/sci/ - Science & Math

>>15995883
Why is this guy so influential? Do everyone use his textbook or something?
I couldn't count the number of time a book/lecture notes/colleague refer to something 's proven on his Vector Analysis or Topology book. It's so weird. And like when I checked, it's true the statements and proofs are only available on his books. Nothing similar or equivalent in other classic textbooks.
Is he just that original pedagogically, like when reorganizing the knowledge to teach?

>>15995883
wtf is a measure?

>>15996177
A non-negative and sigma-additive function from a sigma algebra into the extended real numbers that maps the empty set to zero.

>>15996181
how tf do you map an algebra to the real numbers? isn't an algebra just a set of elements and set of operations on those elements?

What's the strongest form of Determinacy not known to be inconsistent with ZF? I am aware that there are games on sets of reals which are provably indeterminate in ZF alone, so is Real Determinacy the best we can do?

>>15996189
Topology has a different definition of algebra.
A sigma-algebra over a space \[X is a subset \[S\subseteq 2^X\] obeying:
\[X\in S\]
\[S\] is closed under complementation
\[S\] is closed under countable unions

>>15996391
cannot into tex. krumping myself on the morrow

>>15996189
It's the elements of the (sigma) algebra being mapped, silly.

>>15996391
That's just a countably complete lattice, which is an algebra in the other sense.

>>15995883
I'm a fucking noob
I'm reading Hardy and Wrights Intro to number theory and I'm getting filtered by their argument.
[math] q = (2\cdot3\cdot5\cdot\ldots\cdotp) + 1 [/math] gives us a new prime.
So we can say that [math] q < p_{n}^{n} + 1 [/math]
That's fine.
Then [math] p_{n+1} < p_{n}^{n} + 1 [/math]
This is also fine.
Now, [math] p_{n} = 2^{2^{n}} [/math]
I can see that this is true for small primes. I can see that [math] 2^{2^{n}} [/math] grows very fast, whereas I expect [math]p_{n}[/math] to grow slowly.
Next they say [math]p_{n+1} \leq (p_{1}\cdotp_{2}\cdot\ldots p_{n}) + 1 < 2^{2+4+\ldots+2^{n}} < 2^{2^{n+1}}[/math]
I can see why the right inequality is true, and why the left inequality is true. But I cannot intuitively see why the middle inequality is true.
I've tried googling it and I can't find any answers. I've tried asking chatGPT and it doesn't understand what I'm asking.
Is this the right place to ask, or should I post in /SQT/?

Reposting because I fucked up the latex
>>15995883
I'm a fucking noob
I'm reading Hardy and Wrights Intro to number theory and I'm getting filtered by their argument.
[math] q = (2\cdot3\cdot5\cdot\ldots\cdotp) + 1 [/math] gives us a new prime.
So we can say that [math] q < p_{n}^{n} + 1 [/math]
That's fine.
Then [math] p_{n+1} < p_{n}^{n} + 1 [/math]
This is also fine.
Now, [math] p_{n} = 2^{2^{n}} [/math]
I can see that this is true for small primes. I can see that [math] 2^{2^{n}} [/math] grows very fast, whereas I expect [math]p_{n}[/math] to grow slowly.
Next they say [math]p_{n+1} \leq (p_{1}\cdot p_{2}\cdot\ldots p_{n}) + 1 < 2^{2+4+\ldots+2^{n}} < 2^{2^{n+1}}[/math]
I can see why the right inequality is true, and why the left inequality is true. But I cannot intuitively see why the middle inequality is true.
I've tried googling it and I can't find any answers. I've tried asking chatGPT and it doesn't understand what I'm asking.
Is this the right place to ask, or should I post in /SQT/?

>>15996896
fucked up the formatting again, and I just found out they changed how old a post has to be before you can delete it
I meant
[math] q = (2\cdot3\cdot5\cdot\ldots\cdot p) + 1 [/math] gives us a new prime.

>>15996896
Looks like something you could prove by induction

>>15996896
I think you have several notation errors here or what you have written is missing details. Isn't [math]p_n[/math] meant to be a prime so how can [math]p_{n} = 2^{2^{n}}[/math]?

>>15997191
>Isn't [math]p_{n}[/math] meant to be a prime
yes. but i'm retarded and made yet another mistake lmao.
Its supposed to be [math]p_{n} < 2^{2^{n}} [/math]

>>15997181
>Looks like something you could prove by induction
I know it should be true by induction. It just doesn't *feel* true to me, by which I mean it's not intuitive to me. Like the proof I've seen handwaved the induction part to me. So I don't feel like it was proven adequately rigorously.

I think my misgiving is that proving it holds for the the product of the first n primes + 1 may grow faster than 2^(2^n). I don't see how it has been proven.

I'm not taking any math classes. I'm just trying to learn from books on my own

>>15997246
To handwave the induction further in favour of explaining the intuition:
You've accepted that the nth prime is, in general, no greater than 2^(2^n).
It follows from this that the product of the first n primes, then, will be less than [math](2^{2})(2^{4})...(2^{2n}) = 2^{2+4+...+2n}[/math]
Since [math]p_n[/math] grows significantly less quickly than your even powers of 2, adding 1 will not bridge the gap of the product

>>15997246
>I think my misgiving is that proving it holds for the the product of the first n primes + 1 may grow faster than 2^(2^n). I don't see how it has been proven.
I don't think i stated this clearly.
When I see induction proofs, it's usually something clear like:
to find the sum of a f(x) from x =1 to x=n, you can use g(x). Then you prove that adding f(n+1) to g(x) gives you g(x+1). Then you prove your base case. This feels clear to me.

However, induction as used for in this book for this case. It's not clear to me. The inductive step isn't clear.
Maybe I would find a clearly explanation if I read more, or maybe I'd eventually be able to prove it myself. But I thought, maybe I should ask /mg/ and see if anyone can help me be less of a math retard (>inb4 I mean "less of a retard")

>>15997252
>Since pn
>�
>�
> grows significantly less quickly than your even powers of 2, adding 1 will not bridge the gap of the product
The other thing is that I'm not convinced that 2^2^n is always greater than Pn.
Just because I can calculate that it is greater for several million or billion primes, that doesn't prove it is always the case.

>>15996395
so you take a subset of the powerset of some set in which all elements in this subset have a compliment inside this subset and the union of countabl many of those sets in the subset is a set in the subset
then you take this subset and map it to the set of positive reals which also happens to have the infinity inside it for some reason.
why?

Is mathsbot better than khan academy?

>>15997867
Are we supposed to know what this shit is? We read books and go to lectures here.

>>15997791
Do you know the Lebesgue measure? It's a generalization of that.

Is there a nonzero complex entire function that vanishes at m+in for every pair of integers m,n?

>>15997259
Bertrand's postulate says that there is always a prime between n and 2n. From this it is easy to prove that [math]p_n<2^n[/math] which is strictly stronger than your bound. Maybe not that satisfying because you probably haven't studied Bertrand's postulate.

https://arxiv.org/abs/2401.13508

Mochizuki won

>>15998267
For anyone wondering, the answer is yes, look up “Weierstrass factorization theorem”

>>15997259
> I'm not convinced that 2^(2^n) is always greater than Pn.
I’m no expert but, can’t this be proved using the prime number theorem?

>>15998288
>Maybe not that satisfying because you probably haven't studied Bertrand's postulate.
I haven't studied it.
I'll probably just make a note of Bertrands postulate, and make a note of the problem in the book, then see if I can continue the rest of the book.
I don't see much point in getting stuck on this one problem when I might just understand later on.

>>15998505
>I’m no expert but, can’t this be proved using the prime number theorem?
I believe they want to use 2^(2^n) to prove a better upper bound for the prime number theorem. The book is quite old, i don't believe the content has been updated in several decades; however it is a reprint.

I can't find a satisfying solution, and I'm guessing no anon wants to bother trying to write a proof. I'm just going to be (hopefully temporarily) filtered

>>15996177
>wtf is a measure?
The measure of the closed interval [0,1] is 1, right? It is 1 unit long.
The measure of the open interval (2,3) is 1, right? It is 1 unit long and the endpoints are tiny anyway.
The measure of their union [math] [0,1]\ cup (2,3) [/math] is 2, right? Because they don’t overlap.

The formal definition of Lebesgue measure is just trying to capture this intuition. Don’t worry about abstract “measures” and axioms and bullshit like that until you are comfortable with the basic Lebesgue measure and Lebesgue integral

>>15998890
>cup
I mean [math]\cup[/math]

>>15998534
>>15997259
It is easy to prove by “strong induction” using a version of the classic “factorial” proof of the infinitude of primes.

Induction step: Suppose [math] p_k \leq 2^{2^k}[/math] for all k < n. Then [math] p_n \leq 1 + p_1 p_2 \cdots p_{n-1} \leq 1 + 2^{2^1}\cdot 2^{2^2}\cdot \cdots \cdot 2^{2^{n-1}} = 1 + 2^{2^1 + 2^2 + \cdots + 2^{n-1}} = 1 + 2^{2^n-1} \leq 2^{2^n}.[/math]

Others have pointed out that you can get better bounds pretty easily, but this one is nice because it just needs induction and nothing special from number theory.

>>15998890
>and the endpoints are tiny anyway.
that doesn't sound rigorous

>>15999111
That's just some motivation. The precise claim is that the endpoints make up a set of zero Lebesgue measure. Have you considered reading a book?

