Mathematics - 2016-04-12 (page 2 of 2)

user174558

15:18

Hi @MatsGranvik how is your work on the Riemann Hypothesis?

Mats Granvik

@JasonBourne Currently I am working on this sequence: oeis.org/draft/A264736

@JasonBourne Have you seen this? oeis.org/A262725

user174558

@MatsGranvik Nope, looks good. Well, I am only a banana.

Jessy Cat

15:38

0

Q: Find $y \in W_{2}^{1}[-1,1]$ s.t. $\forall x \in W_{2}^{1}[-1,1]$, $f(x)=\langle x, y \rangle$

Consider a Sobolev space $W_{2}^{1}[-1,1]$ with the following inner product: $\langle x, y \rangle = \int_{-1}^{1} [x(t)y(t)+x^{\prime}(t)y^{\prime}(t)]dt$. Let $f(x) = \int_{-1}^{1}e^{2t}x(t)dt$. I need to find $y \in$ the Sobolev space $W_{2}^{1}[-1,1]$ such that $\forall x \in W_{2}^{1}[-...

@JasonBourne, some mean looking guys with guns and a girl with a big face were looking for you...

user174558

@JessyCat Hello Jessica, LOL.

ADG

15:53

Hello

Mats Granvik

16:22

Does anyone know if there has been a change in submission policy at the OEIS? The drafts page is much shorter than it usually is.

Evinda

17:02

Hello @DanielFischer !!!

@DanielFischer Suppose that $k$ football matches are being done and a bet consists of the prediction of the result of each match, where the result can be 1 if the first group wins, 2 if the second group wins, or 0 if we have tie. So a bet is an element of $\{0,1,2 \}^k$. I want to show that $g(k) \geq \frac{3^k}{2k+1}$ where $g(k)$ is the minimum number of bets that is required so that it is sure that at least the second prize will be winned (there will be a bet with at most one wrong prediction).

Frank Booth

17:29

hello

i have a quick question, can someone pls help me with it

can someone help me come up with an example that if a,b,c are elements of a group, it needs not be |abc|=|cba|

i tried a couple of examples, nothing seems to work

user19405892

hi

Eric Stucky

Did you try anything in S6?

user19405892

I need help with this

Let $X^n$ be the $n$-skeleton of a CW complex $X$ with attaching functions $\phi_{\beta}:S^{n-1}_{\beta} \to X^{n-1}$, for all $\beta \in B$, and let $f:X^{n-1} \to Y$ be any free map. Then $f$ can be extended to a free map $\widetilde{f}:X^n \to Y$ if and only if $f \phi_{\beta} \simeq_{\text{free}} c_{\beta}:S^{n-1}_{\beta}$, where $c_{\beta}$ is a constant function.

Frank Booth

@EricStucky I am trying that now

Eric Stucky

Yeah I think [a,b,c]=[(123),(345),(156)] works

Frank Booth

17:39

@EricStucky thanks Eric

Eric Stucky

npnp

user, what is a free map?

('ai Mike)

Semiclassical

17:55

hi chat

Eric Stucky

yo sems

Semiclassical

hiya

how're things

Eric Stucky

haha well I woke up too late to get on my first flight so I had to reschedule for tonight

Semiclassical

ahh

Eric Stucky

Getting shit done though, instead of stuck in an airport, so that's a bonus

Mike Miller

17:59

@EricStucky Hi

Eric Stucky

eya

Kari

On some domain $\Omega$, $f \equiv 0 \implies \nabla f \equiv 0$ right?

In my notes, I've written that a region is defined by some properties, one of which is that $\nabla f \neq 0$ for all $x \in \partial\Omega = \{ x : f(x) = 0 \}$. This is surely wrong, isn't it?

Mike Miller

The boundary is not a domain.

Think about $x^2+y^2-1$.

Kari

So the boundary of the unit ball is all points where $x^2+y^2 = 1$ hence the defining function $f(x,y) = x^2 + y^2 - 1$ is $0$ on the boundary but $\nabla f = (2x, 2y)$ which can't be zero on the boundary.

I'm convinced! Thanks, @MikeMiller :-)

Semiclassical

ugh, why must my mathematica code be so painfully slow

Kari

18:08

Exponential time, @Semiclassical? :-b

Semiclassical

hard to tell. it's a coupled nonlinear PDE

Kari

What are you studying for the PDE to arise, @Semiclassical?

