Hopefully in a year or so I can land a Django / web coding job.

That save button has a lot of server-side code behind it. A lot of stuff had to be coded to get the diagrams stored in a Neo4j graph database in a general (good coding techniques) way.

Thanks @varkor for solving all of the actual CD editor feature with their open source q.uiver.app

Is there a name for the orthonormal basis of analytic functions with the inner product $\left<f,g\right>=\int_0^1 f(x)g(x)dx$ starting with the basis $1,x,x^3\dots$? The first ones are $1$, $2\sqrt{3} x-\sqrt{3}$, $6\sqrt{5}x^2-6\sqrt{5}+\sqrt{5}$...

@PenAndPaperMathematics My personal opinion is that writing by hand is the best way to make websites.

@AMDG with the amount of Bootstrap the above requires, it's best for me to use a $30 tool BSS that exports to Django templates using an export script that I forked on github

@Derivative should that be 1,x,x^2...?

The editing of the style is instantaneous! Hand editing of Python though is correct, I'm afraid you're right: backend = manual coding.

@AMDG BSS lets and you are still required to hand code some CSS and HTML on the Django side. It just helps out a lot I find

if so, then what you're saying sounds like shifted Legendre polynomials. the only substantive difference is that those are normalized in a different way

I'm just saying it is easier to read well-written websites than procedurally placed website elements.

Also Bootstrap5 is enabling me to target all screen sizes! I love web libraries like that

I honestly don't know web technologies beyond raw html, css, and js

@AMDG The BSS drops the learning curve for deploying Bootstrapped HTML/CSS from 3 months to 3 weeks

It's a tool sort of like Webflow but only for frontend coding

What is bootstrap anyways?

And it was only $30, best $30 I ever spent

Bootstrap is a CSS library.

specifically, the polynomials you write can be written as $1,\sqrt{3}\cdot(2x-1),\sqrt{5}\cdot(6x^2-6x+1),...$. if you drop the square roots, then you have the shifted Legendre polys

But Bootstrap Studio targets it

You can code Bootstrap manually, but it's harder, I've tried it than using BSS which does a lot of the restricted forms you can use, for you

Ah, I think I vaguely remember then that I saw it in HS and said, "yeah, nope"

It allows you to target mobile (iPhone 5 say) to large ultra-wide monitors

all with one code

Because the way it works "responsively"

Cool. You can get that with CSS3.

Without it, you'd have to essentially code a lot of what BS is doing for you already

When I say BS I don't mean bullshit, lol, it's quite the opposite

You want to know how much I need in terms of a library for RWD stylesheets?

BSS = Bootstrap Studio, BS = Bootstrap5

What's RWD?

github.com/AdMaioremDeiGloriam/RWD-Flex/blob/master/…

responsive web design

Yes, but you're still ignoring 90% of the features of BS most likely. I didn't code it, but it's not trivial to do all that stuff

My app code will end up smaller than yours for the same (similar) site without it

Please specify said features. We're talking about implementing UIs which, mind you, is part of the inherent design flaws of the web trio (html, css, js).

That's what it does, it does all this fancy web stuff for you so that you can focus more on the critical parts of your code - the code that makes your app unique.

Okay, I'll brb

CSS shouldn't be used to make dropdown menus, for example, yet I can, and it is necessary.

Especially if you want performance

The proper thing for dynamic UI would be to let JS handle all dynamic aspects; CSS ideally is used only for styles (because that's its purpose); and HTML ideally is used only for document structure.

Bootstrap5 usage is using all: JS, HTML, CSS

You still have to code, it's just a library

getbootstrap.com/docs/5.1/getting-started/introduction

I can't find a feature list, but click on all components in the left sidebar if you want to see all the stuff it does. The default style of everything also is amazing

I just want a basic feature overview, that's all. I'm not really all that interested in bootstrap.

I didn't have to style the messages that pop up on my site - they already look like MSE's / SO's popups!

All I had to do was figure out how to show a message dynamically using AJAX, because I didn't want to reload the diagram_editor page

each time you hit save

Well yeah, but anyone could make a template like I did with rwd-flex.css that can do defaults for stuff. As for AJAX, I just use JS as-is, and I wouldn't bother with the overhead of jQuery.

But on page loads the messaging system worked out-of-the-box once you get a hang of exporting from BSS to Django and editing the templates you need to on the Django side

I love jQuery!

