@TedShifrin I was sleeping, what did I miss here?

if I'm using a colormap, which typically explains quantities in relative terms, e.g. temperature, speed, should I still label the colormap as "relative speed", or is that kinda redundant?

well, you can drop the "relative" if you make it clear what the zero baseline is

some authors will also use e.g. +1 -1 etc. on a color bar to indicate some quantity is relative to some baseline

just make sure you make that clear in your presentation and captions and it should be fine

I see @Secret, in my code, I sort the values, create an indexed set to access the color array, and then write plotting code with a for loop. I've never used a range from -1 to + 1, just simply 0 to 1 (so that negative values, e.g. negative torques would map closer to 0, I think).

tandfonline.com/doi/abs/10.1080/…

They don't even define the term precisely. How can you do topology without open/closed sets, filters or whatever?

tandfonline.com/doi/10.1080/0969725042000272834

@Secret lots of people in social science who do mathematics do it in a sloppy way

So when a political scientist starts talking about topology just replace topology with

Topography

ah, no wonder why they go on and on about city layouts

hmmm... so an actual political topology does not really exist as a discipline yet

lol no

Sociology has like 40 subfields but you can probably summarize the whole field in 50 pages

Atleast it’s not as bad as some recent French philosophers who talked about complex numbers as a hidden motive of male patriarchy, since the number i represents the male penis

Am not even joking

so...the runge-kutta method.

why are there commas in the definitions of $k_1, k_2, k_3, k_4$?

as in, $k_1 = hf(x_n, y_n)$?

the output should be just a single number, $y_{n+1}$

(if i understand right.)

Because it has two inputs

the time step and the previous guess

$f(x,y)=x^2+y^2$ for example has two inputs, hence the comma

right but that doesn't mean it has to have two outputs...

it takes as inputs the time step and the previous guess and produces the new guess, right?

or, i suppose, what's the second output?

There isn't, the time step of the simulation is already set at the beginning, thus it remains the same throughout. the y value is the value you want to obtain after you integrate the trajectories, thus there is only one output

...then why are there commas?

you still need the time at the nth step to do the integration, think of runge kutta as partitioning a trajectory into segments dictated by the time step which you integrate them one by one

if you multiply a scalar with a couple of vectors, then add a bunch of vectors, you get out a vector output, not a single scalar output.

@Secret i understand why there must be two inputs. i don't understand how to apply the equations.

oh...wait.

let me ask a different question.

does using the runge-kutta method require knowledge of the exact formulation of the function?

like $f(x) = x^2$ or whatever? or can one use it cumulatively with a set of data points?

You do need to know $f$ in your initial value problem

okay, then i think i understand.

If you only have partial data, then you can sort of estimate $f$ with interpolations between the data points

such as splines and so on

for some reason it wasn't clicking that $f$ required those two, and that the comma was just separating the two arguments. i'm sorry, i can be rather thick-headed at times.

You are not alone. I recall I have to learn most of the numerical analysis from an MIT lecture note from scratch in my honours year in order to do my project

yeah, i'm learning about a bunch of different numerical integration methods on my own for a programming project i'm doing.

simpson method, trapezoidal, cumulative simpson, corrected trapezoidal, now runge-kutta...

the situation i'm programming for has very specific requirements, and i am also not very good at picking up these concepts, making it rather difficult. i'm learning though, and it's very interesting =)

i found a really useful book the other day on the subject which has helped a lot.

anyway, thank you very much for your help @Secret =)

Numerical analysis sounds boring

@Zee it's really not

Actually, anything with computers sounds boring

to each their own, i suppose.

i enjoy programming quite a bit.

But then again , idk anything about it so I can’t judge

you should try it some time.

I enjoy programming too but I don’t like it

I am not a very programming person, I am just forced to write programs so I can do my research

@Zee you enjoy but don't like? what is the difference?

Couse if I end up on a deserted island then I can’t program but I can continoue doing math

It’s a pride thing , my brain is my tool , I can’t rely on anything else

Although by my IQ maybe I should :p

Baam

Luckily, I think kinda like a computer, thus it only take a few headbangs and then I can write the code that will do exactly what I want it to do

albeit inefficient (otherwise I am a master hacker already)

I don’t think you think like a computer , you seem lot have the opposite cognitive style frankly

And I mean that in a good way...