Semiclassical

with a parameter $\Lambda$ for the strength of the nonlinear term. i can do small $\Lambda$ and relatively large $\Lambda<0$, but large positive lambda is just brutal for mathematica

it comes from the KPZ equation but under certain assumptions about weak noise

Kari

Sounds wicked!

Have you heard of spherical harmonics, @Semiclassical?

Semiclassical

i'm a physics grad student. i've had to deal with them more than a few times leading up to now (though mercifully not lately)

Kari

18:12

Do you know of any good books that go into deriving them just from Laplace's equation in spherical polar?

Semiclassical

mostly in the context of electromagnetism and quantum mechanics.

not really. it's one of those subjects i knew once and don't really know now.

Mary Star

Hello!!

Does 0 contains in every prime ideal?

Kari

What's a prime ideal, @MaryStar?

Tobias Kildetoft

@MaryStar $0$ is contained is all ideals

Semiclassical

what's really frustrating about large positive lambda is that you run into issues of numerical convergence.

Kari

18:15

Numerical = pointwise?

Mary Star

@Kari $P$ is a prime ideal iff $\forall a,b\in R$ $a\cdot b\in P \Rightarrow a\in P\text{ or } b\in P$

@TobiasKildetoft Ah ok... Thanks!!

Kari

I'm guessing $R$ is a ring?

Semiclassical

numerical = i do it in mathematica iteratively several times and it doesn't go to a fixed point

Mary Star

@Kari Yes.

Huy

isn't a prime ideal also necessarily not all of $R$?

Tobias Kildetoft

18:16

@Huy yes

Semiclassical

instead, at sufficiently positive lambda it goes to a two-cycle. i can in principle fix that by modifying the iteration appropriately, but it makes things slower

to the extent that each iteration takes upwards of 10 minutes. and evidently I need to be more severe with my 'fix', since right now it still doesn't avoid the two-cycle.

Kari

Sounds brutal!

Semiclassical

it's a pain

Kari

Iterating is bad enough. Fixing them sounds cumbersome!

Semiclassical

well, the idea is simple enough

Tobias Kildetoft

18:20

Hmm, so if $s_{\lambda}(x_1,\dots,x_k)$ denotes the Schur polynomial associated to the partition $\lambda$ and $\rho = (n,n-1,\dots,1)$ then for any prime $p$ and any partition $\lambda$ with at most $n$ parts we have $s_{(p-1)\rho}(x_1,\dots,x_k)\cdot s_{\lambda}(x_1^p,\dots,x_k^p) = s_{(p-1)\rho + p\lambda}(x_1,\dots,x_k)$. I wonder if $p$ really needs to be a prime for that and whether it is easy to prove combinatorially.

Semiclassical

suppose i've got an iteration $x_k=f(x_{k-1})$ for some function $f$. for that to have a fixed point means that $x=f(x)$; to have a 2-cycle means that $x=f(f(x))$. whether they're attracting or repelling takes more detail.

now, suppose i modify my iteration to $x_k=f((1-s)x_{k-1}+s x_{k-2})$ where $0<s<1$

Tobias Kildetoft

(the only proof I know is representation theoretic and it would make no sense for $p$ not a prime, though it also works for $p$ a prime power)

Semiclassical

this preserves the fixed point (since then $x_{k-1}=x_{k-2}$) but not the 2-cycle

the question, though, is which $s$ to do. if $s=0$, then one is back to the original case and nothing has changed. if $s=1$, you've got $x_k=f(x_{k-2}$ which has the same fixed point problem

and plus you have to worry about it making everything much much slower.

Kari

This reminds me of fixed point theory in my differentiation module!

When you finally get it, I'm sure it'll be extremely gratifying, @Semiclassical!

Semiclassical

heh, i'm not surprised, with one big difference: i'm not doing anything rigorously, and the mapping isn't taking scalars to scalars but functions to functions

Evinda

18:41

How can we find this integral: $\int_{|x|=\epsilon} \frac{\phi}{|x|^2} dS$?

$\phi$ is a test function.

TheGreatDuck

Hi

Is there such a thing as a semicircular polar form for the complex number plane?

rather then the full circle polar model?

