the triangles without squiggles that visit them must feel so lonely

those poor triangles were spurnered

@robjohn the bug seems to have been resolved (on my end, at least)

@copper.hat It seems to have been for me, as well. I deleted my meta posts since others were saying it was a duplicate. That is questionable, but it is moot now.

hold on...

@satan29 $\frac12\begin{bmatrix}-\frac{\sqrt{t^1-1}}{t-1}&\frac{\sqrt{t^1-1}}{t-1}\\1&1\end{bmatrix}\begin{bmatrix}\left(t-\sqrt{t^2-1}\right)^n&0\\0&\left(t+\sqrt{t^2-1}\right)^n\end{bmatrix}\begin{bmatrix}-\frac{t-1}{\sqrt{t^1-1}}&1\\\frac{t-1}{\sqrt{t^1-1}}&1\end{bmatrix}$

@robjohn question, do you know any good books for dynamical systems for undegrad?

@EM4 sorry, I don't

thanks :)

Hirsch Smale

made me think of SMERSH for some reason, maybe the recent bond discussions...

For a subspace W of a vector space V over field F, is it true that $fW=W$, where $f\in F$?

I think yes if $f\ne 0$

Because, if $f\ne 0$ then $fw \in fW\implies fw\in W$ (as W is subspace) so $fW\subset W$ and $w\in W\implies f (f^{-1}W)\in fW$ (as $f\ne 0$ so must have inverse in F) and therefore $W\subset fW$. It follows that fW=W.

what does $f \in F$ mean?????

copper, $f$ is an element of field F.

ok, you have two different $F$s

three infact, but one is lower case.

@copper.hat yes. capital eff is field and lower case f belongs to capital eff.

$W$ is a subspace iff $W+W \subset W$ and $\lambda W \subset W$ for all $\lambda$ in the field.

You have F and $F$ and $f$.

ahh, i forgot to put $$ around that :(

@copper.hat yes but $W\subset fW$ is true only for non zero $f$. right?

yes, $f W \subset W \subset {1 \over f} W \subset W$ for such $f$

alright. thanks copper :)

i am very grumpy today.

That sound like a title for a song.

:-). it will tomorrow soon for me so i will need to change the title :-0

"Yesterday, all my troubles seemed so far away...."

Objects in the mirror are closer than they appear.

i liked that in jurassic park

good night/good morning!

the limit as the observer goes outwards to infinity is even that they're already here

is that some monster or something

anyways it cheered me up because it means mathematical fluency is within reach

Which deductive system people use at the meta level?

nevermind that question

I've heard $x^2 - 6x +2$ could be decomposed into linear factors if we use complex numbers

where would the technique for this be discussed?

oh this is the fundamental theorem of algebra

hm... well, although i know it has linear factors, is there a way to obtain them in general?

oh this is literally the quadratic formula, no?

if I find the zeroes I can find the linear factors

never done it with complex numbers tho

(also $x^2 - 6x +2$ isn't a good example because it has real roots)

@shintuku yes

How do we choose n and k here? en.wikipedia.org/wiki/Prime-factor_FFT_algorithm

Or do any work that satisfy the mapping

I know that the LCM of 26 and 35 is 910, but what if I only need an approximation of 35. 3*26=78, which is close to 70. basicly what I'm asking is, is there something else that would be useful? It's for rising demons in a game out of dead bodies.

35 hitpoints=1 demon

26 hitpoints has a hellhound and can be converted to demons

4 (5 with an artifact) has an Imp

obviously 5 is trivial

→ 3 messages moved to ­Trash

why i am being censored

> We don’t tolerate any language likely to offend or alienate people based on race, gender, sexual orientation, or religion — and those are just a few examples. Use stated pronouns (when known). When in doubt, don't use language that might offend or alienate.

Code of Conduct

You weren't being offensive hence why I didn't suspend you but inciting or motivating racist discussion is not something I will allow here. Room owners here and Math.SE mods can overrule me if they want.

what a quality question anyways

what values can $x$ even be in $26*n=35*m+x$, where $m,n\in\mathbb N$?

