/sci/ - Science & Math

https://imgur.com/a/Gzp2Ile

>>14974693
Wtf?

globohomo

What can I concretely do with abstract algebra?

>>14974886
ML-AI things, quantum physics if you really want to shoot yourself

>>14974886
molecules are essentially Rubik's cube on steroids; you can analyze the behavior of a Rubik's cube with group theory...Quora claims that "IR spectroscopy" is essentially group theory

Is there a book on graduate (complex) analysis that assumes I know and makes full use of graduate abstract algebra?

>>14974886
More abstract algebra
It's fun

>>14974886
Cryptography.

>>14974693

>>14975024
woah wtf

Is anyone else here going back to thoroughly study basic math? When I got to more complicated math I struggled and it took me way too long to solve problems because of my shaky foundations. Everyone here likes to brag about how they are currently teaching themselves how to triple integrate p-adic spectral sheafs over riemannian rings or whatever but what about spending long hours grinding through the very basics? Your ability to solve problems in higher level mathematics ultimately depends on your skills in ementary algebra, geometry and trigonometry anyways. This is why I have decided to go all the way back to pre algebra and solve every single problem in every single textbook so that the higher level stuff makes more sense. I wonder if anyone else is doing the same. I would appreciate advice from fellow autodidacts. Especially those who also lost years of schooling due to dropping out/inadequate homeschooling.

>>14975142
The pandemic threw my entire math education entirely off. I’m pretty busy and don’t have hours to go through complex problems, so instead I’m currently memorizing trig identities and other laws day by day using these study sheets:

https://www.eeweb.com/tools/math-help/


Every couple days ago I finally can put my memorization to practice and solve problems maybe for an hour. Eventually hopefully after I feel like I know everything on these sheets I feel like my foundation will be strong enough to finally get into calc 2 and then calc 3.

>>14975166
>I’m currently memorizing trig identities and other laws
That's the mistake I made. You simply cannot learn math by memorization. You must do problems. Confidence is a function of repitition. So if you want to avoid panicking when your final exam finally comes you must do every single problem in the textbook.

>>14975166
If you're an undergrad studying mathematics, then read Axiomatic Set Theory (1960) by Patrick Suppes. You'll learn the basics for everything you need to know.

>>14975278
Please go away. They do not have the mathematical maturity for that subject.

Recently came across the notation C^k to denote class of functions with k continuous derivatives over some domain. 0 is used to represent continuous functions, infinity for smooth, and omega for analytic. Are there any other such classes of functions? Stronger regularity than analytic? Any other values of k used?

Hello /mg/. Here's a new problem. Easier than the last one I shared (the coin flip one) imo because I couldn't solve that one but could this one. Let me know if you find a solution or if you get stuck, I can try to push you in the right direction if I can. Also I won't be able to check the answers or reply until night because I'll be at school. Good luck!

Assuming [math] 1 + 1 \neq 0 [/math] How can I show that if every affine combination of any two elements from a set is included in the set, every affine of any finite number of elements is also included in the set? I know it has something to do with partitioning the sum of coefficients into non-zero sums and dividing, but how can I show such a sum exists?

>>14975142
>This is why I have decided to go all the way back to pre algebra and solve every single problem in every single textbook
finding and filling gaps is fine anon. so long as you're also doing challenging problem sets to prime the pump. you can jump in anywhere though. you might be interested in the series authored by Hung-Hsi Wu:
Algebra and Geometry
Rational Numbers to Linear Equations
Pre-Calculus, Calculus, and Beyond

they're probably an improvement on whatever texts you may have laying about.

>>14974679
undergrad with no knowledge of what these mean, but here's my proof:

First note that with c=1, c'=1 we obtain d(1,1)d(1,1) = d(1,1). Since D is a sector, we have D <= G, so in the group we can cancel and conclude d(1,1) = 1_G. However 1_D = 1_G, so d(1,1) = 1_D and we have satisfied the first condition of a calc.

For the second condition, consider any element in e(cC) = d(1,1cc'). By the definition of a differential, we have d(1,1cc') = d(1,1c)d(1c,1cc') which is an element of d(1,1c)D. So, the given map is a calc.

>>14975142
>>14975563
if you're able to get through the Art of Problem Solving Series after filling those gaps, then i think you'll have the confidence you'll need to continue on.

Can anyone recommend an easy to use software like Desmos to help me visualise objects in Linear Algebra like translate, direct sum, flats, etc.?

>>14975142
I am doing the exact same thing starting from Prealgebra. I dropped out from school at the very beginning of high school.

Is it possible for human brain to comprehend [math]TREE(3)[/math]?

>>14975351
Isn't this just [math]\frac{{12 \choose 5}}{2^{12}} = \frac{99}{512}[/math]? Of 12 steps that either of them do exactly 5 of them have to change the x-coordinates.

>>14975568
I can't find anything wrong with the proof
Here are some remarks as to the meaning:
Sector - abstract writing
Metroid - additive metric measures the length of what is written
Calc - abstract reading/writing

>>14975299
Yes, for example for [math]0 < \alpha < 1[/math] you have [math]C^\alpha([a,b]) [/math] the space of Hölder-continuous functions with Hölder constant [math]\alpha[/math].
Those are the functions such that
[eqn]|f(x) - f(y)| \leq k |x-y|^\alpha[/eqn]
for some constant [math]k[/math] for all [math]x,y \in [a,b][/math].

Between any two Banach spaces you can always find infinitely many interpolation spaces between them. The Hölder spaces above are the interpolation spaces between [math]C([a,b])[/math] and [math]C^1([a,b])[/math].

A small warning [math]C^1([a,b])[/math] is not an interpolation space between [math]C([a,b])[/math] and [math]C^2([a,b])[/math]

>>14975759
Oh very cool thanks. Reading a bit, seems both ideas can be combined to give Holder spaces denoted C^(k,a)

>Khan asks me to try solving a problem before watching how he does it
>can't solve it
>get angry at myself and self-harm because I don't immediately understand how to do any math problem ever
I know it's irrational but I still do it.
I hate being like this.

God I fucking hate analysis. Literally the most uninteresting math in existence.

>B-BUT ARENT SHITTY RESULTS ABOUT SEQUENCES USING NOTHING BUT THE TRIANGLE INEQUALITY SO HECKING COOOOL. I LOOOOOOOOOOOOOOOOOOOOOOOOOOOVE ANALYSIS.

no they arent. And don't tell me that it gets interesting the further you study, abstract algebra was interesting from the start.

>>14975833
>didn't start with the greeks
sorry you didn't go with the historical approach

>>14975833
you seem upset

>>14975821
>post tranny art
>have trannylike psychological tendencies

>>14975467
Wouldn't you just use induction?
An affine combination is just [math]c+\sum_{k=1}^n a_kx_k[/math], right?
Just take out the last term of the sum, and you have an affine combination of the the last term and the previous terms; i.e. an affine combination of two elements in the set. Doesn't matter where you put the c (with this "outer" combination, or with the inner combination), you still have an affine combination of two elements.

>>14974666
Slighlty off topic question bros. I am applying for grad school. Right now my GPA is a 3.76, major(math) GPA 4.0. However after this semester my GPA will drop below a 3.5 (major GPA still a 4.0). Since I am applying right now, this means I only need report my current GPA right? And my transcript that I send will not show the monumental fuck up that was this semester, right?

>>14976049
Affine combination is:
[math] \sum_{k=1}^n a_k \vec x_k; \quad \sum_{k=1}^n a_k = 1[/math]

How do I fix my math that I learned in school? I mean, I did fine upto high school, but I only did so because I usually did so many problems that one of them usually appeared on the test (with different values). But I did not "understand" any of that.

For example, I learned about complex number, and the Argand plane etc. but recently I learned that it is more intuitive to think of complex numbers are real numbers with (phase) rotation to them. This way of thinking suddenly improved my understanding of the Fourier Transform and how the nature of functions in one domain are related to the nature of their transforms in the Tx domain.

How do I fix gaps like this in basic math?

>>14976137
Okay, gotcha.
Are there are any other restrictions on the a_k's?

Here's what I came up with, using the assumption that 2 != 0:
a1 + ... + an = 1, solve for an, and replace in the expression a1x1 + ... + anxn.
You now want to write 1 as a sum of powers of 1/2, like so:
1 = [1/2 + 1/4 + 1/8 + ... + 1/2^(n-2)] + 1/2^(n-2)
I wrote 1/2^(n-2) twice intentionally, instead of 1/2^(n-1).
Call these powers b_k, and c_k = a_k/b_k
Then your expression for a1x1 + ... + anxn becomes the sum of b_k[(c_k)x_k + (1 - c_k)x_n]
Each expression in the square brackets is an affine combination, so it's in the set.
Now you have an affine combination of the brackets terms, of which there are n-1. Use induction.

Pic related for an example with n = 4.

>>14976248
>1 = [1/2 + 1/4 + 1/8 + ... + 1/2^(n-2)] + 1/2^(n-2)
I don't think I can use that. I have to prove it for any field where [math] 1 + 1 \neq 0 [/math]. How can I express the sum like that when the field is integers modulo 3?

>>14976336
.. you can do that...
I specifically used powers of 1/2 BECAUSE we're in a field where 1+1 != 0, i.e. you can divide by 2.

>>14975833
Feel the same way about probability. Man is it fucking boring. I actually enjoy analysis, and I will agree that abstract algebra was the most interesting course.

