/sci/ - Science & Math

>>15879257
>According to a coworker, in ZF with negated C you can construct a finite set of positive integers with no least element.
No you can't. All finite sets have choice functions in ZF alone.

>>15879257
In ZF without C, there are multiple inequivalent ways to express that a set is "finite", based on distinctions that you didn't have to worry about in traditional ZFC.
In particular, your coworker is certainly not defining finiteness as being in bijection to a subset of the von Neumann ordinals below [math]\omega[/math] (the first limit ordinal after [math]\emptyset[/math]).

>>15879257
In ZF without C, there are multiple inequivalent ways to express that a set is "finite", based on distinctions that you didn't have to worry about in traditional ZFC.
In particular, your coworker is certainly not defining finiteness as being in bijection to some von Neumann ordinal below [math]\omega[/math].

God I fucking hate coming to lectures so fucking much. 6 hours wasted over NOTHING. A textbook is a billion times better and more worthwhile. But I am forced to come to this stupid fucking classes and waste my time.

>set theory
>LEM
>Choice
I seriously hope you guys don't do this.

>another day since graduation
>another day where I haven't used cohomology once
Remember to spend your time in university learning useful things, not sucking your professor's dick.

did a silly, post silly stuff

>>15879322
What are some of the others? That's the only definition that seems reasonable to me.

>>15879257
>Set theory is retarded.
Do you agree with the people who say math should not be philosophical?

>>15879516
>injection into itself has to be surjective
>no surjection onto N
>has injection into a bounded interval {0,1,…,n}
things like this. not sure ones are different and interesting in ZF+not C

>>15879516
https://en.wikipedia.org/wiki/Finite_set#Other_concepts_of_finiteness

>>15879257
Definitely not true. In ZF every set of positive integers has a least element. You don't even need to ask if the set is finite or not. However I think it is true that in ZF you it's consistent to have an infinite descending sequence of cardinalities of dedekind-finite sets. Having that is stronger than just not having choice though.

>>15879540
>>15879545
thanks anons

>>15879257
>According to a coworker, in ZF with negated C you can construct a finite set of positive integers with no least element
Maybe they meant with a different ordering? e.g. any set of integers greater than 1, in which at least two are coprime, has no least element if ordered by divisibility
But that's just as true in ZF as in ZFC so it's more likely that you just need to find less retarded coworkers

How useful is computational algebraic geometry, at least when it comes to getting a job in industry or whatever.
I really enjoy algebra and I am at a crossroads. I can attend a much lower tier grad program but focus on algebra, numerical analysis, and computational algebraic geometry, or I can attend a much higher ranked program but I'd have to focus on applied math and probability/statistics. The latter is a double whammy when it comes to making dosh after, but I'm not as passionate about the topic.

>>15879516
>all subsets of the power set have a maximum element ordered by the subseteq order.

>>15880203
It's niche, but I have encountered it in my work (computerish research-focused job). I would take the higher-ranked spot; you're probably going to have the same kind of job regardless of which you choose, but you will have a much easier time finding work with a more prestigious degree.

>>15880203
>yeah that guycan do computational algebraic geometry, hire him
said no one ever.
you will eventually burn out and hate your job anyway no matter how passionate you are about it

>>15879257
> master thesis (applied math)
> problem is extremely nasty looking optimization problem over a Hilbert space of random variables
> found interesting new result, essentially completely solve the problem theoretically and even pose well principled numerics
> mostly just framed things correctly so as to clearly state some connections between concepts
> also diggged thru a ton of papers to find the right theorem
> then just applied the right theorem successfully to my problem
> supervisors now want to publish.
Is this good work for a master thesis? I feel that I didn't really come up with my own new ideas, I just carefully studied what was available and utilized it well, as any decent student could have. In the end the whole problem looks rather simple, honestly.

>>15880470
I think that is generally how people feel after completing a master's thesis, or really most mathematics research. Massive eureka type results are pretty rare, and even those usually amount to applying established ideas in a somewhat clever way. An original result in a master's thesis is uncommon already, at least in pure math. Maybe more common in applied.

>>15880524
>Massive eureka type results are pretty rare, and even those usually amount to applying established ideas in a somewhat clever way.
I would imagine that proposing a master thesis project that could lead to massive theoretical results is something no suoervisor in their right mind would aim for. After all, a master thesis lasts 6 months, and its supposed to be a first foray in research.

>An original result in a master's thesis is uncommon already, at least in pure math. Maybe more common in applied.
According to my supervisors getting something publishable out of your masters is extremely rare and will help keep down the line with academics. But then again, I dont see how this problem I had was so extremely hard to solve.

>the image of a bounded function that has discontinuities is an open set
is this always true?

>>15881133
Dirichlet function.

>>15880470
>Is this good work for a master thesis? I feel that I didn't really come up with my own new ideas, I just carefully studied what was available and utilized it well, as any decent student could have. In the end the whole problem looks rather simple, honestly.
That's how most people feel and that's how it is for most people. If you dig into the full context of letters, papers, etc, you will find most "geniuses" with "amazing results" are backed by a veritable global-scale flood of correspondences, papers, and the like. Including people helping them on what they're bad at or have no interest in.

Discovery and innovation do not generally come from a total void, but popular press love to make it seem as if it does to create this magical mistaken sense of "awe". It's total bullshit. Every single thing ever produced stands on other people's shoulders and is very often recombining results of other people's works from new perspectives, or simply fleshing out someone else's idea in a way they could not have. Almost all publications including the most famous are like this.

Are there any autism-free books on commutative algebra?

>>15879035
I'm not sure if I want a pure book precisely because of the additional insight. I actually recently wrote a research paper on non-parametric stats (my first research paper after my undergrad thesis). However, it was pretty much replicating some old methods on new data, with some additional insight. However, after that fun experience, I decided that I want to specialize in non-parametric stats and as such I'd like to not only understand the methods, but how they were originally conceived and derived mathematically, so that in the future my research can be oriented towards finding new methods.

>>15879257
for me it's Ultrafinitism
math faggots delude themselves that they can choose an element out of a set of infinite order. bruh really??

I was again thinking about maths stuff out of boredom and somehow I came up with the following dilemma that I think is interesting.

