/mg/ - maths general

[Anonymous No.14825203]

Formerly >>14806986

Talk maths.

[Anonymous No.14825246]

Good mathematics is compatible with the assertion that N^N contains no function that can't be turned into a program that actually maps inputs to outputs.

https://en.m.wikipedia.org/wiki/Church%27s_thesis_(constructive_mathematics)

[Doctor Eli Selig !!JQHA6kqyl91 No.14825418]

>>14825203
I have been playing with CHUNGUS numbers, Ilulu, the Maid Library and sometimes Exodia.

[None !obkpuSWh5M No.14825433]

Gonna start finding the dumbest math homework problems on reddit and spamming them in /sqt/, if I see anything funny I'll post it here

[Anonymous No.14825441]

>>14824940
>>14825210
You can generalize this result to when you have two die of even size. (Not necessarily the same size)

[Anonymous No.14825510]

>>14825203
What are the advantages of doing Riemannian geometry with Cartan's formalism (connections and curvature forms) over the classical way?

[Anonymous No.14825511]

OP you dumbass you forgot to post the link to the new thread in the old
No wonder you're a brit you dumbass

[Anonymous No.14825564]

I have a question about probability theory / statistics. It's pretty basic but I haven't worked with these ideas for fucking years.

I have some random variable [math]X[/math] that depends on another one, [math]Y[/math]. [math]Y[/math] takes only 3 possible values but [math]X[/math] takes more. I have a problem where I am concerned with the following: for a fixed [math]Y = y_i[/math] where [math]i \in \{1,2,3\}[/math], and given that [math]X[/math] is one of [math]x_1, x_2, x_3[/math], what is the probability that [math]X = x_i[/math]? What I mean specifically is this:
[eqn]\frac{P(X = x_i | Y = y_i)}{P(X = x_1 | Y = y_i) + P(X = x_2 | Y = y_i) + P(X = x_3 | Y = y_i)}[/eqn]
This value is known for the problem.

What I want to know is: how do I average this expression over [math]Y[/math]? If I just take an expectation value as normal, I get something like
[eqn]\sum_{i=1}^3 P(Y = y_i) \frac{P(X = x_i | Y = y_i)}{P(X = x_1 | Y = y_i) + P(X = x_2 | Y = y_i) + P(X = x_3 | Y = y_i)}[/eqn]
but I feel uneasy about this because we have [math]y_i[/math] in both the numerator and denominator.
Would it make more sense to do something like
[eqn] \frac{\sum_{i=1}^3 P(Y = y_i) P(X = x_i | Y = y_i)}{\sum_{i=1}^3 P(Y = y_i) \left[ P(X = x_1 | Y = y_i) + P(X = x_2 | Y = y_i) + P(X = x_3 | Y = y_i) \right]}[/eqn]
? Does either of these "averages" even make sense?

[Anonymous No.14825607]

>>14825564
[eqn]P(X=x_i | X \in\{x_1,x_2,x_3 \}) = \sum_{j=1}^3 P(Y=y_j) P(X=x_i | X \in\{x_1,x_2,x_3 \} , Y=y_j) = \sum_{j=1}^3 P(Y=y_j) \frac{P(X = x_i | Y=y_j)}{P(X \in\{x_1,x_2,x_3 \} | Y=y_j)} = \sum_{j=1}^3 P(Y=y_j) \frac{P(X = x_i | Y=y_j)}{P(X = x_1 | Y=y_j) + P(X = x_2 | Y=y_j) + P(X = x_3 | Y=y_j)}
[/eqn]

[Anonymous No.14825622]

>>14825607
That makes sense, although I'm interested in something slightly different: there is no "preferred" [math]x_i[/math] that I want to get, I just want to know the value of
[eqn]\frac{P(X = x_i | Y = y_i)}{P(X = x_1 | Y = y_i) + P(X = x_2 | Y = y_i) + P(X = x_3 | Y = y_i)}[/eqn]
on average; the [math]i[/math] is the same for X and Y on purpose. I basically want to know the probability that knowing the value of Y tells me the value of X, on average.

[Anonymous No.14825684]

I'm thinking of a set of ten distinct integers in [1,20].
Your goal is to find a subset of size three.

Any time you show a set of size three, i will tell you if it's a subset or not.

You're trying to minimize the maximal amount of sets you might have to show me.
What's the best strat ?
Can it be generalized for sets of k distinct number in [1,n] and subsets of size r ?

Hint : i have no idea, pls help

[Anonymous No.14825752]

>>14825684
There are [math]{20 \choose 3}[/math] subsets of size three. With each wrong guess you can only eliminate one possiblility so in the worst case you have to try all [math]{20 \choose 3}[/math] subsets no matter what strategy.

[Anonymous No.14825779]

>>14825752
I would agree if i was thinking of three numbers and you had to find them.
But i'm thinking of 10 numbers and you just have to find three of them, so clearly even with the worst strategy possible you'll do better than 20 choose 3.


Eg :
Let's say i have 14 distinct integers in [1,21] and you have to find 3.
You could ask
(1-2-3) (4-5-6) (7-8-9) (10-11-12) (13-14-15) (16-17-18) (19-20-21)
If none of these sets is right, you know that there are exactly two correct number in each of these seven triplets. Then you ask
(1-2-4) (1-2-5) (3-4-5) (3-4-6) (3-5-6)
and you are 100% sure one of these answers is correct.
Only took you 12 tries, which is way less than (21 choose 3)= 1330

[Anonymous No.14825933]

>>14825441
You can generalize further by having the die have any set of consecutive integers, e.g. a 6-sided with -2,-1,0,1,2,3, and a 4-sided with 8,9,10,11.

[Anonymous No.14826001]

Can someone explain to me on the conceptual level what injective modules are?

[Anonymous No.14826068]

>>14826001
It sort of captures the notion of an invertible element without having to actually talk about elements (also we can’t even multiply two elements of a module together in the first place so we need some abstract way of discussing this). Some people say that injectivity measures how much an R-module M behaves like the Z-module Q.
As an example consider two Z-modules, ZxZ and QxQ. Consider the injection from Z into ZxZ via a maps to (0,2a). Also consider the injection from Q into QxQ via a maps to (0,2a).
The image of Z and the image of Q are both sub modules, yet the image of Q spans an entire line in QxQ, while the image of Z only hits every other point on the line. This means that the image of Q has a complement in QxQ allowing us to write it as a direct sum (one of the characterizations of injective)
Since you and I both understand the difference between Z and Q, it’s clear why this happens, Q has invertible elements. But notice that this difference is noticible without even mentioning the multiplication on Q, or any particular element of Q; all we’re talking about here are morphisms.
By the way, this intuition only goes so far, the whole point of abstraction is so that we’re free to write proofs without mentioning stuff like invertible. So make sure your proofs only use the rote definition of injective object.
Also look up the definition of a semi simple algebra, it has some similarities to what we’re taking about here but also some important differences

[Anonymous No.14826150]

>>14825564
What you want is [math]\sum_iP(y_i)P(x_i|y_i)[/math]. If x_i and y_i are all completely dependent then this equals 1

[Anonymous No.14826154]

Every time I want to abbreviate "short exact sequence", my intrusive thoughts tell me to write "SEX".

[Anonymous No.14826197]

>>14826154
kino, using this from now on

[Anonymous No.14826206]

>>14826197
I fear the day I'll have to go up to a board and explain something with short exact sequences, and write "SEX" in big fat letters.

[Anonymous No.14826208]

>>14826206
>>14826197
>>14826154
Faggot.

[Anonymous No.14826210]

>>14826206
You'll be fine if you can do it with confidence and a dead-pan expression to anyone that objects.

[Anonymous No.14826225]

I’m in year one of uni as an engineer and just need to know, is there really no other way to ”get” calculus faster beyond just doing a ton of problems

[Anonymous No.14826234]

1 = 0.999

[Anonymous No.14826240]

>>14826208
Your mom

[Anonymous No.14826254]

>>14826240
?

[Anonymous No.14826267]

>>14826225
Yeah. Once you understand the meaning of infinity and infinitesimal, everything else is just practicing doing exercises.

[https://archive dot ph/uy5Hc No.14826488]

>>14825246
>>14825203

[Anonymous No.14826522]

How many axioms can you take away from ZFC without breaking basic trigonometry?

[Anonymous No.14826580]

>>14826225
Yes. It will be just "this one weird trick to solve the problem!" forever, culminating in calc 4/Diff Eqn where you learn "we literally can't solve these problems unless we punch them and force them into being a shape that fits this one weird trick"

Don't get me wrong, its fantastic, but timed test formats are just testing how many "weird tricks" you've memorized to be able to rapid fire as many problems as possible. Most math departments are too fucking lazy to make lower level courses interesting.

