/sci/ - Science & Math

>>14843605
Listen, you don't need to change the image hash before posting it. I don't use 4chanx and don't filter image hashes.

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

>>14843635
no; v_1=(0,0,1); v_2=(1,2,3); v_3=(1,1,1); v_4=(3,2,1); y=(0,1,0)

>>14843669
wrong.
>n=3
so y only has to be orthogonal to v_1 and v_2. Not sure why Anon included up to v_(n+k)

I’ve been going through all the hartshorne exercises.
My progress has really slowed down now that I’m into the schemes part of the book.
I’m still really enjoying it though, it’s really cool stuff

>>14843782
Nice, I really wish I could learn algebraic geometry. Never took a course in it. Are you typing up your solutions?

>>14843984
Yeah I haven’t taken a course in it either. I’ve been typing up solutions and the doc is at around 130 pages right now.
Luckily a prof offered to do a directed reading program with me in algebraic geometry so i’m trying to get up to speed before we start

>>14843984
>typing up your solutions
Why the fuck would you do that?

>>14843635
yes and you don't need the extra k vectors

>>14843997
I'm not a genius and can't just scribble a few things for a problem and be like, yeah I get it. It helps me to write it all out.

Give it to me straight, bros. What's the best book to learn statistics for AI/ML? I only took Calc1 in college. I must soon learn Multivariable Calculus, Linear Algebra, and Statistics to get a company-sponsored graduate degree in ML. Any advice would be greatly appreciated.

I cannot fathom a hell darker than a world where there exists even a single sexually promiscuous woman who is better at math than me. This is the reason why I study for 12 hours a day. I shall never permit such an tragedy to occur.

>>14843782
hey want to work together on hartshorne? or start a study group? @narodism on twitter (I'd post email but dont want to get spammed)

>>14844001
I thought pencil bros were retarded enough already.

I've recently started self-studying math after not touching it since high school and I have a question. Are you supposed to do every single exercise in a textbook? I'm working through James Stewart's Calculus and looking at it right now, it seems like it will takes ages to do everything

>>14844114
>Are you supposed to do every single exercise in a textbook?
No you retard, do you think highschool teachers and professors assign every single exercise in the assigned textbook?
They assign a problem set and then you can work however much extra you need to make sure you got the concepts down.
Go find a syllabus from a university you wish you could go to and then steal the problem set.
That's what I did for Sheldon Axler's Precalculus, just stole a syllabus from Berkeley.
I'm also guilty of thinking you're supposed to do every exercise too.

>>14844093
The only pencilbro I've seen is that math sorcerer guy on youtube and even then I think he gets a pass for being a boomer.

>>14844129
How is it possible that someone with such atrocious handwriting manages to get a PhD in Mathematics in a top uni.

>>14844114
>do every single exercise in a textbook
If it is a respectable pure mathematics textbook, yes. James Stewart is not one of that.
If you find that all the exercises you are doing are very similar to each other, don't solve them all. If you can already think what the way to derive the solution would be in your head, skip it.

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

>>14844114
>The answers to the odd numbered questions are in the back of the book.
Guess which questions I do?

Reading about Russell's type theory:
>According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property.
Sure, fine, everybody knows what the paradox is, but what is a well-defined property/predicate? What makes it different from a non-well-defined property/predicate? Proof wiki doesn't offer an explanation of what well definition is in the context of set theory or logic, only in the contexts of mathematical objects like mappings and representations and I can't see how the definitions for those to be well-defined can be applied to logic.

>>14844258
The even ones since the solutions are usually only cover the most trivial problems?

>>14844273
>doing questions which you can't get immediate feedback to
Yeah that's real smart.

>>14844229
Where'd he get his PhD from?

>>14844274
>Exercise tells you to prove a statement
>You write a proof for it
>Look in the solution
>There is a different proof given
What does this tell you about your solution?
The answer is of course absolutely nothing. There is a good reason why higher level math books have no solutions for the exercises. They don't serve any purpose.

A solution could only be useful in exercises where you have to calculate something but then you might as well check your answer with Maple or Mathematica.

>>14844281
So pretty much the only reliable way to get feedback on proofs is visiting professors during office hours or just asking online and filtering through the feedback?

>>14844277
I don't know but he mentioned in his grad school video, he managed to get into top 50 or something.

>>14844016
Not gonna happen. My advice is to reconsider your career path.

Anything worse than long computation homework problems...

>>14844297
I took a differential geometry course and the professor gave us computationally-heavy problems (and he puts proof-based questions in the exams).
When I do homeworks, I usually do a sketch of the solution and computations, then rewrite properly into the pages I'm going to submit.
So on the first homework I didn't wanna rewrite the boring computations and thought to myself that this wasn't the point of the course anyways, as it's proof-based, so I wrote "after some long computations" and wrote the final answer.

He didn't like that and deducted marks for it :|

>>14843990
Try uploading your solutions somewhere online to help others

>>14843997
How young are you?

will be studying axiomatic set theory, proof theory, computability theory, and categorical quantum mechanics next academic year
odds on me going insane?

>>14844294
Why not, anon? The offer is not time limited. I thought of applying after I get the prereqs right.

>>14844364
20 and I know [math] \mathrm \LaTeX [/math] because I type my notes in it, but I don't understand why would you type entire solutions of exercises.

>>14844297
>uses a tool made for children to write
>has handwriting of children
Pottery.

>>14844255
If you are eating, sleeping, and exercising healthy, there's not much else you can do except for study. But you should study what is right for your level.

>>14844291
>>14844274
If you intend to do any original work in mathematics, you have to learn how to verify you work yourself.

What must I do(minimum) to call myself a Mathematician? A degree(graduate level? post-graduate level?) Publish a paper?

>>14844941
You should have sex first.

>>14844970
SEX HAS NOTHING TO DO WUTH MATHEMATICS!!

I have a question about Jordan-Holder for modules (if a module has a filtration 0<M_1<...<M_n such that M_i/M_i-1 are simple, then the length and the factors are unique). I know that if a module M has a finite composition series, then so do N and M/N for all submodules N. My question is whether the converse holds, i.e. if a submodule N and M/N have a finite composition series, then so does M. I think this should be true, just double-checking to see if my reasoning is correct: if M/N has a composition series, lift it to M and glue it to the composition series for N, this should be a composition series for M (the consecutive quotients should be simple, I think).

>>14844796
I use a mechanical pencil, pentel orenz nero 0.2mm

>>14844941
Easy mode is get a job with a mathematician title, and at least pass prelims.
Normal mode is publish.
Hard mode is get get a PhD.

>>14844362
I might do that after a lot of proof reading. Not a bad idea

>groomer thread

>>14844608
Typing the solutions is just as fast as writing them down on paper, probably faster honestly.

>>14844986
Yup, that's how you do it.

>>14844608
Typing up solutions means you can copy equations and adjust them appropriately so you don't have to retype the whole thing every time.
You can also do this >>14845065
It also means that it's legible for you and others, and you have access to symbols that might look like other symbols in handwriting, but are quite distinct in typing, or they're hard/tedious to write.

I really got stuck at exercise 2, and I'm not sure whether my solution is good enough, or if there's a more elegant solution to this.

First I proved that [math]F(P)=P'[/math], by saying I can define [math]F[/math] to be [math]F_{PP'}[/math], a translation where [math]F_{PP'}(P)=P'[/math]. So [math]F(P)=P'[/math] exists.

Still, [math]F_{PP'}(Q)[/math] is not necessarily equal to [math]Q'[/math], but since [math]d(P,Q)=d(P', Q')[/math] [math]Q'[/math] has to be a point on the circle that has center [math]P'[/math] and radius [math]d(P,Q)=d(P', Q')[/math].

We can then apply a rotation [math]G_{F_{PP'}(Q)Q'}[/math] relative to [math]P'[/math] and then if we compose the two isometries we get [math]H(x) = G_{F_{PP'}(Q)Q'} \circ F_{PP'}[/math].

Does this work? Feels very messy, but I'm not sure how to optimize this.

>>14845123
Yup, that works.

>>14845111
How is any of that relevant when you're just doing exercises in a book.

>>14845265
This is where the question "how young are you?" comes into play.
All of these are instances of "thinking ahead", which is a concept a 20-year is too young to understand.
It's also why I preemptively asked "how young are you?"; because I was thinking ahead to this moment.

You'll understand once you reach 23 (at minimum). That's when you actually become an adult.

>>14845032
>Easy mode is get a job with a mathematician title, and at least pass prelims.
Retard here, what do you mean?