@Secret sometimes computer scientists use "topology" to describe a "graph", the kind with vertices and edges

if that helps

drives me nuts

Yeah, I only noticed this is recent year, on how hopping around my thinking actually is, which is not like how most of the people I met and my friends said I am organised

As for programming, what I mean I think like a computer is that e.g. when I wrote a command that loops thorugh some entries and filter it in some way, I actually try to visualise what each step the code is doing as if I am the computer and thinking about what to do next

but that kind of habit is kinda thanks to my high school programming teacher who taught me the importance to put flags everywhere to debug code

@Secret actually, one more question about Runge-Kutta

when you say you need to know the function, is knowing the overall differential equation sufficient? because my system is a linked system of differential equations

so you mean your $f$ is defined in terms of a system of ODE and not in a closed form expression?

yes

I have not encountered that case before, my guess is that in integrating $f$ you will end up need to solve a recursive relation in terms of $f$ which will need some kind of root finding method such as Newton Ralpson

hmm, okay.

thanks =)

have a good night.

Nighty

back in chat baby yee ha

@Zee generally applies to all applied math that won't be evident with a bridge collapsing or extinction event, it makes me too angry to say anymore

Hi there! searching for @mercio, but guess he kind of sleeps ...

@Secret you'll find only part of you is computer like, me too, in most cases, I will try to model literally anything in terms of the Kronecker delta function

Hello one and all! I have a question regarding differential operators. If I introduce the differential operator $\delta = y d/dy$. Then what is $\delta^2$. My book says $y^2 d^2/dy^2 + y d/dy$ but I don't see how that comes out that way. Any suggestions?

Never mind! I am losing my mind..

@Rumplestillskin You worked out it was just the product rule?

Hmmm @Alex maybe I haven't worked it out!

@Alex I am not sure I understand where the extra $y d/dy$ term comes from

@Rumplestillskin Apply d/dy to (y d/dy) applying the product rule

@Alex it will be $d/dy + y d^2/dy^2$

Now multiply on the left by y

$y(d/dy)(y\, d/dy)$

Yep so it is now $ \delta (y d/dy) = \delta^2 $

You get that you're done right?

@Zee um what

i need to see this lol

@Rudi_Birnbaum hi

@mercio: Hi! I am slowly getting the picture. There are however two relevant extensions and one principal question. Maybe you again could give me decisive hints? The principal question I added as a comment on the MO (mathoverflow.net/questions/303066/…). And the first and important extension is the case that $V_\sigma$ branches in $(V_\sigma)|G'$.

The other extension would be the case that $V_\rho$ branches into more than two irreps in $G'$. The question then is if there are always two among those for wich the containment holds.

o..o

I'm not sure what you are asking anymore

I mean the one thing is: What happens if $(V_\rho)_{|G'} = V_{\rho_1^{'}} \oplus V_{\rho_2^{'}} \oplus V_{\rho_3^{'} ...}$ are then there always two $V_{\rho_i^{'}}$ and $V_{\rho_j^{'}}$ one can choose from so that $ V_{\sigma^{'}}\subseteq V_{\rho_i^{'}} \otimes V_{\rho_j^{'}}$ holds?

well if $V_\rho$ branches then $[V_\rho \otimes V_\rho]^-$ will also branch into many many pieces.

some of the form $[V_{\rho'_i} \otimes V_{\rho'_i}]^-$ and some of the form $V_{\rho'_i} \otimes V_{\rho'_j}$

if the $V_{\rho'_i}$ have dimension 1 then there is no $[V_{\rho'_i} \otimes V_{\rho'_i}]^-$

(they have dimension 0)

can someone please help me see my gaff here

how do I send jpegs

oh the silent treatment eh well two can play that game

actually no I cant

please help

what's a gaff?

@mercio: OK.

Then what is in case the $V_\sigma$ branches into more than one in $G'$? I conjecture that there should then at least always be one which appear in $V_{\rho_1^{'}} \oplus V_{\rho_2^{'}}$.

(but usually not all)

hmmm

hey everyone. My friend and I've been studying on a problem concerning orthogonal simultaneous diagonalization of symmetric matrices. We think symmetric matrices that commute have a common orthogonal base consisting of eigenvectors. Our attempted proof is here: math.stackexchange.com/questions/2822605/… would anyone mind taking a look at it. Thanks.

