Mathematics - 2024-08-17

Xander Henderson

02:22

@psie What kind of bug? And was it really a bug?

Are you sure it was from the order hemiptera?

nickbros123

@Jakobian yes, and also uniqueness

I mean, I need basis in $V$ given a basis in $V^*$ so that $f_i(\alpha_j)=\delta_{ij}$

As of yet i dont know what $V^{**}$ is

Jakobian

02:47

@nickbros123 I thought that this boils down to the same thing, so its a bit circular

what I said that is

$V^{**}$ are linear functionals on $V^*$

The map which sends $v$ to the linear functional on $V^*$ given by $f(v)(w) = w(v)$ is actually an isomorphism of vector spaces

since $V$ is finite-dimensional

and so we are guaranteed that we can find such element $v\in V$, but the issue is how

and I don't think I really know how, unless perhaps we set up some basis of $V$ and go from there

so what I think is best solution is to set up a basis of $V$ and then set up some linear system of equations to solve your problem

nickbros123

@Jakobian I was thinking the same. Assume some basis, write an arbitrary transformation (in terms of the given $f_i$ and do a change of basis by post multiplying an invertible matrix, and argue about that mattix

Jakobian

well, its just a matter of setting up the linear system of equations

the solution and the "argue about the matrix part" comes after

this is still somewhat complicated thing to do, since given some basis $\beta_j$ of $V$, you need to know what $f_k(\beta_j)$ are

but I think this is necessary, if you didn't know how $f_j$ act on $V$, then you wouldn't be able to solve for $\alpha_j$ most likely

I could attempt to make this "wouldn't be able to solve" more rigorous but I don't see a need to

user535484

04:22

4

Q: Quinn's Uniqueness of Resolutions

I am reading the proof of proposition 3.23, page 288 in https://www.maths.ed.ac.uk/~v1ranick/papers/ends1.pdf, concerning the uniqueness of resolutions. Following the notation in the paper, since $X$ has dimension at least 5 and is resolvable, then crossing with a 2-torus yields a finite dimensi...

psie

10:37

@XanderHenderson these guys, so apparently from the order Diptera :)

porridgemathematics

11:11

@Jakobian yeah I also think it should be true for a.e case, I feel like the proof should involve trying to guess the situation when this expression (the original expression I talked about involving the rational functions with double poles) is maximized at the origin (because we can always reduce to this case via a möbius transformation of the unit disk into itself), at the origin this expression is rotationally invariant, so maybe a good guess would be the symmetric configuration of z_j..

@Jakobian then it should be possible to show for specific “nice” choices of coefficients c_k ,c_k’ that when z_j are roots of unity, the absolute value of the expression at the origin cannot exceed the maximum boundary value. It’s just harder than it appears to actually make this concrete

Btw ignore the subsidiary questions I asked after the first one, they were misguided and don’t help solve the first one (for those subsidiary questions the answer is easy because we are asking if the zero set of an analytic function has measure zero, which it always does)

Basically the hardest part is showing that the symmetric configuration is extremal

Then finally we would have to perturb the “nice” choice of coefficients c_k to get the result for a.e such rational function, this part would also not be easy

I’m tempted to just to write some small script that just checks loads of random cases, does anyone know of a good tool to use to do this? I want to compute the maximum value of an n variable positive function of y_1,…y_n of the form f(x_1,…,x_n)(y_1,…,y_n) defined for (y_1,…,y_n) in [0,2\pi]^n, and the function is a polynomial in the parameters x_1,…,x_n , which are real and between 0 and 1. I’m guessing there are lots of software packages I can use to do this?

Actually it’s a polynomial in 1/x_1,…,1/x_n, but this isn’t important

So I want to be able to compute the maximum for loads of randomly selected x_i’s

With the constraint x_1 + … + x_n =2

Xander Henderson

11:27

@psie Crane flies are not bugs!

psie

they're huge!

yeah probably not a bug then

Xander Henderson

@psie But not bugs.

user 20458579510081670432

12:11

When is a bug not a bug?

A: when it's a feature :P

Sahaj

@user20458579510081670432 if something is a bug, then it's a bug, if its not a bug, then its not a bug. how can a bug also not be a bug?

🐞👾🪲🐛

user 20458579510081670432

That was a play on the two meanings of the word "bug."

A "bug" can also occur in computer programs.

