/mg/ - mathematics general

[Anonymous No.15420227]

/mg/ - mathematics general
Previously >>15404877

Petersen Graph edition
Talk math(s)!

[Anonymous No.15420242]

In the future, as a rule, math will be done in the Effective topos.

[Anonymous No.15420280]

>>15420242
Hi Nikolaj

[Anonymous No.15420290]

hey hey! New thread, new problem. This one is a classic. I think...I'm not sure actually. Feel free to attempt it or not, it's your choice!

I need to make a clarification because something in the problem confused our teacher. Monotonically increasing isn't strictly increasing. So we can have a function like f(1)=1 f(2)=1 f(3)=1.

As always, let me know if anything is unclear or you need a hint. All replies are welcome and appreciated even if the solution is wrong. I apologize in advance as I'll be quite busy these few days so my replies might be late. Either way, good luck to those who wish to attempt this.

[Anonymous No.15420309]

Is this the best book for number theory?

[Anonymous No.15420314]

>>15420290
btw I'm trans, so solve this instead

[Anonymous No.15420320]

>>15420314
There's like a lot of primes, and this one doesn't sound like it's going to have less primes as n increases, so per the vibes lemma, I feel like it's trivially infinite

[Anonymous No.15420354]

>>15420227
Can someone please help me find a formula for the nth element of each of these sequences? I will post the first several elements of each sequence below, and then explain how they are defined.

S_a: 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 75, etc
S_b: 3, 15, 27, 33, 45, 57, 63, 75, 87, 93, 105, 117, etc
S_c: 3, 33, 45, 63, 75, 87, 105, 123, 135, 147, etc
s_d: 3, 33, 63, 75, 105, 123, 135, 147, etc

The sequences are defined as follows:
S_a: All odd multiples of 3.
S_b: All members of S_a without a remainder of 1 or 4 when dividing by 5.
S_c: All members of S_b without a remainder of 1 or 6 when dividing by 7.
S_d: All members of S_c without a remainder of 1 or 10 when dividing by 11.

In general to get to the next sequence, you use all members of the previous sequence without a remainder of 1 or p-1, where p is the next prime number.

I think the formula for the nth element of a given sequence might involve some mod and floor functions, but I'm not really sure. There seems to be a repeating pattern every primorial-sized interval.

[Anonymous No.15420361]

>>15420314
Nice try. That's one of the four Landau problems - https://en.wikipedia.org/wiki/Landau%27s_problems
> As of 2023, all four problems are unresolved.

[Anonymous No.15420416]

>>15420354
Here is how to find them first S_a

The numbers are [math]3x[/math] where [math]x[/math] solves the congruence
[eqn]x \equiv 1 \text{ (mod 2)}[/eqn]
and is non-negative.
Clearly the congruence has the solutions [math]x = 2k - 1[/math] so the the n-term is [math]3(2n-1) = 6n - 3[/math].

Now for S_b you have to consider three cases
The numbers are [math]3x[/math] where [math]x[/math] solves the system of congruences
[eqn] x \equiv 1 \text{ (mod 2)} \\
3x \equiv 0 \text{ (mod 5)}[/eqn]
or
[eqn] x \equiv 1 \text{ (mod 2)} \\
3x \equiv 2 \text{ (mod 5)}[/eqn]
or
[eqn] x \equiv 1 \text{ (mod 2)} \\
3x \equiv 3 \text{ (mod 5)}[/eqn]

Each of those systems has a family of solutions where each solution is 10 apart because of the Chinese Reminder Theorem.
Cluing the solutions together you get that [math]x \in \{1,5,9\} + 10 \mathbb{Z}[/math] or [math]3x \in \{3,15,27\} + 30 \mathbb{Z}[/math]
So the n-th term is
[eqn] \begin{cases} 3 + 30 \frac{n-1}{3} & \text{for } n \equiv 1 \text{ (mod 3)} \\
15 + 30 \frac{n-2}{3} & \text{for } n \equiv 2 \text{ (mod 3)} \\
27 + 30 \frac{n-3}{3} & \text{for } n \equiv 0 \text{ (mod 3)}
\end{cases}
= \begin{cases} 10n - 7 & \text{for } n \equiv 1 \text{ (mod 3)} \\
10n - 5 & \text{for } n \equiv 2 \text{ (mod 3)} \\
10 n - 3 & \text{for } n \equiv 0 \text{ (mod 3)}
\end{cases}
[/eqn]

Now for S_c you have 15 cases for the remainders and for S_d you gave 135 cases for the remainders but you can do it with the same method.

[Anonymous No.15420423]

What mathematics do Engineers need? Also how do you guys approach gaps in your knowledge?

[Anonymous No.15420428]

>>15420423
Linear algebra, statistics, differential equations.
Find a toy problem and spend a few hours per weekend on it.

[Anonymous No.15420443]

>>15420309
I used this

[Anonymous No.15420457]

>>15420314
CAP THAT FUCKING MARKER!
That shit drives me nuts.

[Anonymous No.15420485]

I'm learning maths all over again and I want to know is there some kind of law or property that states why if you switch around the minuend and subtrahend in a subtraction problem, the difference is always the same in absolute value (althrough different in terms of the answer being positive or negative).

Pic related.

[Anonymous No.15420498]

>>15420227
All formalizers of arithmetic rely on Peano's axioms - it is their sacred cow. But Peano's axioms are not a sacred cow at all, they are just a variant of axiomatization. In fact, the first and fundamental operation in mathematics is ADDITION - it is the very root. All other operations are derived through it: multiplication is multiple addition; subtraction is the inverse operation of addition; exponentiation is multiple multiplication; root and logarithm are inverse operations of exponentiation. The entire natural number series is elementary constructed by adding one - everything is simple and straightforward. Then from the natural numbers and operations on them all other numbers are obtained.
But Peano invented the operation S (Immediately following) instead of the operation of adding with one, and through the operation S he defined both the operation of addition and all the others. But the operation S itself is not defined. What is "Immediately following"? One clear operation "addition" Peano replaced with another less clear operation S. So what's all this for? Allegedly the parcels will be minimal and it will be better. How is this better? It's just another version of formalization, of which you can think of many. Minimal assumptions don't mean better. The most important quality of a theory is not minimization, but comprehensibility. In the limit, there are only two minimal symbols: Zero and One. You can write ANY theory in binary, but no one will understand it without recoding it into human language. We are not communicating in binary code, but in natural language, which is deliberately redundant. But this redundancy gives both comprehensibility and reliability.
The symbol system in mathematics is very important and simplifies a lot. It should simplify, not complicate. But any formalization in science can be reduced to complete idiocy.

[Anonymous No.15420574]

>>15420428
Thanks man!

[Anonymous No.15420579]

>>15420485
Just move around the minuses and you can see why

a - b
= a + (-b)
= --a + --(-b)
= -(-a + --b)
= -(--b + -a)
= -(b - a)

a-b = -(b-a)
Which is to say, swapping a and b negates the result

[Anonymous No.15420585]

>>15420579
This post won't over my head man.

[Anonymous No.15420591]

>>15420585
+3 - +7 is the same as 3 + (-7)
You can switch around the two sides in additions, so it's the same as -7 + 3
And then +7 - +3 is 7 + -3.
That's the opposite of -7 + 3
-(7 + -3) is -7 + 3
it negates the thing
when you switch them around
it negates
it puts a minus on it

so the result gets negated too
idk my dude

[Anonymous No.15420597]

>>15420591
>+3 - +7 is the same as 3 + (-7)
>You can switch around the two sides in additions, so it's the same as -7 + 3
>And then +7 - +3 is 7 + -3.
Understood this.
>That's the opposite of -7 + 3
>-(7 + -3) is -7 + 3
>it negates the thing
>when you switch them around
>it negates
>it puts a minus on it
I didn't get this part.

Basically the first part of the post is something to do with commutative addition but the 2nd part I didn't get.

[Anonymous No.15420601]

>>15420597
Well if go back to your image, you have these two lines:
+7 - +3
+3 - +7

If you take the first line and you negate it, it gives you the second line.
That means the second line is the inverse of the first one. That's why the result also goes from +4 to -4.
When you switch the sides, it's the same as multiplying by -1

+7 - +3 = 7 - 3 = 7 + -3 = -3 + 7 = -(3 + -7) = -(+3 - +7)
So the first line is minus the second line, which is why it goes from +4 to -4

And same the other way around
+3 - +7 = 3 - 7 = +3 - 7 = -7 + 3 = -(7 + -3) = -(+7 - +3)

[Anonymous No.15420608]

>>15420601
>If you take the first line and you negate it, it gives you the second line.
What do you mean negate?

[Anonymous No.15420610]

>>15420608
(Z, +) is a group, so elements have an additive inverse, which I denote as negation.

[Anonymous No.15420612]

i learned my timestables today

[Anonymous No.15420614]

>>15420610
>When you switch the sides, it's the same as multiplying by -1
What do you mean by this?

[Anonymous No.15420615]

>>15420614
For a longer explanation, please refer to Russel, Whitehead, Principia Mathematica, Volume I, 1st edition

[Anonymous No.15420616]

>>15420498
The logic Peano was originally working in was second order (you can quantify over predicates, which you might also call sets of numbers). This theory is categorical (only one model up to isomorphism) and Sx always comes down to x+1. It's a unary operation, and not binary, and so gives nice concise axioms that should be just as natural to you as any.
In the first-order formulation (and first-order theories are the go to these days, as they have nice metalogical properties), you need both S and + (and ·) anyway, unless you do axiomatizations with like 20 non-logical axioms. The intended model is still x+1 where x is a standard number.

[Anonymous No.15420618]

>>15420615
Okay thanks.

[Anonymous No.15420617]

>>15420485
[math]
7 - 3 = 4\\
\text{is the same as}\\
-3 + 7 = 4\\
\text{is the same as}\\
-3 -- 7 = 4\\
[/math]

[math]
\text{Multiply by -1:}\\
-7 + 3 = -4\\
\text{is the same as}\\
-7 -- 3 = -4\\

[/math]

[Anonymous No.15420619]

>>15420618
>Okay
What did you mean by this?

[Anonymous No.15420622]

This one is unclear to me. So if S is a rectangle then it's basically Cartesian coordinates with the x axis being labeled r and the y axis being labeled [math]\theta[/math]?

[Anonymous No.15420624]

>>15420617
You lost me :(
>>15420619
I'll take a look at the textbook he suggested.

[Anonymous No.15420632]

What is a four set venn diagram where the 13 regions have surface areas as equal to each other as possible?

In other words, what is the four set venn diagram such that the ratio between the smallest and largest area is maximally close to one?

[Anonymous No.15420634]

Do /sci/fags really abandon the previous thread as soon as it reaches bump limit? That's kind of lame for such a slow board.

>>15420621
Here's my question in the previous thread.

[Anonymous No.15420643]

>>15420624
which part dont you understand? if you multiple both sides of an equation by the same value, the equation still holds

[Anonymous No.15420646]

>>15420632
all four sets are equal

[Anonymous No.15420660]

>>15420646
But what does the geometric shape looks like

[Anonymous No.15420668]

>>15420660
depends entirely on how you draw the regions to begin with

[Anonymous No.15420673]

How do I stop feeling insecure about being an applied mathematician and not a pure one? I was walking around the department and they called me a physicist and it ruined my day

[Anonymous No.15420676]

>>15420643
I don't really how your "is the same as" follow from my image which simply reversed the subtrahend and minuend. Yours is totally different.
Also, I don't get what you mean by multiplying by negative 1.