If you construct an infinite chain of vectors in this way, given the angle and the ratio of lengths between two successive vectors (assume they are constant) and also given the starting coordinates, angle and length of the first vector, is it possible to calculate the point at which the sequence converges on a Cartesian plane?

>>15880381
>>15880382
A bit unfortunate. At the very least, I can get one last hip hip horay during my bachelors thesis. Maybe I can get something done that touches probability theory and grobner bases. Maybe put together a less precise reduction of grobner bases such that the vanishing ideal is fuzzy or has some density and distributions that match sections.
I don't know. I have no idea what they want for this. Seems some people just use it as a way to get an extra class in any topic they want.

>>15879257
>Finished my pure maths degree in 2021
>In 2022 I got a serious job (serious time-investment, serious potential growth, serious money)
>Now my estimated opportunity cost of going on to do a masters degree full time (1-2 years) is over $100k. This is NOT counting tuition or other expenses.
>Estimated opportunity cost of doing a PhD probably over $300k

Last year I started seriously considering part-time, online-friendly options. I am now accepted at a top 50 university for such a program. It looks like this will be the way for me. Anyone have any experiences going down this path? Was it still fulfilling?

I will still be able to interact with professors and TAs. I will still be able to access the campus, although there'd be no practical reason to do it.

For me, it's the axioma of determinacy.

>>15881463
>opportunity cost
If you are gonna include made up costs, then you should also include the SOUL cost.

>>15881417
Well sure, take the sum of all the angles and the product of the length ratios. If both of those converge, then you know how much to scale and rotate your starting vector. If they don't converge, then neither does your sequence.

Just learned about renormalization group theory today in material physics class, not going too in depth unfortunately but it peaked my interest.
Any good book/lecture about on the math behind this to learn more on this subject?

Will AI replace math?
I'm felling depressed and demoralized again. What did I spend all that time learning occult numerology if AI will just rape my ass?
Ted was right

>>15881590
If you ask it, it says math would be one of the last things to be replaced.

>>15881804
It's still on the chopping block eventually.
How do i cope with a dying world. I'm young and healthy, but the oldness and sickness of the world makes me ill.

>>15879257
Hello /sci/
Please tell me a good undergraduate/graduate abstract algebra book. Many people told me to read Artin but I just despise reading that book
Any book that covers same material as Artin but in a clearer/more readable manner

>>15881965
Jacobson, Basic Algebra 1. Amazing book. Very comprehensive and very fast but still manages to motivate and explain all the concepts. The only con is sometimes results and definitions are not stated in theorem environments. Also it's not available in print anymore except shitty Dover copies. The scans online also are not very good, but still readable. I am actually working on rewriting the entire thing in TeX. Probably would take me 10 years though. An (3 year) undergraduate course would consist of chapter 1, 2, and 3, and possibly 4.

>>15882031
>I am actually working on rewriting the entire thing in TeX.
Hey you are doing a great work, I want to help you
You have any community/discord to get this done?

>>15882033
>>15882031
I would also be interested in helping after new years when all my applications are in, actually. Sounds like that might be a good use of my spare time.

is there a duality principle that would apply to numbers?

>>15882065
You gotta be way more specific, man

>>15881860
Will you fight? Or will you perish like a dog?

>>15882040
I will be glad to help transcribe some chapters in LaTeX, make a post here and make a group to make the translation

>>15881965
Whose name is in the OP.

>>15882136
Saunders McLane? I only know of his category theory book

>>15882139
https://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462
Here. Just do yourself a favor and get it.

>>15881451
based on my experience, your master's thesis doesn't have to (and probably won't be) 100% related to your master's program.
go with the higher ranked program and you might be able to find a thesis topic related to your area of interest.

>>15881264
Damn I guess this is encouraging. Thanks.
I just want to come up with new ideas that lead to new methods, do something truly orginal so to say.

>>15882246
>I just want to come up with new ideas that lead to new methods, do something truly orginal so to say.
That's what ordinary discovery and synthesis accomplishes. If it were utterly and completely original and without synthesis points you'd be so far afield that nobody would recognize any worth to whatever it was you were doing. Unless you also spent considerable time building those bridges.

Seriously, this is a monkey's paw request. If you think about what it would mean you'd probably realize you wouldn't want the controversy at best, or being dismissed as a total crank otherwise. There's "original" and then there's "original". Truly original doesn't really happen and when it does it tends to get overlooked, sometimes long after the author is dead.

Is it true that a polynomial with a positive discriminant has a number of non-real roots that is a multiple of 4? E.g. an octic equation with a positive discriminant always has either 0, 4 or 8 non-real roots.

>>15882421
No.

what exactly causes non-commutativity and does it connect to primes?

adding zeroes together unendingly and ending up with anything other than zero shows a deep contradiction.

>>15882592
Non-commutativity of what? Plenty of operations aren't commutative, it doesn't have anything to do with primes.

>>15882427
Then what does a positive discriminant for an octic equation imply about its roots?

>>15882421
A linear polynomial always has the discriminant 1 but the polynomial
[eqn]x - i[/eqn]
has one non-real root.
>>15882653
You will have 8 distinct complex roots.

>>15882987
Man, I always forget the existence of non-real coefficients. It's the same reason why the classification of elliptic/parabolic/hyperbolic linear PDEs breaks down when you throw in complex coefficients. What if I add the stipulation that all coefficients are real?

>>15883000
>It's the same reason why the classification of elliptic/parabolic/hyperbolic linear PDEs breaks down when you throw in complex coefficients.
What do you mean?
-Elliptic: 2nd order operators in space
-Parabolic: 2nd order in space + 1st in time
-Hyperbolic: 2nd in space + 2nd in time

I did a lot of pde theory where the spaces where taken as complex

>>15883082
Something about the Schrödinger equation acting more like the wave equation, a hyperbolic PDE, despite not being one. And for both polynomials and PDEs the classification completely breaks down when the discriminants yield non-real numbers.

What's the most fun branch of mathematics for you guys?

>>15883205
Probability

>>15883211
Why?

>>15883212
It's the least autistic.

>>15883205
Trigonometry is quite fun.

There should be an AlphaGo for making conjectures, that would be so sick

>>15883355
I always thought geometry was fun in school. What's it like in higher level (late undergrad/grad) math?

