/mg/- mathematics general

[Anonymous No.15279472]

Previously >>15260487

Armenian genocide edition

Talk maths.

[Anonymous No.15279493]

Thanks for the new thread anon.

NEW THREAD NEW PROBLEM!!!!
Don't know how hard this one is,haven't attempted it yet. I took it from a combinatorics problems book and it was under the introductory section so it shouldn't too hard hopefully! As always all replies are appreciated. Good luck to all those who want to attempt it! I apologize in advance if I reply late, it's getting late here.

[Anonymous No.15279584]

>>15279493

[Anonymous No.15279642]

Are there any examples of an object which can be proven to exist, yet which are (provably, let's say) incapable of being found, given, specified?

[Anonymous No.15279648]

>>15279642
Godel incompleteness theorem

[Anonymous No.15279658]

>>15279648

Oh, it really is that? For real? Jeez. I must have forgotten the statement of the (first?) theorem and then regurgitated it just now.

I really ought to re-read the Nagel reader on same, I sort of fudged my way through about ten pages of the proof details (their paraphrase of same, not literally what he did I guess).

[Anonymous No.15279689]

WHAT IS SCRIPT THEORY?

Script theory is the study of what is written and translations of what is written. The
basic objects of study are the script space a.k.a. cancellative univalent monoid and
the calc map of script spaces. Univalence is a new property of a binary operator with
a left and right identity: the product of two non-identity elements is a non-identity
element. The calc property of a map is also a new one, preserving the identity element
and binary prefix relation.

WHY STUDY SCRIPT THEORY?

One of the most exciting new objects in script theory is the *link product*. Instead
of developing it directly, I'll motivate a partial definition and then extend to the
full case. Take a script space A and equip the set of unary calc operators on A with
composition of maps, turning it into monoid E. Next, note that A acts on E on the
right by shift, and the shift obeys the chain rule
>(e o f)^a = e^{f(a)} o f^a.
As a result, the link product
>(a,e) (b,f) = (ae(b),e^b o f)
turns the Cartesian product A x E into a monoid. Instead of taking all of E, we can
take any shift invariant submonoid of E, and instead of using a one-to-one inclusion,
we can take a script space C and a monoid homomorphism phi : C -> E such that
1. C is a right A-set
2. phi(c a) = phi(c)^a
3. The action of A on C satisfies the chain rule condition
>(c d) a = ( c phi(d)(a) ) ( d a )
Now we can define the link product in full on A x C as
>(a,c) (b,d) = (a phi(c)(b),(c b) d).

[Anonymous No.15279698]

>>15279493
[eqn] \begin{pmatrix} 1 & 1 & -1 & -1 \\
1 & 1 & -1 & -1 \\
-1 & -1 & 1 & 1 \\
-1 & -1 & 1 & 1 \end{pmatrix} \\
\begin{pmatrix} 1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
-1 & 1 & -1 & 1 \\
-1 & -1 & 1 & 1 \end{pmatrix} \\
\begin{pmatrix} 1 & 1 & -1 & -1 \\
1 & -1 & -1 & 1 \\
-1 & 1 & 1 & -1 \\
-1 & -1 & 1 & 1 \end{pmatrix}
[/eqn]
and the matrices you get by permutating the 4 columns and last 3 rows. That should be
[eqn]18 + 2*144 = 306 [/eqn]

[Anonymous No.15279699]

>>15279698
Oh wait the last two of the matrices are actually the same up to the permutations so it's only 18 + 144 = 162

[Anonymous No.15279700]

Anybody go back to school? I'm 29 and got lucky in tech to never work again so I feel like I should do something actually interesting.

[Anonymous No.15279702]

>>15279689
Let A,B be script spaces and e : A -> B a calc map. The *shift* of e by a <- A is the map f = e^a satisfying the condition
>e(a c) = e(a) f(c)
for all a,c <- A.

[Anonymous No.15279731]

>>15279698
>>15279699
>>15279493
pretty sure you could do this as a SAT problem

[Anonymous No.15279762]

>>15279642
Well orderings of arbitrary sets, choice functions for arbitrary sets, bases for arbitrary vector spaces, uncomputable numbers like Chaitin's constant.

[Anonymous No.15279775]

Tested IQ 116 here. I struggled with math and hated doing it in all of school and was always a “social science kid”, but now in uni I’m thinking of getting into programming. To what degree can I naturally improve my math skill through practice/study?

[Anonymous No.15279800]

Uncomputable numbers are retarded mathematicians fantasy

[Anonymous No.15279818]

>>15279584
Hi anon. I'm sorry but I don't understand why you replied with a picture of Jacques Hadamard smiling. I'd be really happy if you explained.
>>15279698
>>15279699
Unfortunately anon the answer is neither 306 nor 162. But thanks for the attempt!
I think you can definitely get the right answer. Did you make a mistake somewhere in there by accident?
>>15279731
Well I don't know how difficult the SAT is anon, I can't really comment on that.

[Anonymous No.15279925]

>>15279800
What 0 geometry does to a mf

[Anonymous No.15279963]

>>15279493
Row 1, we have [math]{4 \choose 2}=6[/math] possibilities we may freely pick from, each uniquely determined by the position of the +1s.
The same is true of row 2. This gives us 36 possible cases to consider with the first two rows.

Rows 3 and 4, however, are constrained by the first two:
If rows 1 and 2 are identical (six such cases), then rows 3 and 4 are each forced to be their inverse. Because these arrays are uniquely determined by our choice for row 1, there are only six of them.
If rows 1 and 2 are inverses (six such cases), then row 3 is completely free and row 4 is forced to be its inverse. This provides us with free choices in rows 1 and 3, and so gives us 36 more arrays to consider.
This leaves 24 linearly-independent pairs of initial rows. In this case, row 3 is forced to be the inverse of one of the initial two, and row 4 is completely dependent on the first three. This provides us with another 48 arrays to consider in total.

Hence we arrive at 6+36+48=90 such arrays.
Now tell me where and how badly I fucked up.

[Anonymous No.15279972]

>well founded
>well ordered
>ordinal
>lemma
>morphism
>recursively ...
>axiom
>transfinite
>cardinality
>relations
>decidable
>satisfiable
>uncountability
Foundational mathematics is making me lose my sanity, wtf is all of this shit supposed to mean?

[Anonymous No.15280006]

>>15279963
Oh my god... I don't even know where to begin. I really want to tell you where your mistake is anon but that just doesn't seem feasible.
Since there's no mistake, haha!
Nice job anon! Your solution and answer are both 100% correct. Full marks!
I mean, your explanation is very clear. Your answer is of course the right one. The only thing I can criticize you for is saying "Now tell me where and how badly I f**** up." when you've done such a good job. But of course I understand your caution, I can never be sure of answers either.

That's it then. What did you think about the problem? Not gonna lie, I personally didn't actually like it all that much. The answer almost felt a bit disappointing, I don't know why I feel this way. Usually it's the opposite.

Thank you very much again for your time and effort. Have a pleasant day!

[Anonymous No.15280154]

Should I bother reading Spivak(calculus) or just go into something like Zorich? I’m open to other suggestions as well, I am just stuck on what to do.

[Anonymous No.15280248]

How do i calculate [eqn] \frac{3^2\cdot7^2\cdot11^2\cdot\cdot\cdot}{(3^2-1)\cdot(7^2-1)\cdot(11^2-1)\cdot\cdot\cdot} \times \frac{(5^2-1)\cdot(9^2-1)\cdot(13^2-1)\cdot\cdot\cdot}{5^2\cdot9^2\cdot13^2\cdot\cdot\cdot} [/eqn]

[Anonymous No.15280429]

How do i find the fxy and fyx (mixed partial derivatives) of e^(x^2 y)-e^(xy^2 )

[Anonymous No.15280654]

>>15264697
does this mean algebraic geometry is complete?

[Anonymous No.15280680]

>>15279925
pseud

[Anonymous No.15280691]

>>15280680
El goblino filtrado...

[Anonymous No.15280695]

>>15280691
show me your uncomputable numbers then retard

[Anonymous No.15280726]

>>15280248
[eqn]
\lim_{N \to \infty} \prod_{k=1}^N \frac{(4k-1)^2((4k+1)^2 - 1)}{((4k-1)^2 - 1)(4k+1)^2} \\
= \lim_{N \to \infty} \left( (2N+1) \left( \frac{(4N-1)!!!!}{(4N+1)!!!!} \right)^2 \right) \\
= \lim_{N \to \infty} \left( (2N+1) \left( \frac{\pi \Gamma(N + \frac{3}{4})}{\sqrt{8} \Gamma(\frac{3}{4})^2 \Gamma(N + \frac{5}{4})} \right)^2 \right) \\
= \frac{\pi^2}{4 \Gamma \left( \frac{3}{4} \right)^4}
[/eqn]

[Anonymous No.15280744]

question to the anons of this general

after you complete a proof, how good at you at remembering your thought process that lead you there?

[Anonymous No.15280756]

>>15279519
GPT-4 is the new, supposedly much better model you get access to if you pay $20 a month on ChatGPT, so you likely tried out the old, free one, which is GPT3.5 I think.
GPT-4's math capabilities are much better than GPT3.5 but that doesn't mean a whole lot as this ultimately boils down to it making basic arithmetic mistakes less often.

[Anonymous No.15280760]

>>15279519
GPT-4 is the new, supposedly much better model you get access to if you pay $20 a month on ChatGPT, so you likely tried out the old one, which is GPT3.5 I think.
GPT-4's much better at math than the previous one but that doesn't mean much as the previous one was terrible even at basic arithmetic. It's not that bad at recalling definitions and theorems, though.

[Anonymous No.15280762]

question to the anons of this general

after you complete a proof, how good at you at remembering your thought process that lead you there?