Bug (engineering)

In engineering, a bug is a design defect in an engineered system that causes an undesired result. Although used exclusively to describe a technical issue, bug is a non-technical term; applicable without technical understanding of the system. The term bug applies exclusively to a system that is (human) designed; not to a natural system; and that the issue is within the influence of human control. For example, humans have faults but not bugs, and a server crash due to natural disaster is not a bug. In addition to or instead of defect, some use: error, flaw or fault. Engineered systems is a broad...

Sahaj

12:37

Right.

Mad

13:14

i am very confused about the usage of terminology.

We call elements of the dual space Covectors. And define a tensor as a multilinear map $T^s_r$ where s is the number of copies of $V^*$ and then we say s the "contravariant" part.

You would expect, since $V^*$ elements are called covectors, that $s$ is the covariant part?

Nevermind, i understand why they are calling it like that, very tricky

Mad

14:33

i am also struggling to understand the concept of the order of the definition.

is there a fundemantal difference between $ V \times V^* \times V $ and $ V^* \times V \times V$ as domains of definition, i mean, these two spaces are isomorphic as product spaces.

Xander Henderson

@user20458579510081670432 There is also the ambiguity of the term "bug" in everyday English vs the more technical definition used in etymology. In everyday English, many people will call any insect a bug (I've even heard people call spiders and centipedes bugs).

psie

15:39

I'm working an exercise on the completion of metric spaces. We say that a complete metric space $Y$ is the completion of $X$ if $X$ is a dense subspace of $Y$.

> Show that when $Y$ is a completion of $X$, then the inclusion map $X\to Y$ extends to an isometry of $\tilde X$ onto $Y$.

Here $\tilde X$ is the set of equivalence classes of the set of Cauchy sequences in $X$, under the relation $\tilde s=(s_n)\sim(t_n)=\tilde t$ to mean that $d(s_n,t_n)\to0$ as $n\to\infty$. The metric on $\tilde X$ is $\rho(\tilde{s},\tilde{t})=\lim_{n\to\infty} d(s_n,t_n)$.

Question: I don't understand the exercise. What does it mean for the inclusion map $X\to Y$ to extend to an isometry of $\tilde X$ onto $Y$?

flowian

Can anyone suggest literature for motivation of box spaces and asymptotic dimension?

(asymptotic dimension of groups)

zeta space

16:09

Alessandro Codenotti

@flowian as far as I can remember the main motivation for asymptotic dimension is that it is a quasi-isometry invariant, unlike classical notions of dimension

So it is "the correct" thing to consider in geometric group theory

flowian

@AlessandroCodenotti Makes sense, thank you.

I'm looking for some background to fill into my thesis.

Jam

16:25

Can you help me find all ideals of $\mathbb{z}[x]/<x^3-x>$

i can use either the correspondence theorem or the chinese remainder theorem for rings to proz my ring is isomorphic to $ZxZxZ$

Simd

16:36

How can I count the number of distinct weakly connected DAGs on 4 labelled nodes?

porridgemathematics

@Jakobian nvm, i realized what I hoped was true fails spectacularly badly. You can get a huge class of examples where the absolute value of the expression at 0 exceeds the maximum boundary value by taking the Schwarzian derivative of a Schwartz christoffel mapping onto a very long and thin rectangle , I.e with one dimension much larger than the other. This gives a positive measure class of rational functions which for that expression obtains a maximum value only in the interior

But it is likely that for Schwarzian derivatives of maps onto regular polygons, it doesn’t fail.

nickbros123

Any computer geeks around here? I was writing a code to find the root of a function, using a) bisection of interval method and b)Newton Raphson Method. What I noticed is that the bisection method takes 52 iterations to converge, for almost all but a few trivial intervals I give it. why is it like that?

newton method converges anywhere between 5-20 iterations as long as my input number is fairly "slopey" (i.e, its slope is not close to 0)

would it be better to ask this question in computer forum instead of math forum?

Jam

16:53

actually my isomorphism is wrong.

