/sci/ - Science & Math

>>12214123
Not Necessarily.

>>12214123
I'm in abstract algebra and I'm really struggling with the few occasions where I have to do anything to or with polynomials. Essentially, I've forgotten half or more of the algebraic manipulations necessary.
What's the best way to fix this?

What to learn after multi variable calculus?

Can anyone suggest supplemental material or a more advanced text that pairs up nicely with Axler, aka develops stuff without assuming that the field is \mathbb{R} or \mathbb{C}?

>>12214253
Measure theory.
Differential geometry.

>>12214123
Has this been solved yet?

>12214361
Think of the subset without the identity.

>>12214387
Also there is an ambiguity within the notation: when talking about subsets of A, do we talk about the set of elements used by the operation of the group, or the operation of the group itself (aka pairs a \in A ( (a', a''), a''')?

>>12214392
Set theory was a mistake.

>>12214253
real analysis.

>>12214253
computer science

>>12214312
>>12214401
>>12214406
how do I start? what's it about?

>>12214413
Start with point-set topology.

>>12214413
>Measure Theory
How to construct non-shit integrals that you can solve problems with. Leads to applications in probability and lets you prove more convergence theorems. Start with Wikipedia.
>Differential geometry
You are probably accustomed to geometry in the standard euclidean space. Diffgeo let's you graduate to spaces with different properties such as the surface of a sphere and shit. Start with Wikipedia.
>Real analysis
First you need to know about the triangle inequality. Look it up on Wikipedia. You should now prove everything you already know about calculus by mindlessly spamming the triangle inequality. This is real analysis. You can start with Wikipedia.
>computer science
This is the easiest for a beginner. You can start right now by killing yourself.

>>12214361
Can't you just pick a subset that doesn't contain an identity element?

I don't actually know anything about algebra.

>>12214253
linear algebra, real analysis, advanced calculus (Calc on R^n), and elementary differential geometry. they are all natural extensions of what you’ve already covered.
>>12214296
Roman, Lax, Greub, and Dieudonne all have very good abstract advanced lin alg books

>>12214296
forgot Kostrikin and Manin as well. After you’ve done any of these you’d be close to ready for functional analysis provided you know your basic real analysis and a little bit of measure theory.

>>12214123
Let's turn it into another question.
Consider a finite group [math]G[/math], i.e. a set [math]S_G[/math] of size [math]g=|S_G|[/math] and a particular, fixed, group structure [math]*_G[/math].

Here's a list of the first such groups.
https://en.wikipedia.org/wiki/List_of_small_groups

The number of groups per finite cardinality [math]n[/math] is
[math](\sigma_n)_n=[/math] 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, ...
The fractions of those values over [math]n[/math] are
[math](\tfrac{1}{n}\sigma_n)_n[/math] = 1.0, 0.5, 0.333, 0.5, 0.2, 0.333, 0.143, 0.625, 0.222, 0.2, 0.091, 0.417, 0.077, 0.143, 0.067, 0.875, 0.059, 0.278, 0.053, 0.25, 0.095, 0.091, 0.043, 0.625, 0.08, 0.077, 0.185, 0.143, 0.034, 0.133, 0.032, 1.594, 0.03, 0.059, 0.029, 0.389, 0.027, 0.053, 0.051, 0.35, 0.024, 0.143, 0.023, 0.091, 0.044, 0.043, 0.021, 1.083, 0.041, 0.1, 0.02, 0.096, 0.019, 0.278, 0.036, 0.232, 0.035, 0.034, 0.017, 0.217, 0.016, 0.032, 0.063, 4.172, 0.015, 0.061, 0.015, 0.074, 0.014, 0.057, 0.014, 0.694, 0.014, 0.027, 0.04, 0.053, 0.013, 0.077, 0.013, 0.65, 0.185, 0.024, 0.012, 0.179, 0.012, 0.023, 0.011, 0.136, 0.011, ...

But since we fixed the group operation on [math]S_G[/math] in advance, it seems a little more complicated to compute the chance of a random subset [math]S_A\subset S_G[/math] equip it with [math]*_G[/math] restricted to [math]S_A[/math] forming a group. Correct me if I'm wrong and it's simpler.

In the original subgroup-subset variant, we can ask the possibly complicated question given B, I think, since some groups will be more likely to be subgroups than others

For n<100, say, we have all the data to answer all questions (even brute forcing it), so all these have concrete answers.

>>12214123
Let's turn it into another question.
Consider a finite group [math]G[/math], i.e. a set [math]S_G[/math] of size [math]g=|S_G|[/math] and a particular, fixed, group structure [math]*_G[/math].

Here's a list of the first such groups.
https://en.wikipedia.org/wiki/List_of_small_groups

The number of groups per finite cardinality [math]n[/math] is

[math](\sigma_n)_n=[/math]
1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, ...

The fractions of those values over [math]n[/math] are

[math](\tfrac{1}{n}\sigma_n)_n=[/math]
1.0, 0.5, 0.333, 0.5, 0.2, 0.333, 0.143, 0.625, 0.222, 0.2, 0.091, 0.417, 0.077, 0.143, 0.067, 0.875, 0.059, 0.278, 0.053, 0.25, 0.095, 0.091, 0.043, 0.625, 0.08, 0.077, 0.185, 0.143, 0.034, 0.133, 0.032, 1.594, 0.03, 0.059, 0.029, 0.389, 0.027, 0.053, 0.051, 0.35, 0.024, 0.143, 0.023, 0.091, 0.044, 0.043, 0.021, 1.083, 0.041, 0.1, 0.02, 0.096, 0.019, 0.278, 0.036, 0.232, 0.035, 0.034, 0.017, 0.217, 0.016, 0.032, 0.063, 4.172, 0.015, 0.061, 0.015, 0.074, 0.014, 0.057, 0.014, 0.694, 0.014, 0.027, 0.04, 0.053, 0.013, 0.077, 0.013, 0.65, 0.185, 0.024, 0.012, 0.179, 0.012, 0.023, 0.011, 0.136, 0.011, ...[/math]

But since we fixed the group operation on [math]S_G[/math] in advance, it seems a little more complicated to compute the chance of a random subset [math]S_A\subset S_G[/math] equip it with [math]*_G[/math] restricted to [math]S_A[/math] forming a group. Correct me if I'm wrong and it's simpler.

In the original subgroup-subset variant, we can ask the possibly complicated question given B, I think, since some groups will be more likely to be subgroups than others

For n<100, say, we have all the data to answer all questions (even brute forcing it), so all these have concrete answers.

>>12214123
What's difference between group and sets? For sure it's subset, it's not proper group because doesn't contain elements twice but groups doesn't have to so it's a subset.

It was boring. What about trying to reduce some simple integral to abstract algebra next round?

>>12214252
For real coefficients just treat it as a vector space (bc it is one)
Elaborate a bit more on what it is you're struggling with.

>>12214518
group is equipped with an operation. set in general is not.

>>12214518
A group is a non empty set (with binary operations).

>>12214123
am I being dumb or is the OP pic trivial,

take any G, any nontrivial subgroup A, then take B to be a singleton that is not the identity, it's not a subgroup unless b^2=b.

>>12214567
You're mostly correct. However, this part is a bit redundant.
> it's not a subgroup unless b^2=b.

>>12214484
Thank you for the suggestions!
Which one is your favourite?

>>12214567
you need an identity for any group or subgroup

>>12214567
ja es ist selbsvesthendlich

>>12214123
If B is a group then yes.

>>12214641
subgroup of A*

>>12214567
lol nope. read the problem once again

Good morning /mg/!
Let's let go of the low IQ gorilla problems from lost redditors and instead here's a bit more interesting problem: If a function f:R->R is continuously differentiable and uniformly continuous, does it follow that the derivative f' is bounded?