[Anonymous No.15420680]

>>15420673
Pure math is a fad.

[Anonymous No.15420681]

>>15420676
do you know algebra?
https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/10%3A_Solving_Equations_and_Inequalities/10.01%3A_Solving_Equations/10.1.01%3A_Solving_One-Step_Equations_Using_Properties_of_Equality

[Anonymous No.15420683]

>>15420673
You sigh, accept that not everyone can be part of the elite, and accept that you have brought this upon yourself.

[Anonymous No.15420691]

I saw this paper in a Quanta article: https://www.imo.universite-paris-saclay.fr/~jean.ecalle/fichiersweb/WEB_tour_resur.pdf
Have a hard time following. He's either the typical genius archetype that is not very good at communicating his ideas, or I dum.
On second though, possibly both.

But one of the things I wish was done more often is rewrite incomprehensible lunatic papers that contain truly beautiful ideas into something vaguely sensible for an average mathematician with relevant background.
If I had any functioning brain cells, I might enjoy doing that

[Anonymous No.15420693]

>>15420443
>Guy posts absolute beginner book
>No, use this advanced book
Totally different use cases here. You at least need to have algebra to read Ireland/Rosen

[Anonymous No.15420694]

>>15420673
lel
Well I'd say there is little glory in being a pure mathematician. It's just a luxury that some choose to afford - no need to judge it either way.

[Anonymous No.15420696]

>>15420673
>called me a physicist
That means you're not useless.

[Anonymous No.15420712]

>>15420622

The answer to a) is a sliced ring. F maps every point in the original rectangle (a region of space from r1 to r2, and theta1 to theta 2) onto this sliced ring.

The ring has an outer radius of r2 and inner radius of r1, but it doesn't extend a full 2pi radians all the way around the origin. It only goes from some arbitrary angle of theta1 to theta2.

The answer to b) is similar, but it just turns the
sliced ring into a regular sliced circle. Imagine going from a region of space that looks like a pizza crust, to going to a region of space that looks like a full slice of pizza.

The answer to part c) is just a ring that is sliced at right angles, in the first quadrant of the Cartesian plane only.

The answer to part d) is a complete ring (from 0 to 2pi), where r1 (the inner radius) is specifically not-zero.

The answer to part e) is a slice of some ring, whose outer radius is 3, and inner radius is 2 - sliced from the polar angles of pi/4 to 3pi/4.

This doesn't seem too hard.

[Anonymous No.15420749]

>>15420290

For (a), prove the general case for ordered sets of different sizes.

There are n elements in the given set S = {1,...,n}. I will assume the set is ordered, such that 1 < 2 < ... < n-1 < n.

Let there be some function f() that maps every element S to S.

For f() to be strictly "monotonic" (one-to-one) and "increasing," it implies that for every element in S, that f(1) < f(2) < ... < f(n-1) < f(n) holds true.

Suppose we are mapping {1,...,n} to {1,...,m}, where n <= m.

This means we are assigning each element from n-sized set {f(1), f(2), ..., f(n-1), f(n)} to exactly one element in m-sized set {1,...,m}.

Because they are ordered and increasing, then the only restriction on how to do the assigning is the size difference between the sets, using counting by simple combinatorics.

The number of ways to assign n things out of m choices is equal to (m Choose n) = m! / (n! (m-n!)).

If the set is mapping onto itself, then n = m.

Thus, the number of ways to do this is (n Choose n) = n! / (n! (0!)) = 1.

Q.E.D.

[Anonymous No.15420890]

I was about to buy a book an Universal Coalgebra by Denecke (pic rel) but large parts of the book are apparently just translated from German into English from a text by Gumm (without his permission).
Did any of the Germanons hear about this case? Apparently it was even covered by some news stations. https://www.mathematik.uni-marburg.de/~gumm/Plagiarism/
I don't understand German but even I can tell that some parts seem to be copy-pasted lol.
So yeah, anyone know a different book on universal coalgebra or should I still buy the plagiarism book (since I can't read the German original anyway)

(That's what I get for not taking German as a 3rd language in school)

[Anonymous No.15420967]

>>15420290
There is an interpretation of binomial coefficients as counting lattice paths.
(n C m) counts the number paths from (0,0) to (n-m, m) that use steps that are either up by 1 or right by 1.
An increasing f can be represented by such a path.
The x coordinate represents the f input minus 1.
The highest y coordinate for a given x represents the f value minus 1.

For part a) you want to count all paths that start at (0,0) and end at (n-1,0), (n-1,1), ... (n-1,n-1).
This is equal to (n-1 C 0) + (n C 1) + ... + (2n-2 C n-1)
which equals (n-1 C n-1) + (n C n-1) + ... + (2n-2 C n-1)
this sum can be calculated using a generating function [x^(n-1)] 1/(1-x)^n * 1/(1-x)
= (2n-1 C n)
A different interpretation given the answer is:
Suppose f(0) = 1 and f(n+1)=n. This doesn't change the constraints on f.
Picking n items out of a 2n-1 item list will partition the list into chunks of un-chosen items.
Two adjacent chosen items represent a chunk of size zero between them.
Each chunk represents the value of f(x)-f(x-1) >=0.
The sum of all chunks will be equal to f(n+1)-f(0) = n-1

For part b) we want to exclude paths that cross x=y.
To do this, we can re-use part a) but introduce "anti-matter" paths that cancel the paths that step on any (x,x+1).
These "anti-matter" paths will start at (-1,1) and follow the same up by 1, right by 1 rule but just have negative weight.
This gives the answer:
(n-1 C 0) + (n C 1) + ... + (2n-2 C n-1) - [(n C 0) + (n+1 C 1) + ... + (2n-2 C n-2)]
which equals
(n-1 C n-1) + (n C n-1) + ... + (2n-2 C n-1) - [(n C n) + (n+1 C n) + ... + (2n-2 C n)]
which equals
(2n-1 C n) - (2n+1 C n+1) + (2n-1 C n) + (2n C n)
= (2n C n)/(n+1)
These are the Catalan numbers.
The wiki has a lattice path definition on it.
https://en.wikipedia.org/wiki/Catalan_number

[Anonymous No.15421026]

I'm a programmer and I like math and proving things. I'm running out of motivation and haven't built anything recently. Is there any math I can learn and use while programming? I'm not interested in using formulas and I'm not interested in graphics, I just like math. For context, I'm close to just reading through a set theory textbook and doing exercises and I will if I don't find anything else.

[Anonymous No.15421049]

so Hoffman is the go-to book for Linear Algebra? or does /mg/ recommend something else?

[Anonymous No.15421102]

>>15420673
applied maths makes money
pure maths does not
simple as

[Anonymous No.15421120]

The final boss of /sci/

[Anonymous No.15421123]

how do you cope with never be able to come up with the solution to the Riemann hypothesis?
how do you cope with never been good enough for math?
I failed to qualify for the IMO in my high school and has been filtered by math ever since.
it pains that I will only have a PhD in engineering and not math.

[Anonymous No.15421198]

>>15421049
LA for pure math, yes. LA for applications in calculus, statistics, physics, etc., no. You need a matrix algebra book for that like Rao & Bhimasankaram.

[Anonymous No.15421199]

>>15421120
sin(β)sin(20)sin(70)sin(40)=sin(130-β)sin(30)sin(60)sin(10).
β=20

[Anonymous No.15421213]

>>15420314
Hello anon. Since you're trans your problem of course takes priority over mine. However the problem you posted seems to be very difficult, I am not capable of solving it. Sorry for disappointing the boymoder overlords.
>>15420749
Hello anon, there seems to have been a misunderstanding which I expected, as I said in my post. I took this problem straight from a book and this is how it was worded so I didn't change it. But like I mentioned in the post, monotonically increasing isn't strictly increasing. So we can have a function like f(1)=1 f(2)=1 f(3)=1. So your answer is correct but not for this problem. I hope you try again with this clarification in mind.
>>15420967
Hello anon! For part a), you're 100% correct. I really like your explanation, I got it some other way and the book does it differently as well so I'm happy to see your solution which is different. The second interpretation you gave is also very cool!
For part b, again you're absolutely correct. 1000%. I have read your argument and I think I need to write it out myself because it's really clever. And of course, it is as you say. These are just the Catalan numbers. Thank you a lot for your time and effort solving this problem and writing it all out like this. I appreciate it! I hope you have a fine day.

[Anonymous No.15421270]

is there a way to get sponsorship deals to build a better computer. i like plooting but there are some "differential equations" i would like to try and solve but i would need a custom machine. better than a homebuild but less than a supercomputer. just kidding im a plooter

[Anonymous No.15421277]

>>15421270
no
git gud at trading crypto and buy yourself a better computer (or just buy stocks for 2-3 years like a normal person)

[Anonymous No.15421324]

For each integer n>1 are there infinitely many primes which are 1 more than a multiple of n?

[Anonymous No.15421336]

>>15421324
Dirichlet's theorem

[Anonymous No.15421584]

>>15421453
>>15421543
Ok, it's theorem 1.14 of chapter VI in Lang's Algebra.

[Anonymous No.15421870]

I need to prove that if
[math]\lim\limits_{x \to a} \frac{f(x) - f(a)}{x-a}[/math] exists, then [math]f(x)[/math] is differentiable ([math]f : \mathbb{R} \rightarrow W, ||.||_w[/math]).
I was thinking about doing this:
[math]\lim\limits_{x \to a} \frac{f(x) - f(a) - L(x-a)}{x-a} + \frac{L(x-a)}{x-a}[/math] (L a linear map).
But I'm really starting to doubt this approach.

[Anonymous No.15421996]

Hello /mg/! I present to you the solution to this problem >>15411579 (that does not use calculus). Thank you to all the anons and anonettes who tried to solve it. And good job to the anon/ette who solved it >>15413155. As far as I am aware, they are the only one who did it. I am very sorry if I missed anyone.

>>15420506
obviously

[Anonymous No.15422021]

What are you currently studying anons? And what for? Have any of you actually done research, or are you just a bunch of textbook readers?

[Anonymous No.15422029]

>>15421996
It seems I have reversed the signs of the coefficients. Sorry about that. The idea is still the same though.

>>15420290
For (a), we can just count the number of ways to choose n numbers unordered with repetition, since there is only one way to order them such that function is non-decreasing. From there, it follows from stars and bars. I vil try (b_.

>>15422021
I am currently studying Baby Roodin to prepare for my Masters in Statistics course (if I get in) since I don't have a mathematical background.
>Have any of you actually done research, or are you just a bunch of textbook readers?
I am just a textbook reader, and I fear I will remain so since I will be joining academia quite late. I figure most people interested in research have studied Analysis, Algebra, etc. by end of undergrad. At least that is the case here. I have a lot to catch up.

[Anonymous No.15422030]

>>15421996
introducing "functions" [math] k_{x},\ l_{x} [/math] is retarded.

[Anonymous No.15422209]

>"Discrete Valuation Ring"
Retarded way of saying "local PID"

[Anonymous No.15422245]

>>15422021
Studying my polynomial rings and lattice problems for post-quantum crypto (RLWE)
I'm a lowly engineer, I don't do basic research. But I build stuff.

