/mg/- mathematics general

[Anonymous No.15300294]

/mg/- mathematics general
Previously >>15279472

The Man Who Loved Only Numbers edition.

Talk maths

[Anonymous No.15300348]

Sex with math books

[Anonymous No.15300638]

this general should just die, mathtrannies are too attention whorish to post in it

[Anonymous No.15300756]

>>15300294
>>>/lit/21807043
It's over.

[Anonymous No.15300792]

Hello anons

How do I know if SNR is a concave function? What's the best way to prove it?

[Anonymous No.15300931]

>>15300348
Math with sex booths
https://soundcloud.com/djturmix/turmix-on-45-live-from-new-york-soho-radio-ny23112022

[Anonymous No.15300946]

>>15300294
Sum of permutations from 0 to n is??????

[Anonymous No.15301353]

Why does Mergelyan's theorem exist? As in, why wouldn't the Stone-Weierstrass apply?

[Anonymous No.15301392]

>>15301353
Stone-Weierstrass is only applicable for real-valued functions

[Anonymous No.15301441]

Is probability real math?

[Anonymous No.15301445]

>>15301392
Are you thinking of the Weierstrass approximation theorem? There's a version for continuous complex-valued functions on a compact Hausdorff space: if you have a subalgebra that separates points, vanishes nowhere, and is closed under conjugation, then its uniform closure is the entire space of such continuous functions.

[Anonymous No.15301447]

>>15301441
Yes and no, in the same way you can ask the question about physics.
The descriptive parts of those subject can be formalized into a purely mathematical form, but essentially exactly because of this it's loaded with philosophical questions. In probability, it's e.g. the question of interpretation of probabilities ("can/should Bayes rule be applied in cases that go beyond a frequentist/combinatorical/geometrical scenario") and adhoc choices in probability theory. In praxis, pretending everything is essentially like a frequentist scenario and pretending least squares always gives a good metric, usually works. Or the places where it doesn't are too hard to handle.

[Anonymous No.15301450]

>>15301447
The least square being a way to evaluate distance between sample data and regression in decision theory, i.e. the choice of how to map both the data and the model to some real

[Anonymous No.15301511]

>>15301441
Perspective based like anything else. A bunch of faggots here say stuff like calculus is arithmetic and not real math.
My take is just like general linear group in group theory is real math, but Heisenberg group is physics, there are formalized systems in probability (the ones that aren't experimentally derived like some distribution tables) that are real math, but as soon as they're applied they're their own field.

[Anonymous No.15301729]

HEY HEEEY

NEW THREAD, NEW PROBLEM.

I think this time it's actually a super cool problem. It's not hard. In fact I'd say it's really easy BUT of course that may change depending from person to person. But I feel confident in saying YOU SHOULD BE ABLE TO SOLVE THIS!

As always feel free to ask for clarification. I appreciate anyone replying whether they've got a solution or not. If you get stuck, feel free to ask for hints that may take you to the answer.

And good luck to anyone attempting!
>>15301441
here's a probability question(sort of?)! Decide yourself if this counts as math. I don't see why it wouldn't be.
>>15300756
They're shutting down because they started giving out copyrighted books away during the pandemic, right?
Kind of stupid in my opinion, there were already sites for that and they went out of their way to find trouble. I wish they just didn't do that. I don't really see how it'd have serious consequences for math though. What am I missing?

[Anonymous No.15301767]

>>15301447
It's another case of confusing probability with statistics
>>15301441
It is indeed PURE mathematics.

[Anonymous No.15301776]

>>15301729
Sorry anon but that problem is very easy and not interesting at all; unironically high school tier. Please try this problem instead.

[Anonymous No.15301784]

>>15301729
[math]\frac{417}{512}[/math]

[Anonymous No.15301790]

>>15301447
>>15301511
>>15301729
>>15301767
is statistics real math?
>>/sci/thread/11850693#p11851353

[Anonymous No.15301827]

>>15301776
50! binom(100, 50) binom(50, 24) 24!^2?

[Anonymous No.15301903]

>>15301790
Only a part of Statistics is math, that also applied math. But probability is as much of a pure math as is geometry.

[Anonymous No.15301917]

>>15301729
[math] 425 / 512 [/math]

[Anonymous No.15301993]

>>15301776
You're right anon, it's pretty easy. Is it uninteresting though? I don't know, i found it a bit fun but that's how I see most things. And it is actually a high school problem, for real.
The problem you posted seems very fun as well and definitely more challenging. I have classes and an exam today however in the evening I'll be free so I'll give it a try! Thank you for sharing it! It'll probably take a while for me to solve it though if I can at all since I'm not actually good at this.
>>15301784
That is corrrrect anon! Good job! Thank you very much for your time and effort solving this problem. Did you find it fun? I hope you did. Have a nice day!
>>15301917
Unfortunately anon that's not the correct answer. However it's rather close to the answer so I suspect you were on the right path but made a small mistake. How did your solution go? Can you check again and make sure there isn't some careless arithmetic mistake? Regardless good attempt! I believe you can get the right one!

[Anonymous No.15302007]

>>15301993
Yes, I figured out my mistake.

[Anonymous No.15302049]

>>15301993
now i demand for an explanation, which doesn't involve counting, like the one using generator functions, can i get one
also, was the question about getting getting all types of cards from s*n cards, your first question? or were there more to this series that i missed out on?

[Anonymous No.15302065]

>>15301729
If the center is blue then there is no 2x2 red.
That accounts for 256.
If the center is red, you can enumerate based on what the 4 cells adjacent to the center are.
All 4 red (1 way) ~ each corner must be blue ~ 1
3 red (4 ways) ~ 2 corners must be blue and 2 can be anything ~ 4*(2*2)
2 adjacent red (4 ways) ~ 1 corner must be blue and 3 can be anything ~ 4*(2*2*2)
2 opposite red (2 ways) ~ 4 corners can be anything ~ 2*(2*2*2*2)
1 red (4 ways) ~ 4 corners can be anything ~ 4*(2*2*2*2)
No red (1 way) ~ 4 corners can be anything ~ 1*(2*2*2*2)
This gives a subtotal of 1+ 10*2^4 = 161

(256+161)/512 = 417/512

[Anonymous No.15302077]

>>15302007
Very good! Nicr job, I knew you were close!
>>15302049
Eh, for this problem? I don't know any anon. I don't know how it'd go to be honest. I could look for one later tonight though I'm not very hopeful.
>also, was the question about getting getting all types of cards from s*n cards, your first question? or were there more to this series that i missed out on?
No, it wasn't my first. There were many more before. I post them on a discord server as well so if you're interested in seeing them I could send an invite link. But you could also look for them in previous threads. However I'll say the problems I post aren't exactly a special bunch. I usually take them from textbooks.

[Anonymous No.15302082]

>>15302065
Sorry I couldn't reply to you in my previous message, I saw it after posting it.
Yes anon, your answer is correct. Very nice job! Thank you a lot for your time and effort solving this problem. I appreciate it. Did you find it fun?

[Anonymous No.15302089]

>>15302077
>I could look for one later tonight though I'm not very hopeful.
nah, leave it
>I could send an invite link.
i don't know if posting invites on 4chan is a good idea, but if you can, i'd appreciate it, in any case, i hope you post more here, because i'll be staying

[Anonymous No.15302100]

>>15302082
>Did you find it fun?
Yes. These are the best part of /mg/
I enjoy a distraction from the difficult problem I'm working on and just end up staring at for hours.

[Anonymous No.15302119]

I don't know fuck all about math but got a question. Say you pick a random prime and find the distance to the next prime above it and below it, seeing which is closer. Across primes, are they equally spaced or biased in one of those directions? Is that even answerable?

[Anonymous No.15302124]

>>15302119
Solve the Riemann hypothesis then you'll know

[Anonymous No.15302133]

>>15301776
Anon our chemistry teacher is sick so I attempted the problem you posted now and I got 100!/50*25 as an answer(100!/1250). Could you tell me if it's correct? I kinda rushed at it and I don't know if I missed anything. Pic related is my solution. Sorry for the terrible handwriting.
>>15302089
https://discord.gg/5spZEWHZ2u
Here's the link anon. I have shared this here before and I didn't have any problems(well we have problems in the server but the math kind).
>i hope you post more here, because i'll be staying
Thank you a lot. Very kind. I enjoy posting here because of you anon.
>>15302100
>First things first, nice dubs. Secondly
>Yes. These are the best part of /mg/
I'm incredibly happy to hear this anon.
>I enjoy a distraction from the difficult problem I'm working on
That's a very good way to use these I suppose. Thank you very much for the compliment.

[Anonymous No.15302257]

>>15301776
[math]\frac{100!}{26\cdot 25}[/math]
>>15302133
I don't think that's correct.
>>15301993
>Did you find it fun?
I feel like one of those search and rescue dogs at Ground Zero who the firefighters had to hide in the rubble for.

[Anonymous No.15302288]

>>15302133
>Could you tell me if it's correct?
I don't know. I got same as yours.
>>15302257
>I don't think that's correct.
Why please explain your working anon.

[Anonymous No.15302304]

>>15302288
[eqn]\binom{100}{50}50!\binom{50}{24}24!^2[/eqn]
Choose your first 50 numbers. Permute the other 50. Identify the minimum out of the 50, put it last, then choose numbers 26 through 49, and permute. You are left with determining the ordering of first 25 numbers now, so identify the minimum again, and permute the other 24.

[Anonymous No.15302308]

>>15302288
Nah you guys are right.

[Anonymous No.15302350]

>>15302257
>I don't think that's correct
Entirely within the realm of possibilities. I checked again and my answer hasn't changed however of course I might have mistakes.
>>15302288
Well, while it's reassuring you got the same answer I'm a bit dissapointed you don't have the definite answer. Where did you find it anon?
Also thanks again for sharing, it was quite fun. But I'll say that it wasn't really harder than the problem I have posted.
>>15302308
Oh, alrighty then. Well, I hope it is indeed correct.

[Anonymous No.15302424]

How do you anons go about reading books, what is your process from page to page?

Do you keep some problems out in front of you to think about while reading? Do you take notes while reading? Or are you one of those who just reads?