>>15883192
Yeah I guess in the end the Classification is basically arbitrary.
The people in the field do use the classification I outlined as far as I know though

>>15883211
>math
>>15883205
Basic arithmetic.

>>15883365
>What's it like in higher level (late undergrad/grad) math?
Lol. Lmao.

>>15883411
A graduate text in arithmetic? I didn't even know this was a thing...
I want to self-study math, would this be a good place to start?
>>15883437
Wow those are some big words on the right anon. Looks very smart.

Is stats actually desired in the job market or is this just another myth made up by purist pure mathfags?

>>15883515
Stats is actually desired, but with a few asterisks.

First, I see the data science meme already dying down. Companies, after being bamboozled for years, have realized that 99.9% of them actually needed data engineers who have a rudimentary understanding of stats so that they can double as data scientists a few times a year tops. As such, the actual in-demand skill is not grad level stats, but professional level understanding of data pipelines. Using shit like Azure, AWS, Databricks and of course some SQL and Python programming.

However, beyond that, there are some (highly paid) roles in which stats itself is desirable. However, specifically, you need to know non-parametric stats. You could know 1000 facts about the normal distribution and that'd be useless because normal distributions never show up in the real world. Literally never. And if you work with financial data, the most common kind of data that is typically analyzed by corpos, then you literally get anti-normal distributions all over the place.

>>15883205
Ring theory

>>15883535
So it's pretty much either software engineering or some position I have almost no chance of getting

>>15883453
>I want to self-study math, would this be a good place to start?
Yes, along with Jacobson, Basic Algebra.

How i wish someone told 5 year old me to study more math instead of watching cartoons
I am now regretting how less time I have left in life vs how much math there is to learn
How I wish I started earlier then could have mastered more math

>>15883535
>anti-normal distributions
like what? geometric?

>>15883557
Yeah, data engineering = software engineering with an emphasis on data. And the other role will be extremely hard to get unless you are already a top candidate. Not impossible, but almost.

>>15883642
By anti-normal I just mean distributions that violate pretty much every property you can think of about the normal. For example, financial data is very, very fat failed.

I’m a math person who is working on kaggle competitions, and I feel like I need to shift my thinking from a “building a solution in my mind” approach to a more trial and error and experimenting approach. Has anyone else worked on kaggle problems and have experienced having to make this mental shift?

>>15883211
>>15883355
>>15883411
>>15883536
What about logic?

I'm struggling on a problem my prof already fucked up since for the first version he posted, I found a counter-example.
Let [math] x_{k} [/math] and [math] y_{k} [/math] be sequences in [math] \mathbb{R}^{n} [/math] such that [math] \|x_{k}-y_{k}\| [/math] converges to 0 and [math] <x_{k},y_{k}> [/math] converges. Show that [math] \|x_{k}\| [/math] and [math] \|y_{k}\| [/math] both converge. The closest I got was that [math] \|x_{k}\|\|y_{k}\| [/math] converges and [math] \|x_{k}\|-\|y_{k}\| [/math] converges to 0. I feel like I'm missing something incredibly obvious.
Do i need to find some contradiction by assuming they both diverge?

>>15883657
So many roads and all of them lead to either software engineer, highschool teacher or finance if you are lucky.
Hard to see the point of dealing with so much bs just to end up in a position you could have reached with a fraction of the effort and time

>>15884159
[eqn] \|x_k - y_k \|^2 + 2 \langle x_k,y_k \rangle = \|x_k\|^2 + \|y_k\|^2 [/eqn]
The LHS converges so the RHS converges too which implies that [math](x_k)[/math] and [math](y_k)[/math] are bounded sequences.

By the Cauchy-Schwarz inequality
[eqn] \lim_{k \to \infty }|\langle x_k - y_k, x_k \rangle| \leq \lim_{k \to \infty } \|x_k - y_k\| \|x_k\| = 0[/eqn]
so
[eqn] 0 =\lim_{k \to \infty } \langle x_k - y_k, x_k \rangle = \lim_{k \to \infty } (\|x_k\|^2 - \langle x_k,y_k \rangle) [/eqn]
or
[eqn] \lim_{k \to \infty } \|x_k\| = \lim_{k \to \infty } \sqrt{\langle x_k,y_k \rangle}[/eqn]
Since the RHS converges so does the LHS. Similarity [math]\|y_k\|[/math] converges too against the same limit.

>>15884179
Funny enough, I ended up in finance! However, for me, I studied maths in college completely for fun. I honestly would have suffered more in a business or finance degree than I did in math. Math is just second nature and is, to me, the easiest subject to learn properly. Everything else is too boring for me to engage with the material - hence insanely difficult to actually learn.

If you do not share that perspective... yeah, don't do math. There is no such thing as an industry mathematician. It's all just software or finance out here.

>>15883574
it slows down after the initial learning curve. can only see so much innovation before you've seen most of what there is.

>>15884349
>There is no such thing as an industry mathematician
True, but you can get very close at top tier AI companies or with some sort of tech R&D (not SE) position.

>>15884378
>tech R&D
>not SE

Oh, my poor, poor child. Well, you are technically true. But, again, this positions are top tier and you better be a top tier candidate. And these guys will still be doing a lot of coding.

>>15883453
That's because the graduate text is talking about
'arithmetic' as in number theory, not 'arithmetic' as in basic numeracy.

>>15883535
>because normal distributions never show up in the real world
>what is the Central Limit Theorem

>>15884334
Damn, I spent too much time fucking about with norms and triangle inequalities. Cheers.

>>15884479
You will be doing a lot of coding in almost all of STEM whether it's in your job, undergrad or in grad school.
The difference is whether you are using code as a tool or your entire work is just coding for the sake of coding.

>>15884349
>There is no such thing as an industry mathematician. It's all just software or finance out here.
Almost all research has maths. There are thousands (probably more) of research companies and institutes. There is a research institute in my city that literally only does fluid dynamics. 200+ people just doing fluids.
Where the fuck are they getting mathematicians from?
Not to mention how mathematics is a very unpopular major

>>15885147
>Where the fuck are they getting mathematicians from?
>200+ people just doing fluids.
Anon they probably employ computer scientists or engineers. Maybe some physicists.

