serious-discussion

[gleaming panther]

Log base 7 of (2/7*7) +1

Break it into addition

Log base 7 of 2/7 +1 +1

Log base 7 of 2/7 +2

But after you get below 1 it starts getting negative

So since your adding, log base 7 of 2/7 and 2, and log base 7 of 2/7 is negative, you know the answer is between 1 and 2

But 2/7 is closer to 0 than it is to 1

So just estimate a decimal

Like .35

Then put that between 1 and 2

So I guess that answer is 1.35

That's probably wrong

Because I estimated

But it's helpful I guess

The only limitations is that you can't do it to log base 1

But log base 1 has no values

So I guess that makes sense

Hmmmm

Let's see if that was right

,calc 7^x = 14

[fathom swallow]

The following error occured while calculating:
Error: Invalid left hand side of assignment operator = (char 5)

[gleaming panther]

Oof

[serene lance]

log base 7 of 14 is around 11.83 lol

[gleaming panther]

How do I type logs in Latex

Hmmmmmmm

I did something wrong

[tacit magnet]

Its about 1.35 lol

[serene lance]

wait

whoops i wrote log 7 * 14 accidently

[gleaming panther]

Ohhh

Okay

Wait

Is it right?

[serene lance]

yeah ur right

LOL
mb

[gleaming panther]

YEEEEEEEE

I guessed it right

Thats good

To be fair I thought of this like yesterday

So still some tweaking needed

I guess you could use this for values that do work

Like log base 2 of 8

Break 8 into 2* 2 *2

So you get 1+1+1

Which is 3

That cool

[serene lance]

tho 2/7 is about 0.285. not quite 0.35

[gleaming panther]

Huh

I know

I was just guessing

[serene lance]

so this seems very much like a coincidence idk

[gleaming panther]

Probably

This is why I need to fix this

At last it tells you it's between 1 and 2

That's good

It also tells you if a value actually works for that log

Like if it has a whole number value

[serene lance]

can you try to estimate log base 7 of 15 with this

[gleaming panther]

I think so

Just a bunch of decimal

[serene lance]

ill try

[gleaming panther]

Log base 7 of (15/7*7)

Log base 7 of 15/7 +1

Log base 7 of 15/49 +2

So it's between 1 and 2 now

I guess

But it's closer to 1

[serene lance]

i dont think u have to do the third step

[gleaming panther]

Oh yeah

Your right

But if it was like 15/2

Then the next would be 15/4

So we can't really know we can stop just with that step

[serene lance]

log_7(7 * 15/7)
1 + log_7(15/7)
and we know 15/7 is just a little bit over 2 (since 14/7 = 2), and since the base is 7 that means that the answer is between 0 and 1. add one to that and u get that the value is between 1 and 2

so i guess its nice

[gleaming panther]

Yeah

I feel like doing that last step is nicer on the brain

Since you can get all the way to the negatives and then just subtract by two

WAIT

I think we can separate 15/7 into log base 7 of 15 - log base 7 of 7

Nope

That's useless

It just undoes a step

Since log base 7 of 7 is one

But we are subtracting

So you remove one

And go back to the original

I am going to work on this method

To make it more exact

[pastel tusk]

log_7(14) = 1 + log_7(2)

right

[gleaming panther]

Yeah

Just keep repeating the process

Until the log gets below 1

Then you know it's between the outside value and 1 less than the outside value

[gleaming panther]

Actually

I just realized

The value could be anywhere underneath the outside value

Since if the remaking log value is small enough you could subtract more 1 value

[gray glade]

Is it ok if the first time I see the formal limit definition I find it a bit circular? lol

[neat lintel]

what do you think is circular about it?

[gray glade]

idk I just saw a video

Like the epsilon delta stuff tells you the error. So less error closer you get. I get that. What is a limit then. An aproximation so close you get how the function behaves. But how close?

[frigid pebble]

2+2= -5

[gray glade]

I am not saying is wrong I am saying I find it a bit wierd compared to the derivative definition which makes more sense

[river moon]

a limit is a number

[gray glade]

I just find it a bit counterintuitive I guess...

[river moon]

and if the sequence has a limit L then this number would satisfy all this epsilon-delta stuff

it doesn't give you an algorithm to determine for arbitrary sequence what this number is

[orchid badger]

what you mean by cross product

[mystic ivy]

[lapis hull]

I'm writing a presentation on program optimizations for a class I'm TAing (writing cache-friendly code, loop unrolling, etc) and I was wondering if anyone had any ideas for other methods of optimizing code

[pearl moth]

ah. code generation

[lapis hull]

Im looking at my course's book and it has examples, but theyre all pretty trivial and not so interesting

[pearl moth]

low hanging fruit: common subexpression elimination

[gritty heath]

off hand i am thinking you could look at what the O O2 and O3 flags do

and see if you could use those optimizations

[pearl moth]

there's a lot of easily accessible monographs on lcse and gcse

[lapis hull]

id prefer not having to teach too much about convoluted assembly

[pearl moth]

you could talk about branchless programming at a theoretical level

[lapis hull]

ah thats a good idea

[pearl moth]

hyou don't have to get into the grunge of how branch predictors actually wortk, that's probablyt above level for a UG class anyhow

i'm thinking about the optimizations i have to unroll while decompiling stuff

[lapis hull]

tbh i myself dont know how branch predictors work, this isnt a course i have so much knowledge in anyway. they asked me to TA it because they dont have enough TAs lol

[pearl moth]

like one i see in some environemtns (not in C because || and && have execution sequencing as part of their semantics), but this totally happens in SQL query planners) is inverting the order of boolaen operaitons to do the less expensive one first

[lapis hull]

thats interesting

do you have any advice on writing cache-friendly code? the course does focus a bit on the cache

[pearl moth]

are we talking about at the level of what a programmer should do or what a compiler can/will do automatically?

[lapis hull]

what a programmer should do

maybe also a bit about the compiler

beyond just locality

[pearl moth]

the problem with being cache-aware is that caches vary in size

[lapis hull]

or maybe specific clever tricks about locality

[pearl moth]

mostly just the usual, like using structures of arrays instead of arrays of structures

there's a tradeoff involved with that one

[lapis hull]

thats a thing?

[pearl moth]

i think the game developers call it "ECS"

like if you have an array of some structure, what'll happen (in C/C++, at least) is your array will have each structure in series, with the variables of each object arrayed one after another and each object arrayed one after the next

so if your code is going to scan through this array and access one field in each structure, that generates a memory acess pattern that fetches memory "sparsely", which causes a lot more cache line fetches than if you inverted this and had a structure of arrays. then your scan will fetch adjacent memory locations and use far fewer cache line fetches

[lapis hull]

ahhhh i see

[pearl moth]

the downside of this, of course, is that it's annoying to code in C or C++ because their natural model is to have an array of objects. you can write adapter classes, of course, but that sorto f metaprogramming is ugly af in C++ and can only be done manually in C

it's probably possible in C++ with a metric buttload of C++ template magic, buit i would not want to write or maintain that code

i have no idea what this would be like in rust

and in basically any other language you probably won't care

well, maybe ada, but i haven't looked at ada in three decades

the thing is, there's no "right" way to do this, because it's totally dependent on what your "typical" use cases are

[lapis hull]

right

well the idea of the class is to just give some ideas for ways to optimize programming, and this is a great example i think of ways to write something more cache friendly

even if it isnt always usable

[pearl moth]

sorta like why you have to know what your workloads are before you can determine what indices you should add to a sql table are

it's more cache friendly for one specific access pattern (which happens to be quite common)

[lapis hull]

right

but itll still make the students think

which is good

[pearl moth]

we can h ope

[lapis hull]

i think i need to fill up two lectures about this, so if i explain it poorly enough maybe theyll ask enough questions to fill up my time

[pearl moth]

i've just been asked to write a presentation about a particular implemnetation detail of the g++ compiler that is causing a difficulty for the project i'm on now, and i'm mulling over how to do that, so i understand where you are coming from

i am pretty sure i undersatnd the issue but that doesn't mean i can explain in a way someone else can understand

[lapis hull]

haha yeah i get that

explaining is hard

[pearl moth]

in case you're curious: if a dynamically-loaded library include a class that has an inline method that accesses a class-level static variable, the resulting DLL cannot be dynamically unloaded when compiled with gcc for a target architecture that uses the ELF executable file format

since the project i'm on relies on being able to dynamically load and unload DLLs, and we support linux (which uses ELF), this matters to us

[lapis hull]

i dont remember enough c++ to understand this, but what is the project youre working on?