I would like it more if JS was fast

I don't use it much, but a lot of the code examples I seek are written that way

The JS of https:q.uiver.app is very fast. See its repo (click on the logo upper left) and you'll see it's 99% JS

jQuery is excellent in concept but terrible in practice

I'm using that in my site, which makes my site currently 95% JS

> 95% JS

I don't use a lot of jQuery but it makes the syntax easy / standard when doing the basic things I need it

So how often do you manage to get load times for your webpages at 50ms or less?

Yes, github has a display on the repos showing % of each lang

I haven't measured that yet, my site is still local

How do you do that - chrome dev tools?

You can still measure that, and yes, in chromium F12

It's built-in.

Interestingly Quiver (embedded in my site, but not mine - it belongs to varkor here on MSE)

uses no jQuery

With all the JS, I'm guessing something yuge

150ms or more

Maybe. Yes q.uiver.app is very complex I've hacked with its code ui.js / quiver.js mainly

Not a great UX to be in the order of 150ms

I just do the smallest change to hack an API into quiver essentially, so that I can hide its stock GUI which isn't screen responsive and replace with Bootstrap GUI

Their editor quiver almost works on iPhone, even the arrow options panel is viewable

and object / arrow placement works

If you rewrote q.uiver.app do use another library for doing the graph editing and display, that 150ms number would likely go way up

"features: almost works on iPhone!" :P

So it's nice that varkor essentially wrote his own SVG render engine for graphs

CDs (commutative diagrams) are somewhat different than general labeled graphs and how they're default displayed with graph viewing libs out there (there's at least 30 of them)

@PenAndPaperMathematics Like I said, I don't really use libraries. Chances are it'd go down if I rewrote quiver.

Well the bottleneck of my Abstract Spacecraft (the site which uses a local copy of Quiver) will be the Neo4j database searching, not the GUI

So when optimzing you have to find the bottlenecks first (low hanging fruit first principle)

And from my experience with older version of code (Quiver Database I called it), the bottle neck was saving / loading / searching on the server side

I could tell, without looking at the dev tools read outs

Makes me wonder if paying for an expensive database host in particular for Neo4j, would you get higher speed?

You don't necessarily need bottlenecks to optimize something.

We differ in our coding approaches, man!

Bottlenecks are only an incentive to optimize out of necessity. I just optimize everything.

I work from the top-down: GUI - to - backend

You probably work in the opposite dir, lol

J/k. Just saying we don't all code the same way

I work in a manner that is contrary to what most do.

Or at least, that is what I perceive

That's what makes the world interesting. Likening us to Genetic Algorithms that can generate code that solves problems, it's good that we differ there.

I'm not sure what that latter part means

@AMDG do you have a project site up? Want to see it :)

There's a field of genetic algorithms called genetic programming

I do not have a project site at this time, but I do have a discord server for my primary project.

GA's usually have you make some user defined struct for genes. But Genetic programming has the genes be actual programs. Also genetic generative grammars of programming languages.

You take the grammar of the language, usually available in EBNF and generate programs that are usually valid

It's very interest

@AMDG what's your project?

Project Aquinas.

It'll consist of a metacompiler, programming language, and execution environment.

What's the language accepted by the metacompiler?

Any language you define for the metacompiler. If you mean the default language, it'll be the programming language itself.

What's the programming language, can you paste a snippet?

I haven't thought about its syntax too much at this time as I still need to finish the foundation of the project. I only have the semantics.

Tell me more :)

@AMDG

Well first things first, it will not be OOP just to get that question out of the way. It will also not be functional. It will be primarily procedural, but it will essentially contain everything necessary for the design and implementation of any software or program.

So it is a Turing-complete language

Yes

most definitely

What's the purpose?

What does it bring to the table

Also what language are you using?

Host language for your language

It needs to be C++ I think

The metacompiler will let me convert anything to and from any form I define. The execution environment will allow me to run software on any hardware and platform. The programming language lets me program for the execution environment and program the metacompiler.

The host language will be its own language. For now, I'm implementing the metacompiler in C and then it will be rewritten in the programming language that I make.

Well, can it convert English / informal math into formal math (say Lean code)?

You say "anything"

With enough work, you could.

Nice

This seems to have turned into a computer programming chat. Not sure I like that.

So you can start connecting all the disparate languages on the web

Computer Science is a subfield of Math

Um, no.

Yes

It's all math, a lot of graph theory e.g.

Well, programming is not graph theory, is it?

Who's the overlord that says what and what isn't under the umbrella of math?

Well, I'm in charge of the room, so I'm more of an overlord than you are.

It's clearly math, Big-O, Algorithms (Algebra has grobner basis *algorithm") so..