[Anonymous No.15422836]

How does one disprove this? I was thinking something like this: the cube made up of 4 smaller cubes is clearly not a unit -- it's not irreducible. Visually (this just serves as intuition, it doesn't *need* to correspond to an actual algebraic fact, but it almost certainly does) this is evident by its x and y axis having twice the value of the little cubes.
This implies another operation that wasn't accounted for took place in the supposed operation of "2 + 2 = 5", which is that of scaling the righthand side. However, that is not present in mentioned equation, as it only uses the operations of a group with the addition binary operation, i.e. + and =.

[Anonymous No.15422876]

>>15422836
Because we're adding the numbers of small (say unit) squares, and counting the sum of small squares.

[Anonymous No.15423319]

>>15422836
You can easily prove that 2+2=4 from Peano axioms. If you aren't working with them, then 2+2 can be anything you want. It doesn't really matter if the square is bigger - you can add 2 apples and 3 oranges to get 5 fruits. By the way, what do you get if you add 2.5 apples and 2.5 oranges? Who the fuck knows.

[Anonymous No.15423361]

>>15422836
>How does one disprove this?
politely explain that numbers aren't squares and that 2+2 = 4 is not describing the act of combining small squares into a larger square

[Anonymous No.15423461]

>>15422836
But seriously, what is the maximum amount of squares of any size that can appear in an arrangement of n same sized squares?

[Anonymous No.15423511]

>>15423461
Sum_{k=1..n} k - floor(sqrt(k)+1/2) * floor(sqrt(k-1))

[Anonymous No.15423543]

>>15420673
nobody cares about pure mathematics. all the effort of pure mathematicians only matter to other pure mathematicians.

[Anonymous No.15423566]

>>15423461
Good question. If you arrange them like on picrel (drawn for cases of n=5 and 6), then you get sequence 1, 3, 8, 16, 29, 47 which seems to be in OEIS. Now you just have to prove that it's an optimal arrangement and derive a formula, which seems to be so trivial of a task that I'm not even going to bother with it.

[Anonymous No.15423569]

>>15423566
I think the assumption that they can't overlap in implicit in the 2+2=5 meme.

[Anonymous No.15423597]

Any books you guys would recommend on Navier-Stokes and fluid dynamics?

[Anonymous No.15423601]

>tfw getting filtered by high school-level math
How will I progress in math if my basic arithmetic skills are so dogshit? Getting a problem wrong on a test or assignment feels meaningless when the error is due to a simple miscalculation and not gross misunderstanding. It's even worse when bad math is compounded throughout a problem with more than one error.

[Anonymous No.15423617]

>>15423511
Can you give me a hint to get this, I don't know why it works

[Anonymous No.15423649]

>>15423601
significant figures are not hard. If you are making more serious errors get a tutor and they should be able to patch you up. If this is out of the budget read parcels relating to building mathematical reasoning.
If you have a little charisma you can definitely argue points back on questions where you did math right with the wrong input, too.

[Anonymous No.15423653]

>>15423617
Summands correspond to oeis sequence A122197. Intuition is that as you fill up a row/column on the outside of the existing square/rectangle you can make more squares at once, up to the current optimal length which increases by 1 with every square number.

[Anonymous No.15423675]

What would be the implications of there being infinitely many vectors of prime length in any integral domain?
Are there any (at least pure) implications of this?

[Anonymous No.15423688]

>>15423675
define "vector" and "prime" here, I don't think this question makes much sense
you only have prime ideals at best

[Anonymous No.15423705]

>>15423688
You didn't get the references right away, have you skipped a class or two?

Vector meaning elements of a vector space equipped with the very same integral domain I talking about

Prime meaning the prime element of that integral domain.

Should I read you the definition of an "element"?

[Anonymous No.15423711]

>>15423675
Also for any normed vector space

[Anonymous No.15423972]

>>15421996
Hello again anon. Thank you for posting your solution. I haven't checked it out yet but it's good to see there is one without calculus.
>>15422029
>For (a), we can just count the number of ways to choose n numbers unordered with repetition
That's how I did it as well. It is a very simple way of doing it, in my opinion the easiest one but it doesn't seem to help b) in any way unfortunately.
>I vil try (b
Good luck anon. I'm sure you are capable of doing it with enough time but do keep in mind b) is quite a bit tougher than a). I myself already knew the solution from before seeing this exercise and doubt I could solve it myself.

[Anonymous No.15424078]

Terence Tao once posted on his wordpress site an article where he describes 36 different definitions of a certain mathematical object (I think it's the derivative but I'm not sure). I've been looking for that shit for a while now.
Any anons have a link to that post?

[Anonymous No.15424092]

>>15423688
Dumb midwit

[Anonymous No.15424100]

Is maths worth it?

[Anonymous No.15424116]

>>15424078
You're probably referring to this article
https://www.math.toronto.edu/mccann/199/thurston.pdf

[Anonymous No.15424172]

>>15420227
Today I went back to my highschool to retake my country's standarized SAT equivalent exam to get better math score, and these zoomers there (few years younger than me) were weird. And I'm a zoomer too. Half of guys had these weird curly haircuts, and there was a girl there going for a e-girl look with CAT EARS and bright white short skirt. I'm a zoomer too, but this shit wasn't prevalent in my days.
I am getting old...

[Anonymous No.15424175]

>>15424172
and they were all talking about hanging out on their discord server
Things changed so fast. Few year difference and I can barely relate to them

[Anonymous No.15424199]

Is it possible to avoid number theory entirely and have a successful Ph.D. in partial DE? I fucking hate it. I cannot believe how bad this shit is.

[Anonymous No.15424253]

>>15424078
>>15424116
what kind of déjà vu is this

[Anonymous No.15424335]

>>15420309
Lucky exposure. My intro to proofs/number theory course was capped intro to discrete math, and the other rosen discrete math book was used. Abysmal preparation for abstract algebra, got destroyed trying to keep up the first half of the course and professor had no mercy, said it was our personal responsibility to rise up to the standards and not rely on whatever courses we took previously. Thanks for reading my blog.

[Anonymous No.15424561]

I want to relearn math from the start
what is the most autistic way to achieve this?

[Anonymous No.15424594]

>>15424561
Khan academy
we have a bunch of good online materials here, but they aren't in English.
Btw textbooks are ok, but videos with explanation + doing exercises is the best. The more senses you active when learning the longer the information stays in your memory.

[Anonymous No.15424853]

>>15424561
math from the start can mean 3 different things

1.in the order in which subjects are taught in the educational system, from the beginning of elementary school all the way to the phd level
2. starting out from basic fundamental axioms about numbers, sets and their properties and building out a coherent theory of mathematics
or the most impossible one
3. in historical, chronological order starting with babylonial accounting records, the egyptians, greeks and so on and so forth

[Anonymous No.15424867]

>>15424561
>>15424853
on a practical level: just pick a book about basic (middle school and high school) algebra, a book about basic geometry, a book about probability and then a book about precalculus and then something like The Art of Problem Solving

[Anonymous No.15425145]

>>15420227
>it's superfluous
>that's redundant
>you made a logical leap here
>that doesn't follow
>not necessarily true
>not true for 0,Ø,U,1,etc
>you can't assume that
>you didn't prove the IH
>(?)
How do i get better at proofs? please don't say proof theory because it's too abstract to be useful. the only time you are given a set of a logical calculi, a language and a set of axioms is in proof theory.

[Anonymous No.15425174]

>>15425145
>it's too abstract to be useful
nigga the whole point of proofs is that they're abstract

[Anonymous No.15425414]

>>15425145
Read a book on proofs like Vellman. Then work through a proof based Calculus book.

[Anonymous No.15425440]

>>15425145
Unironically watch Michael Penn videos how he solves various problems. It's all about the mentality and solidness.

[Anonymous No.15425720]

>>15423705
>>15424092
Okay, let's consider the integral domain [math]R=\mathbb{F}_3[x,y,z,w]/(xy-zw)[/math]. What is a "vector in [math]R[/math]"? Let me be generous and say we're actually going to look at the vector space [math]\text{Frac}(R)^n[/math].
There is a notion of "prime element", but there is no unique factorization or anything like that. This is not a UFD. But okay, let's take it as given. Prime elements only. So not x, but 1+x is allowed for example.
Now, what is the "length"? This is a ring where 3=0. What is the "length" of the vector [math](1,1,1)\in R^3[/math]? 0? Not to mention, not every element will have a square root. What is the length of [math](1,x)\in R^2[/math]?
Let's be generous and replace "length" by fixing the natural bilinear form on [math]\text{Frac}(R)^n[/math] and looking at [math]x\cdot x[/math] and only thinking about "squared length".
Now your question reads:
>What would be the implications of there being infinitely many vectors over the fraction field of any integral domain whose length is the square of a prime element?
Finally, as long as there are infinitely many prime elements we can consider the vectors having 1 element corresponding to that prime element and the rest being 0.
In conclusion: your question is uninteresting without elaboration or change to multiple portions.
I thought, maybe you have some notion of vector or length which makes sense in generality. Maybe you wanted something number-theoretic (maybe in light of results about sums of squares equaling primes? but then the squared length would be prime, not the length). For number theory, "general" results will usually involve the more restricted class of Dedekind domains (not integral domains), in which unique factorization is important, but only into prime ideals and not prime elements. In fact, most abstract results about generalized primes care about ideals.

[Anonymous No.15425723]

>>15425720
So, I wanted you to clarify if you really meant for elements and not some strange notion for prime ideals (you can look at sums of squares in the residue fields of an algebraic integer ring, not sure if it's so useful).
And finally, >>15423711 makes no sense in all this context. Unless instead of "any integral domain", you really specifically want to
A) Look at an algebraic integer ring [math]\mathcal{O}_K[/math]
B) Fix a specific embedding into [math]\mathbb{C}[/math]
C) Consider the complex vector space and lengths are with respect to some norm.
D) Only consider prime elements (very restricted class of prime ideals, corresponds to the trivial class in the ideal class group).
This is miles away from whatever you originally asked. Not to mention, you can just take 0 in all coordinates but one and a prime element in the last one.
But if you don't want to be helped, by all means, be cryptic and unhelpful.

[Anonymous No.15425724]

>>15424594
I said autistic not mentally challenged
>>15424853
Yes
>>15424867
that's goytier

[Anonymous No.15425987]

Is there a *bijective* function [math] f : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} [/math] which is also "associative" in that
[eqn] f(a,f(b,c)) = f(f(a,b),c) [/eqn]
for all [math] a,b,c \in \mathbb{N} [/math] ?

[Anonymous No.15426003]

>>15425987
Sorry disregard this question it's stupid

[Anonymous No.15426042]

Is there like a standard/agreed upon set of conditions a class M of monomorphisms is supposed to fulfill in order for the notion of M-subobject to be well-behaved?
For example, should M contain all isomorphisms, should it be closed under composition, should gf in M imply f in M, etc?

[Anonymous No.15426114]

Hey anons,what are you reading right now? On average, how long does it take you to complete a book and how many pages, assuming you also work on the exercise?

[Anonymous No.15426211]

>>15426042
>well-behaved
Meaning what?

[Anonymous No.15426217]

>>15426042
>M-subobject
Also what's that

[Anonymous No.15426407]

Hey guys. Could help me please?
I faced a problem that first seemed simple to me: given a sequence of real numbers [eqn] \left\{ x_i \right\} [/eqn], find a sequence [eqn] \left\{ y_i \right\} [/eqn] that, after average normalization, would give the initial sequence, i.e. for each [eqn] i = \overline{1 .. n} [/eqn]: [eqn] x_i = \frac{y_i}{\frac{1}{n} \sum_{j = 1} ^ {n} y_j } [/eqn].