[Anonymous No.15302565]

>>15300294
I found pic related in an AOPS book, kek

[Anonymous No.15302569]

>>15302565
2/5

[Anonymous No.15302571]

>>15302569
3/5

[Anonymous No.15302574]

>>15302571
4/5

[Anonymous No.15302581]

>>15302574
5/5
women know the axiom of life

[believe it or not, I enjoy my life No.15302618]

I can understand believing that. But I won't ever be so pathetic to believe things like that, and simultaneously make no change to my way of life or the people I surround myself with. If you don't belong here anymore, leave. I don't have time for preachers and especially not for the spineless variety

[Anonymous No.15302709]

>>15302350
>I'm a bit dissapointed you don't have the definite answer
I guess you can program it applying the same algorithm with smaller numbers.
>Where did you find it anon?
https://www.isical.ac.in/~admission/IsiAdmission/PreviousQuestion/MStat-PSB-2018.pdf

[Anonymous No.15302824]

>>15301729
2^9=512 possible grids
4*2^5 grids have a red square
4*2^3 grids have two adjacent red squares
2*2^2 grids have two opposite red squares
4*2^1 grids have three red squares
1 grid has 4 red squares
4*2^5-(4*2^3+2*2^2)+(4*2^1)-(1)=95 grids with a red square
(512-95)=417 grids with no red squares
417/512

[Anonymous No.15302875]

>>15302709
You know what, I think I'm gonna instead just assume we got the right answer.
Cool pdf btw! Seems to have some nice problems. How did you come across it and why did you decide to share the problem from it? I'm asking out of curiosity of course, I'm thankful you shared it because it was fun.
>>15302824
And congratulations! Your answer is correct anon! Your solution is of course very clear. Thank you very much for your time and effort solving this problem. I appreciate it. Now if you don't mind, tell me if you found the problem a bit fun.

[Anonymous No.15302876]

[Anonymous No.15302899]

>>15302875
Fairly easy, worth a few minutes. Might be more interesting to see how it extends to larger squares.

[Anonymous No.15302947]

>>15302875
>How did you come across it
It's from an exam I am preparing for,
https://www.isical.ac.in/~admission/Syllabus-And-QP.html (M.Stat. PSB 2018)
>why did you decide to share the problem from it
Because I wanted to compare my solutions to others'.
>because it was fun.
Glad you had fun fren.

[Anonymous No.15302977]

Home Math Companion: Some Damn Fool Thing in the Balkans Edition
Now available in Polish, German, Russian, Hungarian, Swedish, Finnish, Turkish, Slovenian, and Czech

[Anonymous No.15303162]

What are you studying/researching /mg/?
For me, it’s arithmetic of derived categories

[Anonymous No.15303304]

>>15303162
gonna into galois theory this week - somehow skipped over it in undergrad

[Anonymous No.15303669]

>>15303162
but only today

[Anonymous No.15303899]

Does anyone know a good digital keyboard for Android that allows indexes and common math symbols?

[Anonymous No.15303941]

>>15301729
# 0.814453125

import random
def genBoard():
return [[random.randint(0, 1) for _ in range(3)] for _ in range(3)]

def hasSquare(board):
out = 1
if board[0][0]==1 and board[0][1]==1 and board[1][0]==1 and board[1][1]==1:
out = 0
if board[0][1]==1 and board[0][2]==1 and board[1][1]==1 and board[1][2]==1:
out = 0
if board[1][0]==1 and board[1][1]==1 and board[2][0]==1 and board[2][1]==1:
out = 0
if board[1][1]==1 and board[1][2]==1 and board[2][1]==1 and board[2][2]==1:
out = 0
return out

NUM_ITERATIONS = int(1e7)
hits = 0
for _ in range(NUM_ITERATIONS):
board = genBoard()
hits += hasSquare(board)

print('empirical',hits/NUM_ITERATIONS)
calculated = (2**9 - (2**5 + (3*(2**3)) + (3*(2**3)-2) + (3*3*2-1)))/(2**9)
print('calculated',calculated)

[Anonymous No.15304095]

Why is the Weierstrass function not differentiable at [math]x = 0[/math]?

The Weierstrass function is
[eqn]f(x) = \sum_{n=0}^\infty a^n \cos{b^n \pi x}[/eqn]
and if you attempted to differentiate it, you get
[eqn]f'(x) = \sum_{n=0}^\infty -a^n b^n \pi \sin{b^n \pi x}[/eqn]
with [math]ab > 1 + \frac{3 \pi}{2}[/math]. The reason the function is non-differentiable is because the partial sums of the derivatives diverge to infinity. But that does not apply at [math]x = 0[/math], where
[eqn]f'(0) = \sum_{n=0}^\infty -a^n b^n \pi \sin{0} = 0[/eqn]
The same applies when using the definition of the derivative:
[eqn]f'(0) = \lim_{h \to 0} \sum_{n=0}^\infty \frac{a^n \cos{[b^n \pi (0 + h)]} - a^n \cos{b^n \pi (0)} }{(0 + h) - 0} \\
= \lim_{h \to 0} \sum_{n=0}^\infty \frac{a^n \cos{[b^n \pi (0) + b^n \pi h]} - a^n \cos{b^n \pi (0)} }{h} \\
= \lim_{h \to 0} \sum_{n=0}^\infty \frac{a^n \cos{b^n \pi (0)} \cos{b^n \pi h} - a^n \sin{b^n \pi (0)} \sin{b^n \pi h} - a^n \cos{b^n \pi (0)} }{h} \\
= \lim_{h \to 0} \sum_{n=0}^\infty \left[ \left( \frac{ \cos{b^n \pi h} - 1}{h} \right) a^n \cos{b^n \pi (0)} - \left( \frac{ \sin{b^n \pi h} }{h} a^n \right) \sin{b^n \pi (0)} \right] \\
= \sum_{n=0}^\infty \left[0 · a^n \cos{b^n \pi (0)} - 1 · a^n \sin{b^n \pi (0)} \right] \\
= \sum_{n=0}^\infty (0·1 + 1·0)a^n \\
= \sum_{n=0}^\infty (0)a^n \\
= 0[/eqn]
This would mean the function is indeed differentiable at [math]x = 0[/math], regardless of the value of [math]a[/math] or [math]b[/math]. Yet by saying that it is nowhere differentiable, that implies that it's also not differentiable at [math]x = 0[/math]. So there must be some other explanation. What am I missing?

[Anonymous No.15304120]

>>15304095
Made a mistake with the fourth last line, it should be
[eqn]\lim_{h \to 0} \sum_{n=0}^\infty \left[ \left( \frac{ \cos{b^n \pi h} - 1}{h} \right) a^n \cos{b^n \pi (0)} - \left( \frac{ \sin{b^n \pi h} }{h} \right) a^n \sin{b^n \pi (0)} \right][/eqn]

[Anonymous No.15304125]

>>15304095
I think the issue is that you’re swapping the infinite sum with the derivative operator, so underlyingly you’re swapping limits without justification

[Anonymous No.15304129]

>>15304125
It shouldn't matter due to the linearity of differentiation. Or am I mistaken?

[Anonymous No.15304148]

>>15304129
You’re mistaken, that can only be used for finite sums.
To be explicit a sum from n=1 to infinity is defined as the limit as k goes to infinity, if the sum from n=1 to k

[Anonymous No.15304224]

>>15304148
So in the end, the sum is still 0?

[Anonymous No.15304505]

>>15300294
Think about the implications of this.
>if the angle = 90deg cos becomes 0 and sin becomes 1
But that also means we have a triangle where the adjacent is 0 and the opposite = hypotenuse. So it says that a single line is a triangle.

It's true mathematically, because the sum of all the angles is still 180deg, because he have two angles with 90deg and one angle with 0deg. How the fuck can this be? How can a single line be a triangle? But it's legit mathematically.

[Anonymous No.15304605]

Intelligence test. Which one equation produces this curve?

[Anonymous No.15304613]

>>15304505
It is a degenerate triangle.

>>15304605
[math] (x-1)(y-1) = 0 [/math]

[Anonymous No.15304618]

>>15304613
Nope. That made just part of the curve.

[Anonymous No.15304621]

>>15304618
No, it makes the whole curve, but doesn't exclude the negative regions.

[Anonymous No.15304633]

>>15303941
what i did
def has2x2(n):
if (n & 1) and (n & 2) and (n & 8) and (n & 16):
return True
elif (n & 2) and (n & 4) and (n & 16) and (n & 32):
return True
elif (n & 8) and (n & 16) and (n & 64) and (n & 128):
return True
else:
return (n & 16) and (n & 32) and (n & 128) and (n & 256)
count = 0
for i in range(2**9):
if has2x2(i):
count+= 1
print(2 ** 9 - count)

[Anonymous No.15304636]

>>15304605
[math] |(x-1)(y-1)|+|x-|x||+|y-|y|| = 0 [/math]

[Anonymous No.15304666]

>>15304636
Right answer: [math]x^yy^x=xy[/math]

[Anonymous No.15304668]

>>15304666
My answer is better.

[Anonymous No.15304679]

>>15304636
This one is completely correct.
>>15304666
This one looks correct on graphs, but there are still isolated points in the negative quadrants.

[Anonymous No.15304992]

How do you calculate the shaded area?

[Anonymous No.15305003]

>>15303162
In my free time outside of school, differential geometry.

[Anonymous No.15305010]

>>15304992
Let [math]f(x)=\sin(x) - x^2[/math]. Let [math]a,b[/math] be the two roots of [math]f[/math]. The area is then given by
[eqn]\int_a^b f(x) dx[/eqn]

[Anonymous No.15305096]

>>15305010
>let this be that and you have some arbitrary math symbol
>no numbers anywhere
Mathfags can't even calculate the area.

[Anonymous No.15305114]

>>15300792
https://www.ie.cuhk.edu.hk/people/documents/geometric3.pdf

[Anonymous No.15305121]

>>15304992
calculate points of intersection of the two functions, one is obviously at 0 and the other at a point, call it A
then calculate the integral of BLUE from 0 to A and same with RED.
subtract the RED area from the BLUE one

[Anonymous No.15305142]

>>15305096
>how do you
>here's how
>NN...NOOO NOT LIKE THAT YOU WERE SUPPOSED TO SPOONFEED ME MY CALC 2 HOMEWORK ANSWERS NOOOOO
kys

[Anonymous No.15305149]

>>15305142
Because the integral symbol is trivial stuff. Getting an actual number for the area is not trivial.

[Anonymous No.15305162]

>>15305149
What the fuck do you even mean with actual number? The area is certainly not an integer.