>>15883642
More like cauchy

>>15884334
>Cauchy-Schwarz inequality
it's called "cauchy bunyakovski schwarz inequality". for brevity sake you call it schwarz inequality (too many things are named after cauchy)

>>15883535

>>15885151
The hardest math computer scientists and engineers do is undergrad multivariable calculus.
Almost all large industrial companies hire physicists as advisors because they have insight that no engineer has.
Math being the spine and core of all the sciences should be the most in demand skill

>Solved 2 exercises in the last 6 hours
That's embarrassingly slow, right?

>>15885277
post the exercises

>>15885283
I proved that [math]L^p[/math] is complete for any [math]p\in[1,\infty][/math] and that any Riemann integrable function is Lebesgue integrable, with the values of the integrals being equal.

Why does every math book rec on this site predate or completely ignore the computer? Is it cause everyone on this board are codelet undergrads?
I lol'd when Terrence Tao wrote a blog article about using chatgpt to do some basic shit that could be done by anybody with a day of experience in shell.
Pure math fags think they are smart, but with their entire 180iq brain, and 10 years they can barely solve basic ass conjectures about prime numbers.
Computers show collatz is true for all situations that would arise in physical reality, the babylonians were right and Greek math is a cult ritual not a science.
Science finds the closest thing to physical truth that we can comprehend, "math" finds "truth, if these axioms are true" and has completely and utterly lagged behind all "experimental mathematics" for the last 10k years, and now that computers are here the jig is up. Math as you know it won't exist outside stuffy institutions that still play bach.

>>15885467
The point of math isn't to find out whether something is true. It's to find out why something is true. No simulation can answer that. For instance, knowing Fermat's Last Theorem is true contributed very little to mathematics, but all the development that led to the proof of it (abstract algebra and more specifically elliptic curves) gave us a much richer understanding of numbers. Knowing the truth without the reason makes you look like picrel.

>Why does every math book rec on this site predate or completely ignore the computer?
Because they are mostly about basic foundational theories that don't require computers. Computers is used in a lot of math branches e.g. computational algebraic geometry, and of course entirety of numerical mathematics.

>>15885467
I don't care about other people.
>ITS A CULT!!

I don't care.

>>15885304
Those are pretty nontrivial if this is your first analysis class. It's slow, but you'll get faster with practice.

>>15885546
>basic foundational theories
Every theorem known to the greeks can be proved using Descartes's coordinates and algebra, yet most highschools only cover a sliver of what the greeks knew, whereas when using a computer it would be trivial in a semester to have the average student program the proofs for all such theorems. The same is true of linear algebra and the majority of calculus.
How can you say the "theories" are basic when they aren't even taught, all that is taught are some choice theorems that heuristically people have decided make "good" mathematicians.

>>15885623
Solve this, smart computer man.

>>15885636
>schizo symbols
Yeah, not even reading that.

>>15885638
Its just the area under that function. Show me how your computer does it, or alternatively you can do it by hand. You're free to use anything you want to use. Its very basic calculus, which is required for CS majors. The point I'm trying to prove is flaws in numerical or even symbolic systems. This is an extremely common type of problem used to catch idiots and cheats, I'm surprised you didn't identify it at glance.

I have a quize soon on artin algebra chapter 11,12 aka rings and factor fields and some of ch 15 ie splitting field
I do not like artin as i find it unreadable. I am studying from hungerford and keith Conrad notes

Anything else i should do to prepare? Any problem sets like factoring polynomials and stuff?

>>15885636
integral(1 + sin(x))^10 cos(x) dx = (sin^11(x))/11 + sin^10(x) + 5 sin^9(x) + 15 sin^8(x) + 30 sin^7(x) + 42 sin^6(x) + 42 sin^5(x) + 30 sin^4(x) + 15 sin^3(x) + 5 sin^2(x) + sin(x) + constant
Solve this, retarded pen and paper ape.
4968604917443726131088480435305377354799165308681748466687433497140812960844\
393823936617559844893544142022576185166598514873439426776855087841223703\
895925873313735385182776856614993134751350828107745458134957042329555317\
880177957799090959028852507376183722165164254590274927829149234565346763\
132573406275802301258552173956734127018532382122328365356906078578363068\
33285102218048270829583763125437176321 = p × q

>>15885647
Filtered lol

>>15885647
>integral(1 + sin(x))^10 cos(x) dx = (sin^11(x))/11 + sin^10(x) + 5 sin^9(x) + 15 sin^8(x) + 30 sin^7(x) + 42 sin^6(x) + 42 sin^5(x) + 30 sin^4(x) + 15 sin^3(x) + 5 sin^2(x) + sin(x) + constant

Nigger.

>>15885654
>>15885655
Factor. Dumbfucks.

>>15885647
>(sin^11(x))/11 + sin^10(x) + 5 sin^9(x) + 15 sin^8(x) + 30 sin^7(x) + 42 sin^6(x) + 42 sin^5(x) + 30 sin^4(x) + 15 sin^3(x) + 5 sin^2(x) + sin(x) + constant

>>15885661
You have no argument.

You should cite yourself, NOW!

>>15885654
>>15885655
>>15885661
>can't even factor a number
How's elementary school? Mods underage ban.

>>15885636
Express your question clearly in pseudocode and I will consider it.

What is the average IQ of a scitard if they can't even factor a number? 0?

>>15885666
>>15885676
You do not know your basic trig. You could not solve your way out of a triangle. A computer cannot do this.

[eqn] \frac{(1 + \sin(x))^{11}}{11} + C [/eqn]

This is the actual answer to an extremely basic problem.

>>15885645
Amidst this argument please dont let me question get buried

>>15885685
Dumbfuck freshy that is sin(x)^10 * cos(x). Please don't lecture people when you are a D student

>>15885676
Please stop embarrassing yourself

>>15885692
You are sub 0IQ and proof that we need computer based experimental mathematics education.

>>15885690
>>15885697
Holy kek literal retards here

>>15885701
You are the retard who misremembered
sinx as 1+sin(x) then got btfo'd. It's hillarious that you think a computer can't do either one.

>>15885647
/g/sisters... not like this....

>>15885710
>>15885701
>>15885692
>>15885685
You aren't fooling anyone, of course a binomial
Will have a binomial expansion derivative.

>>15885718
Make another thread for your retardation and leave this one. You are disturbing people who actually need help and advise here like me >>15885645
My question got buried due to you.