>>15999116
>Have you considered reading a book?
no. a professor mentioned measures in a lecture and i had a brain aneurysm trying to understand the definition on the wikipedia page

I have to take 2 of these classes. I'm studying part time while working as a software dev.
>measure and integral theory
>abstract algebra 2
>statistics 3
>linear optimization

Any suggestions? All 4 sound very interesting.

>>15999177
My favorite classes so far were topology and probability theory btw.

>>15999141
hi I’m the one who wrote the “tiny” thing. yeah I was just trying to motivate what measures are supposed to do, because it’s actually very simple. then the definition starts to make sense

>>15999177
>>15999182
those are all good topics
>topology
what kind of topology? simplicial complexes, homology, homotopy? or more like: open sets, filters, completely Urysohn’s Lemma? Those 4 courses all sound good, but personally I would stick to
>algebra 2
>measure and integration
because these are core courses in any pure-math programme. the most interesting of the 4 is linear programming imo, but that is a subject unto itself, it doesn’t really lead into other topics except harder kinds of optimization

>>15999727
Based Maimon (where is that guy nowadays anyway?). Wildberger's meandering when someone challenges him to be more specific with his polemics is just exactly what you'd expect from him.

>>15998931
Oh that's a nice straightforward proof, thanks anon

here, shouldn't it be 1 <= i < j <= n, instead of i <= j

>>15999889
i = j would just correspond to an edge starting and finishing at the same vertex no?

>>15999962
ohh yes, it would, i mistook i = j as epsilon
thanks

Anyone have the /mg/ curriculum?

>>16000019

>>16000027
You have read any of those books have you anon?
Post proof or it didn't happen.

>>16000045
>he didn't learn stochastic partial differential equations in highschool
ishygddt ngmi.

>>15999740
>where is that guy nowadays anyway?
Quit academia and quit the internet. I think I read he still answers personal emails. Probably just Perelman-maxxing.

>>15998931
you added the series wrong because theres no 2^0 term, but yeah thats not going to change anything important

>>16000230
My probability teacher says, in all seriousness, that measure theory should be taught in high school.

>>16000941
as a replacement for Riemann integral? people say that sometimes but I don’t think they understand the average HS student *or* the average HS teacher

>>16000027
Contents of the 'Freshman' level would be enough to get masters in pure maths in here. Math teaching level should be pupmed up a little, starting in middle school. We dont even have integrals in hs ffs, dont know about the US but its probably even worse.

i'm writing matura in may how much can i cram in 3 months?
i can ace the basic one but i get 40% at best on the extended curriculum

>>16001302
which country? If you have no problem with the basic one then 3 months would be more than enough to get through the entire extended material from highschool.

>>16001464
poland
the material is fine i just cant come up with solution to many questions or fuck up basic arithmetic like 4*3=24
my idea is to just read solutions for questions a couple of times and try to solve them later across the whole curriculum at once ex. i do geometry 3 times a week along with other things in spaced intervals and then revise these same questions a month later

>>16001485
The trial maturas are usually much harder than the real ones. I'm tutoring on highschool level and some of the extended trial math questions I'm being asked to explain are straight up retarded.
>fuck up basic arithmetic like 4*3=24
Yeah messing up simple things is a problem but just don't get too stressed. You have 3 hours for everything and don't need to get 100% on extended one. I don't know where you want to go to uni but I only had like 40 something and still got easily accepted for maths (niby tylko MS na Politechnice Sląskiej ale jednak xd) and then did good enough to publish my first paper before even graduating masters.

Your method is good, just go over the material chronologically and you'll notice that you are doing the same thing over and over again but with just one new element added in each section.

>>16001510
yeah i really fucking want to get to UJ or UW ideally for CS but that's unrealistic, i need at least 75% for math
>Your method is good, just go over the material chronologically and you'll notice that you are doing the same thing over and over again but with just one new element added in each section.
yeah i came up with it a year ago too bad i haven't even started yet

Uhhhh.... algebraic geomesisters.... our response?

>>16001825
I'd rather live free and be poor than be a cuck for richfags while wearing a gilded collar.

>>15996896
You don't need to prove that it's a new prime. You just need to prove that the technique of multiplying together a set of natural numbers (abcd) and adding 1 gives a number not divisible by any of the numbers.
The simple way is to imagine that the number abcd is congruent to 0 mod a, mon b...mod d and you're making it congruent to 1(thus not divisble by any a, b, c, d).

idk if this is the proper place for this kind of thing but it's math related and I'm a math major so fuck it

>shit happened in my past that led to depressive issues+a tendancy to disassociate from sources of anxiety+stress, i'm being vague
>feel a retarded need to prove myself to the world
>so far in school I've done well, straight A's though it's mostly been worthless geneds along with some less advanced math so I guess it doesn't matter
>took a test for a math class today, spent the past two weeks in a disassociative haze which led to me barely studying
>fucked up on a few questions, will either get a B or C (at worst I believe)
>spend entire day having a meltdown and being a wreck in bed

the point of this pathetic blogpost is to ask if there is anyone else here who has or had the combination of issues I listed (disassociative mechanism that can impact studying+stupid overachiever mentality/cutting achievements down) and if so how did you fix this? this isn't healthy for me, and the more abstract the math gets the more effort will be required, effort that it feels like I have to battle against my brain to perform.

[eqn]\mathbb{E}\left[ \left( X-\mathbb{E}[X] \right) (Y-\mathbb{E}[Y]) \right] =\mathbb{E}\left[ X\left( Y-\mathbb{E}[Y] \right) \right] [/eqn]
what the fuck is this. I'm tripping balls

>>16001825
you have to go back

>go through natural deduction
>man this shit is confusing
>excercises look like black magic
>go through chapter again this time more confident
>cant get a single question correctly anyway
any good recources for introduction to formal proof I feel like Im retarded

>>16002038
>anyone else here who has or had the combination of issues I listed
Yes.
>and if so how did you fix this?
I'll let you know when I do.

>...The familiar path from positive whole number arithmetic all the way to arithmetic on the real numbers (as used in calculus) that many teachers (for the most enticing of reasons!) find so attractive - which is actually not the same as the actual historical development, but never mind for now - goes completely counter to the way arithmetic systems should be developed to be able to meet today's societal needs. Arithmetic (with exponentiation) on the real numbers system is the most fundamental. All other arithmetics are special cases of that. (Full disclosure: the complex number system is the one you need to start with to really do everything you need in the world. But in this column I'm focusing only on the number systems that typically arise in K-8 or perhaps K-12 mathematics. Complex numbers have no good intuitive conception other than a geometric one that is only partially effective, and so have to be developed axiomatically. But they come so late in the educational process, and the step is restricted to students who have already mastered a lot of other mathematics, that there really isn't any need for a "conceptual prop", nor is there any danger of something having to be undone later.)

>One consequence of the 180 degree turnaround is that addition on the positive whole numbers is a special case of addition on the real numbers, and multiplication on the positive whole numbers is a special case of multiplication on the real numbers. And make no mistake about it, real number multiplication most definitely is not repeated real addition.

>The point is, the needs of society made it necessary to produce a single number system that works for all possible purposes. The real number system (strictly, the complex number system) is that system. All other number systems are subsystems. Since the real number system is the one that connects to the real world in the most significant way, it is the one that must be taken as the default.

Mathsisters, our response?

>>16002321
what book are you using?

>>16002048
Literally just use linearity of expectation and cancel terms.

>>16002364
>addition on the positive whole numbers is a special case of addition on the real numbers, and multiplication on the positive whole numbers is a special case of multiplication on the real numbers
what is this supposed to mean
I'm too intoxicated to understand. what about these is special compared to the usual? or is it just the fact that you don't need to worry about signs? because that's pretty shitty reasoning if that's all it takes for it being a "special case"
ohohoho addition of even numbers is a special case of addition because you don't need to worry about odds! same thing for multiplication! we should only be teaching even numbers!

>>16002048
Helps if you substitute A = E[X] and B = E[Y] and then do what >>16002368 says

>>16002370
That's the opposite of what he's saying; his reasoning would be that we shouldn't base everything on adding even numbers, because adding even numbers is a special case of adding reals

why does Ae being R-faithful imply that Ae is also A-faithful?
context not in pic: e is a torsion free idempotent in A giving us the decomposition

>the majority of the people in my math and general stem classes come from relatively wealthy backgrounds and have, without exaggeration, been groomed from infancy to do well in these and other subjects
>private schools, private tutors, specialized counseling, assistance in everything from finding housing to gaming financial aid
Meanwhile, I am here, from a background of nothing. Playing what feels like an eternal game of catch-up that is rigged against me.

>>16002364
>teach real numbers before integers
A silly Lockharts-Lament polemic. Schoolkids already learn fractions and decimals early on, including arbitrarily long long division and repeating decimals, and about pi in geometry, and the difference between counting vs measurement is even baked into basic English grammar. Wtf does he want, Cauchy sequences or some shit?

>>16002468
well from the sounds of it, you got into a selective school. Yeah its annoying to compete against last year’s IMO team or whatever in your calc 1 class but seriously, the fact that youre there with them means you are doing well, and importantly, your program will be a good program and teach you a lot

pic related, its not us

>>16002468
Only children and the naive think that life is fair.

>>16002579
Stop thinking before you post, retard. You post nonsense otherwise. Your brain doesn't work.

>>16002580
esl?