[Anonymous No.14826831]

Take a prime, leave a prime:
6619949901628198243
1267785902967691
2169246882650584664593430294400808481281937925231781963676325511910404738243
4420249025811789781763947444028492132784863602824103145158536402550322884584343933278417349590960683320188564779756788753

[Anonymous No.14826915]

don't steal this prime you fuckers, I discovered it it's mine

[Anonymous No.14826922]

>>14825246
>the intermediate value theorem can be false
Be careful what you wish for, Mr bat cleaner.

[Anonymous No.14826966]

What are some good senior capstone topics I could do in logic?

[Anonymous No.14826991]

>>14826966
Take a look at event calculus? Interpret the Intermediate value theorem in that context?

[Anonymous No.14827017]

>>14826966
Prove ZFC isn't consistent and we've all been studying the empty set.

[Anonymous No.14827032]

>>14826921
>>14826937
Any idea ?

[Anonymous No.14827112]

>>14826915
I got more primes than I know what to do with.
847614981584646707188670230023355009998632012780097528653078349131574045737709911179076468188378702307876474661663627412005186058860707657947742293531053070186484057516238471073992258895415558522445142602387013952822255509817561159718184025253139086497788569956427791326562830448749681

[Anonymous No.14827119]

>>14825779
Yeah I think yours is the best strategy. For sure the first n/3 guesses should be a partition of the set
>>14826225
just grind problems until you get it yeah. you could do calc 1+2 in one week imo
>>14826522
Depending on what you mean by basic trig, it can be done using Euclidean geometry so no ZFC required
>>14826966
If ur a strong student, temporal type theory

[Anonymous No.14827204]

>>14827119
I know cryptography usually multiplies two large primes together as some kind of secret, but aren't there relatively few "large primes"? This would seem to be a weakness. Where does one find a large list of large primes?

[Anonymous No.14827209]

>>14827204
Aren't there a lot of primes in the size range they use? How many primes are between 1024 and 2048 bits long for example?

[Anonymous No.14827254]

>>14827204
There's infinitely many large primes

[Anonymous No.14827318]

>>14827204
Large primes are ten orders of magnitude larger then their security level. Discrete logarithm doesn't use primes, so is more efficient.

[Anonymous No.14827399]

>>14827204
> relatively few
relative being the key word. primes do, on average, became rarer as N increases but the size of N used in cryptography more than compensates for that. we are talking about _huge_ numbers.
> Where does one find a large list of large primes?
now how cryptographic software finds a prime within a certain range is actually quite an interesting topic. they also do not rigorously confirm a chosen number is prime, the tests they use only give a probability.

[Anonymous No.14827542]

>>14825433
If you want to fuck with people instead write a solutions manual to a popular mathematics book such that the solutions look correct but are wrong. This idea is inspired by the fact that I posted my solutions one time to a book to a blog and got yelled at when someone used my work on their homework (it was wrong).

[Anonymous No.14827770]

>>14826488

[Anonymous No.14827912]

>>14825433
Why ruin a thread though? It's no different than spamming BMWF and /pol/ bait.

>>14827542
This a lot better, since it punishes brainlets who just copy solutions they found on the internet.

[Anonymous No.14827930]

differential geometry > topology > analysis > algebra > combinatorics >number theory

[Anonymous No.14828183]

>>14827930
algebra > number theory > combinatorics >>>>>>>>>>>> everything else

[Anonymous No.14828205]

Why is mathchan so bad?

[Anonymous No.14828359]

Is Mathematics something you can learn on your own? If you can, would you be allowed to publish papers or do you need a degree to be credible?

[Anonymous No.14828363]

>>14828359
You can learn on your own and you can publish papers on your and you need a degree to be credible

[Anonymous No.14828376]

>>14826922
It's the price we pay for numerical meaningful results bro

[Anonymous No.14828378]

>>14828205
because /qa/ raids it every third day

[Anonymous No.14828504]

>>14828363
>degree
You don't just need a degree but you need to be actively affiliated with an academic institution.

[Anonymous No.14828510]

How do I quickly learn all math in a month or so? I'm starting a math degree and I don't know any math

[Anonymous No.14828517]

>>14825203
The sequential definition of functional limits make much more sense than the standard one, and should replace the latter. Like if agree.

[Anonymous No.14828522]

>>14828510
Read all the textbooks and do all the exercises

[Anonymous No.14828528]

Can someone post that math chart for autodidacts? It had a guide(with the names of books) on to learn math from basics to post graduate level.

[Anonymous No.14828562]

>>14828528
Probably some retarded list made by an autist. Just follow the wiki dude. It is also a retarded list made by autist, but at least it is organised by topics and gives context for books.

[Anonymous No.14828572]

>>14828562
Twice I posted a book from the wiki and you guys said they wete bad books.

[Anonymous No.14828578]

>>14828528
just read lang

[Anonymous No.14828580]

>>14828572
were*

[Anonymous No.14828594]

>>14828578
Serge Lang? Are all of the books good?

[Anonymous No.14828596]

>>14828594
Yep, every single one of them

[Anonymous No.14828611]

>>14828596
Thanks.

[Anonymous No.14828627]

>>14828510
Read all of Bourbaki
Read all Encyclopedia of Mathematical Sciences books (well over 100 books)
Read Dieudonné's 9 volumes on analysis

[Anonymous No.14828645]

Where do you guys get books from? They are so expensive.

[Anonymous No.14828656]

>>14828627
Looks too difficult. Anything easier? I'm not good at thinking

[Anonymous No.14828658]

>>14828645
thrift shops. Otherwise, ask your professor. Most don't need 99% of those books anymore

[Anonymous No.14828688]

>>14828645
libgen.is

[Anonymous No.14828715]

>>14828572
>bad undergraduate books
No such thing. The books they call bad are the ones they haven't read. I just pick a book written in Computer Modern, because I like the font, and I rarely every go wrong with undergraduate texts. My favourite Linear Algebra book is literally nowhere mentioned in the board.

[Anonymous No.14828749]

Say it don't spray it

[Anonymous No.14828759]

>>14828658
>professor
Not a student.
>>14828688
>libgen.is
You mean, like read it on a computer? Doesn't that kill your eyes?

[Anonymous No.14828916]

>>14828759
>Doesn't that kill your eyes?
Use dark mode.

[Anonymous No.14828952]

>>14828759
If you really want printed books, you can bind them yourselves. They are not that hard and a fun braindead thing to do when you are mentally exhausted. Or you can just be lazy and get it bound professionally.

[Anonymous No.14828990]

>>14828952
Wouldn't it still be expensive?

[Anonymous No.14829002]

>>14827399
>the tests they use only give a probability.
Do they use something other than Miller-Rabin? Very useful test for primes in the hundreds of digits. I'm aiming for hundreds.
I've composed a number ~48 million digits long that has an approximate 30% chance of being prime, but Miller-Rabin can't test something that large with my poor laptop. It's been a while since I looked at it but it only has two powers of two, meaning its still a 48 million digit power.

[Anonymous No.14829048]

I am using a programming language that does not have floating point numbers even for divisions.
I am trying to divide an integer into a 95% and a 5%. To do that I am doing amount/100 * 95 and amount/100*5 to get each percentage. Do these calculations leave a remainder?
Say 180/100 = 1
1*95=95
1*5 = 5
95+5 = 100
There is a literal 80 remainder.
For the numbers I have to work with (integers up to 18 trailing 0s) so a remainder of sub 100 is no big deal.
Is there a way to evade the hassle of having to calculate the remainder?
Thank you in advance.

[Anonymous No.14829052]

>>14829048
>To do that I am doing amount/100 * 95 and amount/100*5 to get each percentage
Do it the other way around: amount*95/100 and amount*5/100

[Anonymous No.14829081]

>>14828990
lulu can do like 700 pages for $15

[Anonymous No.14829083]

>>14829052
180*95 = 17100
17100/100 = 171(correct)
180*5 = 900
900/100 = 9(correct)

But for 183
183*95 = 17385
17385/100 = 173(more)
183*5 = 915
915/100 = 9(correct)
I guess this is better but I cant afford it to be more.
Thank you anyway.

[Anonymous No.14829114]

>>14828990
The cost of mass binding is less than 1% the price of a book. Not only that, you could get much higher quality binding at far cheaper cost. Sewn books are something else. But I am speaking as a third worlder, where service cost is very cheap, so I don't know how much cheaper it will be in US, but seeing the price of books, you'll still probably save up to 70% of the price, maybe 95% if you are doing it yourself

[Anonymous No.14829141]

>>14829083
What are you talking about? It always gets rounded down. It can't be "more" than the real number.

[Anonymous No.14829163]

>>14829141
Somehow it doesnt.
Am I doing something wrong?