I am trying to plot all three of the equivalent expressions for the second Chebyshev function

and I'm off but only slightly and it's very simple its just annoying Im not seeing it immeadiately

I want to post the graph its easier then latec

a gaff is like a problem that is really simple that I'm obtuse to for whatever reason

so anyway how do I post jpegs

@Rudi_Birnbaum I think I am starting to see an interesting question

ive seen ppl post pictures so its kinda of impossible that im not being ghosted intentionally here

fine l8r crack exchange

that if $V_\sigma$ doesn't branch, it shouldn't be in an antisymmetric square of one of the $V_{\rho'_i}$ because it should be invariant by the "action" of $G$, and I suspect $G$ should switch around the antisymmetric squares

at least if $G'$ is normal in $G$ there might be a bit of sense in that

@Adam maybe you can post a link to a picture

yeah I really wanted to phase out facebook being in anyway helpful but u are right ok here it is

facebook.com/…

like ill be able to fix blue and reds difference pretty easy think that's just the summation bounds ive stuffed but black was a really poor guess at what it should be and now I want to know why its correct with a slight adjustment needed

anyway I know its basic as hell I can feel the shame in the force

-sry- I was just in discussion with another colleague.

im sorry im just too tired to do the latex thing im new to it

@Adam You can only upload image files when your reputation passes a certain threshold; I think the threshold is 200 reputation.

oh ok lame

they should just have admin powers to wipe retarded pics at 200

I imagine it is to stop trolls from posting obscene images.

Btw, if you go here, you'll see that the room is somewhat quiet at this time, from the activity graph: chat.stackexchange.com/rooms/info/36/mathematics

oh now that is cool real time chat activity plotting

whatever it is I will be a minority time shedule

ill just cope with the lag\

anyway its no problem ive only just started to study legit PNT so it's expected that ill have a bumpy road for the next 6 monthysd

and I should be sleeping

Wonder if this is legal pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf

but if anyone wants a laugh, remember back to a self help con called "the secret" and introduce yourself to Australian humour

@mercio: " I am starting to see an interesting question" that makes me happy!! I will be away now for about 12h but "I'll be back!", thank you again for interest and help thus far.

@Alex I think you're pinging the wrong person

Nope, it's just, afaik, not possible to selectively ping one of two people with identical usernames.

I think replying to an actual message itself works but I'm not certain

Pinging a message does work for sure yep, I should have just done that ¯ \ _(ツ)_/¯

@Adam ugh

I hate that crap

“The power of positive thinking” is problematic enough when it’s just a statement about individual psychology. But when that turns into a claim about the secret nature of the universe then I entirely lose my patience

@Semiclassical live and let believe

Plus, they tend to try to invoke quantum mysticism as a supposed ‘scientific’ basis for such

And that irritates me to no end

If I’m honest, though, stuff like the secret doesn’t bother so much. It doesn’t have enough traction / influence in the US

By contrast, the prosperity gospel is IMO the Christian equivalent of that. And, to the extent that evangelical Christians are a political influence in the US, it’s not something I can dismiss as just “someone else’s belief”

prosperity gospel?

you'd need to introduce me to the kinds of stupidity in the USA; I don't know them well enough

@LeakyNun let me count the ways...

$\deg f = n$, $\deg f(x^p) = np$?

sanity check

$n < np \implies n(p-1) > 0 \implies n>0 \land p>1$

Basically: material prosperity (wealth etc) as a sign of God’s favor and the depth of your faith

@Semiclassical and what is the influence of this belief?

megachurches exploiting the crap out of the poor

that's just the poor being stupid?

@EricSilva and why would you know about stuff in the US :o

i live in chicago

oh

Obligatory

@LeakyNun this is a really shitty thing to think

@EricSilva please enlighten me

if youre in a situation where literally every image of success in your life is constantly reinforcing a dumb belief youre not stupid to believe it

Question: In the Euclidean space $\Bbb{R}^n$ is it possible for $r > 0$ and two distinct points $x$ and $y$ in it to have $B(x,r) = B(y,r)$?...I don't think this is possible. But even looking at $\Bbb{R}^2$, I have tried several things and have had no luck. E.g., if $x=(x_1,x_2)$, $y=(y_1,y_2)$, and $x_1 < y_1$, then I can choose $k \in \Bbb{N}$ such that $1/k < r/2$ and $x_1 + 1/k < y_1$. With this in mind, I considered $(x_1+1/k,x_2)$ and $(y_1+1/k,y_2)$ to the $r$-balls are distinct...

but got lost in messy computations

I could use a hint.