[Anonymous No.15305228]

>>15305121
>>15305162
to be fair, I don't think there's a closed-form representation of the upper limit
not that that affects the answer to the question, but it does make it a bit harder to actually do the procedure

[Anonymous No.15305467]

[math]A\subset {\mathbb N}\land \forall (s\colon {\mathbb N}\to A). \exists (m\in {\mathbb N}). \forall (n>m). s_n < n [/math]

[math]\implies [/math]

[math] \exists (m\in {\mathbb N} ). \forall (a\in A). a < m[/math]

You should be able to prove this.
In words, the bounded sets of numbers are exactly those sets for which all sequences are eventually majorized by the identity sequence.

[Anonymous No.15305471]

>>15303162

[Anonymous No.15305488]

>>15305467
If [math]A[/math] is an unbounded set then the sets [math]A_k := \{a \in A | a > k \} [/math] are all non-empty which allows you to constuct a sequence [math](s_n)_{n \in \mathbb{N}}[/math] with [math]s_k \in A_k[/math] for all [math]k\in \mathbb{N}[/math]. This sequence trvially is not majorized by the identity sequence.

[Anonymous No.15305493]

>number theory not a pre-req to sign up for abstract algebra
>Course automatically expects you to know essentially all of basic number theory from the dover Andrews book.

I'm getting fucked exactly the same way I did two semesters ago when I took calculus III without taking linear algebra first.

[Anonymous No.15306423]

It is SO fucking over....

[Anonymous No.15306638]

>>15306423
>used wolfram

[Anonymous No.15306784]

How do I choose r ordered numbers from n such that no numbers are consecutive.

[Anonymous No.15306863]

>>15306423
AI will literally solve all of humanity's math problems. You can give it a lost of unsolved math problems from Wikipedia and it will solve them all eventually.

Picrelated was a problem that I invented to test the AI, and it solved it like it was nothing. Despite the fact that the problem probably didn't exist before I invented it. It literally has independent intelligence.

[Anonymous No.15306867]

>>15306423
AI will solve all of humanity's math problems. You could give it a list of unsolved math problems from Wikipedia and it will solve them all, eventually.

Picrelated was a problem that I invented to test the AI, and it solved it like it was nothing. Despite the fact that the problem probably didn't exist before I invented it. It already has independent intelligence like a human.

[Anonymous No.15306934]

>>15306867
an independent intelligence would just look at which square is blank to determine the coordinates and count the number of times each number appears to figure out which is one short

[Anonymous No.15307309]

>gpt-4 is already better than me at math
How do I cope?

[Anonymous No.15307354]

When I am supposed to show "p precisely when q," do I show "p iff q," or "p if q."
>>15307309
It's not. You could have also just gone to wikipedia and copied the answer. It did no thinking of its own

[Anonymous No.15307386]

If you have a random number between zero and one (call it the number X), but you want that the distribution is more biased towards one such that if you double the distance towards one you double the density of points. For example, the probability of X being near 0.7 should have twice the probability of being near 0.35.

I tried taking a square root of X and it seems to produce this effect visually but how do you prove that it is correct mathematically? My head hurts when I'm trying to think of this. Not even homework I was just doing maths for fun.

[Anonymous No.15307455]

>>15307386
You want [math] f(x) = 2 f(x/2) [/math], any line passing through the origin satisfies this, but you want a probability density from 0 to 1, so [math] f(x) = 2x [/math]

[Anonymous No.15307503]

It's probably better to argue in terms of

[math]\lim_{\varepsilon\to 0}\dfrac{ \int_{2t}^{2t+\varepsilon} f(x) {\mathrm d}x }{ \int_t^{t+\varepsilon} f(x) {\mathrm d}x } = 2[/math]

But the answer is the same, given

[math]\dfrac{ \int_{s t}^{s t+\varepsilon} 2x {\mathrm d}x }{ \int_t^{t+\varepsilon} 2x {\mathrm d}x } = \dfrac{s + \varepsilon /(2 t)}{1 + \varepsilon /(2 t)}[/math]

[Anonymous No.15307505]

>>15307455
If you do f(x)=2x, then you're just doubling the interval (picrel) but you're not producing a non-uniform density like I'm trying to do.

[Anonymous No.15307540]

>>15307505
You're doubling the sampled value, not the sampling chance.

If you only got an uniform sampling tool available, then I think you gotta do:

NUM_SAMPLES = 1000
nums = []
for _ in range(NUM_SAMPLES):
x = random(1)
if random(1) < x: nums.append(x)

[Anonymous No.15307544]

In your notation x=a and still b=random(1)

[Anonymous No.15307546]

>>15307540
I mean, this is what I was looking for. Now it sort of looks like halfing the distance from one always halves the density along the x-direction. But I wonder if you could prove that this indeed is right and not just an approximate solution to my problem.

[Anonymous No.15307560]

>>15307540
I mean, this is what I was looking for. Now it sort of looks like halfing the distance from one always halves the density (along the x-direction, the other coordinate doesn't matter). But I wonder if you could prove that this indeed is right and not just an approximate solution to my problem.

[Anonymous No.15307562]

>>15307540
I don't know desmos well but it's quite annoying that it draws some random circle just because you try to use the letter r

[Anonymous No.15307566]

>>15307560
I mean, halfing the distance from zero, not one.

[Anonymous No.15307571]

>>15307560
The post you reply to has my solution here >>15307562

To calculate 'a', I sample a skipping condition 's' and keep 'a' only if a>s.
So twice as large 'a' are kept twice as often.

Since I don't know Desmos, I just map the dropped values to 1. But you can probably make them vanish properly.
Also desmos might have a broader range of sampling functions.

[Anonymous No.15307580]

I tried none, nil and null, and it seems it drops it if I do "n"

Only 4chan would know what n stands for

[Anonymous No.15307581]

no wait, n is the slider window, so it must be far out there, lol. Anyway

[Anonymous No.15307588]

>>15307571
That's all cool and I might use that. But I still wonder if the square root produces the same effect. Because my intuition tells me that it does but I have no idea how to prove that. This is the stuff that keeps me awake at night

[Anonymous No.15307592]

Btw. I think the way you described it is a bit ambiguous.
For x<1/2, it's true that the density f(x)=2x doubles chances as you go away from 0.

But if you instead want doubling as you get closer to 1 (e.g. that chance of sampling around 0.9 is twice as high as sampling around 0.8), then you need another, more complicated approach.

[Anonymous No.15307598]

>>15307588
>This is the stuff that keeps me awake at night
I mean transformation of random variables is a base thing in probability theory.

I suppose
f_X(x) = const/uniform = 1
Y = g(X) = X^(1/2)
h(y) = g^{-1}(y) = y^2
f_Y(y) = 1 · dh/dy = 2y

So if I'm not confused, yes? Haven't checked if all conditions are satisfied for the Jacobian calc.

[Anonymous No.15307687]

>>15307560
I don't think you even understand what you are asking. Frankly, it seems you have very little understanding of probability distributions.

In Desmos, [math] \texttt{random(100)} [/math] produces 100 samples of a standard uniform distribution. So here >>15307505 you have plotted a scatter plot of 100 samples of [math] (2U, V) [/math] where [math] U,V [/math] are [math] \textit{independent} [/math] and identically distributed standard uniform. PLOTTING A SCATTER PLOT OF INDEPENDENT VARIABLES IS STUPID. The whole point of a scatter plot is to visualise dependence between two variables but in this case the [math] y[/math]-axis is pretty much irrelevant.

>>15307560
Now here, you have plotted [math] (U^{1/2},V) [/math]. Here also the same stupidity of plotting independent variables arises. However this plot is very related to my answer because what do you think is the probability density function of [math] U^{1/2} [/math]? Let us find out. Since CDF of [math] U [/math] is [math] F(u) = u [/math], for any [math] u \in (0,1) [/math], we have:
[eqn] P ( U^{1/2} < u) = P ( U < u^2) = F(u^2) = u^2 [/eqn]
Hence the CDF of [math] U^{1/2} [/math] is [math] u^2 [/math], and consequently probability density function, which is the derivative of CDF, is... [math] 2u [/math]. Same as my answer. To plot this in an actually intelligent way, do picrel. Here, you can see that the density of the plot increase as you go further to 1. That is precisely because the probability density function i.e., [math] f(x) = 2x [/math], is a strictly increasing function with respect to [math] x [/math]. So points closer to 1 are more probable. In fact, [math] \textit{probability density} [/math] of every point in [math] (0,1) [/math] is twice that of half the point. Note, the actual [math] \textit{probability}[/math] of every point is 0 for all continuous distributions.

[Anonymous No.15307698]

>>15307687
Actually, this is even more intelligent. I have made the lines very thin, so it is easier to see the dense points.

[Anonymous No.15307753]

>>15307560
>Now it sort of looks like halfing the distance from one always halves the density
Actually [math] U^{1/2} [/math] doesn't satisfy this. I think what you are looking for a random variable with CDF [math] F [/math] that satisfies:
[eqn] \forall a \in [0,1] \qquad F(a) - F(a/2) = 2 F(a/2) [/eqn]
or equivalently:
[eqn] \forall a \in [0,1] \qquad F(a) = 3 F(a/2) [/eqn]
Now, it is not possible to have a CDF like this in [math] (0,1) [/math]. Since the [math] F [/math] must be non-monotonic to integrate to 1, over such a small interval, but it is probably possible for a larger interval.

[Anonymous No.15307783]

>>15307753
>Since the F must be non-monotonic to integrate to 1
What?
A CDF has to be right-continuous, monotone increasing function with [math]\lim_{t \to -\infty} F(t) = 0[/math] and [math]\lim_{t \to \infty} F(t) = 1[/math].

[eqn]F(t) = \begin{cases} 0 & t \leq 0 \\
t^{\log_2(3)} & 0 < t \leq 1 \\
1 & 1 < t
\end{cases}
[/eqn]
will work.

[Anonymous No.15307801]

>>15307687
Uh... yeah I see that f(x) = 2x makes sense when you just want a function to describe the relative density of those points. Of course it is 2x, that's just another way of saying what I said to begin with. But how do you convert between U^(1/2) and f(x) = 2x? Or generalize this procedure? That's what I was thinking about... I don't know anything about probability theory sorry if that was a stupid question lol.

[Anonymous No.15307856]

>>15307801
>But how do you convert between U^(1/2) and f(x) = 2x?