Eric Stucky

What do you have in mind?

(Also, hi)

TheGreatDuck

19:31

Me?

Eric Stucky

ya

Mike Miller

@PVAL Do you know any examples of non-smoothable homology 4-spheres?

Frank Booth

hello

i have a question

PVAL

@MikeMiller I don't think so. Think any of these would have to have infinite fundamental group.

Frank Booth

suppose that G has exactly 8 elements of order then. how many cyclic subgroups of order 10 does G have? i guess for all a, |<a>|=8, so, i have found 8 cyclic subgroups of order 10. are there others?

order 10*

Mike Miller

19:41

Non-cyclic yeah.

Non-cyclic infinite, I mean

PVAL

@FrankBooth You have potentially found less than 8.

How many generators are in a cyclic group of order 10?

Frank Booth

10

?

PVAL

@MikeMiller I don't think I know the condition to realize KS\ne 0 for somewhat complicated groups.

Mike Miller

I don't know it for any groups.

PVAL

well i know it for the trivial group

Frank Booth

19:44

in a cyclic group of order 10, there are 10 elements that can generate that subgroup

PVAL

It's just that the intersection form is odd.

@FrankBooth nope.

Mike Miller

There's a MathOverflow question looking for non-smoothable homology spheres. This could in principle happen in either dimension 4 or in dimensions at least 8 (when the obstruction class isn't automatically zero), though in the latter case the homology sphere is PL.

I thought about it a bit but can't write down examples in either case.

Frank Booth

@PVAL i mistyped i guess, i have found 8 elements of order 10

PVAL

Try again

Mike Miller

It's almost certainly false but I can't see anyone writing down examples.

Frank Booth

19:47

G has exactly 8 elements of order 10. i need to find the number of cyclic subgroups of order 10. for each of the elements of order 10, say, a, |<a>|=|a|=10. so there are 8 cyclic subgroups of order 10

are there others?

PVAL

How many generators are in a cyclic group of order 10?

i.e. how many elements of order 10 are in a single cyclic group of order 10.

Frank Booth

i dont understand it

can you pls elaborate a little

PVAL

How many elements of order 10 are in $\Bbb Z/10\Bbb Z$?

If you need more explanation you should consult your text for the definitions (e.g. order and cyclic).

Frank Booth

i swear i know them

i know the definitions

there are 10 elements in a cyclic group of order 10?

PVAL

how many elements of order 10

Frank Booth

19:55

one element that generates the group

PVAL

and..?

Frank Booth

the identity?

PVAL

I no longer believe you when you said you knew the definitions.

Frank Booth

i swear i do

i do

my head is gonna explode now

can you please help me with this?

PVAL

Calculate the order of all the elements in $\Bbb Z/10\Bbb Z$.

@MikeMiller The answer to your question is certainly either yes or that it is unknown.

Mike Miller

20:08

I agree.

PVAL

I'm sure there's many groups where Freedman's results say nothing about.

Mike Miller

I also feel this about the same question for arbitrary dimensions $n \geq 5$. It seems preposterous that homology spheres would all be smoothable.

Frank Booth

@PVAL are there 5 elements of order 10?

TheGreatDuck

hi im back

drive.google.com/file/d/0B_-_Z6IMqDH9NFdLMkl5Wkh1bVU/…

Mike Miller

@FrankBooth. Can you not just write down all of the elements in a row and what their order is

TheGreatDuck

20:10

Is there such a thing as a model for complex numbers built in this form?

As a semicircular polar coordinate?

PVAL

@MikeMiller Well I am saying this as close to a fact. There are groups with non-sub exponential growth with 0 abelization.

There is no way we have any idea of what goes on in these cases (e.g. s-cobordism controlled s-cobordism all are unknown topologically).

Mike Miller

I agree, but are you claiming that Freedman's result for other groups should imply that if they're homology spheres they're smoothable?

PVAL

Idk.

Mike Miller

Fair

PVAL

I'm saying there might be an example someone could construct, but there is no way anyone could prove the non-existence of such a thing with a non-good fundamental group.

TheGreatDuck

20:13

anyone here familiar with complex number models?

Mike Miller

@PVAL Ok, I see your point.