[Anonymous No.14829190]

>>14829163
>Somehow it doesnt.
I don't get what you mean.

95% of 183 is 173.85, which gets rounded down to 173.
5% of 183 is 9.15, which gets rounded down to 9.
173+9=182, so you lost ~0.5% to rounding errors. You're not going to do better than that with integers.

[Anonymous No.14829192]

>>14825779 (me)
>>14827119
I also think that in this particular case my strategy was optimal but i wonder if there's a way to generalize it ("i'm thinking of k distinct integers in [1,n], you must find x of these numbers, how many questions will you need to ask ?")
Ideally i would like an exact answer, but even an estimate would be really good.

[Anonymous No.14829209]

>>14829190
I am fucking retarded.
I got confused with the 171 from above.
I guess there is no perfect solution and thank you very much.

[Anonymous No.14829238]

>>14829209
If your numbers are small enough (or your integer type is big enough), you can just store all your values scaled up by a factor of 10^n for some n>1. Then you can do your operations on those values and scale them back down only when you need to output a final result. That way you get n extra digits of fixed point precision.

[Anonymous No.14829443]

>week 3 quiz question
>prove squeeze theorem for a multivariable function
bros this is too fast paced for me, I thought I was smart after getting an A in calc II

[Anonymous No.14829484]

>>14827119

Hello, you must be "remiliafag" or "chachakan" . I was wondering if you'd consider adding me on discord or some other platform. I'm a maths student myself and mainly specialised in analysis but I've been trying to explore other areas recently. I think a mathematician with an interest in touhou would be very cool to talk with for me, as I'm one myself.

"anitranny" recommend that I reach out to you in this way. My discord is Ayu#1061, but let me know if you'd prefer to use another platform instead.

[Anonymous No.14829533]

i generally get how you can transform formulas using axioms but i don't get how you can apply a function, yet

[Anonymous No.14829557]

Stupid question, but when people talk about the group D4 and its associated elements (r, r^2, etc.), does the set of elements of D4 the transformations (r, r2^2,...) or is the set of elements of D4 the square with different labels for the vertices (ABCD, ADCB, BCDA, etc.)?

[Anonymous No.14829595]

>>14829443
>write iterative solution to solving the problem of continuity on a 3D plane.
>"Nnoooooo!!! that isn't elegant!!!!"

Heh, nothing personnel.

[Anonymous No.14829640]

>>14829557
D4 is group. So the actions.

[Anonymous No.14829652]

Are there methods improve your Math, or can it be only achieved through rigorous studying and practice? Like playing chess or solving sudoku puzzles? Are those just bullsht?

[Anonymous No.14829686]

>>14828715
Is that better than my favourite LA book? Linear Algebra & Geometry by Kostrikin and Manin.

[Anonymous No.14829776]

Made some more primes just now. Way more than this, these ones are just longer than most of the ones I've made today.

77559578759316840003455838398241670342940533180097335539886427223917316127341185032849547943727959169512774261260121571221529060253767348778701448407821477875711772611896735252091062368694560586014345757748889137

1057260273706526596520806478525268695400202880480792117810488542869237542971534553075108828984841753603448918288693746952498098359102655696219879480298703230945666243882469680647308231967879309217252507587121965933219729585339415958218389174656701465180973850027

[Anonymous No.14829828]

>>14829484
no sorry to disappoint but that's not me lol i just like touhou too, i think most of the regulars here do
>>14828645
1.) international versions (published for chinese or indian audience) of popular books are 10x cheaper
2.) university library (at mine you could easily walk in without being a student)
3.) online pdf

[Anonymous No.14829839]

>>14829002
From what I remember of the original PGP code they used Fermat for an initial very fast test and then some variant of M-R for a more accurate test. I'm not sure any of the methods used will work on the size of numbers you are talking about. They are suitable for numbers with 1000's of digits and they given an answer in milliseconds, which is fine for crypto, but not with millions of digits.

[Anonymous No.14829921]

>>14826922
I don't get it, if you could explicitly construct a function where the intermediate vaue theorem is false then isn't that an L for classical mathematics?

[Anonymous No.14830026]

>>14829839
Yes I can generate primes in the 1k digit range and test them no problem. Miller-Rabin is really quite neat for that.
I've developed my own little test for composite-ness on larger numbers, but it gives way too many false-primes to be useful as a prime detector. Its novel enough I might put it into a paper some day soon, but in its current state it's not terribly useful. I have used it to generate some larger primes it does actually tell you at least one of the factors if it's small enough. Even so I use Miller Rabin to be sure. For example, I tested it against RSA 129 and it detects that as prime.

[Anonymous No.14830325]

>>14829484
Nice redditspacing nigger

[Anonymous No.14830384]

>>14829533
What do you mean when you keep saying apply

[Anonymous No.14830516]

>>14830325
I know right?
One time I saw a professor write a paper and he used Reddit spacing. When I confronted him about it, he said they were "paragrafs". Dude just made up a word on the spot lmao what a Redditor

[Anonymous No.14830614]

Is learning Math just like a Mind sport(Chess, Go, Shogi, etc)? Meaning, you have to learn it when you're young to reach the top, because your brain stops developing when you'e older?

[Anonymous No.14830636]

>>14829921
In constructive analysis it doesn't hold because it sometimes requires you to explicitly construct certain real numbers that are not constructable.

[Anonymous No.14830687]

>>14830614
Math is not like "mind sports" and what you said about the latter is also false

[Anonymous No.14830696]

>>14830687
>latter is also false
Huh? Isn't it pretty much proved that mindsport require childhood practice because of neuroplasticity or something?

[Anonymous No.14830702]

>>14830696
I don't think you understand what neuroplasticity is. Just because the nucleus basalis is not constantly stimulated like in critical phases does not, at all, mean neuroplasticity "stops".
The brain constantly changes itself. The point of view you're talking about has been debunked since the 1960s-1970s; ever since Merzenich came around, more or less.

[Anonymous No.14830718]

>>14825203
Today, the math teacher talked about monoid kek! MONOID! That's hilarious.

[Anonymous No.14830722]

>>14830702
Then what is reason for people who start late being unable to reach the top in mindsports?

[Anonymous No.14830729]

>>14830722
Depends on the individual. You can easily cause chronic inflammation in the brain by eating like shit, taking drugs, etc. in the brain that significantly inhibits neuroplasticity.

[Anonymous No.14830746]

>>14830722
NTA, but I think there's 3 major reasons:
1- most people got exposed to the sport at childhood and everyone that was interested got good.
2- in adulthood you have responsibilities so you cannot devote yourself fully to a sport if you are not already extremely good at it.
3- you simply get more experience by starting earlier.
So you need someone to fall into at least 3 extremely rare classes: switched from no interest to very strong interest, can afford the time and effort and has no other obligations, and has the "potential" to be a world class player. And even then everyone else in the field has at least 10 years of experience over you.

[Anonymous No.14830863]

So, just to confirm, /mg/ unanimously agrees that Serge Lang's books are the go-to for mathematics? I've contacted a seller to buy the complete set, so just making sure.

[Anonymous No.14830889]

>>14830863
please do not just buy a whole set unless you're getting a really good deal and no, the lang books are not the best for every subject

[Anonymous No.14830898]

>>14830863
>>14830889
also where did you find someone to sell you a complete collection of lang books?

[Anonymous No.14830950]

>>14830863
No, theyre garbage. For example his books on diophantine geometry rely extensively on results by Mordell and Siegel, both of whom said Lang's books suck.

[Anonymous No.14831299]

>>14830863
His linear algebra is pretty readable but I haven't really consumed anything else.

[Anonymous No.14831359]

Read Aluffi but it felt a bit shallow on some topics like commutative rings, galois and representation. What text has good chapters on these topics that I can use to supplement?

[Anonymous No.14831362]

>>14831299
>>14830863
Langs undergraduate books are all quite good. Ive read his undergrad analysis and algebra books.

[Anonymous No.14831418]

>>14830384
Like if you took a string and applied a function to it. But for this it's more like transforming formulas that have symbols that represent a function.

[Anonymous No.14831440]

What does helix sound like?

[Anonymous No.14831508]

Does anyone else find Axler's books incredibly ugly? It's the entire reason I have never given his books a chance. Fucking autist typesets his own books, but makes them much uglier than Springer's default typesetting (which is ugly enough as it is with its shitty STIX font).

[Anonymous No.14831540]

>>14831508
>using polynomials
to find the determinant

kino

[Anonymous No.14831709]

>>14830898
I know a dealer who mainly sells educational books. He takes a few weeks but he delivers. He is a bit of a shady guy though, some of the books I purchased from him seemed like fake copies, the content was perfect albeit the paper quality was a bit poor. Not really a problem for me though.