>>12214253
Algebra! I recommend Stillwell's Elements of Algebra.

>>12214671
If you can't solve this in under 10 minutes you don't belong in /mg/!

>>12214747
Good day and good bye /mg/.

>>12214123
I never even finished calc and I know the answer, what

>>12214809
Didn't you hear? That's now officially the dumbest question ever asked on /mg/. It's funny because only a complete retard could seriously ask such a question. That's why we are pointing at him and laughing. He makes us feel smarter, and feeling smart is good for doing maths.

>>12214820
>That's why we are pointing at him and laughing. He makes us feel smarter, and feeling smart is good for doing maths.
Nah, we talk about it because we have nothing else to talk about and nobody puts in any efforts in posting, let alone answering anything

Lemma 2.11 (ii) in Jech states that if a is an ordinal and b is in a, then b is an ordinal. He says that this follows from the definition, which is that an ordinal is a set that's:
1. Well ordered with respect to set membership.
2. Transitive.
I don't see how it's immediate though. The well order of b is immediate but not transitivity. Assume c is in b. We know c is also in a and hence a subset of a, but that doesn't tell us that the subset must be contained in b, right? What if there are elements in c that are outside of b?
I have come up with a proof of this using well-orderedness by looking at the least element not contained in b, I just wanted to check if I'm overthinking it?
Is it really obvious from the definition that an element of an ordinal is well-ordered?

>>12214451
I read the wikipedia page, but now I'm confused: what's they funny €, and why is it always bigger than 0

>>12214988
The funny euro sign is epsilon, and you don't want to use the ugly \epsilon but the cuter \varepsilon ([math]\epsilon\ \varepsilon[/math], respectively). The idea is that [math]\varepsilon[/math] is an upper bound for how far away something is from something else, be it an entry in a sequence or a value of a function from for example a limit. This distance can't be negative, and the idea is that you want to show that this distance can be arbitrarily small, i.e. no matter how small (positive) epsilon you choose, the distance will eventually still be smaller than that.

>>12214671
The answer is so simple and intuitive when you think about it.

>>12214959
Try contradiction. If b<a with a transitive, and b not transitive, then there exists y<b which is not a subset of b, but since b is a subset of a then y is not a subset of a, contradicting that a was transitive. In general, when Jech says something is obvious or by definition, it probably takes a couple of sentences to reason out.

>>12215031
Stop posting anime!!!

>>12214959
Nevermind, it's actually much more simple. All elements of b are also elements of a by transitivity of a. Then just use the fact that a is totally ordered under set membership to get the transitivity of membership for b, since all hereditary members are members of a.
>>12215037
>which is not a subset of b, but since b is a subset of a then y is not a subset of a
This logic does not check out lol.

>>12215113
I guess so, my bad its early here. In my head I was already using the linear order to imagine that an element of an ordinal can't jump over an ordinal that contains it to have an another one.

>>12214575
>>12214567
lmao the absolute state of /mg/. Non-whites shouldnt be allowed in here!

>>12215176
hey... some of our best posters are mongoloids

>>12215206
How do you know?

>>12215206
>>12215219
Oh you mean the finnish tranny lol

>>12214451
>You should now prove everything you already know about calculus by mindlessly spamming the triangle inequality. This is real analysis.
Ahahaha

>>12215224
it's funny because it's true

How would I show that if a function is in [math]H^1(\Omega)[/math], then it’s restriction to a an open subset [math]\omega[/math] of Omega is in [math]H^1(\omega)[/math] ? I can show that the restriction is in L^2 but can’t show that the derivative is in L^2. Any idea ?

>>12214988
>the funny €

>>12215284
who the fuck denotes a set by a lowercase letter. lmao. what is your issue in showing the derivative is in L^2? the derivative should be the restriction of the derivative on [math] \Omega [/math], no?

This is a retard question and prob wrong thread but:

In sudoku, if every row adds up to 45, and every column adds up to 45, and all grids contain values between 1 and 9, is there any possible way for this to not be valid?

>>12214451
>his is the easiest for a beginner. You can start right now by killing yourself.
>>12214413
For computer science, Knuth's Concrete Mathematics is a good start if you want to learn more about algorithms (and it's a damn fine book that goes from intro to proofs into advanced mathematics in the later chapters). The book teaches you a lot about how to *approach* novel problems, which is very important.
What I especially like is that its discrete probability teaches you hard probability problems without measure, so that when you work with more general settings and probability measures, you haven't lost out on much prior intuition (aside from the usual differences when going from at most countable to uncountable probability spaces).

>>12214123
Why the hell is combinatorics so hard bros? I got through baby Rudin, Folland, a bit of papa Rudin, artin, Aluffi, Willard and Munkres, and some of Hatcher.
BUT Van Lint, Alon-Spencer, and Stanley are kicking my ass

>>12215443
Yes. 45 = 1+2+3+4+5+6+7+8+9 = 1+2+3+4+5+6+8+8+8
Take a valid sudoku solution and replace all the 9s and 7s with 8.

>>12215396
Yes it should be that but I can’t prove prove it.

>>12215557
How are you defining the derivative of a function in H^1? Is it some Fourier bullshit or some other way?

>>12215443
Is there any way for what not to be valid?

>>12215581
We define [math]\frac{\partial u}{\partial x_i}[/math] in [math]H^1(\Omega)[/math] as the function v such that [math]\int_\Omega u(x) \frac{\partial\phi}{\partial x_i}(x)dx = \int_\Omega v(x)\phi(x)dx[/math].

>>12215620
For my exercise I just need to show that the restriction of this derivative is equal to the derivative of the restriction and the conclusion will follow but I can’t prove that.

>>12215557
>>12215581
Oh, you're probably just explicitly talking weak derivatives. I think then the idea is just: Let [math] u \in H^1(\Omega) [/math] and consider the function [math] v = (Du)|_\omega [/math]. Then [math] v = D(u|_\omega ) [/math] because for any test function [math] \phi [/math] supported inside [math] \omega [/math], [math] \phi [/math] extends to a test function on [math] \Omega [/math] just by setting to 0 everywhere new, now just test v against phi, extend the region of integration to Omega, use Du's definition, and re-restrict the integral.

>>12215629
>>12215631
Sorry, I got lazy with the Latex because this is an incredibly small manipulation.
>Why can you extend phi
Compactly supported in an open set in R^n, means it vanishes in some neighborhood of the boundary.
>Why can you extend/contract the region of integration
Support of the integrand always contained in little omega.

[eqn](ax + by + cz)^7 = c^7 z^7+7 b c^6 y z^6+7 a c^6 x z^6+21 b^2 c^5 y^2 z^5
+42 a b c^5 x y z^5+21 a^2 c^5 x^2 z^5+35 b^3 c^4 y^3
z^4+105 a b^2 c^4 x y^2 z^4+105 a^2 b c^4 x^2 y z^4+35
a^3 c^4 x^3 z^4+35 b^4 c^3 y^4 z^3+140 a b^3 c^3 x y^3
z^3+210 a^2 b^2 c^3 x^2 y^2 z^3+140 a^3 b c^3 x^3 y z^3+
35 a^4 c^3 x^4 z^3+21 b^5 c^2 y^5 z^2+105 a b^4 c^2 x y
^4 z^2+210 a^2 b^3 c^2 x^2 y^3 z^2+210 a^3 b^2 c^2 x^3 y
^2 z^2+105 a^4 b c^2 x^4 y z^2+21 a^5 c^2 x^5 z^2+7 b^6
c y^6 z+42 a b^5 c x y^5 z+105 a^2 b^4 c x^2 y^4 z+
140 a^3 b^3 c x^3 y^3 z+105 a^4 b^2 c x^4 y^2 z+42 a^5
b c x^5 y z+7 a^6 c x^6 z+b^7 y^7+7 a b^6 x y^6+21 a
^2 b^5 x^2 y^5+35 a^3 b^4 x^3 y^4+35 a^4 b^3 x^4 y^3+21
a^5 b^2 x^5 y^2+7 a^6 b x^6 y+a^7 x^7[/eqn]

Brainlet set theory question. To my shallow understanding, a "large cardinal axiom" is an axiom which states that there exists a cardinal satisfying some property [math]\varphi[/math], such that ZF(C) cannot prove nor refute the existence of a cardinal satisfying [math]\varphi[/math]. Denote the theory obtained as ZF + [math]\varphi[/math].

