Mathematics - 2024-06-19

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user10478

00:01

Is it possible to, if not always then at least for most famous probability distributions, construct a (discrete if appropriate) surface by gluing the PDFs with neighboring parameter values next to each other, and reslice the surface in an orthogonal direction to get another probability distribution? For example, the surface could be the glued Poisson PDFs for each value of $\lambda$, which would then be resliced treating $k$ as the parameter and $\lambda$ as the support variable.

Would this distribution in $\lambda$ adhere to the probability axioms in that case (for each fixed $k$)? Is this basically what Bayesian statistics is doing under the hood by positing a distribution of belief over a parameter?

Jakobian

00:36

@user10478 I'm not sure I understand you well, you're looking at a family of PDFs parametrized by some parameter $\theta$ lets say, $f_\theta$

is that right? And then what do you do?

user 85795

00:56

is not "difficult" a relative
term

Jakobian

@user85795 yes but the point is that I never meant "difficult" in the first place

user 85795

"unpleasant or unattractive" things are difficult

Jakobian

in what sense

user 85795

you avoid them

Jakobian

I don't think thats necessarily the case

usually people avoid unpleasant things, but I don't think the same can be said about unattractive things

user 85795

01:06

Sure, beauty is in the eye of the beholder.

Jakobian

even if this was something one would avoid, that doesn't necessarily make it difficult, as much as it makes it difficult to deal with

it doesn't make it difficult like how a math problem can be difficult, for example

its more psychological

user 85795

yes

user10478

@Jakobian Yeah, so if for example $\theta$ happened to be discrete, your new probability distribution be $Pr(\theta = 1)$, $Pr(\theta = 2), ... and the old variable $f$ used to range over would now be a parameter.

Jakobian

01:32

@user10478 So you're treating $\theta$ as a random variable. I think I've seen that in a statistics class before. But I'm still not sure what you're asking

could you use less words and more symbols/definitions?

this might be better to ask among statisticians as well

user10478

I was just wondering if those probabilities over $\theta$ automatically sum to $1$ and satisfy sigma additivity, so as to constitute an actual probability distribution.

Jakobian

02:00

@user10478 what do you mean? If $\theta$ is a random variable then it will naturally have a probability distribution

I'm still not understanding

user10478

Can I treat a $\theta$ obtained in that way as a random variable?

Jakobian

02:19

In what way though, in what way

you're going further but not clarifying what you mean at all

I don't think the whole talk with surfaces really made much sense, it didn't to me, so I'm still struggling on what you mean

but I get it if you don't want to elaborate because lets say, you don't really know what you want to ask

user10478

03:13

@Jakobian Sorry, I didn't see latest reply right away. Yeah it might be better if I make a picture in Mathematica at a later time and show what I have in mind.

2 hours later…

Soumik Mukherjee

04:45

@EE18 I thought you were asking about the 1/k^2+k question, so my hint about prime between n and 2n was just about k^2<k^2+k<2k^2, bounding the sum. Nothing to do with prime:)

leslie townes

05:21

oh, i dunno. nothing to do with prime?

Soumik Mukherjee

05:33

🥹

1 hour later…

robjohn

06:53

@copper.hat Hey there. Sorry, I've been extremely busy recently.

2 hours later…

psie

08:33

@Jakobian I'm looking at this again and the condition $x\geq 2y$ needs to have a lower bound too I believe. When $X=10$, the possible $Y$ values are $4$ and $5$, but the condition $x\geq 2y$ says that $p_{X,Y}(10,y)$ is nonzero also for $y=1,2$ and $3$. I don't see what the lower bound is right now though.

user image

psie

08:54

@Jakobian maybe this is what you were saying here, not sure. Or simply add $x\geq 2y$ when $x\leq 7$ and $x\geq 2y> 2(x-7)$ when $12\geq x>7$, something like that.

psie

09:30

I wonder how one can rewrite the function then in a concise way with all them conditions.

s.harp

In this mathscinet review mathscinet.ams.org/mathscinet/article?mr=2044031 the reviewer writes:

> this sort of definition is wholly ungeometrical and totally sterile

which I find hilarious

(just to remark, he is actually praising the book quite a lot in the context of the review, I just like the formulation a lot)

nickbros123

10:32

in the proof that "if Ax=0 and Bx=0 have the same solutions, they must be row equivalent" I noticed one thing

user image

in the above, when defining the map K, there are m-r! ways of maping deltas to gammas, so that K stays invertible

this could mean that there are m-r ! invertible matrices such that KA=B?

