they're used interchangably in Neukirch at least

interchangeably

dunno how to spell that word, but you get it

Alessandro Codenotti

Usually either your set is very nice, or it decomposes as a union/intersection of nice sets, or you cry in a corner

Edward Evans

Oh

Hang on, I think my set actually decomposes really nicely

$\lbrace (x_\tau) \in \prod_\tau \Bbb R : \lvert x_\rho \rvert < c_\rho,\ x_\sigma^2 + x_\overline{\sigma}^2 < c_\sigma^2 \rbrace$

Ashe Danni

Hi guys, Newcomer here!
I have a question:
Is cube root and cube root of unity the same thing?

Alessandro Codenotti

11:48 AM

What's $\tau$, $\rho$ and $\sigma$

Edward Evans

$\tau \in \operatorname{Hom}_\Bbb Q(K, \Bbb C)$ and $\rho$ are real embeddings and $\sigma$ are complex embeddings (which come in conjugate pairs)

but they're just indices here tbh, the point is I can decompose $\prod_\tau \Bbb R$ into $\prod_{\rho\ \text{real}}\Bbb R \times \prod_{\sigma\ \text{complex}} \Bbb R$

Masterphile

@AsheDanni cube root is a function which attains generally three different values when applied to some complex number, and cube root of unity is a cube root applied to unity! :D

Edward Evans

so I think that set is just $\lbrace (x_\rho) \in \prod_{\rho} \Bbb R : \lvert x_\rho\rvert < c_\rho \rbrace \cup \lbrace (x_\sigma) \in \prod_{\sigma} \Bbb R: x_\sigma^2 + x_\overline{\sigma}^2 < c_\sigma^2 \rbrace$

Alessandro Codenotti

Makes sense

Edward Evans

and the "volumes" of those guys shouldn't be too hard to calculate.. lol

thanks, I think I just needed someone to say "decompose" at me

Ashe Danni

11:54 AM

@Masterphile I want to know about unity too. But I'll google it. Thank you for your time.

Alessandro Codenotti

Haha you're welcome

Tobias Kildetoft

Anyone else following the Mochizuki stuff a bit like if it were a soap opera?

Balarka Sen

Yeah hahaha

Edward Evans

has something funny happened?

Balarka Sen

inter galactic is published now

Edward Evans

11:58 AM

hahaha

Nice

Tobias Kildetoft

@BalarkaSen Well, accepted for publication

Balarka Sen

ahhh ok

I have been following it like a hawk, got super excited when the whole Scholze-Stick v Mochi thing happened

Tobias Kildetoft

Probably the best place for in-depth commentary has turned out to be the comments section on Peter Woits blog (Not Even Wrong)

Balarka Sen

That was last season though

Tobias Kildetoft

There is even mathematical content in those comments

Balarka Sen

12:00 PM

something happened between the Mochi gang and Fesenko gang right

Tobias Kildetoft

(not that I understand any of it as I am not familiar with $p$-adic Galis stuff)

Edward Evans

Isn't Fesenko part of the Mochi gang?

Balarka Sen

yeah he vouches for the correctness of inter galactic

Tobias Kildetoft

I thought Fesenko was the #1 Mochozuki fanboi

Masterphile

Beastie Boys - Intergalactic

Balarka Sen

12:01 PM

but Mochi's colleague said something about Fesenko

and that created a crack in friendship between the gangs

Edward Evans

oof

Tobias Kildetoft

@BalarkaSen You should take a look at those comments I mentioned. Might be some stuff you understand there

Mats Granvik

@Secret At 24 minutes of this video: youtube.com/watch?v=1eAmxgINXrE

Alessandro Codenotti

@EdwardEvans He is half of the Mochi gang, the other half being Mochizuki himself

Balarka Sen

@TobiasKildetoft hahah as in drama?

i have 0 clue about the math

Edward Evans

12:02 PM

rofl

Tobias Kildetoft

@BalarkaSen As in math

Balarka Sen

log theta data lattice

Edward Evans

Scholze has the best hair so

Balarka Sen

i love the first thing scholze did after fields medal was diss on mochi

he planned it

Tobias Kildetoft

The comments actually seem to be managing to explain some of the ideas without using any of the silly language of the intergalactic papers

Balarka Sen

12:02 PM

Ah

Edward Evans

I'm finding your unironic use of intergalactic instead of interuniversal hilarious

Balarka Sen

Wow Scholze commented there @Tobias

Tobias Kildetoft

@BalarkaSen Yeah, together with several people who actually seem to know what they are talking about

Sha Vuklia

12:42 PM

guys, could someone explain this last step to me?