>>14845265
If you’re 20 you’ve gone through 2 years of college right? Tell me all the math classes you’ve taken.
If you’re thinking of calculus, or some of the very first proof based courses like abstract algebra then I can see why you wouldn’t type solutions

parskip y/n?

>>14845351
>>14845351
There are jobs, usually working for the government, in which your job title will be "mathematician" and only require a 4 year math degree or a masters in math.
Prelims are exams yoh are required to pass to shoe mastery over your specialty or topic of interest. During a PhD, if you completed the required courses and preliminary exams, you can usually abandon the PhD and leave with a consolatory masters.

>>14845318
>>14845360
Not him, why don't you all just answer the question instead of acting like elitist condescending faggots? Absolute nigger behaviour.

>>14845461
No.

>>14845545
The question was already answered here
>>14845111
copying equations, and writing symbols that are hard to distinguish in handwriting are obviously relevant to doing exercises. But he still asks how it’s relevant

>>14845576
>handwriting
If you cannot write symbols, you are probably too retarded to study graduate mathematics. Please neck yourself.
I swear graduate schools should start having handwriting tests to filter the brainlets.

>>14845626
Now show us YOUR handwriting so we can laugh at you.

>>14845626
Regardless it’s still faster to type these symbols than to write them.

>>14845626
Here's Terry Tao's handwriting: https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/hw2-4.JPG
Math isn't something to gatekeep the way you gatekeep your dick from women.

>>14845549
Why? My main concern is that editors might be anal about it.

>>14845644
>"if we can show we would be done" instead of "if we can show we will be done" or "if we could show we would be done"
>evetually
Weird, that's a pretty high volume of mistakes while handwriting.

>>14845654
Follow your editor's template.

>>14845659
>evetually
based

>>14845576
>copy equations
If you cannot figure out a system to quote equations without having to rewrite them, that's another reason you're too retarded for graduate mathematics.

Personally, I never read books in math unless they were hand written.
If the author needs to type a book instead of hand writing each copy then he’s clearly too stupid to teach the subject.
Did I mention I’m very smart and also a massive faggot by the way?

>>14845688
>equates doing personal exercises with books meant to be sold to public
How the fuck did you get so far in mathematics despite being retarded? I thought you'd drop out while in school.

Did you watch any of the 3blue1brown's summer of math exposition this year?
I wanted to try some but there are just a fuck ton to choose (380 videos).
I just want to watch the good ones.

basic module question: if [math]A,B,C[/math] are submodules of [math]M[/math], [math]B\subset A[/math] and [math]A\cap C=0[/math], then [math](A+C)/(B+C)=A/B[/math], right? the map [math]a+B\mapsto a+(B+C)[/math] is well-defined, surjective, and is injective because of the condition [math]A\cap C=0[/math]

iOS 16 changed the MathJax font back to CM, nice.

>>14845742
Replace S with A, and T with B+C and this theorem implies what you’re saying.
Also your direct proof seems to work

>>14845680
>he doesn't know that you manipulate equations to to get results
Why are you talking about graduate math when you haven't even reached middle school?

>>14845714
Here you go (I'm more algebraically/combinatorically inclined):
https://www.youtube.com/watch?v=_DaTsI42Wvo
https://www.youtube.com/watch?v=3gyHKCDq1YA
https://www.youtube.com/watch?v=myKkhVy74V4
https://www.youtube.com/watch?v=ODfYA6nOKjk
https://www.youtube.com/watch?v=pLNZbykPDOA

I didn't see many good ones, but some of these have more to them than what's in the thumbnail, especially that 4th one about binomials of negatives.

>>14845669
This isn't realistic advice. Every journal has its own template. Only if your article is accepted, they might ask you to adjust your article's tex. Usually, that's the job of the journal. At least in math.

>>14845742
Intuitively, you're basically zeroing out everything in C, so C goes away, and zeroing out everything in B, but A still has its own elements that aren't in B, so that stays, and you're left with A/B.

i want to check my reasoning on 2 more things
1) if [math]M[/math] is a module, [math]0\subset M_1\subset\cdots\subset M_n=M[/math] is a composition series and [math]N\subset M[/math] is simple, then there exists a unique [math]i[/math] such that [math]N\cap M_{i-1}=0,N\cap M_i=0[/math] and [math]N\cong M_i/M_{i-1}[/math]. the first part is clear, because the intersections [math]N\cap M_j[/math] are submodules of [math]N[/math], so there is such an [math]i[/math]. for the isomorphism, since [math]N\cap M_i=N[/math] we get that [math]N\subset M_i[/math] and there is a non-zero map [math]N\to M_i/M_{i-1}[/math] (non-zero because [math]N\not\subset M_{i-1}[/math]), therefore it's an isomorphism because both sides are simple
2) with the submodules and [math]i[/math] as above, we have [math]M_{i+1}/(N+M_{i-1})\cong M_{i+1}/M_i[/math]. there is a non-zero map [math]M_{i+1}\to M_{i+1}/M_i[/math] and the kernel contains both [math]N[/math] and [math]M_{i-1}[/math], so there's a non-zero map between the two quotients. right side is simple, left side is too (by assumption from somewhere else in the proof), so the map is an isomorphism
>>14845812
thanks
>>14845882
i have no problem with the intuition, just double checking the reasoning

>>14845626
>handwriting sucks
>buy fountain pen and try to practice with it
>handwriting still sucks
>go back to pilot G2 ballpoint pen
>handwriting looks like LaTeX

>>14846115
it's cause [math]G_2[/math] is a cool Lie group

do we really need the AC to prove ACC=>every non-empty subset has a maximal element? isn't induction enough? pick [math]a_1\in S[/math], assuming there is no maximal element there must exist [math]a_1<a_2[/math], repeat with [math]a_2[/math]. this is inductive, not transfinite

>>14846297
In general if you need to make an infinite amount of choices, you need axiom of choice.
> a_1<a_2, repeat with a_2
In order to actually contradict ACC, we need a chain of infinite length, so we need to repeat this step an infinite number of times. In order to actually have a proof, it needs to have only a finite number of lines/deductions.
>this is inductive, not transfinite
What induction can show, is that for any natural number n, we can find a chain of length n, without a trans finite step, this is different from saying there exists a chain of infinite length, so we don’t contradict ACC

Math = <Faithful functor, Homological set> (post feynman)
Math = Study of numbers and operations (Gauss)
Which do you prefer? Graphs can be considered an operation on a cardinality/matrix btw

>>14846297
Let [math] A_i = \mathbb N \backslash [1,i] [/math]. Use the inductive hypothesis that: [math] \bigcap_{i=1}^n A_i \neq \emptyset [/math]. Therefore, by induction we can prove [math] \bigcap_{i \in \mathbb N} A_i \neq \emptyset [/math]. What is wrong with this proof?

Holy shit why am I so bad at basic arithmetic. It's like my mind's eye glazes over when I try to conceptualize anything relating fractions and decimals. Algebra I can handle, anything else I can handle, but somehow my basic number sense SUCKS

>>14846297
What's ACC?
Not countable choice I guess?

I get the analogy with multiplication but I don't get how I'm supposed to write non-commutative stuff in this system, like [math]GH[/math] is not the same as [math]HG[/math]. So for the table where I have [math]G[/math]'s row and [math]H[/math]'s column, which do I write first?

By the way H stands for reflection through the horizontal axis here

>>14846981
Consider a square around the origin with the corners labled 1,2,3,4 counter-clockwise with the bottom left corner being 1.

Then both any product of G's and H's can be thought of a permutation of the corners.

For example G maps 1 to 2 to 3 to 4 to 1. Or written in cycle notation
G = (1 2 3 4)
and H swaps 1 with 4 and 2 with 3 which in cycle notation is
H = (1 4)(2 3)


Now you can compute any product between them
GH = (1 2 3 4)(1 4)(2 3) = (2 4)
to calculate to permutations just look at the images of any element. Here 1 gets mapped to 1, 3 gets mapped to 3 and 2 and 4 get swapped.

G^3 = G^(-1) = (1 4 3 2)
H G^3 = (1 4 3 2)(1 4)(2 3) = (2 4)

The formulas
GH = HG^3
G^4 = 1
H^2 = 1
are actually already enough to write down products. Use the first formula to move all H's to the left and then use the others formulas to remove all extra G's and H's.

Small typo
H G^3 = (1 4)(2 3)(1 4 3 2) = (2 4)

>>14847024
>>14847030
Thanks, the cycle notation was not mentioned by the book here, so I learned something new.

I've been writing down each product in this matrix formula that the book introduced, and then if two matrices are the same, the two isometries are the same. The task still feels a bit tedious, but maybe with your explanation and some more grinding it will "click".