[Anonymous No.15280793]

>>15280695
"Real numbers" clasically are supposed to model the line, with each "number" representing a point. We do this because since the time of descartes at least we realized that treating points like they are numbers is really fucking powerful. Your confusion arises (and I admit this is a problem with maths education, we don't explain students how things came to be) in thinking that points are supposed to LITERALLY be numbers. Of course, if you draw a line and mark two points with 0 and 1, you can then match alot of points with the integers, rationals, computable numbers, ect. Hence we say things like "the rationals are a subset of the reals" which is an abuse of language that works for our daily lives. But if we were to be REALLY technical about it, we would say that there's a subset of the reals whose structure is identical to the rationals, or in other words, they behave the same, hence we can make our "abuses" without much problem. Having all this in mind, why should we expect all points in a line to be computable?

[Anonymous No.15280826]

>>15279972
>well founded
X a bottom
>well orderer
The order for X also works on the powerset PX
>ordinal
Transitive set of transitive set ("well stuffed", membership models and order)
>lemma
Theorem
>morphism
Composable connection
>recursively ...
Definition uses the object defined
>axiom
Theorem provable without other inference rules
>transfinite
Beyond the naturals
>cardinality
Stratification using bijective functions
>relations
anon, I...
>decidable
One part of an excluded middle disjunction is explicitly provable
>satisfiable
>uncountability
Lack of a counting function with domain beyond the set of naturals

[Anonymous No.15281000]

[eqn] \sum_{i=1}^n i {n \choose i} !n = n! [/eqn]

[Anonymous No.15281005]

[eqn] \sum_{i=1}^n !(n-i) i {n \choose i} = n! [/eqn]

[Anonymous No.15281086]

>>15280726
U a legend, thanks

[Anonymous No.15281105]

>>15280793
>all that garbage gibberish talk
We already have an abstraction for that. It's called [eqn]\infty[/eqn] and all your "uncomputable" crap lies between 0 and [eqn]\pm\infty[/eqn]

[Anonymous No.15281297]

>>15281105
Cope & Seethe

[Anonymous No.15281326]

>>15281105
Uncomputable numbers are numbers that you can't describe an algorithm to output. Basically if you can describe a number then it's computable. It's kind of a mindfuck, like trying to think of a new color.

[Anonymous No.15281465]

we need a reformation

[Anonymous No.15281471]

>>15279702
* What are the important facts?

Notation: Za is the integers
- script space
- a script space is a cancellative univalent monoid
- calc map
- a calc map of script spaces preserves the identity and binary prefix
relation between scripts
- shift of a calc
- FACT: shift of a calc is a calc
- chain rule for shifts of compositions of calc maps
- FACT: there is a formula for computing shifts of compositions of calcs
- link product
- FACT: the link product turns a Cartesian product A x C of a cancellative
monoid A and monoid C together with a homomorphism phi : C -> Calcs(A)
into a monoid
- semiring, semifield, semimodule, semivector space, semialgebra
- Let S be a commutatie, unital semiring. An S-*semialgebra* T is an
S-semimodule
- monoid ring
- example: using the monoid ring Za[A x E] where E = Calcs(A) and A x E
has the link product to enumerate observations progressively as a
matter of calculating event probabilities
- example: bingo time with an n-sided die, using a monoid ring to
calculate the probability of yelling "BINGO" after k throws
- direct product and free product of script spaces
- monoid congruence property of a binary relation on a monoid
- quotient of a monoid by a monoid congruence
- monoid congruence generated by a binary relation

[Anonymous No.15281483]

>>15281471
*_What_are_the_important_facts?

-_script_space
__-_a_script_space_is_a_cancellative_univalent_monoid_
-_calc_map
__-_a_calc_map_of_script_spaces_preserves_the_identity_and_binary_prefix
____relation_between_scripts
-_shift_of_a_calc
__-_FACT:_shift_of_a_calc_is_a_calc
-_chain_rule_for_shifts_of_compositions_of_calc_maps
__-_FACT:_there_is_a_formula_for_computing_shifts_of_compositions_of_calcs
-_link_product
__-_FACT:_the_link_product_turns_a_Cartesian_product_A_x_C_of_a_cancellative
____monoid_A_and_monoid_C_together_with_a_homomorphism_phi_:_C_->_Calcs(A)
____into_a_monoid
-_semiring,_semifield,_semimodule,_semivector_space,_semialgebra
__-_Let_S_be_a_commutatie,_unital_semiring._An_S-*semialgebra*_T_is_an
____S-semimodule_
-_monoid_ring
__-_example:_using_the_monoid_ring_Za[A_x_E]_where_E_=_Calcs(A)_and_A_x_E
____has_the_link_product_to_enumerate_observations_progressively_as_a
____matter_of_calculating_event_probabilities
__-_example:_bingo_time_with_an_n-sided_die,_using_a_monoid_ring_to
____calculate_the_probability_of_yelling_"BINGO"_after_k_throws
-_direct_product_and_free_product_of_script_spaces
-_monoid_congruence_property_of_a_binary_relation_on_a_monoid
__-_quotient_of_a_monoid_by_a_monoid_congruence
__-_monoid_congruence_generated_by_a_binary_relation

[Anonymous No.15281540]

>>15279472
I took Calc 1 and 2 about five years ago. I can do everything up to them easily, but I remember very little of them. I want to return to college in the summer and take Calc 3 as a summer course. What can I do to regain my calculus powers in the next two months? And is Calc 3 harder than 2? Because 2 was probably the hardest class I’ve ever taken in my life.

[Anonymous No.15281590]

>>15281297
dilate
>>15281326
And these by definition lie within infinity. You're just delusional bro ong

[Anonymous No.15281837]

>>15281471
>>15281483
>with a homomorphism phi : C -> Calcs(A)
into a monoid
should read
>with a homomorphism phi : C -> Calcs(A) and a right action of A on C satisfying the linkage conditions [1] into a monoid
[1] see points 2. and 3. in
>>15279689

[Anonymous No.15282073]

since axioms don't have any way to be proven without infinite regrees, and they can be chosen independant of their relation to the physical world, are there any maths books where the existence of god is treated as an axiom?

[Anonymous No.15282169]

>>15282073
https://www.biblegateway.com/passage/?search=John%201&version=NLT

[Anonymous No.15282200]

>>15279472
when you really think about it math is a spiritual activity. it is about understanding the universe and applying its principles towards human achievement. engineering buildings is a good example of applying mathematics to solve problems with overpopulation but the issue is that people forget the spiritual aspects and apply it to everything, even things it should not be applied to like making porn with computers.

anyway, mathematicians are also pretty good at seeing the future because they can draw logical inferences about what will happen given a certain set of circumstances like a growing population and dwindling resources for food and shelter.

[Anonymous No.15282247]

>>15282200
https://soundcloud.com/lucifer-rothschild-363442434/seyyed-hossein-nasr-knowledge-and-the-sacred

[Anonymous No.15282329]

>>15282247
those keywords sound retarded anon, what's the summary b/c i ain't reading all that

[Anonymous No.15282342]

>>15282329
He's a pioneer of Islamic achemical a.k.a. Islamoegyptian science.

[Anonymous No.15282387]

what exactly counts as maths
if i start from any set of axioms and used them to prove statements is that maths no matter what the axioms are?
let's say there was an axiomatic system for physics, would that just be maths then?
or if there was an axiomatic system for natural languages. Would that be maths then, and not just linguistics?

[Anonymous No.15282429]

Just a formula for prime numbers

[Anonymous No.15282434]

Any good books on Iwasawa theory?
Also anything on the arithmetic of abelian varieties? Mumfords book seems more concerned with the alg geo aspect, rather than arithmetic questions, and milne’s notes for it are still incomplete

[Anonymous No.15282436]

>>15282387
ya that's all maths, it's actually impossible to not do maths if you are sentient and can plan more than a few minutes ahead. prediction and projection is mathematical. for example, if population of earth keeps growing without changes in resource utilization then hunger and famine will shortly follow

[Anonymous No.15282551]

Pi approximation

[Anonymous No.15282564]

>>15282551
>calculate pi with sin
genius

[Anonymous No.15282586]

>calculus : computational math :: ???? : theoretical math

wjat do you think

[Anonymous No.15282602]

there is no money in doing foundations

[Anonymous No.15282699]

Some of the hardest most intractable problems that have stood unsolved for centuries are in additive number theory and integer partitions. How TF is addition of numbers so hard???

[Anonymous No.15282813]

>>15282699
Terrestrial monkeys don't really have big brains. Other animals are much smarter than us in many ways. for example, other animals are not the cause of the 6th mass extinction

[Anonymous No.15283101]

I need someone to explain what is going on here, thanks.

[Anonymous No.15283173]

>>15283101
Just to iterate, obviously, the centre is (1/3,1/3,1/3), because 1/3+1/3+1/3=1, but the radius part, I don't understand.

[Anonymous No.15283203]

>>15282602
There's no money in most things, including most math. Even if you pick something more applicable, like differential equations, then the actual math department research doesn't many anybody money, and the aerospace engineers at companies mostly have to do their own new math since what they have to do (mostly numerics) barely connects.

[Anonymous No.15283220]

>>15283173
It's just Pythagoras.
The hypotenuse is the radius of the sphere and the catheti are the radius of the circle and the distance between the centres of the sphere and circle.

[Anonymous No.15283274]

>>15283220
Thank you, but how come 3/9 isn't to the power of 2 if it's Pythagoras?
a^2+b^2=c^2
x+(3/9)^2=2^2
x=sqrt(2^2-(3/9)^2), which doesn't equal the correct answer.