I tried to write it out explicitly in components, but since $L_X$ is a map from $M_n\mathbb R$ to $M_n\mathbb R$, I lost overview concerning my indices

Also, even though $L_X$ is linear, its differential is not given by the matrix $(X^i_j)$ right? that would hold if we had $L_X\colon\mathbb R^n\to\mathbb R^n\colon x\mapsto Xx$

Secret

2:14 PM

twitter.com/CardColm/status/1249038195880341505

Abhas Kumar Sinha

2:26 PM

@Secret heard today morning

moteutsch

2:56 PM

"Choose the continuous unit normal field ν along γ." What does this mean? (Gamma is a simple plane curve.)

Sha Vuklia

3:08 PM

guys, a document I'm reading claims that $SL_2(\mathbb C)$ is simply connected, as it is a matrix Lie group

that seems bold to me; not all matrix Lie groups are simply connected, are they?

ye no, I already found a counterexample

hm, why do they say that then;(

Balarka Sen

Yeah that doesn't make sense; $U(1)$.

Masterphile

@ShaVuklia If a matrix Lie group G is not simply connected, the degree to which it fails to be simply connected is encoded in the fundamental group of G.

Balarka Sen

That's nothing specific to a matrix Lie group, @Masterphile. If a space is not simply connected, the degree to which it is not is always encoded in it's fundamental group!

So I don't see your point

Astyx

That's what the fundamental group is by definition no ?

Balarka Sen

Yeah haha

Anonymous

3:23 PM

I was given an exercise to try to write $A_5$ as a subgroup product of its Sylow subgroups: $\mathbb Z_3, \mathbb Z_5$ and $ V$. I guess could try out all the $3!=6$ permutations, write out the elements explicitly and see which one matches $A_5$. But maybe there's an easier and less ugly way. Any idea?

Anonymous

4:02 PM

Uh, that may not have been a correct interpretation of the exercise. If the subgroup product has to be a (sub)group, the subgroups necessarily commute. I suppose what I've been asked to show is that any element of $A_5$ can be written as a product of elements from its Sylow subgroups. Though I'm not quite sure how to approach that either.

Astyx

What's $A_5$ again ?

Symmetries of the pentagon ?

Anonymous

@Astyx The alternating group of degree 5

Anonymous

@Astyx Yes

Astyx

And $V$ ?

Anonymous

I suppose it's more sensible to think of it in terms of the even permutation group of 5 letters in this context, though.

Anonymous

4:05 PM

@Astyx The Klein 4-group

Knight

@Astyx I need your help. It’s about Fluid Mechanics, should I shoot ?

Astyx

Sure, not sure I'll be able to help though

Anonymous

The Klein 4-group is not really a subgroup of $A_5$ however. It's just isomorphic to the Sylow 2-subgroups of $A_5$.

Astyx

You're likely to have more luck at the physics SE chat

or something like that

Wait, $A_5$ isn't the symmetries of the pentagon

Not according to wikipedia at least

Knight

In Sommerfeld’s Lectures on Theoretical Physics, Vol II, Chapter 2, Section 6, Page 43 we derive an expression for the equilibrium of liquids as $$ grad ~p = \mathbf F$$ Where $p$ is the pressure and $F$ is the exertnal force. Then he writes,
[ The equation above ]includes a very remarkable Theorem: equilibrium is only possible if the external force has a potential, that is, if $\mathbf F$!can be represented as the gradient of a scalar function: $$ \mathbf F = -grad ~U$$ Where the minus sign is prompted by the relation to the potential energy. The existence of the potential function $U$is …

Anonymous

4:10 PM

@Astyx Well, that's true. I was thinking of the dihedral groups. Forget it anyway. :P

Astyx

Anyways, if you know it's written as a product of those subgroups, and if you know the cardinal of the group and the subgroups (which are coprime) you can find the decomposition

But as you said, I'm not convinced the subgroups commute, so just taking the product might be overly simplistic

Anonymous

Product of group subsets

In mathematics, one can define a product of group subsets in a natural way. If S and T are subsets of a group G, then their product is the subset of G defined by S T = { s t : s ∈ S and t ∈ T } . {\displaystyle ST=\{st:s\in S{\text{ and }}t\in T\}.} The subsets S and T need not be subgroups for this product to be well defined. The associativity of this product follows from that of the group product. The product of...