Question: Are there any cardinal [math]\kappa[/math] and large cardinal property [math]\varphi[/math], such that ZF is capable of proving the *existence* of [math]\kappa[/math] (say, [math]\kappa = 2^{2^{\aleph_0}}[/math], whose existence is implied by ZF), but only through ZF + [math]\varphi[/math] can we prove that [math]\kappa[/math] satisfies [math]\varphi[/math]? Or does adding a large cardinal axiom actually create new elements that were never there before, in the world of ZF?

>>12215703
Usually k is defined by phi, so you can't prove that k exist without also proving that it satisfies phi.

>>12215689
Wow. You calculated that yourself?

>>12214123
toxic general

>>12215703
No new cardinals are added. For instance, it is consistent that [math]\omega_1[/math] is a measurable cardinal. Certainly ZF proves [math]\omega_1[/math] exists, but ZF cannot prove there is a measurable cardinal.

>>12215742
Yes.

>>12215473
It's probably because you're not hungarian. You might as well give up.

Combinatorics will be the first field to be automated with AI.

>>12215882
and that's a good thing.

>>12215838
Thanks anon, this is exactly what I'd hoped to hear.

>>12215473
although on a more serious note, you learn combinatorics 100% by solving problems. This is probably the least efficient way to learn because it takes a very long time. A lot of the problems in those books are very difficult and the expectation should be that they take like 3+ hours. When I did combinatorics with van Lint, I think we were only assigned like 3 problems a week and I still had a ridiculously tough time finishing the problems.

>>12215642
Oh thank you so much, I was not using the fact that phi could be easily extended to a test function.

>>12215882
lmao good luck with that

>>12216141
What's so hard about counting?

>>12216145
Everything.

>>12215930
yeah, you're right. We get like 4 problems to do over 2 weeks, where the professor says that you need the full two weeks or else you probably won't get like half of them. With other topics, I'd get a lot of work, yeah, and I did a lot of problems in those courses too, but in neither did I get such a large concentration of "supremely deep insight" questions. Each combinatorics problem set I get is nothing but questions that ask you to think completely out of the box with what you're given.

Also this class has massively improved my impression of discrete mathematics. It's fucking tough as nails and interesting to boot - wish it got more love from the theory building crowd, especially with stuff like matroid theory

>>12216145
counting is hard. Also combinatorics takes a lot of leaps in reasoning - it's not as simple as linear recurrences. Even enumeration is rich with problems for which a general algorithm can't really help. And enumeration is only one (1) part of combinatorics, though it's the most famous. Transfer theorems in complex analysis and combinatorics were made largely for computer science, but aside from that I don't see how AI could automate combinatorics.

>>12215838
>No new cardinals are added.
Are you saying this is, in some sense, always the case like that? I guess not.

>>12216373
It's definitely not always the case. And in some cases it might not even be a meaningful question, there might not be a natural way to say that a cardinal in one model is the same cardinal as one in the other, or the might each have cardinals the other doesn't. If they're both well-founded models you can match up the ordinals of each model with the ordinals of the universe and compare them that way, but not all models are well-founded and without that you're stuck. Also note that every cardinal is countable in a forcing extension, so even the largest cardinals can be turned into tiny ones by forcing.

>>12215838
How is your example any different from the example where the cardinal is aleph_0 and the property phi is that the set of subsets of the cardinal is the successor cardinal (CH)?

>>12216373
I interpreted the dudes questions like is there is cardinal that ZF proves exists, which may or may not have a large cardinal property. As >>12216416 says, this question is a bit weird. because we are not always sure what the cardinals are in some model. It takes choice to even say that every cardinal is an initial ordinal. So I guess I kind of mean that since ORD is absolute, and if every cardinal is an initial ordinal, then no new cardinals are added. Of course different models may see different initial ordinals, as >>12216416 says, by collapsing cardinals.
>>12216444
That is not true, if you mean the set of subsets of aleph_0 is in bijection with aleph_1, which is just CH, which is not a large cardinal property. A large cardinal property usually adds consistency strength. That is, if you assert that phi is a large cardinal property it should be the case that ZF+phi proves the consistency of ZF.

>>12214253
Topology -> Algebra -> Algebraic topology -> Category theory

Axiom of induction is wrong.
Today I can read a certain amount of words in a textbook. I can read one word. If I can read n words, then surely I can read one more word to get to n+1 words.
Now axiom of induction says that today I can read
100000000000000000000000000 words.
Ridiculous.

>>12216534
induction isnt an axiom you retard

>>12216545
>https://en.wikipedia.org/wiki/Mathematical_induction#:~:text=The%20axiom%20of%20induction%20asserts,rather%20than%20over%20individual%20numbers.
>The axiom of induction asserts the validity of inferring that P(n) holds for any natural number n from the base case and the inductive step.

Did anyone solve the question about the bounded derivative?

>>12216534
My friend, this is why we are interested in topological spaces and not topological time. Time is the enemy of induction, but the seemingly infinite space always has room for one more grain of sand.

>>12216569
The proof is trivial.

>>12216594
Not sure if you're joking.

>>12216556
>In second order logic
thats not how you were using it fucking retard
you were using induction on naturals, which is a theorem you moron

>>12216619
You are a moron and a fucking retard, ok?

>>12214451
>You should now prove everything you already know about calculus by mindlessly spamming the triangle inequality. This is real analysis. You can start with Wikipedia.
Were the fuck does this meme come from?
Only some stuff is proved with this.
I literally only recall using the triangle inequality when we studied point set topology and sequences, other than that you would always use some theorem from the topic at hand.

>>12216619
Oh, you want to talk about the naturals, bitch?
https://en.wikipedia.org/wiki/Peano_axioms
Induction might be a scheme, but it's not a theorem.

Consider games with 2 players, win/loss rewards no draws, perfect information. In particular, consider games where theres a finite number of number of moves to be made, and if it's a players turn and there are no moves then they lose.

Obviously then its either a forced win or loss for the first player, depending on the initial game state

What can be said about the 'disjoin sum' of two gamestates? That is, a player can make a move in either gamestate, and the two states do not affect each other. Imagine two separate tic tac toe boards and you can only place an X on one board at a time (an ignore the possiblity of draws; TTT is not a great example).

>>12216735
An axiom scheme, or just an axiom if dealt in second order logic.
Also a theorem in ZFC. These are not mutually exclusive. The anon you're replying to is literally retarded.

>>12216761
That sounds like the stuff conway wrote onag about

>>12216735
>he doesnt know how the peano "axioms" work
so an operation is associative by the group axioms huh?
axiom has different uses you idiot
we prove that the peano "axioms" hold for our definition of the naturals, so that we can use the consequences of the peano "axioms" that we already know
in the exact same way by proving that something is a group by proving it satisfies the group "axioms"

they arent axioms for the naturals, theyre theorems

>>12216774
axiom has two different uses you moron
something is isomorphic to the naturals if it satisfices the peano axioms
we still have to fucking prove them from prior principles to use them on an example dipshit

>>12216649
he’s making shit up. it’s not true at all.