If you want to find the density of a differentiable bijective transform [math] g [/math] of any continuous random variable [math] X [/math] with PDF [math] f [/math]:
[math] \qquad f (g^{-1}(x) ) \left| \dfrac{dg^{-1}(x)}{dx}\right|[/math]
is the density of [math] g (X) [/math]. For functions, which aren't bijections, it is more involved. But the general approach is what I have done here:
>>15307687
>P(U1/2<u)=P(U<u2)=F(u2)=u2

If you want to plot a random variable with CDF [math] F [/math]:
[math] \qquad F^{-}(U) ; \quad U \sim \mathrm{Unif} (0,1) [/math]
is a random variable with CDF [math] F [/math] where [math] F^-[/math] is a right inverse of [math] F [/math] (it always exists). This is called the probability integral transform.

>>15307783
Yes I must have made some mistake then. So the required random variable is: [math] U^{1/ \log_2 3} [/math]

[Anonymous No.15307918]

>>15307687
>it seems you have very little understanding of probability distributions.
What motivates someone to write this?
I'm not the guy you replied to but this seems overly unfriendly.
I don't want to play the politeness police, but the guy you reply to was quite unthankful as well.

Guys get your etiquette up

[Anonymous No.15308037]

>>15307856
I don't think your function is right (if you're saying whenever you double the distance from zero you double the probability density)

[Anonymous No.15308048]

>>15308037
>(if you're saying whenever you double the distance from zero you double the probability density)
I didn't. I said this >>15307753. Or you said this. I don't know what you want.

[Anonymous No.15308972]

Is there a general formula for quintic polynomials [math]ax^5 + bx^4 + cx^3 + dx^2 + ex + f[/math] that have a solution by radicals? Or is there a general formula for quintic polynomials that don't have a solution by radicals?

Additionally, is there a general solution for all solvable quintics that does not apply to unsolvable quintics because the solution has requirements to be applicable that unsolvable quintics do not fulfill?

[Anonymous No.15309061]

>>15300756
For a second I thought they meant "arxiv" and I almost had a heart attack. But then I realized that makes no sense

[Anonymous No.15309242]

Is there a positive integer n , with n not a prime power , such that the ring Z/nZ has exactly 2 idempotents?
(Obviously 0,1 are always idempotents. Also, note if n is a prime power then Z/nZ has exactly 2 idempotents.)

[Anonymous No.15309271]

>>15303941
It's a bit interesting you decided to go with programming anon. It's not what I prefer but it's your solution your preferences. Nice job however I should point out the exam this problem is from doesn't allow calculators, let alone computers.

[Anonymous No.15309341]

>>15301729
Since this problem is quite short I'll post another short problem. The thread is a long way from being over, so. This one is actually surprisingly similar to the last problem in some ways.

Feel free to ask for clarification if anything is unclear to you. But this is really a *solid problem* hahahah. Anyhow, good luck to anyone attempting!

[Anonymous No.15309442]

>>15309341
843

[Anonymous No.15309494]

Unironically how do I get gud at math? I've been a B math student my entire life and am drowning in an engineering degree RIGHT NOW. I'm good at literally everything else but math just makes my eyes glaze over for some reason, it's not even that I'm uninterested, my brain just shuts down. how do I stop being a shitter so I can just NEET and make six figures already?

[Anonymous No.15309521]

>>15309442
Anoooon, your answer came really quickly and I applaud you for that but it's unfortunately incorrect! I don't know how you went about it but I'm sure you had the right idea, maybe messed up somewhere along the way. I'll say that the correct answer is lower than yours so perhaps that might help you reach the correct solution. But regardless thank you for your fast reply. I believe you are capable of getting it right and if you need any hints let me know. Good luck!

[Anonymous No.15309536]

lel. babbies first random variable biasing lessons

[Anonymous No.15309537]

>>15309341
Impossible.

[Anonymous No.15309579]

>>15309537
Anon, sorry but what's impossible? The problem isn't impossible, I assure you. It's not easy per say but it's definitely not particularly hard either. I'd encourage you to try seeing how this problem plays out in the 2-dimensional case and then try to go from there.
I'll post solutions(from the book I took it from as well as my own solution perhaps) before the thread closes so you can see how it's solved but I think it'd be more beneficial if you tried yourself, I think you'll see how it's definitely not impossible in any sense.
>>15309494
Anon I'm not good at math either so maybe don't take me too seriously but I believe you just have to practice practice practice. Is there really a easy way to becoming good at math? What sort of math are you doing anyway?

[Anonymous No.15309625]

i love you dyadic intervals
i love you dyadic squares
i love you dyadic cubes
i love you dyadic 4-cubes
i love you dyadic 5-

[Anonymous No.15310034]

>>15309341
765 but i did what >>15309442 did the first time too

[Anonymous No.15310130]

>>15308972
I thought about this a bit but my algebra is pretty weak.
That said, what you are saying basically amounts to finding a general procedure, using just the coefficients of the polynomial, to determine whether its Galois group is solvable.

[Anonymous No.15310179]

>>15310130
Yeah, come to think of it, I guess that's what I'm asking for.

[Anonymous No.15310186]

>>15301392
The Stone-Weierstrass theorem has a complex version dumbass. (You need closure of the subalgebra under complex conjugation however)

[Anonymous No.15310603]

>>15310034
Hello anon. That's interesting you got the same answer as the first anon. I'm sorry to say unfortunately your answer is also incorrect by a small amount. It's actually a bit less than the correct answer so you probably subtracted something twice. But I'm sure you can fins what it is and add it back!

[Anonymous No.15310904]

>>15305471
what does that mean

[Anonymous No.15310936]

>>15310904
I assume they're using a system with Einsteinian gravity coupled to a Klein-Gordon field

[Anonymous No.15310976]

>>15304129
I'm not an expert on analysis but iirc you need something like uniform convergence (or smth else) to be able to move integrals/derivatives inside a summation

[Anonymous No.15311094]

Any new progress on the subgroup-subset problem?

[Anonymous No.15311131]

>>15300294
So, topology.
Is there anyone who really thinks the "clopen" deal is justified in any ways? The definitions being so ass-backwards is a detriment to the field, because it makes people avoid topology.

Someone should re-label the foundations to use even vaguely reasonable terms.

[Anonymous No.15311214]

Does there exist an n-dimensional exotic sphere (smooth mfld homeomorphic but not diffeomorphic to the standard S^n) for infinitely many values of n ?

[Anonymous No.15311289]

>>15311131
>relying solely on definitions for intuition
Like everything in abstract math, you're supposed to build your intuitive understanding from theorems and examples, not just the single axiomatic definition

[Anonymous No.15311304]

>>15311289
Fair enough, I suppose
Still, at least in terms of not biting off the newcomers, this seems somewhat suboptimal

[Anonymous No.15311523]

>>15311304
You know, you don't *have* to study anything you don't want to.

[Anonymous No.15311548]

Do you have a moment to talk about our Lord and savior super symmetry? And please cool it with the antisemitism

[Jealous No.15311554]

>>15311548
https://vixra.org/abs/1307.0075

[Anonymous No.15311560]

>>15311554
Is this guy serious

[Anonymous No.15311841]

>>15307354
iff
>>15308972
https://mathoverflow.net/questions/22923/computing-the-galois
Galois group algorithms, you care if the output is solvable or not
https://math.stackexchange.com/questions/3916878/example-of-f-in-kx-solvable-by-radicals-but-having-a-root-inexpressible-onl
if you want existence of relatively explicit algebraic conditions for the quintic

[Anonymous No.15311844]

>>15311214
https://en.wikipedia.org/wiki/Exotic_sphere
See the section on Theta_n, for example Wang & Xu 2017 shows odd n > 61 all work I think

[Anonymous No.15311849]

>>15311554
Oh wow, this guy truly is a narcissistic faggot. Moreover, it sounds like Nottale invented his "theory" way before he did.

[Anonymous No.15311857]

>>15304095
>>15304224
https://math.berkeley.edu/~brent/files/104_weierstrass.pdf
In-depth proof with a couple typos. At the point x0=0, the sequences ym=-1/b^m, zm=1/b^m will make the difference quotients bad.
It's true that every fixed difference quotient at any n=k will go to 0. However, each of those go to 0 slower and slower (measured by h getting closer to 0) as k grows. So intuitively, along this sequence the "slope" ends up being dominated by terms with larger and larger k value as h nears 0, and never stabilizes in a meaningful sense (and its sign actually fluctuates), see the middle of page 3

[Anonymous No.15311873]

Say I have a set agents moving in an R^3 space, each with position and velocity vectors consisting of the expected triple representing the dimensions. In a physical sense, taking the norm of velocity vectors returns the 1-D speed of the agent.

I cannot for the life of me figure out what the physical sense for the norm of the position vectors is. I have tried to google it, and gotten no where. None of my colleagues are available to answer this for me, either, or at least they aren't responding to emails.

The reason I want to know is because I have optimized some swarm behavior against certain metrics as control laws in a simulated environment, and I want to analyze the impact on swarm behavior. Getting the total swarm speed by taking the norm of every velocity vector at a given time is obvious, because it shows whether or not the speed of the entire swarm is aligning over time. I can also take the norm of each agent in the swarm at time t, average it, and look at that per time hack.
But I'm not sure how to evaluate the position, because I don't have an intuitive since of the physical nature of the norm of a position vector.

Any help you can give is appreciated.

[Anonymous No.15311875]

>>15311841
also
https://math.stackexchange.com/questions/117849/is-there-something-like-cardanos-method-for-a-solvable-quintic
https://www.ams.org/journals/mcom/1991-57-195/S0025-5718-1991-1079014-X/S0025-5718-1991-1079014-X.pdf

[Anonymous No.15311918]

>>15311844
Awesome, thanks anon

[Anonymous No.15311934]

>>15311873
>None of my colleagues are available to answer this for me, either, or at least they aren't responding to emails.
Kek

[Anonymous No.15311949]

>>15311934
It's fucking late where my colleagues like me a lot okay? ;_;

[Anonymous No.15311951]

>>15311949
>where
Fuck, meant here

[Anonymous No.15312000]

>>15311873
if the origin is not significant in any way it could mean anything
but the norm of the position vector in general is just the distance to origin
so if that's the starting point then that would be the distance from the start
idk what this question is asking desu

[Anonymous No.15312004]

How do I break down (13)^27 * mod 55?