[Anonymous No.14831728]

>>14828205
How does it compare to lambdaplusjs?

[Anonymous No.14831917]

>>14830516
This post isnt an example of redditspacing. Just separating your sentences into paragraphs like this, especially for the purposes of readability, is not redditspacing.

Ironically, most anons think this is, because redditspacing catches out newfags who think it's spacing like this, but since most of this site is newfags and redditors, people who say paragraphs are redditspacing, like you, are just announcing that they're part of the cancer and need to leave.

[Anonymous No.14831921]

>>14831917
That's what I was saying.
It's called irony.

[Anonymous No.14832036]

>>14831508
I just don’t understand how people have a “favorite” analysis book or a “favorite” linear algebra book.
I just don’t see any reason to read multiple books of a subject I already know just to pick a favorite when I could read something new instead

[Anonymous No.14832262]

does each structure in first order logic have one theory for it?

[Anonymous No.14832263]

>>14832036
>I just don’t see any reason to read multiple books of a subject I already know
Undergrad detected

[Anonymous No.14832351]

Are non-logical axioms like axiom schemas, or are they sentences in a theory? Is there one theory for each structure?
I don't want to keep spamming but I don't know how to figure it out other than thinking really hard for a long time

[Anonymous No.14832512]

>>14832263
there you have it anons, to get your phd all you have to do is read 4 different linear algebra books

[Anonymous No.14832560]

Is it possible to have thread specific captchas? Asking for a friend.

[Anonymous No.14832893]

>>14832560
2019*2019 - (2019+2*2018)
= 2019*2018-2*2018
= 2017*2018

[Anonymous No.14832903]

help me bros. I'm becoming wildbergerpilled.

[Anonymous No.14832922]

>>14832903
why is that wrong?

[Anonymous No.14832938]

>>14832903
>>14832922
Wildberger = triggered by the word "infinity"
You wanna limit yourself, that's fine, just don't pretend your way is the only way and spaz out when people disagree.

[Anonymous No.14832960]

Jesus christ how did lang publish so many books.
One of my profs said he was seething that he wasn’t in bourbaki so he decided to publish a book in every subject he could just out of spite

[Anonymous No.14832973]

>>14832938
look, I'm not gonna try to change your outlook, but I still want to say you will look like a lunatic /x/ schizo to me, simple as.

[Anonymous No.14832985]

>>14832973
...is what I say to finitists

[Anonymous No.14833006]

>>14830863
Please don't do this. Lang's first course in calculus isn't that good. Worse than Spivak for a first exposure, because its just too easy.
You should not do this unless you already have a degree in math and they will just serve as references.
I don't care what some clown on 4chan tells you, the school curriculum textbooks are the best textbooks for actually learning topics. Nothing will top Stewart calculus for precalculus and calc 1-3. Gallian is the best text on intro to abstract algebra, baby rudin is the best intro to analysis.

[Anonymous No.14833030]

>>14832985
...how? my outlook translates into an actual state of affairs (we are finite beings, therefore we an only do a finite number of things), while your outlook clearly has concepts that don't map into any state of affairs. I'm just sticking with what i think it's possible. If you could show me how all infinitist concepts map into reality somewhat, I would be happy to change my stance of course. We are scientists after all.
>inb4 what about limits
The thing is, there isn't any infinity involved here. All you are saying when you take the limit is that given an arbritrary error, you can tell me when the distance of the sequence and the limit are within that error. you can do this by constructing a funtion that relates the error with a natural number. So I do believe alot of calculus can be put in a finitist grounding if you are simply careful with our definitions.

[Anonymous No.14833035]

>>14833006
stewart is not the standard for multivariate at good programs. I suggest Vector Calculus, Linear Algebra, and Differential Forms by hubbard & hubbard

[Anonymous No.14833039]

>>14833030
Why don't things that are defined as having no end count as states of affairs?

[Anonymous No.14833041]

>>14833030
>my outlook translates into an actual state of affairs
Is there an end to time, or is it infinite?
Is there a largest number, or are the naturals infinite?

[Anonymous No.14833082]

>>14833039
I don't have a proof that there isn't, but so far I don't know and i don't think anybody knows of something that can be described as never ending.
>>14833041
It is true that the rules that allow us to construct natural numbers make them unbounded (I ca always add +1), so to talk about the largest number would be a mistake. But even then, to reveal more numbers of the sequence, I actually need "work it out", I can't just conceptualize all the natural numbers, as that would not correspond to anything. There just isn't a box with apples map one to one with the natural numbers. I for time I don't know if it's finite or infinite, if it is infinite, then I've been proven wrong, but I don't think any physicist has been able to show it. ( is it even possible?)

[Anonymous No.14833123]

>>14833082
>I can't just conceptualize all the natural numbers
You just keep putting more and more arbitrary restrictions on yourself.
>for time I don't know if it's finite or infinite
Neither do I. I have never seen anything that "lasts forever", nor have any of us checked if there exists anything that does. There very well could be something like that, so you're preemptively putting a restriction on yourself and locking yourself out from a huge part of mathematics.

But even if there isn't... and there's no such thing as infinity in real life, why should anyone block out an entire concept? It's simply fun to work with. Why deny fun?

[Anonymous No.14833162]

>>14832960
>Lang
>not in Bourbaki

[Anonymous No.14833255]

>>14830863
http://www.jinfo.org/Mathematics_Comp.html
>Serge Lang
Yes, support Israel, goy.

[Anonymous No.14833346]

>>14833255
Supporting a Jew != supporting Israel/Zionism

[Anonymous No.14833455]

Need to refresh my knowledge on undergraduate probability really fast. Does anyone have a problem set with answers?

[Anonymous No.14833519]

Rosen or Epp for Discrete Maths? Also nobody told me that enduring bad smells was one of the challenges of college.

[Anonymous No.14833726]

>>14833519
Neither lol both absolutely nogged books read Concrete Mathematics by Graham Knuth Patashnik

[Anonymous No.14833875]

>>14833519
Rosen for the graph theory-directed graphs specifically

[Anonymous No.14833951]

>>14833726
>>14833875
Any books specifically about propositional logic? I feel like Rosen's text is taking me for a ride. Are all of these statements really equivalent?

[Anonymous No.14833960]

>>14833726
>>14833875
>>14833951
Especially the highlighted ones. What in the fuck?

[Anonymous No.14833978]

>>14833960
If P then Q is equivalent to P only if Q because, P cannot be without Q, since P would necessarily mean Q. If it is raining for a while, street is wet is equivalent to saying, It is raining only if street is wet. However street can be wet without it raining, so it does not mean: If Q then P, or equivalently Q only if P.
Now do the rest.

Although consider that propositions are inherently static statements, they are not dynamic. So in reality it is obviously weird to say that street being wet is a necessary condition for it to rain, especially considering that they don't become wet after few minutes have passed. By a necessary condition, they don't mean a source or origin per se, street being wet is not a source of it raining. What it means is that the statement: "streets are wet" must be true for the statement: "it has been raining for a while" to be true. They are essentially comparing the boolean value of the propositions.

[Anonymous No.14834017]

>>14828749
>Therefore there is no set whose cardinality is strictly between that of the integers and that of the real numbers.
But you just said
>Card N < Card Z < Card Q < Card R

[Anonymous No.14834041]

>>14833978
So is intuition by language just completely unreliable when dealing with propositional logic? Should I just think of the statements, operators, connectors, conditionals, etc completely in terms of boolean values? Or I guess in a more general way, mathematically?

[Anonymous No.14834042]

>>14833951
>books
I don't think any particular book will help with this current problem, which is just a list of stupid shit mathematicians will say when they want to be confusing.
>>14833960
>p only if q
means you can't have p without q. so if p, then q.
>q is necessary for p
again, means you cannot have p without q. so if p, then q
>a necessary condition for p is q
ditto

I don't think the author is requiring you to memorize these like there's an exam or something. He's just giving you a heads-up for when you inevitably run into these in the wild -- and, I think, making a point of his own, that p->q is important enough in mathematics that they have multiple ways of expressing it. Like that apocryphal story about "300 eskimo words for snow".

[Anonymous No.14834065]

>>14834041
It is still reliable once you consider that objects you are dealing in math are fundamentally different than ones in universe. They don't have any source they are being generated from, they do not change with time, etc., they are either true or false. That is all. Personally, I do not like the English language equivalents either myself, but unfortunately, math academia avoids logic notation. So you have to get used to them if you intend to read math books and papers. One advise I have is to attach "is true" in the end of every proposition. So instead of:
P only if Q, say P is true only if Q is true
or
Q is necessary for P, say Q being true is necessary for P to be true.
This also helps a lot when using negation.
~P: instead of saying P is not true, say negation of P is true. This helps quite a bit in nested negations

[Anonymous No.14834075]

>>14834065
>One advise I have is to attach "is true" in the end of every proposition.
Did you come up with that or stumble upon it from a source like a professor or textbook?