I just wasn't sure which way I should compose the two functions in the table; if G is a row and H is the column then should I take a look at GH or HG? I cheated and took a look at the solutions, and now I know that the book expects me to evaluate row:G column:H as GoH

What is your favorite calculator? Considering a DM42.

>resisting the urge to laugh every time the TNB frame is mentioned in lecture

>calc III
>STILL getting tripped up over systems of linear equations and finding variable values

How the fuck do I suck SO BAD at algebra 1 shit. Its bullshit, and I'm going to end up one of those retards with a math degrees that looks down on arithmetic because I can't do it.

>>14847236
?????

>latech what? sry prof sex havers like myself pronounce it like latecks.

>>14847109
my laptop

>>14845873
>>14845714
https://www.youtube.com/watch?v=ehmHoz-hIsg (long because it's introductory, so skip ahead)

Do you guys pretty much understand all of these math concepts from just the textbooks without any context from other outside sources?

What do you think about philosophy of mathematics?

>>14847408
>Do you guys pretty much understand all of these math concepts from just the textbooks without any context from other outside sources?
Basically, yeah.
My physics level stopped at undergrad Newtonian mechanics, if that's what you're asking.

>>14847408
sometimes sometimes not. Books generally skip over visual and intuitive explanations. So I sometimes look those up. Topology for example has some very unmotivated definitions.

>>14847420
Perhaps what I said was ambiguous whether I meant understand the concepts, or understand what concepts they're referring to

The real numbers are euclidean. A hyperbolic number line would not distribute the same way.

>>14847449
What do you mean by those two things? Elaborate

>>14847463
Schizo-babble
Like the time a schizo kept telling me to take the "inverse of a fractal"
It doesn't mean shit, they're just stringing words together

Already halfway through Calc1. How do you do fellow math bros? Man, it's fun doing advanced math, isn't it?

Okay so I'm in that weird part of calc one where I am being asked to prove a limit. I apologize by the way for being a dumb phoneposter who can't into LaTeX :(

f(x) = 2x-5
Prove that the limit as x approaches 4 = 3
My approach was to take |x - 4| < δ and δ = ε/2 then notice that 2δ is less than |f(x)-3|
Importantly I didn't take |f(x) - 3| then manipulate it to |x - 4| to show this. (But the other way around)
I talked to a professor and he said that it was a clever proof but I don't see why. The proof he wrote to show me the "normal" approach seems basically exactly the same so I don't know what the compliment has noticed.

>>14847273
https://en.wiktionary.org/wiki/TNB

>>14847696
>>14847236
An easy solution to this is to stop being racist.

>>14847724
Sorry, professor. I'll tender my resignation from Berkeley now

Am I retarded?
[math]\frac{\partial}{\partial t}\frac{\partial \varphi}{\partial \epsilon}=\frac{\partial}{\partial\epsilon}\frac{\partial \varphi}{\partial t}=\frac{\partial v}{\partial \epsilon}\implies \dot{\psi}(t)=g[/math]

>>14847673
>the other way around
The way you are doing it IS the right way. The implication is from [math] \delta[/math] to [math] \epsilon[/math], so you have to start from the former and [math] \textit{imply}[/math] the latter, to make a proper proof. However, in some cases you can start from the latter, provided you are making [math] \textit{if and only if }[/math] statements.
[eqn] \phantom{{} \iff{}}| f(x) - 3 | < \epsilon \\
\iff |2x - 8| < \epsilon \\
\iff |x-4| < \epsilon /2 [/eqn]
When you do more complicated proofs involving squares and general functions, you have to make sure if you are starting from the latter, there are arrows leading the former to the latter. Starting from the latter is generally advised to figure out what [math] \delta [/math] is appropriate.

>he said that it was a clever proof but I don't see why
Because it's not a clever proof, if anything it's a worse one, because it does not show how one would come up with that value of [math] \delta [/math] in the first place.

>>14847830
>provided you are making if and only ifif and only if statements.
Sorry, this is sufficient but not necessary. What's necessary is you make [math] \textit{if }[/math] statements, i.e., there should be an arrow leading back. You can start from wherever you like as long as the arrows follow the right direction.

I'm double majoring in German and maths. Any suggestions on topics for my thesis that involve both?

>>14847844
Gauss

How fucked am I if I'm beginning a maths degree in europe without knowing any math? I can barely recall what a logarithm is

>>14846966
Not him but I think it’s ascending chain condition from the way the contradiction works

>>14847844
Look at some of chomskys work. The idea is to describe a languages grammar formally with set theory. I think there’s a paper by chomsky doing it with english, but I don’t know of one for german.
Could be a challenge with how the word order changes

>>14847887
highschool math isn't so important and you can easily pick it back up again as you need it

>>14847779
It doesn't work like this.
[eqn]\dot \psi = \lim_{h \to 0} \frac{\psi (t+h) - \psi(t)}{h} = \lim_{h \to 0} \frac{\lim_{k \to 0} \frac{\varphi(t+h,k) - \varphi(t+h,0)}{k} - \lim_{k \to 0} \frac{\varphi(t,k) - \varphi(t,0)}{k}}{h} = \lim_{h \to 0} \lim_{k \to 0} \frac{1}{k} \left( \frac{\varphi(t+h,k) - \varphi(t,k)}{h} - \frac{\varphi(t+h,0) - \varphi(t,0)}{h} \right) = \lim_{k \to 0} \frac{v(t, \varphi(t,k),k) - v(t, \varphi(t,0),0)}{k}
[/eqn]
but
[eqn]g(t) = \lim_{k \to 0} \frac{v(t, \varphi(t,0),k) - v(t, \varphi(t,0),0)}{k} [/eqn]

What's a difficult but introductory book on Abstract Algebra? Should assume no prior knowledge on Abstract Algebra, but with difficult exercises.

>>14847887
You're gonna have a really hard time with small things that you should know from middle school I am >>14847268 and I went back for a math degree at age 28. I can do a lot of things, but there are a few very basic algebra concepts that destroy me every time. The less a course requires the use of content from before pre-calculus, the better I do. Courses that are structured towards heavy number cruncthing and computation just fuck me up. Other courses like abstract algebra and discrete math I excelled at. Even in Calculus II, my best grades were in series and sequences.
Get a copy of Lang's basic math and go over the weird concepts you have to learn, like linear equations and logarithms. If you can, take a precalculus course. I did and I barely got a B in it (82%)

>>14847916
There's Serge Lang infamous book. Technically it doesn't require prior knowledge and it's very difficult.
But really you're better off with something easy like Jacobson's Basic Algebra 1+2.

>>14847941
>technically doesn't require prior knowledge
>graduate course

Could I also go straight from calc 2 into papa rudin as I already know integral/differential calculus and complex numbers from middle school?

>Math course in university topic comes up
>everyone having a swell time
>Brit shows up
>"Oy! 'Mericans doing basic multiplication in univesrity! In Engerland we take topology in our second year innit"
>Brit presents coursework from cambridge, oxford, or warwick as an example. Compares it to University of Nebraska, gets mad if you bring up Princeton as US comparison instead

If you compare strathclyde, Swansea or Bath, they are the same as your average american school and nothing like cambridge. FUCK I hate brits and their faggot voices.

>>14847952
Rudin is an introductory real analysis book, you're supposed to go for it immediately after calculus.

>>14847830
Isn't the point that the two are equal, and if it's equal, it's not arrows, it's identities?

>>14847109
The HP15C. Original, not a remake. The new LCD screens aren't as nice.

>>14848012
No one is taking real and complex analysis by rudin immediately after a calculus series. You first take an undergraduate analysis course and undergraduate complex variables course.

>>14847844
Why the future looks bleak

>>14847941
>should assume no prior knowledge
All of those are graduate books.
>>14847952
Stop larping as me, also no.

How come there aren't any meme'd stat/probability books? Is it because it's not pure math?

Any recs for a data analyst guy? One of my seniors asked me a stat related question and while I could solve it with a little googling I wanna be prepared for the next time something like this happens (and it's also a topic I'm curious about).

>>14848052
Graduate book doesn’t mean it’s hard.
Plenty of undergrads take graduate level analysis or algebra courses in their third or fourth years

>>14848026
Oh right, he said "papa" Rudin.
My bad, my brain filters those nicknames out because of how absurdly silly they are.

>extreme value problem system of linear equations with five variable

FUCK YOU FUCK YOU THIS IS JUST A FUCKING TIME SINK FUUKCCCC

>>14848137
Use Karmarkar's algorithm it's much better than the Simplex shit that many intro courses shill.