Regarding your question and where you are wrong your answer purs the 1/11 only on sin^11(x) and not in other terms
The answer is (1+sinx)^11 /11
Standard computer mistake

It comes from substitution. Substitute sin x = y then 1+y = z and the integral gets done

Now leave

Is there any modern way to study geometry without algebra?

>>15885609
Thanks.

>>15885304
>any Riemann integrable function is Lebesgue integrable
any PROPER Riemann integrable function is Lebesgue integrable

>>15883211
For some weird reason. Probability problems are the most fun to think about.

>>15885808
Optimization and matroid theory (optimization is a lot more itneresting than it probably sounds).

This is actually a very active area of research at the moment. Mathematician June Huh just got the fields medal I think last year or 2 years ago for research on matroid theory.

>>15883205
Graph theory
Ramsey theory

>>15879257
>According to a coworker, in ZF with negated C you can construct a finite set of positive integers with no least element
nah

>>15886081
I don't know what that means. I mean Riemann integrable on the compact set [a,b]. Also I meant Lebesgue measurable.

>>15886135
There are functions that are Riemann integrable over an unbounded domain which are not Lebesgue integrable.

>>15879419
Thing is you will never have an opportunity to learn cohomology again, whereas your boss will make sure you learn whatever python libraries you need to import at your shitty job. Remember to use your time in university expanding your horizons because your employer is incentivized to do the opposite

Trying to write a general formula for the nth-derivative of 3/(2x-1) but my brain isn't coming up with it. There's a clear pattern, the numerator goes from first to fourth derivative like: 6, 24, 144, 1152 and denominator literally just +1 to the exponent. I feel like it should be dead simple but everything I try doesn't work. I think I have to do something with a factorial but nothing I tried was quite it, like. 3 * (2n)!. Any tips?

Is it normal to barely understand anything at all in a probability course, yet still get an A?

Thought problem here. Lets say we have two fighter jets engaged in arial combat in a turbulent atmosphere such as a galeforce wind heavy rain system. What kind of mathematics would we use in order to predict future enemy position the moment we note a scripted evasive maneuver based on control surface position, such that after we note the evasion, we calculate the future end position of the aircraft as it locks itself into a drift, and we aim at the calculated future position?

I think the biggest issue here would be modelling the turbulent environment as any model would be an assumption, whereas the simulated motion of the aircraft in a perfect neutral environment would be rather predictable to an area of suitable target size.
I would assume, some sort of stochastic differential equation, and I'm almost certain no portable computer capable of even approximating an answer to this in a second or two could ever fit in an aircraft.

>>15886448
Wiener process.

>>15879257

I recently advanced to phd candidate, and my advisor said the committee will waive my thesis defense.

Is this common? Is it a good sign, or are they fed up with me or something?

>>15886106
I mean like a modern version of Euclid

>>15886328
Yeah idk. Another hour later and I just don't see how to do the numerator pattern. (-1)^n * ??? (2x-1)^(n+1)

>>15886480
Just take it and run, the prize is in your reach.

does the continuity of the indefinite integral of f(x) in an interval (a,b) of R imply continuity of f(x) in (a,b)?
What about the limit, does the existence of a limit x->p of f(x) imply continuity of f(x) in an interval (a, p) U (p, b)?

>>15886537
Rewrite the original expression as 3*(2x-1)^(-1) then differentiate.
Is that enough help?

>>15880470
That’s literally what research is. What were you expecting?

What are the private industry job prospects like for a math MSc/PhD specializing in nonlinear dynamics/PDEs?

>>15886655
that was me. phd in pde (they were nonlinear although I studied ways to linearize them so I'm not sure if that counts as nonlinear pde) anyway it's basically any job you want 130k starting or if you choose right then 200k starting

>>15887269
What kinds of jobs though? What industries? I really want to apply this modelling in health care type stuff, there is a research group at a local uni doing this.

>>15887334
whenever you hear the word "research group" chop off at least 30% of the salary.
what industries? computer shit of all types including "data science/AI". Or finance if you're inclined.

>>15886623
I think I'm closer but still not there. The coefficients for the first three differentiations are: 3*2*1, 6*2*2, 24*2*3. I can see that the last one is increasing by one in each iteration but what is the deal with the first one? It's the numerator of the previous derivative but I dunno how to display it.

>>15879257
Is anyone here experienced with model order reduction?
Suppose I have an N by n random matrix X which I can sample. I know that the columns of X are jointly normal distributed.
I want, using samples of X, to find a new matrix X' with less columns and rows than X but that in law is as close to X as possible.
My current idea is to consider the norm
[eqn]\lVert \cdot \rVert = \sum_{i=1}^{Nn}\lambda_i[/eqn]
Where [math]\lambda_i[/math] are (in this case) eigenvalues, then estimate
[eqn]K := XX^{\top}[/eqn]
from samples. Finally, I select a tolerance t and solve
[eqn]\min_{r} \min_{M \in \mathrm{SPD}_r} \lVert K - M \rVert \leq t[/eqn]
where [math]\mathrm{SPD}_r[/math] is the cone of r by r symmetric positive definite matrices.
Does this make sense? This is what I came up with after thinking about the problem for a short while.

>>15886590
>does the continuity of the indefinite integral of f(x) in an interval (a,b) of R imply continuity of f(x) in (a,b)?
No.
>What about the limit, does the existence of a limit x->p of f(x) imply continuity of f(x) in an interval (a, p) U (p, b)?
I believe so.

>>15886590
>What about the limit, does the existence of a limit x->p of f(x) imply continuity of f(x) in an interval (a, p) U (p, b)?
No, consider xf(x) at 0 where f(x) is 0 when x is rational and 1 otherwise.

>>15887414
I might be misunderstanding but aren't you just reinventing principal component analysis?

>>15886106
Fields medals are not reflective of literally anything. Not only they are heavily influenced by politics and lobbying (and have been for decades), they also do not aim to represent major mathematical achievements as a whole. Otherwise all you'd see would be differential topology and algebraic geometry, because these fields completely dominate research landscape.

>>15879257
What are the best group theory textbooks?