[Anonymous No.14834080]

>>14834042
>>14834065
Thank you bros. I'll have to pick this up again after getting some sleep. I'm less confused now after reading your explanations.

[Anonymous No.14834082]

>>14834075
I came up with it when I realised I kept unnecessarily attaching ==true, whenever I would write an already boolean statement in Java.

[Anonymous No.14834146]

>>14834082
Weird I've never made that connection to discrete math, I knew that assigning a boolean value to a statement was kind of redundant depending on the context but never thought about it that way.

[Anonymous No.14834211]

>>14831440
A slide whistle going up and down?

[Anonymous No.14834226]

>>14833960
I think the "implies" symbol is just supposed to mean in any state that A is true, B is true. So if the possible states are
A is false and B is true or
A is true and B is true,
then in any state that A is true, B is true. so if b is true it means that if a is true b is true (inconcise explanation because I don't get how it works myself exactly)

[Anonymous No.14834229]

>>14833519
>Also nobody told me that enduring bad smells was one of the challenges of college.

You know now! Hell, I recall doing a test in a room
with someone that smells of motor oil sat about
4 feet next to me. He plainly announced to
the others in the room: "Yes, I know I smell bad.
What about it? I'm an engineer!"

So anon, endure it and talk about it afterwards.
Helps out with memory, I think.

[Anonymous No.14834235]

>>14833960
>>14834226
i think the "A only if B" is because
if "in any state A is true, B is true" means
"in any state B is not true, A is not true" so
"in any state A is true, B is true" so
"A only if Q"

and "B must be true for A to be true" could be
"B is necessary for A (to be true)"

tell me which part of that was not perfectly clear so I have more information

[Anonymous No.14834690]

What are some fun constructive proofs? I wanna get into constructive math.
It seems challenging and fun.

[Anonymous No.14834709]

How important is it to actually pursue a Ph.D? I'm about to graduate with an original paper under my belt have an opportunity to crank out several more under my advisor and with grad students. Is there a point where getting a Ph.D is moot unless it's a hard requirement to get a job? I'm still probably going to pursue it, but wondering what the view is on this. How much will it matter if I just start publishing papers after my masters?

[Anonymous No.14834948]

Wtf is a literature review?

[Anonymous No.14835081]

>>14834948
Review of literature.

[Anonymous No.14835087]

>>14835081
hee hee hoo hoo funny my le ebin redditor

[Anonymous No.14835094]

>>14835087
I really don't understand what else you want me to say. It's like asking what is a red ball. A ball that is red, what else could I say.

[Anonymous No.14835100]

>>14835094
>what is an Algebraic Group?
>it's a group that is algebraic
>hilarity ensues

[Anonymous No.14835111]

>>14835100
Not really, because algebraic doesn't have a concrete definition unlike literature and review. A better example would Commutative Group, and if you know what both those terms mean, you know what it means. Unless you are insinuating that you do not know what literature or review means.

[Anonymous No.14835114]

>>14835111
>here's my headcanon

[Anonymous No.14835131]

>>14835111
>I unironically think literature review means "9/10 literature, really good graphics"

[Anonymous No.14835132]

>>14835087
A literature review is a review of the previously published works [literature] on a specific topic.

[Anonymous No.14835143]

>>14835132
So >>14835131

[Anonymous No.14835214]

>>14834948
You collect the important papers, cite them, list their contributions, put them in context, give people are starting point to get into it or yourself a way of contrasting your (future) work

>>14834690
I mean constructive are also always classically valid proofs.
Conversely, roughly speaking, all classical proofs can be expressed constructively.

https://en.wikipedia.org/wiki/Double-negation_translation

Note that assuming the implication [math] (P\lor(P\to Q))\to Q [/math], this implication is a valid antecedent for itself, so
[math] ((P\lor(P\to Q))\to Q)\to Q [/math]
Now for [math] Q [/math] equal to [math] \bot [/math], you've proven
[math] \neg \neg (P\lor\neg P) [/math]
Meaning no consistent logic or theory with basic rules for "[math] \to [/math]" and "[math] \lor [/math]" will prove.
[math] \neg (P\lor\neg P) [/math]

So my answer to your question would be would be "any". Take any theorem you like and find the constructive analog. If you know lambda calculus, it's more or less intuitive what provable on the nose and what not. If you don't, then as a rule replace "there exists" with "it can't be ruled out that there exists"

This section illustrates this point:

https://en.wikipedia.org/wiki/Constructive_analysis#The_intermediate_value_theorem

[Anonymous No.14835213]

>go to masters program, immediately pass two prelims in first semester based on self study of only two textbooks I studied during the undergrad to graduate gap.
>feel like a clown on easy street, as I only have courses left for masters and I'm done.

I should have just applied for a PhD. I didn't believe in myself and am paying my own way through a masters (dad is paying) and now I'm just set to waste my time on easy courses and twiddle my thumbs the rest of the day.
How do I go about requesting an upgrade to a PhD track? Do I have to reapply? Can my department just wave its hands and just make me a PhD track?

[Anonymous No.14835237]

>>14833951
Rosen isn't bad for discrete math, but if you want to focus on that aspect of things without it being too overlooked, try "A transition to advanced mathematics" by Smight/Eggen/St. Andre.

[Anonymous No.14835238]

>>14835214
>You collect the important papers, cite them, list their contributions, put them in context, give people are starting point to get into it or yourself a way of contrasting your (future) work
Thanks

[Anonymous No.14835437]

Hey guys, so I'm a first year at my uni and I managed to pass my first semesters math course with a C and im fairly disappointed with the results and wish to improve to get ay least a B+, ideally an A- or whatever you guys use for marks in uni

I attached a picture that shows what we took and I'd like to ask if you could point me to books that would help me prove beyond a C. Maybe a workbook or something as well? Thank you

[Anonymous No.14835607]

>>14829002
primality testing is in P, but only becomes useful for meme numbers
miller-rabin is still the gold standard

[Anonymous No.14835617]

https://www.youtube.com/watch?v=ft3Zl--hIR0
funny, he's calling out /sci/ meme texts.

[Anonymous No.14835809]

>>14835213
>How do I go about requesting an upgrade to a PhD track? Do I have to reapply?
Why the fuck are you asking us? Go talk to someone in your department, they won't bite

[Anonymous No.14836089]

Thanks bros for helping me out with the if p then q conundrum. I've moved past it and I'm working on the conditional theorems now.

[Anonymous No.14836097]

How do I show that absolute difference between any two distinct rational numbers with denominator bounded above is no less than the reciprocal of the bound?

[Anonymous No.14836107]

>>14836097
You don't because it's not true.

|(1/2) - (1/3)| = 1/6 < 1/3

[Anonymous No.14836341]

/mg/ has the smartest people on this site right? Has to be.

[Anonymous No.14836397]

>>14836341
/sci/ is better than other boards, but being a nerd and developing a passion for solving partial differential equations shouldn't wow you. Smarter than sci are the history and /lit/ guys who have a bit of a grasp about how humans work.
I got my physics PhD years ago but I don't value math and engineering minds over people who read Joyce or the Koran on their own terms, to get an idea of how the currently world came about

[Anonymous No.14836399]

/mg/ is fucking dead.

[Anonymous No.14836402]

>>14836397
Math
Literature
Music
Games

Those are the 4 great arts that every men should try to master.

[Anonymous No.14836417]

any sites/computer programs where I could move planes and points around in 3d and watch how their distance to each other changes, etc.? I feel like it would help my geometric intuition

[Anonymous No.14836444]

>>14836341
/mg/ yes, by a long shot. But you have to look for folks who actually reply to problems being posted and not the mouthbreathing retards (you)

[Anonymous No.14836460]

Studying math just feels like I'm copying information, not learning anything

[Anonymous No.14836461]

>>14836397
Majority of /his/ are just /pol/tards and /k/ucks who can read. They don't read anything beyond Vietnam, Nazis, and Rome. Majority of /lit/ are pseuds who read classics because they want to appear intelligent, or again /pol/fags who can read. Similar for /sci/, but /sci/ at least has /mg/ which is a great filter for brainlets. And the thing about /mg/ and /sqt/ is that you could easily tell which anons are intelligent by their replies, while for /lit/ you can never be sure if they are bullshitting. The most superior board I would say is /out/ because the entire board is a filter.

[Anonymous No.14836466]

>>14836460
You are supposed to do the exercise (yes, even the proofs).

[Anonymous No.14836467]

>>14836460
That's because you're not engaging with the material.

[Anonymous No.14836518]

Is Analysis a Soulslike? Brutally difficult, but the almost universally considered the most enjoyable branch of math, except by autismo number theorists:
>muh prime numbers are so ebin awesome
Shut the fuck up.