What the fuck is a lattice

Anons doing the meme book path, how did you work through Lang? I'm a couple chapters in on basic mathematics and I think I'm making good progress on it but don't know how to really self-assess. Is there a good way to measure understanding in self-study/ensure good pacing?

>>14847887
I started one three weeks ago.
Let's say you have to do a proof where you need to do lots of stuff with fractions, and you can't quite remember how to multiply fractions with each other.
If your immediate instinct is to look up "how to multiply fractions" on google or youtube until you understand it, and then do the problem, you'll do fine.
If your immediate instinct is to give up, or just do the proof wrongly without attempting to relearn the basics, you'll do awful.

[math] {\mathcal P} ( {\mathbb N} ) \approx \omega_1 [/math]

Let M be a matrix such that :
1) The equation MX=0 has a solution X >0.
2) For any i, if i delete the i-th column of M (let's call N_i the resulting matrix) the equation N_i X = 0 does NOT have a solution X>0.
3)There exists a vector v such that for any i, if i add v as a new column to N_i (let's call M_i the resulting matrix) then the equation M_i X = 0 has a solution X>0.

Does such a matrix exist ?

(X>0 means "every coordinate of X is > 0").

>>14848619
M =
[1, -1]
[1, -1]

X =
[1]
[1]

>student comes to me after class
>says he's from another section attending with this one
>even though that's against department policy
>says he wants me to mark him present in his section
So basically he invited himself to this one, did not inform me at all before coming, and now wants me to do whatever he wants.

Fucking hate teenagers.

>>14848703
This matrix does not respect the third condition.
If i delete the second column and add the vector you suggest i get
M_2=
[1,1]
[1,1]
and there's no positive solution to M_2 X=0

>>14848712
Inform him that you can't mark him as present if he doesn't show up on the list. Don't bother yourself to do anything else about it. If he wants to miss class by attending a class he's not signed up for, he should be marked as absent and face the consequences for it himself.

>>14848723
>>14848619
>>14848703
Also i forgot to say that but i'm looking for a matrix of full row rank.

>>14848727
That's what I did.
And I ignored him by doing other stuff so I don't get into some kind of argument with him.
Department policy, end of story.

>>14848723
>add the vector you suggest
I did not suggest any vector, what are you on about?

>>14848729
[1,0,-1]
[0,1,-1]

[1]
[1]
[1]

>>14848759
Well then what vector do you suggest ?
Ok sure the matrix checks the first two conditions but you still have to be give me a vector that will satisfy the third condition.

>>14848712
Damn, you must work at a big state school.
I attended a small college for my undergrad, and the department knew me by name and allowed me to go to any lecture I wanted. There also wasn't an attendance requirement, just exams.

>>14848619 (me)
>>14848729 (me)
If someone finds a solution that is not of full row rank it looks quite easy to deduce a full row rank solution (just delete some rows...) so i guess that does not really matter.


>>14848761
Ok so there is a positive solution to the equation MX=0, if i remove (any) column there is no more positive solution, so far so good.
But you also need to be able to find a vector v such that it can "replace" the column you decided to delete so that there is a positive solution to M_i X =0


(just to clarify : the positive solution to "MX=0" does NOT need to be the same as the positive solution to "M_1X=0" for instance)

>>14848770
>department knew me by name
So does mine, seeing as how I'm always up in their faces whenever they do shitty things, AND due to me getting 4.0 in every course I took so far.
>allowed me to go to any lecture I wanted
In grad and upper-level undergrad, you can audit courses here. But for stuff like calc, there's just too many students to handle. I teach like around 170-180 students (in different sections).
>state
Not everyone lives in the US dude

>>14848786
>In grad and upper-level undergrad, you can audit courses here. But for stuff like calc, there's just too many students to handle. I teach like around 170-180 students (in different sections).
Holy cow. My Calc II course was 30 students. Only 8 math majors in my year (2014)

>upper level online course with uncaring professor.
>no lectures, everything is on webassign
Fuck I hate these people. Its rutgers btw, dogshit university.

I am too stupid to learn how to use a graphing calculator for anything other than basic arithmetic. Its so damn time consuming to work out every problem on paper to find critical points and solve integrals. No idea what CAS is, but my calculator supposedly has it and it's supposed to work wonders.

>>14848921
RTFM

>>14848921
Nice blog, dude. How do I remove it from my RSS feed?

Found out stewart calculus 1st edition was made in the early 2000s.
For you more knowledgeable or older anons, what have been the standard calculus textbooks throughout the last 100 years?

>>14847952
No.
But Lang's Algebra serves as a good reference. It's good as a companion book. My undergraduate courses used Dummit & Foote but I found the exercises lacking sometimes. Lang has great exercises. I did a lot of them because Dummit & Foote would leave me feeling like I only half grasped the subject despite its wordiness. So it's worth picking up just for the exercises.

Basado
https://youtu.be/HWAnsUSK2yQ

>Mathematical Calculi Consist Exclusively of Intensions and Extensions: Given that we have invented only mathematical extensions (e.g., symbols, finite sets, finite sequences, propositions, axioms) and mathematical intensions (e.g., rules of inference and transformation, irrational numbers as rules), these extensions and intensions, and the calculi in which they reside, constitute the entirety of mathematics.

>Given that a mathematical extension is a symbol (‘sign’) or a finite concatenation of symbols extended in space, there is a categorical difference between mathematical intensions and (finite) mathematical extensions, from which it follows that “the mathematical infinite” resides only in recursive rules (i.e., intensions). An infinite mathematical extension (i.e., a completed, infinite mathematical extension) is a contradiction-in-terms

>Given that the mathematical infinite can only be a recursive rule, and given that a mathematical proposition must have sense, it follows that there cannot be an infinite mathematical proposition (i.e., an infinite logical product or an infinite logical sum).

>Since there are no infinite mathematical extensions, irrational numbers are rules, not extensions. Given that an infinite set is a recursive rule (or an induction) and no such rule can generate all of the things mathematicians call (or want to call) “real numbers”, it follows that there is no set of ‘all’ the real numbers and no such thing as the mathematical continuum.

>Given the non-existence of infinite mathematical extensions, Wittgenstein rejects the standard interpretation of Cantor’s diagonal proof as a proof of infinite sets of greater and lesser cardinalities.

>>14848954
>"DUDE ITS ALL COMPACT"
The absolute state of finitists

http://www.brooklyn.cuny.edu/web/academics/schools/naturalsciences/undergraduate/math/majors_details.php?major=074&div=U&dept_code=60&dept_id=90&mode=data

>mfw too late to switch to a regular mathematics degree
>diploma will always say computational mathematics

Its over...

>>14848939
Before Stewart? Thomas' Calculus I think. Before that I'm not sure. I think Courant's Differential and Integral Calculus was more for 'advanced' students at the time. I really like it though.

>>14848980
probably better for industry, and I doubt grad schools care that much

>>14848331
do every exercise

>>14849103
Already doing that. For some reason it doesn't feel like I'm getting all of it. Maybe I just need to take it a bit slower and check my understanding every few paragraphs.

>>14849134
I'm pretty sure Basic Mathematics covers most of Precalc so you could just grab practice/actual exams from actual universities that are posted online and see how you do.

TIL that m is BEFORE n in the alphabet

>>14849239
It does not. It mostly covers the range of situational algebra 1 content that is easy to forget. Chapter 2 is literally just linear equations in various unknowns.
If you legitimately need an actual precalculus review, get the stewart book, its massive and thorough.

>>14849461
>ch, nn, rr, no longer in the alphabet as letters.....

is topology just "continious geometry"?

Where the fuck am I supposed to start with this question: Compute the generating function E(t^Y) t>= 0, and Y = X + Z, where X and Z are random poisson variables

>>14848954
HOLY BASED

>>14849461
At least difeq does M dx + N dy

>>14849844
[eqn]
E(t^Y) = E(t^{X+Z}) = E(t^X t^Z) = E(t^X) E(t^Z)
[/eqn]
Assuming that X and Z are independent. Otherwise you can't split the product like this.


[eqn]E(t^X) = \int_{\Omega} t^X dP = \sum_{k=0}^\infty t^k \frac{\lambda^k e^{-\lambda}}{k!} = e^{-\lambda} \sum_{k=0}^\infty \frac{(\lambda t)^k}{k!} = e^{- \lambda + \lambda t} [/eqn]
So
[eqn]E(t^Y) = e^{(\lambda_1 + \lambda_2) (t - 1) } [/eqn]
where [math]\lambda_1, \lambda_2[/math] are the parameters of X and Z.