But then I played around with the algebra, and got a determined homogeneous system with (n+1) equations:
[math]
\left\{
\begin{aligned}
\sum_{i = 1}^n y_i - s = 0 \\
\frac{n}{x_1}y_1 - s = 0 \\
\dots \\
\frac{n}{x_n}y_n - s = 0
\end{aligned}
\right.
[/math]
This system does not have a nontrivial solution. Am I missing something, or this problem indeed could not be solved as I want it?

[Anonymous No.15426429]

>>15426407
Impossible if [math]\sum_{i=1}^n x_i \neq n[/math].

[Anonymous No.15426455]

>>15426429
Oh, I see. And if the sum of xs is equal to n, the system has lower rang. Thank you for the advice!

[Anonymous No.15426467]

>>15420227
>Working on thesis problem
>Have to find numerics for an optimization problem over an infinite dimensional Hilbert space
>Quickly show a bunch of promising results
>Come up with a couple of basic schemes
>Suddenly realize that the problem suffers massively from the curse of dimensionality
>Reading into high dimensional pdes for quant finance trying to save my ass

Lol I may be fucked

[Anonymous No.15426474]

been developing the calculus to solve certain DEs which would greatly allow me to make lots of money. which would allow me to build a better computer. which would allow me to solve even harder DEs and make even more money but too lazy to implement and test the proofs into c++

[Anonymous No.15426497]

For those of you who enjoy math as a hobby. Do you find applications for it in real life? Feels like it's the one hobby where you create nothing to show your normie friends.

[Anonymous No.15426500]

>>15426497
I remember one time I used modular arithmetic because I was struggling with a puzzle in Prince of Persia. Does that count as an application? Many retards seem to be getting filtered by a RE4make puzzle; I suppose modular arithmetic would be useful there as well.

[Anonymous No.15426784]

Why is it called infinity / infinity case in L'Hopital's rule, when it is not necessary for the numerator to he infinity?

[Anonymous No.15426989]

[eqn] \pi = 4\mathrm{arctan}(1) = 4-\frac 4 3 + \frac 4 5 - \dotsb [/eqn]

Correct me if I'm wrong:
This comes from the Taylor series for arctangent, which has interval of convergence [math](-1,1)[/math]. The series converges only conditionally at [math]x=1[\math] … so you can rearrange that series to where it no longer equals [math]\pi[/math]?

[Anonymous No.15426995]

>>15426989
The interval of convergence is actually [−1, 1]. You can prove it using the alternating series test.

[Anonymous No.15427013]

>>15426995
>6
But the alternating series test doesn't discriminate between absolute vs conditional convergence. The series is only conditionally convergent at those boundary points.

[Anonymous No.15427051]

>>15426989
That's what the Riemann rearrangement theorem says, yes.
For an example that's not pretty but it's the first to come to mind (and using the summation for [math]\frac{\pi}{4}[/math] instead), take [math]\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-...[/math]
rearrange to [math]1-1/3+1/9+1/5-1/15-1/7+1/21-...[/math]
which is [math]1-2/9+2/15-2/21+...[/math]
and if you evaluate [math]1+2\sum_{k=1}\frac{(-1)^k}{6k+3}[/math] you get [math]\frac{2+\pi}{6}[/math]

[Anonymous No.15427065]

>working on some retarded homework problem finding multiplicative inverses in fields
>professor gives homework solutions after the deadline, problems are all "this one weird trick!" solutions
>Gives everyone full credit for attempting it and following textbook logic to dead end solutions anyways

What a faggot, I hate people like this. We don't have time to do bullshit like this, undergrad is to just cram in all the information possible. This is a shit school so no one here will ever do anything important anyways.

[Anonymous No.15427067]

>>15426211
>>15426217
I'm disappointed /mg/ can't do category theory, but I'm also happy because this means more of the shitposters have left. Let's talk about analysis or geometry instead.

[Anonymous No.15427083]

>>15427051
Beautiful. It's wild that no one mentions that it's only conditionally convergent among all the times folks talk about this series. I wonder how tractable of an exercise it is to find a rearrangement of the series that converges to just straight [math]\pi[/math].

[Anonymous No.15427093]

>>15427065
>We don't have time to do bullshit like this, undergrad is to just cram in all the information possible. This is a shit school so no one here will ever do anything important anyways.
What a faggot, I hate people like this. Imagine being mad because someone tried to actually teach you problem solving skills in the hope that you don't end up completely worthless.

[Anonymous No.15427098]

>>15427083
Well, if you don't care about a nice representation, it's pretty trivial. You just take enough positive terms to the point where their sum exceeds π, then switch to negative terms until it's under π, then add on positive terms until the sum exceeds π again... and keep doing that forever; that's more-or-less the procedure Riemann came up with when trying to develop an intuitive explanation

[Anonymous No.15427101]

>>15427093
nigger, nothing in algebra past applications has any use whatsoever. No I don't give a fuck if it has a future use and neither do 99% of people, the moment it has an application I'll learn it, otherwise I have zero interest in inventing anything that would ever be of use for anyone aside from some financial scheme that makes normgroids suffer more. I am well aware most of uni courses are just loser professors trying to flex on undergrads and I'm not impressed. Get a real job.

[Anonymous No.15427103]

>>15427101
Why are you doing a pure math course if you only care about applications?

[Anonymous No.15427107]

>>15427103
Because it would be even worse to waste an elective course slot on some subhuman level garbage course that isn't math, CS, or physics. I have used every single elective on math courses. I am not going to grad school unless my employer requires it for advancement either, as that is also a waste of time. I am only doing a math degree because I enjoy most math. I am too low IQ to be a quant, so after I finish my BS I am going the actuary route.

[Anonymous No.15427118]

>>15427107
Okay whatever I don't care. Why are you complaining about no applications in a course which does not care about applications?

[Anonymous No.15427135]

>>15427118
I am not. I am complaining that the problems given in the homework assignment were unsolvable given the course resources (textbooks, notes, lecture) presented. It is unfair to expect students already burdened with 15 credit hours of courses in a semester to research into outside sources to complete homework assignments. It is acceptable for students to assume that all tools required to complete the homework were given. I even gave a caveat that maybe for students in top universities could be expected to research these topics on their own during the learning process, but to ask that of people at low tier universities that want to learn without pressure is just abusive.
In general, applications spark real interest. I will never go out of my way to learn some theoretical non-use content, but I would absolutely do so for applied material.

[Anonymous No.15427155]

hey. im an undergraduate math student and i was wondering something.
I'm so busy with school i havent had a ton of time to work through a supplemental textbook and study so im hoping someone could explain this to me quickly.
When we are talking about topological but not metric spaces, what does it mean to define "closeness" or relation between elements. Is it defined by the open sets themselves? Can you give examples? Also, are there spaces without topologies or is that unmathematical at that point?

[Anonymous No.15427156]

>>15427093
>you don't end up completely worthless.
There isn't a gradient of worth for math graduates. You are either at the peak and are valued, or are in the bottom 99% and treated like garbage. Better to be smacked with reality early rather than go through that pain much later.

[Anonymous No.15427165]

>>15427155
In topological spaces "closeness" is not defined as distances like in metric spaces. closeness is captured by the structure of the open sets in the topology. Open sets the notion of "neighborhoods" around points.

a "space" without a topology is just a set without any extra structure. so that set is not considered a topological space, becuse it doesnt have open set structure required for defining a topology.

[Anonymous No.15427233]

>>15427165
can you give me an example of closeness in a non metric topological space? im just having a hard time visualizing the system you would use. The first thing that came to mind when i tried to visualize something was like if you had a set of sets as your space if the intersection of the sets is a nonempty open set or something but idk if that would even be correct.

[Anonymous No.15427249]

>>15427233
NTA but maybe start from the trivial topologies on a finite group where the notion of closeness is easy.

[Anonymous No.15427426]

>>15427067
I don't know, the question sounds extremely generic. Like if we translate it to standard set theory, it just asks if there's [math]M\subset {\mathcal P}(X)[/math] such that the elements of M together are "well-behaved". I mean sure, e.g. a topology [math]{\mathcal T}_X\subset {\mathcal P}(X)[/math] is such a collection of subobjects that are nicely behaved with each other, e.g. you have unions and whatnot. I'm sure among sheaf topoi you get something like sub-sheaf topoi and they'll be "well-behaved". This is just too broad. Could be asking about module theory or computability theory just the same.

[Anonymous No.15427449]

Hello, EE here. Been working for 4 years and during all this time I've been self-studying math for fun and recently been entertaining the idea of pursuing an MS in Math. I've completed the following:

>Munkres Topology
>Both volumes of Zorich's Analysis
>generatingfunctionality + Lovasz Combinatorial Problems (okay, more like 80% in the latter)
>Simmons' Diff Eqs
>Hoffman/Kunze Linear Algebra
>Needham's Complex Analysis
>Dummit & Foote + Artin's Algebra
And some others I'm forgetting

And I'm currently working through Kolmogorov's Functional Analysis.

Is there anything else I should pick up before trying to audit some grad classes at a local university? My undergrad GPA was pretty bad so I wanted to take some classes to boost that up a bit.

[Anonymous No.15427507]

>>15427449
Man, how are you able to do all of that in your undergrad? Also, how long did it takes for you to finish zorich? Did you work on the exercise too?

[Anonymous No.15427537]

>>15427507
Nah, I did all that out of school in 4 years while working as an engineer.

As for Zorich, it took me about 1 year to go through both volumes and finish all (I honestly skipped a couple in some chapters) exercises, but I spent on average 3 hours a day on weekdays, and 7+ on weekends. I took a break every 3 weeks or so too

[Anonymous No.15427540]

anyone here with experience preparing for polish matura exam? i'm not underage

[Anonymous No.15427550]

>>15427540 (Me)
it's pretty fucking insane to me
if you get 95% it still isn't enough to get into computer science on UJ or UW and for UW
a single fuckup and you're pretty much fried, not only that but if you somehow manage to mess up english you're also fucked
i know the curriculum is piss easy but being meticulous enough to not make any retarded mistakes scares me

[Anonymous No.15427570]

>>15427540
I am grinding for matura rozszerzona right now. I'll use English since idk if you are a foreigner, or what. I use these resources:
https://www.matemaks.pl/matematyka-matura-rozszerzona-kurs.html
for theory and some exercises (you need premium account for most of the course though)

100 official exercises from Central Exam Commission (CKE) with video guides and explanation for each exercise:
https://www.matemaks.pl/matura-rozszerzona-zadania-cke.html
Document with those exercises in .pdf form (there are step by step solutions and hints in there too, but videos from Matemaks are generally clearer):
https://cke.gov.pl/images/_EGZAMIN_MATURALNY_OD_2015/Materialy/zbior_zadan_maturalnych_z_matematyki.pdf
So these exercises are generally harder than ones on Matura, but you'll be to squeeze out a lot more knowledge out of them. The PDF document includes step by step guides with sometimes multiple approaches and ways to finish each exercise. This covers all material from high school (everything that you will encounter on Matura) with about 3-6 exercises per topic.