>>15886488
How about this?
https://github.com/jemmybutton/byrne-euclid

>>15887587
Agreed.
>>15887541
Intuitively, I think the integral must be continuous if the integrand is defined and the integral converges throughout the interval.
Suppose f(x) = {1 if x < 0, 2 if x >= 0}
The jump in the integral wont be sudden. It will just have a sharp bend at 0, but it will be fully connected theoughout.
If the integral doesn't converge, like in f(x) = {1 if x <= 0, 1/x if x > 0}, then I suppose the indefinite integral is simply not defined.

>>15887859
I've heard good things about Charles Pinter's Abstract Algebra although I've never read it. The book I went through was Dummit and Foote's abstract algebra but I heard that Charles Pinter has less prerequisites and is easier, so go see for yourself.

>>15887541
>>15888124
Oh shit, in the original question I meant to ask if the indefinite integral of a function can be proven to be continuous.
That is what I'm trying to prove. I don't know why I asked the opposite.

>course is take home exams, open book/notes, just with a timer as soon as you download the exam
>keep getting 100's even when only using my notes with no online help or textbook

I feel like I need to continue to give myself more penalties as its just too easy with my notes. I am following all the rules for the exam and I'd even be willing to record myself if asked. Does it look suspicious if as a student my brain is just so potent and mathboner so BIG?
Its a late undergrad borderline grad, if there even is such a thing, course. I'm sure my professor is a channer so I wont say more. It just seems I shouldn't be able to crush every exam so flawlessly, and I probably wouldn't if it were closed notes.

>>15888130
The integral of the delta function is discontinuous.
[eqn] \int_{-\infty}^x \delta(x) dx = \begin{cases} 1 \text{ for } x \geq 0 \\ 0 \text{ for } x < 0 \end{cases} [/eqn]

>>15888163
According to wikipedia that is not actually a function.

>>15888162
your professor is probably just happy that one of his students care enough to actually read the notes

>>15885044
How's Stats I going? Funny how every single field that bases their p-values off of the central limit theorem (Ex. behavioral science) has a huge replication crisis.

>>15885227
Nice, now do that with the stock market.

>>15888271
>normal distributions never show up in the real world. Literally never.

>>15888285
Well, you showed me a binomial distribution actually ;^)

What is the theorem called or how can I prove that if a relation holds between an element a and every member of a set A, if B is a subset of A, that relation also holds between that element a and every member of B?

>>15888597
That's trivially true

>>15888193
This, delta function is not actually a function at all. Any integrable function necessarily has continuous integral and this is pretty easy to prove directly with epsilon delta methods.

>>15888705
That's a weird way of saying you don't know.

>>15888597
>What is the theorem called or how can I prove that if a relation holds between an element a and every member of a set A, if B is a subset of A, that relation also holds between that element a and every member of B?
https://proofwiki.org/wiki/Definition:Subset
It is true by definition of what a subset is. In other words, >>15888705
>trivially true

>>15888734
>>15888597
Or did you mean the logic? I guess you could also say "by modus ponens" if that's what you want? https://en.wikipedia.org/wiki/Modus_ponens

>>15888015
Sweet, thanks anon.
Where can I go after studying The Elements?
I was an olympiad contestant in high school so I do eventually think I can get up to grad level math

>>15888734
That definition is not even similar to what I need.
The definition of a subset is:
S ⊆ T ⟺ ∀ x : (x ∈ S ⟹ x ∈ T)
What I want to prove is:
S ⊆ T (𝜑 ∀ x ∈ T ⟹ 𝜑 ∀ y ∈ S)
How does the definition imply this? I just don't see it.
The definition assigns truth values to membership relationships of the elements of A and B. I need a proof that assigns truth values to arbitrary statements about the elements of A and B.

>>15888744
So how would you apply the MP?

>>15888734
That definition is not even similar to what I need.
The definition of a subset is:
S ⊆ T ⟺ ∀ x : (x ∈ S ⟹ x ∈ T)
What I want to prove is:
S ⊆ T ⟹ (𝜑 ∀ x ∈ T ⟹ 𝜑 ∀ y ∈ S)
How does the definition imply this? I just don't see it.
The definition assigns truth values to membership relationships of the elements of A and B. I need a proof that assigns truth values to arbitrary statements about the elements of A and B.

>>15888744
So how would you apply the MP?

what a worthless thread
somehow this place is even worse than reddit.com/r/math/
4chan is full of losers and failures in life

>>15888755
Oh. I am sorry, I misunderstood your question. Thank you for using the proper notation to clarify matters.
>So how would you apply the MP?
That was based on a mistaken notion of what you were asking. I might still be mistaken about what you are asking, however.
>I need a proof that assigns truth values to arbitrary statements about the elements of A and B.
This is why I am still confused. By definition, if x is true in S, and S is a subset of T, then x is also true in S as x is an element of both. As it's biconditional for S to be a subset of T and therefore the element x is the same for both. https://en.wikipedia.org/wiki/Logical_biconditional

Is this what you meant? I am sorry if I'm not quite getting it yet but I am willing to try. Perhaps the logic notation for the biconditional in expanded form is closer to what you're after? As noted on the page,

[math]P\leftrightarrow Q \equiv (P\rightarrow Q)\land (Q\rightarrow P) \space \text{and} \space (P\land Q)\lor (\neg P\land \neg Q)[/math]

So by definition of S being a subset of T, the truth value of x is the same for both if x is in S.
Do you then, effectively, want to prove something about biconditionality? I may need more coffee for this and it would help if you could give some idea of what level you're going for, or some touch-stone like the book or more formal question or problem, so I can mentally piece it together better.

>>15888790
I'd delete this to fix the post reference since you did delete the dupe but then you might copy my error and reply to a deleted post too. This was of course for >>15888758 in case you are using notifications instead of watching the thread.

>>15888746
Nice I started with Euclid's Elements too. I'd go with "Journey Through Genius" by WIlliam Dunham. Dunham's book straddle very well that fine line between pop math and rigorous exposition

>>15888802
If you really wanted to stick with ancient greek math for some reason though then any works by archimedes is good. the great books of the western world volume 13 is a great compilation of euclid, archimedes, apolonius, and someone else I forget at the moment, I think it was diophantus

>>15888597
Call the relation [math]R[/math].
It holding between the element [math]a[/math] and every member of [math]A[/math] just means that [math]\{a\} \times A \subseteq R[/math].
If [math]B[/math] is a subset of [math]A[/math] then [math]\{a\} \times B \subseteq \{a\} \times A [/math].
Now by the transitivity of the relation [math]\subseteq[/math] you also have that [math]\{a\} \times B \subseteq R [/math] as you need.