>Go thought extreme mental anguish daily
>endorphin rush when I see the capital letter A or number greater than 90 on assignment
Still feel like an idiot.

>>14849593
Geometry can already deal with continuous stuff.
A better (and more common) name is "rubber sheet geometry"

>professor says "not you" to me multiple times per class when asking questions

>>14849949
Meaning he asks the class questions, and rejects you when you attempt?

>>14849946
yeah but I mean that in the sense that differential geometry got its name from

Losing my mind over how to prove that pic related is true when x is any real number (not necessarily > -1) and r is an even number. Can't find any proofs online.

Every time I think I have a solution, I sit for 20 minutes before realizing I made a wrong assumption.

>>14850015
>"Can't find any proofs online"
>pic has name

>>14849957
Yeah
He knows I already know the answer lol

>>14850015
Bro just let x = t-1 then show
t^r >= 1 + r(t-1)
This is equivalent to showing that
t^r - rt + r - 1 >= 0
Differentiate wrt t to find turning points at
r t^(r-1) - r = 0
ie, t = 1 because r is even
t^r - rt + r - 1 evaluates to 0 at t = 1. Since t=0 gives r - 1 > 0 and t = 2 gives 2^r - r - 1 > 0 for even r > 0 you are done because both sides of the root are positive.

>>14850031
it's easy to find proofs that assume x>-1. But not with the assumptions I listed.

>>14850015
Just use the binomial theorem?

>>14850040
if r is even isnt it trivial for x <= -1 since (1+x)^r is positive and 1 + rx <= 1 - 2|x| < 0
and like you said x > -1 is a common proof for all r

>>14850015
Let
f(x) = (1+x)^r - 1 - rx
We have to prove f(x) >= 0

For r = 0
f(x) = 0 >=0
Now for r unequal to 0 just differentiate
f'(x) = (1 + x)^(r-1) - 1
which is only 0 for x = -1.
It's a minimum since the second derivative in 0
f''(0) = r (r - 1) >= 0
(Either r>=2 or r<=2. In both cases it's positive)

So the unique minimum is in 0 and f(0)=0 therefore f(x) >= 0.

>>14848921
You're better off doing just that. Fancy calculator operations are extremely cumbersome to the point where its faster to just do it by hand.

https://www.youtube.com/watch?v=VyQpylN0FiE&ab_channel=ScottCollins

Look at this fucking mess. Dude took like 3 minutes to solve easy problems.

>>14850075
>f'(x) = (1 + x)^(r-1) - 1
f'(x) = r ((1 + x)^(r-1) - 1)

>>14850069
I just came to the same realisation.
Yeah, if we show it for x>=-1 the usual way with an induction proof, we just have to show it for x<-1 now.

(x-1)^r is obviously positive when r is even.
1+rx is negative since r>=2 and x <-1. Thus pic related is true

>calc 3 is way easier than calc 2 which is way easier than calc 1
what the H

>>14849895
Somehow, it seems so simple once you see it done.

Thanks a lot, any idea on how I can review questions like this? Not too sure what too search for.

Hey, not usually browsing /sci/ but I feel like this might be a good place to ask.

I am trying to figure out a way to uniquely identify each possible one billion digit integer using the least digits possible in base 10 (or another human-compatible base).

My intuition tells me that using an n-dimensional space with each element of the space containing one integer would be efficient.

Any clues? Are there any resources on "condensing" integers like this?

>>14850221
>I am trying to figure out a way to uniquely identify each possible one billion digit integer using the least digits possible in base 10
You need to use numbers with a billion digits. There is literally no better way (in base 10).
You have 9 options for the leading digit, then 10 options for each digit after that. That's 9*10^(billion - 1), or roughly 10^billion. (unless you meant UP TO a billion digits, then it's just straight up 10^billion)

What's the relationship between F[[x]] and the localization of F[x] and (x)?
Is there an inclusion/isomorphism?

>>14850163
And tensor calculus on manifolds is even easier

>>14850347
localization of F[x] AT (x)

Do there exist infinite non-isomorphic fields?

Newfag here, How I know that I'm learning from the right source?

>>14850497
just follow one of the many very helpful picture guides :)

Victor Isai Mazariegos Harvard University 9-14-2022
Hilbert’s 1st Problem The Continuum Hypothesis Theorem
The computability of natural to transcendental numbers relies on the existence of the aleph-null, which is the rational difference between a change in y over a change in x per at least XY.
Relation: computable number, aleph-null, and natural to transcendental numbers Function: cos x/ ln x
Key Ratio: 6
Proof
The transpose of the similitude of numbers over the plotting of coordinates over an enumeration becomes easily deceptive at the constant affix of any irrational number. Therefore all irrational numbers are not included in the foundation of the computable numbers. Moreover if the foundation of computable numbers is strictly rational then the constitution of computable numbers is in algebraic NP-complete form for entropy-based and those not of entropy. For example the magnetic turn of the sun is “entropic” but does not rely on a completely finished continuum of “outer-space” and other types of “space.” By this way aleph, which is meant to be smallest number is thereby Z rational numbers. Null in part is therefore the real numbers and anything else the Transcendental numbers based on the obstinate fact that if rational numbers take the sum of any quadrant then the “transcendence” is at outer-spaces, not inner “mazes.” If we return to calculus here the straightest curvoid as in curve and constant cos x over natural logarithm of x contrast this logic in the symbol key ratio 6 by depicting, being, and telling quite straightforwardly the truth that matter that undulates over matter that grows fast will always be the start of the next glove, clover, or skyline.

>>14850486
Yeah, Z/pZ.

>>14850513
On the scale of cringy spam bot attempts, this gets a 2/10.

>>14850486
R,C,Q,F_p(X),Q(X,Y)...
And infinitely many more.

>>14850213
>Somehow, it seems so simple once you see it done.
that's because most of the time it is, there isn't much to it. all you need are two basic measure-theoretic lemmas
1) if [math]\mu[/math] is a measure on [math]\Omega[/math], [math]f:\Omega\to\Sigma[/math] is measurable, [math]\mu_f[/math] is the pushforward (i.e. [math]\mu_f(A)=\mu(f^{-1}(A))[/math]) and [math]X:\Sigma\to\mathbb R[/math] is measurable, then [math]\int_\Sigma X\,d\mu_f=\int_\Omega f(X)\,d\mu[/math]
2) if [math]\nu[/math] has density [math]f[/math] w.r.t. [math]\mu[/math], i.e. [math]\nu(A)=\int_Af\,d\mu[/math], then [math]\int_\Omega X\,d\nu=\int_\Omega Xf\,d\mu[/math]
here's the application of both to your question: to say [math]X[/math] is Poisson-distributed is to say [math]P_X[/math] has density [math]f(k)=e^{-\lambda}\frac{\lambda^k}{k!}[/math] w.r.t. the counting measure [math]\zeta[/math]. then [math]E[t^X]=\int_\Omega t^X\,dP\stackrel{1)}{=}\int_{\mathbb Z}t^k\,dP_X\stackrel{2)}{=}\int_{\mathbb Z}t^kf(k)\,d\zeta=\sum_{k=0}^{\infty}t^kf(k)[/math] and the rest anon above has already explained (last equality holds because the integral w.r.t. the counting measure is the series of the function's values). this approach is maybe a bit high-brow, but i believe it's good to see it once, because it then becomes completely engrained in memory and comes naturally
two more probabilistic results that are often important are that if [math]X,Y[/math] are independent, then
1) [math]E[XY]=E[X]E[Y][/math]
2) their joint distribution [math]P_{(X,Y)}[/math] is the product of [math]P_X,P_Y[/math] (i assume you know what product measures are)

>>14850501
Where?

>>14850630
there you go
happy hunting :^)

>>14850262
Thanks
That's too bad.

>>14850619
To be honest anon, I think I might've forgotten the measure-theoretic lemmas or never went over them in my probability theory class. Could you elaborate a bit more on it? I'm a bit confused on why we integrate to find the expectation since both x, and z are random variables.