[pearl moth]

dfhack, a modding framework for dwarf fortress. links in profile

https://github.com/DFHack/dfhack/issues/4301 is the issue related to this problem, but i've been asked to provide a writeup that explains this in more detail

[tribal slate]

if we have some binary equation G of A, B thats like is this true A+G(A,B)=A+G(1,B)?

in binary

[pearl moth]

what do you mean by "in binary" and what is "G"

[reef carbon]

what is a "binary equation"

[wooden path]

Hi guys! I'm a beginner in math(not completely but I want to revisit it from scratch) and so I want to apply for an university this year and I wonder what kind of things I should know before attending university classes. I'm targeting for UK universities for Computer Science degree. UK is not my native country and I think that the math taught in their school also quite differs from mine. Can someone tell me what are the things in math that every student must be familiar with or send some sort of roadmap? Any help will be greatly appreciated! thanks in advance))

[neat lintel]

First of all, I would start by saying maybe I would be right and maybe I would be wrong

Secondly, in my opinion you don't need to do Math all over again from scratch because for some reasons such as you're not gonna do a math degree although that you will still need math in CS, and with your knowledge of math you could enroll in the CS uni and take your courses and if you feel you don't understand something, go and study it, it would be better than doing everything from the beginning that some of it you might need and some not

Thirdly, which I highly recommend talk with email the uni of the prerequisite of knowledge you need to have

if the uni doesn't get back to you, speak with someone is at the same uni

or people have the same education system such as England for CS

[wooden path]

I see so I don't need to study from complete scratch but I think I need to know at least school math because UK requires foreign students to study 12-grade in which they definitely teach some math so I thought it would be easier for me to study if I have some math background but I really don't know what sort of math they teach so I'm in confusion

[wooden path]

nice idea I will give it a try thanks!

[neat lintel]

I don't know what's math for 12-grade but Khan Academy has some beginner's stuff

you could check it out

[wooden path]

aha thanks)

[round schooner]

Both the ACM and the Computer Society of the IEEE have model computer science curriculae. You should check those out. In Europe, they probably expect you to have already taken a calculus course intended for mathematicians, physicists, and engineers. Once in the program, you should expect a combinatorics course including computability and a calculus based algorithms course. As for calculus. Last I heard, the UK uses Newtonian notion while North America uses a version descended from Liebnitz.

[sly flare]

Are math related coding questions allowed?

I dont see anywhere where anything about that is specified

Im currently making a 3D renderer and im having a really hard time so i came to here for help

[river moon]

if it is a math question ,yes

you are more likely to get answers on coding in the coding servers

[sly flare]

Dang..

Embed fail

[sly flare]

Yea like thats ever gonna happen

I hope this server is more helpful though

[river moon]

most of the people who do math have never written any code or touched rendering

[sly flare]

They dont know how camera perspectives work?

[river moon]

nope

[sly flare]

Sigh....

And its precisely that im having issues with

[sly flare]

How can you be so sure though?

[river moon]

you don't need to know how to code to study math

[sly flare]

I mean yea but you should at least be able to know how camera perspectives work

Its infact an extremely easy equation

[river moon]

and rendering is such a specific thing to learn as well, like not even every programmer worked with rendering

you want a person who knows math, knows coding and worked with rendering

[sly flare]

I have a person in mind but theyre close sourced

😔

Arghh

[river moon]

I think criver on this server knows something about it

[sly flare]

criver?

[river moon]

but I'm pretty sure he's been saying that how stuff is done in rendering software is backwards to how it's done mathematically

[river moon]

it's a nickname

[sly flare]

And who is this criver?

[river moon]

somebody who may or may not answer your question, at least he seemed like he fits all of these

[sly flare]

Im asking so i can contact him...

Can you give me his discord username?

[river moon]

just use search on this server

criver

[sly flare]

Maaan and i thought you said it was just a nickname; not his actual discord name

Though its best to ask him in a help channel right?

Ill just take that as a yes

[fervent pebble]

i am fairly sure not even the uk uses newtonian notation given that his integral notation is just drawing a box around the function

[next schooner]

UK does not use Newton or Leibniz notation, we use whatever the lecturer feels like using given the context

[burnt ledge]

Hahaha

The UK uses Newton and US uses Leibniz

so true

[pearl moth]

lol

i've seen US instructors use both newton and leibniz in the same expression

[fervent pebble]

people actually unironically use box to integrate?

[pearl moth]

i can't say i've ever seen anyone use newton's notation for integration

[fervent pebble]

i was worried 😭

[pearl moth]

he had four different notations for integration, none of which anyone uses with any regularity

[fervent pebble]

$\fbox{f(x)} = F(x)$

hm

wtf bot where did u go

hmmmmmm

[fresh comet]

what is box

[fervent pebble]

like a box

like

[fresh comet]

??

[fathom swallow]

valley

[fervent pebble]

that

box notation

he also had the vertical bars

[pearl moth]

one of the ways newton had to notate integratn was to enclose the integrand in a box

also used a vertical bar or tick mark above the integrand (usually only when the integrand was simple, such as just x)

[fresh comet]

wth is that

Never seen it before lol

[pearl moth]

the box notation is a bitch to typeset

[lapis hull]

my geodiff course used newton, leibniz, and lagrange

[fervent pebble]

yeah i js cannot figure out how to get the dot inside

$\fbox{\dot y}$

[lapis hull]

this is cursed and not just because you forgor mathmode

[fathom swallow]

valley
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[fervent pebble]

oh my god why dont u work

[lapis hull]

\fbox{$\dot y$}

[fathom swallow]

sqeral

[fervent pebble]

wtf is mathmode

[lapis hull]

fbox is a textmode macro

[fervent pebble]

wow fuck this

[pearl moth]

\fbox falls out of math mode

[fervent pebble]

it is??

wild

[lapis hull]

youre gonna have to learn tex to become an elite-level colleger

[fervent pebble]

so would i have to do $\dot \text{\fbox{y}} = y$

[fathom swallow]

valley
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[fervent pebble]

welp

[pearl moth]

no

stahp already

[lapis hull]

$\dot{\fbox{$y$}}=y$

[fathom swallow]

sqeral

[lapis hull]

you just forgor braces

[fervent pebble]

i dont like this game anymore

scary

time to sleep

[spice ore]

Hey guys im a high school student with an interest in math and i would be thrilled to dive into topics related to topology, complex analysis and differential equations. My question is where do i start? I'm not sure. I started leaening about set theory and related topics but like what would be the main topics i have to cover?