[Anonymous No.15283285]

>>15283274
because it's not [math]\sqrt{(1/3+1/3+1/3)^2}[/math], it's [math]\sqrt{(1/3)^2+(1/3)^2+(1/3)^2}[/math]

[Anonymous No.15283293]

>>15283274
[math]b=\sqrt{(1/3)^2+(1/3)^2+(1/3)^2}=\sqrt{1/9+1/9+1/9}=\sqrt{3/9}[/math]

[Anonymous No.15283295]

>>15283274
b is the distance between the centres, which is sqrt((1/3)^2+(1/3)^2+(1/3)^2)=sqrt(1/9+1/9+1/9)=sqrt(3/9).
would have done this with math formatting but for some reason the tags didn't feel like working today

[Anonymous No.15283326]

>>15283295
What you're telling me is... I'm stupid, thanks for the help, gonna try to unstupidfy myself now. Have a great weekend!

[Anonymous No.15283401]

If two partition of observations have the same OLS estimates, does it need be that the entire set of observations should have the same OLS estimates.

[Anonymous No.15283465]

can someone please explain epsilon-delta proof of continuity to retard please?
i understand that you have to find a delta in terms for some epsilon. What i dont get is how to finalise the proof, what is it supposed to look like after i found a delta?

[Anonymous No.15283501]

>>15283465
>what is it supposed to look like after i found a delta?
Then you are supposed to write QDE

[Anonymous No.15283569]

>>15283465
For a real function [math]f[/math] to be continuous at [math]x = a[/math] means that no matter how small an interval [math]Y[/math] on the y-axis centered at [math]f(a)[/math] you take, you can always find an interval [math]X[/math] on the x-axis centered at [math]x=a[/math] for which EVERY point in that interval that is in the domain of [math]f[/math] gets mapped in [math]Y[/math].

Notice how this assertion just asks for any interval [math]X[/math] with the said property. Any interval contained inside of [math]X[/math] will also satisfy the property (of its members being mapped inside [math]Y[/math] if they're in the domain of [math]f[/math]) and it doesn't matter if there's a bigger interval than [math]X[/math] that satisfies the property. All it suffices is to find a suitable [math]X[/math].

A common and very useful formalization of the above statement translates the [math]Y[/math] interval with the interval [math](f(a) - \varepsilon, f(a)+\varepsilon)[/math], where [math]\varepsilon[/math] represents the variation of [math]Y[/math], and the interval [math]X[/math] with the interval [math](a-\delta, a+\delta)\setminus\{a\}[/math] (notice why the behavior of [math]f[/math] at [math]x=a[/math] is actually irrelevant blah blah).

Therefore, the number [math]\delta[/math] means the same than finding the interval [math]X[/math] we previously talked about. Once you've managed to prove that interval's existence, the continuity is then proven.

[Anonymous No.15283577]

>>15283465
>>15283569
So, in short, once said [math]\delta[/math] is found that suits a given [math]\varepsilon > 0[/math], all you gotta do is take any [math]x \in (a-\delta, a+\delta)\setminus\{a\}[/math] and prove that its image lies in [math](f(a)-\varepsilon, f(a)+\varepsilon)[/math]. This IS continuity at x=a.

These set memberships obviously translates to inequalities of real numbers for which you can use different tools to prove the aforementioned result.

[Anonymous No.15283598]

Is there a consistent nontrivial injection from all transfinite ordinals to the naturals?

[Anonymous No.15283701]

>>15283577
Thanks, that clears it uo a bit. I guess i am still struggling with proving/showing that the found delta has that property? Its probably just a trivial formality to you but im still unsure

[Anonymous No.15283706]

>>15283598
In what theory?
What's a trivial injection?
Clearly the naturals N nicely inject into any infinite ordinal, so assuming Schröder-Bernstein only countable ordinals can inject into N.
If you don't adopt the axiom of infinity, then you can make N the class of all ordinals, and then the answer is yes for a class function.

[Anonymous No.15283712]

For [math] R [/math] a commutative ring, is [math] R [/math] the only (up to isomorphism) [math] R[/math]-module [math] M[/math] satisfying [math] M \otimes_R M \approx M[/math] ?

(The stupid questions thread >>15263241 has reached the bump limit by the way)

[Anonymous No.15283718]

how the shit does classification of surfaces fit with uniformisation?

[Anonymous No.15283723]

>>15283701
The condition [math]|f(x) - f(a)| < \varepsilon[/math] will often give you the clues. Assuming an additional condition like [math]|x-a| < 1[/math] could make things easier, so you end up taking [math]\delta = \min\{1, \varepsilon\}[/math].

This may sound like absolute nonsense to you, but really, try searching for examples of proofs of this kind online, come back to this reply and you'll see.

[Anonymous No.15283809]

>>15283712
Sorry this is answered for instance here:
https://math.stackexchange.com/questions/413316/modules-which-are-isomorphic-to-their-tensor-product

[Anonymous No.15283863]

anyone who calls maths a science should be culled. wikipedia should delete this article, it tarnishes the names of these fields.
>https://en.wikipedia.org/wiki/Formal_science
Maths is not subject to the unrigorous nonsense that is the scientific method.

[Anonymous No.15283867]

>>15283712
The tensor product of any module with itslef a countable number of times should work I believe.

[Anonymous No.15283875]

How is equality defined or axiomatized?

[Anonymous No.15283899]

>>15283875
Over typical logics, equality is a binary predicate symbol. We won't call a symbol equality symbol if it doesn't fulfill a bunch of properties that you can find on wikipedia too (e.g. reflexivity, forall x. (x=x), or symmetry forall x,y (x=y => y=x)). In a framework with terms, proven equality then allows you to do term substitution. How to prove equality of two entities depends on the theory (their axioms). Often it's defined by some form of Extensionality, meaning you postulate a sort of quantity by also describing what is observable about them and what not, and you judge two objects to be equal when these and only these observable features provably match. For example, for sets, you generally adopt the axiom that they shall be equal as soon as they have the same members. Or for functions, you often say they are equal if the same inputs result in the same output. In other frameworks, you might for example have intentional equation, in which e.g. the functions g(n):=(n+1)^2 and f(n):=n^2+2n+1, which are extensionally equal over the naturals, are not intensionally so. Apart from extensional equality, there's also induction-type principles that lead to equality. And generally many axioms define objects by ruling out potential model by adding more conditions. So tl;dr it depends on equality of what. For the minimal assumptions about it I had mentioned, just check
https://en.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms

[Anonymous No.15283929]

Is there a rank ≥2 vector bundle which cannot be decomposed as a direct sum of two nontrivial vector bundles?

[Anonymous No.15283931]

>>15283929
If not, what about a rank ≥2 vector bundle which cannot be decomposed as a direct sum of two nonzero vector bundles?

[Anonymous No.15283960]

>>15283718
I don't understand it deeper than this, but here is an outline of how Uniformization follows from Classification of surfaces and the Riemann mapping theorem.

- Fact 1: A smooth closed surface is orientable iff admits a conformal structure iff admits a complex structure, and morphism between surfaces of conformal structures coincides with one of complex structures, etc. This can be proved "directly"

- Fact 2: The sphere and the plane are the only simply-connected surfaces (without boundary). This is where classification of surfaces is used.

- Claim: every Riemann surface diffeomorphic to the plane is biholomorphic to either the open unit disk or the complex plane.

Proof: Let M be a Riemann surface diffeomorphic to the plane. First because M is diffeomorphic to the plane we can cover M by a single coordinate chart which we can also choose to be holomorphic; this provides a holomorphic embedding of M onto an open subset U of the complex plane. If M is not biolohomorphic to the plane, then U is not all of the complex plane; applying the Riemann mapping theorem, then U is biolomoprhic to the open unit disk, whence M also is.

- Using the Claim and Fact 2, the Uniformization theorem follows.

[Anonymous No.15283971]

>>15283960

Actually there is technically one more piece remaining, which is to show any Riemann surface diffeomorphic to the sphere is actually biholomorphic with the Riemann sphere.
This should be possible after picking coordinate charts missing e.g. the north and south pole respectively, and "deforming" these charts slightly to be holomorphic, as in the proof above

[Anonymous No.15284198]

>>15283929
>>15283931
I’ll assume you’re talking about topological vector bundles.

Consider the tangent vector bundle on the 2-sphere. It can’t be decomposed into a direct sum of two trivial line bundles, since we could then choose two sections which form a basis at each fiber, which violates hairy hall theorem.
But one can use clutching functions:
https://math.stackexchange.com/questions/1923402/understanding-vector-bundles-over-spheres
to show that every line bundle over the 2-sphere is trivial.
Therefore the tangent vector bundle can’t be decomposed at all, for that decomposition would result in a direct sum of two trivial line bundles as mentioned before.

[Anonymous No.15284212]

>>15284198
Nice one, thanks anon.

[Anonymous No.15284216]

Can an (open) annulus be the intersection of two contractible open subsets of the plane?

[Anonymous No.15284237]

Another question:
Up to homeomorphism, is [math] \mathbb{R}^n [/math] the only contractible noncompact topological manifold of dimension n?

[Anonymous No.15284252]

Bros, any book to enjoy math again? I can't cope with my disgust for math, and i need to get my undergrad, pls help, bros.

[Anonymous No.15284256]

>>15284252
How can one be disgusted with math?

[Anonymous No.15284261]

>>15284252
How can one be "disgusted" with math? What is there in it to be disgusted by?

[Anonymous No.15284263]

>>15284237
https://en.m.wikipedia.org/wiki/Whitehead_manifold

[Anonymous No.15284280]

>>15284261
I don't know, for the last year i hate everything related to maths. I liked them the first 3 years of undergrad, but suddenly i just stop liking them, Now is a pain to study.

[Anonymous No.15284283]

>>15284263
Perfect, thank you

[Anonymous No.15284291]

>>15284280
If you don't enjoy it, and you don't have any other good reason to study it, then why bother?

[Anonymous No.15284297]

>>15284291
Good question. But mainly because i don't want to disappoint my mother. She has put so much effort in me, so it wouldn't be fair for me to drop everything.