>>14850860
in the subsequent i assume you know the basics of measure theory, e.g. measure space, measurable function, integral w.r.t. a measure aka the Lebesgue integral. otherwise you have no business studying probability theory or at the very least won't get far doing it
the setting of probability theory is a measure space [math](\Omega,\mathcal{A},P)[/math] such that [math]P(\Omega)=1[/math]. a random variable [math]X[/math] is just a measurable function on this space (most of the time with values in [math]\mathbb R[/math], since those are the ones you can integrate). the expectation [math]E[X][/math] of a real-valued random variable is just its integral [math]\int_\Omega X\,dP[/math]. when we say a variable has a certain distribution we're saying that we know the measure [math]P_X[/math] (defined by [math]P_X(A)=P(X\in A)[/math])
>I'm a bit confused on why we integrate to find the expectation since both x, and z are random variables.
we integrate because the expectation of a variable is *by definition* a certain integral. if you're tasked with calculating [math]E[t^X][/math] as above, by definition this is [math]\int_\Omega t^X\,dP[/math]. now this doesn't get you anywhere because you don't have an explicit description of [math]\Omega[/math] or [math]X[/math], you don't actually know what values [math]X[/math] takes on what inputs (so it's not like calculating [math]\int_0^1x^2\,dx[/math]), but what you have is a description of the measure [math]P_X[/math] (it's a Poisson distribution and we know how that behaves) and you have the 2 lemmas that are basic calculation rules. so you apply all this to the expression above and transform it into [math]\int_{\mathbb Z}t^kf(k)\,d\zeta[/math]. this last integral is more akin to something like [math]\int_0^1x^2\,dx[/math] in that you can calculate it directly, which we do
cont'd

>>14850527
How so? Someone else comment

Linear algebra question:

Suppose there's a 5x5 matrix A, with column vectors a1, ... , a5, and a3 = pi(a1) - 7(a2) + 1/4(a5).

Does the homogenous system Ax = 0 have non-trivial solutions? (and why)

If the a3 = pi(a1) ... part were instead x3 = pi(x1) ..., then I would understand, but I don't see how there being a column vector that's the sum of other column vectors helps me here.

>>14850906
>>14850860
cont'd
>Could you elaborate a bit more on it?
what exactly do you wish me to elaborate on? if you know basic measure theory these are elementary integration results the proofs of which you can find in any book or work out for yourself with the monotone convergence theorem and indicator functions
if you don't know any measure theory then i'm afraid all of what i've said in the post above will likely be gibberish to you and elaborating on the lemmas won't be of help, so allow me to try something else. if what i've mentioned above hasn't come up in class, i assume the expectation of a random variable was defined in 2 cases, discrete and continuous. in the discrete case it might've been defined as [math]\sum_{n=0}^\infty nP(X=n)[/math] and in the continuous as [math]\int_\mathbb{R}xf(x)\,dx[/math] where [math]f[/math] is the density of the variable. if so, you might also have learned a calculation rule like [math]E[h(X)]=\sum_{n=0}^\infty h(n)P(X=n)[/math] for the discrete case and [math]E[h(X)]=\int_\mathbb{R}h(x)f(x)\,dx[/math]. it turns out that this definition of the expectation is ad hoc and there is a a general definition that uses the Lebesgue integral, namely [math]E[X]=\int X\,dP[/math]. with this, the preliminary definitions i gave for the expectation are just the 2nd lemma, while the calculation rule i gave above is the 1st lemma. in any case, the two lemmas are very useful calculation rules because 99% of the time there is no concrete description of [math]\Omega,X[/math] or [math]P[/math], so you don't really have a way of calculating something like [math]\int_\Omega X\,dP[/math] directly. the lemmas reduce this expression to an integral w.r.p. to some *known* measure, which simplifies things immensely

>>14850931
>If the a3 = pi(a1) ... part were instead x3 = pi(x1) ...
You're gonna have to explain how "a3" and "x3" are distinct, because we don't know what this notation is. Is "x3" the third row and "a3" the third column?

>>14850962
x3 would be the third element of the solution column vector x for the system Ax = b. a3 I assume is just the column vector of the matrix A from what was the first part of the problem, but I said that x3 would work better because I believe that would imply there are free variables and that the system has infinitely many solutions.

>>14850977
Ok cool.
The system DOES have non-trivial solutions, since a3 is a linear combination of the other vectors.
This means that the rank of the matrix is at most 4, so the nullity is at least 1, giving you a non-trivial solution.

>>14850982
We haven't gone over rank or nullity, at least yet, in my class so I don't think I could cite those; is there any alternative explanation? The part b to this is whether matrix A is invertible - so it would not be invertible because A is invertible if and only if Ax = 0 has a only a trivial solution. (Thanks by the way)

>>14850982
Hmm, is rank just the number of free variables? That's the terminology we use. And nullity the number of fixed variables? My question is that how can a column vector being a combination of other column vectors imply that the system has non-trivial solutions. I would understand if an solution vector element (x3) was a combination of other solution vector elements plus a constant (so x3 would be a fixed variable), but how/why does that extend to column vectors?

I'm almost finished with my math undergrad and I still sometimes count using my fingers. Is this common or am I a retard

>>14850992
>>14850995
Explicitly, you can use this column vector as a solution:
[-pi, 7, 1, 0, -1/4]^T

You can easily see it from the equation a3 = pi(a1) - 7(a2) + 1/4(a5). Put everything on one side to get pi(a1) - 7(a2) - (a3) + (1/4)(a5) = 0
So I just took the coefficients and used them (their negatives, but same thing really).

Ax = -pi(a1) + 7(a2) + (a3) + (0)(a4) + (-1/4)(a5) = -pi(a1) + 7(a2) + pi(a1) - 7(a2) + (1/4)(a5) + (-1/4)(a5) = 0, as required.

>>14850945
Thank you elaborating, it makes more sense. Haven't covered these topics in my probability course so I'll start there. This course isn't actually a probability course, but a finance course, so I guess they expect some knowledge I don't have atm.

>>14851025
Ah, that's cool! OK, so that directly gives a nontrivial solution that's a sum of column vectors to the system then. Very cool. Thanks again, you're a saint!

>be fancy $200 calculator
>can solve literally all of pic related by just plugging in variables
>ask it to integrate sin(x)^3
>explodes

Even wolfram and symbolab trips over itself with stuff like tan(2x)^10 sin(x)^3

What the fuck is going on? Is it too much to ask to have a decent calculator?

>>14851149
Some functions simply have no elementary integral.

>>14851136
No problem
Check out 3Blue1Brown's Essence of Linear Algebra series to get a better intuition

>>14850660
Hmmm...Last time I posted this chart, /sci/ said it was shit.

>>14851311
It still is. It’s a joke chart, no one would read 13 books on “foundations”

>>14851311
>>14850660
>>14850722
>>14851343
>[math] \text{Velleman, D.J. } ( \oldstyle{1994}). \textit{ How to prove it.} [/math]
Do everything.
>[math] \text{Montgomery, H.L., Niven, I., & Zuckerman, H.S.} ( \oldstyle{1957}). \textit{ An introduction to the theory of numbers.} [/math] AND [math] \text{Long, T.C. } ( \oldstyle{1965}). \textit{ Elementary introduction to number theory.} [/math]
Will help with writing proofs. Solve everything upto congruence. Then skip unless you find Number Theory interesting.
>[math] \text{Artin, M. } ( \oldstyle{1991}). \textit{ Algebra.} [/math]
Will help getting used to abstraction. Do as much as you can.
>[math] \text{Friedberg, S.H., Insel, A. J., & Spence, L.E. } ( \oldstyle{1979}). \textit{ Linear algebra.} [/math]
You should do the whole thing, will prepare you for multivariable analysis. Watch 3blue1brown's series on Linear Algebra first.
>[math] \text{Enderton, H.B. } ( \oldstyle{1977}). \textit{ Elements of set theory.} [/math]
Gives context to Analysis and most of modern mathematics.
>[math] \text{Steele, J. M. } ( \oldstyle{2004}). \textit{ Cauchy-Schawrz master class.} [/math]
Prerequisite for Calculus/Analysis.
>[math] \text{Amann, H., & Escher, J. } ( \oldstyle{2005}). \textit{ Analysis.} [/math]
Watch 3blue1brown's series on Calculus first.
>[math] \text{Abbott, S. } ( \oldstyle{2000}). \textit{ Understanding analysis.}[/math]
Do this if the former is too difficult, or if you want to get to the meat of Analysis faster.

Working through Velleman's How to Prove It 3rd ed

Let P stand for the statement "I will buy the pants"
Let S stand for the statement "I will buy the shirt"
What English sentences are represented by the following formulas?
[math]\neg (P \wedge \neg S )[/math]

Is it correct for me to say
"I won't both buy the pants and not buy the shirt."

>>14851487
Yup.
But a more natural way to say it is "if I buy the pants, then I will buy the shirt". (Think it through and check the possibilities to verify it)

>>14843594
I have a question for math majors, graduates, and anons working in academia. Did you date any women who were pursuing the same degrees/ jobs? If so, how were they? I really want a pic rel gf bros. tfw no makise kurisu gf.