[river moon]

real analysis and linear algebra

[spice ore]

Any good book recommendations?

I would like something that's not so difficult to understand

[oak quest]

Technically you will require knowledge of proof writing before diving deeper into these subjects

[spice ore]

So anything beyond basics of Calculus is required to know proof writing?

[oak quest]

I'd recommend "Visual Group Theory" by Nathan Carter if you're interested in getting a exposure of advanced (undergrad) math

[spice ore]

Thanks much

So this book requires proof writing skills i guess?

[oak quest]

Also you can check out the video playlist "Proofs" on the channel Michael Penn, he has followed "Proofs" by Richard Hammack in those lectures, also the book is available for free online(in pdf version obviously).

[oak quest]

Not much, especially in the beginning I'd say zero

[spice ore]

Ok nice

[storm sage]

this is so terrible lol

[oak quest]

I'm also a high school student and tried reading analysis, algebra books and didn't understand anything, basically you need to have a certain level of mathematical maturity before you try to learn these subjects on your own using books and online resources alone

[vivid halo]

pain

yeah let's just write 9 boxes around this

[muted totem]

What's that?

[oak quest]

Some kind of measurement I guess that's what integrals do, right?

[muted totem]

Yeah but it's integrating over weird stuff

[mystic ivy]

sure is

must be some sort of witchcraft

[oak quest]

I know about general linear group, the O notation (landau symbol) amongst all those weird objects

[muted totem]

I thought that was the orthogonal group

[oak quest]

Lmao maybe it is

[muted totem]

In this context at least

[oak quest]

Ohh nvm you're right, it is that

[vivid halo]

integral expression for a certain part of an L-function (generalizing Riemann zeta)

[neat lintel]

looks like shit

[muted totem]

Rather scary

[vivid halo]

it's just kind of tedious and notation heavy but the actual ideas involved are not that bad

like the most absolute baby case of this type of integral is something like

$Z_\infty(s,e^{-\pi x^2})=\int_{\mathbb{R}^\times}|x|^se^{-\pi x^2}\mathrm{d}x=\int^\infty_{-\infty}|x|^{s-1}e^{-\pi x^2}\mathrm{d}x=\pi^{-\frac{s}{2}}\Gamma(\tfrac{s}{2})$

[fathom swallow]

nGroupoid

[vivid halo]

it's the same story in general but your Gaussian function gets more complicated, there is more shit to integrate over, you have to do it one matrix coefficient at a time, the answer is slightly more complicated, etc

[mystic ivy]

omg a based \mathrm{d}x enjoyer

[vivid halo]

[mystic ivy]

my life has never been as joyful

[neat lintel]

in less than two sentences sell me on upright d instead of just dx at the end of my integrals

[vivid halo]

\mathrm{d} reads more like an operator, d reads more like a variable

[mystic ivy]

looks better and is correct

[muted totem]

Good exercise for the fingers

[neat lintel]

i buy it

[vivid halo]

like it's not really that big a deal I just think it looks better and is technically more correct (even though the alternative hardly sacrifices understandability)

more keystrokes though yeah

[neat lintel]

\newcommand...

[vivid halo]

I don't use macros

[neat lintel]

\frac{\partial}{\partial x} over and over

i would kill myself if i didn't have macros

[vivid halo]

I mean if you're having to type the same thing a lot you can just use copy paste

idk I just don't like having to define a bunch of commands

it's definitely bad if you're collaborating

[deep mango]

Now that i use the physics package I use \dd mostly for the better spacing

[neat lintel]

i only ever use like one or two commands

and it's the partial derivative shit

[deep mango]

Before I had a macro to do hacky spacing

[vivid halo]

I occasionally define commands if it's like

some actual typesetting hackery or new symbols

[neat lintel]

simply don't collaborate

[vivid halo]

but I don't like defining macros for existing commands

[neat lintel]

\newcommand{\bR}{\mathbb R}
\newcommand{\bZ}...

[vivid halo]

I have a nice command which is like, for Hom sheaves you often typeset this in a silly way like \mathcal{H}om

and you can't write \mathcal{Hom} because it doesn't typeset right

but I have a command that makes it so \mathcal{Hom} typesets as \mathcal{H}om

[neat lintel]

$\mathcal{H}om$

[vivid halo]

or any \mathcal for that matter

[fathom swallow]

person2709505
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

rakko

[neat lintel]

this?

[vivid halo]

yeah this is a common thing that people do for Hom sheaves

sadly there is no lowercase mathcal

and mathmode text looks close enough

you just have to hack \mathcal to do the right thing

$\mathcal{H}\mathrm{om}$ looks kind of weird imo

[fathom swallow]

nGroupoid

[vivid halo]

and it's very clumsy to type

[dusty thorn]

Hey can anyone tell me if this is correct or not:
The quantity of real numbers between any two values is technically greater than the total quantity of rational numbers despite both being infinite because you cant create a bijective function that relates the two, as it will never be surjective

[river moon]

I made a command \prox{a}{b}{c} to typeset $prox_{a}^{b}(c)$, used it a few times in a document

[fathom swallow]

Transparent Elemental
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[river moon]

[tawdry pecan]

-any two non-equal values
-what do you mean surjective? Are you talking about R->Q or Q->R? a function R->Q can be surjective, but a function Q->R cannot

And it being surjective isnt really an explanation for why you cant make a bijective function

[wooden path]

Thanks for information! I will check them out

[neat lintel]

ask @leaden torrent

is this time cube 2.0?

[leaden torrent]

you werent banned, you were expelled from the advanced channels for incoherent spam

if you want i can ban you instead

so do i

i dont see how, i have a PhD in math

not computer science

[tawdry pecan]

cohere into a different server pal

that was not worth the time it took you to type it sorry

[jovial ember]

Me = chair monkey

[leaden torrent]

what is the stabilizer of 5 in the cyclic group of order 15?

you have 10 seconds to answer before i ban you

[tawdry pecan]

no dont ban him this is funny

[neat lintel]

let him cook

[tawdry pecan]

the answer is trivial

sotrue

chatgpt speaks through this man beautifully

like a trumpet in the hands of angels

[leaden torrent]

no, chatgpt would construct sensible english phrases

this is home grown

truly a work of art

[tawdry pecan]

his parents must be sturdy fellow

[neat lintel]

@neat lintel what do you think about the axiom of choice

[neat lintel]

The drug he’s doing is home grown for sure

[tawdry pecan]

LMAO

[neat lintel]

do you reject or accept it?

[leaden torrent]

we can tell

[jovial ember]

Bro decided not to replace the word “of” with ()

[tawdry pecan]

Ah yes, the Axiom of Choice! An elegant solution to many problems in mathematics, scholars everywhere sing its praise.

lmfaooo

[neat lintel]

Same, I understand the instinctive nature of things just by recognizing simple inputs

[tawdry pecan]

ur a markov chain stfu

[leaden torrent]

no QUESTIONS
no MASTERS
no GODS

[neat lintel]

That’s like low level text prediction engine

[tawdry pecan]

yessir like the shit in your phone keyboard

[leaden torrent]

anyway fun's over

[tawdry pecan]

awww bye willy i hope u learn to socialize

[leaden torrent]

RIP William, PhD (2024-2024)

[neat lintel]

The Discord bot GenAI uses this exact algorithm

[tawdry pecan]

discussy will remember you )i promise)

[tawdry pecan]

lmfao

[peak tide]

phd in the math

[neat lintel]

William P.H. Dingledong

[jovial ember]

He will come back stronger

[tawdry pecan]

and we will be ready

[jovial ember]

He will have 2 P.H.D. Next time

[tawdry pecan]

i did not plan for this

[leaden torrent]

tragic that Z/5Z is the trivial group now...