2 hours later…

Jakobian

12:13

@psie no, I'm not saying this is always non-zero for $x\geq 2y$.

Just that its equal to certain probability involving $U_1, U_2$ for those $x, y$. Which can and will be zero for most $x, y$

@psie split it up because the events are either disjoint or equal depending on if $x= 2y$ or not

Xander Henderson

12:53

Oh, heck yes! A Zelda game with playable Zelda!

'Bout damn tme.

nickbros123

I wonder how one discovered an object like compact sets

Im basically done with compactness section of rudins basic topology chapter, and, aside from knowing its cool properties, I dont rly have a "feel" for this concept much more than the definition

Xander Henderson

@nickbros123 Being kind of hard on Platonism, myself, I don't think about "compactness" as having been "discovered", so much as "invented". The idea of compactness is kind of natural in real analysis---think about how many theorems rely on a function being defined on some closed, bounded interval.

The extreme value theorem for continuous functions, the mean value theorem, the most basic definition of an integral, etc.

In this context, "closed and bounded" turns out to be a nice property. So it is reasonable to ask how much that concept can be extended. Do the theorems work for any closed and bounded set, or just intervals? What happens if you drop one of the conditions? How about in higher dimensions? Or in more radical spaces?

Once you start asking those kinds of questions in the context of more general topological spaces, the idea of compactness is more-or-less natural. It gives you some idea of closed and bounded, in spaces where "bounded" doesn't make sense (e.g. because there is no metric).

nickbros123

that makes sense

I can see how "every open cover has finite subcover" kinda simulates a "bounded" criteria

Jakobian

13:16

@nickbros123 See Heine-Borel theorem

it was motivated by problems in analysis

Joe

There is some discussion of the intuition behind compactness on a thread from MSE. I find the idea of compactness being a topological generalisation of finiteness very compelling and useful. That being said, I don't think it is an easy concept to "discover" – the history of compactness would also seem to suggest this.

nickbros123

@Jakobian very interesting!!

Thorgott

13:27

@nickbros123 the reason for this is that you are reading Rudin

nickbros123

@Thorgott what else would you rather I read

Jakobian

@nickbros123 the modern definition of compactness is due to Vietoris, see his paper: Vietoris, L., Stetige Mengen, Monatsh. f. Math. u. Phys. 31 (1921),173-204.

I believe the translation of the title of the paper would be "continuous sets"

that's where he provides various equivalent formulations for compactness, still using countable covers though

correction: apparently Vietoris didn't formulate it in terms of open covers

Joe

@nickbros123: You might want to try Analysis I and Analysis II by Terence Tao. I think it's better suited for self-study of analysis than Rudin. It includes more motivation, at least

Jakobian

Vietoris defined compactness using filter bases/nets

nickbros123

13:43

@Joe will have a look. IDK if its some kind of stockholm syndrome or inertia from having spent many a day working through Rudin, but I kinda like it

Jakobian

well, general way in which it went is that, as far as I understand it, people were first thinking of compactness in terms of sequences, then tried to move away from it, although countability was still persistent, and then finally moved to arbitrary open covers

it wasn't like a smooth process

Joe

@nickbros123: I definitely think it is a very elegant book which is also a decent reference.

But it is impossible for a mathematics book to be everything and anything. In the case of Rudin, I think what he gains in succinctness and elegance he loses in motivation

Jakobian

user image

user image

nickbros123

@Jakobian is that a paper by bourbaki or is it in their analysis book?

Jakobian

@nickbros123 their topology book

Soumik Mukherjee

13:53

What is meant by [0,1] compact spaces?