>>16002586

>>16002468
Good job, and don't worry about it. Dont have such a big chip on your shoulder either. They're people just like you, just be friends and don't hold it against them. There are many people worse off than you ad well. Life is too short to have the kind of mindset you're sharing, especially for something objectively inconsequential like mathematics.

.Suppose [math]\mathcal{X}[/math] is a finite set along side the measure [math]\mu[/math] with [math]\mu(\{ x \})>0[/math] for all [math]x \in\mathcal{X}[/math].

Then it should be the case that
[eqn]
\int_{\mathcal{X}}^{} f(x) \, d\mu(x) = \sum_{x \in \mathcal{X}}^{}f(x) \mu(\{ x \}).
[/eqn]

Is this correct? Apparently is not correct in my solution to a problem and I need to do a sanity check on each step. Just a Yes/No from a measure theory bro would help a lot.

>>16002717
Geytard

>>16002468
sounds like you have a great network to rely on in the future at least
just curious how far nothing are we talking about? druggy trailer park trash or what

>>16002732
looks ok. more context would help us troubleshoot, but my first guess is that maybe the topology on X is not the expected one (“the discrete topology”), or maybe mu is sub-additive instead of additive.

>>16002580
don't change your name faggot, you're ruining my filter

>>16002926
Why would you filter the smartest here? Goonpopm'gee

>>16002038
Work out till your exhausted, every other day. Your body hasn’t been producing enough serotonin, and, your diet is likely nutritionally stagnant and poor.

Working out pretty often would fix both those issues.

Also, write down your emotions throughout the day. Alot of your emotions, and therefore your very thoughts, are cyclic, and manifest at set times in the cyrcadian rhythm regardless of context.

Alternatively, drinking alcohol should stop the visions, or at least make them quite ignorable.

I’ve got a bunch of voices in my head, and I dissociate a couple times a day. Alcohol just turns all of that right off, and so too does post workout fatigue. Again, I suspect it’s caused by lack of serotonin, lack of proteins, and lack of vitamins. I know alot of my more intense hallucinations suddenly stopped once I began eating alot more proteins every day. And eating vitamins improves my focus significantly.

Hey there.

I'm sorry for the very basic question, but can you guys please explain what the difference between "associativity" and "commutativity" is supposed to be.

I cannot see a difference. Do putting brackets around something not just change the order of evaluation like you're re-arranging them?

>google it
I've tried.

It's the same answer every time, and the answer is the most literal description possible.

It comes from highschool math teachers and I don't think they have any idea what this means or the implications of it are. It's clearly something big.

>>16004201
>Do putting brackets around something not just change the order of evaluation like you're re-arranging them?
They do not. Associativity is changing the order you EVALUATE them in, while commutativity is about changing the order you DO them in outright.

For example, a+(b+c)=(a+b)+c by associativity. Whichever pair you evaluate first, you're still ultimately adding a to b to c.
Commutativity would amount to adding c to b to a. This may be a little hard to grasp, because addition is commutative as well so it's a moot point. Let's demonstrate the difference with a real-life example of noncommutativity.

Grab a book or something. Literally anything where each side is distinct.
We're going to be rotating it. I'll refer to a 90-degree rotation about the vertical axis as Z, a 90-degree rotation about the left-right axis as X, and a 90-degree rotation about the towards-away axis as Y.
Apply X to the book, and then apply Y. Remember how it's oriented. Then return to how you had it originally, and then apply Y and then X. You're not going to get the same position, because we've changed the actual order of the operations: it's not commutative.

But if you were to apply Y and then Z, and call the net rotation A... then if you applied X and then A (i.e. [math]X \cdot ( Y \cdot Z )[/math] ), you would get the same thing as if you applied X and then Y, and then Z ([math](X \cdot Y) \cdot Z[/math]), because while 3D rotation isn't commutative, it is still associative. Again, you aren't actually changing the order you ultimately carry them out in, just the order you're going about it in.

>Do putting brackets around something not just change the order of evaluation like you're re-arranging them?
They do not. Associativity is changing the order you EVALUATE them in, while commutativity is about changing the order you DO them in outright.

For example, a+(b+c)=(a+b)+c by associativity. Whichever pair you evaluate first, you're still ultimately adding a to b to c.
Commutativity would amount to adding c to b to a. This may be a little hard to grasp, because addition is commutative as well so it's a moot point. Let's demonstrate the difference with a real-life example of noncommutativity.

Grab a book or something. Literally anything where each side is distinct.
We're going to be rotating it. I'll refer to a 90-degree rotation about the vertical axis as Z, a 90-degree rotation about the left-right axis as X, and a 90-degree rotation about the towards-away axis as Y.
Apply X to the book, and then apply Y. Remember how it's oriented. Then return to how you had it originally, and then apply Y and then X. You're not going to get the same position, because we've changed the actual order of the operations: it's not commutative.

But if you were to apply Y and then X, and call the net rotation A... then if you applied Z and then A (i.e. [math](X \cdot Y ) \cdot Z[/math]), you would get the same thing as if you applied Z and then Y, and then X ([math]X \cdot ( Y \cdot Z)[/math])
), because while 3D rotation isn't commutative, it is still associative. Again, you aren't actually changing the order you ultimately carry them out in, just the order you're going about it in.

Sorry about the double reply, I botched the standard I was going with for order halfway through.

>>16004250
Why am I like this?>>16004201
At least you don't have to worry about feeling like the dumbest guy in the thread

I don't want to go to graduate school...
I'm just too tired of this shit, no wonder they give you guys 300k starting
how do I get a job with a math BS?

>>16004201
Here are some examples:
>addition is commutative
2 + 3 = 3 + 2—both are equal to 17
>addition associative
(1 + 2) + 3 = 1 + (2 + 3) —both are equal to 6
> division is NOT commutative
2 ÷ 3 ≠ 3 ÷ 2 —these equal 0.66… and 1.5 respectively
> division is NOT associative
1 ÷ (2 ÷ 3) ≠ (1 ÷ 2) ÷ 3 —these equal 1.5 and 0.166… respectively

>>16004369
>2 + 3 = 3 + 2—both are equal to 17
I mean 5… oh my god lol

>>16004201
commutative - can swap the order i do things without swapping pairings
associative - can pair things in any way without swapping order

associativity allows you to write 1+3+5 and have it be unambigious, instead of having to write either (1+3)+5 or 1+(3+5)

commutative allows you to swap the order in which you do things (1+3)+5 = 5+(1+3)
without touching the ways elements are paired

EXAMPLE:
consider the operation # on the real numbers defined by x#y = xy + 2
clearly # IS COMMUTATIVE: x#y = xy + 2 = yx + 2 = y#x for all real numbers x and y
NOT ASSOCIATIVE: (1#1)#2 = (1+2)#2 = 3#2 = 6+2 = 8
but 1#(1#2) = 1#(2+2) = 1#4 = 4 + 2 = 6

for an example of something that is associative but not commutative, go read about matrix multiplication

>>16004369
This has got to be one of the most retarded posts in all of /sci/, and not even because you miscalculated 2 + 3.

>>16004372
We've all had brain farts before with basic math but I have to ask anon, what kind of mental gymnastics were going on in your head to get 17?! No matter the twisted logic I can come up with I can't even get close.

>>16004454
I would have to assume that he did (2+3)*3+2 like a reprobate because I have no clue what else it would have been

Lim x->0 1/x =girlfriend

How do I find all solutions for 1/(x^a) = (x^a)/x? The obvious one is a=0.5, x=2 as well as all x=1, but where do I go from here?

1 = x^(2a-1)
ln 1 = (2a-1) ln x
Let y = 2a -1
Ln 1 = y ln x
Y = ln 1 / ln x

Wait ln 1 = 0 so 0 = y ln x

>>16005176
>>16005174
kek you retard

Thanks lads.
>>16004369
>>16004446

>>16004250
>it's like rotating an object
Fascinating.

Is that kind of like algebraic topology?

These are questions from someone that doesn't know anything about math but sometimes gets spontaneously interested and looks into some random topic for hours at a time.

>>16005201
Same here, more of a lawyer, but got into math because I enjoyed calc 2

Rolling to go to the library

Try 2

Do chain map homotopies "work" in the same way as regular homotopies? For example, if i want to prove that f composed with k is homotopic to f' composed with k' (f being being homotopic with f' and same for k), can i prove it in the same way as proving composition preserves homotopies like in topology?

>>15995883
Despite being so commonly used early on, I can not think of a single proof of the Liebnitz rule that doesn't involve measure theory or other relatively advanced math.

>>16005201
>Is that kind of like algebraic topology?
In some capacity? You run into a lot of noncommutative operations in abstract algebra in general. They're so common in group theory especially that there's an entire term (Abelian) just to describe groups with ones that ARE commutative

>>15995883
Any Project Euler chads in here?

If not you should try. I'd say it helps avoid the one-sided mental development that comes from only doing proofs.

Are there infinitely many positive integers n such that 3^n is 1 more than a power of 2 ?

Also, what is the smallest integer n≥3 such that 3^n is 1 more than a power of 2 ?

How do you find the domain of the derivative?

>>16005449
No. If n>8 then either n or n+1 has a prime factor >3 . See https://oeis.org/A002072

>>15995883
Unpopular opinion. Set theory is being worked on the wrong represented states of abstraction.