I know that (13)^27 mod 55 = (2197)^9 mod 55 but I don't know where I should go from there.

The final answer should be 7 but, like I said, I don't know how to prove it.

[Anonymous No.15312006]

>>15312000
Thanks for the feedback, that actually answers my questions and explains my confusion about it. There's no reference point in my simulation besides the swarm itself, so the norm here is totally useless.

[Anonymous No.15312024]

>>15312004
you can reduce 2197 mod 55 and continue the computation
or, compute 13^27 mod 5 and mod 11 (you should be able to do this) and then use chinese remainder thm

[Anonymous No.15312026]

>>15312004
(13)^27 mod 55
= (2197)^9 mod 55
= (39*55 + 52)^9 mod 55
= 52^9 mod 55
= (140608)^3 mod 55
= (2556*55 + 28)^3 mod 55
= 28^3 mod 55
= 21952 mod 55
= 399*55 + 7 mod 55
= 7 mod 55

[Anonymous No.15312036]

>>15312026
52=-3
(-3)^9=(-27)^3=(1/2)^3=1/8=56/8=7

[Anonymous No.15312101]

>>15310976
And in this case the sum goes to 0 so it should satisfy whatever convergence condition is needed.

[Anonymous No.15312112]

>>15312101
this is false >>15311857

[Anonymous No.15312166]

>>15309271
He did actually solve it 'without' programming - look at the 'pure' calculation value at the bottom.

I think he was just trying to show that he could get the equivalent answer empirically i.e. by just 'brute forcing' it, generating 1e7 random boards and getting the ratio of 2x2 red boards and the 1e7 iterations

[Anonymous No.15312176]

>>15311857
>>15312112
Just saw that post. So if I'm reading this right, you're saying that it's impossible to have a continuous everywhere function that's differentiable at only one point and nowhere else?

[Anonymous No.15312181]

>>15311841
>>15311875
Now this is what I was looking for. Especially the pdf link. Thanks

[Anonymous No.15312185]

Addition to >>15312176
(Or for that matter, a function that's continuous at only one point and nowhere else)

[Anonymous No.15312614]

>>15312166
I see now, I didn't actually read it all so I missed it. Thanks a lot for pointing out anon. And thank you to that anon who wrote code so now we see that the result holds empirically as well. Very cool!

[Anonymous No.15312737]

>>15312176
>>15312185
no. but in this particular case, it is not differentiable there

continuous only at 0: f(x)=x*g(x) where g is always +-1 and in each rational open interval we choose a point to be +1 and a point to be -1. Check this is continuous precisely at 0

differentiable only at 0: same thing with f(x)=x^2g(x). check it is continuous and differentiable at 0, but is not even continuous anywhere else.

[Anonymous No.15312953]

>>15312737
I see. So what about an everywhere continuous function that is only differentiable at one point?

[Anonymous No.15312962]

We want to compute 13^27 mod 55 . This is an exercise in using the Chinese Remainder Theorem:

Using Chinese Remainder Theorem , we have that Z/55 is isomorphic to the direct product of the rings Z/5 x Z/11 , the isomorphism Z/55 --> Z/5 x Z/11 given by sending n mod 55 to n mod 5 , n mod 11 .

So 13 mod 55 corresponds to ( 13 mod 5 , 13 mod 11 ) = ( 3 mod 5 , 2 mod 11 ) .
So 13^27 mod 55 is sent to ( 3^27 mod 5 , 2^27 mod 11 ) .

To figure out 3^27 mod 5 : we know Z/5Z is a finite field and hence its invertible elements are 1,2,3,4 mod 5 , forming a cyclic order-4 group. ( Any finite subgroup of the multiplicative nonzero group of a field is cyclic. ) Directly we see 3 is a generator. Hence in Z/5Z we have 3^27 = 3^(6(4) + 3) = 3^3 = 27 = 2 .

For 2^27 mod 11 , we know 2^5 = 32 = -1 mod 11 , so 2^10 = 1 mod 11 , so 2^27 = 2^7 = - 2^2 = -4 mod 11 .

So finally 13^27 mod 55 corresponds to ( 2 mod 5 , -4 mod 11 ).

We want to find m such that m is 2 mod 5 , -4 mod 11.

We know 5a + 11b = 1 where a= -2 , b=1 . Hence 5a=1 mod 11 and 11b=1 mod 5 .

Hence m = (11b)(2) + (5a)(-4) is 2 mod 5 and -4 mod 11 .

So m = 22 + 40 = 62 , which is 7 mod 55 .

However I agree this is faster:
>>15312026
>>15312036

[Anonymous No.15312972]

>>15312953
https://math.stackexchange.com/questions/618166/continuous-function-with-precisely-one-point-differentiability

[Anonymous No.15312982]

>>15312953
https://math.stackexchange.com/questions/618166/continuous-function-with-precisely-one-point-differentiability

It is actually very easy, if you already have a function continuous everywhere and differentiable nowhere

[Anonymous No.15312996]

>>15312972
>>15312982
Okay, that works. I see why the Weierstrass function is not differentiable at x = 0 now. Thanks.

[Anonymous No.15313006]

>>15312982
Now my intuition wants to compose these to try to control precisely where it's differentiable and how many times, but I'm not sure I understand what the general formula would look like.

Is it possible to have some kind of group where the elements are Weierstrass + n differentiable points at x={0, 1, ..., n-1}?
Can I make any kind of algebra out of adding and removing differentiable points?

[Anonymous No.15313022]

>>15313006
>try to control precisely where it's differentiable and how many times, but I'm not sure I understand what the general formula would look like.
Read the stackexchange answer, it is quite easy if you're doing it at finitely many points. Work through the steps if you're unsure

[Anonymous No.15313027]

>>15313022
It's the inverse operation that breaks my brain a bit.
I'm not sure how I could have a closed group, if I start dividing by powers of (x-n)
Or any kind of nice structure, really

[Anonymous No.15313030]

>>15313006
Also why composition?

[Anonymous No.15313038]

>>15313030
Suboptimal word choice on my part.

[Anonymous No.15313039]

>>15313027
Sorry I thought you're just looking for a function continuous everywhere but differentiable at exactly n points.

The answer would be multiply by (x-x_1)^2 (x-x_2)^2 ... (x-x_n)^2

[Anonymous No.15313153]

if zfc was inconsistent, could it be proven that it was inconsistent.

[Anonymous No.15313270]

why do professors insist on throwing contradiction proofs at everything, even when there are perfectly fine direct proofs of a theorem?
is this supposed to be of some kind of pedagogical value or do they not know any better themselves?
pic related

[Anonymous No.15313337]

>>15313153
Yes. Come on though, it won't happen, and even if it did we could move on somewhat easily

[Anonymous No.15313589]

Anyone here working outside of academia?
The recent lay-offs make me think that the tech industry is going down.

[Anonymous No.15313618]

>>15313589
I'm in tech. Not a mathematician exactly, but I do cryptography de facto, so I'm currently learning about polynomial rings in an attempt to have the slightest clue.
Tech is mostly laying off people to counteract the insane hiring spree of 2019.

[Anonymous No.15313630]

>>15313270
The are guiding their students away from constructive proofs that take more time to grade

[Anonymous No.15313818]

>>15313589
Looks like this is already true when you are granted
[math]{\mathcal P}(A\cup B)\subseteq {\mathcal P}(A)\cup{\mathcal P}(B)[/math]
right?

Through
[math]\forall (a\in A). \forall (b\in B). \big( \{a,b\} \subset A\lor \{a,b\} \subset B \big) [/math].

Now I agree that a direct prove is nicer than one by contradiction, but in this case I can play devils advocate and say that the direct proof is just uncluttering the notation, while the prove by contradiction makes one ponder about separation axioms.

>>15313153
The way you asked the question is a bit tautological. If being inconsistent means that the theory is capable of proving an inconsistency (say 0=1), then to "prove" it inconsistent we just have to show this inconsistency.
Consider a scenario in which it was both inconsistent and not demonstrably inconsistent. If it's not demonstrably inconsistent, then it is not capable of proving an inconsistency, wouldn't it? Or are you talking about practicality?

What's more, since ZFC (or even HA) is capable of encoding its own prove calculus, if we have established an inconsistency of ZFC, then we can also immediatenly give a proof of the inconsistency of ZFC via ZFC itself (by explosion).

[Anonymous No.15313821]

>>15313270
>>15313818

[Anonymous No.15313885]

>>15313818
Mhm, to finish this off with my two-point strategy there, I think one must assume

[math]A=B\ \lor\ \exists(a\in A). a\notin B\ \lor\ \exists(b\in B). b\notin A[/math]

to be decidable.
I might be wrong tho.

[Anonymous No.15313910]

(which holds assuming [math]A=B[/math] is decidable and assuming [math]\neg(\forall a.P\land \forall b.Q)[/math] implies [math](\exists a.\neg P\lor \exists b.\neg Q)[/math])

[Anonymous No.15313931]

Can anyone learn mathematics?

I'm British and did it at school up to GCSE level, where I got a B in my exams.
For Americans, that's like getting a B at 10th Grade / Sophomore level math.
That was about a decade ago and I didn't study outside of school and my school was a shithole where you didn't really get to study while in class either.

Is it possible for me to teach myself now, using books and youtube etc, to actually get gud and learn to a significantly more advanced level? Or is it a case of there being a hard limit that people are just too dumb to learn above?
Is there any one book you'd recommend as a starting point?

[Anonymous No.15313945]

>>15313818
Is it possible for a consistent theory to prove another consistent theory is inconsistent?

[Anonymous No.15313962]

>>15313931
People who taught you were idiots, you have it pick it up mostly by yourself, university might help but not sure with the current state of it

[Anonymous No.15314007]

>>15313945
PA+notConsistent(PA) vs PA
under ZFC both these are consistent theories but you would agree the first "proves the other inconsistent" (but it's important to realize that you are translating English to math in a certain way when you say that)

[Anonymous No.15314029]

>>15313945
was about to write this, but it's essentially the same as above me

[Anonymous No.15314030]

>>15313945
>prove another consistent theory is inconsistent?
wouldn't this necessarily make the latter theory inconsistent, not consistent, to begin with??

[Anonymous No.15314048]

>>15313270
>>15313910
[math]A \cup B \subseteq A \cup B[/math], so [math]A \cup B \in P(A \cup B)\subseteq PA\cup PB[/math]. In the [math]A\cup B\in PA[/math] case we get [math]B\subseteq A\cup B\subseteq A[/math]. Similarly the other case gives [math]A\subseteq B[/math].