Though if I’m honest it’s not even prosperity gospel as such that bothers me, as much as the broader belief that poverty /wealth is a mark of moral failing/ virtue

@user193319 draw a picture

hint: think about the line connecting $x$ and $y$

Prosperity gospel is more risible but less pervasive

@Semiclassical is that belief common in the US?

On the part of Republican politicians it sure seems to be

especially when youre in a crappy situation where there are very few recourses that could reasonably lead out, you take the options that people present to you, it's hope commodified to people who dont have access to it, it's exploitative and to blame it on the poor themselves is ridiculous

@EricSilva I see

@Semiclassical *the protestant ethic*^tm

@Semiclassical they must have forgotten the story of Job :P

Mos def

seriously, can Christians be consistent in their beliefs

if you want to be wrong, at least be consistent about it

@EricSilva that points to another thing I dislike about stuff like homeopathy etc: people investing money into it instead of stuff that actually has a basis for success

There’s a lot of room to criticize modern medicine/healthcare/pharmaceuticals

But if your response is “don’t trust drugs, trust homeopathy” then you’re a quack

i guess the thing that i think is bad about it is how everything is mystified

And given how important it is for stuff like cancer is to be treated/detected early, that kind of thinking is not merely delusive but actively dangerous

it's like people get into thinking that all the ills of modern society can be cured by like folksy wisdom

Ya

it's all coelho speak

Coelho?

paulo coelho

the writer

Ah

he wrote the alchemist which is like the pinnacle of self-help mysticism

the tagline of the book is like literally something like "if you want something hard enough the universe will help you get it"

Bleh

As poetry, I don’t mind that stuff

"coelho" isso e uma palavra?

means rabbit

but it's a common name

As a supposed insight into the nature of the universe, though, it can **** off

eu posso entender "LH", mais o "coe" me parece estrano

interessante

@Semiclassical take it from someone who read the book in 2 languages: he's a shit prosaist too

Particularly when it picks up a moralist dimension

Then you get into victim blaming / shaming

cuniculus > *cuniclum > cõello > coelho

interesting

the "o" was nasal!

internalizing the broader ills of society as individual neuroses is like, the milieu of late capital

I really don’t get how supposed Christians go down that route tho. As was noted earlier, the book of Job should be warning enough against that

@Semiclassical I'd say that Christianity in the US isn't about studying the bible thoroughly; more like believing what your pastor tells you

and believing what other Christian tells you

@TobiasKildetoft hi

@LeakyNun Hi

I proved that group homomorphisms preserve identity: would you like to have a look? @TobiasKildetoft

To be fair, the USA is big enough that it’s not homogeneous

Man, who puts an application deadline at 10 am?

sure

@Semiclassical probably because it's actually broader social forces that are just dolled up in the garb of christianity and not actually "a christian phenomenon" or w.e.

@TobiasKildetoft ^ there you go

I'll buy that

asshole atheists are also susceptible to self-help mystic garbage

The bible as a convenient pretext for other beliefs etc

@EricSilva interesting

Which tbh I agree with at some level: I don't consider the bible to have any inherent authority

@LeakyNun That looks horrifying

I get enough such diagrams in the paper I am reviewing right now

@EricSilva wiki says there's only 3% / 10% atheists in chicago :o

But they want to both use the Bible selectively and as a purported authority

@TobiasKildetoft lol i'm sorry :P (no i'm not)

And that deals with the $2$-categorical version of coalgebra and comodule objects in a category

@LeakyNun sounds about right

@EricSilva yeah, but they don't sway as many elections :/

@Semiclassical sure my point was more that it's not localized to christianity

in a well-ordered set, there necessarily exist immediate predecessors/successors, right?

oh, sure

@GFauxPas predecessor is not necessary

it's just that Christian examples are by far the dominant ones in the US context

why not?

@GFauxPas Only successor

can you give an illustrative example pls

@GFauxPas pick any ordinal $>\omega$

of one where there isnt predecessor

take two copies of the naturals and put one after the other for an example

Protestants in the US place a lot of stock in knowing the Bible and being able to reference it; I think personal study is quite often encouraged

I was raised Catholic and we don't care about that though lol

like there are lots of people who arent hardcore weird christians who are also like "but making ur bed every morning gives u magical powers"

Or just add a single element to the naturals which is larger than all of them

oh then the second $0$ has no predecessor?