PVAL

I should mention though that I don't know offhand how to construct groups which grow quickly with 0 abelization, but I'd imagine that people have constructed similar things.

TheGreatDuck

Hello?

Mike Miller

@PVAL Sure, that doesn't sound surprising.

TheGreatDuck

@EricStucky This was what I had in mind. drive.google.com/file/d/0B_-_Z6IMqDH9NFdLMkl5Wkh1bVU/…

Eric Stucky

20:17

Yeah I have no idea what that picture is supposed to be telling me :/

PVAL

@MikeMiller I'd imagine the fundamental group of the Alexander Horned ball is already an example.

though that probably isnt finitely presented.

Eric Stucky

Unless you mean that, like $-a-bi$ is supposed to be identified with $-a+bi$ ?

TheGreatDuck

No, those are engles and radii

Eric Stucky

(The point you label as $-A$ does not appear to be $-A$, is what I'm getting at.)

TheGreatDuck

@angles

so the negation of that point in that direction is the point mirrored across the x-axis

in other words, negation is mirroring

square root cuts the angle in half

and it only exists on a semicircle

Eric Stucky

20:20

$y$-axis, presumably? Or doesn't it matter?

TheGreatDuck

is this just an angle or just the ramblings of a madman? I remember reading it a while back. Not sure if it is true

x-axis yes.]

Eric Stucky

If it doesn't matter, $x$-axis is easier, since then $-x=\overline x$.

TheGreatDuck

*y-axis

damnit

Eric Stucky

:P

TheGreatDuck

XD

i was curious if anyone has done a polar model with only half a circle...

or if it's even realistic

(It seems to break every rule with complex numbers but I'm not real sure.)

Eric Stucky

20:23

It depends on what its for

It shouldn't work as a number system, no.

TheGreatDuck

from what I understand it's just a model of them. XD

oh ok

probably just someone making a fatal flaw then'

Eric Stucky

Eh hold on

TheGreatDuck

(making negation mirror across y rather than mirror across origin)

thanks

?

Eric Stucky

Are you only interested in the unit circle, or do you want this to work for all complex numbers?

TheGreatDuck

this is a model of the complex number system

where there is an infinite number of "signs" ranging from negative to neutral to positive

Eric Stucky

20:26

But you're interested in all the numbers, not just the signs?

(the "signs" are collectively called the unit circle, btw)

TheGreatDuck

from what I'm understand the person is saying that complex numbers consist of a sign and a magnitude

or to use the polar model

angle and radius

Eric Stucky

yeah. Some trouble at zero but otherwise true.

TheGreatDuck

for some reason they made negation mirror across y-axis

and restricted themselves to 0 to pi radians

not sure why

other than potential mistakes in the model. :/

Eric Stucky

Well, it depends on what it's for

TheGreatDuck

well... As far as I'm aware it's a model for complex numbers

like how the number line is for real numbers

Eric Stucky

20:28

If we call the nonstandard negation $\lnot$, then $-y$ and $\lnot y$ have the same $x$-coordinate.

TheGreatDuck

they never really said a "use"

PVAL

Well it has multiple things with $a=-a$. Which is a problem.

Eric Stucky

o.O

PVAL: you mean $\lnot$ is not injective?

TheGreatDuck

That's the "neutral sign"

where it is neither negative nor positive

it is its own negative and it's own positive

Mike Miller

You sound like a crank.

Eric Stucky

20:30

Mike, I know this person :/

TheGreatDuck

@MikeMiller I did not write this model. It was something I saw a while back and now that I /know/ complex numbers I'm wondering if this is something legit or the work of a crank.

im leaning towards crank

Mike Miller

Sure, I wasn't paying much attention to the conversation. But the discussion of your neutral sign sounds like garbage is all.

TheGreatDuck

shrugs might be useful for magnetism

PVAL

Well usually to see if somethings written by a crank you shouldn't have to do any actual examination of the actual mathematics.

TheGreatDuck

good point

but if I remember correctly, this was in an online textbook.

and called "degrees of sign"

Eric Stucky

20:33

Wouldn't happen to be John Gabriel, would it? :P

PVAL

For instance if your acknowledgement section is replaced by " Please nominate me for the Abel prize."