[Anonymous No.14836537]

>>14836461
I have too little respect of people who learned math or physics at this point, I've been around them for 16 years. They are only smart about their little subject and their opinions beyond that are as normies as any other person. Maybe that's true in general. I have the hope we'll read people are a but more cautious about their judgements

[Anonymous No.14836722]

>>14836460
Just convert everything to be geometrically intuitive and it'll make sense. Had to do this with models for multivariable differentials, and I never forgot afterwards.

[Anonymous No.14836739]

>>14836518
analysis is so fake and gay. Combinatorics is where the demi-gods are.

[Anonymous No.14836890]

>>14836739
Combinatorics is a bunch of completely unrelated topics in a trenchcoat pretending to be a field

[Anonymous No.14836897]

>>14825203
discord.gg/2FTEjQVs

[Anonymous No.14836949]

>>14836897
What's that?

[Anonymous No.14836986]

>>14836949
A link to a CP marketplace. Don't click the link.

[Anonymous No.14837034]

>>14833006
>Gallian is the best text on intro to abstract algebra
Get fucked

[Anonymous No.14837077]

>>14837034
Let me guess, I have to use some ancient textbook like Miller's elements of modern abstract algebra? why can't you just be normal? do you think you know better than all the math departments that chose Gallian's text? Why are you so contrarian?

[Anonymous No.14837085]

What comes after differential geometry?

[Anonymous No.14837102]

>>14837077
>Let me guess, I have to use some ancient textbook like Miller's elements of modern abstract algebra?
No.
>do you think you know better than all the math departments that chose Gallian's text? Why are you so contrarian?
Look at you being a good little boy and doing what you're told and not questioning the shit you get served up.
Should we also not question the blatantly shit Wiley textbooks that abound in undergrad education and clearly give a poor quality education? After all, who are you to question your superiors? If they say you need the 23rd edition of the latest calculus fad textbook, plus the code so you can submit your assignments online, who are you to think there might be a better alternative? They know better than you because they're in charge.
Your stupid argument aside, Gallian is bad because it is essentially the same flavor of textbook as these mindless calculus textbooks which have flooded the market. It focuses far too much on being a cookbook and gives you too little proper perspective of what the subject actually is, why it works, what the essential ideas that underpin the theory are. The book will fall out of fashion and be forgotten in a decade or so.

[Anonymous No.14837261]

Which is the most based way to read it?

[Anonymous No.14837267]

>>14837261
jeece of ef

[Anonymous No.14837273]

>>14837261
goff

[Anonymous No.14837298]

Which Dover Books would you guys say are good?

[Anonymous No.14837340]

>>14837261
An autsist wrote this sentence

[Anonymous No.14837374]

>>14837298
A Book of Abstract Algebra

[Anonymous No.14837434]

>>14837374
Which one specifically?

[Anonymous No.14837449]

>>14837434
That's the title.

[Anonymous No.14837458]

>>14837374
seconding this one

[Anonymous No.14837490]

How to get started honing my high school tier math skills for my nurse entrance exam?

[Anonymous No.14837512]

>>14836890
and that's why it's based. It reflects real life. It's messy, you have to wing it by really out there
gigabrained arguments, it's super complex, but there's real beauty in it. Meanwhile analysis tries to pass an image of beauty and order, but in fact this is very artificial and fact, with many results not having computational meaning. It's sterile and for limp dicked dorks.

[Anonymous No.14837524]

>>14837512
>modern world of engineering vs card tricks
hmmmm

[Anonymous No.14837529]

>>14837524
>analysis
>modern world of engineering
Yeah, because engineers have to rudin and attend measure theory classes (they have to watch out for the banach tarski paradox right?????). Just because you guys leech off calulus doesn't mean what you do is meaningful. fuck off.

[Anonymous No.14837536]

>"the proof of Fubini's theorem is an exercise left to the reader"
>don't do the exercise
>pop quiz, one question
>show the proof for Fubini's theorem

[Anonymous No.14837539]

>>14837529
It is through developments in analysis that we find applications in engineering in order to accomplish things in the real world. Attending classes isn't meaningful. You should get your ego checked before it crashes into self harm once you realize you will never accomplish anything in academia. Worried for you anon.

[Anonymous No.14837567]

>>14837539
all analysists could die right now and and the modern world would continue to function. Engineers can teach things and discover things by themselves you know? and I'm not in academia, I have an actual job.

[Anonymous No.14837594]

>>14837567
You're coming from the perspective of things already built. If all analysts died post 1910, we would be very far behind where we are today.
If all combinatoricsters in history never existed, we would not notice a difference.

[Anonymous No.14837597]

>so stupid I have to still use tables to solve sections of multivariable partial derivatives
>get lost and forget what the bigger problem that required this even was.

What can low IQ bros like me even do?

[Anonymous No.14837634]

Ahhhhhhhhh these are so boring and I still got it wrong

[Anonymous No.14837642]

>>14837634
immediately I can tell you your answers for the partials, dp/dv and dp/dw are wrong, as you have 10 in the denominator. Its the sqrt of 10. The rest of your computations were correct.

[Anonymous No.14837680]

>>14825203
>It's easy to see/come with/ obtain
>proof is left as an exercise
why some books do this? at least give me some hints, I don't have a life time to spend in redoing all the damn work.

post related >>14835401

[Anonymous No.14837688]

>>14837680
Ok, I'll post it here to get a little attention.
It's not homework btw. I'm self studying, just that. The problem is to show both expresions are equivalent:

[eqn] \frac{\lambda s^{\alpha-1}}{1+\lambda^2\pi^2} \frac{d}{ds} \left[ \int_s^1 \int_0^u (u-s)^{-\alpha} (u-t)^{\alpha-1}t^{1-\alpha} f(t) dtdu \right] [/eqn]
and
[eqn]\frac{f(s)}{1+\lambda^2\pi^2} + \frac{\lambda}{(1+\lambda^2\pi^2)s^{1-\alpha}(1-s)^{\alpha}} P.V.\int_0^1 \frac{(1-t)^{\alpha}t^{1-\alpha}f(t)}{t-s} dt[/eqn]
and I'm fuckin stuck.

[Anonymous No.14837890]

>>14837490
welp
https://youtu.be/GAN-jgzYsIo

[Anonymous No.14837994]

>>14837688
What book are you studying out of?

[Anonymous No.14838078]

>>14837994
Singular integral equations - Estrada, Kanwal, chapter 3, pp 80-81

[Anonymous No.14838092]

>>14837512
Combinatorics is irrelevant because they can be solved using computers now.

[Anonymous No.14838144]

>>14838092
Analysis is irrelevant because Wolfram

[Anonymous No.14838181]

>>14838144
Nope.

[Anonymous No.14838214]

>>14838181
Isn't this the game we're playing? Just saying the dumbest shit?
Go on, it's your turn.

[Anonymous No.14838382]

>>14836518
Is number theory autistic or schizo? Zeta function stuff blurs the line.
>"Are the imaginary numbers in the room with you right now?"
>>14835607
>miller-rabin is still the gold standard
Good to know.
What's the hardware limitation on Miller-Rabin? Processor, memory, a combination of the two? I found a nice, fat 2,700ish digit prime yesterday and it took the about three minutes to get to k = 40.

[Anonymous No.14838454]

>>14838214
Wolfram can't prove a single theorem in Real Analysis, a proof based course. Combinatorics is entirely just enumerating things, which is what computers are made for. The entire subject of combinatorics is dedicated to things which would have been too difficult to calculate before computers.

[Anonymous No.14838456]

>>14838382
Cantor set is more schizo desu

[Anonymous No.14838512]

>>14838456
You're right.
Here, have some primes:
1434209047681093231633997244967410944214411075790067553804261067003618899169067680945942547
1434209047681093231633997244967410944214411075790067553801095137113264090559283766337135757
1434209047681093231634006270141471153403851329029569213321280671563007798127765965783648423

[Anonymous No.14838527]

>>14838382
> What's the hardware limitation on Miller-Rabin?
CPU - it's a polynomial-time algorithm. Memory is insignificant.

[Anonymous No.14838575]

How is it proved that the trigonometric functions are continuous?

[Anonymous No.14838588]

>>14838575
You have prove that sums, products and quotients (provided the function in denominator is not 0) of continuous functions are continuous. Then you prove the exponential function is continuous. Then the identities

[eqn]\cos(z) = \frac{e^{iz} + e^{-iz}}{2} \\
\sin(z) = \frac{e^{z} - e^{-iz}}{2i} \\
\sec(z) = \frac{1}{\cos(z)} \\
\csc(z) = \frac{1}{\sin(z)} \\
\tan(z) = \frac{\sin(z)}{\cos(z)} \\
\cot(z) = \frac{\cos(z)}{\sin(z)}
[/eqn]
tell you that they are all continuous in their whole domains.