[Anonymous No.15284571]

>>15284297
math is the only thing that has kept me alive this entire time anon. math is a spiritual activity, it's not about grades or school or even logic, it's about understanding the structure of one's mind and how it interacts with the universe.

if you don't see this in math then you will not see it anywhere. i don't even know what motivates people who study any other subject. oh you're studying engineering to be a good slave of the system, that's nice, i studied my inner spiritual world and became a better person

[Anonymous No.15284622]

>>15284216
No. If A and B intersect in an annulus, then one of A or B (or both) won't be simply connected (the hole in the middle of the annulus must contain a hole in either A or B)

[Anonymous No.15284648]

Rate this shitty meme I found

[Anonymous No.15284650]

>>15284571
this is math general, not "pushing your leftist political agenda general" not "i'm a lonesome crybaby, please hug me general"

[Anonymous No.15284652]

>>15281540
Two months is plenty of time to practice calculus again up to where you stopped. If you get stuck, reference a trig or algebra book. Most US-based calculus 3 courses are building calculus into higher dimensions (mostly 3 since most calculus courses in US are engineering focused).

[Anonymous No.15284653]

>>15284648
1/10

[Anonymous No.15284894]

How would you make a function that grows slowly at first but eventually, for some n, exceeds TREE(n)?

[Anonymous No.15284954]

>>15284650
> brainlet faggot
What did you study in your spare time anon? Was it jerking off to hentai or where you actually doing something other than stroking your wiener

[Anonymous No.15285254]

>>15284894
[math]f_n(k):=k\text{ if } (k < n)\text{ else } \big(\mathrm{TREE}(k)+1\big)[/math]

[Anonymous No.15285319]

>tfw dyscalculic
i am a worthless subhuman,

[Anonymous No.15285323]

why do i exist i am a worthless retard

[Anonymous No.15285376]

>>15285323
You exist because your father wanted to spread his seed

[Anonymous No.15285384]

>>15285319
Pure mathematics do not deal with calculations involving arithmetic.

[Anonymous No.15285400]

>>15285384
number theory is all about calculations involving arithmetic, and it's the purest mathematics in existence that isn't schizo shit

[Anonymous No.15285497]

>>15279493
Here's a cute way to do this.

First fix two columns where the first row has ones - 6 ways of doing this. Then split into two cases - in one, there's some other row that also has ones in the same columns. In this case, the three rows after the first look like one repetition of the first row and two repetitions of its negation - 3 ways of ordering these in total.

In the second case, we have (1 -1), (-1 1), (-1 -1) show up in some order in the columns we're looking at for the three remaining rows. 6 ways of ordering these in total. Once you fix an order, there will be two ways of ordering what happens in the remaining two columns, independently of the first two. Total: 6*3 + 6*6*2 = 90.

[Anonymous No.15285539]

I don't get it. What happened to the negative sign on the right hand side?

[Anonymous No.15285553]

>>15285539
-1/(b-a)=1/(a-b)

[Anonymous No.15285782]

>>15285539
there are two y's because it's y^2, I put the minus sign on the wrong side but you get it

[Anonymous No.15285785]

>>15285376
He succeeded, but that doesn't mean I have to deal with this overcomplicated bullshit existence

[Anonymous No.15285841]

>>15285400
was grothendieck schizo?

[Anonymous No.15286063]

>>15279689
Cumulative Algebra

It is a remarkable fact that for the script space a.k.a. cancellative univalent monoid, we can define
the free product without appealing to a quotient. As a result, we can easily prove that
1. the free product of two or arbitrarily many script spaces is a script space, and
2. projection onto any subset of factors of a free product of script spaces is a monoid homomorphism
and we are hard pressed to avoid the "obvious" idea that any calc map e : A -> B of script spaces can
be extended to a calc operator on A * B by giving e(c) = e(pi[A](c)) for all c <- A * B, where pi[A]
is projection of A * B onto the first factor.

This gives rise to the study of the category of unary calc operators on cancellative univalent monoids,
with objects (C,e) such that C is a script space and unary operators e : C -> C satisfying the calc
property. A morphism phi : (C,e) -> (D,f) is a monoid homomorpihsm C -> D that commutes with the calcs,
i.e. f o phi = phi o e.

Related, we can define the free product of two calc maps e : C -> D and f : G -> H of script spaces,
k = e * f where k : C * G -> D * H as the unique map satisfying
- k(c g) = k(c) k(g)
- k(g c) = k(g) k(c)
- k[|C] = e
- k[|D] = f
and this definition can be extended to arbitrarily many calc maps.

[Anonymous No.15286080]

>>15286063
second to last line, add: for all c <- C and g <- G

[Anonymous No.15286181]

Fellas i need help on this: Is the answer (a/2)^2/3 ??

[Anonymous No.15286182]

>>15279493
Add 1 to each entry then divide by 2.
You get an adjacency matrix where each vertex has in-degree 2 and out-degree 2.
Dividing by two again gives a doubly stochastic matrix.
>Birkhoff polytope
>Convex combination of permutation matrices
>Draw some loops and dots
>???
>51

[Anonymous No.15286777]

>>15283960
bit late but i found this.

[Anonymous No.15286944]

>>15279493
Add 1 to each entry then divide by 2.
You get an adjacency matrix for a digraph where each vertex has in degree 2 and out degree 2.
There are 8 graphs on 4 vertices that satisfy this condition (ignoring vertex labels and edge directions).
When you add the labels and directions back you get 90.

[Anonymous No.15287024]

How is bottom the standard matrix? I dun undastand

[Anonymous No.15287034]

Lads, starting abstract algebra tomorrow. I'm finally here. I read a few chapters ahead to prepare. Still don't think my proof writing is up to par. Its gonna be tough.

[Anonymous No.15287047]

recommend me avenues of going deeper into number theory, particularly algebraic (interested in learning class field theory)
currently finishing up Samuel, reading a little Neukirch, but it's tough. any good recs on local fields? i've gotten the basics of p-adics from Gouvea

[Anonymous No.15287090]

>>15287047
Why are you interested in learning CFT? Its not really something you learn just to learn. If you dont have a real goal I would say dont bother until you do, and I mean a specific question not just "i want to learn CFT"

anyways, the best way to learn CFT imo is to follow some certain historical goals associateed with it that interest you most. A few are:
>generalize reciprocity
>calculate class numbers
Both of these trails start in the late 18th century (really they start with Gauss). Learning theory without knowing why the theory was even needed in the first place has never worked for me.

[Anonymous No.15287097]

Guys a bit of a weird question, what are the chances of annals of mathematics accepting a very short paper proof of Dirichlet's theorem using basic general topology?

I think I might be delusional for even attempting but I saw some short papers published there in the past.

[Anonymous No.15287109]

>>15287090
>Why are you interested in learning CFT?
because
a) i view it (perhaps inaccurately) as what comes after ANT (not that i've mastered it already)
b) i'm fascinated by the Kronecker-Weber theorem and similar results
>Learning theory without knowing why the theory was even needed in the first place has never worked for me.
i am fundamentally in agreement with you, although i wouldn't go so far as to say that it has never worked for me

[Anonymous No.15287481]

>>15284571
>i don't even know what motivates people who study any other subject
https://en.wikipedia.org/wiki/Alexander_Grothendieck
>after several years at the IHÉS, Grothendieck seemed to cast about for new intellectual interests. By the late 1960s, he had started to become interested in scientific areas outside mathematics. David Ruelle, a physicist who joined the IHÉS faculty in 1964, said that Grothendieck came to talk to him a few times about physics.[b] Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics.[41]
did he fall from grace anon?

[Anonymous No.15287494]

>>15285497
Hello anon. Very nice job! Cute indeed.
Your answer is of course correct and the way you went about it is pretty cool, very short, and it seems clear. I'm quite in a hurry right now so I haven't read it carefully but yeah, it seems quite good! Thank you a lot for your time and effort, I hope you found this entertaining. I appreciate your solution!
>>15286944
Woah! That's different.
>Add 1 to each entry then divide by 2.
So basically instead of working with 1's and -1's, we work with 1's and 0's? And then you can view the matrix as a adjacency matrix for a digraph where each vertex has in degree 2 and out degree 2? That's brilliant anon. I have to leave home now but when I'm at the library I'll try to read your solution carefully and try to do it myself.
This solution definitely takes the cake in terms of being unique I'd say. Thank you a lot for your time and effort. I appreciate it! Oh and of course I forgot to say your answer is 100% correct, it is 90 indeed.


Again, I congratulate both of you for your correct answers and cute/unique solutions.
Have a pleasant day!

[Anonymous No.15287529]

>>15287024
T(x,y,z) = 1x + 1y +2z

[Anonymous No.15287627]

>>15287481
Towards the end of his life he went full schizo but his correspondences during seclusion were still very lucid and insightful.

Most mathematicians eventually branch out and away from pure math so he's not special in that sense. Every mathematician eventually thinks about applying their craft to more mundane and pedestrian problems and they're often successful. Turing is the most famous example but there are many others that made fundamental discoveries in different fields after their work in pure mathematics.

[Anonymous No.15287636]

>>15285841
Most definitely. Almost every great mathematician eventually loses their mind.

[Anonymous No.15287810]

>>15287636
>Almost every great mathematician eventually loses their mind.
Name one

[Anonymous No.15287848]

why do people call the proof that there is no rational number who's square is to a proof that root 2 is irrational? That would only be the case if you could prove the existence of root 2 in that axiomatic framework. So it is not universal.

[Anonymous No.15287851]

>>15287810
Cantor

He went so crazy that he started to believe that infinities of different sizes exist.

[Anonymous No.15287852]

>>15287848
>to
two

[Anonymous No.15287957]

>4 day old math general, still only 135 posts
how come the math general is so slow? must be an unpopular, boring topic nobody cares about.
/sfg/ fills up like 500 posts a day on a slow day.

[Anonymous No.15288011]

I'm an engineering student, so I have no clue how to tackle this. Basically, I have a pipe with a 2D velocity profile of some liquid going through it, that I want to convert to 1D. In the horizontal parts, I can neglect the y-velocity and in vertical parts I can neglect the x-velocity. But I don't know what to do with the corners.