>>14976336
>>14976339
And by the way, in mod 3, 1/2 = 2.
Multiply both sides by 2, and you get 1 = 4, which is true mod 3.
In general, explicitly, in a field with an odd characteristic p, 1/2 = (p+1)/2.

>>14976339
>>14976353
Oh okay, I don't really know Abstract Algebra. Thanks.

>>14976391
A field is basically a set with the four basic operations: Addition, subtraction multiplication, division.
And they work in the same way (commutativity/associativity/etc.)
And you can divide anything by anything, except by zero, of course.
Saying 1+1 != 0 means 2 != 0, and so you can divide by 2.

>10 years ago, John was 5 times older than Steve. John is now 3 times older than Steve. How old is John?

Can someone solve this? I had a test and it bothers me that I couldn't figure this out. I'm a front-end web monkey so I don't fucking know math, but I had to do a test for some reason

Hey, you, ya fucking nerds, invent more useful equations for us engineers.

>>14976415
(J-10) = 5(S-10)
J = 3S

Solve.

>>14976418
Finish the ones on your plate.

>>14976415
x - John's age
y - Steve's age

>10 years ago, John was 5 times older than Steve
(x-10) = 5*(y-10)

>John is now 3 times older than Steve
x = 3*y

two linear equations, two variables

>>14974886
i don't know the actual details, but modern cryptography is 99% number theory and algebraic geometry, which in turn are 100% algebra

>>14975142
I'm going to community college to take precalculus in the spring. I learn better when herd instinct kicks in, so I think I'll do better at cc than on my own.

>>14976415
John is 60.

>>14976456
Algebraic geometry is geometry.

>>14976480
Define a Groebner basis using a compass and straightedge.

>probability analysis class
>world problem about deductibles, premiums, and expected profit.
>Zero fucking idea of what a deductible or premium is, there is no additional information in the problem explaining what they are or how they work
>get a zero

I hate how EVERYTHING in this gay planet is enveloped in some word or language trickery, which just serves to obfuscate the actual mathematical content inside.

>>14974695
this one doesn't do anything for me

>>14976497
>define a CAT(k) space with compass and straightedge
>define a spin group with compass and straightedge
>define a tangent space with compass and straightedge

>>14975024
OPEN UP YA OPEN UP YA OPEN UP YA OPEN UP YA

>>14976523
If you wanna define those too, then go ahead.

Are all flats in a vector space over a field of order 2 necessarily subspaces?

mathematica vs matlab vs octave vs sagemath vs maple?

>>14976781
No. All singletons are flats; less trivially, in F2^2 {(0,1),(1,0)} is a flat

>>14976415
>I don't fucking know math, but I had to do a test for some reason
This isn't math; it's an IQ test

>>14976806
Matlab if you want a job

>>14974693
>>14975024
what causes this?

>>14976837
Not really, there were no shapes or patterns it was just math. But mostly percentages and ratios which were easy. And a few impossible questions like how much would water level rise if in a pool if you threw a cube in it

>>14976806
MATLAB if you actually do stuff in the real world.
Today I was running a 30 million dollar wind tunnel and processed my data with MATLAB

>>14975351
>>14975649

12C5 paths that connect the two points. They would meet after 6 steps.

So (12C5)/(2^6 * 2^6)

>>14976439
>Finish the ones on your plate.
I'm the one making the demands here, nerd. Don't make me beat you up and take your lunch money.

>>14976520
It's somewhat unclear how these guys work, and if you wanted to do vision research with free guinea pigs, 4chan is where you'd find 'em...both pic related and that one are not mine originally, it seems to have something to do with building up some extra saccade detection hardware in the eye, essentially bionic drugs that grow into your body from the computer screen...nothing new for the Space Invaders and Galaga generation
SI and Galaga are important because they rewire the way your brain uses your hands and eyes; these are the same, but you're also using your eye movements as your hand, as well as your back and the "chicken" part of the brain that maintains a consistent frame of view
so it's eye-hand vs. eye-back
so you should be aware of your breathing and your back and your saccades, and maybe listen to a breath meditation or some progressive trance

so what do I do if the second derivative test is inconclusive for a point?

I think I might be stupid.

I'm doing this word problem right (not asking for help on this particular problem). It's a combined rates problem where I'm trying to find the speed of the current in a river with only the speed of the swimmer and the knowledge that they swam 4.5km with the current and 3km against the current in the same amount of time.

The solution Khan Academy uses was so genius. I spent like an hour staring at my paper going nowhere scribbling down the same shit over and over because I had no idea what to do. I've watched the combined rates example video 3 times now and I still had no idea what to do. I think I might be retarded. I almost self-harmed again.

I hate myself.

We're just posting Openstax as a joke to dick over people who can't google and come here looking for elementary textbook recommendations, right? I skimmed their precalculus book and it isn't very good, even by the standards of books that have colorful pictures.

>>14977822
It's okay, anon. Not everyone is meant to be smart.

>>14977822
>I spent like an hour staring at my paper going nowhere scribbling down the same shit over and over
Given
>scribbling down the same shit over and over because I had no idea what to do.
You're most likely tunnel-vision'ing on the problem, and overlooking something due to your frustration. It's not an issue of you literally being unable to solve it ever. It's just an issue of you not allowing yourself to, because you're getting worked up.
Be objective with yourself, and discern if there's basic knowledge gaps that you need to fill, or if you simply need to try another one. Your monkey brain wants to avoid stressors at all costs (not understanding), and will deter you away from the act. Reject the reaction, and work out the problem slowly.

>>14977822
You invested an hour and were victorious; you wouldn't be upset about investing an hour in Super Mario Bros. to achieve a victory.
Just to test this theory of mine, I'd like you to do an "experiment" on yourself with a sample size of 1.
Play Super Mario Bros. on the OpenEmu emulator.
Start playing and take a baseline sample.
Record your performance (score, time spent on each level, etc... and use a stopwatch too, don't just go by the time in the game)
Now invest an hour in achieving goals in the game. The goals are given to you, so it's just one thing after another, not like math where you have to constantly find new problems.
If you are well-slept and focus on the game, you should be able to improve, as long as you're enjoying the process to a certain extent; if you hate the game, you probably won't improve.
If you still hate yourself after improving at Super Mario Bros, after achieving goals in the game, after investing an hour, then it's an attitude problem, you're going to hate yourself no matter what sort of skill you develop, even if you can achieve goals after investing small amounts of time, less than the time it takes to watch an ordinary feature-length film, no less

what study methods do you guys use

>>14977822
My biggest problem with math is translating a real-world or verbal problem into a math problem. I usually cannot do it unless it's very simple arithmetic.

Math intelligence is a specific kind of intelligence, either you have it and you're Good Will Hunting or you don't and you never will. I definitely do not have it, though if I can memorize an algebraic process I can replicate it. But it means almost nothing to me so I soon forget it again.

>>14977928
Pay attention in class

>>14976415
>10 years ago, John was 5 times older than Steve. John is now 3 times older than Steve. How old is John?
John is the red bar, Steve is blue bar. Black bar is the added 10 years.

John has 60 years

>>14977928
no need to study if you aren't a weeb

>>14975142
I've been thinking about doing it. Probably not so thoroughly, but maybe go through Basic Mathematics by Serge Lang and Khan Academy other specifics. I'm in a PhD program and noticing all too frequently the things I lack, because I wasn't really interested in math until undergrad, so anything before calc 1 shaky.

>>14976086
As long as you submit your applications early enough, then yea schools won't see this semesters grades. The only school that'll see it is wherever you go, because you'll need to send in an official transcript once you're done with undergrad.

>>14975649
Hey fren. Nice job. You're absolutely correct.
>>14976930
Hey fren. Nice job to you also. You're absolutely correct as well!

Thank you both for your time.

>>14977761
Insert an algebraic error to end up with a derivative that is non zero, and get partial credit.

>>14974823
This is a globohomo approved thread with its open source crap. Best math books are those not written for free.

>>14978357
I just looked up the graph on WA and figured out it was a saddle point, but still annoying

>>14977761
Double check algebra and then test different values near the critical point/think about the function logically (or just graph). Often you can tell whether it's a maximum or a minimum just by looking at the function and if you start getting really confused then it's probably a saddle point.

>>14977761
Take higher derivatives until it works:
https://en.wikipedia.org/wiki/Derivative_test#Higher-order_derivative_test

>>14978403
I think they're complaining about multivariable functions, as their text is presenting the Hessian determinant here. Taking higher derivatives like this is only valid in the single variable case.

That being said, I know I saw some analogue does exist where it's valid for arbitrary Banach spaces, but I can't be assed to look it up.

>>14977761
Honestly, your best approach for this is to graph it, plug in some points around the critical point to see what's happening, or think about things logically.

>>14978445
For instance, f(x,y)=(x+y)(xy+xy^2). (0,0) is inconclusive for the second derivative test, but for x=y=ε>0, f(ε,ε) = (2ε)(ε^2+ε^3) > 0, and for x=y=ε<0, f(ε,ε) = (2ε^3)(1+ε^4) < 0, so it's a saddle.

How do I show union of two flats, not subset of each other, is a flat only if one is a translate of the other?

How do I show union of two flats, not subset of each other, over a field of two elements, is a flat only if one is a translate of the other?

>>14978531
What's a flat?

>>14978610
Affine subspace.

>>14974666
Having a real hard time understanding a proof.