[neat lintel]

but what IS the stabilizer of 5 in the cyclic group of order 15

[tawdry pecan]

my guess was 3 like an idiot idk what stabilizer is lmao

[leaden torrent]

the set of elements x such that 5x = 5

[tawdry pecan]

that makes sense

[leaden torrent]

so in the case of the cyclic group of order 15, that'll be the elements that are 1 mod 3

[tawdry pecan]

wait so i was fucking

wait nvm i was close

its 5 then i think

[leaden torrent]

uh, the subgroup has cardinality 5

[tawdry pecan]

yeah i keep thinking thats what ur asking

[leaden torrent]

its {1, 4, 7, 10, 13}

[tawdry pecan]

reading is not my strong suit

yeah

what can you use stabilizers for? like, why worry about them

[leaden torrent]

i am yanking your chain

[tawdry pecan]

can you split the group up or collapse it into a smaller one

[leaden torrent]

the cyclic group of order 15 is additive

not multiplicative

[tawdry pecan]

AAGH

[leaden torrent]

so the stabilizer is actually just 0

[tawdry pecan]

LMAO

[leaden torrent]

so they were right that it is, in fact, trivial

!!!

[tawdry pecan]

were they right on purpose tho???

[leaden torrent]

(this is true for all stabilizers in finite cyclic groups)

[neat lintel]

Fool

[tawdry pecan]

ngroupoid looking down on me shaking their head rn

yeah no shit its not multiplicative theres no inverse for 0

[leaden torrent]

(Z/15Z)^* would be a wild ass "group"

[tawdry pecan]

ughhhh

[leaden torrent]

lmao

[leaden torrent]

orbit-stabilizer theorem mostly

[tawdry pecan]

wow you can use them for a theory invented because of them lmao

[leaden torrent]

orbit-stabilizer is kinda rank nullity for groups

[tawdry pecan]

rank nullity is where a matrix has multiple identities, right?

I havent taken a lin alg course

[leaden torrent]

i mean, rank nullity for groups is just the first isomorphism theorem

but like, orbit-stabilizer lets you do similar things to rank-nullity

[tawdry pecan]

yeah I think id need to watch a group theory intro vid to understand that

[leaden torrent]

express the size of a group in terms of an element's orbit length and its stabilizer

[tawdry pecan]

oh that sounds neat

[leaden torrent]

this fact gives you the Lemma that is not Burnside's

which is important in combinatorics

[tawdry pecan]

combinatorics gets real scary real fast lol

[leaden torrent]

stabilizers are also just kind of a useful structural fact of a group though

theyre a convenient notion in a bunch of places

[tawdry pecan]

hmm I can see that

[leaden torrent]

to be clear, this notion is used when referring to group actions specifically

stabilizers arent very useful when the group acts on itself lmao

[tawdry pecan]

the group can what???

[leaden torrent]

hence why "stabilizer of 5 in a cyclic group of order 15" is a meme question

[tawdry pecan]

how is that not referring to the group action

[leaden torrent]

well its just that

if the group is acting on itself

thats literally just the group structure

if you have ab = a in a group, what can you conclude about b

[tawdry pecan]

its identity i think

[leaden torrent]

right

so any such stabilizer will be trivial

[tawdry pecan]

ohhhh

i get it

[leaden torrent]

we only care about them when the element in consideration comes from another set

and the group is acting on that set

[tawdry pecan]

ohh ok

I didnt know you could do that even lmao

I thought groups were like

[leaden torrent]

the typical example you might have is like

[tawdry pecan]

really just contained in themselves

[leaden torrent]

imagine a set composed of different colourations of the 6 sides of a cube

so maybe you have 4 colour choices or whatever

and you consider the group action of the rotation group of a cube on this set

this group action corresponds to just... rotating the cube

so the stabilizer of an element will be all of the rotations that don't change the resulting sides

[tawdry pecan]

mhm

[leaden torrent]

for example, if the net of your cube is like:

      [green]
      [red]
[blue][yellow][blue]
      [red]

180 degree rotations along one of the axes will result in the cube being unchanged

[tawdry pecan]

yeah

[leaden torrent]

but all other rotations will change which face is which

so the stabilizer of this particular element is the "trivial" rotation and a 180 dgree rotation about the green-yellow axis

(unless im having a brain fart and missing one)

this is a contrived example but you can see why this might be a useful thing to talk about

e.g. lets say you want to count all possible colourings of a cube up to rotational equivalence

[tawdry pecan]

so then why isnt that a group in itself? why is it applying actions to a different set?

[leaden torrent]

so we dont want to double-count a cube colouring if we can just rotate an existing cube

[leaden torrent]

the group itself is the rotations, the set it's acting on is the set of "colourings"

[tawdry pecan]

I see

[leaden torrent]

rotations have a group structure since we can compose them associatively and undo them

[tawdry pecan]

so the group and set have to have some kind of shared structure then?

[leaden torrent]

in theory you can combine whatever you want, but in practice yes we'll be concerned with objects that are naturally related

[tawdry pecan]

I see

[leaden torrent]

wikipedia gives this as an example in fact (though it uses 3 colours, not 4): https://en.wikipedia.org/wiki/Burnside's_lemma#Colorings_of_a_cube

[tawdry pecan]

yeah I think I understand the stabilizer part pretty well now, thanks

[surreal bison]

...why?

[leaden torrent]

well i mean, if you just take Z/15 and replace + with *

[tawdry pecan]

I'm interested in this idea of applying groups to sets, though. How does that end up being visualized in the general case?

[leaden torrent]

obviously (Z/15Z)^× does not have 15 elements

[surreal bison]

You said (Z/15Z)^*.

[leaden torrent]

yes, *

not ×

[surreal bison]

Both x and * to me means the group of units.

[leaden torrent]

i have always seen the group of units denoted with ×

¯_(ツ)_/¯

not saying your convention is wrong

just havent seen it

[surreal bison]

[leaden torrent]

the term for this is "group action" and theres a lot of literature on it

[surreal bison]

A whole lot!

[fervent flame]

@jagged forge got everything set up, the brightness settings are wonky, but im getting 10+x fps on valorant LMAO

[leaden torrent]

some would argue that it's the whole purpose of groups, in fact

[fervent flame]

600-800

[leaden torrent]

(i would be among those "some")

[tawdry pecan]

ok, ill have to check that out once I get enough initial understanding

[leaden torrent]

i mean if youve seen like

[surreal bison]

I wouldn't be. I'd say it's half the purpose.

[leaden torrent]

the common examples of groups

permutations of a set, rotations or symmetries of a geometric object, etc

most of them feel like structural observations that arise when working with something else, no?

[surreal bison]

Permutation group on n elements acts on a set with n elements.

[leaden torrent]

permutations of a set are naturally, yknow, paired with a set

your group action is just permuting a given set

[surreal bison]

Any finite group is a subgroup of the permutation group, so, there's that

[leaden torrent]

yeah this fact is a lot more obvious if you think in terms of group actions

to the point of almost feeling trivial

[surreal bison]

OH!