[Anonymous No.15288082]

what did whitehead mean by this approach

[Anonymous No.15288133]

Was asked to prove this for a class. I succeeded, but my proof is quite gross. I can post a pdf if anybody would like to see but I'm curious if anyone has a different approach (surely nicer-looking than the one I took). Maybe something with pure trig. I always just use complex exponentials and derive trig shit rather than actually learning trigonometry.
[math]
\int sin((n+1)x)cos^{n-1}(x) = -\frac{1}{n}cos^n(x)cos(nx)
[/math]

[Anonymous No.15288204]

>>15286944
Really neat solution, anon. I do not know graph theory, but I am looking at some Wikipedia pages. I have a question: How do you know how many "ways" exist for the undirected graphs drawn to "become" digraphs? Did you count manually?

[Anonymous No.15288337]

>>15288133
Prove it backwards:
[math]\frac{d}{dx} \Bigl[-\frac1n\cos^n(x)\cos(nx)\Bigr] = -\frac1n\bigl(-n\sin(x)\cos^{n-1}(x)\cos(nx)-\cos^n(x)\cdot n\sin(nx)\bigr) = \cos^{n-1}(x)\bigl(\sin(x)\cos(nx)+\cos(x)\sin(nx)\bigr) = \cos^{n-1}(x)\sin((n+1)x). [/math]

[Anonymous No.15288400]

how much of mathematics is really just learning notation

[Anonymous No.15288428]

anyone know what this fucker is used?, it must be for transfinites, i know that much
https://unicodeplus.com/U+2138

[Anonymous No.15288430]

>>15288400
not much, you just pick shit up along the way, is what i found

[Anonymous No.15288467]

>>15288428
according to Simple Wiki it's used to denote the fourth (?) infinite cardinal, but I have never ever seen this

[Anonymous No.15288468]

>>15287848
you can prove the existence of root 2 in that axiomatic framework (presumably ZFC with reals as dedekind cuts or cauchy sequences)

[Anonymous No.15288473]

>>15287097
Very high if it's a legit proof, shorter papers are better for both the journal and the reader
but if you're the one who made the thread on it your proof is unreadable nonsense

[Anonymous No.15288479]

>>15288473
I know, I edited it so now it's quite short, it's funny though I tried to do that with Acta Mathematica but they told me they don't deal with proofs for old, classical and already known theorems, I think the same goes for any more prestigious journal like the annals which is just out of scope for me.

[Anonymous No.15288486]

>>15288468
yes but it depends on what axioms are taken.

[Anonymous No.15288494]

>>15288486
>presumably ZFC

[Anonymous No.15288496]

>>15288479
You should put a preprint on ArXiV

[Anonymous No.15288507]

For proof of problem 1, why not just state that given the assumption that a|b and b|c, then by definition a is a factor of b and b is a multiple of a, therefore there exist an m as an element of integers such that a(m)| c. Seems like a much faster way to prove this unless I have any holes in what I'm writing

[Anonymous No.15288522]

SAFE AND EFFECTIVE

[Anonymous No.15288526]

>>15288522
>29, MIT, Jew
don't care

[Anonymous No.15288530]

>>15288507
you're just rewording the proof and keeping a step implicit (you still have to prove am|c implies a|c), that is in no way "faster" and doesn't cut down on any of the actual logical steps involved

[Anonymous No.15288537]

>>15288337
[math]
sin(x)cos(nx)+cos(x)sin(nx)=sin((n+1)x)
[/math]

Wouldn't have been able to pull that out my ass but that's what I was looking for. I solved it backwards as well but ended up finding and proving a recurrence relation between the left and right sides implied by the integral, kek.

[Anonymous No.15288540]

>>15288530
Gotcha. Sorry, by faster I guess I meant flows naturally from my head to the paper. It seems I can use that as a starting point, review, and expand until there are no implicit steps remaining and I should be somewhat okay....

[Anonymous No.15288550]

>>15288540
you do you, but imo there's no use to reinventing the wheel in such situations - sometimes there really is a "best" proof

[Anonymous No.15288552]

>>15288204
I approximated it first.
Since total in degrees equal total out degrees, you know each graph consists of just cycles.
A permutation in S4 consists of just cycles and contributes 1 in and out degree for each vertex.
There are only 5 types of permutations if you ignore vertex labels and digraph directions.
I was looking at all ways to use 2 permutations to satisfy the graph criteria.
Since each directed edge can only be used once, you know a valid pair of permutations cannot map the same input to the same output.
Put another way (perm1)^(-1)*(perm2) cannot contain any 1-cycles.
Put another way, (perm2) is contained in (perm1)*{4-cycles, 2 2-cycles}
To get an upper bound on the final answer I just counted all ordered pairs of permutations with that property (this is a double count of unordered pairs):
Sum[ |(perm1)*{4-cycles, 2 2-cycles}|, perm1 in S4] = |S4|*|{4-cycles, 2 2-cycles}| = 24*9
Half of this gives 108.
The amount of overcount depends on how many different decompositions into permutations each digraph has.
>>15286944
Top left can be written as (4) + (4) OR (2,2) + (2,2)
Middle left can be written as (1,1,1,1) + (2,2) OR (1,1,2) + (1,1,2)
Bottom middle can be written as (1,3) + (1,3) OR (4) + (1,1,2)
This gives an overcount of 3+3+12 = 18.

[Anonymous No.15288555]

>>15288537
[eqn]\cos((n+1)x) + i \sin((n+1)x) = e^{i (n+1) x} = e^{ix} e^{i n x}= (\cos(x) + i \sin(x))(\cos(nx) + i \sin(nx)) = (\cos(x) \cos(nx) - \sin(x) \sin(nx)) + i (\cos(x) \sin(nx) - \sin(x) \cos(nx)) [/eqn]
Now compare the immaginary parts of both sides.

[Anonymous No.15288557]

>>15288550
Oh I'm not trying to reinvent proofs. I am trying to understand (should be refresh....) proof writing a bit more as I am a high anxiety level type of person and am taking an abstract algebra course. Now reviewing chapter 0 of gallian (again) and making sure I cover all content to its maximum extent before going forward. I am painfully aware that my success so far in math has only been due to effort and not any actual skill or smarts, so I'm trying to compensate even harder in this course.

[Anonymous No.15288563]

>>15288467
>>15288428
Dalet/daled is the fourth letter of the Hebrew alphabet, so like you might see ℶ as the sets starting with cardinality 2^א, just take two more steps, past ג to get there.

[Anonymous No.15288574]

>>15288552
>>15288204
Nvm. Wrong question (I went too big brain).

1-cycles and 2-cycles don't need directions added since they are redundant.
If there is a distinct node start with 4, if there is another distinct node multiply by 3, etc. If there is a 3 or 4-cycle with ambiguous direction, multiply by 2 and just add a direction. Basic combinatorics stuff. I kinda just do it on autopilot so I can't really give an exact recipe.

[Anonymous No.15288578]

>"No, there will be no examples. You have been given the theorem so you should be able to learn from that and solve everything."

[Anonymous No.15288582]

>>15288578
Midwit.

[Anonymous No.15288606]

>"No, there will be no examples. You have been given the shitpost so you should be able to infer from that what I meant and agree with everything."

[Anonymous No.15288615]

>>15279493
I'm late to this thread, but since I haven't seen this approach in the replies: you can prove that at least two of the columns are complementary to each other (meaning, one has [math] 1 [/math]'s where the other has [math] - 1 [/math]'s and vice versa). If that weren't the case, then every pair of columns in an array would have at least one [math] 1 [/math] in the same row - but that would mean there are at least [math] 6 [/math] horizonal pairs of [math] 1 [/math]'s in such an array, when there are clearly just [math] 4 [/math]'s (one for every array).

This means that two of the columns in the array are complementary to each other, which implies that the other pair is complementary as well. Now we just cover two cases, whether the two pairs are different or not. Together we get [math] 3 \times 24 + 3 \times 6 = 90 [/math] arrays and as a nice corollary, there are exactly [math] 3 [/math] combinatorial [math] (4,2,2) [/math]-designs.

[Anonymous No.15288618]

>>15287851
settle the debate anon: was cantor ashkenazi

[Anonymous No.15288621]

>>15288615
*when there are clearly just 4 (one for every row).

I'm typing this on my phone.

[Anonymous No.15288685]

>>15286080
The semifree product of two calc maps of script spaces.

Let C,D,E,F be script spaces and g : C -> D, h : E -> F calc maps. Let phi : C -> E be
a monoid homomorphism. The semifree product of g and h by phi is k : C * E -> D * F where,
informally, k keeps instances of objects g and h, sends message c to g and phi(c) to h
when sent c <- C, emits g(c); sends message e to h when sent e <- E, emits h(e). There
exists a unique calc map k that satisfies

- there is a bijection p : g^C x h^E <-> k^{C * E}
- for m <- C, we have p(g^c,h^e)^m = p(g^{mc},h^{e phi(c)}) and p(g^c,h^e)(m) = g^c(m)
- for q <- E, we have p(g^c,h^e)^q = p(g^c,h^{eq}) and p(g^c,h^e)(q) = h^e(q).
Proof. Proof? You need proof? You should be doing C programming (if you're a real man)
or object oriented programming (students) at this point.
https://www.youtube.com/watch?v=vqYbWk6rwxg
https://soundcloud.com/walter-corral/essential-mix-desyn-masiello
https://www.mixcloud.com/brent_borel/the-whoop-project/
I'm getting drunk. I refuse to comment on Scheme vs. Python for the students.
It's a sore subject.
>t. listens to people wax pedagogical over 6.001

[Anonymous No.15288851]

which field of mathematics attracts the most autists or otherwise mentally ill people

[Anonymous No.15288855]

>>15288851
Which level of attraction are we talking.
I think Specialization is still pretty much a cultural thing, many will become mathematicians in fields that available advisors are already working in.