>>12216761
>Obviously then its either a forced win or loss for the first player, depending on the initial game state
Is it really obvious?

>>12216791
if it's not obvious then its at least well known

I also should clarify that I am asking what is the state of the composite game based on the state of the component games.

I'm not positive, but I don't think its a simple as knowing the W/L value for each. It's more nuanced than that

>>12216802
>if it's not obvious then its at least well known
lol. You said it's obvious, you no longer think so? Do you even know the proof?

>>12214518
Groups come with an Associative Binary Operation.

People often like to repeat the meme that in maths, you understand, not memorize. That if you actually understood what was going on, there's no need to remember anything.
How true do you find this to be in higher maths?
I often forget the proofs of theorems. Yet I was pretty sure I understood them when I read them.

>>12216831
>How true do you find this to be in higher maths?
>I often forget the proofs of theorems.
You will remember and understand theorems as you keep using them. Stuff like compactness becomes very innate.
Just keep exposing yourself to the same things over and over, and it'll stick without needing to memorize by rote.

>>12216831
Somewhat true. Remember, when you're proving something always go back to the definitions/axioms. Also, when you are doing hw problems, always ask yourself what the problem gave you. Ask yourself what the problem implied.

>>12216791
It is obvious. Either player 1 must be able to force a win, or player 2 must be able to force a win based on the first move of player 1. If it were not the case, then either the game must go on forever, which is excluded because it was said to be a finite game, or there must be some uncertainty at some point in the game and that is excluded because it was said to be a game with perfect information and deterministic states.

>>12216761
Two games includes the possibility of each player winning one of the games. 1) if a player can force a win from move 1, then the first will use their move to force the first game, while the second will force game 2. However if move number 2 is the decisive one then the first player will always lose both games.

>>12216784
>idiot
>moron
this is /sci/, can you tone it down a little

>>12214671
Is there a quick and nice solution to this? I can only do it by contradiction.

>>12216946
Post your proof. I thought it's false.

Ok so, we equip a countable set with a well-ordering and whoop de doo there is a canonical way of turning this into the group Z based on the structure the well-ordering gives, right?
Now what if we take a well ordered set of uncountable cardinality and do the same thing. Can we have an uncountably long version of Z??? What the fuck?

>>12217067
Well, you can have uncountable rings based on ordinals, using the surreal numbers. But that method gives large countable ring for large countable ordinals, I don't think I know a natural way to turn ordinals into rings that gives Z for every countable ordinal. I think whatever method you have of turning a countable ordinal into Z will use the bijection with N explicitly, so trying to do it for an uncountable ordinal will just get stuck instantly.

>>12214671
I think sin(x^3)/x might work as a counterexample, its derivative is 3 x cos(x^3) - (sin(x^3))/x^2, which is continuous and unbounded, and I have numerical evidence that the 0.5-holder coefficient is finite(approx 2), which is sufficient for uniform continuity. I need to actually prove the last part but I'm too sleepy, I'll do it tomorrow.

Anyone can give me pointers on this basic introductory math class problem?

(TRUE or FALSE)
Let x and y be integers:

Then 12 | 4x + 5y <=> 12 | y - 4x

I'm meant to know how to prove this properly and I can kind of guess the answer is yes, but I have no clue how to properly prove it.

>>12214252
Looks...

There's addition, that means you count them together, then multiplication, which means you repetitevely count same number in a row, exponentiation is multiplying in a row,
tetration is exponetiating in a row.

Then there's some thing like 3^n = 3^n-1 * 3 flip to it.

That's all I know.

>>12217422
Suppose 12 divides y-4x. Then y-4x = 12n <=> y=4x+12n, for some integer n. Now 4x+5y = 4x+5(4x+12n) = 4x + 20x + 60n = 24x + 60n = 12(2x + 5n). Try doing the similar for the other direction.

How does this book compare with Stein and Shakarchi's books on real analysis and complex analysis?

>>12217475

Thank you, anon.

So, in a similar way, I would write:

Suppose 12 divides 4x + 5y such that 4x + 5y = 12k for some integer k. Then:

4x + 5y = 12k <=> 4x = 12k - 5y

Therefore
y - 4x = 6y - 12k = 6(y-2k).

Since 12 | 4x + 5y <=> 2 | 4x + 5y, it follows that y is an even integer such that y=2p for some integer p.

Thus,

y - 4x = 6(2p - 2k) = 12(p - k)

And since (p - k) is an integer, 12 divides y - 4x.
----------------

Would you perhaps also be able to give me a hint regarding one last problem?

True or false:
Let n be a natural number such that (7n+2) is divisible by 15. Then, (4n+9) is divisible by 5.

Showing that 15 divides into 7n + 2 such that 15k = 7n + 2 doesn't seem productive here. I'll try giving it another attempt later on when I'm a bit less tired, but if you have any hints I'd like to hear them.

Thanks again for the help on the previous problem

>>12217653
That seems correct, yes. For the second problem, let's use the fact that 5 divides x if and only if 5 divides 3x (prime numbers 3 and 5). Suppose 7n + 2 = 15k. Then 3(4n + 9) = 12n + 27. What can you do with this?

>>12217696
Does it feel good to hand hold noobs?

>>12217702
Yes.

>>12214671
Obviously just the typical fast oscillating decaying function, like sin(x^2)/x^2 or something.

>>12217592
Stein and Sh'karchi are based. Rudin is a meme.

>>12217067
look up the long line in topology

Are there any topological games you could actually play? I tried defining a game which kind of has the rules of go but you play on a topological space but it's pretty shit

>>12217887
Why?

>>12217696
Thank you, that cleared things up greatly.

Without writing the whole proof formally, I assume one can just suppose that:

15k = 7n + 2 <=> 2 = 15k - 7n

And since 5 divides a number x iff 5 divides 3x, we know that 5 | (4n + 9) <=> 5 | (12n + 27)

So,

12n + 27 = 12n + 25 + 2 = 5n + 15k + 25
(By "plugging in" the 2 from before)

Factoring 5 shows that 5 divides 12n + 27 = 3(4n + 9) meaning 5 divides 4n + 9.

>>12218484
Or what I had mind was 12n + 27 = (7n + 2) + (5n + 25) = 15k + 5n + 25 = 5(3k + n + 5), but the result is the same.

I feel like an absolute retard reading this book
I spend forever on every sentence and still do not understand what they are trying to say

is the book bad or am I dumb?
I notice its typesetting is pretty subpar

>>12217883
d/dx((sin(x^2))/x^2) = (2 x^2 cos(x^2) - 2 sin(x^2))/x^3
That derivative is bounded. You'll need go be a little more careful than that.

>>12217422
Adding the divisions you get 12 | 6y so y must be even. So in general this doesnt hold

>>12218610
>didn't put comma after hence.
This is amateur hour.

>>12218610
They're saying that given a point (x0,y0) on a curve, we can always find the slope at that point. Using this fact, we can use Euler's method to get an idea of the value of y for x values relatively close to x0. Euler's method is linear approximation with more autism.

>>12218672
There's books I'd really like to read but I just can't because they are written by some Greek or Spanish guy with zero regards for english grammar. And I don't mean merely commas, I mean sentences that are well intended but very hard to read.

>>12217067
>well-ordering
You mean total order?
(In the Z case, well-order works too, but that seems unnecessary)

>>12218610
Looks alright to me. Maybe you should get some sleep.