Jakobian

the book from analysis you're referring to is from 1949 I think

user image

so it means every compact Hausdorff space is a closed subspace of $[0, 1]^\kappa$ for some $\kappa$

there is this characterization that any Tychonoff space can be embedded into $[0, 1]^\kappa$ for some $\kappa$ by using continuous maps $f:X\to [0, 1]$

Tychonoff cube

see the property that the Tychonoff cube is universal for the space of compact spaces

@Jakobian general topology, not algebraic topology

Jakobian

14:12

as another example, realcompact spaces are $\mathbb{R}$-compact

there is also $\mathbb{N}$-compact spaces in literature that I know of

and although this is definitely an interesting concept, I think it more natural to embedd $X$ into $\mathbb{R}^\kappa$ using all bounded continuous $f:X\to \mathbb{R}$, rather than just $f:X\to [0, 1]$, since the former has the benefit of making the connection between $\beta X$ and the ring $C^*(X)$ more explicit

Perhaps rather than $\mathbb{N}$-compact spaces, there should be considered $\mathbb{Z}$-compact spaces as well, so that there might be some connection drawn between them and the ring $C(X, \mathbb{Z})$.

Soumik Mukherjee

Thanks!

Thorgott

14:39

@nickbros123 depends on what you're trying to learn

Xander Henderson

@Joe As much as I like Tao's book, I would actually recommend against trying to work from both Rudin and Tao at the same time (I would recommend against trying to learn out of Rudin at all, probably). Once you pick a text, I think that it is generally best to work through it as a text, until you feel that you have gotten out of it what you want to get out of it.

Switching between books can often leave gaps, as the approaches may be different.

And I think that there is value in working through a single text, doing the exercises, and figuring things out on your own. If all you care about is finding some proof, the internet has millions. The goal should be to work out those proofs without looking them up somewhere else.

Jakobian

I'd say that it is a good idea to focus on one main text and use others only as reference when struggling, but generally speaking analysis books are self-sufficient

Xander Henderson

@Joe Motivation and any kind of exposition. Also, I hate his proof of the Mean Value Theorem. And the wheels kind of fall of the book in the last couple of chapters.

Joe

Yeah a lot of people don't like how he treats functions of several variables. I've never read the chapter, so I can't really comment. However, I did find it useful to learn other areas of mathematics before learning any kind of rigorous calculus on $\mathbb R^n$, i.e. I went wider rather than deeper. That way, I was able to read books on analysis that could lean on tools from topology and algebra.

By contrast, if you just read from Rudin, you would encounter Stokes' Theorem before even seeing a precise definition of "vector space", which is ... weird

2 hours later…

PM 2Ring

16:46

I've been playing with the gyroid, "an infinitely connected triply periodic minimal surface discovered by NASA scientist Alan Schoen in 1970". The true gyroid is parameterised by elliptic integrals (of the first kind), but there's a close approximation in terms of circular functions: $\cos(x)\sin(y) + \cos(y)\sin(z) + \cos(z)\sin(x) = 0$

Here's the approximation: sagecell.sagemath.org/…

Here's the true gyroid, which I adapted from code by Parcly Taxel gitlab.com/parclytaxel/Malibu/-/blob/master/malibu/minimal/…

sagecell.sagemath.org/…

1 hour later…

Jakobian

17:55

Lucifer, Satan, and the Devil

The Jewish position of "God is responsible for both good and evil" seems a lot better than the Christian one

In particular there is no problem with "God is everything that's good" or whatever Christians say, there is no problem with God being good yet not stopping evil from happening

also interesting how Christians just merged all those "evil" beings into one

@GratefulDisciple

Better in the sense of being more believable

Peter

18:16

@Jakobian It is better , yes , much harder to be refuted. However , there are other good arguments against any god , no matter whether it is an always good god.

copper.hat

there's always Pastafarianism if you want a tasty alternative

@robjohn Hope busy means some fun.

Jakobian

@Peter definitely. I believe the concept of "there has to be a creator" is just your usual personification that humans do.

its one of those things with which religious people try to convince you

we like to see ourselves in things other than ourselves, that is nature

Peter

18:39

@Jakobian Correct. Since money is essential for us , it is also essential for god although god does not need money. Exactly what you say , they project all their wishes and needs to a hypothetical being. And I think , people only believe in a paradise because they wish it would exist. But why then the belief in the hell and the wish that other people arrive there ? This is what I cannot understand.

Jakobian

Even if you think about creation, why does the world need to be created? I think even the physical theories like Big Bang don't talk about creation of the universe, only of its expansion

I.e. I think Big Bang theory permits the universe to have always existed

nickbros123

@Thorgott ...Analysis ?