Context is c0 semigroup theory and abstract differential equations.
The proof shows that for any [mat]z_0[/math] in the domain of the infinitesimal generator [math]A[/math] of the c0-semigroup [math](T(t))_{t \geq 0}[/math] then the function [math]z(t) = T(t)z_0[/math] is the classical (once continuously differentiable) solution to the initial value problem:
[eqn]\dot{z}(t) = Az(t) \hspace{2mm} z(0) = z_0[/eqn]

The argument I am looking at defines a function [math]\zeta (t) = T(t-s)z(s)[/math] and shows that for all [math]s \in (0,t)[/math] we have that [math]\zeta[/math] is differentiable with derivative 0, thus constant and this shows that the proposed solution is indeed the unique classical solution to the IVP.

I dont get it, why does this show that the solution is unique?

>>14976523
i accept your concession

>>14978531
>>14978632
"over a field of 2 elements" is not an assumption in the question, but something to prove.
What you're trying to prove is this:
"If (the union of two flats is a flat) and (neither is contained in the other), then (|F| = 2) and (they're translates of each other)"

It's best if you write down what each assumption implies, and what it is you're trying to prove, in terms of elements.

>>14978697
You think you know better than me what I am trying to prove?

>>14976900
Archimedes knew the answer in 3rd century BC.

>>14977822
Figure out why you couldn't answer. Hone in on the exact point.
In my experience, the problem is usually with interpreting the units like speed and distance and how to make intuitive sense from them.

>>14978748
Bad bait.

>>14978759
If you can't answer the question, shut up. I know what I am asking, and what I am asking is what I wrote. Keep your fanfiction about my post to yourself.

>>14978772
If you knew what you were trying to prove, you would've proved it already and wouldn't be here.

>>14975024
>>14974693
I think I'm gonna throw up

>people over 100 years ago had to look up values in sine tables
>year is 2022, people still have to look up cumulative Poisson probabilities from a physical book table

Are you fucking kidding me.

>>14979039
You only do that in low-tier classes.

/sci/ plays /mlp/ shortly >>14979025

https://mathjobrumors.com/thread/137/page/4#p69

join us. it is /sci/ lite. free and anonymous

Can someone give me quick rundown of modern geometry? Algebraic, affine, projective, differential, what do all these mean?

>>14979168
how is this sci alternative better or worse than mathchan, 7chan sci or the deepweb sci boards? They all suck.

>>14979169
The more sheaves you have the more geometry it is

>>14979180
You can replace sheaves entirely if you work with Grothendieck topoi

>>14974689

>>14979177
What the fuck is MathChan? Banned me for failing a captcha it never asked.

>>14979217
we can also define the *complete n-halt*
>Hal^n e = {c <- C: hal^i e^c = 0 for all i < n}
Challenge: find C,D, and e such that Hal^3 != hal^3

god bless professor leonard

>>14975833
it does actually get better, it gets interesting when you get to analysis on manifolds and then that open path to know about manifold theory which then kind of overlaps with abstract algebra (algebraic topology). But yeah, I agree, abstract algebra seems more fun at first glance

God I hate physics
>so you have 142 farts per meter cube area in the z axis upwards reverse direction levagraged over the fulcrum of 10 sneeze square gravitons (with no friction), how many snoorps are sdorfed that day?
>computed at 14.2, but we round up to 15, and and that point 100 is a good enough answer for our models (less than the speed constant of half the distance to the sun of two dogs asleep)

Somebody please recommend me a good book on stochastic optimisation.

>look up old exams of functional analysis
>"give an example of a complete space, and an incomplete space"
>20 points
>out of 100
What the fuck

I want to write a program that converts SVG files into G-code for a laser engraver, and that means both drawing outlines AND rastering the fills. If I want to do the solid fills by just drawing concentric outlines (much better than a simple raster for small vertical lines), I need to solve this vector problem.

Given arbitrary vector coordinates forming a corner A, B, and C, given equal spacing of the new line r, what kind of equation do I need to figure out D? I've looked into using trig, but going to and from polar messes with my head so if there's a way of just doing so with vector algebra that would be incredible.

After that I'll need to figure out if the new outline causes an overlap, or how different outlines will intersect one another.

>>14975279
Bluepilled

>>14977928
>buy a book
>read it, do the exercises
>buy the next book

>>14976415
they are both born in this instant

>>14979651
wait maybe y = y' + m (x - x') is the way to go

>>14979651
Okay, I'll assume from the drawing that you want
two right angled chevrons with a separation of r
from the sides. A small square of length r can be
drawn between the two apices. With B being at
the origin (0,0), D is along the diagonal of the
square whose length is [math] r \sqrt{2} [/math].
So, D is located at [math] (0, - r \sqrt{2}) [/math].

Now, for each point in the upper chevron [math] (\pm a, -a) [/math],
they become position vectors [math] \langle \pm a, -a \rangle [/math]
and those points are dragged down to a spot [math] r \sqrt{2} [/math]
under them. So, the equation for now becomes [math] \langle \pm a, -a \rangle + t \langle 0, -r \sqrt{2} \rangle [/math]
with t between 0 and 1 to allow for laser sweeping.

Would this be fine or can it be improved some more?

>>14979780
Not right-angled. Arbitrary coordinates, with arbitrary angle between.

https://en.wikipedia.org/wiki/Pythagorean_triple#Geometry_of_Euclid's_formula
Did it really take until 1998 for someone to figure out the unit circle proof?

>>14979861
I doubt the citation would be implying that it's the first resource

>>14979889
if you read the citation, the guy proposes it like it's new and I can't find any older references to it

>>14979651
Uhhh, it's kinda getting there? I'm rotating the unit vectors of each line by 90 degrees anticlockwise, then multiplying it by the spacing constant, and adding that to the location of the corner point, in order to set the offset of the lines when calculating their intercepts via >>14979771. I'd expect that would cause it to be inside the lines of the shape if it's an anticlockwise coordinate list, and outside if it's a clockwise coordinate list, but that square is giving me shit. Even avoiding the edge cases by slanting all the lines in it keeps that same offset bullshit.

Turns out right angles means i have a bunch of infinities and 0s as my gradients, which isn't great when I add and subtract them. So I just clipped them to 1E-6 and 1E6 respectively.

>>14979861
https://math.stackexchange.com/questions/210190/history-of-the-study-of-rational-points-on-the-circle

>>14979780
>>14979802
I got it now, skew chevrons with the angle at the
apex being arbitrary and arbitrary coordinates.

Per the illustration >>14979651 let point B be
something like [math] (a,b) [/math].
Then point A is [math] (a-k_1+k_1 \cos(\theta),b-k_1 \sin(\theta)) [/math]
and point C is [math] (a+k_2+k_2 \cos(\theta),b-k_2 \sin(\theta)) [/math],
where [math] k_1, k_2 [/math] are random scalars
and theta is between 0 and 180 degrees.

But each point has to be dragged down to
account for the spacing by r, so by trigonometry
the amount to be dragged down is [math] r \sqrt{2 \over {1-\cos(\theta)}} [/math].
So point D can be found by this amount dragged
down and the resulting equation for each point
in the skew chevron is:
[math] \langle a \pm k_1 +k_1 \cos(\theta),b-k_1 \sin(\theta)\rangle + t \langle 0,-r \sqrt{2 \over{1-\cos(\theta)}} \rangle [/math]

What do you think?

>>14980005
Would that also work for angles greater than 180? The outline of the shape I want to trace will have both concave and convex corners. When I calculate theta in the first place, that would mean an inverse trig function, which can get ambiguous with angles greater than 180 IIRC.

How to show that a face of a cone is a cone (polyhedral geometry)?

>>14980005
>>14980108
Using inverse trig functions they do get ambiguous
(not injective) past 180, which is why they have a
defined domain. In that case, do 360 minus any
angle past 180 to get the apex angle of a down
pointing skew chevron. The k scalars would both
be negative so they point in the right direction.

I feel I made an error in the vector equation
because the coordinates A and C are two
different positions.
The TWO equations are:

[math] \langle a-k_1 +k_1 \cos(\theta),b-k_1 \sin(\theta)\rangle + t \langle 0,-r \sqrt{2 \over{1-\cos(\theta)}} \rangle [/math]

[math] \langle a+k_2 +k_2 \cos(\theta),b-k_2 \sin(\theta)\rangle + t \langle 0,-r \sqrt{2 \over{1-\cos(\theta)}} \rangle [/math]

>>14980108
>>14980145
If possible, can you show the results of implementing
the two formulas so we know we're on the right track?

>>14977761
Use extreme value theorem, consider a compact space. Verify the interior points within the set, then consider the boundary points, then consider the points outside the set.

>studying linear algebra
>it's the elementary operations chapter

>>14978531
Let [math] \mathbf z + \mathbf a = \mathbf b[/math] for some [math] \mathbf a, \mathbf b [/math] exclusively in [math] A, B[/math] respectively. Show that [math] B = \mathbf z + A[/math]

>>14978697
I assume he has already shown that the only way their union can be a flat in a field containing more than 2 elements, is iff one is contained in the other. So now he is trying to prove the necessary condition for when the field contains only two elements. Hence the assumption.

>>14980159
idk i think i'm too much of a brainlet for all this intersection bullshit
so i'm gonna implement a rastering move that rasters in like 8 different directions, hopefully that's good enough
just wanna get my damn laser slapped onto a 3d printer working

I ploot myself today. to see if i still graph. the periodic function. the only thing thats real (and imaginary).

sorry i know you hate plooters

>>14980264
>>14980159
I know the formulas are a lot, but each part of it
means a direction which can be lasered to form
the shape. The goal of it, of course, is to
describe how the laser moves. If my explanation
can match up with your laser's operation then we
have a working system. Maybe you would work with
parametric, for example, if the vector doesn't do?