[leaden torrent]

like clearly its a sub set

[surreal bison]

Because g acts on G!

[leaden torrent]

because it has to be

and then it makes sense that its a sub group or else shit would go really wrong in the action

[surreal bison]

So G->S^n sending g to g•x!

Wait, I already knew this

[tawdry pecan]

namington is so good at explaining you learned it twice

[surreal bison]

I don't think I connected this to actions.

That's what I got

[surreal bison]

i learn things much more than twice

[tawdry pecan]

let me goof around 🥺

[leaden torrent]

group actions are frequently the most natural way to "visualize" facts about groups, the issue is that theyre harder to approach at first since they feel more abstract

[surreal bison]

Actions of a vector space such that each element acts linearly (that is, send each element to an invertible matrix) is called a representation. These are pretty important basically because linear algebra is nice and actions are important.

[tawdry pecan]

yeah I just meant if there's a nice way to visualize it in any case, eg some sort of graph representation

[leaden torrent]

well, graphical representations are group actions

when you e.g. draw a particular polygon to use as an example of a dihedral group

[jagged forge]

wtf is 10+x fps

[leaden torrent]

youre really depicting that dihedral group acting on that polygon!

[fervent flame]

10x fps

my b

[leaden torrent]

but as mathematicians, we're used to abstracting things so we typically skip the step that connects the group to the graphical depiction

[tawdry pecan]

yeah I get that groups often have intuitive representations, but im looking for a more general one, yeah

[leaden torrent]

well groups are a very very general notion so youre not going to find any graphic that suitably describes all of them

[tawdry pecan]

damn

[surreal bison]

...what's a graph representation?

[tawdry pecan]

like, think DFAs

nodes and labelled directed arrows

[surreal bison]

...discrete finite automata?

[leaden torrent]

for finite groups you can create multiplication tables

which is basically as specific as a DFA graph

[surreal bison]

directed finite acyclic blank?

[tawdry pecan]

dicrete finite automata sorry

multiplication tables boring tho :/

[surreal bison]

seems like my guessing works.

so...what's a discrete finite automata?

[tawdry pecan]

uhh you have an accept state right

[surreal bison]

is it like a Markov chain but without the probability?

[tawdry pecan]

and a start state

yeah

kind of like that

but you have an input

[surreal bison]

So, we have a directed graph, a start and an accept, and an input deciding transitions?

[tawdry pecan]

and you consume acharacter and move along the arrows

[leaden torrent]

potentially multiple accept nodes, but yes

[tawdry pecan]

for a deterministic one yeah

also yeah could have multiple accepts

[surreal bison]

...so, is the finite state space what prevents this from being a Turing machine?

...and this is the thing that regex is, right?

[leaden torrent]

finite state space and lack of a stack

[tawdry pecan]

umm I wouldnt say the finite space is what limits its cimputational power as much as its inabillity to store stuff

yeah

[surreal bison]

How is not every finite state machine an example of this?

[tawdry pecan]

and regex is a regular language yeah

[surreal bison]

isn't a stack a infinite state mechanism?

[leaden torrent]

nope

you can have finite computation with a stack

see pushdown automata

[tawdry pecan]

you have to increase the size of the dfa a lot faster than if you have a stack

[leaden torrent]

but youre right that that isnt a "finite state machine"

[jovial ember]

Is this the Wolfram stuff

[leaden torrent]

chmonkey...

[jovial ember]

Cellular automata

[tawdry pecan]

that stuff is fun too

[leaden torrent]

to clarify the terminology here

[jovial ember]

Anything stupid I say is satire and on purpose

[leaden torrent]

"finite state machine" means deterministic finite automata U nondeterministic finite automata

this like

matters a priori but doesnt matter in practice since these are equivalent

the proof of equivalence is nontrivial though

[surreal bison]

...nondeterministic finite automata?

[leaden torrent]

one direction is easy, the other is annoying

[leaden torrent]

yeah; in a deterministic finite automata, the transition function is, well, deterministic

so in a state x and an input i, f(x, i) takes on a single state value

[tawdry pecan]

for deterministic, each symbol in the language has one arrow out per node

[leaden torrent]

but in a nondeterministic finite automaton, f(x, i)'s output is a set

[surreal bison]

Ah.

[leaden torrent]

additionally, we begin to accept empty transitions

f(x, ε)

[surreal bison]

Like a nondeterministic Turing machine!

[leaden torrent]

where we transition arbitrarily without reading input

right, same exact distinction

like nondeterministic turing machines, it's harder to model but much easier to prove things with

like with turing machines, we have that NFAs and DFAs are "equivalent" in the sense that you can construct a DFA that accepts the same input set any NFA does, and vice versa

one direction of the proof is immediate

the other is "obvious" but requires a bit of work (mainly to handle empty transitions)

you get superexponential blowup in the size of the automaton

the interesting thing is that, if you add a stack that you can push to and pop from, you no longer have equivalence

[tawdry pecan]

oh is it superexponential for dfa's?

[leaden torrent]

nondeterministic pushdown automata are strictly more powerful than deterministic ones

ie there exist languages recognized by an NPDA that cant be recognized by any DPDA

[leaden torrent]

i think so because of epsilon transitions?

i think its just exponential if not for those

[tawdry pecan]

ohhh yeah

[leaden torrent]

but epsilon transitions fuck things up since you have to account for those "repeatedly" for each possible transition out of the state

[tawdry pecan]

yeah they annoying

[leaden torrent]

i dont remember the details though so i might be wrong

its certainly at least exponential

[tawdry pecan]

yeah def

I just remember seeing somewhere that the worst case was like, 2^n or smth

but that couldve been w/o epsilons

[leaden torrent]

that might be true

idk

theres a bunch of numerical facts about this stuff that no one actually cares about in practice since "combinatorics of models of computation" is like, 3 layers removed from anything that matters

[tawdry pecan]

lmfao

[leaden torrent]

like you can bound the minimal size of the pumping constant as a function of the number of states

i think its like n + 1 or something

[tawdry pecan]

that sounds useless yeah LOL

[tawdry pecan]

So what does this mean computationally? Are NPDA turing complete?

[leaden torrent]

nope

[tawdry pecan]

woah

[leaden torrent]

theyre finite

like the stack can grow arbitrarily large but the underlying state space is finite

[tawdry pecan]

how many classes of computation like that are there then?

[leaden torrent]

whereas in a turing machine, the state space is the stack, so thats not an issue

[tawdry pecan]

yeah

[leaden torrent]

i dont know how youd quantify that

[tawdry pecan]

youd have to make models that are all provably computationally different from each other

[leaden torrent]

wikipedia provides this hierarchy but it only features models people care about

im sure theres like

"in between models"

that are stronger than PDAs but weaker than TMs

[tawdry pecan]

yeah thats what i wanna know about

that sounds like an interesting topic

[leaden torrent]

and obviously theres models of computation stronger than turing machines

namely, the notion of a mathematical function

which is not restricted by needing to "compute" things using a "procedure"

[tawdry pecan]

ah yeah, like the busy beaver function

[surreal bison]

Or any function that's not computable

[tawdry pecan]

I never thought about functions like that

[surreal bison]

Like any noncontinuius functions

Or the function that tells you if a Turing machine halts

[leaden torrent]

well i guess calling it a "model of computation" is ambitious lmao

[tawdry pecan]

yeah

[leaden torrent]

since its neither a model nor does it do computation

but my point is that you can certainly construct more general things than a TM

[tawdry pecan]

its also still bounded by recursive enumerability in practice id think

but yeah in theory theres stuff stronger

[leaden torrent]

fwiw you may be interested in https://en.wikipedia.org/wiki/Chomsky_hierarchy

its not quite what youre after i dont think

but its close

[tawdry pecan]

yeah ik about that

ill have to come back with a more consice definition

I feel like there's a notion of computation that goes beyond accepting languages but I cant put my finger on it

[leaden torrent]

the important thing is that "type-0", i.e. languages produced by a turing machine, can be produced by a set of general rules mapping any string of symbols and terminals to any other such string

so if youre still able to visualize computation as rules for matching strings (ie as grammars)

turing machine is the most general possible

[tawdry pecan]

I see

Do you know of a more general idea of computation than accepting or rejecting languages?