Jakobian

@nickbros123 why the question mark? Your original question was about topology

Thorgott's question was reasonable

Peter

@Jakobian The big bang theory was invented by a priest , our bad luck is that some observations seemed to back it up. When observations against the big bang theory occured , instead of giving it up , the dark matter was invented :( In reality , the universe was not "born" and won't "die".

Jakobian

I don't know. A lot of physicist are for it, it might be reasonable. It won't matter to me if its true or not anyway

its true that the one who came up with the theory was religious and so he had his own reasons to advocate for it

but this doesn't discredit it

PM 2Ring

19:06

@Jakobian Big Bang theory says that space & time as we know it began ~13.7 billion years ago. BB theory talks about what happens at times t>0, but it doesn't talk about t=0 itself, because in pure General Relativity you get an unremovable singularitt at t=0. Perhaps a quantum gravity theory will avoid that problem. There's some info at physics.stackexchange.com/a/136861/123208 & the various linked questions.

One option is that there's a discontinuity, and there is no actual t=0. (If you use logarithmic time, t=0 corresponds to negative infinity). A related notion is that the very earliest instants of time may not be well-orderable (due to quantum uncertainty), so for times on the order of the Planck time, it's not possible to define causal sequences.

@Jakobian Indeed. OTOH, it did slow down the initial acceptance of BB theory (to an extent), by people who thought BB theory was too motivated by religious belief in a creation event. At the time, the dominant paradigm was an eternal steady-state universe.

Jakobian

@PM2Ring very interesting, so if BB did happen, there might have been some weird shenanigans due to quantum mechanics at small times $t$

could it be then said that ~13.7 billion years ago was, most likely the origin of space and time, the universe as we know it, but not necessarily the universe itself?

PM 2Ring

Right. But even if we do create a consistent quantum gravity theory, it will be really hard to empirically verify it. Standard GR works superbly in regimes we can investigate. We need quantum gravity for stuff we can never observe directly, like the core of a black hole, and the state of the universe within a few Planck times of t=0.

We're pretty confident about BB theory for times after a picosecond or so after t=0, because we can investigate those energies in our particle accelerators. But even a particle collider the size of the galaxy can't get to the energies involved at the Planck time.

Peter

19:23

The big bang theory is , as much more of the cosmology , so-called "mainstream physics" , not reasonable , it is a linear regression to an assumed event billion years ago with no evidence that it happened. This is like claiming that the USA will once have $10^{12}$ people. Even if the universe was more compact in the past (not the space , but the matter contained in it) , this does not mean that it was concentrated in a point.

PM 2Ring

We know the early universe was quite uniform, to ~1 part in 100,000. That's the scale of the inhomogeneity in the CMB (cosmic microwave background). That radiation was released when the universe became transparent, ~370,000 years after the BB.

Presumably, there's also a CNB, the cosmic neutrino background, which was released around t=1 second (IIRC). But we can't detect it: our best neutrino detectors need neutrinos with kinetic energy ~1 million times greater, and even then they only detect ~1 neutrino per billion, or so.

nickbros123

@Jakobian why would I learn topology from rudin

Jakobian

@nickbros123 to learn analysis

what were you trying to ask me because I don't understand

you were learning topology from Rudin so its normal to think that you might have wanted to learn more topology

nickbros123

Right, yeah thats possible

But I was intending to learn analysis

PM 2Ring

@Peter There's plenty of evidence for the BB. Eg, the observed spectrum of the CMB, and the isotope ratios of light elements are consistent with the theoretical predictions. But you're certainly entitled to maintain some scepticism. Plenty of cosmology is rather speculative, without a very firm foundation.

@Peter The universe is probably infinite in its spatial extent. And if that's the case, then it's always been infinite. However, there is a density singularity at t=0. But that doesn't mean you can claim that the universe was a point at t=0. But as I said earlier, BB theory doesn't try to talk about t=0. :)

nickbros123

19:43

@Jakobian Tbf, I kinda already knew Rudin's topology is bare minimum and can't learn actual topology from that, A) considering (even in metric space context) I have a proprietary metric space book (which deals metric space results topologically rather than with epsilon delta) which itself goes for 300+ pages B) it's metric spaces, not the general thing

Nordine Lotfi

I'm just wondering but, given we can make a recursive sum function into a closed form expression like: n * (n + 1) // 2 how come we can't do the same for recursive multiplication?

I mean, without using iteration like with the closed form version of recursive sum

Xander Henderson

@NordineLotfi I don't understand your question.