Anonymous

> A lot more can be said in the case where S and T are subgroups. The product of two subgroups S and T of a group G is itself a subgroup of G if and only if ST = TS.

Anonymous

You mean you're not sure whether the subgroups commute in this context of $A_5$?

Astyx

Yes

Knight

4:13 PM

Astyx please ping when you get free :-)

Astyx

And also not sure about my claim about there being something singular with groups that are the product of their subgroups, but according to your quote I'm right on this point

@Knight I think he means the pressure has to be the same all around the liquid

single value, as in the pressure is constant on the surface of the liquid

Anonymous

Well, it shouldn't be too hard to check if $V, \mathbb Z_3, \mathbb Z_5$ pairwise commute.

Knight

@Astyx But he was talking about $U$ and it’s about $U$ that he said has to be single valued

Astyx

Yes, but in the equilibrium you can take $U=p$

I think

Knight

I think in equilibrium $ U + p = constant $

Astyx

4:20 PM

In equilibrium, the pressure is constant around the fluid

So U has to be as well

Knight

Okay he meant that potential energy at all point in the fluid is same?

Astyx

Yes

That is my understanding

Then again, I'm not an expert

Knight

Thank you 😊

Secret

4:57 PM

@MatsGranvik oh yeah, surreals, game theory and nimbers are very intimately related

Jaakko Seppälä

5:11 PM

What would be an example of a function that is bounded, piecewise defined as a rational functions, differentiable and $f'(x)>0$ always?

Astyx

5:24 PM

rationnal function meaning it only take values in rationnals ?

Tobias Kildetoft

@Astyx Usally it means quotient of polynomials

Astyx

Thank you

Leaky Nun

how do you say it in french?

Fonction rationnelle

En mathématiques, une fonction rationnelle est un rapport de fonctions polynomiales à valeurs dans un ensemble K. En pratique, cet ensemble est généralement R {\displaystyle \mathbb {R} } (ensemble des réels) ou C {\displaystyle \mathbb {C} } (ensemble des complexes). Si P et Q sont deux fonctions polynomiales et si Q n'est pas une fonction nulle, la fonction f = P…

Astyx

indeed

Astyx

5:36 PM

You could probably take ${1-{1\over x+1}}$ for $x>0$, $-1-{1\over x-1}$ for x<0, which gives you a monotone piecewise rationnal function that looks like a sigmoid

David Reed

Am I just awful at searching or is there actually no proof of A5's simplicity on this site. Want to make absolutely certain before I take the time to type one up

Astyx

2

Q: Generalizing the proof of simplicity of the alternating groups

The standard proof for the simplicity of the alternating groups consists of three claims. If $n \ge 5$ then (1) all elements of $A_n$ are products of 3-cycles, (2) all 3-cycles in $A_n$ are conjugate, and (3) any nontrivial normal subgroup must contain a 3-cycle. Thus any nontrivial normal subgr...

David Reed

5:53 PM

@Astyx That question was unanswered. The one I was thinking of putting up uses the combinatorial argument where you determine the # of elements of certain orders in A5 and then use lagranges theorem to determine the possible sizes of subgroups and show that each possible normal subgroup would have to contain more elements than its order using the fact that if $H$ is normal to $G$ and gcd$(|x|,|G/H| = 1$ then $x \in H$.

Astyx

Can't find anything

Apoorva Anand

6:32 PM

Hey y'all, can you please check my question out stackoverflow.com/questions/61176175/… :)

Ted Shifrin

6:50 PM

@Astyx No, $A_5$ is the symmetry group of the icosahedron/dodecahedron. The dihedral group of order $10$ is the symmetry group of the regular pentagon.