[Anonymous No.15288873]

>>15288851
Set theory. There was a literal autistic guy in most of the classes I took as a freshman and he was constantly harrassing professors, like when they were doing proofs he demanded everything be derived from the ZFC axioms. I haven't seen him since so the retard must have failed all of those classes lol.

[Anonymous No.15288878]

Odd numbers aren't whole numbers

[Anonymous No.15288910]

[eqn]
0 \notin \mathbb{N}
[/eqn]

[Anonymous No.15288935]

>>15288873
you mean disrupting the class??
how did the professor react

[Anonymous No.15288940]

which field of mathematics attracts the most sexy/engineer/creative/built or otherwise conceited/narcissistic people

[Anonymous No.15288947]

>>15288578
>>15288606
>t. remembers this guy give the "just whip the examples out of the fanny pack" speech

[Anonymous No.15288956]

any other fourier analysis chads in the thread?

[Anonymous No.15289008]

If everything in math is logically connected, how have even simple sounding questions not been answered yet?

[Anonymous No.15289207]

>>15289008
If everyone in humanity is biologically connected, how have you not had sex yet?

[Anonymous No.15289209]

Which Millenium Prize problem is the next most likely to be solved?

[Anonymous No.15289245]

so you can make up all sorts of axiomatic systems, but is there a way to know if they reflect the real world? surely there must be some way. the universe is logical isn't it. so it's rules should be able to be found logically. Instead of having such discussion as X exists, no X doesn't exist, and all sorts. You can say pretty much anything exists in an axiomatic system, but how can you prove they exist in the real world space and time

[Anonymous No.15289248]

>>15289209
Yang-Mills or Navier-Stokes
Literally impossible to say but I'd guess these

[Anonymous No.15289418]

Sages of /mg/, I invoke your wisdom! Having read the Preface of Munkres' Topology, I'm of the idea that I can read through it without out having to go through any formal proof book or Calculus book, have any other anons tried this? Or for those who've just read the book before, is it really true? Can I really just study this book with precalc knowledge?

[Jealous No.15289473]

>>15289209

[Jealous No.15289477]

>>15289473

[Anonymous No.15289550]

If I do p/q surgery on some knot, how can I tell if my resulting manifold is Seifert fibered? I can't find a straight answer anywhere.

[Anonymous No.15289557]

>>15289418
You won't get far without an understanding of proofs. If memory serves, some problems use topological properties of the real numbers you will have seen in a calculus course to invoke things like intermediate value theorem. Nothing crazy, I don't think much calculus is directly necessary. You should still take calculus first.

[Anonymous No.15289864]

>>15288615
>>15288621
Hi anon. First things first, you're not late to this thread by any means! An anon is never late nor is he early, he replies precisely when he means to. I keep replying to the answer to my problems until the end of the thread.

Secondly your answer is correct of course. And your solution is amazing. Love it. I think you went about it quite cleverly. I have heard of combinatorial designs before but I actually never checked them out, might be time to do so.

Thank you very much for your solution. I appreciate the time and effort you gave to it.
>I'm typing this on my phone.
me too, quite often. Though not right now.

Have a nice day!

[Anonymous No.15289927]

>>15279472
>Armenian genocide
never happened. we live on a planet teeming with armenians, why are there so many if they were genocided?

[Anonymous No.15289943]

>>15289927
cool it with the antisemitism

[Anonymous No.15289998]

>>15282434
milne's book on elliptic curves, and arithmetic of EC by silverman

also check out algebraic curves: the brill and noether way

[Anonymous No.15290002]

Supposing this Langlands/perfectoid stuff isn't schizo and there is a fundamental duality between algebraic topics and geometric topics, what is the algebraic equivalent of n-polyominoids? To help clear things up if you haven't heard of them the 2-polyominoids are dominos, triominos, tetraominos and other compounds of squares, the 3-polyominoids are compounds of cubes etc

[Anonymous No.15290335]

>>15289209
https://aiimpacts.org/resolutions-of-mathematical-conjectures-over-time/
whichever one was posed/has been around most recently

[Anonymous No.15290546]

Abstract algebra is kicking my ass. Not even the actual course, but my professors notes. I've never taken number theory and it does feel like I should have covered some before taking course. Most of it just seems incomprehensible to me and overly technical with terms I've never heard of.

[Anonymous No.15290556]

>>15290546
Yes, the memorization of definitions is quite stupid but it must be done

[Anonymous No.15290620]

>>15280429
Take the derivative with respect to x, then the derivative of the result with respect to y. Remember that y is treated as a constant in the first derivative, so your first term would be 2xye^(x^2)y. Rinse and repeat for fyx.

[Anonymous No.15290624]

>>15290556
Is that enough? This is the first course where I feel this overwhelmed and inadequate.

[Anonymous No.15290747]

>>15288910
[math]0 \in \omega[/math]

[Anonymous No.15290768]

>>15290335
The Poincaré conjecture was the second oldest of the bunch though. And P versus NP is technically the newest, but it's the least likely to be solved any time soon, by the looks of it

[Anonymous No.15291378]

Prove [math]{\mathbb R}[/math] has a subset of size [math]\aleph_1[/math].
We working in [math]\mathsf {ZF}[/math].

[Anonymous No.15291439]

>>15279472
Could some kind anon give me a vector calculus problem? Maybe something with Stokes or divergence theorem. Been self teaching and I want to see if I can do it. I would like to be held accountable by a random anon to solve it.

[Anonymous No.15291578]

>>15290624
sufficient??
Anon said it was necessary.
Besides, it's pretty hard so something like SuperMemo is definitely helpful
I mean, super retarded ideas like
- write down each definition you need to know such as group, ring, unit, &c. on a piece of paper, one definition per line, then in another column the page number on which it's defined and the line number of the page where it's defined, and then a note to help you recall the definition
Then, have a stack of 3x5 notecards handy and write the defined term on the top line and the body of the definition below.
Also write the defined term on the reverse side. Now you can do flash card study.
The SuperMemo spaced repetition tells you how your brain and memory work; you NEED to READ this to understand the modern psychology of memory!
It's pretty simple: memory training is strength training. You get your gains when you rest and force your brain, body, and memory to adapt to training according to the SuperMemo protocol.
There are no nerds anymore, only jocks.
https://en.wikipedia.org/wiki/SuperMemo
https://www.youtube.com/watch?v=k3fz6CC45ok

[Anonymous No.15291587]

There are infinitely many Fermat primes.
There are infinitely many odd perfect numbers.

[Anonymous No.15291643]

can you debunk wildberger

[Anonymous No.15291676]

Any of you lot knowledgeable in differential equations?
>>15291649

[Anonymous No.15291712]

>my dad asked me to help him out with a business thing with my math skills
>did it super easy
I feel really good about myself. It was super basic algebra but he was impressed and I like it when my dad is proud of me. It makes me feel weird though I don't know. It's uncomfortable. I don't want to think about this.

I think I might be autistic.

[Anonymous No.15291856]

>>15290768
I may be changing my answer after realizing I was incorrect, but there are 2 effects: something older being likely to be in the middle of its existence and therefore continue longer; and things of this class being exponentially(?) less likely to last a longer time.

I didn't re-check what it said the half-life of conjectures was supposed to be, either. But this is also not a question of logic, but like a soft science.

[Anonymous No.15291877]

Can anyone explain why if f and g (both L1 on the unit circle wrt lebesgue) have the same fourier coefficients they are equivalent a.e.? This seems glaringly obvious but I hate analysis.

>t. brainlet

[Anonymous No.15291910]

>>15289557
What about the briefing on Logic at the begining of the book, is it not enough?

[Anonymous No.15291953]

do we have a pure math guide?

[Anonymous No.15291974]

My proof writing skills are absolutely terrible, I neglected them, had easy professors until now, and I'm finally paying the price and will be exposed as a fraud.

[Anonymous No.15292088]

>>15291910
In a formal sense, if you were to completely internalize it and apply it perfectly, then sure, it's enough, but humans need practice. You're welcome to throw yourself into it if you like. I'm telling you that unless you have experience writing proofs yourself and understand how the way they go, you will be largely unable to solve the problems in Munkres.

[Anonymous No.15292127]

whats up fellow plooters. made this one last night drunk. kinda like it. I wish i could get sponsored by someone to build a better computer so i could solve harder shit. anyways, no one cares so no sense in getting all bent out of shape over it

[Anonymous No.15292164]

>>15292127
stop drinking, fag

[Anonymous No.15292172]

>>15292164
thanks for the criticism. but your advice is not very helpful

[Anonymous No.15292183]

>>15292172
okay, fuck you
SAY you stopped drinking
lie to us, motherfucker

[Anonymous No.15292185]

>>15292183
OH!
And take one (1) dose of LSD.
Just in case you have
>muh alcoholism
420blazeitfgt

[Anonymous No.15292189]

>>15292183
jackson pollock was a notoriously drunk person and his art sells for millions of dollars
>>15292185
sorry i dont take orders from random angry strangers that insult me

[Anonymous No.15292196]

>>15292189
DUDE
LSD
you haven't read about the research into the one dose LSD anti-alcoholism trick
fuckin' drunk

[Anonymous No.15292198]

>>15292196
I fucking demand
I absolutely fucking demand you do the research into LSD vs. alcoholism.
ONE
>FUCKING
DONE

[Anonymous No.15292199]

>>15292196
thats illegal and irresponsible. also you have anger issues which is way fuckin worse than fermented sugar water

[Anonymous No.15292204]

>>15292198
^DONE^DOSE

[Anonymous No.15292205]

>>15292199
LSD prohibition is bullshit.
LSD was made "illegal" by the alcohol industry.
LSD isn't actually illegal.
LSD is only detrimental to alcohol industry profits.
Alcohol industry is corrupt.
Alcohol industry is politically corrupt.
Alcohol industry tried to make LSD "illegal"

[Anonymous No.15292208]

>>15292199
LSD is great medicine. The people that don't like LSD are wimps.
They are mental wimps.