>>16005479
You take the derivative and find the interval of possible values for x that don't cause a division by 0 or create an imaginary number, which you can see in your photo

Am taking real analysis and was wondering:

is it possible to construct a bounded sequence w/ an infinite number of subsequences which converge to different points? This is trivial if it's unbounded because you can just do

(1, 2, 1, 2, 3, 1, 2, 3, 4, .....)
But I can't think of anything similar if it's bounded

>>16005863
nvm I just figured it out, am retarded

>>16004454
>>16004459
it’s simple, I started with 8+9 or something and went back and changed it
>>16005449
hint: try writing out the first few 1,3,9,27,… mod 16 or mod 32 and see where it starts looping.
>>16005864
it happens

I am getting filtered real hard by multivariable calculus. Even something as simple as the chain rule explodes into a large equation.
The hard things from single variable calculus become near impossible. The epsilon delta definition of continuity and limits for multivariable functions (especially vector valued one) is pretty much incomprehensible

>>16002038
In your experience how much of an exam you need to know to get a passing grade? Other anons in Uni (more specifically European Unis), how much % of the exam was a C grade in your first semester?

>>16006098
You don’t really need to know epsilon delta definitions to do lower-division multivar calc

>>16006479
I don't know what's the standard curriculum for multivariable calculus around the world, but it's a part of the curriculum here and some earlier exams did ask for epsilon delta proofs for the last question (last question is always the hardest one and required for an A).

What are some examples of groups in nature?

>>16006946
The prototypical example of a group being physically real is SO(3), which describes rotations of an object in 3D space

>>16006946
S^1, the (unit) circle
It describes rotation and their actions on itself in one plane.

>>16006946
besides stuff like the integers [math]\mathbb{Z},\ \mathbb{Z}[x][/math] the polynomial ring/group and stuff like that, the canonical example is group of symmetries of some geometrical object. For example a cube has 24 ways you can rotate it (48 if you allow mirror images) and these rotations when done in order behave like a noncommutative group operation

>>16006098
>The epsilon delta definition of continuity and limits for multivariable functions (especially vector valued one) is pretty much incomprehensible
The definition is pretty much exactly the same for both of those and for regular unary functions

I'm studying multiple view geometry for a week and so far so good, but there is a one example that I don't fully understand (I wasted half a day trying to figure). Pic related.
The example states:
> "a', b' and c' have coordinates 0, a' and a' + b' which may also be expressed as homogoneous vectors."
Obviously it could not be just (a', 1), (a'+b',1), right? First it would no more be a homology and second it won't have any sense.

Or should I just take random line other then 1 that not necessary intersects other line at a point (0,1) and only make sure that proportions (0, a', a'+b') are correct?

why do some mathfags look down on statistics? a (non meme) undergrad in stats has more (required to graduate) measure theory than a pure math undergrad and more (required to graduate) optimization and PDEs than an applied math undergrad

>>16008342
because its still applied math and not pure platonic shit

I'm trying to get my head around a problem involving compact sets and norms.
So far I'm supposed to show that for a given norm [math]N[/math] of [math]\mathbb{R}^{n}[/math] there's a positive number [math]C[/math] such that [math]N(x)\leq C||x||[/math] for all [math]x[/math] and that [math]N[/math] is continuous. That's all fine, no problem. Then I'm given the set [math]\mathbb{S}_{N}:=\{x\in\mathbb{R}^{n}:N(x)=1\}[/math], which is clearly the unit sphere defined by the norm. By continuity of [math]N[/math], the set is closed. However I can't show, given the previous information that it is bounded, obvious as it seems when I look at it. I can see that if I have an open ball centered at 0 and with a radius of [math]2/C[/math] then [math]N(x)\leq 2[/math], but I can't find a way to show that's the case for all [math]x\in \mathbb{S}_{N}[/math]. However, the next problem is to show that there exists some positive [math]C'[/math] such that [math]C'||x||\leq N(x)[/math] then problem solved trivially. If [math]x\in \mathbb{S}_{N}[/math] then [math]C'||x||\leq N(x)=1[/math] so [math]x[/math] is in the closed ball of radius [math]1/C'[/math]. Is there a way without the last part is did my professor mess up the order?

I want to work through Zorich’s Analysis . I have taken l calc 1-3 and understand basic proofs. However, I also work and have full time classes. Would it be most efficient to just skim while reading just to get an idea, then focus on doing the problems/using the theorems, or would it be better to meticulously go through each theorem and try to prove it myself regardless of how long it takes? I’m leaning toward the former.

Thanks in advance for any input.

>>16008390
Yeah, probably do the former first, you can go back and do the second later.

>>16007292
a is the origin, so a on the coordinate to each ordered pair functions as a partial coordinate shift relative to y which is defined as the same point at all coordinates then x functions only to shift the length of the projection in each ordered pair.

anyways, just made this shit up...can some nonretard pls help

>>16008356
yeah it sounds like the questions are probably just out of order. the other possibility is that there is a general theorem about continious functions that says inverse image of a compact is all ways compact, but this seems more like a question where your supposte to use the definitions directly. I think its just out of older

>>15996896
Hope I get the latex right.

for the last part you don't even need induction I think
from [eqn] p_n < 2^{2^n} [\eqn]
let's use something different from n, like i going from 1 to n, to index the primes in the [eqn] p_1 * p_2 * ... * p_n [\eqn]
[eqn]
p_i < 2^{2^i}
[/eqn]
so for example
[math]
p_1 < 2^{2^1} < 2^{2^2}} \\
p_2 < 2^{2^2} < 2^{2^4}}
[\math]
and so on

so if you multiply these primes p_i and their assigned powers of 2 that are larger by the above inequality you get:
[math]
(p_1 * p_2 * p_3 * ... * p_n) < (2^{2^1} * 2^{2^2} * 2^{2^3} ... * 2^{2^n}) = ( 2^2 * 2^4 * 2^8 * ... * 2^{2^n} ) = 2^{2 + 4 + 8 + ... + 2^n} = 2^{2^{n+1} - 2} < 2^{2^{n+1}}
[\math]
now that's without the + 1 part in the p_1 * ... p_n product
so let's say you have these [eqn] (p_1 * p_2 * ... * p_n) [\eqn] and for each i: [eqn] p_i < 2^{2^i} [\eqn] then since [eqn] (p_2 * ... * p_n) > 2 [\eqn] so if you put instead of the [eqn] p_1 [\eqn] a larger number since it's a product the whole thing will be bigger than if you just add one, so what follows is that
[eqn] (p_1 * p_2 * ... * p_n) < (p_1 * p_2 * ... * p_n) + 1 < (2^{2^1} * p_2 * ... * p_n) < ( 2^{2^1} * 2^{2^2} * ... * 2^{2^n}) = 2^{2 + 4 + ... + 2^n} = 2^{2^{n+1} - 2} < 2^{2^{n+1}}
[\math]
the last part is the same as in the one without the + 1 but I put it there for completeness
in there is both the middle inequality and then the right one also

>>16008563
for fucks sake lol

>>15996896


[eqn]p_n <2^{2^n}[/eqn]
let′s use something different from n, like i going from 1 to n, to index the primes in the product [eqn]p1∗p2∗...∗pn[/eqn]
so
[eqn] p_i< 2^{2^i} [/eqn]

so for example
[math]
p_1 < 2^{2^1} < 2^{2^2}} \\
p_2 < 2^{2^2} < 2^{2^4}}
[/math]
and so on

so if you multiply these primes p_i and their assigned powers of 2 that are larger by the above inequality you get:
[math]
(p_1 * p_2 * p_3 * ... * p_n) < (2^{2^1} * 2^{2^2} * 2^{2^3} ... * 2^{2^n}) = ( 2^2 * 2^4 * 2^8 * ... * 2^{2^n} ) = 2^{2 + 4 + 8 + ... + 2^n} = 2^{2^{n+1} - 2} < 2^{2^{n+1}}
[/math]
now that's without the + 1 part in the p_1 * ... p_n product
so let's say you have these [eqn] (p_1 * p_2 * ... * p_n) [/eqn] and for each i: [eqn] p_i < 2^{2^i} [/eqn] then since [eqn] (p_2 * ... * p_n) > 2 [/eqn] so if you put instead of the [eqn] p_1 [/eqn] a larger number since it's a product the whole thing will be bigger than if you just add one, so what follows is that
[eqn] (p_1 * p_2 * ... * p_n) < (p_1 * p_2 * ... * p_n) + 1 < (2^{2^1} * p_2 * ... * p_n) < ( 2^{2^1} * 2^{2^2} * ... * 2^{2^n}) = 2^{2 + 4 + ... + 2^n} = 2^{2^{n+1} - 2} < 2^{2^{n+1}}
[/eqn]
the last part is the same as in the one without the + 1 but I put it there for completeness
in there is both the middle inequality and then the right one also

>>16008568
oh nvm I see someone else already answered I guess the same thing

so I guess I can at least ask if someone could please tell me how to make the latex less spacey when posting here?

>>16008568
1) You can delete posts
2) You can preview the Latex.

>>16008576
>make latex less spacey
you’re hitting enter a lot even outside of your math tags. some browsers seem to render math tags with their own extra newlines too, if that what you mean then theres no fix, you would have to petition the Chink to put down the ah pian and fix it

>>16006222
at my uni here in Czechia the passing grade is E and the scales usually go like
50 - 59 % E
60 - 69 % D
and so on by increments of 10 upwards

>>16008596
The [eqn] tag seems to add extra spaces, the [math] tag doesn't.