Nordine Lotfi

@XanderHenderson sorry, let me make it clearer: given a recursive sum function (let's say in Python, that would be: pastebin.com/raw/pJjfds6m) and the closed form version would be n * (n + 1) // 2, I'm wondering if an equivalent expression exist but for recursive multiplication (same as the Python version but using *= instead of += )

(here I uploaded an iterative version since it's easier to read then the recursive version which use list slicing)

Xander Henderson

I read math, not python. I am not going to try to parse your code. Can you please ask the mathematical question you are interested in, in the language of mathematics?

That being said, $n!$ can be expressed in terms of the Gamma function.

Which means that you can express things like $\prod_{j=m}^{n} j$ in terms of Gamma.

Nordine Lotfi

ah, yeah already knew about the gamma function. I can approximate it using gamma(N+1), N being the natural number. But this is just an approximation I think, isn't there a similar expression like with the recursive sum example I provided?

Xander Henderson

19:53

No, it is not an approximation.

$\Gamma(n)$ is precisely equal to $(n-1)!$.

Jakobian

@nickbros123 compact and connected sets are all that matters for most people

that's how I look at it. Everything else is optional

Peter

After having read good and convincing articles about the issues with modern cosmology , I am very careful with all this mainstream physics. Cemented ideas can still be utter wrong. But I think, this is not the right place to discuss this. This site is about math.

Nordine Lotfi

@XanderHenderson got you, might have misunderstood then :) Thank you

Jakobian

@Peter well, "Associated with Math.SE; for both general discussion & math questions alike." as per channels description. This chat is for math, but not only or math. Discussion about, say, types of bread, is actually welcome here

@NordineLotfi this is not very clear question, what is and what isn't a closed form is a subjective question. Usually we can try and define what it means. For example, for roots of polynomials we say that generally speaking, polynomials of $5$th degree are unsolvable, but that means they are unsolvable by radicals. So this sort of "closed expression" is provably impossible for all such polynomials. But you can still find a closed expression in terms of elliptic integrals...

Here Xander showed you that $n!$ can be written in terms of Gamma function, which has definition as an integral, so that might be somewhat satisfactory for you. The question about if it has a certain, well-defined type of expression in case that isn't satisfactory, is most likely very hard whatever you decide on to call "a closed expression" similarly how its hard to show that only polynomials of degree $\leq 4$ are solvable by radicals

i.e. for the question to be objectively answerable, you would need to provide a proper definition of what it means to be a closed expression, that is, a definition not subject to your or other people opinions

of course that would be a definition only for the purposes of the question itself

Soumik Mukherjee

20:09

Anyone watching Euro 2024?

nnabahi

Problem: Take the number line and randomly select a point to "zoom into". As an example, zooming into 0 takes ...(-3)(-2)(-1)(0)(1)(2)(3)... to (-3)(-2)(-1)(-1/2)(0)(1/2)(1)(2)(3)... so that the distance between 0 and 1/2 becomes 1. This is still a valid metric space, but the question is to quantify how different it is to the regular number line
for this, zooming in gives the number halfway between the number you're zooming into and the numbers adjacent to it

copper.hat

disastrous showing for Ireland in UEFA 2024

Jakobian

@nnabahi if I understood you well, this should be a uniformly equivalent metric, so the properties are basically unchanged

say, you have a function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x) = 2x$ on $x\in [-1, 1]$ and linearly extended to be continuous and linear with slope $1$ on $[1, \infty)$ and $(-\infty, -1]$ with $d(x, y) = |f(x)-f(y)|$

believe this is what you're trying to do

Soumik Mukherjee

2

Q: Euro2024-inspired scoring problem

Motivation. The Euro 2024 soccer football championship is in full swing, and the male part of my family are avid watchers. Right now the championship is in the group stage where every group member plays against every other member exactly once. I got the impression that the games in the groups get...

co.combinatorics infinite-combinatorics recreational-mathematics sumsets

copper.hat

@SoumikMukherjee there was a lot of discussion about whether Ireland's best strategy was to lose against the NL

Nordine Lotfi

20:22

@Jakobian you're right, sorry for the misunderstanding. I'm just not really versed with the math notation or specific terminology as you can see :)

nnabahi

@Jakobian This seems right. I'm a bit confused since wikipedia says "[uniform] equivalence requires that there is a single set of constants that holds for every pair of points in X." So at every blow up, there's a new set of constants defined for it?