>>14980281
The hurdles with the concentric method that are dissuading me are:
>figuring out what the inside and outside of shapes are
>cutting out self-intersections
>cutting out intersections between seperate shapes
That last one is the real problem, because it means I can't just arbitrarily iterate on a shape-by-shape basis, rather I have to do one layer on every shape, then check for intersections as I construct every single concentric ring. Then I have to combine the intersecting lines into their own rings, which completely changes the ideal order with which I draw them. I simply have no clue how I'd tackle that, the nontrivial order of operations is a travelling salesman problem.

>>14980281
>>14980315
It seems that the laser would always be on as it
draws the figure and fills it in, right? Could it be
programmed to stop and move elsewhere to
start drawing again?

From what you will say, it's possible to have an
answer to your hurdles. Try to go at it one
hurdle at a time. Perhaps some anon here
would have a clever solution, if possible.

>>14980367
>Could it be
>programmed to stop and move elsewhere to
>start drawing again?
Oh yeah it is, but reconstructing all the new polygons is nontrivial. The intention is to carve away paint on a piece of raw circuit board material such that acid can dissolve the copper, so it's the area outside the shapes I need to erode. Here's an example, there would need to be concentric rings running around the outside of all the green shaded areas, with a separation of 0.1-0.5mm or so, until they fill in the entire black area.

And I just realised that those green areas aren't just a single shapes, but multiple overlapping polygons, which will make this even harder. For a simple raster, I won't just be able to count intersecting lines to figure out whether I'm inside or outside, rather I'll need to do a proper geometric test on each individual polygon. For concentric rings, it will basically be impossible without first unifying all polygons.

>>14974666
Is
>A set is a collection of definite, distinguishable objects.
a good definition for a set?

>>14980409
>>14980367
Hmm...this is more complex than I thought.
If you're really looking for a formula to have the
laser handle the various polygons and the
intersections they make, some parametric formulas
for shapes would do and lots of it. And a CAD
to help plan out the shapes you'll draw.

I'm not sure of what else to suggest in a
practical math perspective as, like you say,
you might be butting into the traveling salesman
problem.

>>14980272
what software do you use to plot complex functions? I know plotting all 4 dimensions at once isn't possible, but I am looking for something that can plot the independent complex variable with real and imaginary part of the function separately.

Like, for example, this function:
[math]H(s) = \frac{100}{s^2 + 10s + 100}[/math]

I want to plot the magnitude plot.

>>14980462
A set is a collection of null sets. Best definition.

>>14975142
I’m going through a ”foundations of math for teachers” book from the library, basically serves as a crash course starting with aritmethic through to statistics. Probably going to borrow more equivalent books afterward. I wouldn’t have to be an autodidact if Swedish schools weren’t shit but c’est la vie.

>>14980717
but null sets aren't distinguishable

>>14980261
Thank you.
>>14978697
Learn Anon. This is how you reply.

Im having trouble to understand what part b) of that exercise wants me to do.
Could some one give me an example that helps understandinng the generalisation?

>>14980852
If [math]I = \mathbb{N}[/math] then [math](\delta_{ij})_{j \in I} [/math] which is often written as [math]e_i[/math] is just the sequence whose i-th entry is 1 and all the other entries are 0.

[math]\langle \left\{ (\delta_{ij})_{j \in I} | i \in I \right\} \rangle[/math] is then the set of all linear combinations of the [math](e_i)[/math] so it's vector space of all sequences of elements in [math]\mathbb{K}[/math] where all but finitely many entries are zero.

>>14980899
Thanks, this aligns with my understanding.
But as I said, im not sure what exactly I have to prove here.
Do I have to prove that the set of all linear combinations of the (ei) is a generating set for the direct sum of K?

Also for my general understanding. The n-tupel (0,0,0,0,...,0,3,4,5,....,0) is an element of the direct sum of Vi? Say if the cartesian product was RxRxRxRx......xR

>>14980833
>Learn Anon. This is how you reply.
We're reaching unprecedented levels of second-hand disdain.

>>14980833
Nobody forced you to spaz out. You did that on your own.

>>14975142
I don't know your level but yes. You should master a subject before `continuing on' to something more advanced. Having a solid basis will be so so important when later on you need to have a years worth of definitions in your head for a new topic.
Depending on your area you should study either group theory or analysis until you can dream it all.

>>14975833
Interesting, for me it's the opposite. I only find a topic fun or interesting if I know that it will lead to something applicable. So I enjoyed basic analysis because it opens the way to convex optimization and the like, whereas anything beyond the definition of a field felt pointless to me.

>>14976516
That seems like a terrible class, but you will learn what a deductible and a premium is when you turn 18 and have to pay for your insurance.

>>14977761
Keep the point in mind and compare it to any optima you find later.

>formalists
>"NOTHING matters, its just a bunch of symbols just a game, meaningless just like every human life, we are just here physically for no reason then we die for eternity, just like everything you work on!"

>platoist
>"There is meaning beyond the symbols, there is an idealized, perfect form of what you can analyze, just like your own human soul, we are all worth something greater, brother"

Why would anyone be a formalist? You can't prove either system from inside the system, so its just a matter of what mindset you want to carry for the rest of your life.

>>14981185
>we are just here physically for no reason then we die for eternity
>There is meaning beyond the symbols [...] just like your own human soul, we are all worth something greater
In this contrast, what you've admitted here is that, in your eyes, religion/spiritualism is your way of coping.

>>14981188
Yes, I wanted to convey that as the main contrast between both ways of thinking. I don't think its possible to truly be a formalist unless deep down you don't care about mathematics or yourself.

>>14981191
>I don't think its possible to truly be a formalist unless deep down you don't care about mathematics
I don't care about mathematics; I care about how much fun it can give me.
>or yourself
I do care about how much fun/enjoyment I can have in life. Is that the same?

>>14975821
this is a result of the problem inherent to the way mathematics is taught. example:

a teacher gives a pre-schooler a basic addition problem, 1 + 4. the pre-schooler writes out "14" as the answer. the teacher draws a big red "0" over the page. "F!" "Failure!", the child subconsciously vows to never study this painful subject.

alternative universe: the answer is not correct, but clearly displays that the child fundamentally understands what addition is, directly combining two numbers together into one. instead of marking the question on whether or not it's correct, it should be marked on how logically sound it is. although numerically the answer is wrong, it shouldn't be given a failing grade because it displays critical thinking skills that can be further blossomed with some gentle guidance.

take this with a grain of salt, as i'm an undergrad who's trashed my study more times than I care to admit because of me being autistically stubborn on problems I don't understand. although I still have my issues, I recognize that the instinct to hate yourself because you got a question wrong is totally unnatural to the way mathematics is meant to be done; you're meant to fuck around and find out, and mistakes are a sign that you're making progress. re-wire your brain to associate failure with pleasure for the same reason we pick the box with the question marks all over it rather than the laundry machine.

leave your ego out the door anon, Terrence Tao grasped group theory and fields when he was 8, you're not gonna impress anyone by getting a streak of right answers on Khan. just enjoy the process

>>14981197
I don't know. Its not like either of them can be proven to be actually correct, and both groups contribute to advancements in mathematics. For some reason I just take offense at the formalist way of doing math, and how that applies elsewhere in life.

>>14980695
that looks like a laplacian function.
x = the real part of the function
y = the imaginary part
s = a complex functional plane
use e^is = cos(s) + i*sin(s) eulers identity

if you can graph a circle you can graph that function. hint it looks like some kind of cone with an infinity pole somewhere

i just use c++ and graph to a bitmap which i convert over. I do this because its simple and i can write a lot of bitmaps fairly easily to make animations if i want

>>14978679
Your definition of zeta seems wrong, here zeta(t)=T(t-s)z(s)=T(t-s)T(s)z_0=T(t)z_0=z(t). I feel like you (or the author) meant to take a difference between two proposed solutions and show that that function is identically 0. That usually shows uniqueness.

>>14979039
ppois() in R

>>14981185
>thinking X feels better than thinking Y therefore you should think X
The absolute state of platonist shilling in 2022.

>>14981296
Well, another program to download. Been trying to program the function in my calculator so I can use it on the fly the same way I have a program for permutations/combinations, but I'm too stupid.
I think there is some R studio think I can get for free with a student account, so looks like another program language to halfass learn learn... C++, matlab, wolfram mathematica, maple, mintab, and now R...

>>14981304
Can you make an objective distinction as to why Y is better than X? If not, it all comes down to preference.

>>14981305
It's free and open source you know. Also, why would you 'learn' it beyond what you need it for (statistical tables)?

>>14981314
I have a huge backlog of things I have to learn, followed by things I want to learn.

>>14981304
taking a journey > not taking a journey
simple as

>>14981320
Penis longa, vita brevis

>>14981269
Idk about the Laplacian function but it's a transfer function of a second order system with 0.5 damping ratio and 10 natural frequency.

The poles exist as conjugate pairs with real part = -1.

I have no idea what yours pic rel is actually representing.
Pic related is what the magnitude looks like at s=-1

>>14981354
my pic is only slightly related. in that you cant actually "plot" imaginary. you just make the x axis the real part. and the y axis is the imaginary part.

i was slightly wrong. its related to laplace transforms which are related to fourier transforms. which comes back to circuits with resistive, inductive, and capacitive elements.
resistor being the dampening part
the capacitor and inductor combo being what drives the frequency.

which looks something like sin(stuff)*exp(-someamount*stuff)

what your graph tells you is that if you drive the circuit at some frequency (negative or positive one) it wont dampen out

anyway my rambles probably dont help you much. but i could ploot some more versions of your function later if you are still around

Retard here. I am writing a program and i need some math help, there is this one math thing i need to do and i suck shit and don't know how to do it.