TheGreatDuck

I only made the connection to complex numbers cause square root of negative one is the unit length along the neutral axis

@EricStucky the name sounds familiar

Lol he's an actor

notorious

How do you negate this sentence: If James gives a presentation on Thales, then Kevin knows why a triangle inscribed in a circle will have a right angle opposite a side that is a diameter. ???

Jinyuan gives a presentation on Thales, but Kevin does not know why a triangle inscribed in a circle will have a right angle opposite a side that is a diameter. ???

TheGreatDuck

Hold on I'm thinking. :)

The truth table for implication is:

FF = T

FT = T

TF = F

TT = T

what you want is FFTF

notorious

P^~Q

TheGreatDuck

20:43

thank you

i had a mental block for some reason

so it would be

James gives a presentation.... And Kevin does not know why a triangle inscribed....

notorious

ok

wait

TheGreatDuck

I'm not going anywhere. :)

notorious

Do you never have to negate the statements themselves?

TheGreatDuck

You negate the part after then

notorious

a circle will have a right angle opposite a side that is a diameter

TheGreatDuck

20:45

by saying "Kevin does NOT know"

the statement is saying what Kevin knows

notorious

ok

TheGreatDuck

therefore, the negation is that Kevin does not know

think about this

Bob knew 2+2 = 4

the negation would be:

Bob did not know 2+2 = 4

OR

notorious

Bob did not know 2+2 != 4

is wrong I see

TheGreatDuck

Bob though 2+2 was not equal to 4

however the second implies bob made a connection excluding 4

so I'd go with the first due to possible ambiguity

notorious

double negating just gives the original meaning of the statement pretty much

TheGreatDuck

20:47

yes!

it's called the double negative

Like when someone says yes to "do you mind X"

PVAL

It is false that if James gives a presentation on Thales, then Kevin knows why a triangle inscribed in a circle will have a right angle opposite a side that is a diameter.

notorious

that's cheating

TheGreatDuck

@PVAL he wanted the not distributed.

lol

not cheating

Just.... Smart-Alek-y

notorious

You can technically start every sentence with "it is false that..."

TheGreatDuck

^ :p

notorious

20:50

You can use 'but' instead of 'and'? it means the same thing right?

TheGreatDuck

no

think about grammar

the dog chased the cat, but the cat stood still

the dog chased the cat and it stood still

it's not quite the same meaning

notorious

the dog chased the cat, and the cat stood still

TheGreatDuck

the first carries the insinuation that the cat should run

notorious

the dog chased the cat, but the cat stood still

They are not logically equivalent?

TheGreatDuck

it's different

because but isn't pure logic

whereas and is

notorious

20:53

How would you write the dog chased the cat, but the cat stood still, using logic and statements like P and Q?

TheGreatDuck

But always carries a bitter taste. It essentially says "this is false. It should be true"

but has no logical meaning

it's an and that tells the reader it should be true yet it's false

notorious

That's weird cause my teacher sometimes uses it instead of and in negations

TheGreatDuck

like adding a footnote in an equation

meh

grammer and logic mix in weird ways

Eric Stucky

Yes, that's a metalogical feature, noto

If you are negating something, then in your brain you first had to assume that the thing was true

TheGreatDuck

They probably just thought that "but" didn't sound as esoteric

Eric Stucky

20:55

(not logically of course)

(but in order to see what it being false means, you first think about what it being true means)

TheGreatDuck

well it's just that the statement P & Q shouldn't by P but Q

as no assumption is made in a stand alone expression

it's a connotation adding by the English language

notorious

If James does well on the quiz, the Michael does well too.

James does well on the quiz, and Michael does not do well.

TheGreatDuck

James does well on the quiz and Michael does not do well

notorious

James does well on the quiz, but Michael does not do well.

TheGreatDuck

the but could be used or the and could be used.

notorious

20:57

All the time?

TheGreatDuck

It's really a matter of wording choice that sounds appropriate

yes

when converting logic?

notorious

yes

and writing it in english

TheGreatDuck

My last two posts got reversed

it was a rhetorical question

writing in English is different'

like I said, but is a connotative meaning

so you might need it in English

converting logic to English will always make a grammatically correct statement

they just might sound very weird and esoteric

is this logic class part of math or programming or what?

notorious

I guess I'll just always stick with "and" even though it might sound a little weird.