[Anonymous No.15314123]

>>15314030
Roughly, he inconsistency statement that's added (in an ad hoc way) is logically patched together with formalizations of proof, and it involves some existence claims that can't really be unpacked to get an actual proof in the other theory.

>>15314048
Looks reasonable.
The P(A u B)=(P(A) u P(B)) is quite the ruse, given we only use (A u B) in (P(A) u P(B)).
Maybe the goal was the lemmas we made along the way.

[Anonymous No.15314498]

Cutting fuel injection hose...can I just use a Dremel with a wheel ? Is an angle grinder with a cutting wheel better ? I want a smooth cut as much as I can do. It's just going to go over a quick fitting but still. Don't want to fuck up fuel.

[Anonymous No.15314532]

>>15314123
>involves some existence claims that can't really be unpacked to get an actual proof in the other theory.
ah, thanks, really appreciate it

[Anonymous No.15314551]

>>15314532
In PA+notConsistent(PA) vs PA
Lets say we r in ZFC looking at these 2
"provability" of shit in these just means a list of derivations
By completeness u can make models for both
the model of PA+notConsistent(PA) has a "number" x which is a "valid proof" of number y (godel number of "notConsistent(PA)" wrt ur encoding)
So models truth interpretation says notConsistent(PA) true
But ZFC doesnt believe this
Cuz x isnt a "standard number" ie not in the ordinal omega which is the "standard model of PA" in ZFC
So x is some object in some large shit model that has the standard model as a submodel but x is outside
So u cant turn x into a proof in ZFC that PA actuallys inconsistent, that ZFC believes
If x was in omega u could but u cant
Also btw u know PA+notConsistent(PA) is consistent (to ZFCs knowledge) by incompleteness+bcs ZFC can prove PA consistent

[Anonymous No.15314556]

>>15314551
this is way beyond my level of set theory or proof comprehension, but again I def take your word for it.

[Anonymous No.15314571]

>>15314556
https://www.logicmatters.net/tyl/
chapters 1-3,5-7, and if u want 9
not how i learned logic but ok guide

[Anonymous No.15314661]

>>15312006
Try taking differences of position vectors to get vectors whose norms are distances between agents.

It's good to note that the norm is "useless" here because the origin doesn't mean anything to you, but the position vectors are just as arbitrary.

[Anonymous No.15314796]

>>15310603
alright i 100% real side str8 up on god gots this fr fr this time no cap......... 766

[Anonymous No.15314893]

>>15314796
Anooon, you're gonna make me cry ;C
It's still wrong! It's not the right answer. It's very close though. You could tell me what you're doing, maybe I could point it out. Or I can post the answer as well, I think you're close enough it could be counted as correct.

[Anonymous No.15314898]

test post.

[Anonymous No.15314918]

>>15309341
150+324+375-6-75-3+3 = 768

[Anonymous No.15314979]

>>15314571
thank you for this mate

[Anonymous No.15315007]

>>15314918
And here it comes, the correct answer!
Nice job anon! Textbook solution, really concise... maybe a bit too concise? But I get what you did! So great job!
Thank you very much for your time and effort solving this problem. I hope it was fun for you. What did you think of it btw?

[Anonymous No.15315410]

>>15300294
Hi /mg/. I am a moron (and a CS student). I've heard people refer to Set Theory as "the language of mathematics" or as "the foundation of mathematics" in much the same was a lambda calculus is for computing.
My questions is, is it possible to represent based arithmetic (addition, subtraction, multiplication, division, exponentiation, root, and logarithm) as set operations? I genuinely don't know if what I'm asking is possible, but I can see that you could for example represent addition with just union, if you use the lengths of the two sets as the two numbers being added and the length of the result as the sum.
Also, recommend books on set theory (preferably introductory).

[Anonymous No.15315434]

>>15315410
Everything defined in mathematics under the ZF-C set theory is a set, including arithmetic operations. So excluding, the fundamental axioms, logical symbols, etc., it's all sets.
>moron (and a CS student)
In set theory, this is called a tautology

[Anonymous No.15315441]

>>15315410
>I've heard people refer to Set Theory as "the language of mathematics" or as "the foundation of mathematics"
https://www.youtube.com/watch?v=U75S_ZvnWNk

[Anonymous No.15315524]

>>15315410
No one used or needed set theory for most of the history of mathematics. Thankfully logicians and set theorists are a dying breed of kike phonies that tried to set themselves up as creating the "foundation" of mathematics after everyone else already did the hard work.
So no, you don't need set theory at all for real math, its just superflous.

[Anonymous No.15315557]

Man, modern algebra is such a stupid fucking subject. The simplest concepts and topics possible made as convoluted and "beautiful" as possible. Pure mathematicians are such worthless faggots. Who focuses in this field? For what purpose? Why can't analysis Chads stop winning?

[Anonymous No.15315569]

>>15315557
This post was typed by hands that eat corn in spirals
t. corn row gang

[Anonymous No.15315579]

>>15315557
I used to feel that way too, but then someone, a real rockstar, told me that "Nuclear spaces enter representation theory due to Schwartz kernel theorem."

[Anonymous No.15315728]

If you just need to make statements restricted to integers, do you prefer the notation:

x is element of [a, b)
or
a =< x < b

I am not sure what to prefer. Using inequalities to denote ranges is extremely common and often more clear as well; on the other hand, it might give a false impression what is being communicated. (This is only important for the you in 6 months who doesn't want to waste brainpower trying to logically understand what is being said, so you can just skim a page of your notes, etc.)
Personally, semantically, I prefer to keep inequalities comparisons.

On the other hand, I know it's not part of their definition, but an interval to me just screams to communicate a dense order like the reals. The inequality notation has the advantage that it puts more emphasis on the "haram" end points, rather than the interval set being the important thing to be considered. We are not dealing with real analysis here, after all, but just toddler compsci shenanigans.

[Anonymous No.15315736]

>>15315728
...for [math]x\in\mathbb{Z}[/math], such that [math]a\leq{x}<b[/math]

[Anonymous No.15315773]

>>15315007
straightforward exercise in PIE

[Anonymous No.15315794]

>>15315410
>is it possible to represent based arithmetic (...) set operations?

First of all, formal logic itself is enough to provide a framework for simple stuff like basic arithmetic, it's called formal number theory. You can even do shit like group theory with classical formal first or second order logic alone.

On the other hand, as for your set-theoretic related question, yes. It is a bit convoluted though but that's obviously going to happen. I'll be assuming ZF axioms (not all of them are necessary though!) and standard notation. So you start with the empty set [math]\varnothing[/math] and go like this

The idea is the following
[math]0 := \varnothing [/math]
[math]1 := \{\varnothing\} = \{0\}[/math]
[math]2 := 1 \cup \{1\} = \{\varnothing, \{\varnothing\}\} = \{0, 1\}[/math]
[math]3 := 2 \cup \{2\} = \{0, 1, 2\} [/math]
etc. Think for a while about why this makes sense.

The Axiom of Infinity assures a set that contains this "sequence" exists, and some other axioms assure you can properly define the set of EXACTLY these "special sets" we will call "natural numbers. We denote the set of natural numbers [math]\mathbb{N} [/math].

The proper definition of a (set-theoretic, obviously) function can be very convoluted so let's just take that for granted. Since [math]\mathbb{N}[\math] is a well-defined set there's no problem in defining the following function
[math]s: \mathbb{N} \to \mathbb{N}[/math]
[math]s(n) = n \cup \{n\}[/math]

We now define addition for all natural numbers n and m in the following manner.
[math]+ : \mathbb{N} \times \mathbb{N} \to \mathbb{N}[/math]
[math]n + 0 = n[/math]
[math]n + s(m) = s(n+m)[/math]

The pair [math](\mathbb{N}, +)[/math] satisfies all expected rules of addition. If you still find this construction entertaining after all this shit then I recommend you to read Enderton's Elements of Set Theory.

[Anonymous No.15315817]

>>15315736
>[math]x \in \mathbb{Z}[/math]
You mean [math]x \in \mathbb{R}[/math], right?

[Anonymous No.15315825]

>>15315817
Never it's only integers, I'm fucking blind.

So to >>15315728, neither really communicates that x is restricted to integers. You need [math]x \in \mathbb{Z}[/math] for that. And if it's being restricted to integers, [a, b-1] or a ≤ x ≤ (b-1) would give a clearer boundary.

[Anonymous No.15315833]

>>15315557
>modern algebra
Literally what? The last algebra theorem was that you can compute a Grobner basis.

[Anonymous No.15315842]

>>15315825
>in the integers you have to say [math] x\leq b-1[/math] instead of [math]x <b[/math]

[Anonymous No.15315845]

>is it possible to represent based arithmetic (...) set operations?

First of all, formal logic itself is enough to provide a framework for simple stuff like basic arithmetic, it's called formal number theory. You can even do shit like group theory with classical formal first or second order logic alone.On the other hand, as for your set-theoretic related question, yes. It is a bit convoluted though but that's obviously going to happen. I'll be assuming ZF axioms (not all of them are necessary though!) and standard notation. So you start with the empty set O and go like this

The idea is the following
0:=O
1:={O}={0}
2:=1∪{1}={O,{O}}={0,1}
3:=2∪{2}={0,1,2}
etc. Think for a while about why this makes sense.

The Axiom of Infinity assures a set that contains this "sequence" exists, and some other axioms assure you can properly define the set of EXACTLY these "special sets" we will call "natural numbers. We denote the set of natural numbers with the symbol [math]\mathbb{N}[\math].

The proper definition of a (set-theoretic, obviously) function can be very convoluted so let's just take that for granted. Since [math]\mathbb{N}[\math] is a well defined set there's no problem in defining the following function

[math]s: \mathbb{N} \to \mathbb{N}[/math]
[math]s(n) = n \cup \{n\}[/math]

We now define addition for all natural numbers n and m in the following manner.
+:N×NN
n+0=n
n+s(m)=s(n+m)

The pair (N,+) satisfies all expected rules of addition. If you still find this construction entertaining after all this shit then I recommend you to read Enderton's Elements of Set Theory.