I'm quite aware of Gröbner bases, and I even know how to spell it.

I don't do umlauts - I'm not German

Have your fun. Goodbye for now.

There's an adjoint functor between your category and mine

That's okay :D

Adjoints are everywhere man

Curry-Howard correspondence - programs are proofs and proofs are programs

@PenAndPaperMathematics Well yeah... if I cared enough about said languages. I'm only interested in supporting them... to get rid of them because they're garbage and are all full of design flaws. None of them adequately capture the substance of a programming language, hence why I'm making one.

There's only a handful of languages that I like off the top of my head, and the greater of them is the more flawed. I like C and lua.

Of course by handful I mean two

Anyways, looks like Ted is getting agro'd so if you want to discuss more, we should probably head over to another chatroom or chat elsewhere such as on discord.

@Semiclassical nice, thanks

Hi everyone

hi

bon something or other

What identities are available for real numbers (or any subset of R) for floor functions?

Specifically I want to know what identities are available for various permutations of floor/ceil/round for the elementary operations.

e.g. floor(a)floor(b) vs ab; floor(a)b; floor(b)a; floor(ab); etc.

It seems that he is giving a strange more intense flu. We have to be careful

wha

@AMDG en.m.wikipedia.org/wiki/Floor_and_ceiling_functions here we have some properties

Anyways, I'm wanting to know because I realized that I can compute floored divides to arbitrary inputs even if I only have a strict subset of ratios, in this case $[2^0, 2^{-n}]$ up to integers $x \leq 2^{2n}$.

Thanks, Alex

It's based on a simple modular identity of $x$ for any given modulus, but in this case modulo $2^n$: $x = 2^n \lfloor 2^{-n} x\rfloor + x\bmod 2^n$.

We have for example that $\lfloor x\rfloor \leqslant x \leqslant \lfloor x\rfloor +1$. or maybe that $\lfloor x+n\rfloor =\lfloor x\rfloor +n$ with $n$ integer.

I don't understand

What's the question?

prove that $1+{1 \over 2}+\cdots+ {1 \over n}$ is never an integer for $n\ge 2$.

If we restrict the domain so that $\lfloor 2^{-n} x\rfloor \leq 2^n$, then according to the above identity and based on the properties of division for integers, $\frac{x}{y} = \frac{2^n \lfloor 2^{-n} x\rfloor + x\bmod 2^n}{y}$, however, in my case, I want the floored quotient. Now, I can compute floored quotients for $x\in [2^0, 2^8]$, so how would I rewrite the RHS in terms of floored quotients to compute the floored quotient of x and y?

copper: ugh, did ted put that one in?

If I try to use only the floored quotient of one of the coefficients for the first term, I do not get correct results. It's close, but it needs another step.

You can see what I mean with cell 33 here: desmos.com/calculator/hgjecddlq9

@leslietownes no, my latest distraction ia801603.us.archive.org/13/items/…

but i am still grinding through Ted's questions, hoping to hit rings by the end of the week.

copper: oh. i think that problem is pretty tough for a first go-round. often books provide 'hints'

i am slowing raising my level of awareness, but not competency

alan baker doesn't seem to give hints

pete clark's answer at math.stackexchange.com/questions/2746/… (which links to a set of course notes) is the standard semi-conceptual approach. he motivates it pretty well.

but, he works for some crank discredited university, so take everything he says with a grain of salt

A question: an "answer" only the answer is a hint or a solution? And is it permitted in MathSE? A few hours ago I was challenged for giving an answer to a question from a user asking for a hint. I gave the answer in the comments.

:-). i notice an answer by a Bill D. His answers always look enticing but I can rarely follow.

alex there are one or more threads on meta about this. the last i checked there was something of a split of opinion.

i love many of bill's answers. i am less in love with some of the non mathematical content

@Alex use your judgement. remember the idea is to help the OP

he seems to have strongly expressed opinions.

isn't georgia a former soviet holding?

Thank you

copper: he may have more jumpsuits than you do.

I know that $f\sim g\implies \int f, \int g$ both converges or both diverges. If I'm studying the convergence of $\int_{0}^{+\infty}f(a,x)dx$ for some $a$ unknown and I find a function $g$ such that $f\sim g$ so the last result can I obtained the $a$ for the convergence of the integral?

depends on what $\sim$ means

$f\sim g$ in the sense $\displaystyle f\underset{+\infty}{\sim}g\iff \lim_{x\to +\infty}\frac{f(x)}{g(x)}=1$.

what does $f \sim g$ mean when $f$ has an unknown $a$?