TheGreatDuck

is this part of a programming class?

notorious

21:00

no

geometry class

TheGreatDuck

hmm

are you familiar with programming cause I thought of a clever analogy

notorious

Euclidean Geometry. Just started this week

Kind of I only took 2 programming class. I know C

TheGreatDuck

tharts good enough

english is to logic as a library is to c

but happens to be one of those extra functions

and it acts ridiculously similar to and

but it isnt

weird huh?

notorious

I guess it makes sense

TheGreatDuck

Anyway I'm gonna head off

cya around

notorious

21:03

Ok thanks for the help

How do I negate this? I have no idea: There is no point common to all lines.

I think: There exists a point common to all lines

But I'm not using Quantifiers

Kari

21:24

That seems right ...

Mary Star

Does it hold that $\mathbb{Q}[x,y]/(x)=(\mathbb{Q}[x][y])/(x)=(\mathbb{Q}[y])[x]/(x)=\mathbb{Q}[y]‌$ ?

( (x) is an ideal )

PVAL

Try the obvious maps/

Check they are isomorphisms.

Frank Booth

21:52

hi

I have a problem. suppose H,K are subgroups, and HK= (hk, h in H, k in K), KH=(kh, h in H, k in K). I need to show that KH is a group iff HK=KH. can you help me fo the forward proof?

Oh I am sorry I solved it thanks dont worry about it :D

Jessy Cat

22:38

1

Q: Find $y \in W_{2}^{1}[-1,1]$ s.t. $\forall x \in W_{2}^{1}[-1,1]$, $f(x)=\langle x, y \rangle$

Consider a Sobolev space $W_{2}^{1}[-1,1]$ with the following inner product: $\langle x, y \rangle = \int_{-1}^{1} [x(t)y(t)+x^{\prime}(t)y^{\prime}(t)]dt$. Let $f(x) = \int_{-1}^{1}e^{2t}x(t)dt$. I need to find $y \in$ the Sobolev space $W_{2}^{1}[-1,1]$ such that $\forall x \in W_{2}^{1}[-...

analysis functional-analysis sobolev-spaces inner-product-space integral-equations

user174558

Hi @JessyCat. Why are you studying Sobolev spaces?

Akiva Weinberger

23:03

Is there a concept of a nonabelianization of a group? Like, given an abelian group $G$, it would be the largest group whose abelianization is $G$ (if such a notion makes sense)

So a sort of inverse of abelianization

anon

took me since yesterday to figure out I have to type [fragile] as an option for \begin{frame} to get tikzcd to work inside beamer. still don't know why it wouldn't recognize &s, don't know why this fixes it.

user174558

Hi @anon.

anon

heya. long time yadda yadda.

user174558

I am still alive.

Adeek

hi @anon

PVAL

23:10

@AkivaWeinberger Well such a group $G$ should have to satisfy $G \cong G \times P$ for any perfect group $P$. I'm not sure if any such $G$ exists.

Adeek

long time no see

PVAL

@Akiva In fact there should exist perfect groups of arbitrary cardinality.

anon

@JasonBourne same

user19405892

hi

anon

hi

Akiva Weinberger

23:11

Oh, right. Thanks, @PVAL

Adeek

that is nice

user174558

@anon Are you finishing undergrad soon?

anon

iunno

user174558

I am still trying to get better.

anon

have you tried X, Y and Z?

user174558

23:13

Well, I think I have tried what I need to try.

user174558

Chris has disappeared, prob working on her book.

bolbteppa

@JessyCat might help math.stackexchange.com/a/460809/82615

Paradox 101

How do I write $[T]$ if $T(a,b,0,0)=(a+b,b,0,0)$ in the basis $(e_1,e_2)$?

Akiva Weinberger

23:32

How many connected components can the complement of a homeomorphic image of $\Bbb R$ in the plane have? My guesses are either $3$ or $\infty$

Mike Miller

23:47

do you have a single example where it's 3?

Akiva Weinberger

@MikeMiller $r=2+\tanh(\theta)$ for $\theta\in\Bbb R$. It's an infinite spiral that fits between the concentric circles $r=1$ and $r=2$.

Note that it can't be extended to a homeomorphic image of a closed interval; its limit points are those in the union of those two circles.

Mike Miller

sure

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$Mathematics$ Mathematics