So basically i have a simple flat plane with points on it, each point has X,Y coordinates. like 1,2 or 4,1 etc. I know coordinates of all the points.

The black points have X,Y coordinates relative to the outer plane (the red one) and all the green points have y,x positions relative to the inner plane (the black one)

Now i take the whole black plane and rotate it 180 degrees on the x axis, move it up or down a little bit on y axis and i even roate it a bit on the z axis. (but it always remains a 2d flat plane).

The red plane never moves.

I now measure the new coordinates of the black points relative to the red plane, and find out how their new X,Y coordinates, or rather how much the black plane moved.

Knowing how much these two points moved and also knowing i rotated the black plane by 180deg on x axis how do i calculate the new positions of all the green points?

To make it more clear what i'm after:
I have CNC mill and i drill two alignment holes, (the black holes) into a piece of wood. then i need to flip the wood and drill rest of the holes, but because i flipped it the wood moved so the holes will be misaligned, so i need to tell the mill where rest of the holes are supposed to be.
I can measure by how much the wood move by measuring how much the alignment holes moved.
Please help frens, i am too stupid to come up with the needed way to calculate this.

>>14981374
>but i could ploot some more versions of your function later if you are still around

Please plot the phase function too if you can, one with the function's phase as it is, and one with phase shifted by -180 degrees. It would be cool to see what really happens when phase if overdriven to make the system unstable (i.e., negative damping).

Solved the first problem in basic mathematics by lang in lean.

we did it boys. we're math now

Computers connect to the platonic realm where the answers to all math problems lie. The connection is laggy sometimes which explains load times.

>>14981559
Isn't this "proof à la Dali"?
https://www.youtube.com/watch?v=hpEzV91ohOg
I'm not defending a Fitch derivation of this fact
in fact, for illustrating the proof of this fact, there are typically two main stylistic themes
(1) the use of derived versions of the associative and commutative law that let you replace subexpressions with their equivalent values; this fact is usually not stated as a theorem but rather as a "quasi-bullshit semantic-syntactic trick" because the vast majority people who learn it aren't sophisticated enough to figure out what's really going on or are, but the tradition is to teach the version that works for the vast majority of people and then years later the truly dedicated logicians and model theorists get to know the knowledge not intended for non-mathematicians
Let me be blunt: the standard proof of this fact is supposed to exercise the page layout functions of your proof typesetting engine, and your proof typesetting engine has no page layout capability
moreover, you aren't even making reference to the fact that x + -1 * y = x - y even though it seems pretty clear that the theorem is using the subtraction operation and the proof only makes reference to the addition operation?
So, even though your computer accepts the proof, it's ugly and it sucks
https://www.youtube.com/watch?v=sIRDCR8xSO0
GUILTY!

[math]\circ\colon(({\mathbb N}\to{\mathbb Q}^{\mathbb Q})\times({\mathbb N}\to{\mathbb Q}^{\mathbb Q}))\to({\mathbb N}\to{\mathbb Q}^{\mathbb Q})[/math]

[math]f\circ g := n\mapsto q\mapsto f(g(n)(q))(n)[/math]

>>14981666
If you put this proof into lean you would get lots of assistance in seeing how each step works. Second, the proofs in lean are not at all like the proofs you would normally write, they are very abstract and hide huge amounts of details from the mathematician or dumb nigger (In my case).

Also, I can't get parenthesis to work in Lean. Don't know why but they just don't work. It will not accept
(a - b) + (c - d) = (a + c) + (-b - d).

Parenthesis just get stripped for no reason.

>>14981677
So, the thought process here is:
(1) is the use of composition operator symbol standard or non-standard (I have no clue after staring at this equation for hours)
(2) Currying is a CS-ism that arguably makes the comment un-parsable nonsense, and if it is, then any mathematician worth his salt would immediately rewrite what's going on in terms of maps f,g: Na x Qu -> Qu
>p = f(n,q) and r = g(k,s)
(3) it looks like you have a typo regarding the order of arguments to f, and regardless, the whole point of rewriting this is to excise your implied suggestion that math notation incorporate Scheme-style function application, which happens to conflict with standard notation for multiplication in Na and Qu
(4) you're having too much fun, and the fine folks over at /r/zen can straight you out, kid
(o_<)
https://www.youtube.com/watch?v=NgUd8riJRak

>>14981677
(5) overloading function composition operator with point-wise composition operator could result in ambiguous parses of math expressions without context, and even with context, the argument for notationally distinguishing composition from point-wise composition remains
(6) presumably you want to add more conditions and then figure out for which subsets A of Maps(Na x Qu,Qu) does it hold that a point-wise two elements of A results in an element of A, for example
>whether or not f(n,q) <= f(k,q) for all q <- Qu and n,k <- Na, n <= k
in other words, the search is on for examples of submonoids of Maps(Na x Qu,Qu) that are closed under this binary operator, point-wise composition, and of course the simplest of these will be the cyclic submonoids...and I suppose pointing out that point-wise composition is associative would be helpful in reassuring us that this is indeed a monoid

>>14981688
>dumb nigger
Someone get that negro a cup of tea. Please!
::snaps fingers::

am i going mad or does this not make sense? take [math]N=1[/math], then the determinant should be [math](-1)^{M+1}\det A[/math], not [math](-1)^{M}\det A[/math]

>>14981992
nvm, i'm a retard, it's [math](-1)^{M+2}\det A[/math]

What was the book that took you the longest to read and how long did you read it?

>>14982079
I had a dragonlance chronicles book that was almost 2000 pages, some special edition bullshit. Took me about a month.

>>14981848
Hardy was real depressed as he got older.

>>14982424
I would head on over to /lit/ or /his/ if you want to discuss that

Can I become better at math by just hard work, even if I suck in some basics stuffs? I have an opportunity to do a major in maths in the best college of my country.

>>14974666
Somebody give me a list of books that will teach me everything from pre-algebra to calculus in order rn

>>14975142
I don't know how to multiply or divide without a calculator.

>>14975142
I'm doing the same thing anon
>10th grade drop out
>Wanting to take GED and move on to college
>Decide to relearn everything from the ground up and go further than I've ever gone before

How many states does this Turing machine have?
What numerical format is used to represent and work with, prove things about, the number of states this Turing machine has?
>tfw thinking about Turing machines that use other Turing machines to calculate their state changes

>>14982530
Shut the fuck up you phd ass motherfucker

>>14981053
Nobody forced you to make a fanfiction of my post because you could't answer it. If you can't answer just don't reply.
>ah yes a person solving an affine geometry question probably doesn't know what implication means, I better explain him, I am so intelligent

>>14981185
Math is problem solving. If you want to talk about your low iq pop philosophy, go to >>>/lit/

>>14982565
See? Spazzing for no reason.
You got issues.

>>14977761
Try this question anon, consider the function to some restricted curve and try to compute part (c)

Any idea how to prove this integral equals zero using complex analysis? I can do the finite cases simply by invoking the residue theorem, with a contour that includes the poles at z = 0+ni, but I'm not certain how to do it with this infinite product in the denominator.

I'm pretty certain it should evaluate out to zero. I need to use complex analysis here, I can't just do the partial fraction decomposition method sadly.

>>14982697
Dude...

x^2 + n^2 >= n^2
The product of n^2 goes to infinity
Its reciprocal goes to what? Integrate that.

>>14980467
Well I'm using KiCAD to draw the circuit itself, but it can only output files in a few formats. I can't build the CAD around the manufacturing method, it has to be the other way around. I don't think trying to make an addon/plugin for it would be any easier. I think I'll have to split it into steps:
>convert all curves into a series of line segments
>convert overlapping objects into single objects
>rasterise from 4 different angles

>>14982732
The point is I'm trying to do this with the residue theorem. Or, more particularly, I'm trying to do this with a Mobius transformation, mapping the real axis to the unit circle and the poles of the function to a set of points along the line segment between the origin and the point (0, -1) in the complex plane. I would then perform a keyhole integration around my arc for completion.

>>14982755
>>14980467
There you go, those are very good steps to work
on. For the first step, I think you can imagine
putting some points inside a curve and making
straight lines to each point from start to end, right?
More points in the curve makes a better curve.

>>14982480
>just by hard work
no, you also need to like math and learning. If you think you can manage that, then go for it

This definition of determinant is incorrect.
Am I the only one that gets it?

Take n=2 then det A = a_12a_21-a_11a_22

if i have some 2d object made from 4 points, like in the pic, can be a square for example (but it can be any shape made from 4 points)

And i move and rotate the square (shape and dimensions of the square don't change) and i know the new positions of the 3 points of that square (after i moved and rotated it) and put them into a second matrix, how do i calculate the position of the 4th point, so that the shape of the square is unchanged?

>>14982913
det(A) = s(1,1) a(1,1) a(2,1) + s(1,2) a(1,1) a(2,2) + s(2,1) a(1,2) a(2,1) + s(2,2) a(1,2) a(2,2)
= sgn(0) a(1,1) a(2,1) + sgn(1) a(1,1) a(2,2) + sgn(-1) a(1,2) a(2,1) + sgn(0) a(1,2) a(2,2)
= a(1,1) a(2,2) - a(1,2) a(2,1)

>>14974666
I downloaded a few of the books in op's picture. They're not bad but they don't have the answer keys in them. Do I need to have a profile on openstax to see the answers?