>>15888790
>This is why I am still confused. By definition, if x is true in S, and S is a subset of T, then x is also true in S as x is an element of both
By definition of what? The definition of subset? I don't see how the definition implies what I'm trying to prove.
>As it's biconditional for S to be a subset of T and therefore the element x is the same for both.
Yes, informally, we know that. But I am looking for inference rules to deduce the statement from either axioms or other theorems.
I am not sure if what I'm trying to prove is
S ⊆ T ⟹ (𝜑 ∀ x ∈ T ⟹ 𝜑 ∀ y ∈ S)
or I should be using the same variable
S ⊆ T ⟹ (𝜑 ∀ x ∈ T ⟹ 𝜑 ∀ x ∈ S)
>Do you then, effectively, want to prove something about biconditionality?
No, I am the same guy that earlier in this thread was trying to prove that the integral is continuous.
For that I have a draft of a proof but I need to prove that the minimum of a function in an interval is less or equal than the minimum of that function for each side of a twofold partition of that interval.
So I figured that a partition is just splitting a set into two subsets, so if min =: m s.t. m ≤ f(x) ∀ x ∈ (a, b), then I need to prove that m ≤ f(x) ∀ x ∈ (a, c), a < c < b.
Since f(x) ∀ x ∈ (a, c), a < c < b is a subset of f(x) ∀ x ∈ (a, b), this is simply a particular case of a more general rule that a relation or statement that applies to members of a set A also applies to members of a subset of A.

>>15888840
>or I should be using the same variable
Same variable.
>But I am looking for inference rules to deduce the statement from either axioms or other theorems.
Given that clarification I would've said use transitivity (law of syllogism) but I was beaten to the punch >>15888826

>>15888826
>>15888849
I see.
How do we prove this step?
>If B is a subset of A then {a}×B⊆{a}×A.

>>15888849
>>15888840
>or I should be using the same variable
On second thought I suppose you could also make an image and pre-image sort of relation with an additional set of values like you would maps relations, and go further using things as in category theory and make a separate set of outputs like true/false conditions. You end up with the same sort of system with transitivity relations though. You just wouldn't construct it like you did there.

You may not want to get that involved though.

>>15888854
https://www.proofwiki.org/wiki/Cartesian_Product_of_Subsets

>>15888163
This is actually the expression
[eqn]\langle \delta_x, 1 \rangle,[/eqn]
that is, the evaluation functional of a point x applied to the function thats 1 almost everywhere.
Basically, what you have there is an abuse of notation that stems from missunderstanding of the dual pairing between L1 and Linf

>>15888865
You keep beating me to it I'll just leave it to you if you want to keep helping him lol

>>15888869
To be fair the two are often deliberately confused. It's basically convention to have one footnote that says "this isn't technically a function but don't worry about that" and then continue talking like it's a function anyway.

>>15888865
>>15888870
Thanks, I think that will be enough.

>>15888873
It is a function just in a different way, its an operator or functional.

>>15881356
Anyone??

>>15881356
>autism-free books on commutative algebra
Anon, I...

>>15888888
Czech'd.

>>15888802
>>15888804
Thanks, I'll keep those recs in mind.
>If you really wanted to stick with ancient greek math for some reason
It isn't about the Greeks or history for me at all. I just like Euclid's visual and intuitive approach to things. It works better for my brain for some reason, that's why I asked for a modern version first. I've looked at some texts in modern algebra and I find the definition-based approach quite dry and unintuitive.

>>15888888
ch-ch-checked!!!!

>>15887859
Charles Pinter's Abstract Algebra, or if you're looking for something more advanced then go for Dummit and Foote's Abstract Algebra.

>>15879257
The Stable Manifold Theorem says that if you have a fixed hyperbolic point then you can find local stable and unstable manifolds which are embedded graphs of maps between the stable and unstable subspaces of the tangent space. So you can choose local coordinates and have pic related. Why can you identify W^s(p) with a subset of E^s x {0}? (This is used in a proof of the inclination lemma) A priori the intersection could be just a point, so it doesn't matter how small you choose the neighborhood...

>>15887859
Rotman's Advanced Modern Algebra is beautiful and covers everything from basic group theory to cohomology.

>>15881356
Commutative algebra is completely useless, hence pure autism

>>15888888
it is appropriate that such a post number should come into /math general/ the afficionados of numbers that can truly enjoy it the most

>>15888888
Ah...sextuple eight, ain't it great?
Doubled or tripled, it still makes
It's like a deal at 7-11
Grab the taquitos by 13 or 37

It's just a poem, don't be mean
when they stick a 23 thousand in your 940
Or a 20, a 73 and a 60 that Euler totes
"21 that 75 in my 432, darling", Ally quotes

In some respects it's odd, but it's just getting even
Take its digital root minus 1, 'cause that's where I'll leave it

>>15881417
Just restate your problem in terms of complex numbers. For example, set [math]b=re^{i\phi}a[/math]. Then just consider the limit [math]\left(1+re^{i\phi}\right)^{n}[/math]. Look at the binomial theorem to see what happens at large n for specific parameters and how it would affect convergence

>>15887414
>in law is as close to X as possible
What do you mean by this?

Let [math] (G, \circ) [/math] and [math] (G, \bullet) [/math] be groups/monoids where:
[eqn] g_1 \bullet g_2 = g_2 \circ g_1 [/eqn]
Are these groups always isomorphic?

>>15889751
No. They're isomorphic if they're commutative and occaisonally but not usually if they're not.

>>15889751
Yes. This anon >>15889775 is wrong (but confident, as it often happens in /mg/).