Can anyone help me with this? math.stackexchange.com/q/4212962/876009

Hello, any idea how to find the boundary of a distribution please? Can we find boundary if a distribution in 2d and 3d or we can find it for n-dimensional data as well?

@SAJW find the minimum by isolating $x$

since $m,n \in \mathbb{N}$ you can tell the minimum

oh, I forgot n>m

so obviously x=26n-35m, but what now?

the 26 can be some other values, but I first want to understand one case (hellhounds->demons)

also the quickest method is to let n always be a multiple of 35... then x is 0

but at the start you don't have the firepower to kill your own hellhounds in that big of numbers fast

maybe the solution for demonfarming is to simply wait until you can in this game

map the gaps that $x$ can't occupy in $\mathbb{N}$

you have to begin doing it manually, writing a couple of iterations of the sequence, then see if there is some combination of $n,m$ after which there are no more gaps

@hyper-neutrino No overruling from me.

why is it bit.ly, will it steal my kidney if I open it

I have opened the link repeatedly and can confirm that I have some lumps where my kidneys should be

meh

i could formulate it in this way: x=26n mod 35?

jserv, i read it as whether the grant is successful in its aims. otherwise the separate data about "whether the grant fails" is a little odd (a grant not awarded cannot fail)

That's good reasoning, Leslie, thanks. That does mean that with only a 5% chance of success, the final answer is extremely low

grant=1000?

The grant is 5M

oops, should better learn to read

@SAJW $26\cdot31=35\cdot23+1$

so $x$ can be any positive integer

it can actually be any integer

(or 31 cases in our example, since 1 is the best besides 0)

@SAJW The extended euclidean algorithm. See this answer

woah that's nice

$26\cdot4=35\cdot3-1$

so $x$ can be any negative integer

this says that $3\cdot35-4\cdot26=1$ which is $-3\cdot35+4\cdot26=-1$ add the homogeneous solution $26\cdot35-35\cdot26=0$ to get $23\cdot35-31\cdot26=-1$ which can be rearranged to $31\cdot26=23\cdot35+1$

I'm now willing to admit I've spent far longer than 20 minutes on this problem, and I'm almost certain I've got it wrong. Can anyone please point out any obvious errors? i.ibb.co/XscV365/Screenshot-2021-07-30-195201.png

Edit: phrasing in the last sentence should be overall social ROI, not just social sROI if successful: i.ibb.co/DtfRQ2h/Screenshot-2021-07-30-195459.png

@hyper-neutrino I came all the way here to invite you to the Cafe chat for a new word-search game. All others here, @robjohn, @TedShifrin, @leslietownes, @copper.hat, Everyone is welcome, in fact!!

Also, note that the linked chatroom in my last post is Cafe and Tavern on the math.se, in case that might perk users' interest! ;D

Hang on, I think I've got one. I've flipped DALYs being recovered and DALYs being lost in one of my steps

Hello

random growth is easier to understand on random surfaces than smooth ones. The randomness in the growth model speaks, in a sense, the same language as the randomness on the surface on which the growth model proceeds.

cool!

@geocalc33 what models are you using for random growth?

@amWhy All the way here! I hope that that trip didn't take up an inordinate amount of your day!

@robjohn Awww, Let me pretend I ran a marathon to get here, at least. ;D

when i play boggle with my daughter the rule is that she only wins if she gets at least twice my score. word search is her forte not mine :-(

@copper.hat There was a Boggle game in the tea room at Fine Hall. That is about the only place I've played Boggle.

@robjohn i have not been to Princeton yet...

@copper.hat Hah! I like that. Well I'll convey, all the way over here to this chat, there are bonus points for answering why INTERNATIONAL SPACE STATION seemed relevant, now, to be a search phrase???? I.e. anyone here anything of late about it??

the russian module pushed the ss out of the way delaying the boeing flight?

Now who can't come up with a four or more lettered word from the all caps phrase.

or was that a joke? i'm out of sorts today

@copper.hat Yay!! You win. Yes, and it took took the station out of alignment, so took some nifty work from the ground to bring it back. See, do not underestimate yourself!!