>>14982961
How about you read a real book?

>>14982961
You can use Maple to check the answers but if you understood the content you will be able to tell by yourself if your answers are correct.

>>14982968
>>14982966
You're both retarded but in different ways.

>>14982961
Click on the book you are looking at and scroll
down to table of contents on the left side.
Click on that and look through to see answer key
or solutions in the chapter listings.

>>14982974
The table of contents lists those sections but the PDF I downloaded from openstax is missing a hundred or so pages at the end. I assumed they want you to have a profile and interact with the website to fully use their free textbooks. Or maybe i'm the retarded one.

>>14982976
>>14982974
Just print them out from the website.

>>14982956
Stupid, stupid me...

>>14982978
It's looking like user error. Two of the three do have an answer key. I bet I closed the tab the PDF was saving from before it finished composing itself.

>>14982978
>>14982983
If that's the case, try again.

>>14982944
anyone? i assumed it would be some simple solution, have i accidentally reached the limits of math?

>>14982973
Why would anyone read these books is beyond me.

>>14982944
If you know the new positions [math](x_1',y_1')[/math], [math](x_2', y_2')[/math] and [math](x_3', y_3')[/math], then by using "homogeneous coordinates" (i.e. adding a third coordinate equal to 1), your transformation can be encoded as a 3*3 matrix:
[eqn] \begin{pmatrix} \cos\theta & -\sin\theta & t_x \\ \sin\theta & \cos\theta & t_y \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \end{pmatrix} = \begin{pmatrix} x_1' & x_2' & x_3' \\ y_1' & y_2' & y_3' \\ 1 & 1 & 1 \end{pmatrix} [/eqn]
So you just need to invert and solve for the three free parameters, to obtain your transformation matrix (called the "rigid-body traransformation matrix" or "Euclidean transformation matrix", in case you want to look it up further). Then you can apply it to your last point, or any other points that you have lying around.

>>14982944
You only need 2 points to figure out the rest actually.
Let [math] \mathbf X = [\vec x_1 \dots \vec x_4][/math] be the original points, and [math] \mathbf Y = [\vec y_1 \dots \vec y_4][/math] the transformed ones. Then find [math] \mathbf A \in \mathbb R^{2 \times 2} [/math] such that for some:
[math] \mathbf A[ \vec x_i \; \vec x_j] = [\vec y_i \; \vec y_j] [/math]
where [math] (\forall \alpha \in \mathbb R)(\vec y_i \neq \alpha \vec y_j)[/math]
Then calculate [math] \mathbf A \vec x_4[/math]

>>14982944
Let [math] \mathbf X = [\vec x_1 \dots \vec x_4][/math] be the original points, and [math] \mathbf Y = [\vec y_1 \dots \vec y_4][/math] the transformed ones. Then find [math] \mathbf A \in \mathbb R^{2 \times 2} [/math] such that:
[math] \mathbf A[ \vec x_i \; \vec x_j] = [\vec y_i \; \vec y_j] [/math]
where [math] (\forall \alpha \in \mathbb R)(\vec x_i \neq \alpha \vec x_j)[/math]
Then calculate [math] \mathbf A \vec x_4[/math]

>>14982489
bump

>>14983207
unbump

>>14982489
It's literally in OP
It's all free too

Or you can refer Khan Academy

>>14983115
the matrix i posted is 2d it's coordinate parts x,y
isn't the thing you posted meant in a way where x coordinates are original ones and y are the new ones? since it's (x1,y1) = coordinates for one point before movement and then new values for (x1,y1) after the movement for the single point.
Sorry if i'm being retarded, i am way out of my paygrade here

>>14983029
so i plug in the original coordinates for the 3 points into the first matrix, then destination coordinates into the second matrix on the right, the calculate the resulting final matrix out of that, which is the "transformation matrix" that can then be applied to find out the 4th coordinate?
How would i apply the transformation matrix to the the coordinates of that 4th point?
This is the first time i'm using matrices and i am literally doing the khan academy course on matrices to understand how they work properly, since i desperately need to solve this to finish my project so thats why i keep asking these stupid questions.

>>14983347
>matrix to the the coordinates
*to get the

>>14983332
>matrix i posted is 2d
There is no such thing called 2d matrices, so I have no idea what the you are asking.
[math] \vec x_i \in \mathbb R^2 [/math] is the [math] i[/math]-th coordinate before rotation. So, for example [math] \vec x_i = (1,-1)[/math].
Similarly, [math] \vec y_i \in \mathbb R^2 [/math] is the [math] i[/math]-th coordinate after rotation.
[math] \mathbf A [/math] is the matrix that rotates vectors in [math] \mathbb R^2 [/math], which you can find by solving [math] \mathbf A [ \vec x_i \; \vec x_j ] = [ \vec y_i \; \vec y_j ] [/math], where [math] \vec x_i [/math] and [math] \vec x_j [/math] are two linearly independent vectors .
You are supposed to find [math] \vec y_4 = \mathbf A \vec x_4 [/math].

>>14983347
>so i plug in the original coordinates for the 3 points into the first matrix, then destination coordinates into the second matrix on the right, the calculate the resulting final matrix out of that, which is the "transformation matrix" that can then be applied to find out the 4th coordinate?
That's right, the equation is of the form [math]TX=Y[/math] where you know the matrices X and Y, so you can solve for [math]T=YX^{-1}[/math], which is the "transformation matrix" that I was referring to.
>How would i apply the transformation matrix to the the coordinates of that 4th point?
You can always interpret the matrix [math]T[/math] as a function that takes a vector [math]x[/math] as input, and returns a vector [math]Tx[/math] as output. (In other words, you "apply the matrix to a vector" by multiplying it on the left. For example, [math]T \begin{pmatrix} x_4 \\ y_4 \\ 1\end{pmatrix} = \begin{pmatrix} x_4' \\ y_4' \\ 1\end{pmatrix}[/math] calculates the transformed position of your last point.)
In this case [math]T[/math] is a square matrix, but if it was a general m-by-n matrix, then the input [math]x[/math] would be an n-vector, and the output [math]Tx[/math] would be an m-vector. In other words, the interpreted function would have domain [math]\mathbb{R}^n[/math] and codomain [math]\mathbb{R}^m[/math].

It's understandable to be a bit confused if you're seeing this interpretation for the first time, but it's standard and you'll surely see it again if you continue working with matrices (mathematically speaking, the nice property of [math]T[/math] is that it's linear in its input arguments), so it's definitely worth understanding.

>>14983389
>>14983389
I have made here the assumption that you have two linear independent coordinates, but if you don't (you will always have one if its a non-degenerate polygon with more than 3 sides, then you can solve the for the matrix:
[eqn]
\begin{bmatrix}
\cos \theta & - \sin \theta \\
\sin \theta & \cos \theta
\end{bmatrix}
\vec x_1
= \vec y_1
[/eqn]
This approach will of course require you to solve the angle, and hence know the values of trigonometric inverse functions.

Does anyone know how to prove q6? Progress so far: only eigenvalue of A is 0 and its arithmetic multiplicity is n

>>14983519
What is the dimension of A(R^n)?

>>14982530
So, for example, a Turing machine with 256^10^9 states gives you a gigabyte worth of 'Gödel storage space' if you like in the base-256 representation of the state number, and the tapes could be split to make it less than a Turing machine: one tape for reading input and one tape for writing output, both tapes are half infinite and can only rest or increment the cell number
This is more like a Markov chain than a Turing machine, however, with the input tape cells being taken as i.i.d. samples from a simple Bernoulli process or fuzzing a parser

>>14983519
Let's do an easier case, and see if you can modify this to fit your need.
It's easy if you know that A^(n-1) is not zero. Call this matrix B.
Then you have a vector v such that Bv is nonzero.
You can prove that (A^k)v forms a basis for R^n, for k = 0 to n-1. (How?)
Once you do that, since it's a basis, you can write the matrix A in this basis, which will be upper triangular with 0's on the diagonal. (Why?)
Since we found a basis where the condition holds, then A is similar to this matrix.

Try to focus on the key idea here, and figure out how to loosen it.

>>14983519
A has only zero as an eigenvalue and it's similar to it's Jordan Canonical form

Fucking triangles, how do they work?

if i have a triangle where two corners have the following (x,y) coords:
A = 0,1
B = 5,1
C = ?

And i know all 3 side lengths of the triangle, how do i calculate C?

>>14983786
Let C=(x,y) then

x^2 + (y-1)^2 = b^2
(x-5)^2 + (y-1)^2 = a^2

Two equations and two unknowns so it's easy to solve. Take the difference of the equations
10x - 25 = b^2 - a^2
x = (b^2 - a^2 + 25)/10
y = 1 + sqrt(b^2 - x^2) = 1 + sqrt(100b^2 - b^4 - a^4 - 625 + 2 a^2 b^2 - 50b^2 + 50a^2)/10

>>14983398
That's only for rotations. OP is asking about translations as well, which are not covered by your matrix.

>>14980462
There is no "good definition" for sets because there is not a formal definition at all, at least in modern, first-order flavor notation. But that green text is a good intuition. Read Bourbaki's Summary of Results first lines, from their Set Theory book. In some sense, the "definition" of sets is embedded in the axioms of set theory indirectly, which is tantamount to say that the cumulative hierarchy of sets is the closest to a mainstream definition of the set concept mathematicians have, like anon says >>14980717

>>14981559
>>14981666
>>14974666
How come there is not a computer-assisted mathematics general on /sci/ or /g/?