I've got a fun combinatorics problem here.
[math]X=\{1,2, \cdots , n\}.\\
Find\ |\{f \in X^X | \forall x \in X\ \exists k_{f,x} \geq 0 : f^{k_{f,x}}(x) \leq m \}|[/math]
X^X is just the functions from X to X. f^k means k iterations of f.

>book uses real life examples like from physics
>don't understand a single thing
>book uses the most pedantic abstract language possible
>understand it perfectly
is it autism?

>>16009174
>understand it perfectly
"understand"
I guarantee you can't solve a single problem. Your "understanding" is just your delusion.

>>16008342
Have you read a statistics book? I don't think statisticians understand the topic that well either. The books basically contain a collection of techniques and methods that seem to work. It's like physics but even worse because at least physics use experimental data, statistics use models out of your ass.

>>16005513
>>Are there infinitely many positive integers n such that 3^n is 1 more than a power of 2 ?
>No. If n>8 then either n or n+1 has a prime factor >3 .
Maybe I'm being dumb, but why does this mean there aren't infinitely many such n ? Or am I misunderstanding

>>16009174
It means you don't have physical intuition and so you can't be an engineer

>>16009455
nta but if you got your hypothetical [math] 3^n + 1 = 2^m [/math] then by the other claim either [math] 3^n [/math] or [math] 3^n + 1 = 2^m [/math] has a prime factor larger than 3, but the former only has 3 as a prime factor and the latter only has 2, which is a contradiction with the claim that for every number above 8 either it or it plus one has a prime factor larger than 3

>>16009484
oh oops I got it backwards, you got your
[math] 2^m + 1 = 3^n [/math] then either [math] 2^m [/math] or [math] 2^m + 1 = 3^n [/math] has a prime factor larger than 3, but the former only has 2 as a prime factor and the latter only has 3 as a prime factor so that's a contradiction with the original claim

What is the fundamental theorem of calculus? It in the fact that the rate of change of the area under the curve f(x) is equal to the height of the curve. Why is this true? The area increases by dx * f(x) after moving dx to the right: so dy/dx = f(x) dx /dx = f(x).

>>16009548
It's just the telescoping series trick.

>>16008599
Thank you for answering! I hope you have a nice sunday! This puts my heart at rest

>>16009954
no problem and thanks, a nice sunday to you too! may I ask why you wanted to know?

>>16009053
thanks for the question, it is fun. a “when u see it” style question.. I think I have an answer.
>>16009548
yes that is the way to look at it at least in one direction.
>derivative of the integral is f
there is also the other direction
>integral of the derivative
but not all textbooks put both directions.. it also is not a formal proof but you can distill a formal proof out of it if you are careful

red pill me on knot theory. any decent books to start with?

>>16010701
>I think I have an answer
Unless you have some clever argument, I doubt it.
It took me a while to do the gymnastics to get to the answer but I may have missed some clever shortcut. The answer is pretty simple, though.

>>16010953
give the number then faggot

>>16010001
Well, I just did my first math exam and was anxious. There were 6 questions and I answered 4 very well, but I had trouble answering 2.

>>15995883
This is gonna sound silly but I'm not a math guy and I just need a quick counsel.
On a scale of 1-10, 1 being basic math and 10 being the highest level of math what is the rating of picrel

>>16011109
>defines
nuclear physics is not my specialty, but on its face, this looks like kook shit.

>>16011115
So it's just nuclear physics equations?
I assume kook means crazy people stuff. Can I just ask about this last one? If it's the same thing then that's fine. Thanks.

>>16011103
Find it ;^)

Can this be separated so I can integrate both sides?

[math]\frac{dy}{\sqrt{ds^2+dy^2}}=s[/math]

>>16011179
I'll add, for the case m=1 you'll need a named result.
I'm hardcore so I proved this result my own way.
To do the m>1 case you can use the named result with some extra steps or just use a generalization of the named result directly.

Not getting the same answer for the second derivative in example 5

For any two points x and y in [math]\mathbb{R}^{n} [/math] we can describe a line between them with [math]\lambda x+(1-\lambda)y [/math]. Does the equation [math]\lambda x+(1+\lambda)y [/math] extend this line? I'm trying to use this to show that some point is on the boundary of a set with x in the set and y on the boundary, but I can't figure out a way to find some point in any open ball around x that is in the exterior. Are there any tricks for finding such points?

>>16011565
Then you have made a mistake in your calculation. Both derivatives are correct.

>>16011580
>>16011580
You might notice the two occasions I went against that as marks. Once where I bit the berry, and a second time that was more like a love bite unknowingly. To check what it was. But no more. They're not annoying.

>>16011571
In a vector space, y + (x-y) gives me x. x-y is the vector we add to y to get x. Now if I take any multiple of the vector x-y, say λ(x-y), it goes parallel to x-y. Adding this to y, I get y+λ(x-y) = y(1-λ) +λx. Not the same thing as (1+λ)y+λx, which would imply that I added a multiple of y+x. Letting λ = 1, you can see that we would just get 2y+x, which is a completely different vector.

I’m a little confused by what you mean on the next part.

>>16011580
Sorry I got it now, maybe that might have been more of stupid question

>>16011596
Cheers, it was just some thought i hadn't checked properly from fatigue. It's some introductory topology stuff. I just need some methods for showing that a set i cooked up is the boundary of another set. The idea was to use a line based on two points, one in the boundary, the other in the set. I'll post it tomorrow if I don't get anywhere.

What is "infinity"?

I don't think I have any problem actually -learning- maths, but I'm not sure how people -retain- math fluency when you're not actively using it/learning new stuff. How do you guys do it?

For example, I speak two languages, but I can retain fluency by consistently reading/writing/speaking them. This is fairly doable, but if I try to do something similar with maths, am I meant to do Algebra/Trig/Geometry/Calc/etc... problems daily...? weekly? That sounds like a terribly inefficient thing to do. I'm not in school/academia, and I work a full time job. At least when I use a language, the main purpose isn't to just use the language, but to communicate using the language i.e. it's ancillary.

I mean, it's not that I don't use any maths. I could use some to design a more efficient algorithm, find the probability of something, look into some financial concepts/products, etc... every so often as they come up at work or some personal project. But there's no way in hell that I'm consistently having to solve such a wide breadth of math problems on a regular basis.

One friend told me that he doesn't retain jack shit and just re-studies a subject if he needs to use it, but he's in academia. I don't really have much of a reason to "brush up" on any particular maths field for any practical purpose in my daily life. Do I just resign myself to "eh, I could relearn it if I want to but don't need to lmao"?

>>16012002
>How do you guys do it?
The same way I retain how to ride a bike.

So what book should I read if I stopped paying attention after algebra 1 and I've forgotten pretty much everything?

>>16012002
Doing drills in every subject you've ever learned is overkill, you can indeed just re-study it as needed. But solving some interesting problems/puzzles regularly will help keep your mathematical reasoning sharp.

>>16012002
You generally forget the most high level things first. A guy who only learned HS-level algebra might just retain arithmetic. A guy who learned point set topology and pdes might retain calculus, etc.
My advice would be to try to "learn high" in whatever field is particularly interesting/useful to you, then you'll still retain the lower level stuff

>>16012040
Unfortunately I'm not a gigabrained genius with photographic memory.

>>16012058
I mean I guess it just boils down to this if I want to do everything.

>>16012085
This makes sense. Yeah, I could probably manage studying some subjects out of interest and coincidentally maintain some of the easier maths as well. My biggest problem is basically losing mathematical intuition after not using it for a long time, so this might be the most feasible option for me, thanks anon.

>>16012109
Do you need photographic memory to ride a bike?
Maybe you are using the wrong part of your brain to learn math.

>>16011240
Here's how I did the m=1 case.
The combinatorics to get the recurrences are fairly easy. You just need to be careful.
The generating functions and complex analysis are how I got Cayley's formula.

>>16012190
I don't even understand what you're implying.

Yeah, I can "ride a bike" (arithmetic) after years of not doing so. But if I want to do it at a relatively high level like bmx or long distance cycling(calculus, linear algebra) then I'll need to train/relearn after taking a break.

Or are you telling me you can just take a 7 year break from cycling and not get btfo in a long range cycling race?

I'm talking about retaining, not retraining/relearning, which I already acknowledged in previous posts.

>>16011189
i would try rewriting s as cos theta, y as sin theta

>>16012238
>Or are you telling me you can just take a 7 year break from cycling and not get btfo in a long range cycling race?
Depends on how good you were. If you were a good cyclist, then yes, there will be things that are ingrained in you that you won’t forget. If you sucked then you’ll still suck.

>>16012247
Alright, big brained anon. You win the internets this time.

My dumbdumb math and physics postdoc friends need to relearn stuff every so often, and are obviously not as smart as you. I'm even less smartypants since I'm no PhD, so I'll just say I suck at maths and need to relearn after neglecting some mathematical concepts for a few years.

>>16012238
>take a 7 year break from cycling and not get btfo in a long range cycling race?
No. I literally mean like not needing to think about it. It is just natural. There is no trivia to forget because you know it like you know your middle name.

I can understand forgetting more elaborate stuff like the specific conditions in obscure theorems or complicated proofs but for most stuff, even if there is a gap, you can just derive the result from what you know by heart. Math is probably the easiest subject to remember because of that. It is like an error-correcting code where you can reconstruct the whole from a few parts.