[Anonymous No.15292209]

>>15292205
honestly anon i would have preferred to talk to someone who like art or math and isnt obsessed with LSD. but you do you. if you would like to give me some opinions on the artwork i posted im fine with that otherwise im kinda over this conversation
>>15292208
seriously you are obsessed. this isnt the thread for that

[Anonymous No.15292212]

>>15292199
The only reason why you think LSD is either "illegal" or "irresponsible" is because you've been brainwashed by the alcohol industry.

[Anonymous No.15292216]

>>15292209
Fuck you and your alcohol industry shilling.
Fuck alcohol.
Fuck drunks.
Fuck you.
Get the fuck out of here, drunk.
Take your fuckin' LSD, asswipe.
You're a goddamn fuckin' asshole for putting up resistance when I tell you to take your goddamn fuckin' LSD, motherfucker!

[Anonymous No.15292221]

>>15292216
people like you are the reason people like me drink. pipette that onto your blotter paper and ingest it

[Anonymous No.15292233]

>>15292221
We're going to pass a new law that essentially says alcohol dealers are terrorists.
And then we're going to make LSD legal.

[Anonymous No.15292236]

>>15292221
If you want alcohol, you have to deal with a terrorist.

[Anonymous No.15292239]

>>15292233
again i would like to remind you that this is a math thread and talking about illegal drugs over and over is selfish behavior. i tried to post math and get some feedback but instead i got you hijacking the topic and insulting other people. so im going to try again later

[Anonymous No.15292246]

>>15292239
Okay, fuck you.
Your "illegal drugs" narrative is bullshit.
Get the fuck out of here.
I have no fucking time for your bullshit anti-Egyptian anti-alchemical narrative.
You're a fucking idiot, and you're a fucking addict.
You fucking idiot. Stop drinking, and who gives a shit about this plebbit tier artistic effort you want to subject us to.
Like
Go to /r/generative.
Just go there.
Get them to look at it.
We care about YOU, motherfucker.
We fucking hate the alcohol industry.
We KNOW we fucking KNOW the alcohol industry made LSD illegal because the alcohol industry is a bunch of greedy idiots.
LSD cures alcoholism, i.e. excess alcohol industry profits.
Alcohol industry is in the crosshairs, motherfucker
>TAKE
YOUR
>L
S
>D

[Anonymous No.15292253]

>>15292246
The concept of "illegal drugs" is about one drug (i.e. alcohol) engaging in political corruption in order to deny the efficacy of another drug (i.e. LSD)
It is 100% pure anti-medical political corruption, and it is evil.

[Anonymous No.15292295]

[Anonymous No.15292296]

>>15291953
sure primary school:
1 - numbers, monoids and groups
2 - rings and fields
3 - modules and topology
4 - commutative algebra and category theory
5 - homological algebra and introductory algebraic geometry
6 - guided research
7 - guided research
you graduate upon publishing your first paper, failing that, you get kicked out and apply to mit

highschool: oh shit we forgot to teach you analysis edition
8 - real and complex analysis
9 - curriculum reforms have you take euclidean geometry like other highschool students
10 - functional analysis
11 - research
12 - research

uni: now time for REAL math
1: programming, stats and financial math
2: programming, stats and financial math
3: programming, stats and financial math
4: programming, stats and financial math

300k starting yw
low caste will say this isn't feasible

[Anonymous No.15292363]

>>15292295
Home Math Companion: Temperance Edition
Special Journaling function for recording your drunken revelry
https://mp3.hardnrg.com/kemkidd/kemkidd-helix.mp3

[Anonymous No.15292594]

> chad GPT-3: gives correct answer in a matter of seconds
> students: toil their asses for 30 min, most answers incorrect

[Anonymous No.15292602]

>>15292594
I've never seen GPT-3 solve a math problem correctly.

[Anonymous No.15292607]

>>15292602
the only ones where it can the answer already exists. it can only regurgitate text it has parsed.

[Anonymous No.15292635]

>>15292607
Of course it gives incorrect theoretical answers.

But ask it to calculate partial derivatives of a function and see for yourself.

[Anonymous No.15292643]

>>15292635
That's the point. It doesn't calculate. It can only cut & paste from all the input documents fed into the model.

[Anonymous No.15292647]

>>15292635
Why not just use Wolfram Alpha?

[Anonymous No.15292770]

>>15292296
funny

[Anonymous No.15292949]

Can someone explain how he went over this? It's supposed to be a proof of irrationality for any non-integer square root of positive integer n.

[Anonymous No.15293118]

I'm stuck

[Anonymous No.15293126]

>>15293118
Wait, I think I found the solution but now I'm more lost than before.

[Anonymous No.15293181]

>>15293126
Always leave a rope behind you, so you can figure the way back out.

[Anonymous No.15293199]

>>15293181
Yeah, I just figured what I did wrong just now. 17m is for the case of n =1
For n+1 I should be solving for a another variable, such as k.
The solution should just be k = m + 18^n

Need a schizo meme of "I HATE PROOF BY INDUCTION"

[Anonymous No.15293260]

>>15293199
I find it interesting an analogy can apply both to the real world and to solving a math problem.

[Anonymous No.15293271]

>>15293260
Drawing really long arrows (ropes) pointing back to previous locations in the problem where things connect is usually frowned upon though.

[Anonymous No.15293304]

>>15293271
I was thinking of (mentally) retracing steps.

[Anonymous No.15293578]

I what courses would I encounter the fundamental theorem of arithmetic and be required to prove it?

[Anonymous No.15293588]

>>15293578
Number theory is the main one.

[Anonymous No.15293640]

As there are a few textbook lists for math degrees posted on /sci/ frequently, is there one list that focuses on textbooks that aren't bloated with idiotic "real world" examples, homoerotic hero worship with full page portraits of some faggot like Galois, or history lessons that have nothing to do with the mathematical content I am trying to learn?
I've found the less applied and the more theoretical the content becomes, the less information dense each page of the book becomes as it gets bogged down by this shit. At least, that is how its been with textbooks required for courses and pushes me to find an alternative and get rid of the required text.
Luckily nothing yet has been as offensive and worthless as undergraduate physics texts, which could really be replaced by a formula sheet as nothing is being proven and hardly anything is derived from something else besdies "uhhhhh experiment told us dis number so is true now"
Yes I'm mad, as bloated texts make very poor reference books.

[Anonymous No.15293660]

>>15293640
if you want a reference book, just say so from the start nigga. instead of having a whole ass bloated post whining about irrelevant shit, you coulda just googled
>good reference book on x
and ignore any suggestion that says lang

[Anonymous No.15293675]

>>15293660
>nigga
kys worthless niggerified piece of shit

[Anonymous No.15293685]

undergrads >>15293675 just move on a different level, they really do

[Anonymous No.15293687]

Is there a good reason not to postulate that [math]{\mathbb R}[/math] is the union of a countable set of countable sets?

[Anonymous No.15293688]

>>15293660
Thanks, I was just rageposting.

[Anonymous No.15293790]

>>15293687
Yes, there is a good reason for that.

[Anonymous No.15293888]

>>15293790
choice fetishism?

[Anonymous No.15293951]

>stupid kid digs
>"look daddy I dug a hole!"
>me: "Actually....."

How do you explain this in an ELI5 fashion? (He's 4)
I tried to help him and he got angry and said I just made a tunnel.

[Anonymous No.15294194]

>>15293687
hot pic, but she's a slut
break her fucking neck
Hedonic Emotivism

So, the first thing you want to know, since I know you're subject to a Gödellian condition, is the esssential novel question, the question on the top of every thinking person's mind, what is the psychological breakdown of the author when confronted with the text? What effect did the writing process have on the final product? To what extent did previous stages of the product affect later stages? Is the entire thing essentially a degenerate criminal scam?

Hedonic emotivism doesn't give a shit. It gets rid of all of that. And for those nerve cases, we have a solution, but it's below, you have to read. Yes, we do the ghost treatment. We do the ghost-nerve fuck you German idealism fuck you Immanuel Kant fuck you Georg Hegel and FUCK YOU GEORG CANTOR FUCK ALL THE GEORG

Fuck George Washington. FUCK THE USA! YOU ARE ALL FUCKING GLOWNIGGERS AND YOU KNOW IT! GODDAMN! I HAVE HAD IT WITH YOU! I AM AT THE "ELEPHANT MUST" OF ORWELL'S FUCKING SHOOTING A FUCKING ELEPHANT, MOTHERFUCKER! There is no elephant. I am not here. I am pure loving kindness & purefucking bliss.

Fuck the Buddhist Priests.

The necromancers are okay. I like the negroes. Don't make this about me. I certainly don't give a fuck about you, so don't make this about you, either, pal. Fuck the niggers. Blacks are okay.

I can make my Macintosh computer talk to me. I often do this, you know, for neural-therapeutic reasons. Fuck them women. I love women. Treat the women well. Love the women. Bliss.

[Anonymous No.15294243]

>>15294194
You guys aren't quick.
I'm shilling for my favorite book, motherfucker.
I'm shilling for the Oakland Raiders, too!
https://archive.org/details/lizardmusic1/lizardmusic1.mp3

[Anonymous No.15294320]

>>15294194
I'm not smarter now.

[Anonymous No.15294588]

I hate trig identities

[Anonymous No.15295219]

Tooker is in pain.

His blog posts are getting really visceral lately. His "implants" are no longer just in his genitals but also on his EYES as well? I'm worried he's going to hurt himself

What could be wrong with him? Is it psychosomatic or is there really something going on with him?

[Anonymous No.15295884]

what even is mathematical maturity?