>>14984109
>How come there is not a computer-assisted mathematics general on /sci/ or /g/?

Proving even trivial things in a proof assistant is non-trivial.
Documentation isn't good
Learning materials aren't good
It's like learning how to prove things twice.
Tactics are not fun to use.

It has some use in teaching induction. Like I think it's the most intuitive way to teach it.

>>14984109
https://en.wikipedia.org/wiki/Non-surveyable_proof

>>14974886
Programming language semantics: Strings, trees, and many other data types are examples of free algebras; and streams, lazy lists, interactive processes, and many other "dynamic" data types are cofree coalgebras.

>>14984152
>In the philosophy of mathematics, a non-surveyable proof is a mathematical proof that is considered infeasible for a human mathematician to verify and so of controversial validity. The term was coined by Thomas Tymoczko in 1979 in criticism of Kenneth Appel and Wolfgang Haken's computer-assisted proof of the four color theorem, and has since been applied to other arguments, mainly those with excessive case splitting and/or with portions dispatched by a difficult-to-verify computer program. Surveyability remains an important consideration in computational mathematics.

>>14984109
The reputation of computer-assisted proofs is that mathematicians that engage in this sort of snake oil are
(1) outside the Anglosphere
(2) work directly with the Anglosphere mass media, going around the Anglosphere math community
https://en.wikipedia.org/wiki/Kenneth_Appel
https://en.wikipedia.org/wiki/Wolfgang_Haken

>>14984152
>>14984163

99% of people who touch a proof assistant won't even get close to proofs which are too complicated for mathematicians. It's likely you won't even get past undergraduate mathematics in lean.

>>14975166
Use an SRS like Anki to memorize identities more efficiently. It will make doing problem sets much less frustrating, since you eventually won't need to flip through your notes. Anki also serves as a kind of database that you can query quickly.

>>14975821
It's okay to not know something. It's not okay to place blame - just go find a reference for the information you need. Carry on. This will take time.

>>14984166
>too complicated for mathematicians
You have to remember the order of events: German mathematicians went to the Anglosphere press and said,
>you have to say the English are racist anti-German assholes for refusing to accept the validity of BEEP when our black box goes BEEP instead of BOOP because BOOP means the proof has an error and BEEP means the proof is fine, you won the war, stop being such assholes and accept our proof, you POOF!

>>14975821
>self-harm
You're telling strangers on the internet that you're self-harming.
We don't believe you, we think you're part of a psy-op, and we want you to leave and tell the CCP to stuff their damned pay-ops up their yellow asses

>>14976806
Why not just learn python?

>>14984191
One day I will decipher what this schizo gibberish means

>>14984263
she's talking about the first 4-color theorem proof and the the dismissal of the mathematical community towards its validity

>>14984268
Who still doubts the proof?

>have a system of 9 linear equations with 9 unknowns
>use octave with symbolic package to solve it
>takes 10 minutes
>oh wait i am using old versions, lets update and hopefully see some improvements
>no solution even after 2 hours
what the fuck could be the cause of this?
ported the code to python and it also shits the bed
would sagemath be better even on windows?

>>14984274
>The proof has been one of the most controversial of modern mathematics because of its heavy dependence on computer number-crunching to sort through possibilities, which drew criticism from many in the mathematical community for its inelegance: "a good mathematical proof is like a poem—this is a telephone directory!" Appel and Haken agreed in a 1977 interview that it was not "elegant, concise, and completely comprehensible by a human mathematical mind".[9]
From https://en.wikipedia.org/wiki/Kenneth_Appel
Most mathematicians don't have wiki pages that retain insults to their work after their death
This one does
Why is that
Why is insulting this dead man's work of public interest, why is it notable, why do Wikipedia editors think it's notable?

>>14984343
Just trust the kernel bro. So what if it's 500000 lines of C++ code.

>>14984274
Haken'n'Appel hackin' Kallisti's Golden Apple
To the Fairest
But what would appears't—
A TELEPHONE DIRECTORY?

>>14984291
>symbolic
That might be the issue.
Symbolic computations are expensive, and that's the whole point of numerical computations; to avoid the expenses.

How did you ty to solve it? Did you try to find the inverse? Because if you did, then the determinant requires you to calculate 9! terms, each having a product of 9 numbers.
And then the adjugate is a matrix of 9x9 determinants of 8x8 matrices.
And if you're doing symbolically, there's probably fractions involved, which have to be reduced using the GCD.
Maybe you can see how easily this can get out of hand.

>>14984361
>just shit on your own reputation because I told you to
no
you can use truth tables of ZFC
finite industrial control logic uses truth tables
infinite algebraic logic uses ZFC
Academia controls ZFC rate of development
Industry controls truth table rate of development
look up
>coq proof of false
https://github.com/clarus/falso
This proof was a notable event in the history of automated proof assistants
The link to HN makes an important online primary source available as a query that is easy to remember; if only all of the notable CS events were so neatly documented by primary sources then the world would be a vastly different place

>>14984263
It's a reference to the scene where they're all
>MAJOR TOM: So, yeah, we're up in space and this fuckin' computer says it has a fuckin' bug
>GROUND CONTROL: Reading you loud'n'clear, boys, that fuckin' computer is full of shit, there ain't no motherfuckin' bug in that pile of wires flying your tin can, now hear?
https://getyarn.io/yarn-clip/8bbcac90-2d01-4e6b-867e-fec8a2e0fd19
>I say again, "in error predicting the fault"

>>14984404
if you start watching at +1:05 then you'll get all of the narrative content related
https://ww3.putlockers.li/movie/2001-a-space-odyssey-1968_eG1RSFQ3Z2YrUlU9/watch.html

>radon-nikodym
>lebesgue decomposition
>jordan decomposition
>besicovitch's covering lemma
>vitali's covering lemma
>3r covering lemma
>absolute continuity
>uniformly continuous
>mutually singular
>σ-compact
>σ-finite
>semifinite measure
>hölder continuous
>lipschitz continuous
>absolutely convergent
>conditionally convergent
>uniform convergence
>almost uniform convergence
>convergence in measure
>seminorm
>pseudometric
>borel-cantelli
>hölder inequality
>minkowski inequality
>jensen's inequality
>weak convergence
>weak-* convergence
>cauchy-schwartz
>monotone convergence theorem
>dominated convergence theorem
>infinitely differentiable
>continuously differentiable
>analytic
>holomorphic
>chebyshev's inequality
>hausdorff metric
>hausdorff measure
>hausdorff space
>locally compact space
>σ-algebra
>positive measure
>complex measure
>signed measure
>chebyshev's inequality
>urysohn's lemma
>stone-weierstrass
>tychonoff's theorem
>arzelià-arscoli
>parallelogram inequality
>riesz representation theorem
>riesz representation theorem
>riesz representation theorem
>bessel's inequality
>parseval's identity
>cauchy completion
>separable
>convex
>outer measure
>carathéodory's construction
>vitali-carathéodory theorem
>borel set
>borel measure
>regular measure
>borel regular measure
>radon measure
vocabulary spills i'm ill, plus matic

>>14985231
>Anal-ysis
Gay

>>14985236
>uniformly distributed measure
>semicontinuity
>hardy-littlewood
>metric density
>total variation

is it hard going from learning from lectures to learning from books

>>14985246
>learning from lectures
No such thing.

>>14985246
not particularly hard, but learning from books puts more of the burden on you to sort out the important things from the unimportant things. It's hard to do at the undergrad level because you still mostly shouldn't be worrying about where the ideas came from originally or how they fit together in some bigger picture

Help me out /mg/, I don't get the exact usefulness of modules in rings. I'm currently going through the the basic concepts, and it's easy enough, but why are we studying modules anyway? Does it help us investigate the structure of rings? Or perhaps allows us to classify finite rings? What is even the purpose of a ring R over the same ring R? It's not even that difficult a subtopic, but I just don't get it. What exactly is our end goal with modules?

>>14981289
I think the argument is that after taking an arbitrary solution, you see that for all t it takes the form T(t)z_0, thus T(t)z_0 is the unique solution.
Its a very weird argument that's why I had some trouble getting my head around it. All uniqueness arguments ive seen thus far do what you describe instead of this.

>>14975024
>>14974693
when i squint my eyes they stop moving

>>14985437
> you still mostly shouldn't be worrying about where the ideas came from originally or how they fit together in some bigger picture
There's no point in time where you shouldn't be looking for the big picture of ideas.

>>14985246
When it comes to pure math, books and lectures aim different things. Lectures aim to quickly explain you the syllabus so that you pass the exams. Books help you learn how to think about the subject. Books at a higher level deliberately leave out things and let you figure things out on your own, because that's the point of it. You are encouraged to go back and forth and proceed at your own pace and deeply think about the implications of what's written. There are still books which are pretty much lectures but in text but that's mostly at early undergraduate level or on applications.
>>14985437
What do you mean by "unimportant" in math? Do you mean stuff out of syllabus, that's pretty easy to check.

>>14985231
Have an analysis exam soon, and I fucking HATE how much memorization there is, with NO motivation.

I'm having trouble completing the square.

[math]2x^2+x=2(x^2+\frac{1}{2}x)=2(x+\frac{1}{4})^2-\frac{2}{8}[/math]

it looks to me like you need to subtract 2/8 because 1/8 is multiplied by two and that's how you stay equal to the original expression, but when I checked my shit on wolframalpha it said -1/8 at the end. what am I not seeing?