[leaden torrent]

for comparison, PDAs accept rules for matching any nonterminal symbol to any string of terminals and symbols

this is why these grammars are called "context free"

the rules can only consider the symbol itself

not its "neighbours"

[tawdry pecan]

yeah

that's what parsers deal with usually iirc

[leaden torrent]

no

[tawdry pecan]

oof

[leaden torrent]

the problem is that turing machines are like

really really fucking powerful

[tawdry pecan]

yeah

[leaden torrent]

so even if you introduce a lot of structure trying to exceed them

turns out that a turing machine can model that

very frequently

[tawdry pecan]

I feel like intuitively, theres a sort of "strength" measurement that could be applied more generally to things like functions, programs, logic systems and statements etc

[leaden torrent]

and anything that cant be modelled by a turing machine is usually called "noncomputable"

like a noncomputable function or real number or whatever

[tawdry pecan]

yeah

[leaden torrent]

which kind of like... definitionally restricts "models of computation"

to "turing machines, and weaker variants"

i am not sure if theres a way to generalize the notion of "computation" past turing machines but still feeling like, yknow, computation

like, idk, some sort of model "between" a turing machine and something powerful enough to solve the halting problem

[tawdry pecan]

yeah that seems a bit "axiom of choice"-ey to me lol

Im really just interested in seeing how powerful you could make a computation system before it becomes impossible to decide halting in it

or if there is a limit even

[leaden torrent]

i guess part of the problem is that

if your model of computation can model, like, mathematics

and we can express its rules "sensibly" (i.e. recursively enumerably)

you run into goedel incompleteness

which basically bounds its complexity by a turing machine

[tawdry pecan]

yeah really quickly too

like, peano arithmetic early lmao

[leaden torrent]

trying to find something more powerful than a turing machine while not being able to do mathematics feels like a fools errand

[tawdry pecan]

but i guess that's assuming you allow unbounded sizes of numbers

[leaden torrent]

like idk how i'd formally express that that's impossible

if its even possible to do so

but im sure theres some way to say "yeah this aint gonna work"

[leaden torrent]

we could try to ditch this requirement, but then actually talking about this becomes impossible

so its no longer a "model" of anything lmao

[tawdry pecan]

yeah

[leaden torrent]

that said

i am very much a nonexpert

[tawdry pecan]

Im not really looking for a single model though, I'm talking about a sort of measurement of powerfulness

[leaden torrent]

i am sure smarter people than i have thought about this

[tawdry pecan]

yeah

[leaden torrent]

yeah im not sure youll get much better than the chomsky hierarchy

[tawdry pecan]

damn

[leaden torrent]

like there is an entire field of computability theory

to be clear

but it doesnt go past, yknow, computable

so turing machines AFAIK are an upper bound

(though maybe theres some people doing computability theory in more general settings, again im not an expert)

[tawdry pecan]

I dont want to go past it either really

i wanna like, measure the "percentage" turing complete something is lmfao

[leaden torrent]

i mean theres turing degree but i dont think its quite what youre after

[tawdry pecan]

glancing at the wiki page it looks suprisingly close to what im thinking about

where instead of talking about accepting languages its a general desicion problem

[leaden torrent]

accepting languages is a general decision problem, but i get what you mean

[tawdry pecan]

yeah i suppose

Id really have to flesh out my thoughts more on this, im sure when I do ill realize im trying to do something literally impossible lmao

[leaden torrent]

im also suspicious that the notion you have in mind is totally ordered

[tawdry pecan]

mmm not really

im thinking more of like, uhhh, well shit i dunno

yeah ill get back to you on that

gn for now

[ripe needle]

@fresh comet

[edgy cedar]

https://www.youtube.com/watch?v=xvFZjo5PgG0

guys this is hard

see if you can understand this video

[kindred oriole]

ykpwbtzrgz
boring data structures class going on. join if you wanna and blast some music or smth
google meet

[neon oriole]

Meow

[rocky shuttle]

Meep

[visual breach]

Peep

[pearl moth]

yeep

[tired kettle]

bad idea to withdraw from a course after being admitted to a masters?

[pearl moth]

depends on why you want to withdraw

[lapis hull]

Basically I have two functions:

void int_iter(int len) {
  for (int i = 0; i < len; i++);
}

and

void long_iter(int len) {
  for (long int i = 0; i < len; i++);
}

I would expect int_iter to run quicker or at the same speed as long_iter, yet long_iter runs substantially quicker on my machine. Any idea why?

I looked over the compiled assembly, they're both identical save loading using a dword vs qword

[pearl moth]

not without looking at the assembly

also, most compilers would optimize the loop out entirely if you specified any level of optimization at all

[lapis hull]

Thats what i would think, weirdly mine didnt

Maybe I should try running this on a friend's machine

[pearl moth]

lemme get these up on godbolt

compiled with msvc lastest no optimizatoin both of these generate identical machine code

the machine code for the functions is line for line identical

gcc 10.5 compiles them differently though

[lapis hull]

hmmmm

[pearl moth]

the loop in the int version is mov eax, DWORD PTR [rbp-4] cmp eax, DWORD PTR [rbp-20] jge .L4 add DWORD PTR [rbp-4], 1 jmp .L3 while the loop in the long version is mov eax, DWORD PTR [rbp-20] cdqe cmp QWORD PTR [rbp-8], rax jge .L8 add QWORD PTR [rbp-8], 1 jmp .L7

https://godbolt.org/z/bs688fE4K

i'd be curious how you did profiling of these, since those loops are going to be screamingly fast and timing them is going to be difficult

the loops themselves should tit within at most two cache lines, and the data references should also fit within at most two cache lines, so really you're just testing the branch predictor

once the L1I and L1D caches are fresh, nothing is going to get in the way of the execution engine zipping around that loop as fast as it can

and really what will determine how fast it spins is the branch predictor correctly predicts "branch" on the jge instruction (since it's going to be a branch every time except the last iteration)

[lapis hull]

I set len to 10000000 and had it measure the average over 100 runs using getrusage

[pearl moth]

that's probably measuring OS overhead more than actual app peformance

[lapis hull]

but its consistently quicker

[pearl moth]

void int_iter(int len) {
    for (int i = 0; i < len; i++);
}

void long_iter(int len) {
    for (long int i = 0; i < len; i++);
}

int main(int argc, char** argv)
{
    int_iter(1000000000);
    long_iter(1000000000);
}```

that output above is from vTune, intel's profiler

they're essentially the same

[lapis hull]

hmmm

[pearl moth]

int_iter uses 209.031 ms, long_iter used 203.608ms

[lapis hull]

i tried profiling using unix time, now theyre essentially the same

[pearl moth]

of course, this is using msvc's compiler., which generated identical assemblies iirc

[lapis hull]

thats a little concerning because i profiled the rest of the material for my class similarly

[pearl moth]

gcc doesn't generate identical assemblies, so i'd except maybe slight differences

i don't have the infrastructure in place to do HW profiling on linux (it's not that it can't be done, i just am not set up to do it)

and i don;t offhand know how many i-slots cdqe requires. i'd be really surprised if it's more than 1, but would have to look

[lapis hull]

though i dont understand why profiling with getrusage consistently shows that long is quicker

i-slot?