Nordine Lotfi

also, as far as I tried, the solution with gamma presented earlier is actually not always exact, e.g.: after some specific size of integer, it start to be just an approximation, with a specific error rate.

Jakobian

@NordineLotfi No, the formula $\Gamma(n+1) = n!$ is true in the mathematical sense, the two are literally equal. Your computer might be giving wrong value for $\Gamma(t)$ depending on how it calculates it, but its not because math is wrong, its because computers work this way

most likely the program you're using, language, whatever, package, doesn't give an exact value to $\Gamma(n+1)$

Nordine Lotfi

@Jakobian right, from a theoretical point of view, I do agree, it is 100% exact. But on any CPU, GPU, basically any hardware or software stack, unless you can afford infinite floating precision (which will use more memory and computation), it's going to always be an approximation at some point when using it on large integer.

@Jakobian I can decrease the error rate, but only through increasing the floating precision

but that's only temporary if I'm working with very large number

copper.hat

all very fuzzy

Jakobian

20:28

that's outside of the scope of what mathematician usually cares about

nickbros123

I wonder how computer would do the gamma integral. does it just give value from the stirling approx?

Jakobian

@nnabahi what exactly are you asking me

copper.hat

@nickbros123 math.stackexchange.com/questions/19236/…

Jakobian

the constants should be the same as long as the "zoom" is always by the same amount. So they work uniformly for all "blow up points"

nickbros123

user image

Jakobian

20:32

in fact it shouldn't be hard to find what those constants are explicitly

nnabahi

@Jakobian (-2)(-1)(0)(1)(2) ->on 0 (-2)(-1)(-1/2)(0)(1/2)(1)(2) -> on 1 (-2)(-1)(-1/2)(0)(1/2)(3/4)(1)(1 1/2)(2), to say that the zoom on 1 blows the right side up by a different amount than the left

copper.hat

sweet god

Nordine Lotfi

@Jakobian that's fair, didn't think about it from that perspective :)

nnabahi

@nickbros123 this is gorgeous

Nordine Lotfi

@nickbros123 gamma(N+1) is actually much more accurate than Stirling approx. Also faster depending on the implementation of gamma

I tried it yesterday with a bunch of different other alternative and implementation

Jakobian

20:36

@nnabahi I think for any discussion to be had here, you need to stop using numbers and begin using actual formulas and symbols that make whatever you are trying to do well-defined and precise

copper.hat

so many things evaporate when formalised

Jakobian

like doubts about what is being talked about

copper.hat

the number of convex questions had dwindled, is ai/ml losing popularity?

if i listen to music with mandarin words on youtube, then all the ads i get are in mandarin. was hoping for more subtlety with prediction

my overall life experience is that people skip the basics/axioms. unfortunate. then they become teachers and propagate the vagueness.

not everyone, of course.

indeed, i think i may have skipped peano, etc.

Jakobian

@copper.hat trends.google.com/trends/… dwindling a little, it seems, but stable

copper.hat

ohh, that's cute!

nnabahi

20:48

Take $\mathbb{R}$ and the usual metric $d(x,y) = |x-y|$, and begin by considering only $\mathbb{Z}$. Pick a point to zoom in on, say $x$. This imposes a new distance on $\mathbb{R}$, where $d(x,\frac{x+1}{2}) = 1$ and $d(x,\frac{x-1}{2}) = 1$, and otherwise distances are preserved. Both $\frac{x+1}{2}$ and $\frac{x-1}{2}$ are now part of our set to choose from, $\mathbb{Z} \cup {\frac{x-1}{2},\frac{x+1}{2}}$, and the process is iterated.

copper.hat

huh?

nnabahi

What can I clarify

Jakobian

what do you mean by considering only $\mathbb{Z}$

$x\in\mathbb{Z}$?

nnabahi

the starting set of points I want to zoom in on are only from $\mathbb{Z}$, and then points are appended to $\mathbb{Z}$

copper.hat

So, you are trying to define a distance on the dyadics?

Jakobian

20:55

you can't change only distance from $x$ to $\frac{x\pm 1}{2}$ because of triangle inequality

that won't be a metric then

nnabahi

I'll try a picture, otherwise I'll be back later when this is more properly formalized

copper.hat

it might be easier if you describe what you are trying to achieve

i'm not a fan of reverse engineering something that is not engineered in the first place...