[Anonymous No.15315855]

>>15315845
>Think for a while about why this makes sense.
Shrug. There are many of ways to do it. Maybe [math]n=\{0,\ldots,n-1\}[/math] is especially appealing to you, I dunno. It has its charms. But the reason it is popular is that you can define addition and multiplication using it, which is outside the scope of just “think for a while” imo.

[Anonymous No.15315864]

>>15315855
No it's popular because of ordinals/transitive sets being nice

[Anonymous No.15315871]

>>15315842
It's way easier to see 10 ≤ x ≤ 19 than 10 ≤ x < 20.

[Anonymous No.15315881]

>>15315864
ordinals predate von neumann numerals by several decades
>>15315871
ok retard

[Anonymous No.15315885]

>>15315871
Your variant is much less generalizable. It wasn't stipulated that the endpoint necessarily has to be an integer itself.
It's perfectly common to talk about the integers between e.g. 3 and 6.7. Your solution would give a wrong result.

[Anonymous No.15316100]

do you think theirs some deeper understanding to be gained of the nature of axioms
there must be some meaning
some reason why the world works the way it does
something more than just assumptions
some higher truth

[Anonymous No.15316115]

>>15315410
simply put: every concrete mathematical thing is a specific set, every abstract concept is set-theoretic.

[Anonymous No.15316139]

>>15316100
Try setting the axioms a different way and seeing what happens.
You might even find something that's broadly incompatible a physics compatible with life. Which by anthropic principle gives you a very unsatisfying way to say "that's just the way it is". But you can't really argue with it.

[Anonymous No.15316364]

>>15316100
I think that comes down to "why are there things that remain distinguishable even as time passes" and "should and can the modus ponens have a justification"? pic related

>>15315410
To represent and compute the addition of two numbers, you need little. Having defined equality of sets and the notion of intersection, you only need the operation mapping two sets x and y to a set x u {y}.
This is more modest than a set of all numbers in which such operations are validated. The latter is necessary, but not sufficient to have a model of arithmetic: While it gives you a means of adding any two natural numbers, it doesn't mean you have represented addition (+ in the signature of the theory) as a function in the theory, which is technically what you ask. For this you need some form of Replacement and the Recursion theorem.
https://en.wikipedia.org/wiki/Recursion#The_recursion_theorem
The recursion theorem is proven by mathematical induction, which classically follows from full Separation (but also from the existence of all countable function spaces and union.) The last few comments are just very technical details.
You can ask me about the log on the reals, but whatever by now

[Anonymous No.15316402]

How are parametric equations turned into regular equations?

[Anonymous No.15316759]

>>15316402
If x and y are each functions of u and v, then if take any function f(x,y) and write
f(x, y) = f(x(u, v), y(u, v)), where on the right hand you resolve x and y in favour of u and v, then you got an equation.
In your case, it's just an "accident" that the right hand side is independent of t.
If you would replace sin(t) with sin(t^2), it wouldn't happen.

[Anonymous No.15316790]

Book recommendations for CAD programming/maths plz

[Anonymous No.15317042]

Is there a field that studies (the syntax side) of logic abstractly just based on its language and a consequence relation [math]\vdash[/math]. For example, stuff like "a maximally consistent theory is complete" can be proven just using the abstract properties of [math]\vdash[/math] such as monotonicity etc (using suitable definitions of "consistent" and "complete" that don't rely on the existence of specific connectives)

[Anonymous No.15317130]

>>15317042
probably proof theory

[Anonymous No.15317622]

I'm a mecheng who wants to get better at theoretical math just for fun, maybe to improve my visual thinking and help myself better program something to visualize stuff at work for mechanical systems.

I have a limited (several years out of uni, so I'm rusty) understanding of theory up until PDEs, multivariate calculus and linear, abstract and tensor algebra, as well as numerical analysis and continuum mechanics.

Does anyone have a infograph of what fields to study to understand different aspects of mathematics?

[Anonymous No.15317626]

>>15317042
I'm not sure what you contrast this too. If you do logic syntatically, then any interpretation is just a bonus on top. That bonus motivates what theorems are worth studying, but don't really affect the proofs.
Anyway, there's
https://en.wikipedia.org/wiki/Abstract_rewriting_system
but I wouldn't say it's about studying chains of symbols that we'd read as logical deductions. But again I think logic itself already does what you ask for. I can tell you at least 20 logics that people studied in depth.

[Anonymous No.15317830]

They mean [math]C^0(\Omega)[/math] with the supremum norm, correct?

[Anonymous No.15317861]

>>15317042
you may find relevant stuff on a Metalogic book

[Anonymous No.15317991]

>>15315569
I actually do eat food in spirals. Even as a small child I ate my yogurt from a cup in a spiral strip mine fashion. Interesting meme, is there more on this?

[Anonymous No.15318011]

>>15317622
evan chen napkin has a diagram which covers an okayish portion of math
do not recommend the book
>>15317991
corn in spirals = normal people and analysts
corn in rows = freaks and algebraists

[Anonymous No.15318210]

>>15318011
yeah, I also really dislike Chen's approach

[Anonymous No.15318227]

>my math degree doesn't require analysis

[Anonymous No.15318242]

How do I go about constructing an analytic function that is everywhere continuous, nowhere differentiable, and is strictly increasing?

[Kari Salgado No.15318258]

>>15318242
>analytic function that is everywhere continuous, nowhere differentiable
I'm not sure you know what analytic means.<div class="xa23b"><span class="xa23t"></span><span class="xa23i"></span></div>

[Anonymous No.15318260]

>>15318242
You don't. A monotonic function is differentiable almost everywhere.

[Anonymous No.15318270]

>>15318011
>>15318210
I'm >>15317622, what is wrong with his book?

So, you suggest following his diagram of what to study but you think his way of explaining the terms or how he deals with the material is wrong? Or too shallow?

[Anonymous No.15318292]

>>15318258
>>15318260
Oh, never mind then.

[Anonymous No.15318301]

3blue1brown has admitted that his videos are popmath and not educational.
https://youtu.be/UOuxo6SA8Uc?t=201

[Anonymous No.15318440]

>>15318301
>mathematician: rigor
:|

>Mathematician: intentionally abstruse gatekeeping
:o

[Anonymous No.15318457]

>>15318440
mathematicians don't care about rigour
if they did, they would understand foundations and not outsource it to philosophy

[Anonymous No.15318459]

>>15318270
Read chapter 0, has the diagram and explanations of each subject included
His approach is kind of a narrow "get to the point where I can define the relevant objects", which is more obvious in the advanced chapters (where there are also lots of typos)
So, less rigor and less comprehensive reach/context outside what is going to be addressed literally in the text
Not to mention he's missing important stuff like (general) commutative algebra and Lie Theory and (proper) Fourier analysis and functional analysis and PDE theory (distributions, tempered distributions, elliptic PDEs, Sobolev spaces)
It's more of a primer, if you're confused learning something it can serve as reference

[Anonymous No.15318474]

Can I start Algebra with Aluffi?

[Anonymous No.15318488]

do automata theory

[Anonymous No.15318502]

>>15318474
Dummit and Foote

[Anonymous No.15318506]

>>15318502
Why?

[Anonymous No.15318553]

>>15318502
I fucking HATE this retarded nigger board and big headed fucks that recommend graduate and meme textbooks for every case. Guy is asking "can I start algebra", so no, that is a terrible choice. He should go with either Saracino or Herstein.

[Anonymous No.15318586]

>>15318553
Get through Dummit and Foote and you are set. You will actually learn something.
If you want "easy" or "inclusive", go to Reddit.

[Anonymous No.15318706]

Has anyone tried to do functional analysis in soloway's set theory? Just curious about how that would look like.

[Anonymous No.15318775]

let's assume that the kinds of things we want to be true, actually are true
is there a bigger scam in modern mathematics than axioms, the basis for the foundation of modern mathematics itself

[Anonymous No.15318828]

1,1,2,1,2,3,1,1,2,3,4,2,1,5,7,5,6,5,1,1,4,7,3,3,7,5,6,7,8

──────────────────────────────────────────────────────────
*23571113171923293137414347
──────────────────────────────────────────────────────────
123571113171923293137414347
24610142226343846586274828694
369152133395157698793111123129141
481220284452687692116124148164172188
51015253555658595115145155185205215235
6121830426678102114138174186222246258282
7142135497791119133161203217259287301329
81624405688104136152184232248296328344376
91827456399117153171207261279333369387423
1020305070110130170190230290310370410430470
1122335577121143187209253319341407451473517
1224366084132156204228276348372444492516564
1326396591143169221247299377403481533559611
1428427098154182238266322406434518574602658
15304575105165195255285345435465555615645705
──────────────────────────────────────────────────────────
HI /mg/ have you seen the above sequence before? Starting at 2 it's the shortest path to each higher number in the multiplication table of primes and naturals.
I haven't seen this on oeis (but some fairly long subsequences do bring up results.

[Anonymous No.15318834]

>>15318828
Here's a cleaner looking table.

[Anonymous No.15318896]

>>15318828
Can't say I've seen it. You can try to make the
contribution to OEIS and see if it's good to post
there.

[Anonymous No.15318943]

>>15318834
Follow up I've looked at shortest, longest paths with and without diagonals, number of each shortest and longest paths and still didn't find anything on oeis, seems odd for such a seemingly basic object to be ignored, seems like (((they))) don't want me to know somthing.

[Garry Reed No.15318963]

How would math be different if you added another even prime number lets say $ that isn't ordered.
so 3*$ is an even number divisible by 3 and $, but not by 2.
Would there be more even numbers than odd? or would it still be half-and-half
Here even means 2n+1 or $n+1<div class="xa23b"><span class="xa23t"></span><span class="xa23i"></span></div>

[Larry Sneed No.15318980]

>>15318963
What a fucking stupid question. This is no different from just adding a whole different set of natural numbers with different characters so it wouldn't change anything.

[Anonymous No.15319142]

I’m trying to show there doesn’t exist a C > 0 such that for all compactly supported test functions, one has sup |Hess(u)| \leq C sup|laplace(u)|. My idea was to take u to be the convolution of the Greens function |x|^(2-n) with a mollified version of the indicator function of a cube. Anyone think this can work ?

[Anonymous No.15319153]

>>15318301
You probably shouldn't have linked the video. It makes it really easy to listen to the next 10 seconds where he says something different.