[pearl moth]

getrusage is not using a HPET. you're measuring times that are barely largeer than the system timer ersolution here

timing a race with an hourglass

[lapis hull]

but i would expect it to sometimes show int being faster then

[pearl moth]

https://www.agner.org/optimize/instruction_tables.pdf

cdqe does appear to be one slot on every major processor revision

a naive analysis would suggest that the long version should run slower because it has four instructions in the loop instead of three, but with modern processors it rarely ever works that way

[lapis hull]

four instructions?

[pearl moth]

superscalar architecture means that the mov, cdqe, and cmp execution can overlap

including the conditional jump

oh hm, this is actually don't branch not branch just noticed that

oh i bet i know

it's probably due to the ALU being optimized to do 32-bit operations more efficiently

[lapis hull]

but long is 64

and its performing long quicker

[pearl moth]

sorry

[frozen vector]

[pearl moth]

well, same thing

the issue is that when the alu does a 32-bit operation it's required to clear the top 32 bits of the register

this is architecturally mandated

[lapis hull]

yeah

[pearl moth]

the alu itself is at least 64 bits wide (it may actually be 128 bits wide)

so you actually get a performance penalty for downsizing

(also there are like nine alus in a single processor core, i'm simplifying)

[jagged forge]

unrelated, but do simd registers just dispatch to multiple alus at the same time

[pearl moth]

lemme look up the semantics for 32-bit MOV

i know 32-bit arithmetic ops clear the high half of the 64-bit register, not sure about MOV

[pearl moth]

probably depends on processor features

[lapis hull]

mov does too

[pearl moth]

the spec is frustratiningly vague

it discusses claering the upper half of the register only for moves to segment registers

oh i should redo my windows test with long long, i forgot that long on windows is only 32 bits

[lapis hull]

oh

[lapis hull]

well fuck me

[pearl moth]

i am running a more extensive test right now, 1 billion times in each loop, repeated 1000 times with the int loop first and 1000 times with the long loop first. this is tkaing a while

should be about 800 seconds clock time, or ~12 minutes?

i have 10 cores so it's not really a problem lol

[lapis hull]

thank you so much!

[pearl moth]

the reaosn for reversing the order in half the tests is in case there's a cache priming effect

[lapis hull]

yeah

[pearl moth]

overclock me baby

cpu speed is 4.52 GHz

ooh 4.65 GHz

ah just finished

the int version uses inc eax while the long long version uses inc rax

[lapis hull]

damn

right

[lapis hull]

inc rax takes more time?

[pearl moth]

yes and no

answering that would require that i do microprofiling and look at instruction retirements

remember that this is a superscalar processor, it executes multiple instructions overlapping

so what you're really seeing there is where the pipeline stalls were

[lapis hull]

ah okay

[pearl moth]

the stalls in inc eax were probably due to having to wait for a replacement register to be available, presumably because they were all allocated to holding results not yet written to memory

id' be curious how many of the writes actually made it to memory as opposed ot being retired before being written

[lapis hull]

i think this getting to be beyond my paygrade

i tried running it using gprof, and im still seeing long running significantly faster

[pearl moth]

sure, you may well get different results with a different compiler

or with a different processor, i've got an i5-12600k which is alder lake, a different microarchitecture may perform differently

tbh this shit is complicated

[lapis hull]

i just wanted to give a simple reason to my students

[pearl moth]

i am not an expert on intel microarchitecture, i know a bit here and there but not nearly enough to explain everything going on at that level

[lapis hull]

i appreciate the help nontheless!

thank you very much

[pearl moth]

np that was actually interesitng and it's good to practice with the toolset

[gusty pelican]

guys I had an idea (though I don't know if it already exists)

though it probably does tbh

what if we did the cauchy sequence definition of real numbers, except without the converging requirement?
a "excauchy" real number (extended-cauchy real number) would be an equivalence class of sequences from N to Q using your usual equal equivalence (for every epsilon there being an omega or something such that for all natural numbers gamma bigger or equal to that omega |g_(gamma) - f_(gamma)| <= epsilon)

so like (this is a bit informal but whatever) EC(0, 1, 2, 3, 4....) would be equivalent to EC(0, 2, 2+1/2, 3+1/3, ....) but not equivalent to EC(1,2,3,4,5....) or EC(1,4,9,....)

wait would multiplication be an issue?

like (5+1/5)5 being 26 while 5 *5 is 25

uh

well addition would work atleast

is there a way to make multiplication work?

wait could you do this with asymptotic equivalence instead of normal equivalence?
well then you need to deal with the problem of 0 uh

also addition might become a problem in that case

sorry for using the discussion page btw

[alpine verge]

So essentially two of these sequences are equivalent if the difference-sequence converges to 0 (using some preciser notion of convergence) ?

[gusty pelican]

well my definition of convergence works for that
the problem is dealing with multiplication

since even though EC(0,1,2,3,4,5....) and EC(0,2,2+1/2,3+1/3....) are equivalent
I'm pretty sure EC(0,1,2,3,4,5....)EC(0,2,2+1/2,3+1/3....) would actually be (EC(0,1,2,3,4,5....))^2 + 1

since for example
5(5+1/5) is equal to 26 instead of 25 and 6(6+1/6) is equal to 37 instead of 36

[alpine verge]

Use "exponentially bounded" sequence (with basis r fixed above 1) and then restrict to convergence below this exponential factor, I'd say.

That would be a major restriction though.

[gusty pelican]

exponentially bounded like <= Ke^(thetax) or something? (sorry i'm not familiar with that term)

[alpine verge]

yeah. along those lines, like with polynomials and their radius of convergence.

[gusty pelican]

this problem has nothing to do with how fast it grows though?

[alpine verge]

My idea was just to impose other constraints, so that multiplication is well defined.

on second thought this wouldnt work out.

a = (a0, a1, a2, ...)
b = (b0, b1, b2, ...)
(a * b )n
= sum_k<=n BinCoefficient (n,k) * a(n-k)b_k
<= 2^n max{c_ar_a^n,c_b *r_b^n}
So each multiplication would blow up the "difference" by 2^n max{cr_a^n,c*r_b^n}, which AFAIK can only work if we use a cauchy sequence that decreases faster (asymptotically etc.) than all exponential functions, maybe inverse tetration or something.

[gusty pelican]

by a^(n-k) do you mean a_(n-k) by the way?

[alpine verge]

I made a mistake, by a^n I mean the radius r_a, so that a_n < c*r_a^n

[gusty pelican]

also what is r_a^n ?

[alpine verge]

well now I define r_a so that a_n <= c*r_a^n

analogous for b.

This only works for sequences for which such an r_a exists.

[gusty pelican]

by the way why max? It's a_(n-k) times b_k after all

[alpine verge]

just an easy to work with upper bound.

[gusty pelican]

no I mean wouldnt it be 2^n (c_a * (r_a)^n) (c_b * (r_b)^n)?

[vivid socket]

How do I solve, 1 (4/5) ?