>>12218687
that makes sense
I think what bothers me is the notation and the words used/not used

I get confused by whether/when y and y_0 are functions or numbers and what it means to use a single number y_0 in place of y in f(x,y)
I just feel fatigued by trying to understand if what I think I understand is what they meant

>>12218736
y_0 tends to be a constant specific value most of the time, it would be very rare for it to represent a variable.
for just y, you just have to be careful and consider the context

>>12216534
>then surely I can read one more word to get to n+1 words.
non sequitur

>>12219056
It's an empirical fact. I have never observed a situation where I could not read one more word. Every time when I stop reading I know I could have read one more word.

I found an efficient estimator of the mean mu of a multivariate normal distribution with know variance (identity matrix), the efficient estimator I found is the arithmetic mean. How would I find all functions depending on mu such that an efficient estimator exists?

>>12219111
>every time I've considered a number, it was much smaller than googolplex
>I have never observed a situation where I couldn't add one to it and stay smaller than googolplex
>it's an empirical fact
>by induction every number is smaller than googolplex

Here's an attempt at a generalization of the notion of [math]\sigma[/math]-algebras. Is this concept familiar to any of you?

Let [math]\kappa \ge \aleph_0[/math] be some cardinal, and [math]X[/math] some set. Call [math]\mathcal{K} \subseteq 2^{X}[/math] a "[math]\kappa[/math]-algebra" if [math]X \in \mathcal{K}[/math], [math]\mathcal{K}[/math] is closed under complements, and for all [math]\{E_i\}_{i < \kappa} \subseteq \mathcal{K}[/math] we have that [math]\bigcup_{i < \kappa} E_i \in \mathcal{K}[/math].

There's a good chance this definition is completely useless of course. I was just thinking that perhaps [math]\sigma[/math]-algebras can be viewed as a special case of a more general set-theoretic structure, defined as above or similarly. Perhaps the same way one defines a measure as a countably additive function acting on a [math]\sigma[/math]-algebra, one might define a suitable analogue as a [math]\kappa[/math]-additive (in the sense of cardinal arithmetic) function on a so-called [math]\kappa[/math]-algebra. Am I completely retarded?

>>12219276
It makes sense, the trouble is that there's just not many uncountably additive measures unless you're willing to assume some pretty strong large cardinal axioms.

>>12219276
Look up complete lattices.

>>12219276
Isn't this just a k-complete boolean algebra?

>>12219339
Yes, it is.

Imagine we have some 2d manifold embedded in 3d space. We pick an angle and project it orthogonally onto the plane. We paint the areas with the manifold all black and the rest white.
Now we notice that from all every angle we choose the result we get is a (full) circle.
Does it follow that the manifold is a sphere?

need help understanding absolute value. please share good material on it. in particular does x-y <= |x-y| hold?

>>12219554
>need help understanding absolute value. please share good material on it. in particular does x-y <= |x-y| hold?
Yes. a<=|a| for all real numbers a.

>>12219521
no. consider a sphere and an arbitrary manifold contained in the interior of the sphere.

>>12219588
Forgot to mention the manifold must be connected.

>>12214253
probability / statistics

>>12219610
then I don't know (but I think it's true)

>>12219629
By the way by a sphere I didn't mean homeomorphic to a sphere. I meant that the embedding itself is a rigid sphere in 3d.

>>12214123
im so embarrassingly bad at integration, any books that get you good at it fr, mb also some stuff on lebesgue-stieltjes integration next to just rieman

>>12219658
like is inside interesting integrals good?

>>12219521
>(full) circle
it's called a disk nigga
take the centroid C of the manifold
take any point x on the the manifold
rotate so that x-C is parallel to the plane
project
distance is the radius of the circle, r
do the same for all x... the distance is always r

>>12219666
I didn't say the circle has the same radius under every projection, Satan.

>>12219666
>>12219673
Also why would the radius of the circle coincide with the distance? Why would the center of the circle be C?

>>12219521
>We pick an angle and project it orthogonally onto the plane
Clarify

>>12214253
How do I get into cryptography? What areas of math should I focus on?

>>12219699
number theory, probability, and combinatorics but mostly number theory

>>12219709
oh and a lot algebra as well

>>12219694
The manifold is a subset of R^3. Any subset can be orthogonally projected to a given plane in R^3. So the manifold can be projected orthogonally in a plane.

>>12219673
it's left as an exercise to the reader to prove that if the projection is always a disk, then one can prove that the disk must alway be of constant radius (tip: arrive by contradiction, showing that there exists a point x such that it has two different distances from C)

>>12219677
what do you mean?
you show that all x have equal distance from a point C

>>12219723
additional tip: use a fixed point theorem

>self studying linear algebra because it turns out the course I took at Uni was corrupted by engineers
>hit a bad patch in my understanding of inner product spaces
>I keep getting problem after problem wrong
>Figure out what the fuck I'm doing wrong after each problem and Feynman it out
>literally run out of problems
Do I just wait 3 to 6 months, hope I forget how to solve them and try again?

>>12219723
>(tip: arrive by contradiction, showing that there exists a point x such that it has two different distances from C)
Are you sure this works? Or are you just blindly guessing? I've no idea how to prove this.
>you show that all x have equal distance from a point C
How? None of what you asserted is justified.
>distance is the radius of the circle, r
I see no reason why this should be true.

>>12219752
Can't you just search for problem sets online?

>>12219771
all the ones on inner product spaces I keep getting are of extraordinarily poor quality and generally lack a solution for me to check my work against

What does "surgery on 3-manifold" mean in simple terms?

>>12219613
Not math.

i used to suck at math now im a natural at it should i give up

>>12219821
You should learn proper grammar.

>>12219821
you should make a rational decision

>>12216831
Total meme desu spread by autists. A better way to say it, is you learn math by doing math. The more you are exposed to infinite sequences, the more you understand their proprties. The more examples of groups you are exposed to, the more you understand, and the more applications of groups to weird problems, the more you understand. The autists who like to quote this are more then likely memorizing, but in a very subtle way, where they memorize a trick or an application instead of the definition.

>>12220039
>applications of groups to weird problems
Examples?

What is the good book on number theory.

>>12220068
Ireland & Rosen

Is anyone here good at statistics? I have a couple of general questions.

i want to go over middle school, high school and uni maths again
does the wiki have good resources for doing this?

>>12220151
>middle school, high school
Khanacademy

>>12220163
Not rigorous.

>>12214123
take B as the empty set

>>12220103
which one is who? do you have their instas?

>>12220163
ok i'm aware of khan academy but i think i would like textbooks better idk

>>12219694
learn linear algebra ffs

Brainlet here

"If C has supremum s, then $D=\{x\in\mathbb{R}|-x\in\C \} has infimum -s$ "

How the fuck do I prove that -s is the greatest lower bound?

>>12220249
Tex isn't showing

>>12220253
{math}D = \{ -x \mid x\in \big(C\cap {\mathbb R}\big) \}{/math}

except [] instead of {}

>>12220253
{math}D = \{ -x \mid x\in \big(C\cap {\mathbb R}\big) \}{/math}

except [] instead of {}

[math] D = \{ -x \mid x\in \big(C\cap {\mathbb R}\big) \} [/math]

>>12220249
>>12220253
>>12220298
x->-x is an isomorphism between the relation x<=y and the opposite relation (x<='y iff y<=x).

>>12220253
{math}D = \{ -x \mid x\in (C\cap {\mathbb R}) \}{/math}

except [] instead of {}

[math] D = \{ -x \mid x\in (C\cap {\mathbb R}) \} [/math]

>>12219752
>>12219774
Download books and do the problems there?