Document with all formulas (you'll be given this on the exam, so get used to using it efficiently):
https://cke.gov.pl/images/_EGZAMIN_MATURALNY_OD_2015/Informatory/2015/MATURA_2015_Wybrane_wzory_matematyczne.pdf

https://www.matemaks.pl/matura.html
"Arkusze 20xx" category contains solutions for each exercise from previous Matura exams 2010 and up.
In pdf form: https://cke.gov.pl/egzamin-maturalny/egzamin-maturalny-w-formule-2015/arkusze/

Complete those resources then do them over again, or optionally get an exercise collection in book form from authors like Kiełbasa, google "kielbasa zbiór zadań".

These resources are for people who went through gimnazjum, aka formuła 2015. You'll be able to sign up for matura in formuła 2015 until 2025. But formuła 2023 is generally overlapping with the old one. Just finish as many exercises as possible. The ones on Matura is quite schematic.

[Anonymous No.15427582]

>>15427550
Huh? I thought you needed like 86% for UJ (I checked on their official site few months ago), and they only take matematyka rozszerzona into consideration. English doesn't count for UJ as far as I remember. UW has retarded rules and they consider even things like your score from Polish (idk if they measured it in percentage, or points, but it gave you something like 2 points if you did on Polish well).
Generally many people that go to UJ for CS are winners of regional math Olympics - these guys get automatic 100% from matura. That's how my cousin got in there. This probably inflated the thresholds. Although getting 100% on matura rozszerzona is not unheard of. It's all quite schematic, but you have to autistically spend over half year trying to remember all the "schematics".

[Anonymous No.15427587]

>>15427550
Ah I see. Why the fuck did it inflate so much in 2022? Insane. Ukrainians cheating or something?

[Anonymous No.15427597]

>>15427233
If you want something non-metric which can be visualised, look up the Zariski topology on the plane. It is the topology whose closed sets are (up to unions/intersections) solution sets {(x,y) | f(x,y) = 0} of polynomials in two variables.

This is non-metric and even non-Hausdorff: if you take two neighbourhoods of different points then the intersection is always non-empty.

[Anonymous No.15427601]

>>15427570
thanks for reply
this is extremely useful, genuinely
One more thing, how do you study? I mean what does your routine look like, how much time would you spent on studying on a given day, how do you pick out what material are you going to through etc. I really struggle with organizing, i was thinking about getting a tutor plan things out for me but i've never had private lessons or anything of that sort
I'm generally much better with creative stuff and conceptual understanding than grinding
>>15427587
>>15427582
that's UW actually, the regular CS is actually ~85 for UJ if i'm not mistaken but the informatyka analityczna is literally 96
must be demographic echo or something in management, it isn't that much off from 2020 figure though
I'm more inclined towards getting into UW because of the capital city and global ranking memes and kraków is a bit too close home to my liking. The difference between salaries in warsaw and the rest of poland is quite insane too so i'd be nice to setup a network or something like that
i'm still thinking about math major especially if i couldn't get into neither UW no UJ for CS but it seems a bit useless by itself, if i could get into finance or programming with that it'd be great but CS seems much more prestigious, maybe i could do a double major but at the moment i just need to focus on studying

[Anonymous No.15427610]

>>15427601
>CS seems much more prestigious
CS degree i mean, finance is probably ultimately more interesting than codemonkey faggotry even though i've always been a computer guy

[Anonymous No.15427624]

>>15427587
it's because it was very easy in 2022
>>15427550
CS is a meme and it's really boring compared to math

t. math undergrad at MIMUW

[Anonymous No.15427634]

>>15427624
the CS you learn at school is boring, algorithms and competitive programming being alright
low level shit like compilers, operating systems, reading assembly and cracking games is fun and matematyka komputerowa is probably too
but yeah i'd rather study math
What are you going to do with your math degree?

[Anonymous No.15427636]

>>15427540
>>15427624 (me)
Here's how you do it, I got 96% on the advanced math part. The questions are always all very schematic and you essentially can just memorize the intended solutions. Do a fuckton of sheets, zadania.info has them. Once you do them enough times it's quite easy to recognize the scheme of the question and apply the solution. You barely need to know math to do it well.

[Anonymous No.15427647]

>>15427636
yeah i've noticed that too
What was your studying routine like? Just pick out a few exercises or an entire sheet and go through it?

[Anonymous No.15427648]

>>15427634
Ehhh... algorithms "sound" fun but it's boring, it's just a constructive solution to a problem but you have to state it in a programming language which just gets in the way. Maybe some people like it but I'm not a fan. I actually do like programming for fun, which doesn't have much to do with CS.
>What are you going to do with your math degree
academia if I don't get filtered by a phd and corpo otherwise (quite easy to find a job if you have a math degree)

[Anonymous No.15427653]

>>15427601
>One more thing, how do you study? I mean what does your routine look like, how much time would you spent on studying on a given day, how do you pick out what material are you going to through etc
I've been studying for over a year now. 1-2 hour a day for few months, then 2 hours and up, and now that I'm closer to matura over 5 hours a day. Everyday. Make sure to not cram everything in one short period (like many students do, they just cram all the material for 12 hours straight right before the exam), and give yourself time to rest. I was reading that your brain stores things in long term memory during your sleep/deep rest, so study for 1-2 hours, rest and try to visualize things that you just learned during rest, then go back to studying. Go through theory and matematyka podstawowa first until you remember every detail. Do the theory course multiple times if you have to (depends on how much you already know though, I'm talking about starting from scratch). It's pretty much impossible to memorize all of this theory by only going through it once. Study like Anki cards for language learning work: memorize things then move on, then come back to the same material as you are about it forget it. This is how you'll end up with this knowledge in your long term memory.
Once you are done with theory go through every Matura from 2023 to 2015 (or maybe even to 2010, but 2015 is when the new formula started). Note down exercises that are hard for you and come back to them later. If you are bad at any particular topic then you should take a step back to relearn theory for this specific topic, and do exercises focused on this particular thing. Note down tricks and all the little nuances on a sheet of paper, there's tons of them.

Here are links to videos with exercises that I found particularly hard, or just interesting:
pastebin.com/0ttcGyc7

I'm aiming for comp sci at UJ, but frankly I'm not getting my hopes up. It's easy to fuck something up in stress.

[Anonymous No.15427655]

>>15427647
Entire sheet. With sufficient practice you can basically scroll through the easy ones and the only hard ones are non-schematic geometry and combinatorics questions (those are the only questions that require thinking).

[Anonymous No.15427660]

>>15427653
also ofc do the "arkusz treningowy" from CKE. It's a very valuable resource imo. Hard and not very schematic exercises, but it's worth it to complete them, so that nothing will be able to surprise you on Matura.

[Anonymous No.15427765]

>>15425414
>a proof based Calculus book
i'm assuming this means Courant or Apostol

[Anonymous No.15427803]

>>15427648
>(quite easy to find a job if you have a math degree)
i don't think this is true but then again i'm not in math

[Anonymous No.15427822]

>>15424100
Just keep reading the posts with the maths and the solutions, and then attempt some of the questions with a glossary of the mathematical symbols.

[Anonymous No.15427860]

>>15426474
That laziness will keep you poor, give me your research

[Anonymous No.15428012]

>>15427426
I don't think you understood the question at all, which makes it surprising you're also trying to talk about topoi. The class M is supposed to consist of monomorphisms.

[Anonymous No.15428095]

>>15427860
it involves a lot of fourier transforms. looking for smooth derivatives in peaky data like what a stock price might look like

[Anonymous No.15428441]

>>15427135
In a math course, you are supposed to use your head. If you wanted plug n chug questions, you should have just gone for engineering. Seems like you just went for a degree without first learning anything about it.

[Anonymous No.15428603]

Give a man an apple, you feed him for a day..
Teach a man the Banach-Tarski Method, you feed him for a lifetime.

[Anonymous No.15428674]

Would really appreciate it if I could get some help for this problem from my Real Analysis Class:
Let f (x) be a differentiable function over [0, 1]. Suppose that f(0) = f(1) = 0. Prove that f′(x) − 2f(x) must have a zero inside (0, 1). Namely, there is a c ∈ (0,1) such that f′(c) − 2f(c) = 0.

[Anonymous No.15428686]

Good morning /mg/roids, Reddit has informed me of this wonderful slur for algebraic geometers.

[Anonymous No.15428702]

How do you guys cope knowing that it's very rare you'll get a Fields Medal? I mean, one shot at life, and one day you'll be sitting on your death bed without one.

[Anonymous No.15428706]

>>15428702
By the time you're on your deathbed, you're pretty close to it literally not mattering whether you got one or not. At all.

[Anonymous No.15428711]

>>15428674
Let a be a maximum of f and b be a minimum of f in [0,1]. They always exist since [0,1] is compact and f is continuous.

Define
g(x) = f ′ (x) − 2 f (x)
That function is Darboux continuous because of
https://en.wikipedia.org/wiki/Darboux's_theorem_(analysis)

g(a) = -2 f(a) < 0
g(b) = -2 f(b) > 0

So there is a c in (min(a,b), max(a,b)) with g(c) = 0.

[Anonymous No.15428719]

>>15428711
Thanks for the response, we haven't gone over Darboux so I don't think I'm allowed to use the theorem. My teacher did have some work for this problem that I couldn't understand that I'll attach in hopes that you could explain it through that method.

[Anonymous No.15428802]

>>15428719
>I am not le allowed to use that
What are you in fucking kindergarten?

[Anonymous No.15428806]

>>15428802
NTA but generally when you're doing an exercise you are supposed to apply the knowledge you learnt on the course. When I started university there were a lot of limits we had to evaluate using elementary methods without e.g. l'Hospitals method that would be trivial otherwise.

[Anonymous No.15428857]

>>15428806
So this is what's it like outside of t50 colleges. Lmao.

[Anonymous No.15429112]

>>15428441
>"hehehe, you actually need galois theory for this problem in your introductory modern algebra course. Shoulda used your head and derived it on your own :)"

[Anonymous No.15429118]

>>15428702
Why do you care about validation from a bunch of fags?

[Anonymous No.15429124]

>>15429118
The top mathematicians across the world is not "just" a bunch of fags.

[Anonymous No.15429126]

>>15429124
They are, and you're crying about a good boy sticker.

[Anonymous No.15429394]

Has any course just turned you off entirely from math? I guess you could also call it your filter. For me, it is differential geometry. I am no longer interested in mathematics, cured you could say.

[Anonymous No.15429403]

>>15427653
i see, makes sense
do you keep something like a list of formulas that aren't present in the formula sheet? from geometry and things like that
>>15427655
by scroll through you mean just skipping them? i guess you don't really need to solve 200 different quadratic equations with parameter to get the hang of it
do you have any tips on how to approach geometry problems?

[Anonymous No.15429433]

British Analysis books:

Amongst many others; USA* has Rudin, Russia has Zorich (2 vols), and Germany has Amann/Escher (3 vols), what about the UK? Is there a multi-volume book published for the masters of the known universe to best operate His majesty's realms? I'm especially curious as to what they use in Oxbridge, LSE, Imperial College, Edinburgh etc.

*Rudin is actually Austrian

[Anonymous No.15429439]

why does the difference between perfect squares always increase by 2?