[Anonymous No.15319163]

>>15318706
refs in
https://arxiv.org/pdf/2010.15632v1.pdf

[Anonymous No.15319174]

>>15319163

[Anonymous No.15319200]

Hello, where can I learn about how order theory and category theory can be used to model induction and recursion (and their duals)? Some keywords from random papers and lecture notes I've come across: least pre-fixed points and greatest post-fixed points, closure operators, initial algebras and final coalgebras etc

[Anonymous No.15319365]

>>15318963
this makes no sense. evens other than 2 are not prime, and 3 times even is always even, regardless of the model of PA/whatever
this is an intrinsic consequence of the definitions of even and prime, you have to change lots of defs from the ground up to get to that point

[Anonymous No.15319601]

>>15318553
Dummit and Foote is an undergraduate textbook though?

[Anonymous No.15319627]

>>15318775
>actually are true
no such thing

[Anonymous No.15319863]

>>15319200
Ask GPT-4 about it, it's better at explaining that any single person here. And you can ask follow up questions.
Just mind the fact that it's confidently wrong sometimes, very much like people here.

[Anonymous No.15319887]

if you write a program.
>take 1000 partitions
>randomly chose one of the 1000 to fill that spot
>keep doing that until you detect a collision
>aka how many times does it take until you get a second hit
>if you keep doing it over and over then average all the tries
>it appears to converge to one over phi
>or 618
>i dont know what that implies

[Anonymous No.15319890]

>>15319887
https://en.wikipedia.org/wiki/Probability_density_function#Absolutely_continuous_univariate_distributions

[Anonymous No.15319905]

How do I find [math]f'(x)[/math] if [eqn]f(x) = \int_a^b g(x,y)~dy[/eqn]

[Anonymous No.15319936]

>>15319905
Exactly how you find derivatives.
Calculate
[eqn]f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} [/eqn]
If [math]g \in C^1(\mathbb{R} \times [a,b])[/math] you can just pull the limit inside the integral.

[eqn]f'(x) = \int_a^b \frac{\partial}{\partial x} g(x,y) dy[/eqn]

[Anonymous No.15320221]

The more math I study the more I hate it. Math was a mistake.

[Anonymous No.15320334]

>>15319887
That image looks like it should be the front cover of a top
notch, /mg/ approved, calculus (or other math topic) textbook.

[Anonymous No.15320335]

>getting C's in abstract algebra homework

[Anonymous No.15320392]

>>15320221
how

[Anonymous No.15320432]

How do I solve [math]x^{1/3} = \ln{x}[/math] and [math]x^{1/e} = \ln{x}[/math]? IIRC the first equation has two solutions and the second equation has one solution.

[Anonymous No.15320470]

>>15320432
I'd use Mathematica/Wolfram before asking here

[Anonymous No.15320494]

>>15320470
I did. Every site I tried I have to pay to see the solution.

[Anonymous No.15320499]

>>15320470
>>15320494
I mean the working to obtain the solution.

[Anonymous No.15320551]

>>15320432
Lambert W

[Anonymous No.15320555]

>>15320432
for the second
e^t>=t+1 for all t with equality only at t=0
plug in t = (log x)/e-1
x=e follows

for the first
u = (log x)/3-log3
e^u=u+log 3
in the previous solution t+1 was tangent to e^t (at t=0), which is convex
this line is shifted up by (log 3)-1
so two solutions, one with u>0 and one with u<0

[Anonymous No.15320580]

>>15320551
>Lambert W
God I hate those.

>>15320555
Thanks anon.

[Anonymous No.15320585]

>>15320551
For the first
let x = e^(3*y).
you get e^y = 3*y
rearrange to get -y*e^(-y) = -1/3
-y = W(-1/3)
x = e^(-3*W(-1/3)).
There are infinitely many solutions (one for each branch of W).

For the second use the same process to get x = e^(-e*W(-1/e))

[Anonymous No.15320593]

Speaking of Lambert W, do I need it to solve the intersection between the line and curve obtained from plotting y^x = x^y?

[Anonymous No.15320611]

>>15320392
The people that study it and "teach" it are just awful people, more concerned with looking smart and "flexing" on students than anything else. I'll just switch majors to something filled with nice and practical people. I don't want to spend the rest of my life around anyone that studied math or physics.

[Anonymous No.15320614]

>axiom schema of specification
is this really the best people can do

[Anonymous No.15320658]

>>15320614
you can use NBG set theory if it makes you feel better but it's a conservative extension of ZFC
or you could use Quine's NF but it's not really popular

[Anonymous No.15320713]

>define A as the proof to the statement

The absolute fucking state.

[Anonymous No.15320838]

>>15320614
It's too permissive.
Go with
https://en.m.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory

[Anonymous No.15322633]

[math]\forall (x\in S). x\in S\implies f(x)\subset S[/math]

Give a name for this property of some fixed [math]f\colon X\to X[/math] and [math]S\subset X[/math].

[Anonymous No.15322651]

>>15322633
S is an invariant subset under f.

[Anonymous No.15322666]

>>15322651
Not sure if 'invariant' is exactly the right word, but I like the general thinking.
I would have probably tried to characterize f with a name, but speaking of S is an idea.

[Anonymous No.15322701]

>>15320838
>pronounced /ˈkrJpki ˈplɑːtɛk/,
Well obviously
I mean that's just the normal way to say it

[Anonymous No.15322966]

>"uh no, don't rely on your textbook. My course notes are your bible, the internationally acclaimed textbook which is a classic standard is not suitable for this course, unlike my mid tier university slop filled with errors that hasn't been fixed in 6 years."

Memes for this feel?

[Anonymous No.15323044]

I hate lean
I hate lean
I hate lean

I spend more time trying to figure out how to do trivial shit than trying to actually solve problems.

[Anonymous No.15323138]

>>15323044
Use mathlib and the automation tactics to make trivial things less trivial. Set.subset_inter_iff should make that proof pretty quick.

[Anonymous No.15323142]

>>15323138
make trivial things *more trivial.

[Anonymous No.15323208]

>>15323142
No, it's more things like me not being able to understand tactics . For example, the "assume' tactic is supposed to add a hypothesis without needing a proof term, but when I try to add an assumption it tells me it failed because it wasn't a conditional statement. I have no idea what that means as other theorems use "assume" on statements which are not conditional.

[Anonymous No.15323218]

>>15323208
I can't tell if you're on Lean3 or Lean4 but assume is not a standard tactic in Lean4. I'd recommend updating if you haven't. If I recall correctly, assume works the same as intro where if your goal is a conditional, then you can make the antecedent of the conditional into one of your assumptions. If your goal is P -> Q, then 'assume/intro h' will give you a hypothesis h : P and make your goal Q. If you use 'assume' on something that isn't a conditional, then that's probably because it is actually a conditional just it's not immediately obviously written out as one.

[Anonymous No.15323221]

>>15323218
what tactic lets me just add an assumption without having to prove it?

[Anonymous No.15323225]

>>15323221
have h : t := sorry will give you h : t automatically proven. You'll obviously have to go back and fill in 'sorry' with a proof. You can also use tactics like library_search to see if what you're looking for is already in mathlib.

[Anonymous No.15323232]

I hate this homework. What a god awful course. I don't have infinite time, I just want to learn what I have to learn and get the fuck out to the next course.

[Anonymous No.15323242]

>>15323232
Tried talking GPT-4 through it already?
It probably won't get the proof right on the first try, but you can keep talking to it to explain things to you and get ideas, even if it doesn't have the answer immediately

You should probably try harder yourself, but whatever.

[Anonymous No.15323248]

>>15323242
GPT-4 is clownishly bad.
I was lazy and tried asking it to check various matrices that were subsets of a general linear group and verify if they were subgroups, and even the definitions used were wrong.
Its like a shitty calculator, by the time you finish typing stuff in, you already solved it in your head anyways. Except with GTP-4, it gives you a wrong answer. I would say its only useful if you need to ask it definitions.

[Anonymous No.15323250]

>>15323248
need to ask it specific* definitions.
Anything that requires nested thinking will be incorrect.

[Anonymous No.15323255]

>>15323248
Yeah, I mostly use it to explain definitions in a nicer way than Wikipedia and low-quality books
It's regularly wrong, but in my experience you can tell when it's wrong, because it just contradicts itself in dumb ways or stays obviously stupid shit

There's tricks like "let's take this step by step" and "you are a renowned, expert mathematician" to make it simulate better.

It's bad. But somehow I'm still a lot more productive with it than without, so

[Anonymous No.15323276]

>cant rotate dihedral groups in his head as 3D shapes
>resorts to algebraic manipulation

What are some other indicators of low IQ?

[Anonymous No.15323443]

Just to be completely sure, the arrow notation here just means picking an element from the R_q randomly, right?

[Anonymous No.15323453]

>>15323443
If it's standard CS notation, it means that the variable a is set to the value R_q, where the latter is presumably a randomly sampled value

[Anonymous No.15323466]

>>15323453
Should've probably mentioned that R_q is a ring (of some sort).

[Anonymous No.15323645]

what conditions need to be fulfilled by two ordered sets so that one can always turn functions between them into monotone functions

[Anonymous No.15323995]

>>15323248
I agree with this. It's advanced enough for
rubberducking purposes, let alone being a parrot.

[Anonymous No.15324152]

Reading through Apostol Vol I. right now; how the fuck does this simplification work?

[Anonymous No.15324462]

>>15317622
>Does anyone have a infograph of what fields to study to understand different aspects of mathematics?

[Anonymous No.15324715]

>>15324152
first equality: by definition probably
second equality: combine the sums into one and simplify
[math]f(x_k)(x_k-x_{k-1})-f(x_{k-1})(x_k-x_{k-1})=(x_k-x_{k-1})(f(x_k)-f(x_{k-1}))=\frac{b-a}{n}(f(x_k)-f(x_{k-1})[/math]
third equality: if you need this you're terminal

[Anonymous No.15324723]

>>15324462
>Optional but Recommended
Only Problem Solving and Introduction to Statistics are of any interest in purple, cut out the philosophical bullshit. Start with mathematical logic or set theory for that stuff.

[Anonymous No.15324999]

New thread!
>>15324994

[Anonymous No.15325021]

>>15324462
lots of great books missing and many of these introductory ones are far too elementary.

[Anonymous No.15325302]

>>15324462
>introduction to
>a first course in
All these books should be burned and their authors hung drawn and quartered. A book should teach a subject from first principles, build upon them sucessively by chapter, and ultimately be definitive.