>>12217887
I agree, love the SS texts. I was just wondering what Rudin's approach to the subject is like. From what I understand though none of his books are as canonical as baby rudin is, so I'm not bothering with papa and grandfather rudin

>>12220249
Prove that -s is a lower bound. Indeed, if x is in D, then -x is in C, so s >= -x, but then -s <= x.
Now prove that if t is another lower bound, then t <= -s. Well if t is a lower bound for D, then -t is an upper bound for C (similar to what we just did). Then s is the least upper bound for C so s <= -t. But then -s >= t.

Nobody has any ideas on the sphere problem?

stupid questions here

*Is there any formal study on whether it's possible to solve an integral or not?
*If you determine that it isn't possible to solve, is there a way similar to "completing the square" where you add an infinitesimally smal term (i.e. 0) which makes it solvable?

>>12220249
Use the fact that for all [math]\varepsilon > 0[/math], there exists [math]c \in C[/math] such that [math]c > s - \varepsilon[/math].

>>12220518
https://math.stackexchange.com/questions/265780/how-to-determine-with-certainty-that-a-function-has-no-elementary-antiderivative

>>12220518
https://en.wikipedia.org/wiki/Risch_algorithm

>>12214671
>\sqrt{x}

>>12220575
He did specify [math]f:\textbf{R}\to\textbf{R}[/math].

>>12220594
Why is /sci/'s math environment so shit?

>>12220598
>there was a time when serious mathematicians used to come here

>>12220481
It doesn't sound too hard, first show that the points on the edges of the disks must form a sphere, and then the sphere must be the whole manifold because adding more stuff inside either makes it disconnected or makes it stop being a manifold.

>>12220646
They still do.

>>12220694
>first show that the points on the edges of the disks must form a sphere
This sounds very hard

>>12220750
you are not a serious mathematician

How long before complex analysis starts to become intuitive? The first results are quite striking and yet seem to fall out of thin air.

>>12216831
"Young man, in mathematics you don't understand things. You just get used to them" - Von Neumann

>>12221037
How is it so far? I'm thinking of taking it next semester. How does it compare to real analysis, if even comparable?

>>12220214
Is this bait? How much has /mg/ fallen?

>>12214123
If B is finite, non-empty, and closed, then yes it's a subgroup of G.

>>12219521
No. You can picture deforming a torus so that there is no straight line going through the central hole.

If you instead consider intersection with each affine plane then it is true.

>>12221118
thats a weird peepee

>>12221072
I can't really say. Class is moving very slowly–a month in and we're only through 2 chapters in Stein's book (the first of which is basically preliminaries). So far I'd say the problem sets (taken from Stein) have been less fun / interesting than Rudin, but the material otherwise is more interesting than basic undergrad real analysis.

>>12221072
I'm taking it right now and I think it's a lot more interesting than real analysis, although I've never been exposed to much with complex numbers beforehand. It's a lot more computational based than real analysis, although there are still plenty of proofs.

>>12221308
What topics are you guys on right now? We just covered Cauchy's integral formula.

>>12214123

As stated, this is trivially false (in general). Suppose that the group G is the integers, and the subgroup A is the evens. Now, B is merely a subset of G, the evens. Pick B = {0, 2, 4} as a counterexample, where the set is equipped with the same addition operation of the original group, G. B fails the group operations with respect to itself (non-closure), using same addition operation, ordinary addition.

I suspect that the exercise is meant to get us to think about distinctions between groups and sets as-such, which entail the operation to be used in the case of a group. Someone please let me know if I am mistaken. Further, I used a banal finite example, and the question probably implies "natural" infinite counter-examples. The crux of course is that thingy need only be a subset of thingy, while other thingy is a subgroup of thingy. In between all this is the question of the appropriate group operation, hence my proposed counter-example.

>>12221485
We've just gone through all of this. Moving onto Meromorphic functions this week.

>>12216619
>induction on naturals, which is a theorem
lol

Can I learn Calculus while not being up on my algebra?
Relearning this shit is boring

>>12221764
technically yes but 99% of all calculus problems involves algebra so you're kinda buttfucked

>>12221771
I know the simpler stuff
Really if I run into something I forgot I'll just look it up

>>12221764
Computational calculus is extremely algebra reliant. Having TAed several calc classes, algebra is where 90% of errors are made.

>like doing differentiation
>hate integration
>trying to pick upper level math courses
>"differential equations"
>neat!
>it's just integration at the fucking time
Fuck this bullshit!

>>12221914
T R I C K E D
A G A I N

>>12221914
What did you think the 'differential' of differential equations meant?

>>12221914
>hate integration
brainlet detected

>>12221924
And don't even get me started on there being zero coverage of scatter or box and whiskers graphs in "Graph theory"

>>12221931
Equations where you differentiate to solve.

>>12221933
>integrate by parts
>integrate by parts
>... More integrate by parts
riveting stuff

>>12221942
you have never done an interesting integral, and I guarantee you cannot even derive the reduction formulas you used for the trig product integrals.

>>12221944
>he thinks that's interesting
Cute

F-friends, I need some enlightenment. Consider the following problem. Take any [math]f\in W^{k,1}((0,1))[/math] (Sobolev space with K derivatives). I extend that function as follows
[eqn]\widetilde{f}(x)= \begin{cases}
f(x) & x\in{(0,1)} \\
0 & x=1 \\
f(2-x) & x\in (1,2)
\end{cases} [/eqn]
It's easy to show that [math] \widetilde{f}\in W^{k,1}((0,1)\cup (1,2))[/math] and the weak derivatives of [math]\widetilde{f}[/math] in [math](0,1)\cup(1,2)[/math] look exactly as you think the look. For a.a. x in [math](0,1)\cup (1,2)[/math]
[eqn]\widetilde{f}^{(n)}(x)= \begin{cases}
f^{(n)}(x) & x\in{(0,1)} \\

(-1)^nf^{(n)}(2-x) & x\in (1,2)\end{cases} [/eqn]
What about the weak differentiability on [math](0,2)[/math]? By using partitions of unity I was able to show that [math]\widetilde{f}[/math] is in [math]W^{1,1}((0,2))[/math] but I cannot show that the higher distributional derivatives of [math]\widetilde{f}[/math] on [math](0,2)[\math] are indeed given by locally integrable function. Any body has an idea how to show it, or can anyone provide an counterexample?

Most kino integral coming through: [math]\int_{-\infty} ^{\infty} \exp(-x^2)dx = \sqrt{\pi}[/math]

>>12220594
>\sgn(x)\sqrt{|x|}

>>12214123
[math] B = A \setminus \{ e \} [/math]

>>12221988
You sure are stupid.

>>12221984
And what's the derivative of that at zero?

I hate proofs so goddamn much. Is this a bad sign?

>>12221118
That will produce slightly dented circles when the entrance is on the edge of the circle, so it doesn't work. It does have projections homeomorphic to a disk, but I don't think that's what he meant.

>>12222005
>You sure are stupid.

>>12222072
It's a sign you don't belong in maths. Because proofs is literally 99% of what maths is.

>>12221955
I don’t think math is interesting, but being disinterested in useful integrals is a sign of low iq. Physicists understand and appreciate them, probability theorists, engineers, and everyone in applied math. You just aren’t very bright and wouldn’t know what to do with them.

>>12222143
Applied math is a sign of low IQ.

>>12221972
Based

>>12222222
wtf there was no post at
>>12222221

>>12222075
You're right I think

>>12222223
Why did you even think to check that?

>>12222222
Anon, I.....

>>12222222
witnessed.

>>12222222
indeed

Any books with probability tricks? Just to have fun with and get the brain boiling for a bit? (and only secondarily to learn something and get new intuition)

>>12222005
He's right though?