@MikeMiller really

or the second $1$ wherever you want to start

right

@GFauxPas exactly

In ordinal's language Tobias's examples are $\omega+\omega$ and $\omega+1$

cool thanks :)

@MikeMiller me 2 but my parents made me read it a bunch

I only know about the first two ordinals, haven't studied any contexts where I needed more

Yeah @Leaky. That ofc doesn't mean their interpretations jive with the way I view the world, and those are surely informed by their pastor and Sunday school and whatnot

$\Omega$ and $\Omega \cup \{\Omega\}$

Uhmmm the first two ordinals are $0$ and $1$

infinite ordinals I mean

um

i dunno, what are they called

the two I stated

Well than that's $\omega$ and $\omega+1$ (or $\omega\cup\{\omega\}$)

catholic vs. protestant is a bit different insofar as protestantism was largely framed around the bible itself as a source of authority rather than the church

$\Omega$ is sometimes used instead of $\omega_1$ for the first uncountable ordinal

i.e. if you want to know what you should do, read the bible yourself rather than ask a priest

in both my analysis book and my topology book, they don't need anything bigger than $\Omega \cup \{\Omega\}$, it's not a general ordinal theory book

so, thats what I know

for now :)

Well those are pretty big, way bigger than $\omega$

It goes both ways, though. Consider the kerfuffle lately when the US Attorney General defended the zero-tolerance policy at the border with a bible verse

Well rather I should say

the kerfuffle not being "wait why is the US attorney general using a bible verse as a source of legal justification"

they're not really dealing with ordinal theory at all except for what you need to do things like induction

Fair, I'm surprised they even need $\Omega$ in an analysis book

but rather the fact that said policy seems to go against the spirit of christianity...oh, and that the verse had previously been used to justify the institution of slavery

@Semiclassical several Christian Republicans spoke out strongly against him on that

to build the Borel hierarchy?

well it's in "optional material"

yeah

and yes

I see, makes sense

on the grounds, though, that it wasn't actually consistent with Christianity

good point

which on the one hand I'm glad to see them recognizing it as such

Christianity isn't consistent with itself, so there's that

it follows that 0 = 1

but on the other hand it's crazy that such appeals are seen as pertinent to a policy question

yeah agreed

that's the problem of theocracy

well the US shouldn't be a theocracy

I wouldn't say it's a theocracy, but it's sure got some theocrats

the problem with attributing morality to god is that it solves no problem and creates more problems

@Semiclassical maybe not so crazy given us history

eh. it can be disturbing without being surprising

also Leaky, aren't we all inconsistent with whether we use $0$ or $1$ for identities of various structures?

lol

@GFauxPas o_O

mind=blown

i mean it's disturbing how much we've never very consistently applied "enlightenment ideals" that are supposed to be like the bedrock of american society but that's kind of par for the course

good luck separating the church and state in the US

let's see how many Christians remember the 1st amendment

that's always been problematic tho

i mean, take Jefferson

what about him

on one hand, he was very much an advocate of enlightenment ideals, in particular the distinction between church and state. on the other, he was a slaveowner and not a particularly nice example of one

Of course, this requires people to actually bother getting the history correct

and there's quite a few people on the right who have tried to distort that to their own ends

David Barton being an obvious example

@Semiclassical i mean i agree, this was kind of what i was saying anyway

sure

@Semi i think really i just wanna parrot horkheimer and adorno re: enlightenment regression into mythology

I've the impression that I should read Habermas eventually?

i havent read him but he's on my list of people to read at some point in the next decade

what is the problem

is it socio politics

or is it moral dilemma

just deconstruct it my dudes

the intrusion (slash pre-existence?) of religious thinking into American politics in particular

power dynamics my dude

it's just a male conspiracy

Foucault of my haters

For me it's a question of what should one take from the Enlightenment and what should be considered as antiquated. Jefferson is a good example of that

@Semiclassical yo read dialectic of the enlightenment

everything is interpretable as anything my dude

Author?

Jacques Derrida

it's what i name dropped

my frankfurt bois

horkheimer and adorno?

horkheimer and adorno

ye

mmkay

Is (the ability to program) essential to doing mathematics?

that question probably has nontrivial time dependence

rather the answer does

maybe the ancients had ability to program but never knew that