>>14851628
It's possible but don't think they are gonna be any more attracted to you because you are also in academia. Actually, they are gonna be less attracted to you if you are in a mathematical science, because women there are probably gonna be rare, and so get hit on by everyone around them. This generally leads them to have prejudice against those in the same field. Moreover, everyone knows academics don't make money, so family oriented women would gravitate more towards men working in the industry.

>>14851628
My brother in law double majored in CS/math, and his wife was in one of those programs with him. He is also a bishounen looking guy, and that is what his wife is into as I've seen some of the drawings.
Im short, if you want an attractive woman with similar interests, just be a male model.

>>14851628
I "dated" (fucked) interns and other students in my vicinity.

>>14851124
if you're "mathematically mature", Schilling's "Measures, Integrals & Martingales" is a great intro to measure theory and integration that assumes no background beyond real analysis

>>14850906
Not sure if you can sue subsequent as a noun

>>14851124
I really doubt you need measure theory for anything in your finance course, or the question you asked.

>>14851837
Stop trolling. Something like Casella & Berger is a much better fit for the questions he's asking.

I was studying some lecture notes on ring theory, and just before the section on Goldie’s theorem they mentioned that “Artin-Wedderburn deals with semiprime + DCC, we are now going to look at what happens with semiprime + ACC”. I had a think about this last night and I had a revelation:

If a ring is semiprime, then it is right Artinian iff it is left Artinian, so the concept of semisimple Artinian and the Artin-Wedderburn theorem are left-right symmetric. However, even in a semiprime ring, right Noetherian does not imply left Noetherian and vice versa. This is the reason why quotient rings exist separately on each side, and why Ore’s and Goldie’s theorems have to be formulated separately on each side.

I had never realised that Goldie’s theorem was essentially just the “semisimple Noetherian” equivalent of Artin-Wedderburn. It’s funny how you have these moments lying in bed a year later, and not when you’re actually studying the material.

>>14851858
not trying to troll him, dumbass. i'm not saying he NEEDS to know this, but if he wants to, Schilling's a great resource

>>14851862
any ring theory resources you could recommend? i've been meaning to learn the basics (radical, Artin-Wedderburn, central simple algebras, Brauer group, etc.)

>>14851863
Good resource for stuff he didn't ask for maybe.

>>14848619
anyone ?

>>14851409
Thanks.

>>14851858
Is there a way to come up with a solution without using measure theory lemmas like the person who answered my question did?

Would Casella and Berger help me better understand these questions? I looked through the table of contents and most of the content that's relevant to me I feel like I've already learned.

>>14852071
I don't remember Statistics much, and I am yet to learn measure, but I am fairly certain I would be able to answer your question if I could recollect my knowledge from undergrad. Casella & Berger would help you I am certain (if you do the exercises). If I recall correctly, there's only like one theorem in Casella & Berger whose continuous case requires measure theory for proof, and they skip it since its out of scope. There is a VERY cursory treatment of measure theory in the first chapter, if you could even call it a treatment. Chapter 2 is related to your question.

>>14852071
>>14852083
However martingale, which Casella & Berger does not talk about, but the other book does, is quite important for Finance and uses Measure, but I doubt your course requires it; it may have it as an elective.

>>14852071
Everything in this >>14849895 solution is straight from the definitions of poison distribution, expectation and independence.

>>14852071
>Is there a way to come up with a solution without using measure theory lemmas like the person who answered my question did?
I'm the person who answered with measure-theoretic stuff, sorry if that confused you. yes, you can make do perfectly well without it. X is a discrete variable, its expectation is defined as \sum_{k\geq0}kP(X=k). since X is discrete, so is t^X and using the rule i mentioned above, i.e. E[h(X)]=\sum_{k\geq0}h(k)P(X=k) (this is proved in all probability courses, even ones that don't use measure theory) you get E[t^X]=\sum_{k\geq0}t^kP(X=k). since X is Poisson, by definition this means P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}

>>14852123
>>14852098
ahhh Okay, thank you for clarifying, I was getting really nervous about this lol.

>start watching 3b1b "essence of calculus" video to hype myself up for calculus since I'm working towards it
>he starts talking about the area of a circle
>not area of a disc
yes, I felt compelled to pause and stop watching

>>14852171
Watch Wildberger to get finitypiled
https://youtu.be/xYPw2gY_3PI

Holy fuck, Dedekind cut is such a badass name for a concept. Maybe I'm easily impressionable, but this gives me immense motivation.

>>14852293
Check out the Cox-Zucker Machine
https://en.wikipedia.org/wiki/Cox%E2%80%93Zucker_machine

>>14852171
He literally uses area of discs to calculate the area of a circle though.

>>14852293
Terminal dedekind basis is a badass name for anything

>>14852293
for me, it's anything with Hilbert's, Gauß's or Wedderburn's names, they've got a good ring to them

>>14852307
he uses the area of concentric rings (annuli). but he keeps talking about the area of the circle, instead of the area of the disc

I'm going back through math from the absolute basics, what books do you lads recommend? I've got all 3 of the strayer upton practical arithmetic books and basic mathematics by Serge Lang. I'd like to go as far as possible with math.

>https://arxiv.org/abs/2209.04919
> Truth, or Mere Beauty?
> High energy physics features many ingenious tools for extracting finite results from formally divergent expressions. This brief note argues from a new perspective that all such formal infinities are meaningful markers of new physics. As such, they deserve to be explored in detail -- and even the simplest quantum field theories have more of them than is commonly thought.

In this paper this guy argues that standard mathematics needs to be replaced by finitism and this will eventually fix problems in theoretical physics. The examples here uses are fairly elementary, any comments?

>>14852629
>I've got all 3 of the strayer upton practical arithmetic books
Never heard of this shit in my life lmao
>basic mathematics by Serge Lang
There's a playlist of a guy going through the entire book
https://www.youtube.com/playlist?list=PLMcpDl1Pr-viA25VUkHNmcUkWx9usPgyb

If I were you, just enroll in math courses at your local community college and see how far your interest takes you.

>>14851865
Lam's noncommutative rings textbook

>>14851865
Sorry I never used textbooks when I was at uni.

how does x^2=0 => x=0 imply that there are no non-zero nilpotents in a ring? e.g. if x^3=0, how would we go about showing x=0?

>>14853048
Because if n in x^n = 0 is even, you can halve it and get x^(n/2) = 0
If it's odd, you multiply by x and halve it.

For example:

x^5 = 0 (given)
x^6 = 0 (multiply both sides by x)
x^3 = 0 (halved)
x^4 = 0 (multiply both sides by x)
x^2 = 0 (halved)
x = 0 (halved)

Adding 1 to an positive odd number then halving is guaranteed to give you a smaller number than what you started with, unless you start with 1.

>>14852629
https://youtu.be/OnBHg6cUyuo

>>14853048
>>14853063
Actually, it can be done in an easier way:
Multiply by x until the exponent is a power of 2, then halve until you reach x.

>>14853268
Many such cases.

Also, the blacking out is really smart if the @ is not blacked out

>>14853268
>>>/pol/

>>14844016
Statistics for ML is a meme. If you want to study AIML, study functional analysis. Statistical learning is really just a meme never been useful

>>14853063
>>14853101
oh fuck, i'm fucking retarded. thanks, anon

Ma boi Kripke just died.

Do you adopt the Kripke schema?
>For every statement ϕ holds: There exists a binary sequence a, such that ϕ holds iff there is some n such that an=1.

Task is to prove that a triangle that has three 60 degree angles has equal length sides.

I feel like this proof just tacitly assumes that the line perpendicular to [math]\overline{PQ}[/math] passing through [math]M[/math] intersecting [math]\overline{PQ}[/math] at the point [math]N[/math] cuts [math]\overline{PQ}[/math] in half.

I mean intuitively it's "obvious" that if you start from bisecting the angle at [math]M[/math] and draw the line to [math]\overline{PQ}[/math] or if you start from drawing the perpendicular bisector of [math]\overline{PQ}[/math] you'll get the same thing. But I feel like there should be a way to prove this without kind of assuming the very thing you are trying to prove. I can see that there is probably some way to link the angle defined by two rays with the length of the segment on a given line (defined by the two points where the rays intersect the given line), especially with one of these rays fixed, but I just have no idea how with just this intuitive toolkit that's provided here.