>>12222263
yes, he is right.

Is it feasible to learn maths from undergrad up using only formal proofs and proof assistants? I know there's a lot of classical results being formalised in coq, but they're usually based on postgrad material, and I'm talking about LEARNING.

>>12222282
That sounds like a neat project.

>>12222291
Yeh, I've already been messing around with coq, but the accessible lit on it (like benjamin pierce books) is almost exclusively classical CS focused.
Among popular works, I've only seen HoTT actually use proof assistants for assisting with OG math proofs and not for formal verification and the like. Then again, it expects you to know cat theory and alg topology beforehand, and I don't want to be the undergrad category theorist.

>>12222303
I've used agda more than coq, it seems nicer to me.

>>12222282
No, it sounds just as retarded as this
https://en.wikipedia.org/wiki/New_Math

>>12222350
But that failed because, while the chances to find someone interested/smart enough for math in highschool AP classes are already abysmal, looking for them in grade school is downright retarded. But it was the 50's so they probably didn't know better.

>>12222355
>while the chances to find someone interested/smart enough for math in highschool AP classes are already abysmal, looking for them in grad school is downright retarded
what the fuck m8

>>12222319
I haven't heard much about other proof assistants (other than obligatory Microsoft NIH shilling), I assume it's because there's even less development being done on them than on coq.

>>12222361
das rite, real mathematics are done on vixra by kindergarten dropouts

>>12222068
Think of the function mapping from R\{0} to R. I don't think this is possible with a function mapping from all of R.

>>12222391
It is definitely easier, on R\{0}, but that's not the question. I think sin(x^3)/x might work on all of R

Hi /mg/!

If someone is interested, there is an OATS talk in roughly 1½ hours. https://sites.google.com/view/nialltaggartmath/oats
>Title: Transfer systems and weak factorization systems
Then some papers someone might enjoy:
>Rational homotopy type of mapping spaces via cohomology algebras
https://arxiv.org/pdf/2010.04579.pdf
>Real K-theories
https://arxiv.org/pdf/2010.04590.pdf
>Abelian Ideals and the Variety of Lagrangian Subalgebras
https://arxiv.org/pdf/2010.04358.pdf
>Prescribed virtual homological torsion of 3-manifolds
https://arxiv.org/pdf/2010.04271.pdf

>>12222222
Checked. What a time to be alive!

>>12222421
>Seminars on Mondays
Algebraists truly are deviants.

What's a good non-meme CS undergraduate thesis topic if I am in a shit university? I'm currently going through Tao and Baby Rudin, and also Artin.

I'm quite interested in any CS research, as long as it has rigorous maths.

>>12222222
CHECKED.

>>12221971
I don't really recall the exact definitions, so please correct me if I'm wrong.
Set [math]f(x) = x[/math]. Clearly [math]f \in W^{k, 1}((0, 1))[/math] for any [math]k[/math], and [math]k = 2[/math] in particular.
[math]\tilde{f}[/math]'s first derivative should be the Heaviside step function and the second derivative is the Dirac delta, which can't be given by a locally integrable function.
Repost these questions in /sqt/, I barely open /mg/ nowadays.

>>12222487
Great! That was exactly what I needed! Thank you!

>>12222222
checked

>>12222222

Assume Someone has a Textbook and a solution guide to the problems in the book. Furthermore, assume that the student has no one to consult with or check his answers for him, which means he will have to self evaluate. Next, assume this person takes the proper amount of time and care to finish a single exercise section within the textbook without rushing with a good faith attitude to learn the material. Thanks to the solution manual, the student's answers can be partitioned up into correct and incorrect answer. Finally, when he finds he has an incorrect answer, he spends time explaining (on paper) why he got the answer wrong and how to arrive at the correct solution.

Question 1: Since the student has had a look at the correct solutions, they are fresh in his mind. How long should he wait before he attempts to solve the set of incorrect solutions again?

Question 2: What correct answers to incorrect answers ratio should the student set as a benchmark for themselves before they move on in the text?

Question 3: Assume the student has set a correct answer to incorrect answer ratio, and has discovered that, after finishing an exercise section, that he has failed to meet his benchmark, what should they do at this point?

If X1,...,Xn is iid following a multivariate normal distribution of unknown mean mu and known variance Identity matrix. I know that the arthmetic mean is an efficient estimator of mu. For which function f can I find an efficient estimator of mu ? For which function f I can have a non biased estimator of mu ?

>>12221764
no

>>12222222
septs confirm

>>12222892
>For which function f can I find an efficient estimator of mu
wtf does that even mean?

>>12223088
Oh sorry I mean f(mu)...

>>12214123
hell no??? B isn't necessarily closed so it's obviously can't always be a subgroup of G.

>>12222682
>plz spoonfeed me how 2 learn

>>12222682
q1 ask yourself what you did x days, find an x that is the closest to your current day and add a week just for safety
q2 50/50, back before the advent of scantrons, answering 50% of the content correctly was the general go to for most schools as their metric for competence, 70% only arose recently due to needing to deal with multiple choice questions + over 200 students.
q3 you're kinda fucked, either advance on and hope you pick up competency as you complete the book or just give up

>>12219699
Just get a phd and do research at that point. Do you really want to spend your entire life destroying freedom globally?

>>12222222
HOLY

>>12223332
It would be the morally correct decision to devote one's life to saving Americans from themselves. Maybe not the fun or fame-bound one, but heroes don't always wear capes.

>>12222421
>>12222432
Thanksgiving no less

>>12222222

Surveychad here

What are the typical courses (including electives) one takes in an undergrad?

can someone post the "terence tao in anime club" image

>>12223455
Calculus, Linear Algebra, Real analysis, algebra, topology, complex analysis makes up the bare minimum at most US universities.

>>12219699
>>12219709
Number theory is important, but most of modern cryptography research requires a lot of ring theory and algebra in general. Here's a good shortlist
>analytic and algebraic flavors of number theory (not elementary)
>computational aspects of the above
>theory of NP-completeness / hardness
>algebra (groups, rings, number fields, field theory, Galois, etc.) because how the fuck do you do lattice / post-quantum without algebra
>graph theory and combinatorics (expanders are used in the isogeny research)
>etc
Crypto is a big field whose theoretical aspects can touch a large amount of math, but the above is the minimum for what you should look at if you want to read research in the field. Note that the inclusion of analytic number theory means you should be comfortable with analysis as well.

>>12223488
Complex analysis is pretty optional these days - beautiful subject, but I think schools prefer you have 2 semesters of upper div honors / early grad real analysis and 2 of upper div honors / early grad algebra more. Topology is nice though.
Also most introductory complex analysis courses end up being a "calculus but with complex numbers" course rather than a proof course proper.

Serious question
if [math]\frac{1}{\infty} = 0[/math], and derivative ratios like [math]\frac{dy}{dx}[/math] are a ratio of infinitesimals (i.e., [math]\frac{1}{\infty}[/math]), doesn't that mean dy/dx is literally just 0/0, and calculus is just solving for the defined, but indeterminate quantity 0/0?

>>12223528
The [math] 1/ \infty [/math] is not well defined and sort of meaningless. You're thinking of [math] \lim_{n \to \infty} 1 / n = 0[/math], which is well defined. As for dx and dy, I would say don't worry about it until you study more about measure theory and other types of integration.

>trying a problem in group theory
>have no fucking clue what to do

>>12223587
Consider a subset without the neutral element.

>>12223598
I don't think that'll work.

>>12223608
Use the classification of finite simple groups.

>>12223561
>1/∞
>1/ \infty is not well defined and sort of meaningless.
but how? it's defined right there, and zero is perfectly well defined. We are talking about the signless +infty= -infty, right?