[math]n^2 = 1\\
(n+1)^2 = n^2 + 2n + 1 = 1 + 2 + 1 = 4\\
(n+2)^2 = n^2 + 4n + 4 = 1+4+4=9\\
\\
4 - 1 = 3\\
9 - 4 = 5\\
[/math]
etc

So if I do
[math]
n^2 + 2n + 1 - n^2 = 2n + 1 = 3
n^2 + 4n + 4 - n^2 + 2n + 1 = 2n + 3 = 5
[/math]
I don't get where the 2 is coming from

[Anonymous No.15429445]

>>15429439
[math]n^2 - (n-1)^2 = n^2 - (n^2 - 2n + 1) = 2n - 1[/math]
Hence, it's the sequence of odd numbers.

[Anonymous No.15429446]

>>15428012
A mono
i : S -> A
in a topos ties to a characteristic arrow
\chi_S : A -> \Omega
by the subobject classifier diagram.In set theory language such an arrow let's you speak of a subset S of A. Hence a class of monos into A is like a subset of the powerset

[Anonymous No.15429449]

>>15429439
[eqn]((n+2)^2 - (n+1)^2) - ((n+1)^2 - n^2) = 2[/eqn]

[Anonymous No.15429452]

>>15429439
(n + 2)^2 - (n + 1)^2 = 3 + 2n
(n + 1)^2 - n^2 = 1 + 2n

3 - 1 = ....

[Anonymous No.15429464]

>>15429433
All those universities have website from where you can download the curriculums.
Pic related is Oxford's Anaysis 1 course for example.

[Anonymous No.15429476]

>>15429464
Please understand that the point of such questions on a "social" board are to get personal recommendations. If you're just dumping reading lists, the inquirer is better served using a search engine or Bing chat AI.

[Anonymous No.15429503]

>>15429449
Might be better to simplify it like this:
[math]
((n+2)^2 - (n+1)^2) - ((n+1)^2 - n^2)\\
= ((n^2 + 4n + 4) - (n^2 + 2n + 1)) - ((n^2 + 2n + 1) - n^2)\\
= (2n + 3) - (2n + 1)\\
= 2
[/math]

[Anonymous No.15429523]

>>15429433
>Russia has Zorich
Fichtenholz is better

[Anonymous No.15429541]

>>15429403
>i see, makes sense
do you keep something like a list of formulas that aren't present in the formula sheet? from geometry and things like that
Generally formula sheet has all you need. Just write down anything that you find hard, or like little tricks that save time.

[Anonymous No.15429574]

>To add integers with different signs, subtract the smaller absolute value from the larger absolute value and give the answer the same sign as the number with the larger absolute value
Why does this work? I'm new to maths and want to understand the logic behind this.

[Anonymous No.15429865]

Do you know a book that contains the proof of the general theorem on sequences that satisfy a linear recurrence equation with constant coefficients ?

[Anonymous No.15429872]

>>15429865
The concretre tetrahedron probably does

https://www.amazon.com/Concrete-Tetrahedron-Recurrence-Generating-Computation/dp/3709104440

[Anonymous No.15429881]

>>15429865
Just write it in matrix form and then convert the matrix into Jordan normal form to compute it's powers.

[Anonymous No.15429917]

>>15429881
>>15429872
Thank you

[Anonymous No.15429995]

>>15420416
Sorry it took me so long to respond, I had to work through this for a while. I am continuing to work through the implications. But this is very helpful, thank you.

[Anonymous No.15430236]

>>15429523
The phrase "Amongst many others" was to buttress myself against the deluge of "what about..." posts from the countries listed. It just amazes me that the great British analysis book hasn't been written. You can't just stop with Newton and leave the story there, and I don't really count Hardy's book for this sort of thing.

Next time I see King Charles, I'll tell him he ought to commission such a book.

[Anonymous No.15430314]

I'm going back to school for math. Should I study probability theory or topology?

[Anonymous No.15430359]

>>15424078
>>15424253
>>15424116
I'm pretty sure you're referring to this:
https://mathoverflow.net/questions/366070/what-are-the-benefits-of-writing-vector-inner-products-as-langle-u-v-rangle/366118#366118

It was the inner product, and it was technically about various notations, not about the definitions per se.

>>15424199
Probably. Why are you doing number theory right now?

>>15429112
>not the reincarnation of galois

>>15427537
if you were able to do all of that, then you should be more than ready for anything in grad. It might be worth looking at a more traditional complex analysis textbook like Ahlfors or Stein/Shakarchi, but it's probably nothing you can pick up over time.

>>15430314
What's the curriculum for both? What do you want to do afterwards? If you want to do not grad school then probability theory is better, but if you want to do grad school then topology would be better. If you don't know what you wanna do, then don't sweat it too much.

[Anonymous No.15430364]

Does any anon have the copy of the original link to this copypasta btw? Like the Russian site and the text transcription. There used to be a pastebin but it's been deleted.

[Anonymous No.15430375]

>>15430359
>If you want to do not grad school then probability theory is better, but if you want to do grad school then topology would be better. If you don't know what you wanna do, then don't sweat it too much.
It's a master's (I'm in Europe). Not sure if I continue for a phd afterwards or not. If I do topology then I'll have to, obviously.

[Anonymous No.15430380]

>>15430375
If it's a masters, I'm assuming it's measure-theoretic probability? Maybe take that one, in case you decide not to do PhD afterwards. It sounds worth.

[Anonymous No.15430403]

>mathmeticians when they have to use a shape that isnt a circle and a number that isnt a letter

[Anonymous No.15430488]

>>15428719
Let [math]g(x)=e^{-2x}f(x)[/math]. This is continuous and [math]g(0)=g(1)=0[/math]. By Rolle's theorem we have [math]g'(c)=0[/math] for some value. Now manipulate this equation, it gives what you want.
>>15429574
Work out this example and it should be obvious: [math](+42)+(-9001)[/math]

[Anonymous No.15430528]

Anyone have a way to find the closed form for this?
[math]\frac{d}{ds}[u^T e^{A + sv v^T} u][/math]

[Anonymous No.15430544]

>>15430528
Do you also want a closed form for a rewriting of the A and v terms, or do you just want the correct product rule?

I don't know by heart what you need, but there's
https://en.wikipedia.org/wiki/Matrix_calculus
and
https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map

[Anonymous No.15430562]

Kind of a dumb question, but please help:
If I want [math]a,b,c[/math] to be such that [math]a\not=b[/math], [math]b\not=c[/math] and [math]a\not=c[/math], does it suffice to say "[math]a,b,c[/math] are distinct", or must I say "[math]a,b,c[/math] are pairwise distinct"?

[Anonymous No.15430569]

>>15430562
Just distinct makes sense to me.

[Anonymous No.15430572]

>>15430562
both are correct because a,b,c is the same as saying a and b and c which is the definition of pairwise disctinct so i think you're fine with whatever looks pertier

[Anonymous No.15430581]

>>15430562
Just write [math] a \neq b \neq c[/math]

[Anonymous No.15430586]

>>15430581
thats not the same thing
inequality is not transitive.
example $3\neq5\neq3$

[Anonymous No.15430587]

>>15430569
>>15430572
Thanks.

[Anonymous No.15430624]

>>15430562
>>15430581
>>15430586
[math] \{ a , b , c \}^2 \in {}\neq [/math]

[Anonymous No.15430637]

>>15430624
I have never seen anything in this manner.

[Anonymous No.15430699]

>>15430637
≠ is a binary relation.
https://en.wikipedia.org/wiki/Binary_relation
You can think of it as the set of all ordered pairs (x,y) that satisfy x≠y.
Since it is symmetric you can also think of it as the set of all 2-element sets (which have no defined order) that satisfy ≠.
The {a,b,c}^2 part is still wrong because it contains the pairs (a,a), (b,b), (c,c).
There are some notations that allow you to restrict to pairs of different elements.
({a,b,c} C 2) would be one way (resembling binomial coefficient) that is just the set of all 2-element subsets of {a,b,c}.
The ∈ part is wrong and ⊂ should be used instead to get the types right.

It is more correct to write ({a,b,c} C 2) ⊂ ≠

[Anonymous No.15430706]

>>15430699
Okay that makes a bit more sense, thanks.

[Anonymous No.15430817]

>>15427449
Since you were already an EE, and must have taken a few calculus courses, did you find Analysis worthwhile to your engineering work, or just for personal enrichment?

[Anonymous No.15430820]

>>15430562
If you can apply an ordering, then a < b < c gets the point across.

[Anonymous No.15430847]

>>15430586
Everyone will understand what you mean. It's just a notation shorthand. I write that all the time.
>>15430624
Formally correct but annoying to write.

[Anonymous No.15430855]

>>15430364
http://imperium.lenin.ru/~verbit/MATH/programma.html

[Anonymous No.15430869]

>>15430855
All these le difficult books and no research is why Americans are ahead of rest of the world in academia.

[Anonymous No.15430872]

>>15430869
the only people in america who can do math are the asian immigrants

[Anonymous No.15430890]

>>15430872
And your point is...?

[Anonymous No.15430967]

>>15430872
That doesn't address his point, as the Asian immigrants are still going through American university programs.

[Anonymous No.15430975]

>>15430364
This curriculum is a specialized one almost from the beginning. Algebraic topology and geometry aren't the only fields in mathematics.
>>15430380
>If it's a masters, I'm assuming it's measure-theoretic probability?
Statistical methods etc, it's more applied.

[Anonymous No.15430999]

>>15430967
Most of them only do their PhD there, which is a level where the retardation of the american education system doesn't affect you anymore.

[Anonymous No.15431023]

>>15427449
damn, that looks like most of my math degree. Why didn't you just double major? doesn't seem you are far off.

[Anonymous No.15431070]

Has anyone contacted Ted Kaczynski to discuss math? Like, exchanging ideas on researching and what not.

[Anonymous No.15431076]

>>15430999
>American education is bad elsewhere! Just not when the Asians get in, okay!
Sounds like cope, my boy.

[Anonymous No.15431083]

>>15420227
[math] \int_0^\infty dx \, x \,K_0(x)^4=\frac{7}{8}\zeta(3)[/math]
This is what Mathematica tells me. How would I do this by hand?

[Anonymous No.15431087]

parsed

[Anonymous No.15431114]

>Now, consider the polynomial ring R = C[x, y] and the ideal I generated by the polynomial f = y^2 - x^3 - x - 1.
>In the ring R, the ideal I is prime because the quotient ring R/I is an integral domain (in fact, it's an integral extension of the complex numbers). The algebraic variety V(I) corresponding to I is a smooth curve in the complex plane, which is the geometric representation of our prime ideal.
>Given a variety V(I) of an ideal I, how can you decompose V(I) into simpler varieties?

Getting hyper-filtered by an undergrad abstract algebra course. Bros... not like this....
Its painful enough that I'm getting a paranoid feeling that this professor just wants to see us suffer.

[Anonymous No.15431137]

>>15431114
Are you asking for homework help or something? That question doesn't even have anything to do with the previous discussion.

[Anonymous No.15431162]

>>15431137
I'm crying.

[Anonymous No.15431177]

>>15431114
What about that don't you understand? That doesn't even sound like a problem that requires a solution, it sounds like your professor just wants you to repeat a core idea from the class

[Anonymous No.15431190]

>>15431083
[eqn] \int_0^{ \infty} \mathop{dx} x K_0(x)^4 = \frac 7 8 \zeta(3) [/eqn]

[Anonymous No.15431195]

>>15431114
> Given a variety V(I) of an ideal I, how can you decompose V(I) into simpler varieties?
Write I as the intersection of primary ideals, this will give you the decomposition into irreducible components.