I'm trying to study the nature of the convergence for $a>0$ for $\displaystyle \int_{0}^{+\infty}f(a,x)dx$ with $f:x\mapsto \frac{\ln(1+x^{a})}{x^{3}}$.

Finding the value of $a$ for obtained the convergence

i'm tired. i am making more mistakes than usual

$\frac{x^a}{1+x^{a}}\leq \ln(1+x^a)\leq x^a$ for all $x^a\in \mathbb{R}$

is there a question there?

Is that correct?

On $]0;+\infty [$ we have $\frac{\log(1+x^{a})}{x^{3}}\leqslant \frac{x^{a}}{x^{3}}$ can I use that fact for the solve the problem?

yes to ${1 \over 1+x} \le \log(1+x)$ for $x \ge 0$.

true for $x \in (-1,0]$ as well.

Uhm

i am missing an $x$ above

Well I understand that part

what is your question

Setting $f(x)=\frac{\log(1+x^{a})}{x^{3}}$ and $g(x)=\frac{x^{a}}{x^{3}}$, Can I study for what values of $a>0$ we have $f\underset{+\infty}{\sim}g$ for ensure convergence of the improper integral $\displaystyle \int_{0}^{+\infty}f(x){\rm d}x$?

@Alex that only works for $a$ that ensures equivalence.

😞

I had my suspicions

since behaviour around $x=0$ seems to be the issue, maybe you can use $x-{x^2 \over 2} \le \log(1+x) \le x$?

Hello

How to prove that the limite of $n^2-n+1$ is $+\infty$

if you write it as (n - 1/2)^2 + 3/4 it's clearly larger than n - 1/2 whenever n is larger than 2

which might help you choose an N given an M, blahblah

david, i am not sure that 'converges faster than' has a meaning. if it does have a meaning, it would probably not be said that sum .9/n^2 converges faster than 1/n^2 although .9/n^2 < 1/n^2 for all n

Hmm exactly right? It’s just that my textbook made a similarly vague statement

what does converge faster than mean?

Courant claims that’s the integral of a power series converges faster than the original power series as the coefficients of the integral are smaller than the original series

@copper.hat I have no idea; intuitively for me it means the series or sequence approaches it’s limit value in less steps

But I haven’t learnt a formal definition if there is one

@Vrouvrou surely you can see that $n^2-n+1=n(n-1)+1$ ? for $n \ge 2$ we have $n^2-n+1 \ge n+1 \ge n$.

What would be the cokernel of $T:\Bbb Z^2\to\Bbb Z^3$ by $(a,b)\mapsto (6a+2b,4a+8b,2a+4b)$ as a direct sum of cyclic groups?

what's the buzzword? smith normal form?

granny smith normal form

cosmic crisp normal form

im out of puns as you can ytell

A priest, a minister, and a rabbit walk in to donate blood. The rabbit says, I think I might be a type-O.

I had to look it up. Further proof of decreptitude.

@Vrouvrou By AM>= GM, $n^2-n+1=\color{red}{(n^2+1)}-n\ge \color{red}{2n}-n$

If $a,b$ are positive integers, then $(ab)!/(a!)^b$ is an integer

@love_sodam look at this answer which shows that that is divisible by $b!$

@robjohn Cool. I was thinking about Lagrange's theorem on $S_{ab}$.

@love_sodam This is a multinomial coefficient, so it is an integer.

$\left[x_1^ax_2^a\dots x_b^a\right]\left(x_1+x_2+\dots+x_b\right)^{ab}=\frac{(ab)!}{a!^b}$

@copper.hat a by-product is cousinmatter.

it is dangerous once removed from a vacuum

speaking of/to a vacuum...

o/

o/

jellyfish

that, sir, is one mean looking fu manchu @robjohn thus making it a mean Fu Manchu Grinch squared

What is ratio of 250/3800?

The percentage is 0.0658 but what ratio?

A bilinaer form on R^n has the following contstruction due to $M_{ij} = A(e_i,e_j)$ where as $ A $ is the bilinear form and $M$ the matrix. so then we have a diagonal matrix obviously. then it should be for "$A(\vec{a},\vec{b}) = \sum_{i=1}^n M_{ii}a_i b_i $ but it is often you find written in textbooks# $ =\sum_{j=^1}^n \sum_{i=1}^n M_{ij} a_i b_j $ Why is that

After finding ratio, what would be a proper amount of first number (250) for amount of new second number (1000)?