Wwwwuuut

Is a left invariant vector field allowed to vanish somewhere? Intuitively, I'd say no because the zero would be pushed around by the group action and make the vector field a constant zero. But I'm not entirely sure.

>>16012556
it has to be if you identify the tangent space at the identity with the left invariant vector fields as you do?

>>16012053
What's your goal? Probably just go through khan academy and its exercise.

>>16012528
>page 128 of 136
>no context
What is the trouble with this page, friend?

>>16012747
It’s from a calc review for a random math book I found. I cannot even follow what the first integral represents

>>16012002
I've heard spaced repetition (e.g. anki) is good

>>16012769
It's already that

>>16012556
You're right, if a left-invariant vector field (on some Lie group) vanishes somewhere then it vanishes everywhere.

You should really prove this yourself as an exercise, to get more familiar with how Lie groups work.

>>16012767
Do you mean Green’s Theorem? It’s a main theorem of multi variable calculus, if your course covered it you would remember it. But in short what it is saying is that if the 2d function f(x,y) is smooth enough, then integrating f over the interior of a shape is the same as integrating f over its boundary (the idea is that each interior point gets cancelled by the nearby points… only the boundary points don’t get cancelled on all sides).

does jerking off help you do math

>>16012817
yes

>>16012822
I do my best to not love Wittgenstein but in his basedness he always comes roaring back

>>16012795
Is that covered in Calc 3(multivariable)?

>>16012795
That doesn't sound right. Consider the function f(x)=1. The integral of it on a square of side a is a^2, but the integral on the boundary is 4a.

I've read multiple proofs of the Recursion Theorem https://www.proofwiki.org/wiki/Principle_of_Recursive_Definition and it just doesn't "click". Should i go back to read Deadekind himself?

>>16013537
Is it the proofs that don't click or the statement itself?

>>16013292
excuse me your right. it is supposed to be the integral of the curl (a kind of vector rate-of-change) of the function. im dumb. the way I think of Greens theorem (and Stokes theorem which is the same thing in 3D instead of 2d) is like flows in the water. the same amoubt of water flows into my round fishing net as flows out of it… what happens inside the fishing net does not matter very much, it can go straight or be turbulent, it can do a somersault or whatever, doesnt make a difference to the amount of water passing through my round net as long as the boundary conditions are the same
>>16013537
maybe say more about your goals? reading the Masters is always great and absolutely do it if you want. I think the page you linked is needlessly abstract for your purposes. this “recursion theorem” is a constructive companion to mathematical induction—recursion is for defining things, induction is for proving thigns. if I told you f(0)=1 and f(n+1)= nf(n) you would recognize that as the factorial function. the theorem is just saying formally that this definition is good enough to define a function from the naturals to the naturals, and only one function i.e. it is not ambiguous

Let G be abelian and g be an element of G, is G/(G/<g>) isomorphic to <g>?

>>16014082
Yes. Map each g^k in <g> to its coset g^k.G/<g>. Show this is well defined and a morphism. It's trivially surjective and if g^k is in G/<g> then k=0, so it's injective as well.
You use the abelianity of G to see that every subgroup is normal, so the set of cosets of G/<g> is a group and the group operation agrees with the quotient map.

I have a question. This was in a programming contest, and I don't understand the logic.
The question gives a number [math]x[/math] and you have to find out if it can be represented as [math]11a+111b[/math]
The solution is to find [math]z[/math] which is the remainder after dividing [math]x[/math] by [math]11[/math]. Then, the number be represented as required if and only if [math]111z \leq x[/math]
I do not understand why that is the case. Could you please explain?

>>16014101
something is weird about this statement. for example it seems z < 11 always. Ans then of course 111z < 1221. But there are x bigger than 1221 that can be written in the required form. So somethingis missing but I cannot tell what.

>>16014108
The problem is originally meant to be dynamic programming on a and b. But the tutorials and solutions do not use dynamic programming and instead use this.
But I am not convinced by this at all.
This code is accepted:

int x;
cin >> x;
int z = x % 11;
cout << ((111 * z <= x) ? "YES\n" : "NO\n");

I though of x = 11a+111b+c, and we have to check if c is 0.
The remainder z would be b+c, and 111*z would be 111b+111c, and it being lesser than x, ie
111b+111c<=11a+111b+c
110c<=11a
seems to have no indication on whether c is zero or not.

>>16012822
masturbating and masturbating to porn are different things
I really wonder what people like nietzsche would've said about porn

>>16014101
11 and 111 are coprime since 1=111-11*10. You can represent any number x with a = -10x and 111 = x.

>>16000986
Probability theory exposits measure theory in a different way. A discrete set with a probability mass function satisfies the axioms of a measure. The measure of a set of points is the probability of those events, etc.

>>16002732
It's not correct, the RHS side will be zero unless the measure mu is a Dirac measure i.e. a sum of dirac deltas for each x in X
integra f(x) sum_i p_i delta(x - x_i) dx = sum_i f(x_i) p_i
It helps to think about your integral in very simple terms i.e. take a smooth f(x) and think of the Reimann sum and how you can take it arbitrarily close to zero

>>16005148
The solutions are a = 1/2 and x = anything, or x = 1 and a = anything. It's equivalent to
-a Log(x) = (a-1) log(x)
from which you can see these are the only two solutions.

>>16009316
>The books basically contain a collection of techniques and methods that seem to work.
Neither physics or statistics are merely based on recipes that "seem to work", they are just sometimes presented that way in introductory books whose readers may not be able to understand proofs

>>16011109
To understand this you would first take 1 year undergraduate course on quantum mechanics, then take a 1 semester course at the introductory graduate / advanced undergraduate level where the general theory of QM is applied to the specific systems here
My guess is that this text is dealing with quantum chemistry. Chemists like to use dense, ugly notation because they like to labor.
In terms of the mathematical sophistication it's just linear algebra / functional analysis
From 1-10 I would give it a 4. My scale is based on "number of years since first learning calculus".

>>16014299
what is 10

>>16014610
Inter-universal Teichmüller theory

I can see how 2 implies 1, but i'm having trouble proving 1 -> 2. I have gotten to "A'' is infinite cyclic so A/A' must also be infinite cyclic", but i dont really know how to go from here

>>16014760
A'' = Z since every infinite cyclic group is isomorphic to Z
Since epsilon is surjective, A = A'''xZ since it needs to be infinite and abelian. Epsilon just ignores the A''' part.
The kernel of epsilon is (A''',0)
The image of mu is (A''',0)
A' = A'''
The element (e,1) of A'''xZ = A is the element you need.

>>16014857
There might be a more rigorous way to state things but the proof should basically capture this idea.

>>16014760
Also, A doesn't necessarily need to be infinite cyclic. It could have more than one factor of Z (or Cn) which makes it not cyclic (cyclic means it can be generated by 1 element).
At most you can say A is infinite abelian.

>>16014857
How does surjectivity of epsilon imply that A = ZxZ?

https://mathoverflow.net/questions/435110/consequences-of-kirti-joshis-new-preprint-about-p-adic-teichm%c3%bcller-theory-on-th/463562#463562

scholzebros?

>>16014082
G / (G / <g>) isn't defined since G / <g> isn't a subgroup of G

Whatever happened with Per Enflo's supposed construction for the invariant subspace problem? It seems to be mostly ignored, so I assume people don't agree. Anyone know anything?
https://arxiv.org/abs/2305.15442

>>16014082
>>16015866
To elaborate, if G is finite and H < G then G/H is isomorphic to some K < G, and there your question makes sense and is true. But for example, it doesn't make any sense to talk about Z / (Z / 2Z), since there is no subgroup of Z isomorphic to Z / 2Z

Does GPT-4 help with or can it even solve problems from your college level math homework?

>>15995883

- Is there a smooth function [math] f : \mathbb{R} \rightarrow \mathbb{R} [/math] such that the function [eqn] g(x) = \left\{\begin{matrix}
f(x)\cos(1/x), & x\neq0 \\
0, & x=0 \\
\end{matrix}\right. [/eqn]
is smooth at x=0 ?

- More generally, is there a smooth function [math] f : \mathbb{R} \rightarrow \mathbb{R} [/math] such that for any *bounded* smooth function [math] h(x) : \mathbb{R}\setminus\{0\} \rightarrow \mathbb{R} [/math], the function
[eqn] g(x) = \left\{\begin{matrix}
f(x)h(1/x), & x\neq0 \\
0, & x=0 \\
\end{matrix}\right. [/eqn]
is smooth at x=0 ?