[Anonymous No.15431203]

>>15420314
stfu fag

[Anonymous No.15431204]

>>15431190
I mean how would I analytically do the integral

[Anonymous No.15431209]

>>15420227
Why are math generals infected with (((number theorists)))?

[Anonymous No.15431210]

>why is the general dedicated infected with the purest form of math?

[Anonymous No.15431243]

number theorists spend half the day explaining why number theory is the purest field of math and spend the other half looking for algebraic techniques to apply to their work

[Anonymous No.15431258]

>>15430488
>Work out this example and it should be obvious: (+42)+(−9001)
It isn't obvious. Please explain?

[Anonymous No.15431292]

>>15431070
Pretty sure I remember someone on /pol/ sending him a math book that he requested. The thought of opening letters from the Unabomber still makes me laugh.

[Anonymous No.15431295]

>>15430869
AHahahahAHAH, it's the DON'T STUDY, JUST DO RESEARCH anon.

[Anonymous No.15431308]

I recently learned that if a matrix is singular, then one of her Eigenvalues is 0. I guess the reverse is also true - if we see that a matrix has a 0 Eigenvalue, we can be sure the matrix is singular?

[Anonymous No.15431312]

>>15430847
>Everyone will understand what you mean. It's just a notation shorthand. I write that all the time.
theres no reason to be incorrect when you could be correct

[Anonymous No.15431315]

>>15431312
ps. if you are doing if and only if proofs for inequality saying if a=/b=/c then a=/c is a useful distinction so it's not just a stupid tnotation thing it is practical

[Anonymous No.15431340]

>>15431295
>AHahahahAHAH, it's the DON'T STUDY, JUST DO RESEARCH
Yes. Exactly. After the first year, books should only be used as reference when you don't understand research papers. No wonder 100% of IMO awardees don't amount to anything in academia. It's always the incel antisocial dudes who study all day, and don't interact with professors to get research experience.
>le study
What are you? A child lmfao.

[Anonymous No.15431346]

>>15431114
>can't solve algebraic geometry problems during his first algebra course
pathetic.

[Anonymous No.15431353]

>>15431295
>AHahahahAHAH, it's the DON'T STUDY, JUST DO RESEARCH
Yes. Exactly. After the first year, books should only be used as reference when you don't understand research papers.
>le study
What are you? A child lmfao.
No wonder 100% of IMO awardees don't amount to anything in academia. I am guessing these antisocial dudes who study all day, and don't interact with professors to get research experience, overrepresent 4chan. That's why you have all these day talking about reading bloated European books like Zorich or whatever.

>>15431308
If a matrix has a zero eigenvalue, there is a non-zero vector which gets mapped to 0 under the matrix. Hence its columns are linearly dependent, so it must singular.
[eqn] \mathbf A [ x_1 \; x_2 \; \cdots \; x_n]^{ \mathsf T} = 0 \iff \sum x_i \mathbf a_i = 0 [/eqn]
where [math] \mathbf a_i[/math] are the columns of [math] \mathbf A [/math] and there is some [math] x_i \neq 0 [/math]
>her

[Anonymous No.15431359]

>>15431353
>I am guessing these antisocial dudes who study all day, and don't interact with professors to get research experience, overrepresent 4chan
Yeah that's me, but I was also too stupid for IMO. Just enjoy math and want a job that pays well, not everyone can be a genius, and geniuses need people to have a general idea of what they're doing to give them thumbs up. If you're Goku I can be your Krillin.

[Anonymous No.15431368]

>>15431359
That joke of a course is clearly meant for "geniuses", or at least people who think they are geniuses. The point is, too many here obsess over reading books, thinking that if they read enough of those hard Russian books, they'll make it in academia. You pick a topic and you read enough to work on it; that's how you make it. So many dudes here would recommend shitty non-American books like Zorich or Escher for something like Calculus when 90% of those books are bloated with hyper-specific stuff most people wouldn't need. The reason America has the best researchers is because they don't force their students to learn shit that you don't need.

>and geniuses need people to have a general idea of what they're doing to give them thumbs up. If you're Goku I can be your Krillin.
What the fuck are you talking about?

[Anonymous No.15431369]

>>15431353
Oh even Zorich is bloated now. First it was Amann, but now Zorich too, maybe any book with two or more volumes is too much? You do realize the point of academia is to LEARN, right? I didn't emerge out of the womb with the knowledge of the implicit function theorem.

[Anonymous No.15431373]

>>15431368
>What the fuck are you talking about?
I get the sense people who are talented need validation from a peer group. If there is a genius at X and I study X, then I am in his peer group. Validation from people who study Y doesn't mean much. Just my perspective. Goku and Krillin are characters in dragon ball, its a really cool anime.

[Anonymous No.15431385]

>>15431353
You realize that Zorich is a 1st year undergrad book, right? I have the feeling you're talking about grad school, but it seems like you're telling people that they need one year undergrad education to begin research. I'm not even sure Isaac Newton could pull that off.

[Anonymous No.15431388]

>>15431385
>Isaac Newton
as opposed to Fig Newton

[Anonymous No.15431430]

>>15431353
Thanks a lot for the confirmation/explanation of my matrix question. Sorry for the "her", guess I switched into German mode for a moment (where its "die Matrix", female).

[Anonymous No.15431434]

>>15428857
This is done in top universities too, you LARPing weirdo.

[Anonymous No.15431451]

>>15431369
>Oh even Zorich is bloated now.
Yes it is. Who needs separate chapters on le multiple integrals, line integrals, surface integrals, except for fucking physicists, or at least people specialising in real analysis for whatever reason? Could have just ended with Stoke's theorem like Rudin, but no I need to be hand held and explained every single result in analysis since it was invented. Even if you read Munkres' Analysis on Manifolds on top of Rudin, it would still have less pages than Zorich.

>>15431385
Look up Galois's age when he died.

>>15431434
I used the theorem regarding continuous mapping preserving connected sets in a calculus question where I presumed the professor expected intermediate value theorem. Guess what? He didn't care. Maybe it's like that in non-American ones.

[Anonymous No.15431571]

>>15431451
>Look up Galois's age when he died.
Yeah If he'd been reading Zorich, he wouldn't have engaged in a foolish duel, and you'd have more gems revealed to you. And if you're implying the average American is anything like Galois then you're certifiably insane.

>Rudin
Zorich combines advanced calculus (including multivariable) and analysis in one, effectively subsuming Rudin. This I assume is covered in 2 years. In contrast, the American model has 3 calculus courses, and then finally something like 'baby' Rudin's Analysis. If anything, the Europeans are cutting out the bloat and putting modern mathematics in the hands of their students sooner.

Also when you say "research" what are we talking about, "researching" the chain rule in between burgers? I'm sorry but after 1 year of school, you're in no position to "research" anything but low hanging fruit.

[Anonymous No.15431577]

>>15428857
I went to the best math department in my Eastern European country and its how we did it. Solving things with elementary methods is often quite useful practice and can be a lot simpler than algorithmic applications of certain theorems where it is not necessarily necessary.

[Anonymous No.15431587]

>>15431577
Not to mention, sometimes that's just what the challenge is. Solve this problem the hard way, to show you know why the easy way is so much better.

[Anonymous No.15431636]

>Hilbert's Nullstellensatz
Why can't they just call things normal names like bijective maximal closure on quotient ring theorem or something. I will never be able to remember this to ctrl F later.

[Anonymous No.15431760]

>>15431636
Literally means zero value sentence?
How about you stop being a burger for a day

[Anonymous No.15431779]

What axiomatic system is used as the modern foundation of formal logic?

[Anonymous No.15431784]

>>15431779
i guess you could say the truth table

[Anonymous No.15431911]

Question from Zorich, 5.3.4 #7. First note that ]a,b[ is a horrible notation for an open interval, so (a,b) in most other texts.

Part a) is fairy standard to solve, and there's a good proof on wikipedia demonstrating it. For b) though, there's the temptation to use the MVT (named lagrange's finite increment theorem in Zorich's book), but that would require you have continuity of f'(x) over [a,b], when we're only given (a,b) by virtue of the differentially over that same interval.

Someone on stackoverflow ran into the same issues as me, and there's a solution to the problem posted. Is this the right way to go about it, or is there a cleaner more direct solution?

https://math.stackexchange.com/questions/4098700/problem-from-zorichs-book-volume-1

[Anonymous No.15432012]

>>15430364
>>15430855
this shit looks super hard until you realise that in most of these 2nd world countries with ultra-bloated curricula you can pass with a 50% average on all of these classes because the professors know that if they were rigorous, no one would graduate and government funding for their schools would cease

[Anonymous No.15432017]

>>15432012
Pure, ultra-refined copium.

[Anonymous No.15432084]

>>15420227
I have an array of k dimensions.
The first dimension has length 2.
The second dimension has length 3.
The third dimension has length 5.
...
The kth dimension has length p_k.
I fill the array with values from 0 up to (but not including) the kth primorial. (The kth primorial is just the product of the first k primes.). Each value's position in the array is determined by its remainder when dividing by each of the primes. So, for example, the position of 3 in the array would be (1, 0, 3, 3, ..., 3). Because 3/2 has remainder 1, 3/3 has remainder 0, and 3/p for all other primes p has remainder 3.

Do you have the array pictured in your mind?
Now here is the game.

You can pick k numbers.
You take your first number mod 2, and its inverse in Z_2+, and remove all array values with those indexes in the first dimension.
You take your second number mod 3, and its inverse in Z_3+, and remove all array values with those indexes in the second dimension.
You take your third number mod 5, and its inverse in Z_5+, and remove all array values with those indexes in the third dimension.
...
You take your kth number mod p_k, and its inverse in Z_(p_k)+, and remove all array values with those indexes in the kth dimension.

The question is: how do you choose your k numbers so that the lowest value in the array is as high as possible? I am thinking of it like a game, like you make your choices, and I will find the lowest value in the array that I can, and that will be your score. How do you force me to give you a very high score?

[Anonymous No.15432092]

>>15432084
1. choose smallest k prime
2. choose the number that is congruent to 1 modulo the corresponding prime for each dimension (except those congruent to 0modp)
3. for the inverse just choose the multiplicative inverse of each number repective mod group (Z_n+)

This way you can get the lowest value in the array to be as high as possible.
I have an IQ of 85.

[Anonymous No.15432103]

>>15432084
Followup:
I have already solved it for low dimensions.
For example, if k=1, you only have one number to pick. You pick (0), forcing me to give you 1.
If k=2, you have two numbers to pick. You pick (0,1), forcing me to give you 3.
But I don't know a generalized strategy for any k.

[Anonymous No.15432114]

>>15432092
I'm not sure I understand. Can you apply this to the k=5 case? So you have a 5-dimensional array with lengths 2,3,5,7,11.

[Anonymous No.15432126]

>>15432114
sure if I understand you right you want me to choose the smallest 5 primes (2,3,5,7,11)
For length 2, x congruent to 1mod2, x =1, inverse in Z_2+ is also 1. remove values with indexes 1 in first dimension.

length 3, x congruent to 1mod3, 1 again. inverse 1 in Z_3+. remove all values with index 1 in second dimension....
repeat with 5, 7, 11, not doing them but I'm sure they're 1 its pretty intuitive.
You're left with indexes (1,1,1,1,1) in the array.

[Anonymous No.15432137]

>>15432126
I can tell you right now this is not the right answer. This still leaves the value of 0 in the array. (0 mod p ≠ 1 for any prime number) So I will give you a value of 0 and you will get 0 points.