I want to find out how much ml of liquid (250) I must use for amount of powder (3800 but this time for 1000).

if you use 250 ml for 3800 gramm of powder (or whatever ) then if you want to keep the ratio constant, you will use x ml = 1000 gramm now you have two equations you can solve

Dont forget to write the units, or the question is meaningless

Nvm the question i asked i have my answer i did that wrong.

@Boris_yo 250/ 3800 = x/1000

yes

How much of milligrams of liquid I should use for 1000 gramm of power?

Did not i just answer you?

I could just divide 250 by 4?

That'll be fine. Thanks.

250 is to 3800 as x is to 1000

65.8

@Boris_yo you could, but you wouldn't get the correct answer

where as $\alpha,\beta$ are constants and could be any reell number, where as $\gamma $ is the determenant

Obviously $x,y$ need to be bigger than zerp, because if either $\alpha,\beta =0$ then the second relationship wont hold if it is not the case. how do you move on from here?

I am denoting $x = A_{11}, y = A_{22}, z = A_{12} =A_{21} $ because of symmetry and $ \vec{(\alpha,\beta)} $ such that we get equation $i$ from teh determenant of $2x2$ matrix and $ii$ from $A(\vec{(\alpha,\beta)} ,\vec{(\alpha,\beta} ))$

sorry in the second relationship we need a + instead of a minus

Can anyone tell me what is wrong with my answer here: math.stackexchange.com/questions/4348297/…

@Semiclassical I think I can describe that spherical orientation system I was trying to describe before now.

ok

Well the intuition at least is that to describe a point in $\Bbb{R}^2$, one needs two 1D real number lines. Likewise, to describe a point in $\Bbb{R}^3$, one needs two 2D number lines or 3 1D real number lines.

Let's define up and down to be the z axis

Now let's take a unit sphere and center it on this z axis such that the center is $(0, 0, 0)$ in $\Bbb{R}^3$ for the sake of this exposition.

Now pick any of the other two axes--we'll choose y--to define a 2D plane.

is this plane the xz-plane, or the yz-plane?

yz-plane

kk

Draw a unit circle at $(0,0,0)$ which exists in the yz-plane, and choose some angle $\alpha$ relative to some starting point in the yz-plane that is also on this circle--we'll choose the point $(0, 1, 0)$ to be associated with an angle of $0\pi$.

so something like pi/2 would be (0,0,1) ?

Yes, for that circle.

err plane

Now draw another unit circle centered at $(0,0,0)$ such that it passes through the points associated with $\pm\alpha$ on the first circle. This second circle now defines its own plane relative to the yz-plane, though it could be aligned to a set of axes accidentally.

I encountered a problem when reading this post: math.stackexchange.com/questions/1570288/…. I know that the notation $(f_*\mathscr O)_{f(x)}$ means the colimit of the directed system $\{\mathscr O(f^{-1}(V)): V\ni f(x)\}$. But why is it called a stalk? I've only seen definitions of stalks at points in the topological space $X$. Is there a more general definition in which case the notation $(f_*\mathscr O)_{f(x)}$ makes more sense?

Then a second angle $\beta$ defines a point as some distance traveled along the second circle relative to the point defined by $\alpha$.

Thus we have a 2D coordinate system which maps to all unit vectors in $\Bbb{R}^3$ modeled geometrically as a point on a sphere.

$f(x)$ is a point in the topological space $Y$ and $(f_{\ast}\mathcal{O})_{f(x)}$ is the stalk of the sheaf $f_{\ast}\mathcal{O}$ on $Y$ at that point

so, here's what seems like an equivalent description. start with the points (0,1,0) and (0,0,1), and make a rotation of angle alpha around the x-axis. the point (0,0,1) now comprises a new axis: rotate the other points through an angle beta w/r/t this new axis

Sounds about right. You put it far better than I ever could.

Also of note here is that it works regardless of whether or not the two circles are perpendicular, but keeping the constraint that they must be perpendicular certainly would simplify calculations.

@Thorgott Aha, I see. I mistook the pushforward sheaf. I pictured it as a sheaf on $X$ because its values are in $\mathscr O_{X}(f^{-1}(V))$. But it's constructed on $Y$! Thanks for your answer.

unfortunately, this still runs into a problem. suppose for simplicity that your object points in the (0,1,0) direction to begin with, so we don't need to employ any rotations to begin with

say, let's take it to be my hand pointing in the (0,1,0) direction with palm facing down (the (0,0,1) direction)

how do i tell that apart from my hand pointing the same direction but palm facing up?