@copper.hat No joke. You dun gud!

i am only confident about things that do not matter :-)

on the positive side, i am going to meet my daughter for a working trip in ireland shorly

both of us will be working remotely, not sure how that will work out.

@copper.hat Wow! That's awesome!

she is more disciplined than me

@copper.hat Maybe she'll inspire you!

it will be fun. a bit challenging socially as we are usually there on vacation

she always inspires me :-)

i try not to pollute her with my fears :-)

. o O ( intentional passion pattern )

looks like bubbles :-)

$$.\,\tiny{\text{o}}\, \text{O}\, \large{\text{O}}\, \Large{\text{O}}$$

i was always a nervous scuba diver, but i loved watching the bubbles go up :-)

Can anybody help me understand why the mobius strip and holed cross cap are homeomorphic? I am not getting it very much- there is self intersection in the holed cross cap but no self intersection in the mobius strip...

theres a self intersection only when you immerse it in R^3

the actual topological space that is a cross-cap has no self-intersections

think of a klein bottle, which has no "self-intersections", but if you try to see it in R^3 you have to have self-intersections to draw it.

im following ...

the way to visualize the actual topological space is to treat it as a subset of R^4 = R^3 x R, the last coordinate representing a color coordinate; so the space is a colored cross-cap, but the two sheets which intersect have different color coordinates (blue and red, respectively)

this means they do not intersect in R^4. two points in R^4 = R^3 x R(color) are the same iff they have the same location in R^3 and have the same color

this is the cross cap in R^4.

if you forget the color of points you get the cross cap in R^3.

which has self-intersections.

Hm - then can I ask if a Mobius strip in R^3 is in fact homeomorphic to a cross cap in R^4 or a hold cross cap in R^4

Holed cross cap*

Hi, a @Balarka

a topological space does not sit inside R^3 or R^4. its an entity independent of where its embedded. a mobius strip is homeomorphic to a punctured cross cap.

hi @Ted

Ted would you be interested in helping me with some general topology today ?

Balarka excels at this!

Okay then...

Try to fold a Mobius strip along its center circle, all the time imagining that the strip can go through itself

You'll end up with the punctured cross cap

i dont think you're remembering what you read correctly

a mobius strip with a disk attached along the boundary circle is RP^2

nothing to do with being in R^3

because this webpage mathcurve.com/surfaces.gb/planprojectif/planprojectif.shtml

also says something about 'classical constructions.'

where does it say about "intersecting a mobius strip with iteself"?

If you would please look at the very bottom in the second figure.

thats a disk with self-intersections

not a mobius strip intersecting itself

Second to last of them - one's captioned A Möbius strip (with self-intersection, but there is a boundary, a face)

thats a cross-cap with a disk removed. that is, as we discussed, homeomorphic to the Mobius strip

well im not sure then how the mobius strip is homeomorphic to this holed cross cap, which doesn't intersect itself because it is in R^4, yet there is a mobius strip with 'self-intersection'...

it is nonsense to talk about self-intersection of topological spaces

these are diagrams that lets you visualize the topological spaces, not the topological spaces itself.

like i said, you'd understand what's happening if you try to solve this exercise:

get some pen and paper and tape, cut out some strips, do the experiment. these are spaces, not presheaves, you cannot understand them by just philosophizing about them

@shintuku I was just reading an article and thought that sentence was interesting

I'd say the mobius strip with self intersection there is an immersion of a holed cross cap then am i right??

i wonder if there is a program that will let you do these experiments virtually (& easily)?

@geocalc33 i liked the idea, does the article have examples?

copper thats what i think sometimes *_ *

@fiapresheaf Yes, this is correct. "Immersion" in the loose sense of the word (like immersing a bottle in a tub of liquid), yes.

@shintuku no examples but I can give you 2 keywords for further investigation: SLE-curve, random geometry

thanks!

random surface

SLE stands for Schramm Loewner Evolution

found the wikipedia article