Astyx

Hey @Ted ! Yes I figured as I read the wikipedia page. It's been some time since I last studied finite groups

Ted Shifrin

My favorite proof of simplicity of $A_5$ (which I learned from Mike Artin) is using the symmetry group of the dodecahedron to see how many conjugates each order of element has. I'm not so fond of twiddling with permutations.

Astyx

I remember proving it last year but I don't remember how :p I should probably reread that sometime

moteutsch

"Choose the continuous unit normal field ν along γ." What does this mean? (Gamma is a simple plane curve.)

Trying to learn a little Geometry. They said this in a proof of Hopf Umlaufsatz.

user76284

7:27 PM

Is the proof-theoretical ordinal of the $n$th set of the Grzegorczyk hierarchy $\omega^n$?

for all $n < \omega$.

topologicalorientablesurface

8:07 PM

How do I show that 17 is not reducible in $\mathbb{Z}[\sqrt{10}]$?

Leaky Nun

@topologicalorientablesurface by solving $x^2-10y^2=0$ in $\Bbb Z/17\Bbb Z$

Secret

To be tested later: Going from a integral back to a sum by redividing

Leaky Nun

the existence of a non-trivial solution is equivalent to the condition $\left(\dfrac{10}{17}\right) = 1$

and then it is reciprocity time

topologicalorientablesurface

Hm.. I haven't taken a number theory course, and so im not supposed to use reciprocity. @LeakyNun

Leaky Nun

do it by hand

topologicalorientablesurface

8:34 PM

Sure, so I got $a^2\equiv 7(mod 10)$ now is it true that $a^2\leq 10$?

sorry, I mean $a\leq 10$

topologicalorientablesurface

8:53 PM

okay ,so

$289=N(a)N(b)=(a^2-10b^2)(c^2-10d^2)$

First case is $a^2-10b^2=17$

which means $a^2\equiv 7 (mod 10)$

now what

domocar1

9:21 PM

I'm searching for a formula that would do the following:
4 map to 5 , 5 map to 6 and 6 map to 4. I need to use mods. Tried my best, haven't been so lucky.

Ted Shifrin

@topologicalorientablesurface So what are the squares mod 10?

Way too vague a question, @domocar.

domocar1

Let me explain

Leaky Nun

x+1 - 2.5(x-4)(x-5)

interpolation reacc only

nitsua60

@ApoorvaAnand I'm not much of a math.se regular (regular lurker, though!), but I feel like the questions of "where did my program go wrong" and "how does their solution make sense" are pretty distinct. I know on my site I'd want to see that split into two.

(I, for instance, would feel perfectly comfortable answering the "how does their explanation make sense" question, and can't even take a stab at diagnosing your python. This illustrates the problem: by linking two not-necessarily related questions, you're limiting your answerer pool to those who're proficient with both, not either.)

Ted Shifrin

@nitsua: Although often people will write an answer just to one when this happens.

Just shows your point is perhaps better taken.

domocar1

9:26 PM

So my best shot was (((x+4)%6)+3)%3)+3 which maps 4->5 , 6->4 , but 5 -> 3. I want 5->6.

Hopefully I explained it well.

Alessandro Codenotti

Hi @Ted

Ted Shifrin

Hi @Alessandro, demonic one.

Leaky Nun

2.5 = 5/2 = (-2)/2 = -1 (mod 7)

so (x+1 + (x-4)(x-5)) % 7

Ted Shifrin

@domocar: What you've written down makes no sense.

Leaky Nun

why doesn't this work

Ted Shifrin

9:28 PM

At least, mathematically it makes no sense.

domocar1

@TedShifrin Does the symbol % confuse you ? I am programming, that's why .

So math in programming

Ted Shifrin

No, I know what that means.

nitsua60

@TedShifrin =)

Ted Shifrin

But to a mathematician what you are writing down is not well-defined. Working mod 3, 1 is the same as 4, but the numbers 1+3 = 4 and 4+3 = 7 are certainly different numbers.

domocar1

Yeah I get that

Leaky Nun

9:30 PM

how about (x+1+2(x-4)(x-5))%7

Ted Shifrin

If you do stuff mod 3, 4, 5, 6 and the same as 1, 2, 0.

domocar1

But is it possible to achieve what I want though ?