[Anonymous No.15296038]

>professor hates me
>I'm stupid, so I inadvertenly became the guy that asks for everything to be proven back to base axioms just to make sure I dont make any mistakes or assumptions

Its me. I'm that stupid faggot now

[Anonymous No.15296153]

>>15295884
having gotted used to things

>>15294588
generalize them. it's fun and there's a million directions to take them

[Anonymous No.15296184]

>>15294588
It's all complex exponentials. That's literally all it is

>>15295884
- Knowing what kinds of results / problems are interesting and for what reasons
- Knowing more about how different subjects are related to each other in larger contexts

[Anonymous No.15296236]

but... it can be a function, its just not onto. What am I missing here?

[Anonymous No.15296246]

>>15296236
What is [math]g([3]_2)[/math]?

[Anonymous No.15296248]

>>15296246
in Z_3 that would be 0? I think I get what you're saying here, that with the definition of the function here, 0 =! 3 ?

[Anonymous No.15296425]

Would you say learning to code is essential for a Math major in $CURRENT_YEAR?
I am the only one from my class that properly got to work with academia, even if not directly related to Math (AI), meanwhile everyone else is either unemployed or learning to code. I was a mediocre student but did a research in ML and a internship as an undergrad.
I think you have to be truly special to do pure math and nothing else right now...

[Anonymous No.15296445]

>>15296425
No, I don't see what it has to do with it. You can find employment with or without a degree. Its better to just study whatever you are passionate about and makes you happy. If you hate coding, you'd hate devoting your life to it just so you can buy a bigger TV or something stupid like it.
>truly special to do pure math and nothing else right now...
Even an average student has a guarantee to get a decent job working for the government, or a cozy commission in the military by virtue of having a STEM degree. There is no lack of work for math majors unless you're really picky or something.

[Anonymous No.15296469]

>>15292949
What step is unclear? It seems pretty straightforward.
1. Choose k with k <= sqrtn < k+1
2. Assume for sake of contradiction sqrtn = p/q, i.e. it is rational (with p,q>0).
3. Now (nq-kp)/(p-kq)=sqrtn (to see this plug in p=qsqrtn on the left)
4. 0<p-kq<q (this is the same as k<sqrtn<k+1 which is true since n is not a square)
5. Now we have a fraction with smaller denominator than before.
6. Keep doing this forever. This isn't possible, contradiction. Equivalently you can start with q that is minimal (reduced form fraction)

[Anonymous No.15296510]

Best language to learn for maffs?

[Anonymous No.15296543]

How do you guys take notes? Do you just make a note of a chapter's important theorems and definitions with the real important ones copying a proof down?

My notes end up looking like a re-write of the book chapter I'm reading.

[Anonymous No.15296631]

Is there a good reason not to postulate that R has no countably infinite subsets?

[Anonymous No.15296692]

>>15287047
Instead of Neukirch you may want to try Janusz book.
The simplest book that introduces into CFT is Cassels's "Local Fields", after that there are many books with more or less a focus you may be interested in, some are Milne's notes, Guillot's "Gentle introduction", some chapters of Lang's and Jacobson's algebra.
I also recommend studying homological algebra since it opens a lot of doors: like Serre's "Local Fields", Artin-Tate's "Class Field Theory" and some chapters of the Cassels-Fröhlich "Algebraic Number Fields" colloquium.

[Anonymous No.15296700]

>>15296631
you mean besides the existence of such sets as the naturals, the integers, the rationals, etc.?
no, not really

[Anonymous No.15297246]

>>15296543
yeah but I write most proofs down in my own language. I find that I'm way too lax with what I'm willing to accept if I just read a proof, I have to write it out and justify the steps that aren't immediately obvious. Of course this is assuming the proof wasn't presented to me by someone that went through the steps on a board or something, in that case I just note down the theorem and maybe the main idea of the proof.

[Anonymous No.15297287]

Since chatgpt is shit and its recommendations is shit, you guys are my last hope.

Could you recommend me lecture aeries or books where power series and taylor series are rigourously derived in probability spaces? Or even generally in spaces with general measure.

[Anonymous No.15297728]

>>15296692
thanks, anon. i was really hyped about reading Guillot, but i really don't dig how he introduces and treats p-adics (maybe the other chapters are better). i'll try Cassels, although i hate the way it's typeset

[Anonymous No.15297747]

Out of nowhere, I’ve had a desire to go back and learn geometry. Can anyone recommend a book/pdf to use?

[Anonymous No.15297763]

>>15297747
Springer

[Anonymous No.15297862]

>>15297747
Coxeter.

[Anonymous No.15298025]

what do you think

[Anonymous No.15298112]

>>15279472
Any German, Swiss or Dutch bros here that have done oral exams before?

How do you feel about them? I can't make up my mind on them after taking almost exclusively oral exams for all my classes at my mathematics master at a German uni.

[Anonymous No.15298121]

>>15297747
>go back and learn geometry
Im assuming this means you know more advanced math? Rees' 'Notes on Geometry' is a great book then. you dont need to know that much either some linear algebra and the rest you can pick up as you go.

[Anonymous No.15298128]

>>15298112
fuck it im making a thread, this board needs some discussion anyways

[Anonymous No.15298348]

>>15279472
Alright /mg/, help me out. Two graduate schools have accepted me.
School B
>Stronger in analysis, applied/computation topology and TDA, optimization and ML
>All their algebraists have died over the past few years
>Slightly pozzed -- tons of jews, nonwhites and women, but jews are concentrated in TDA (how typical) while the analysis group in entirely white
School A (current institution)
>Stronger in differential geometry, shape analysis, algebraic geometry, homological algebra, homotopy type theory, and mathematical physics
>No jews, all non-whites are chinese or upper-caste indians that speak perfect english, and the women are not ugly grotesque monsters
School A is ranked about 20-30 positions higher, close to home, and my SO lives there. School B is a couple hours away from both SO and home.

[Anonymous No.15298355]

>>15298348
CORRECTION:
School B is the one ranked higher, closer to home & SO. School A is hours away from both.

[Anonymous No.15298375]

>>15298348
School B is a dead end for your professional development. You're better off going straight to industry now over school B, unless you're okay with over signaling leftist leanings and prostrating yourself before those groups. Even then it seems you're done for if your goal was to get a job in academia.

[Anonymous No.15298481]

>>15298375
Thanks for your input. You sound immature and insincere, and are most likely telling me what you think I want to hear. I wouldn't be surprised if you were purposefully trying to get me to go to the worse school.

Maybe I'm selling School B a bit short. Their analysis group is very strong, with a focus on PDE's, optimization, machine learning, operator theory, information theory, and non-commutative algebra & geometry. These all seem like very fun subjects, as opposed to type theory, shape analysis, and commutative algebra at School A. Also, half the faculty of School B are Russian, which is a big plus. School B receives about 40% more funding in mathematics from the NSF, despite having the same number of faculty.

[Anonymous No.15298493]

>>15298348
You have a SO?
And you want to study math?
Just take a job as a high school math teacher.

[Anonymous No.15298497]

>>15298493
Only reason to teach high school is for the 15-18yo girls. Nothing to be gained for a guy about to get married.

[Anonymous No.15298529]

>>15298497
1. take $$$
2. give $$$ to wife
3. fuck 15-18 yo girls.
4. Profit? Fucking PROFIT? You give a fuck about PROFIT? Just open up a bible and start reading a random page after you cum in the teenslut to dissipate the sense of corruption & perversion

[Anonymous No.15298533]

>>15298529
oh
your wife has to cook dinner too
nice curry and Asian dishes, too

[Anonymous No.15298535]

>>15298529
Nobody is fucking 15-18 yo girls as a high school teacher.

[Anonymous No.15298536]

>>15279472
[math]
2 + 2 = 4
[/math]

[Anonymous No.15299064]

>>15298536
2 + 2 = 4

[Anonymous No.15299176]

>>15298481
>Their analysis group is very strong, with a focus on PDE's, optimization, machine learning, operator theory, information theory, and non-commutative algebra & geometry.
What? These are all HUGE very different fields, some of them not even analysis

[Anonymous No.15299220]

botwillacceptanything built my hotrod
https://www.youtube.com/watch?v=egPR5MuE7u0
...something about some damn fool thing in the balkans

[Anonymous No.15299484]

https://writings.stephenwolfram.com/2023/03/chatgpt-gets-its-wolfram-superpowers/

Its SO over....

[Anonymous No.15299488]

I feel kind of stupid here. Shouldn't this equation be P = 1/sp (f_nl - f_fl)?

[Anonymous No.15299489]

>>15297287
>books where power series and taylor series are rigourously derived in probability spaces
what do you mean by that

[Anonymous No.15299528]

Fuck all them formulas and gayyyy shit
Math is about Endurance!
What's the biggest number you've counted up to?

[Anonymous No.15299880]

>>15299528
20

[Anonymous No.15300297]

>>15300294

I made a new thread. First time doing it so I hope I didn't do anything wrong.

[Anonymous No.15300311]

>>15284571
There's nothing spiritual about math. It's all demonic, i.e. man-made semiotics and apophenia.

[Anonymous No.15302394]

How would you solve:
[math]\vec{x'} = A\vec{x} + \vec{p}[/math] where [math]\vec{p} = \begin{pmatrix}
1 \\
5
\end{pmatrix}[/math]
I know how to solve [math]\vec{x'} = A\vec{x}[/math], but don't know what to do with the constant vector.

[Anonymous No.15302418]

>>15299528
0, rest follows from natural induction.

[Anonymous No.15302659]

https://www.theguardian.com/us-news/2023/mar/24/new-orleans-pythagoras-theorem-trigonometry-prove

Apparently two teenagers proved the Pythagorean theorem using only the law of sines (which I guess itself doesn't rely on the Pythagorean theorem... didn't know that). I can't find the proof, and the news articles I'm finding are just saying that the teens are SAYING that they have proven it. Am I wrong to assume that the proof is most likely invalid?