Jam

17:22

math.stackexchange.com/questions/4959674/… if any1 is up for some algebra.

zeta space

17:50

is it possible to convert rotation to diffusion?

zeta space

18:02

If I had to try to start solving that I would take some unique solution to a diffusion pde on a region in the plane, represent it with constant time slices (planar picture) and build a section (of a fiber bundle) of these slices, viewed as line bundle in R^3 with a prescribed geometry s.t. a rotation of the bundle over the base projects directly onto the diffusive pde solution

The question is what use would this be?

and is "projection" even a valid "conversion" technique here

perhaps its more kosher to say generalized diffusion and planar diffusion

Jakobian

@psie It means that there is a map $\tilde{X}\to Y$ which is an isometry and such that its restriction to $X$ (where we treat $X$ as the, up to equivalence, constant sequences) is the map which is the inclusion of $X$ into $Y$

冥王 Hades

18:52

@psie I threw a bug in boiling hot water once for daring to bite me

Shaun

19:05

I can't find a simple question: how does one show a cardinality is infinite iff it is idempotent, not 0, and not 1?

Jakobian

If $\kappa$ is infinite then $\kappa^2 = \kappa$, this is standard and depends on axiom of choice

If $\kappa = 0$ or $\kappa = 1$ then $\kappa^2 = \kappa$ as well

psie

@Jakobian thanks!

@冥王Hades oh no, poor bug...reminds me of a story once my grandmother told me of how she cooked her first and only lobster. She cooked it alive in boiling hot water and the lobster started giving away a screeching sound which she remembers to this day.

Shaun

What about the other direction,@Jakobian? And thank you.

Jakobian

and if $2\leq \kappa < \aleph_0$ then $\kappa^2 \geq 2\kappa \geq \kappa+1 > \kappa$

Shaun

Thanks! :)

Jakobian

19:13

now since either of these $3$ possibilities has to hold, since the order on cardinals is linear

the inequality $\kappa+1 > \kappa$ for finite $\kappa$ follows from finite ordinals and finite cardinals being the same

Shaun

This is helpful, @Jakobian, but I was hoping for a question on here I could cite.

in In the search of a question, 27 secs ago, by Shaun

in Mathematics, 13 mins ago, by Shaun

I can't find a simple question: how does one show a cardinality is infinite iff it is idempotent, not 0, and not 1?

Jakobian

Ah. Find a question for reference. This is a different matter

well, would you be content with the proof that $\kappa^2 = \kappa$ for infinite cardinals alone?

Soumik Mukherjee

@Shaun this?

Jakobian

also a reference from a book would be theorem 1.5.11 in Introduction to cardinal arithmetic by Holz, Steffens, Weitz

Shaun

19:29

@SoumikMukherjee Thanks :)

Soumik Mukherjee

:)

psie

@Jakobian do you know if the metric on the completion $Y$ is the same metric as on its dense subset $X$? I am unsure how to solve the exercise if it isn't. I believe the map that sends the equivalence class of $(s_k)$ to the limit of $(s_k)$ is an isometry from $\tilde{X}$ to $Y$, but to verify this, I need to know the metric on $Y$ I think.

The metric on $X$ and $\tilde{X}$ are known, but that on $Y$ I am unsure.

leslie townes

19:45

what is Y?

@Jam math.stackexchange.com/questions/3200710/… is a related exercise with R in place of Z. it uses (an appropriate generalization of) the chinese remainder theorem. does it help?

Jakobian

@psie what do you mean by the same

equivalence class $(s_k)$ to limit of $(s_k)$ in $Y$, yes

just assume $Y$ has some metric such which restricts to the metric on $X$

$d_0:Y\times Y\to [0, \infty)$, $d_0\restriction_{X\times X} = d$

the point is that the metric on $Y$ is somewhat of a mystery

and what you're proving is that $Y$ must essentially be $\tilde{X}$

Soumik Mukherjee

@leslietownes en.m.wikipedia.org/wiki/Y

psie

@Jakobian ok, right, I am proving uniqueness up to isometry sort of (if I understand the exercise correctly), but how would one verify that this map is an isometry?

leslie townes

soumik: oh, i see. a degenerate case of the phoenician waw. many thanks

psie

I guess we want to check $d_0(s,t)=\rho(\tilde{s},\tilde{t})$, where $\rho$ is the metric on $\tilde{X}$.

Soumik Mukherjee

19:51

I am glad to help:)

Jakobian

@psie yes, uniqueness up to unique isometry

actually there is unique uniformly continuous such map, and it happens to be an isometry

the whole uniqueness talk is as if I'm attempting some kind of word salad

it's probably very confusing

well what do you know already about $\tilde{X}$

psie

@Jakobian it is complete

Jakobian

did you check its a metric space, complete, $X$ is dense in it?

psie

the map $x\mapsto\tilde x$ maps $X$ to a dense subset of $\tilde X$, indeed

冥王 Hades

@psie I hate bugs, I feel no empathy towards them.