Leaky Nun

this works

Ted Shifrin

Leaky is writing something that at least makes sense.

domocar1

@TedShifrin I apologise

Ted Shifrin

9:30 PM

You don't need to apologize.

nitsua60

(I live a life of luxury: RPG has a low enough rate of QPD and an engaged-enough base of power users that things like that are routinely handled kindly as normal site business.)

Ted Shifrin

LOL, @nitsua: I have no idea what all those acronyms are.

domocar1

@LeakyNun are you sure ?

if I take x=4 , I get 2 don't I ?

oooh

im sorry

nitsua60

@TedShifrin Rocket-Propelled Grenade and Qualified Public Depository. Or maybe Role-Playing Games and Questions Per Day. I sometimes get mixed up =)

domocar1

@LeakyNun Thanks , I was brainfarting

Ted Shifrin

9:34 PM

I can see why you'd get mixed up :D

topologicalorientablesurface

@TedShifrin how do I calculate them?

$x^2 (mod 10)$ means remainder, right?

do I need to break to cases

even x and odd x

?

oh, what am I saying

its values ranges between 0 and 9

Ted Shifrin

9:59 PM

@topologicalorientablesurface And you might notice something about 1 and 9, 2 and 8, 3 and 7, etc. Can you explain?

Apoorva Anand

@nitsua60 The answer to "where did my program go wrong" (I thought whilst posting the question) depended on "how does their solution make sense" because one would have to understand their solution and then read my program and figure out where it went wrong. I now realize I didn't consider those who could help me with the latter. It'd be awesome if you could help me make sense of their solution, either here, or there as a comment or an answer. Or even a PM

s.harp

Witten will give an online colloqium tomorrow

thats kind of funny

web.math.ucsb.edu/~drm/WHCGP

Salech Alhasov

10:35 PM

Hello all!

Astyx

10:57 PM

chat.stackexchange.com/transcript/message/54063100#54063100 @JaakkoSeppälä

Shaun

Hello all :)

I'm studying topos theory for fun.

I want to do it justice.

In essence, I suppose, I want to avoid ending up a crank because of studying this stuff insufficiently.

That is, without due guidance & care . . .

I've ordered a number of textbooks in the area.

I want to, perhaps, if possible, take formal tuition from someone working in the area; I'm considering a taught Master's degree once I've finished/failed my current PhD.

What do you - any of you - think about topos theory?

Where might I be able to do such a Master's degree?

EnjoysMath

@Shaun I'm impressed

I have the book by Dover publications

but it's over my head

What I've found more attainable as book-finishing goal is

by the same author

And it's called "Sketches of an elephant"

If you email me at fruitfulapproach

(gmail)

I can send you a djvu copy of the book

But before you can tackle the first few pages you need a firm grounding in adjunctions / monads

Shaun

11:19 PM

Thank you, @EnjoysMath. It's over my head too! I have read Goldblatt's book and I'm struggling with Mac Lane and Moerdijk's. I'll send you an email shortly. (Thank you!) Back in 2013, I was introduced to adjunctions via Turi's Edinburgh notes.

@EnjoysMath: I've just sent you an email.

EnjoysMath

@Shaun I know some adjunction math. Let's study together

Sent you an email

I studied adjunctions from "Cats & Sheaves". Basically you start with $\text{Hom}_C(X, R(Y)) \simeq \text{Hom}_D(L(X), Y)$ and you can derive the unit / counit existence by using naturality diagrams and composing functors

I'm actually working on a program to edit category theory diagrams called BananaCats. It's far from finished. I was going to incorporate logic, but my lack of type theory understanding prohibits doing this honestly

So will probably just make it a diagram drawing tool with features like Functoring etc

It is actually a more complex problem because not only do you have to handle the logic, you have to make sure your categories are "adhesive, M-adhesive, or whatever" so that graph rewriting makes sense. Couldn't find a way to make it easier, though I do understand now how Double Pushout (DPO) rewriting works for graphs.

@Shaun tac.mta.ca/tac/reprints/articles/12/tr12.pdf

another good book

I'm liking Goldblatt's book though. It's right at my level

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$Mathematics$ Mathematics