>>16016068
Oops the h(1/x) in the 2nd part should be h(x)

https://www.youtube.com/watch?v=7vc-Uvp3vwg

Can someone just watch this and confirm to me that this makes little to no sense

I get used to watching little snippets of math and science like this. All of it blends together. When I actually stop and hone in on something like this I sometimes find that my gut feeling is correct and that this is nearly impossible to make sense of

>>16016075
(You)

I have no education but my background is in signal processing, so I know all the words and I get the general idea of what it's putting down

Here's what this 1 minute video establishes:

1. A wave function is a universal aspect of physics that permeates the universe
2. A wave function also shows up in quantum mechanics. Things pop in and out of existence, but they do it adhering to sinusoids. This is never explained specifically
3. There's a fundamental property that has to deal with everything sinusoidal. For this example we chose a sine wave, literally the simplest thing possible and the one thing that we can know for certain
4. A sinusoid is constantly accelerating, this is never explained
5. A sinusoid is constantly accelerating, but it's exponential half the time and inverse the other half. So it goes up and then goes down again. It's very unique
6. There is an entire branch of math called "calculus" that revolves around looking at curves and discrete points on curves, this is never brought up but maybe it should be
7. There exists things that look like waves that are not waves at all. The term is probably just "signal", but this is never brought up
7b. There exist things that look like waves that are waves but are expressed discretely. The term is "wavelet", but this is never brought up
8. The "position" of a wave could either be the amplitude or the phase, idk

>>16016075
(You)

If anyone can explain what the uncertainty principle is supposed to be I'd appreciate it

I have faith that it's not that *hard*, it just gets completely lost when you're staring at a blackboard with squiggly lines on it for 40 minutes before your brain decides that a more productive use of your time is thinking about what you want from McDonalds

>>16016068
f=0 :-)

>>16016094
Oh I'm dumb lol

>>16016096 Corrected version:

- Is there a smooth function [math] f : \mathbb{R} \rightarrow \mathbb{R} [/math] such that f(x) has no zeros except at x=0,
and the function [eqn] g(x) = \left\{\begin{matrix}
f(x)\cos(1/x), & x\neq0 \\
0, & x=0 \\
\end{matrix}\right. [/eqn]
is smooth at x=0 ?

- More generally, is there a smooth function [math] f : \mathbb{R} \rightarrow \mathbb{R} [/math] such that f(x) has no zeros except at x=0,
and such that for any *bounded* smooth function [math] h(x) : \mathbb{R}\setminus\{0\} \rightarrow \mathbb{R} [/math], the function
[eqn] g(x) = \left\{\begin{matrix}
f(x)h(x), & x\neq0 \\
0, & x=0 \\
\end{matrix}\right. [/eqn]
is smooth at x=0 ?

>>16016102
>Reminder: /sci/ is for discussing topics pertaining to science and mathematics, not for helping you with your homework or helping you figure out your career path.

>>16016105
Reported you for not discussing math and science

>>16016108
This isn't a fag board either.

>>>/lgbt/

>>16016116
Can you solve the problem or not faggot?

>>15995883
For someone who has completely forgotten algebra, but has graduated college and would like to relearn algebra and learn the higher maths , what would be a good college level textbook to get that covers all that? I'm thinking going back to get my Masters, not in math, but the GRE test is heavy in math and most graduate programs require that for admission.

>>15995883

If f(x) is a single-variable real function defined on an open interval containing 0,
and f is analytic at x=0,
and f(x) is an even function,
then does f have a local extremum at x=0?

>>16016102
For the first one, maybe try something like exp(-1/x^2); 0 at 0.

>>16015741
I used A = A ' ' ' x Z
A''' is just some abelian group.
You know there is a copy of Z there since A'' = Z is a homomorphic image.

Let R be a not necessarily commutative ring with unity. Is there an example of a simple R-module whose underlying abelian group is not simple?

>>16016330
I forgot to add I'm most interested in the case when both R and the module are finite.

>>16016330
Any non-prime finite field

>>16016330
>>16016361
Note a finite Abelian group is simple iff it is prime cyclic.

So, you’re just asking for rings with underlying Abelian group finite but not prime cyclic.

An example is the ring Z/nZ for n not prime

>>16016598
Oops disregard this, I’m dumb and didn’t read your question carefully.

>>16016598
No, those aren't simple modules -- any nontrivial proper ideal gives you a nontrivial proper submodule

>>16016604
Yea like I said in >>16016601
I didn’t read their question carefully

>>16016227
Thanks; by the way, would exp(-1/|x|) also work?

>>15995883
What's an example of a commutative ring R, and two nonzero torsion-free modules A,B over R,
such that Hom_R (A,B) = 0 ?

>>16016658
R = B = Z, A = Q

>>16016677
Oh of course, thank you.

Let me ask a new question then:

What's an example of a commutative ring R,
and two nonzero torsion-free finitely-generated modules A,B over R,
such that Hom_R (A,B) = 0 ?

>>16016684
context?

>>16016699
?

>>16016684
R=B=Z[x]/(x^2), A=Z (There is a unique R-module structure for Z). Alternatively, R=B=Z[x], A=Z (There are countably many R-module structures for Z, but all of them work)

>>15996181
They can be signed real or complex valued, in which case their total variation over the whole sigma algebra is finite. It's pretty neat how Rudin does this tradeoff in complex analysis by developing complex measure and has no use for the argument of a complex number.

>>16014268
It is correct though. His set X is finite so any function is a simple function, and the integral of a simple function is just his RHS.

Is there a way to randomly generate a set theory or abstract algebra problem? Or really randomly generate any area of math, problem

>>16017332
A cloud/over-complex cube.

>>16017333
>cloud/over-complex
What does that mean

>>16017334
Clouds are born of the expansion of the universe. They feed from the over complex cubic nature of space and time.

Can someone come up with an elementary problem similar to alien numbers but more interesting and math related?
https://open.kattis.com/problems/aliennumbers

>>16016102

>>16016227 's [math]f(x)=\exp(-1/x^2),~f(0)=0[/math] works because all of its derivatives' decays overcome the blowups of all derivatives of [math]\cos(1/x)[/math], which are no worse than rational functions.

For the second part to be a generalization, you probably mean fixing h and finding f. You also don't need h bounded. Let [math]s:\mathbb{R}\backslash\{0\}\to\mathbb{R}[/math] be any smooth function that vanishes only at 0. Put [math]f(x)=s(y(x))[/math], where
[eqn]
y(x) = \sum_{n=1}^\infty \frac{x^n}{1+h^{(n)}(x)^2}.
[/eqn]
Something like that.

>>16017518
Well, shit, it's too late to fix the typesetting

>>16017474
Does anyone have any other sites like this or projecteuler (or leetcode)

in compound tiling, if you had to tile a square of some unit square dimension with various shapes, one of which is a 1x1 unit square, is there a way to figure out the minimum number of 1x1 squares that would need to be used to fill the gaps

>>16017580
or i guess a better way to put it would be the minimum number of holes you could make in the square and still have it be tileable with the shapes

Guys pls help a compsi brainlet with unique polynomials

>>16017628

>>16017645
carmine >>>>>>> nessa

>>16017760
Yrp

I hate memorization. Can someone explain why

x=f(t)
y=g(t)

Area under curve = int ( g(t) f’(t) dt)

>>16017957
You mean the area enclosed by the curve, assuming it's a simple closed curve, oriented clockwise (for what you've written)

It's by Green's Theorem

>>16017957
Int( y(x) dx )
Int( y( f ( t ) ) dx / dt dt )
Int( g( t ) f’(t) dt )

>>16018036
Indeed.

Inside the first integral we have the derivative of the area function with respect to x; next we find the derivative of the area function with respect to t using the chain rule x = f(t).

How are holomorph functions in C and gradient fields in R^2 are related?

I am unable to see the significance of the mean value theorem and where it would be used.
it already requires differentiablility in an open interval and it doesn't say where the derivative is equal to the secant of the interval. it barely says anything about the function

Do cars accelerate more at the end high end of a gear when RPMs are the highest? It feels that way.

>>16018221
look at a power/torque vs rpm curve
torque is how hard the engine can turn (pulling weight)
horsepower (= torque * rpm / 5252) is how well the engine can accelerate
you can tow more at peak torque, and accelerate fastest at peak power

>>16018093
You use to calculate bounds of functions.
[eqn] \min_{t \in [a,b]} |f'(t)| |b-a| \leq |f(b) - f(a)| \leq \max_{t \in [a,b]} |f'(t)| |b-a| [/eqn]

Let's say you want to bound [math] \log(1 + x)[/math] for non-negative x then the MVT gives the bounds
[eqn] \frac{x}{1 + x} \leq \log(1 + x) \leq x [/eqn]

I made enough money to never need to work again. Should I study math?

I'm starting college (Mathematics) at the mid of this month. Someone have some tips? What should I know before starting? What should I do to retain more knowledge?

Suppose X is a subset of a group G. Is X a subgroup of G?

>>16019506
How much are we talking about?

>>16019559
Not unless X is a group itself with the same operation.

>"Okay, but what does that have to do with the conjectural framework of derived algebraic geometry and its implications on moduli problems?"

>"I am just coming to that. You said you wanted a serious conversation, not a talk show. You see, in the preliminary consideration of the Grothendieck-Riemann-Roch theorem, we must first establish the foundational principles of cohomology and its derived categories, which naturally leads us to the interplay between stack theory and the étale topology. In the context of three-point incidence geometry, this necessitates an extensive discourse on the axiomatic approach to multiplicative identity and affine connections within a non-Euclidean framework, which is intrinsic to the homological conjectures in algebraic K-theory."

>>16019570
/g/ here

Seems that set theory is heap-allocated because both the structure and elements themselves are dynamically sized.

>>16019845
isn't all of math heap allocated?

>>16019846
god I hate CS retards.
>Heap :(
>Partially ordered binary tree :)

>cookie :(
>Client-side state information token :)

>cloud computing :(
>distributed computing resource service :)

>thread :(
>Concurrent execution operation segment :)

The core of CS content is just dumbed down and informal crap for gaymers.

>>16019942
>Partially ordered binary tree :)
Sounds like forcing desu, making me think CS anon is actually onto something with his comparison

>>16019947
I wasn't addressing his point, just the CS nomenclature used, and going on a sperg tangent. The first time this happened to me was when I was told to click on the "hamburger icon". I absolutely lost it and haven't recovered since. Just call it a drop down menu, literally anything else!!