[Anonymous No.14838597]

>>14838588
Can I prove using only the unit circle definition?

[Anonymous No.14838609]

>>14838597
Nevermind, I guess coordinate geometry needs to be built from Analysis first. Is Differential Geometry the subject that builds geometry from Analysis?

[Anonymous No.14838636]

Is the set that contains all bijective mappings from N to N uncountable?

[Anonymous No.14838647]

>>14836518
>except by autismo number theorists
What about analytic number theory though?
Apostol was an analytic number theorist and he wrote some good books about calculus and mathematical analysis.

[Anonymous No.14838652]

>>14838382
numbers of C are as valid as numbers of N. If you don't accept the validity of imaginary numbers, you must deny the validity of natural numbers, as well

[Anonymous No.14838661]

>>14838636
Yes.
Riemann rearrangement theorem says that for every real number [math]x \in \mathbb{R}[/math] you can find a permutation [math] \sigma \in \text{Sym}(\mathbb{N}) [/math] such that
[eqn]\sum_{n=1}^\infty \frac{(-1)^{\sigma(n)}}{\sigma(n)} = x[/eqn]
So there must be at least as many permutations on [math]\mathbb{N}[/math] than real numbers.
[math]|\text{Sym}(\mathbb{N})| \geq |\mathbb{R}| [/math]

[Anonymous No.14838663]

>>14836537
If you had been around historians for 16 years you would probably feel the same about them.
Nobody is omniscient and everyone comes off as a brainlet about everything other than his narrow specialization.
Even if you try to be somewhat encyclopedic like Will Durant you will only scratch the surface and never gain any deeper understanding.
The choice is always between breadth and depth, and you have to choose depth if you wanna make a living. Making money out of breadth is a lot more difficult.

[Anonymous No.14838665]

>>14838652
>f you don't accept the validity of imaginary numbers, you must deny the validity of natural numbers, as well
Why? No I dont.

[Anonymous No.14838675]

>>14838665
My professor said so. He also said that you don't have to define any more number ranges after C. I can't explain you why. I'd like to know the reason, as well.

[Anonymous No.14838681]

>>14838661
Interesting, thanks.

[Anonymous No.14838721]

>>14838663
>If you had been around historians for 16 years you would probably feel the same about them
yeah, pretty sure that's true. greener grass. I'm not saying I'm not just coping

[Anonymous No.14838729]

>>14838636
I think there's a way in which we can make this a lot tighter:

There is no surjection from N to the set set N->{0,1}, which corresponds to the decidable predicates, making it uncountable in this sense.
And I'm sure we can find many nice injection of that set into your set of bijections.

For example, given an p : N->{0,1}, construct b_p with b_p(n) equal to just n when p(n)=0, but for every consecutive values u, v where p is 1, swap the values.
For example, if p starts out with the values
0 0 1 0 1 1 0 0 1 0...
Then b_p is the bijection on N given as
0 1 4 3 2 8 6 7 5 9 ...

[Anonymous No.14838736]

>>14838609
What the fuck is differential geometry???
>inb4 muh manifold
I don't know what that is.

[Anonymous No.14838864]

>>14838527
Thanks for the help.
You can have some primes too.
934759478251288624231991757273892934807823592400873690544642066178158375069847522387188866611273091674571
934759478251288624231991757273892934807823592582396433947666526948865167580426980519940407816125106140211
17

[Anonymous No.14838880]

>>14838512
>>14838864
No one cares about your stupid fucking primes autist.

[Anonymous No.14838941]

>>14838880
I've solved them though.
I know where to find primes.
Don't even need to solve the Reimann Zeta function. Although I sort of get what the 1/2 means now and why 1 is undefined.

[Anonymous No.14838942]

I did stats and was OK at calc in high school. (20 years ago). What books can I read to get a good foundation of maths given my current level to build on

[Anonymous No.14838951]

>>14837273
fogoff

[Anonymous No.14838977]

>>14838942
You start reading an introductionary textbook like pic related and do all exercises.

[Anonymous No.14839009]

>>14837273>>14838951
My school math teacher used to say fog and gof lol. I didn't realise the o is an operator until I started doing Analysis.

[Anonymous No.14839052]

>>14838942
>[math] \text{Velleman, D.J. } ( \oldstyle{1994}). \textrm{ How to prove it.} [/math]
>[math] \text{Niven, I., Zuckerman, H.S., & Montgomery, H.L. } ( \oldstyle{1957}). \textrm{ An introduction to the theory of numbers.} [/math]
>[math] \text{Artin, M. } ( \oldstyle{1991}). \textrm{ Algebra.} [/math]
>[math] \text{Rao, R.A., & Bhimasankaram, P. } ( \oldstyle{2000}). \textrm{ Linear algebra.} [/math]
>[math] \text{Enderton, H.B. } ( \oldstyle{1977}). \textrm{ Elements of set theory.} [/math]
>[math] \text{Amann, H., & Escher, J. } ( \oldstyle{2005}). \textrm{ Analysis.} [/math]

This ought to give you a good base to start from. The first two chapters of the second book should be enough. You could skip most of the third as well, as long as you familiarise yourself with Groups, Rings and Fields. Whatever chapter you do, do each and every exercise, except for the last one. The last one is going to be much more brutal compared to the others. You can completely skip the second to last one, but I would not suggest it.

[Anonymous No.14839054]

Is the set of sequences in K a commutative ring?

[Anonymous No.14839062]

>>14839052
This list completely skips over probability and statistics though. I don't know much about the subject but I suppose you could try this:

>[math] \text{Blitzstein, J.K., & Hwang, J. } ( \oldstyle{2014)}. \textrm{ Introduction to Probability.} [/math]
>[math] \text{Casella, G., & Berger, R.L. ( \oldstyle{1990}). \textrm{ Statistical Inference.} [/math]

[Anonymous No.14839070]

>>14839054
Yes. You can view sequences as formal power series so it will be isomorph to K[[X]]

[Anonymous No.14839073]

>>14839052
This completely skips over probability and statistics though. I don't know much about it but you can try these:

>[math] \text{Blitzstein, J.K., & Hwang, J. } (\oldstyle{2014}). \textit{ Introduction to Probability.} [/math]
>[math] \text{Casella, G., & Berger, R.L. } ( \oldstyle{1990}). \textit{ Statistical Inference.} [/math]

Also, I forgot to fucking italicise the titles fuck.

[Anonymous No.14839113]

>>14839052>>14839073

I am gonna do it right by APA this time.
>[math] \text{Velleman, D.J. } ( \oldstyle{1994}). \textit{ How to prove it.} [/math]
>[math] \text{Niven, I., Zuckerman, H.S., & Montgomery, H.L. } ( \oldstyle{1957}). \textit{ An introduction to the theory of numbers.} [/math]
>[math] \text{Artin, M. } ( \oldstyle{1991}). \textit{ Algebra.} [/math]
>[math] \text{Rao, R.A., & Bhimasankaram, P. } ( \oldstyle{2000}). \textit{ Linear algebra.} [/math]
>[math] \text{Enderton, H.B. } ( \oldstyle{1977}). \textit{ Elements of set theory.} [/math]
>[math] \text{Amann, H., & Escher, J. } ( \oldstyle{2005}). \textit{ Analysis.} [/math]

[Anonymous No.14839184]

>>14839070
Is multiplication on the set of divergent series really commutative? I thought you could only multiply series that converge absolutely. Could you shed some light on the matter?

[Anonymous No.14839216]

can anyone give an example of such thing?

[Anonymous No.14839251]

>>14839216
[eqn] C_2 \triangleleft V_4 \triangleleft S_4 \\
C_2 \not \triangleleft S_4
[/eqn]

[Anonymous No.14839400]

Is the set of all sequences a field?

[Anonymous No.14839442]

>>14838675
C is algebraicly closed

[Anonymous No.14839470]

>>14839184
K[X] is not the same as finding the sum of a sequence. In this case each x^n is mostly a placeholder for the nth term of the sequence. So an infinite polynomial might converge for some x and diverge for some other x but it doesn’t matter because we’re not using x to substitute numbers

[Anonymous No.14839476]

>>14838609
>>14838736
Answer this question FAGGOTS

[Anonymous No.14839477]

>>14825203
I started struggling to get some concepts in abstract algebra and it's stressing the shit out of me. Ive never struggled to get something. Am I getting filtered? Is this it? Will I never be a mathematician?

[Anonymous No.14839479]

>>14839477
>>14838777

[Anonymous No.14839528]

>>14839400
Depends on how you define your operations.
You can equip literally any set with the field operations in a suitable way and get a field.