Jakobian

20:06

okay so given a Cauchy sequence $(s_n)$ of $X$, you want to send it to $f((s_n))$, the limit in $Y$ of $s_n$

and now suppose $(s_n)\sim (t_n)$. Then check that $f((s_n)) = f((t_n))$

hence this defines a unique map $g:\tilde{X}\to Y$ given by $g([(s_n)]_\sim) = f((s_n))$

did you check that?

@冥王Hades and they probably don't feel any empathy for you too :P

psie

@Jakobian is $f$ the map $x\mapsto\tilde x$, where $\tilde x$ is the equivalence class of the constant sequence $(x,x,\ldots)$?

冥王 Hades

@Jakobian Precisely, which is why it bit me. Why should I not fight fire with fire then, right?

Jakobian

@psie lobsters aren't bugs though

psie

yeah no :) they're more noble

Jakobian

@psie no

$f$ is the map from the set of all Cauchy sequences to $Y$

冥王 Hades

20:11

I remember at a kitchen where a lobster was watching another lobster being cooked. Must've been a horrifying site.

Jakobian

the equivalence class of the constant sequence $(x, x, ...)$ is an element of $\tilde{X}$ so you're pretty far off about what $f$ is

psie

@Jakobian I will check this. Give me some minutes and I'll write something.

If $(s_n)\sim (t_n)$, then $d(s_n,t_n)\to 0$ as $n\to\infty$. Intuitively I want to conclude they have the same limit, but I don't know if there's something more to motivate that $f((s_n)) = f((t_n))$.

Jakobian

@psie How about you move away from the area of intuition and into the area of formal proof

Let $s$ be the limit of $s_n$ in $Y$ for example. Can you show that $s$ is the limit of $t_n$ in $Y$?

This is the same thing but phrased differently

psie

@Jakobian well, this is where my brain asks about the metric on $Y$, because we define convergence of a sequence to a limit in some space with respect to the metric of that space

Jakobian

what does it mean for $t_n$ to converge to $s$ in $Y$

$d_0$ is a metric on $Y$

you don't need to know "what it is"

how you should use intuition is to form a proof

allow it to direct you, but not to dictate what is true and what is not

psie

20:35

I think I need a break and think some more about this.

Jakobian

yeah maybe it's hard to grasp all at once

psie

@Jakobian regarding your comment here though, I get what you are saying, but is an equivalent way of putting it that there's an isometry $h:\tilde{X}\to Y$ such that $h \circ e = i$? Here $e:X\to \tilde{X}$ is the map $x\mapsto \tilde{x}$ and $i:X\to Y$ the inclusion map. Maybe that is what you said?

Jakobian

@psie yes

psie

ok, cool 👍

Jakobian

this is basically what I said, although people usually don't pay much attention between distinguishing $X$ and copy of $X$ in $\tilde{X}$

and what is also to be checked that it really is a copy of $X$

but I imagine you did that already

psie

20:43

yeah

Mats Granvik

Math's Fundamental Flaw

►

Towards the end of this video there is a demonstration of the Game of life being undecidable because you can run the game of life "on" the game of life recursively. 29:30 onwards.

Xander Henderson

21:03

@MatsGranvik Note that Mat SE moderator Asaf Karagila consulted on that video.

Diffusion

22:08

If the Cesaro means $\frac{1}{N}\sum_{k=1}^N a_k$ are bounded in $N$ with $a_k$ positive, must $a_k$ be bounded?

Jakobian

22:29

sorry I didn't see that $a_k$ are positive

We have $$\lim_{N\to\infty} \frac{\sum_{k=3}^N \log(\log(k))}{N} = \lim_{N\to\infty} (\log(\log(N+1))-\log(\log(N))) = \log\left(\lim_{N\to\infty} \frac{\log(N+1)}{\log(N)}\right) = 0$$

and so $\log(\log(k))$ for $k\geq 3$ consists of positive numbers, is increasing and converges to $0$ Cesaro, but is unbounded

Jam

23:43

@leslietownes no cause Z is not a PID like R and also you cannot use chinese remainder theorem cause x+1 and x-1 are not coprime

leslie townes

23:53

jam: but can you at least use the CRT to get partial information? e.g. what about the ideals generated by x and (x+1)(x-1)?

jam: Z[i] is a PID, isn't it? math.stackexchange.com/questions/443842/…

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$Mathematics$ Mathematics