Guessing that this is something that's very simple to show/prove analytically with coordinates and/or trigonometry, but I have no idea how to make this work without those.

in order to define the ring of polynomials [math]R[x_i:i\in S][/math] with variables indexed by an arbitrary set [math]S[/math] is it enough to consider [math]R[M][/math], where [math]M:=\mathbb{N}^{(S)}[/math] is the free monoid over [math]\mathbb{N}[/math] and [math]R[M][/math] is the monoid ring? that is, [math]M[/math] is the monoid of functions [math]\alpha:S\to\mathbb{N}[/math] with finite support (these correspond to all possible monomials) and [math]R[M][/math] is the ring of functions [math]f:M\to R[/math] with finite support (linear combinations of monomials). does this work or do we need some considerations with the AC if [math]S[/math] is infinite?

I have finished these recommended books, what should I follow up with?

Undecided in choosing between 2 advisors. (shitty college)

One is in algebraic geometry, which is the area that I want, but he seems... more computational/applied? I'm not sure, but in class he seems kind of not-as-rigorous as I'd expected.

The other is kinda renowned in the region, but he's in commutative algebra, which is what algebraic geometry is built on. I haven't taken a class with him, though he seems solid from what I've been told.

Thoughts?

>>14854611
>Wild Burger
Follow up with meds

Just took the math gre, what are your guys’ experiences with it

>>14854886
Quite tougher than the old/practice exams I found

>>14854930
I also found it more difficult than the old tests let on. Maybe it has to do with their getting rid of negative points for incorrect answers?

reccommended companions for evans PDE book for more practice problems?
need to be exceptionally proficient with all the simple pdes before my research placement in november

>>14855098
Try Renardy & Rogers 'An Introduction to Partial Differential Equations', I like that book.
There's also Fritz John's book which might be a bit more applied but I like it also, has lots of interesting stuff and some nice problems.
And of course Folland's PDE book as well.

>>14854886
>too fucking early
>long as hell
>harder than my dick

The test really covers every class you took and
any class you may not or would have taken.
Some questions are rather simple or at least
workable for about 10 minutes of your time.
Others need like one or two tricks you probably
never heard of to solve the problem; maybe
a topic you missed? You can only guess so much.

My grades aren't the best (460 or so), but some
colleges overlook that, so good. Also, one day,
the test sites in my city were full so I ended
up going to a high school in another state to
reach there by 8:30 AM. I had to leave my place
from 4:00 AM. Get a damn pastry to tide you
because the test is so long. God, I want to kick its ass...

>>14855112
thank you anon. will look for these on libgen tonight

any reccommended texts on universal algebras or even specifically for partial algebras? Even if its in terms of meromorphic functions or graphs or what have you I'm real curious what the state of affairs is since any of the math encyclopedias i've read don't give a clear description of methods relating to that stuff and its been difficult tracing papers

>>14855119
I think you'll like Renardy & Rogers but if you don't it has Appendix A which lists some similar books under 'Basic Graduate Texts', also it has nice suggestions for further reading if you want to get more advanced. So at least have a look at that.

>>14848166
Poset with sup and inf analogue for any two elements on the order.

>>14855116
>>14854886
>Others need like one or two tricks you probably never heard of to solve the problem
Here's one simple to state that I got on the exam, but hard to solve:
How many surjective functions are there from a set with 6 elements to a set with 4 elements?

Sounds simple, right? :^)

I've been in these threads every single day for an entire year now. Finishing calc III in a few weeks, and so far have only A's in math courses. Thank you for keeping me interested in tons of topics and studying every day. I love you guys.

>sent a quiz announcement
>highlighted the important parts for all the ADHD fucks
100% certain that half of them won't have read it.

>>14855198
>>14855116
Oh god...that's it. This is the shit that'll piss someone off.

>>14855467
>>14855198
Hindsight 20/20 solution: Use the principal of inclusion-exclusion.
I'm having a hard time writing this in words, but basically the solution is just: 1*4^6 - 4*3^6 + 6*2^6 - 4*1^6
Each term represents subsets of size k. The left coefficient is how many subsets there are of size k (which is just a binomial coefficient), the k^6 is how many functions there are (not just surjective) to a given subset of size k.
The expression evaluates to 1560, which I recall was the answer I ended up guessing.

>>14853898
>Kripke died
Feels bad man. He's in a better frame.

>>14853960
>feel like this proof just tacitly assumes...
It doesn't. The only facts about N that are needed are:
>a. N is on the line [math]L_{PQ}[/math]
>b. the angle is 90deg
These two facts, plus the 30 degrees fact, are good enough for the rest of the argument, which is only about the angles between the lines e.g. [math]L_{PM}[/math] and not really about the line segments e.g. [math]L_{PM}[/math].

Why does an N exist satisfying (a) and (b)? I'm guessing it is in your book, but the answer is closely related to the "reflection" construction that the proof also relies on. Draw two circles like in the picture. One of their intersection points is M, and there's another one we can call M'. Now N is the intersection of [math]L_{PQ}[/math] and [math]L_{MM'}[/math] which are perpendicular. (And this is true for any triangle, not just equilaterals. Also I'm skipping some kind of autismo details but you mentioned an intuitive toolkit so I hope this is adequate.)

>>14855660
>the lines e.g. [math]L_{PM}[/math] and not really about the line segments e.g. [math]L_{PM}[/math]
I mean the line segments e.g. [math]\overbar{PM}[/math]

What are the most schizo math fields out there?

>>14855674
Number Theory

>>14853960
The proof uses neither the angle bisector nor the perpendicular bisector, but the "dropped perpendicular" construction. That is, it uses an N satisfying two things:
a. P,N,Q are collinear
b. [math]L_{MN}[/math] and [math]L_{PQ}[/math] are perpendicular
To construct N, just draw the circles in pic related, find the other intersection M', and take the intersection of [math]L_{MM'}[/math] with [math]L_{PQ}[/math]. This works for any triangle, not just equilateral ones.

Now the argument continues: "By the known theorem..." etc. Just stay aware of the difference between [math]L_{PQ}[/math] with [math]\overline{PQ}[/math].

>>14853697
The absolute state of /sci/. Screencapping this to laugh at you pseuds with /g/

>>14855702
you're a homosexual pedophile who masturbates to children's cartoons

>>14855687
What do people have against number theory? It's nice.

>>14855660
>>14855665
>>14855690
Thanks man! I think I got a bit too focused on the line segments, but refocusing on the lines like you said made it all click, now it's all clear. I appreciate the diagram as well.

>>14855707
>>14855687
Probably with how many schizos it attracts due to Collatz, Goldbach, FLT, etc.

>>14853697
Statistically the only pussy you've touched is your mom's

>>14855716
Oh, the people claiming to solve unsolved problems? They don't typically study it seriously though do they?

Here's a fun exercise:
This function I've written in Python creates a list of the inverses mod a prime p (inv[0] = 0 to keep the list at length p); e.g. when p = 7, then inv[3] = 5.
See if you can understand how it works.

It's a cool application of the division algorithm.

>>14855513
>>14855467
That's looks about right. A good amount of people,
even me, knows inclusion-exclusion. But one may
not think of that at first glance or first ten minutes.

This trickery is the stuff that makes one a true
math major...you'll go mad and you'll understand
later. I'm even disappointed they're aren't as many
recent old tests to try other than the massive
GRE books you have to buy or from 20-30 years
ago.

>>14855707
Only a schizo could find some higher meaning in prime numbers.

>>14855882
I think you confuse Number Theory with Numerology.

>>14855702
>comes here for advice
>is given what seems to me to be correct advice
>haha bros look at this absurd response, ML involves very serious statistics it doesn't just boil down to linear algebra and numerical analysis
Are you really going to take the chance people will actually laugh at you when they see the screencap?

>>14855733
I’m confused. The identity seems to be that if p=qa+r, then [math]-aqr^{-1}=1[/math] (all mod p)which seems untrue in general.

>>14855608
kek

>>14855945
ok, I'm sorry. I thought you were trolling me like the jerk who recommended me meme books a few years ago. Thanks for the advice, anon

>>14856051
What's a meme book?

if(digits)
thread.next(theme)=="Cirno";
GOTO 10

It would be nice if I had an un/underemployed autistic friend to study physics with all day.

>>14856090
Unlucky.

New thread >>14856127

>>14855965
Not quite, but you've got the right idea.
Try again.

>>14855716
>>14855707
>>14855882
I just want to know why my algorithm that uses prime numbers to generate other, larger prime numbers works. Everything I read online says what I'm doing isn't possible. And yet, I am able to generate primes using primes.
I can also factor smaller composites using an algorithm that would normally be impossible to use for such a purpose.
If I'm schizo, why does it keep working?