>>16019942
got I hate math retards.
>Group :(
>Associative invertible binary relation with identity :)

>topology :(
>Power set sublattice closed under infinite join :)

>>16019958
Now you're thinking like a category theorist :)

mathematicians, the most inclusive of mind,
too blind to see the etymology of bind,
now blankly staring and filling with despair,
as the initiate draws the circle with the same area as the square,

enranged with emotion at the last verse,
for it is the vesica that causes this curse,
the mathematician asks why not can I see,
"for it is the vesica that causes this inverse",
answers the initiate, as he smiles with glee,

with a straight edge the initiate measures the vesica's degree,
to find a ratio with the square root of 3,
with knowledge of geometry and a mind unorthodox,
the initiate concludes that the shape is a box,

"use a straight edge to measure the degree",
"and you will understand that placing your mind in a box",
"is why you can't see",
answers the initiate, as he smiles with glee.

>>16019968
>Category :(
>Monoidoid :)

>>16019970
you = Σ>>16019970
:^)

>>15995883
Is there a coherent way to map sequences of values (genetic sequences) into integers in a way that makes sense, e.g. longer sequeences are mapped to larger integers, such that it is injective? My first impulse would be to assign each nucleotide with a prime number and then raise it to the power of how often that base is found in the sequence, and then take the product of the numbers.

>>16019514
Do the work, anon. Most people don't do the work.
https://math.stackexchange.com/questions/33656/whats-better-strategy-to-handle-tons-of-formulas-definitions/33987#33987
Stop shitposting on 4chan, and do a lot of problems.

>zfc is defined in first order logic
>first order logic is defined with sets
how does this work?

>>16020297
How are sequences currently represented?

>>15995883
What's a good calculus I and II textbook with exercises? I don't wanna pay will download PDF.

bit of a weird question, but can anyone recommend some good french-language textbooks that have english translations ?
i'm learning french and currently at the intermediate stage, would like to kill two birds with one stone by à la fois studying maths and studying french
ideally anything relating to : algebraic topology, commutative algebra, (basic) algebraic geometry, (advanced) mathematical logic (literally any of the branches), category theory
merci !

>>16020409
it's a mindfuck, but to simplify things we essentially use an informal meta-language to define everything, the idea being sets and logic capture the limits of this informal meta-language

Wolfram tell me that the limit as n goes to infinity of [math] n(e(\frac{n}{n+1})^{n}-1) [/math] is 1/2, which I believe, but I can't prove it. I've been trying to use the usual logarithm trick to pull it apart, but that doesn't give me anything tangible that I can see. Is there some special trick to this?

>>16020798
You've typed that wrong. The limit is not 1/2

>>16020817

Hello,
I currently go to an extremely small local university, and I am unsure of whether my courseload is shit-tier or average. It's definitely not great, I'm not Russian or German lol.
Alongside that, I have to complete some sort of senior project this year, but I have no idea that interests me. The most laughable thing about all of this for me is not taking a DE class.
I half-ass learned it when I had to take a test over all course offerings even though 2 upper-level math courses are only offered every semester.

I've taken/taking:
Calc Series
Linear Alg
Statistics
Numerical Analysis
Intro Complex Analysis
Abstract Algebra
Problem-Solving Seminar (assuming it's a proof class, for most).
Senior Seminar

>>16020821
mb. wolfram misparsed my input.

>matrix multiplication is not obtaining matrix A matrix B times (which I find the most annoying from how I understand the word multiply, even if that wouldn't really make sense), it is not commutative and does not even use x as its symbol since that's a cross product. It's just AB
this irritates me, so what is it that you people consider multiplication because I clearly do not know what it is when 2 * 3 is two 3 times but AB is a weird anticommutative step by step operation using addition, which I guess repeated addition is also technically multiplication but it doesn't make sense to me why it's called matrix multiplication.
This is like that time when I was confused by R^2 (Real number R, not coefficient of determination) appearing in my linear algebra book and it wasn't all real numbers to the power of 2 but a 2‐space. Same notation, different meaning, wtf.
What have all of you read that I haven't?

>>16020887
this is almost certainly bait, but
1. the word multiplication is commonly used for an associative operation that distributes over addition (which is a common word used for an associative, commutative, invertible operation).
2. multiplication of real numbers is already not repeated addition anymore
3. If it helps you can consider matrix multiplication as an extension of multiplication of real numbers, for example 2x2 matrices over the real numbers contain the real numbers in a natural way as the subring of matrices of the form r.I where r is a real number and I is the 2x2 identity matrix.
>What have all of you read that I haven't?
don't be so angry about what names people give to things man. Your linear algebra book definitely defined what they meant by R^2.

>>16020887
You shouldn't feel too bad about it, because it's not really something they ever explain.
Informally, a matrix can be thought of as a way to transform some object (a vector or another matrix) into another. Matrix multiplication, to that end, is defined as it is because it represents compositions of such transformations.
As an example, if you apply [math]\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}[/math] to a 2D vector, you will end up with that vector rotated 90 degrees counterclockwise. If we apply it again, we end up with the original vector rotated 180 degrees, so we want to define matrix multiplication in such a way that [math]\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}[/math] multiplied by itself gives [math]\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}[/math]. Attempting to make this work, and to make similar compositions work, will ultimately force you to arrive at matrix multiplication as it is defined.

You're right in that "matrix multiplication" is not a very intuitive term and something like "matrix composition" would be much better in that regard. The reason it's called multiplication, to not go into too much of a tangent, is essentially that it is distributive over matrix addition (i.e. A(B+C) = AB+AC)

>>16020887
>>16020919
Matrix multiplication is most naturally defined from the dot product, which I assume you're familiar with since you mentioned the cross product.
It also happens to distribute over (vector) addition, hence why it's a "product", even though you might find it stranger, since the dot product of two vectors (from the same vector space) is a scalar from whatever field the vector space was defined over.

You can read this essay about "repeated addition" here:
https://web.archive.org/web/20221218055625/https://www.maa.org/external_archive/devlin/devlin_0708_08.html
I don't agree with a lot of his points, since calling the real numbers the "default" way of thinking about numbers (and neglecting to mention that they're constructed from more primitive systems) is retarded. Still might be an interesting read.

>>16020933
>It also happens to...
Dot products, that is.

>>16020887
Also worth mentioning that the notation [math]\mathbb{R}^2[/math] comes from another kind of product -- the Cartesian product.
[math]\mathbb{R}^2 = \mathbb{R} \times \mathbb{R} = \{(a,b) ~~|~~ a \in \mathbb{R}, b \in \mathbb{R}\}[/math]
For finite sets, the cardinality of the Cartesian product of two sets is the product of their cardinalities, hence why it gets the name "product".

>>16020798
You can taylor expand and use some `standard' asymptotic tricks ([math]e^x=1+x+O(x^2)[/math] for small [math]x[/math]).
You have [math]n(e(1+\frac1n)^{-n}-1)[/math].
Consider [math](1+\frac1n)^{-n}=\exp{(-n\log{(1+\frac1n)})}=\exp{\sum\limits_{k\geq1}\frac{(-1)^k}{kn^{k-1}}}=\exp{(-1+\frac{1}{2n}+O(n^{-2}))}=e^{-1}(1+\frac{1}{2n}+O(n^{-2}))[/math].
Substituting this in the starting formula you get precisely [math]\frac12+O(n^-1)[/math].

Hey peeps. I'm doing some stuff with information entropy, I'm trying to calculate entropies of transformations of a given distribution. I know what the distribution is before and after the transformation, and I'm trying to find an efficient way to find the entropy difference without calculating both entropies directly. Is there any work on this I could reference?

>>16011106
Don't worry about it, that happens. Exam stress sucks and just that combined with being under time pressure might make it considerably easier to mess stuff up. The main thing you should focus on is understanding the material - if you can later come up with the correct answers, or understand them at least and know why and where something was wrong and be able to apply it in your further studies, that is good.
The goal is to learn and understand the material and ideally be able to apply it elsewhere when needed, exams just turn out to be a good way to force people to put in the work to get there.

>>16020704
French math is easy. You don't need English translation.

Does nobody else think it's absolutely crazy that you can solve systems of linear equations represented by circulant matrices with a discrete Fourier transform? Is there a way to extend this to other matrices?

>>16020955
Cheers, I must have messed up when I tried that. On a similar note, i'm trying to find some equivalent at infinity of [math] (1+\frac{1}{n})^{-n^{2}} [/math], I'm at my wits end with this.

>>16021794
With pretty much the exact same steps you can find that [math]e^{n-\frac12}\left(1+\frac1n\right)^{-n^2}\to 1[/math].

>>16021838
I should take a day off, you're right. We got [math] exp(-n^{2}ln(1+\frac{1}{n}))=exp(-n+\frac{1}{2}+o(1))=exp(-n+\frac{1}{2})exp(o(1))=exp(-n+\frac{1}{2})(1+o(1))=exp(-n+\frac{1}{2})+o(e^{-n}) [/math]. Which in turn implies that [math](1+\frac{1}{n})^{-n^{2}}\sim_{+\infty} exp(-n+\frac{1}{2})[/math]. I dunno, [math] exp(o(1))=1+o(1) [/math] just didn't click in my head, thanks.

Humanbros... It's over...
https://youtu.be/AayZuuDDKP0