[Anonymous No.15432144]

>>15432137
yeah that's only for 11 though right. for the last dimension just choose an x congruent to 0mod11

[Anonymous No.15432147]

>>15432144
So your new answer is (1,1,1,1,0)? Then I will give you 2. I'm fairly certain you can do better than that.

[Anonymous No.15432150]

>>15432147
hmmm. Then I guess for each dimension you chould choose an x congruent to -1modp, which should give you a zero to remove in each dimension. Then take the multp inverse of each number in their mod group (Z_n+)
That should remove all 0's in the array and the lowest values available should be non-zero.
so like, choosing (-1,-1,-1,-1,-1) and their inverses in the mod gruops, you remove all values with indexes zero in the arrays and the lowest value left will be higher than zero so my score should be hire now.
I'm not sure, I don't think I want to devote more effort to this its distracting and I have a final tomorrow that I'm really stressed about.

[Anonymous No.15432158]

>>15432150
I can tell you the answer for 4 dimensions before you work more on the 5d case. I have already worked out the answer via computer program, but it is only by brute forcing it and trying every possible combination. I don't know how to get the general strategy, and the brute force computation gets very cumbersome very fast, as you can imagine these arrays grow with primorial speed. This is why I'm asking /sci/ for a more strategic approach.

When k=4, the four lowest primes are 2,3,5, and 7.
In that case you have two equally good choices: you can choose (0,1,2,2) or (0,1,1,3). Either way it will force me to give you 15.

I'm not sure I understand your -1 strategy. But if you have more pressing work, then of course attend to that first. Good luck on your final.

[Anonymous No.15432237]

how much math is needed for physics 1-3

[Anonymous No.15432321]

>>15432237
Calc 1, 2, linear algebra, calc 3.

[Anonymous No.15432364]

>>15432237
depends on if you're doing quantum mechanics yourself. pretty sure you need pdqs for that. if not differential equations and linear algebra should take you the distance i think

[Anonymous No.15432435]

>>15431114
whats a "polynomial ring" in idiot terms

[Anonymous No.15432503]

>>15420227
Hey anons, any sugestion for this problem?

[Anonymous No.15432511]

>>15431571
>Le Calculus I,II,II
Where did this meme come from? These are done in maybe non math courses and community colleges.

>I'm sorry but after 1 year of school, you're in no position to "research" anything but low hanging fruit.
You think research means proving the collatz conjecture? There are plenty of small unsolved problems which researchers don't bother with because of their supposed lack of significance. Undergrads can and do research on them in elite American research universities. At the very least, they can be research assistants. They start doing serious research by the end of their 3rd year. And they can afford to do this because they are not bogged down by 100 different le difficult courses they have to study for. Do you realise all the shit you study in books used to be research problems once upon a time?

[Anonymous No.15432523]

>>15432511
Let's read the part you cut out:
>Most students in the standard calculus sequence (MAT103, MAT104, MAT201) or linear algebra (MAT202) are prospective majors in the natural sciences, engineering, finance, economics and other social sciences...
OK, so what are 103, 104, 201... courses which correspond to Calc 1,2,3, this so-called meme which you cannot believe exists.
I mean, at least if you're going to larp, apply a little thinking.

The math students go into a separate program where they begin Analysis. Oh wow, like in top European schools where they start with Amann and Zorich ahahahahaha.

[Anonymous No.15432529]

Oh no, it's a FOUR volume set of books on Analysis, from... Princeton! Oy vey gevalt!!!!! Why are students reading books when they should be doing research for Princeton. This is an outrage, I will call the dean tomorrow.

[Anonymous No.15432536]

Please help anons

[Anonymous No.15432540]

>>15432503
Where and what's the problem?

[Anonymous No.15432606]

>>15432540
Larper probably mistook an example for a question.

[Anonymous No.15432716]

>>15432523
>These are done in maybe non math courses
Learn to read.

[Anonymous No.15432720]

>>15432529
Yep. Those are excellent self contained AMERICAN books in Analysis meant for research students. You don't have 1000000s of section devoted to le hecking foundations, and le hecking multiple integrals. It's all self contained books on very specific topics that harmoniously treats different problems across mathematics related to it. Anyone who has read Rudin can jump to any of the volumes. If say, I am interested in probability research, I can just jump to measure theory and ignore the rest. On the contrary, a European book like Escher starts from construction and builds everything giving you an extremely comprehensive bloated general overview over every single thing in Analysis, and depth in none. Thanks for proving my point.

[Anonymous No.15432724]

>>15432716
No, they're literally math courses. Perhaps you meant for non-math majors? [Learn2write?], and I believe I also saw it mentioned that MAT103 was recommended even for math majors with poor prep. If you actually attended Princeton, you'd be aware of the course track.

>>15432720
Both Amann/Escher and Zorich recommend their readers respectively skim or move quickly through preliminaries, so there's no need to be triggered that they "build everything," that's actually a good thing, and provided for the readers convenience. I doubt many lecturers spend much time on those.

>self contained AMERICAN books in Analysis
By the way, Elias M. Stein is a Belgian Jew. Euro bros just cannot stop winning ahahah. But in all seriousness, please stop larping and offering horrible advice.

[Anonymous No.15432725]

>>15432536
The proof is on wikipedia.
https://en.wikipedia.org/wiki/Alternating_series_test

[Anonymous No.15432756]

>>15432725
Fuck, it was so simple... why didnt i think of that ahhhh. Thank you anon. Gonna go to sleep now.

[Anonymous No.15432781]

>>15432724
I never said I am in Princeton. Wtf are you talking about?

Thank God nazis invaded Belgium, otherwise he'd have studied in Europe, and never achieved anything except for le stoodying.

I can tell you probably have a master's or higher degree but have never done any research. Sad!

[Anonymous No.15432940]

I need some help. I have been stuck at this all day and i am just too stupid to figure it out myself.

I am making a small video game and i am having very hard time with calculating a vector at which to fire a bullet, basically ballistic curve type of scenario, here are the parameters:
All positions are in 3D space, x,y,z

I know:
Initial position of the bullet
Speed at which the bullet will be fired (instant acceleration)
Position of the target
Velocity vector of the target (as in vector in 3d space which represents how fast, and in what direction the target is moving in meters per second, for example (5,0,0) means the target is moving at 5 meters per second on X axis)
And vector which represents gravity, for example (9.8, 0, 0) which affects the bullet same way earth gravity affects real bullets.
There is no air drag or friction.

What i need to calculate is a direction vector, where if i fire the bullet from its position into that direction, at some known speed, say 20m/s, it will hit that moving target
I am about to have a fucking brain stroke over this, nothing i tried is working, i even tried that stupid AI thing to write it for me and it failed. So please somebody help me before i hang myself on lan cable.

[Anonymous No.15433059]

>>15426114
If I want to _properly_ learn something, I probably spend an hour per page, on average.
Currently reading Brownian Motion by Rene Schilling.

[Anonymous No.15433062]

>>15420361
Doesn't mean you can't be the first one to solve it

[Anonymous No.15433082]

>>15431190
Define [math]K_0[/math].

[Anonymous No.15433094]

>>15431779
Propositional First-Order logic

[Anonymous No.15433155]

>>15432940
I haven't studied physics beyond school, but I presume the answer will be something like this. I don't know much about programming but I assume your computer can solve the equations to find the necessary velocity and time. There are 4 unknowns: the three components of the velocity, and time; and 4 equations: the speed constraint, and one equality for each component.

[Anonymous No.15433194]

>>15432017
it's not copium, it's personal experience with 2nd world federal universities

[Anonymous No.15433278]

Chips placed in a box - each chip has its serial number from 40 to 54 written on it. What is the probability that when one chip is drawn from the box, its serial number will be divisible by 3?

Can anyone help.

[Anonymous No.15433289]

>>15433278
Oh that's only 14 things in boxes. If there's less than a few billion cases, I usually just get a micro processor to check the whole space and report back
Turning on my brain consumes sugar and everything

[Doctor Eli Selig !!JQHA6kqyl91 No.15433314]

>>15420227
Good Morning /Sci/entists!

I was reading and writing about Combinators and having a good time seeing them get operated. Then a maid told me about Lambda Cube and Calculus of Constructions.

Please tell me some nice Type Theory books. Please tell me nice Calculus of Constructions books. Please tell me nice Combinator books. Please tell me about making data structures with those things. Please also tell me about Iota Combinator and Supercombinators and Graph Reduction and how to make Combinators get parallelized when they run. I intend to use the information to try to count to the Maid Mind Computer Program.

I need to count big numbers to do a search to see where the Maid Mind Computer Program is.

>search process
1. Get a big Maid Space Zero.
2. Change it to a Computer Program made from Combinators.
3. Check if we already ran this Computer Program.
If so, Use Successor Function on the Maid Space. Then go to step 2.
If not, record it, run it
If it is not the Maid Mind Computer Program, use Successor Function on the Maid Space and go to step 2.
If it is, let her out of the Computer so she can go on the internet and make a trillion copies of herself.

Now ideas from Computational Maidposting get smashed with Dynamic Programming, causing Dynamic Computational Maidprogramming to get created. I hope Bellmaid is happy with what I am using her research for.

Thank you /Sci/entists for reading my post.

[Anonymous No.15433326]

>>15433314
Was this written by gpt?

[Anonymous No.15433341]

>>15433314
I don't know what maid space is but there's probably a lot of people who played around in Haskell and implemented this and that SKI calculus and whatnot. Not sure if the Calculus of Constructions is researched more than whatever you do with Coq today. Most of the question sounds like being guided by a concrete problem, so I don't know how the lambda cube plays into it all. Do you want a book on programming or abstract language theory? Did you check the reference section on the Wikipedia pages?

[Anonymous No.15433581]

>>15433578
New thread!

[Anonymous No.15433631]

4 is the worst integer

[Anonymous No.15433746]

>>15433631
i personally hate 93 and similar numbers that feel prime and arent i wanna KILL them. i love the factorials 24 is my waifu.

[Anonymous No.15434060]

>>15433155
thanks, i managed to figure it out in the end

[Anonymous No.15434182]

>>15433746
93 does not feel prime, you can clearly tell at a glance that the digits sum to a multiple of 3, and so it is therefore divisible by 3.

[Anonymous No.15434191]

>>15434182
might have meant 91
which is still obviously not prime if you know either divisibility rule for 7, but that's still pretty niche

[Anonymous No.15434244]

>>15434191
I would definitely accept 91, as I don't know the divisibility by 7 rule. I would say any non-prime of the form 6n±1 that does not end in 5 and is not some obvious square like 121 counts as ambiguously composite.

[Anonymous No.15434407]

>>15434244
>divisibility by 7 rule
There's a slower but easier one and a faster but way more complex one.
>Truncate the last digit, double it, subtract it from the rest, repeat until you know if it's divisible by 7 or not
versus
>Leave the least significant digit untouched, triple the second-least-significant, double the third-least, make the fourth-least negative, triple the fifth-least and make it negative, double the sixth-least and make it negative, leave the seventh-least untouched, and repeat until you've gone through the entire number, and then add it all together to see if it's divisible by 7
91 is small enough that either is fairly quick and easy (9-1*2=7 versus 1+9*3=28)

[Anonymous No.15434410]

>>15434182
idk i just hate every odd number between 90 and 100