[Anonymous No.14839538]

>>14838675
>you don't have to define any more number ranges after C
He might mean that "you don't need to add any more numbers to C (like you did to R to get C) to solve polynomials".
You can extend C in any way you like, nothing is stopping you. But if you want a FIELD, and you want it to have solutions to all polynomials (with coefficients from it), then you can get even bigger fields than C to do that (and there are even smaller ones), but you also don't HAVE to.

[Anonymous No.14839559]

>filtered by complex analysis

[Anonymous No.14840086]

>>14838636
No. $N^2$ is countable.

[Anonymous No.14840149]

>>14840086
Read the question properly. The answer is yes.

[Anonymous No.14840175]

>>14840086
Why are you assuming N^2 is isomorphic to the set of all bijections from N to N? Midwit as fuck.

[Anonymous No.14840199]

>>14840175
>the set of all bijections from N to N?
is a subset of [math]N^2[/math]

[Anonymous No.14840208]

>>14840199
A single FUNCTION from N to N is a subset of N^2.
The set of ALL functions from N to N is much larger.

[Anonymous No.14840221]

don't people usually denote the space of bijections between infinite sets like [math]A^B[/math] because for each of the |A| elements in A there are |B| elements to send it to

[Anonymous No.14840237]

>>14839113
>do it right by APA
You'd still be missing the publisher names and edition numbers if you wanted to do it APA style.

[Anonymous No.14840251]

>>14840221
No. A^B is used to denote the set of all functions from B to A, not just bijections.
And the cardinality of this set is |A|^|B|.
So if we're talking about A = B = N, then the set of functions from N to N has cardinality |N|^|N|, much larger than |N|^2.

[Anonymous No.14840253]

>>14840221
B^A is the notation for functions from A to B.

If there exists any bijection, then in set theory that exactly means |A|=|B|.

[Anonymous No.14840266]

>>14828504
Here’s your guy actively associated with an academic institution >>14828749

[Anonymous No.14840285]

Analysis
Analysissy
Anal sissy

[Anonymous No.14840321]

>>14840208
You're right. I mixed a function from [math]\mathbb{N}[/math] to [math]\mathbb{N}[/math] up with the set of all functions.

[Anonymous No.14840386]

>>14838736
>What the fuck is differential geometry???
For niggas that peaked in calc III

[Anonymous No.14840430]

How do I proof the irrationality of 2^(1/2)?

[Anonymous No.14840481]

>>14840430
Be greek

[Anonymous No.14840493]

>>14840481
whoops, sorry. I meant 2^[2^(1/2)]. This is a more interesting question, I figure

[Anonymous No.14840540]

Any ideas for
>>14825684 ?

[Anonymous No.14840640]

>>14840493
2^2^sqrt(2) is the same as e^log(2^2^sqrt(2))

[Anonymous No.14840645]

>>14840640
Whoops where I wrote sqrt(2) I meant 1/2

[Anonymous No.14840810]

>>14830863
Lang is a meme. Read Rudin or die trying.

[Anonymous No.14840818]

>>14834948
A literature review is where you prove that you actually researched the questions in your paper to check that you weren’t working on already solved problems.

[Anonymous No.14840819]

>>14836397
Math PhD here, agree

[Anonymous No.14840821]

>>14825203
She looks like me.

[Anonymous No.14840822]

>>14836739
I did my PhD in analysis but I agree with this now. Analysis was attractive to me because it seemed to be the most frontier breaking field of math, in the sense that algebraists are just abstracting analytical results to their abstract fields or whatever. I still think this is true, but I also think combinatorics has more fresh ground. There’s a reason why you can take almost any combinatorics problem, slightly rephrase it, and have an entirely new research problem.

[Anonymous No.14840824]

>>14837085
This

[Anonymous No.14840826]

>>14837298
All of them. The Applied Analysis book discusses a lot of important topics in a way you can understand with pen and paper that today no book dares mention without giving you computer code.

[Anonymous No.14840887]

>>14840386
so NONE of you know what is differential geometry. You all are just a bunch of pseuds aren't you.

[Anonymous No.14840914]

>Applications of algebra:
>Yeah so here's how we'll use this stuff to answer previously asked questions
>Applications of analysis
>Here's how we'll use this to solve a problem we just created right now on the spot and useful nowhere else

[Anonymous No.14840917]

>>14837085
I'm contemplating hitting diff geo by selfstudying, what should I have to restudy prior to do it?

[Anonymous No.14841017]

Suppose I have a stochastic process where all of the elements have a stationary distribution yet with undefined mean, variance etc, can this process still be called a martingale or even a local martingale? Eg suppose each element has a Cauchy distribution where the location and scale parameters don't depend on time.

[Anonymous No.14841314]

WHAT THE FUCK IS DIFFERENTIAL GEOMETRY YOU FAGGOTS

[Anonymous No.14841352]

>>14841314
Classical Differential Geometry is study of curvature of curves and surfaces in R^3.
Then some people got the idea that it should also be possible to define curvature for objects in R^n and other spaces that behave similiar to R^n which are called Differentiable Manifolds. That is what the modern differentail geometry is about.

[Anonymous No.14841572]

>>14841352
Does it talk about trigonometry and angles, and how they are defined using ZFC?
inb4
>it's just series LMAO
Nope, that's an equivalence not a definition.

[Anonymous No.14841611]

>>14841314
Study of curvy stuff.
>how curvy is this curve
>how curvy is this surface
etc.

[Anonymous No.14841641]

>enjoying myself
>remember I have office hours in 15 minutes
>not enjoying myself

[Anonymous No.14841713]

I love to go to all the old book shops in my city and buy any old math text book I find and do all the questions in them.

Any other anons do this? any tips?

also what the fuck is curvature and homology? I got a book on it by luck but I cant find a solid connection from the books i own to it.

[Anonymous No.14841737]

>>14841713
>curvature
Self-explanatory

>homology
How can you tell a sphere and a torus (donut) apart?
Well obviously just by looking, one has a "hole" and the other doesn't.
What if you can't visualize the two objects? What if you're trying to prove that two 4th dimensional or higher objects are not the same? You can have two completely different definitions for the same objects and wouldn't know it, so how can you find out?
One way is like the above, using holes.
Homology is a way to detect holes in topological spaces and a way to tell apart spaces.
It was later abstracted and you can have homology groups of many other mathematical objects.

Here you go (change "simple" to "en" in the link if you want a deep dive): https://simple.wikipedia.org/wiki/Homology_(mathematics)

[Anonymous No.14841808]

>>14841737
this is really cool, thank u anon

just bought munkres topology hopefully ill be able to use this

[Anonymous No.14841812]

>>14841352
>>14841611
How is it different from Differential Topology. They all sound the same.

[Anonymous No.14841917]

Getting filtered by directional derivatives and gradient vectors. Send help

[Anonymous No.14841998]

>>14841917
The gradient is just a multivariate tangent line; finding the gradient vector is just finding the vector which describes this tangent ‘plane’ - that is, the vector orthogonal to the gradient. A directional derivative is simply a vector within this tangent plane that has a corresponding size to the taste of change in that direction

[Anonymous No.14842522]

The first gulp from the glass of group theory turns you into a discarder of most symbols used in math, but at the bottom of the glass, recreational category autism awaits.

[Anonymous No.14842545]

>>14841998
I just watched a professor leonard video explaining it visually and now its incredibly obvious. This has been the case for pretty much every class I've taken so far. If there were diagrams for every type of problems when constructing the proof, I could cut my study time in half. This isn't how the world works though, so I try to learn the way most books are written, which takes longer. Thanks though.

[Anonymous No.14842758]

>>14842522
(:

[Anonymous No.14842884]

>>14833006
How about Artin for algebra

[Anonymous No.14842981]

Can someone explain what [math] ∃x∈ [2,4] s.t. x < 7 [/math] means? I'm taking proofs for the first time

[Anonymous No.14843040]

>>14842981
It says "There's some number x between 2 and 4 (inclusive) that is less than 7".
In formal terms: "There exists x in the closed interval from 2 to 4 such that x is less than 7".

[Anonymous No.14843156]

New thread?

[Anonymous No.14843223]

>>14843156
Yeah, new thread time.

[Anonymous No.14843370]

>>14839528
That is false. There is no field with 6 elements.

[Anonymous No.14843376]

>>14843370
Meant to say infinite set.

[Anonymous No.14843489]

I consider a list of n+k vectors in R^n that i will call v_1 ... v_(n+k).
Suppose y is orthogonal to v_1 and has a positive dot product with all of the others v_i.
By making successive rotations, can i change y so that it is now orthogonal to (at least) n-1 of the v_i ?

[Anonymous No.14843619]

>>14843594
>>14843594
>>14843594