[alpine verge]

No, we are only working with products of n numbers. But I left out the constants c_a, c_b.

This entire thing is very arbitrary though

[gusty pelican]

Also I'm not sure how it avoids this problem, though proving that this problem exists would be more complicated I guess

(of course you would divide the sum by 2^n first)

[alpine verge]

My idea was that if we assume that the difference a-b, scales inversely to the tetration to the power of n (with n being the index.), it would still do so even if the sequence is element-wise multiplied with an exponentially growing sequence.

All of this feels very arbitrary though.

[gusty pelican]

oh hmm

I see what you are thinking now

[alpine verge]

Maybe we should start at your definition again.

[gusty pelican]

ok here are my requirements

1. The sequences can be anything. Actual cauchy sequences, diverging to infinity, diverging to - infinity, going 0 1 0 1 0 1 forever, literally any function from N to Q (though the excauchy real numbers would be defined with equivalence classes of these sequences)
   Number 1 might have to be restricted a bit (like in your definition) but whatever
2. addition and multiplication are both well defined under the equivalence class
3. for all actual cauchy sequences a, b, c
   a = b => a (extended =) b
   a + b = c => a (extended +) b (extended =) c
   a * b = c => a (extended *) b (extended =) c

I've noticed that these things might have the property of having a !=0 b!=0 ab = 0 being possible

(0, 1, 0, 1....) * (1, 0, 1, 0....) would be (0, 0, 0, 0) uhh....

[gusty pelican]

hmm I can see this working, though I don't know why you need the sum

[alpine verge]

that was a mistake.

I think we shouldn't use the element-wise product, but instead:

(a*b)_n = sum_k a_k *b_n-k

[gusty pelican]

rule 3

I want to have it "not be different" for real number inputs (since this is supposed to be an extension)

[alpine verge]

https://en.wikipedia.org/wiki/Cauchy_product

i.e. this product respects the product of the limits of cauchy sequences.

[gusty pelican]

this is a convolution though

[alpine verge]

yeah

[gusty pelican]

like I still want (3,3,3,3,3....) * (3,3,3,3,3.....) to be (9,9,9,9.......) (or something equivalent to that or something)

[alpine verge]

Ok then we stick to the element-wise product for the time being.

[gusty pelican]

but under your definition it would be
(9,18,27,36,....)

well maybe that over n could work

or n+1 I guess

[alpine verge]

So right now we have {N --> Q} / {(a,b)| a-b is a cauchy null sequence}

We could try to define a metric.

[gusty pelican]

defining a metric sounds a bit problematic

like how would you measure the distance between (0,1,2,3,4,....) and (0,1,4,9,16,...) using a real number

[alpine verge]

We just make something up. Obviously all the L_p metrics fail in this case.

I have no idea.

Lets try something else.

[gusty pelican]

hmm ok
the sqrt of
sum from 0 to infinity of
({if -1<=a_n<=1 a_n, else 1/a_n} * (1/2^n))^2

hmm wait

this is a bit problematic

ignore this

it doesn't fulfill the x != y => d(x) != d(y) condition (sequences)

[alpine verge]

I was thinking maybe |(a0,a1,a2,a3,...)| = r so that a_n <= c*r^n

But this only work for sequences that grow at most exponentially.

its also not linear so nvm.

[gusty pelican]

is it the minimum r for that?

[alpine verge]

yeah. The infimum tbp.

[gusty pelican]

ok makes sense

[alpine verge]

I have a feeling that we wont find a metric for {N-->Q}

[granite bridge]

what about two excauchy sequences are equivalent if

1. they are cauchy and equivalent as reals
2. they are not cauchy

i think that would satisfy all your properties

[gusty pelican]

that ruins the whole point
that's literally just R with one more extra number

[alpine verge]

The equivalence is well defined by saying that their difference is a cauchy sequence converging to 0. I think.

[granite bridge]

i know it ruins the point

[gusty pelican]

well some semblance of the sequences getting "closer" to each other yeah

[alpine verge]

I think cauchy sequences fit the bill perfectly.

[gusty pelican]

I was thinking about using asymptotic equivalence, but I would have to remove 0's (in the sequence) and it would have some of the 0-sequences be "not-equivalent"
also it would cause a problem with addition

[granite bridge]

my point was mainly that you need another property here to make them not satisfiable by the projective line

[gusty pelican]

fair

though tbh my idea is a bit vague in the first place

[alpine verge]

Is this supposed to be an axiomatic definition?

[gusty pelican]

yep

[gusty pelican]

I guess you can consider this the fourth condition

ok I'm not gonna force this one since it probably won't be feasable, you can ignore this one to be honest, but some waying of removing non-zero pairs (a,b) that when multiplied get zero (ab = 0)

[alpine verge]

Thats why we shouldnt stick to the element-wise product and use the cauchy product instead.

[gusty pelican]

hmm

we are using the cauchy product / n+1 right?

[alpine verge]

wdym n+1?

[gusty pelican]

(so that it coincides with rule 3)

[gusty pelican]

the sum from 0 to n sums over n+1 things

I'm having 0 be in N btw

[granite bridge]

i dont think it matters

[gusty pelican]

eh yeah

[granite bridge]

as n -> infinity it wont make a difference

[gusty pelican]

well it does matter if you start from 0 to not get /0 but yeah

[alpine verge]

yeah

[gusty pelican]

also I can see it perhaps mattering with infinitely growing sequences

[granite bridge]

like which

[gusty pelican]

what I mean is like the difference between the sum/n and the sum/n+1 not converging to null

[alpine verge]

The problem we have is that our definition of equivalence does not work under multiplication with an infinitely growing sequence.

[gusty pelican]

also ab = 0 while a!=0 and b!=0 is problematic too, though it's acceptable since this isn't exactly the only extension of real numbers with that property anyway

nevermind

ignore that comment

actually I want to check if that is possible or not under the cauchy sum

[alpine verge]

I'm looking into that also

Havent found anything yet.

my gut tells that its not possible.

[granite bridge]

(1,-1,1,-1,...) * (1,1,1,1,...)

unless i misunderstood the cauchy product

[gusty pelican]

not exactly

you get (1, 0, 1, 0, 1, 0....)

which is not zero

[gusty pelican]

here's the definition of "equivalance" we're using right now btw

well the epsilon has to be bigger than 0 of course

and omega is a natural number

[granite bridge]

what no you get (1,0,1/3,0,1/5,0,...)

if its over n+1

[gusty pelican]

1*1 + (-1 * -1) + (1 * 1)

uh

1x1 + (-1)x(-1) + (1)x1

is 3

divide by 3

get 1

[hardy grove]

=3?

[gusty pelican]

(divide by 3)

[hardy grove]

3/3 =1

[granite bridge]

[alpine verge]

We should stick to the proper definition of the cauchy product and not divide by n+1.

[hardy grove]

I’m correct right

[gusty pelican]

yep

[granite bridge]

well then (1,1,1,1,...)*(1,1,1,1,...)= infinity

which is not ideal

[alpine verge]

No thats good

(1,1,1,1,...)^2 = (1,2,3,4,5,6,7,8,...)

[gusty pelican]

I don't think dividing by n+1 is what's causing the problem here though

[alpine verge]

The unit under the cauchy product is (1,0,0,0,0,0,0....) IIRC.

shit its equivalent to 0.

[granite bridge]

looks right

[gusty pelican]

lol

ok so we're sticking with /n+1 then

[alpine verge]

nope that will get us nowhere I think.