>>12219810
Youre so wrong

>>12223651
Measure-theoretic probability is math, statistics is not. If you're confused, it's because statistics departments kind of died and got reabsorbed into math. CS fared better on its own, and we'll have to wait and see whether data science can make it.

>>12223639
You can define infinity as 1/0, but it behaves differently from real numbers, isn't all that useful, and is not the only kind of infinity there is. dy and dx can be seen as infinitesmials in non-standard analysis or synthetic differential geometry, but those both have their own different notions of infinitesimal which you have to be careful with.

>>12223587
Post the problem so we can laugh at you.

>>12223734
Are there any groups whose growth rate is faster than polynomial but subexponential?

>>12223757
Ouchie, this one's a toughie. Do you know the answer?

>>12223678
>but it behaves differently from real numbers,
have there been no attempts to use it algebraically? The most common issues I see are indeterminate results ([math]\frac{0}{0}[/math], [math]\frac{\infty}{\infty}[/math]), which are no sin of their own, and errors arising from the questionable assumption that [math]\infty + 1 = \infty[/math] (while this is a reasonable assumption when +inf and -inf are divergent, when the two converge at 1/0 it no longer makes sense.)
>isn't all that useful
left as an exercise for the reader

>>12223772
I think it's supposed to be yes, but I can't figure out how to make one and I don't want to just cheat by looking it up.

>>12223790
Well, I won't spoil it for you, but if you eventually want something, check this out.
https://arxiv.org/abs/math/0607384

can you guys teach me how to do quadratic equations

>>12223842
Complete the square
x^2 + ax + b =0
(x+a/2)^2 = x^2 + ax + a^2/4
So
x^2 + ax + b = (x+a/2)^2 + b - a^2/4 = 0
so (x+a/2)^2 = a^2/4 - b
So x+a/2 = +-sqrt(a^2/4 - b)
so x = -a/2 +- sqrt(a^2/4 -b )

>>12223788
>have there been no attempts?
what you're defining isn't part of the real numbers at all. You're doing something that violates the field structure of the real numbers. Either way
>indeterminate results
you can meaningfully define your way out of these in other sets (see wheel theory) but it makes no sense in the usual construction.
>infty + 1 = infty assumption
okay, now I *know* you don't know what you're talking about. There are not only different types of infinite sets, but we do arithmetic differently on ordinals than we do in cardinals. You're trying to express that the union of any infinite set with a single element does nothing to change its cardinality, which is true, but if we understand \infty to be the first infinite ordinal, this is absolutely not true.
>left as an exercise
it's *not* useful because it breaks all the usual properties that *are* useful. You can verify for yourself that considering a / 0 for any a in ring R cannot be part of the set, or else things get fucky.
Again, you could think of situations where it may be useful to consider such a thing, but they aren't compatible with the usual ring / Dedekind cut properties of the reals.

>>12223842
practice factoring. Many quadratic equations can be solved by noticing how to factor it.
If not, just use the quadratic formula.
In practice, people don't complete the square explicitly - they do it implicitly by applying the quadratic formula (whose derivation involves completing the square)

>>12223797
Thanks! :)

>>12223858
>You're doing something that violates the field structure of the real numbers.
I don't know what you mean, all I see is closure of division over the reals. The behavior should be no more controversial than the behavior of zero.
>different types of infinite sets,
I don't understand how their existence is a counterargument. Assuming that infinity actually means infinity, and therefore MUST always be the largest number, only makes sense when the two infinities diverge. Their existence, from what I can see, appears archaic, unfruitful, and complete tangential to the behavior of the convergent infinity, 1/0, which has potential to become well-defined.
I think it needs to be said that, for [math]\pm\infty[/math], "infinity" is actually a misnomer.

also, [math]\frac{a}{0}=\infty[/math], the same way [math]a*0 = 0[/math]. this should not be controversial.

Considering that most of the math community leans more to the left, if we slipt the causes between mainly prevalent community culture pressure versus mainly the nature of someone naturally drawn towards mathematics, what percentage would you give to each?

Is it more a matter of playing the current social game, or is it more a matter of natural factors independent of short-term social context?

>>12224031
Pretty sure that it's mostly nature. Mathematicians tend to put a lot of trust in authority and be conformist. The prevailing dogma in the academia and the media is the left and they just go along. I strongly believe most of them have never even heard an opposing viewpoint that's not a caricature from the left.
Of course, community culture pressure contributes too. There are also some hiding their power level but it's not a lot.

>>12224031
Excellent bait my friend

>>12224063
>Mathematicians tend to put a lot of trust in authority and be conformist.
?

>>12224063
>Mathematicians tend to put a lot of trust in authority and be conformist
Mathematicians are pretty consistently found to be open minded

>>12224074
Bait for what? Causing chaos? I'm genuinely interested in what /mg/ thinks about this.

Personally, I think most of them leaned left over time due to the cultural pressure/influence of universities. They incorporated it little by little. But that's a hypothesis, maybe I'm wrong.

>>12224063 raises a point, some of them just go along with the most seemingly prevalent and official narrative because they're naturally not invested in formulating original political opinions from scratch.
But I don't think those are most of them, the ones that are more politically vocal seem to be very sure of coming up with their positions themselves.

>>12224122
>Mathematicians tend to put a lot of trust in authority and be conformist
>Mathematicians are pretty consistently found to be open minded
These are not mutually exclusive.

>>12224101
I'd like to point out your data samples are nearly exclusively from fucking twitter and that's ignoring the fact that the Mathematician and other member categories have no given source given whatsoever

In the future just report, don't respond to it.

>>12224197
new
>>12224197

>>12224063
>the current dogma is left
>left is author and conformist
Mate if the current political climate feels """"""leftist""""""" to you, you don't know what you're talking about. It is at best liberal socially, but fiscally the US is and always has been conservative.

>>12224144
So, as far as you're concerned, you'd rather not have the risk of potential chaos that the subject can bring than allowing it to be discussed?
Just trying to understand here.

>>12224238
[math]\text{politics}\neq\text{math}[/math]

>>12224244
Right, but we do talk about aspects of the life of a mathematician or of a math student that aren't strictly just mathematics.

But I understand the feeling of repulse. There's so much forced political bullshit everywhere that some people just want to completely get away from it.

>>12224144
spergs are so easy to bait it almost isn't funny.
Almost.

>>12224141
If you read the full series of papers they first show that their system accurately predicts big five personality traits (apparently a standard personality test in psychology) from Twitter. If I recall correctly their prediction system is more accurate to the test they trained to than other big five tests are.
This graph is just an application of that.

I think the original paper is
>M. L. Kern et al., The online social self: An open vocabulary approach to personality
But it's pretty well known to work, for example see
>M. Kosinski, D. Stillwell, T. Graepel, Private traits and attributes are predictable from digital records of human behavior
>G. Park et al., Automatic personality assessment through social media language
>H. A. Schwartz et al., Personality, gender, and age in the language of social media: The open-vocabulary approach

This is just a particular example that I remember because I saw a talk by the author. I'm sure you could find more direct personality data that says the same thing. Or if you are right then you should be able to find some evidence that says mathematicians are conformists.

>>12224139
>conformist
>open minded
These are opposites.
Also in the future just a tip friend you can click on people's posts so you don't accidentally reply to the wrong one!

>>12224341
uh oh

>>12224141
This was the top result I found googling "mathematician big five personality":
>https://www.sciencedirect.com/science/article/abs/pii/S0191886913006132
>Among Big Five factors, Openess was the best predictors of test performance
There's a bunch of papers that I can link you after you debunk this ones methodology. I would love you to post some of your own evidence though!

>>12222222