>>14985640
(x + 1/4)^2 = x^2 + 1/2x + 1/16

>>14985640
[math]2x^2 + x = 2(x^2+\frac{1}{2}x) = 2(x^2+2(\frac{1}{4})x) = 2(x^2+2(\frac{1}{4})x + (\frac{1}{4})^2 - (\frac{1}{4})^2) = 2((x+\frac{1}{4})^2 - (\frac{1}{4})^2) = 2(x+\frac{1}{4})^2 - 2(\frac{1}{4})^2 = 2(x+\frac{1}{4})^2 - 2(\frac{1}{16})[/math]
Hope it's clear now.

>>14985647
>>14985685
Thanks a lot! Figured I was making a basic mistake somewhere but I just couldn't notice. It's all clear now.

can someone please check if this is correct? i am getting weird results

Basically if you take any triangle, and you know coordinates for 2 corners and lengths of all 3 sides, this should then calculate the x,y coordinates of the 3rd corner.
(Or rather 2 sets of coordinates because the 3rd corner of the triangle has 2 possible locations where it can be)

Who is the most based mathematician right now? and who WAS the most based?

>>14985798
Assume the coordinates of C then apply the distance formula for AC and BC. Two variables two unknowns. You'll get a quadratic equation for one of the coordinates to imply it can be in either -x or -x direction.

>>14976859
It must be something to do with confusion over where the light is coming from. In the first one the light source for each oval is similar to the ones around it but gets less correct as you get further away and it's the further away ones that move. It would make sense for perception of position to rely on light source.

>>14985865
I know what the equation outputs, i just need to know if the equation in the picture is correct or not, because it sometimes gives me wrong numbers as a result.

If [math] B[/math] is a non-null matrix and there are non-trivial solutions to:
[math] B [x \; y \; 1]^{ \mathsf{T}} = 0 [/math]
then [math] \operatorname{rank} B = 1[/math]
How can I prove this without separating the third column of the matrix?

>>14985903
Okay never mind I just figured it out. Clearly, some [math] [x \; y \; 1] [/math] is in its null space, and since [math] B \neq 0[/math], then some [math] [x \; y \; 0] \neq 0 [/math] is also in its null space. Hence nullity is 2, and it follows. This is correct?

>>14985903
False.
[math]
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 1 & 1
\end{pmatrix}
[/math]
Non-trivial solution: https://www.wolframalpha.com/input?i=%7B%7B1%2C+0%2C+0%7D%2C+%7B0%2C+0%2C+0%7D%2C+%7B0%2C+1%2C+1%7D%7D+*+%7B0%2C+-1%2C+1%7D
Rank: https://www.wolframalpha.com/input?i=rank%28%7B%7B1%2C+0%2C+0%7D%2C+%7B0%2C+0%2C+0%7D%2C+%7B0%2C+1%2C+1%7D%7D%29

You forgot a condition or something.

>>14985913
Yes I did. That's why I deleted it. The condition was [math] e_1 Bx = 0 \iff e_2 Bx = 0[/math]

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

I'm trying to parametrize the surface of a sphere as a quadratic rational bezier spline surface. The parametrization has the form [eqn]S(u, v) = \frac{\sum_{n=0}^2(\sum_{m=0}^2 B_n^2(u)B_m^2(v)P_{nm})}{\sum_{n=0}^2(\sum_{m=0}^2 B_n^2(u)B_m^2(v)W_{nm})}[/eqn] with control points [math]P[/math] and weights [math]W[/math]. [math]B_n^2[/math] is the n-th bernstein polynomial of order 2. through symmetries i can "simplify" the task of finding the weights to finding the roots of the coefficients of a large polynomial (not the roots of the polynomial itself) giving me probably several possible solutions to parametrizing different parts of the sphere from which i can easily identify the part i'm after. however i have no confidence in my ability to solve this manually, so what is a good easily scriptable computer algebra system that i can use without buying or stealing some commercial enterprise/research shitware

inb4 maxima, i tried it and it's just too wacky for me

>>14985599
>equicontinuity
>pointwise bounded
>uniformly bounded
>hahn-banach
>uniform boundedness principle
>open mapping theorem
>closed graph theorem
>egoroff's theorem
>luzin's theorem
>cauchy condensation test

>>14984377
Making a small enough kernel should be a huge research project

Statistics and linear algebra have a very stable and loving relationship.

>>14986282
How about you tell me about the relationship between linear algebra, vector calculus and Fourier tools?

I'm taking a stats class now, after completing Calculus 1-3. My brain has been fucking RUINED by calculus. I want to use calculus tools for everything and it keeps getting me into trouble and wrong answers, also everything is too "easy" and there isn't enough computation or effort required for each problem, its honestly just pissing me off.

>>14986456
>life after calculus

>>14981744
>it looks like you have a typo regarding the order of arguments to f
True.
Also, true, overloading \circ was a bad idea.
Thanks for starting at it for hours, you efforts were not lost.

[math]\bullet\colon(({\mathbb N}\to{\mathbb Q}^{\mathbb Q})\times({\mathbb N}\to{\mathbb Q}^{\mathbb Q}))\to({\mathbb N}\to{\mathbb Q}^{\mathbb Q})[/math]

[math]f\bullet g := n\mapsto q\mapsto f(n)(g(n)(q))[/math]

It's taking the diagonal in sequences of rational function, thus giving you the means to compute the function of Cauchy sequences to chosen precision.

[math]\pi_n(q) := \prod_{j=1}^n \frac{ 4j^2 }{ 4j^2 - 1 }[/math]
(Constant in q, actually a real number, representative of a Cauchy real)

[math]\lim_{m\to\infty}\pi_m = \pi[/math]
(Wallis product formula)

[math]\exp_n(q) := \sum_{k=0}^n \frac{1}{k!} q^k[/math]
(Exponential series)

[math]\exp_-\bullet\pi_- = n\mapsto q\mapsto \sum_{k=0}^n\frac{1}{k!} \left( \prod_{j=1}^k \frac{ 4j^2 }{ 4j^2 - 1 } \right)^k[/math]

[math]\lim_{n\to\infty}\exp_-\bullet\pi_- = q\mapsto e^\pi[/math]

>physics professor starts getting angry that all my assignments solve to ten decimal places instead of rounding, I give answers as the analytical vice numeric solution if possible, and I use degrees instead of radians

>"No! You have to use radians, not just in harmonic motion but even in an abstract way it MUST be radians."

I'm just ignoring this retard and turning in all my work in degrees. I do not care for his clown opinion.

>>14986905
>degrees
clown opinion

also you appear to have autism

Is it the truth?
Is it fair to all concerned?
Will it build goodwill and better friendships?
Will it be beneficial to all concerned?
>“Two blocks west of the Chat, in a teashop called the Jarre de Thé, Case washed down the night’s first pill with a double espresso. It was a flat pink octagon, a potent species of Brazilian dex he bought from one of Zone’s girls.
The Jarre was walled with mirrors, each panel framed in red neon.”
https://vimeo.com/189426019
https://www.youtube.com/watch?v=CXBk7yy2AqE
https://www.mixcloud.com/ivopospisil9/vintage-cool-by-radio-1-prague-tea-jay-ivo-no-299/
>>14986711

>>14986905
Listen anon, you need to work with people so they work with you. Even twelve year olds know this much.

>>14986711
what does these things allow you to calculate? i recognize about 73.4% of the notation

>>14987176
the concatenation of functions defined as converging series

In a post shared to her Telegram channel, all Simonyan wrote was "Good evening, 1962."
https://www.mixcloud.com/OLDSCHOOLINIZIODITUTTO/dave-ralph-live-at-love-parade-berlin-8-25-1999/
https://ia802208.us.archive.org/17/items/BernaysEdwardL.CrystalizingPublicOpinion1923noOCR/Bernays%2C%20Edward%20L.%20-%20Crystalizing%20Public%20Opinion%20%281923%29%20%28no%20OCR%29.pdf
https://soundcloud.com/d3nd3d3u77/chosen-few-nasenbluten-rave-on-the-eye-tech-07-07-95

>>14987205
look man im just a plooter i dont understand your deep pure math theory shit. i know about stuff like fourier series and thats about it.

pic related is the first few terms of the fourier series for a square wave. for example

>>14987220
waht do you use for plooooting?

>>14987220
Spivak - Calculus 3/e
Hoffman and Kunze - Linear Algebra 2/e
I can't tell if this problem 33 is related or not to your picture

>>14987254
easybmp and c++ on a linux box using gcc

along with some self written code to make it all work.

if you download the easybmp and follow the examples you can get it to work in an hour or so

>>14986282
Linear Algebra is a whore.

how long does it take to learn latex

>>14990720
Watch:
https://youtube.com/playlist?list=PLQTQDG8nyMPjlHdquBvv6f745YSY2Jylb
And then rewrite a few dozen pages of notes or book in TeX, looking up tex.stackexchange when you have a problem. A week or two of this is enough to learn LaTeX just for utilitarian needs. If you want to make efficient attractive documents with consistent typesetting, it will take you a few months of practice and exploring.

>>14990808
thank you

>>14991073
new thread

>>14991981
Thank God someone posted this before that dumb nigger posted a duplicate /mg/ again

>>14991989
Why would I post a new one?
The dumb Brit finally learned his lesson: To link the new thread in the old.
That's what I wanted from the start, and he's consistently doing it now.