Jakobian

I think the point is to iterate this countably many times and see what we get

also I think it should be $x\pm \frac{1}{2}$ and not $\frac{x\pm 1}{2}$

copper.hat

that would make some sense

this exemplifies (to me, at least) the difficulties in communicating remotely. 5 mins and a white/black board would converge quickly

nnabahi

sorry how do I send pictures here? like nickbros123

copper.hat

21:03

i use imgur

but there may be more appropriate ways

nnabahi

imgur.com/a/7XvGrFN

so that the distance between any 2 ticks is 1

Jakobian

there is upload button on the right side

next to send

copper.hat

so you are defining a distance on the dyadics?

nnabahi

I don't know what a dyadic is

but if that's what I'm talking about I'll go do my reading

Jakobian

the picture doesn't explain anything

copper.hat

21:06

n/2^m

Jakobian

@nnabahi rational number of the form $k/2^n$

nnabahi

oh then yes

in the final step of the picture, the distance between 7/8 and 1 is 1, and the distance between 2 and 3 is 1

otherwise again I'll go make it more precise and maybe try later

Jakobian

Ah I get it

copper.hat

$d(-1, {7 \over 8})$ would be?

nnabahi

-1/2, 0, 1/2, 3/4, 7/8, so 5

copper.hat

21:10

you need the triangle inequality to hold

Jakobian

well anyway, what is your question

we're discussing some kind of construction, but there is no question as far as I know

@copper.hat as I see it, its set so that $d(x, y)$ counts the number of points in $[x, y]$ minus $1$

but this is a distance on $\mathbb{Z}$ plus whatever dyadics we have added

nnabahi

My question is to find a way and "quantify" how different this space is to the normal number line. For example, in a random walk on the number line with step values of 1, you'd always end up an integer distance away from 0. Here though, if you zoom in on 0 a bunch of times, it would take forever to get to 1, so you've made less progress

Jakobian

and so if we iterate this construction what is this supposed to lead us to

copper.hat

i may have misunderstood, but i thought $d(0,1) = 1$, etc.

nnabahi

excuse my very imprecise wording

Jakobian

21:13

@nnabahi how does this give you a number line

copper.hat

like i say, formalising is where the work is...

nnabahi

@copper.hat that's what I'm currently learning

Jakobian

as far as I see it, distance isn't the same even on $\mathbb{Z}$ per each step of construction

so we can hardly iterate it to define it on dyadics

this is a problem

nnabahi

Ah, because my random walk question, I'm switching between the "local metric", where d(7/8,1) = 1 and the regular metric on R?

In any case this has been a very helpful exercise in being clear with my language.. Thank you guys

Jakobian

I don't see how to go on and do all those things you are claiming could be done

and you don't know either it seems

so I don't see much point, unless you think about it further and actually manage to formalize it

I don't think this will be fruitful but who knows

copper.hat

21:20

mathematics is a dangerous siren

Xander Henderson

It looks a little like you are trying to (locally) snowflake the real line.

nnabahi

We'll see,, I'm not sure about fruitful but I think I can do some of this more rigorously

Xander Henderson

I'm not quite sure what it would mean to "locally snowflake" a space, but I'm not sure what else to call it.

1

Q: Why the use of the term "snowflaking"?

I've seen a few places in the literature (in particular in fractal geometry) where we consider a metric space $(X,d)$, and then for $0 < \epsilon < 1$ define a new metric space $(X,d^{\epsilon})$ where: $$ d^{\epsilon}(x,y) = d(x,y)^{\epsilon} $$ This new metric space is called an $\epsilon$-snow...

analysis metric-spaces terminology fractal-analysis

copper.hat

might help to spread the dyads out into $\mathbb{R}^2$.

Jakobian

@XanderHenderson cool definition, I might use it

copper.hat

21:22

i thought it was a political label...

nnabahi

@XanderHenderson that is beautiful and definitely adjacent to what I was doing

As a quick final thought, if you "zoom in" to every even or odd number, you get the regular number line but with every distance spread out by 2, again very imprecise language give me a few days :)

2 hours later…

robjohn

23:29

@copper.hat Some fun, but a lot of medical things for my wife and I and also for our dogs.

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