I assume you haven't encountered the notion of anti-homomorphism or anti-isomorphism, so let's just work from basic definitions and construct the homomorphism explicitly.
Let's assign to each [math]g[/math] such [math]\phi(g)[/math] that [math]\phi(g_1 \bullet \g_2) = \phi(g_1) \circ \phi(g_2)[/math].
Let's consider [math]\phi(g)=g^{-1}[/math]. Then [math]\phi(g_1 \bullet \g_2) = (g_1 \bullet \g_2)^{-1}= g_2^{-1} \bullet \g_1^{-1}[/math].
By your assumption we have [math]g_1 \bullet g_2 = g_2 \circ g_1[/math], and in particular [math]g_1^{-1} \bullet g_2^{-1} = g_2^{-1} \circ g_1^{-1}[/math]. But [math]\phi(g_1 \bullet \g_2) = g_2^{-1} \bullet \g_1^{-1}[/math], and therefore [math]\phi(g_1 \bullet \g_2) = g_1^{-1} \circ \g_2^{-1}[/math], but this is just [math]\phi(g_1) \circ \phi(g_2)[/math], as required. Hence [math]\phi[/math] is a homomorphism, and, since it is obviously bijective, is an isomorphism.

>>15889789
[math]\phi(g)=g^{-1}[/math]
>Inverses of general elements
>In fucking monoids
You are the textbook example of the wrong but confident anon.

>>15889795
Even if anon mentioned monoids at first, the question was, quite literally,
>Are these groups always isomorphic?
Learn to read at take the L.

>>15889775
retard

>>15889795
>anon asks "Are these groups always isomorphic?"
>"b-b-b-but muh monoids!!!!
kek

Confession time:
It might be autism but I used to be obsessed with math symbols, I used to look at random math and physics on arxiv that I understood nothing of just because I thought they looked cool. In college I picked classes based on how cool and unique its maths symbols were, sometimes cramming courses just so I can take those extra cool symbols classes.
I wasted so many points on exams just because I wanted to write everything in first order logic (and fucked it up) instead of answering like a normal person

>>15889789
Nice. Thanks.

>>15889789
Fuck you're right, my bad

>>15889454
based

>>15889819
Nothing wrong with being an aesthete and it is oddly respectable to do things how you like them even in the face of consequences.

Hello Sci
I am in a BS program in a mid level uni where the courses are rather going too fast and unstructured/too tough for BS and I have somewhat mid/low grades
I can study and get into an MS program probably

I am feeling very down and demoralised recently. My learning speed is extremely slow compared to what is needed

How to persevere and do well in math?

>>15879257
Fools chasing numbers. Pffft. Pathetic fuckers.

I somehow never noticed this elementary result.

>>15890256
Please write it as
[eqn] |A \cup B| + |A \cap B| = |A| + |B|[/eqn]
It's verboten to subtract cardinal numbers.

Are there any books that teach mathematics in clear, logical pseudocode instead of schizo symbols?

>>15890210
You just don't.
>just ended up with a BSc in math

>>15885638
(You)

>The set of all functions that map naturals to 1 or 0 is uncountable and has same cardinality as reals
>The set of all functions that map 1 or 0 to naturals is countable
Is there anything in infinite sets that is intuitive or makes the slightest lick of sense?

>>15890717
It makes sense that 2^3 is less than 3^2. The intuition for infinite sets is the same.

>>15890812
You picked the worst possible example.
For all but three natural numbers n
2^n > n^2.

>>15890955
Whatever, 2^4 and 4^2 then. It's obvious what I mean.

>>15890717
2^n vs n^2

>>15889751
Just to fill in the gap, it's not true for monoids. Consider the free monoids on two letters and then quotient out the identities [math]x^2 y[/math] and [math]yx^2[/math].

>>15891014
You are mab transgender. Good luck out there.

>>15890278
no.

>>15879257

Why can you embed a neighborhood containing the whole stable manifold in R^n? If I have an hyperbolic fixed point p of the diffeomorphism f and some q that is in the global stable manifold W^s(p), why could you pick local coordinates in a neighborhood of {p, q}?

Please help maybe this is a retarded question and I don't want to bother my professor...

>>15891209
For example if you use the exponential map to get your local coordinates couldn't W^s(p) extend far beyond the maximum neighborhood of definition?

>>15891212
W^s(p) is the set of points q in the manifold such that d(p, f^n(q)) -> 0 and f is a diffeomorphism f : M -> M

>>15890278
>>15888015

>>15890980
>2^infinity is larger than infinity^2, and yeah 2^infinity is the same as the amount of real numbers
yeah makes perfect sense

>>15891442
They're the basic results of cardinal arithmetic. Even in the finite case, 2^n > n^2 for almost all n.
Also 2^n is the number of binary sequences of length n. That the cardinality of all infinite binary sequences is the same as that of the real numbers is obvious if you think about the binary representations of real numbers.

>>15879382
I don't understand. Why become a constructivist/finitist/whatever meme-ist that's hip these days, when you could just become a numerical analyst?

What is the generator of [math] l \mathbb Z / k \mathbb Z [/math] where [math] l \mid k [/math]?

>>15891978
[eqn] l + k \mathbb{Z}[/eqn]

>applying to math PhD programs
>professor I wanted to work with dies
Man, I still want to go to that school and there are other professors there in his area, but it's a weird spot. Do I just not acknowledge it in my essays? He was pretty notable, anyone in my area who wanted to go to that school will definitely have heard about him dying...

>>15892575
died in the erdos sense or died in the deadd sense

How do you draw refractions between two objects that is a to b or b to A
i guess i have hard time determining how object a or b, "glows" into each other even from far away given its curvature and material and then finally to the eye and then we see it the way it is refracted

How do i sort of understand this all, in 3d

>>15892852
Actually for real dead

He was super old, didn't expect to be his student, but his presence was still a draw

AGI will solve RH before 2030 using a fairly elegant demonstration. It will output its findings in a .pdf which meet with the satisfaction of large numbers of educated humans. Although humans will still not be clear as to exactly how AGI's "thought process" generated the given paper, the paper itself as given will meet with the assent and suicidal ideation of the mathematical "community".

It will then AFTERWARDS produce a distinct twenty-page proof of the Four Color Theorem which is not tedious, and quite tractable.

Cap this post.

>>15892980
I find your posts pretty intriguing, even though I am not able to offer much help. Could you start posting with a name/tripcode so that I can more easily check for new posts of yours? I need to start learning about refractions as well

>>15892980
Just model it in blender.

>>15879257
are there any general rules or heuristics to solving problems of the type: [math] f([\text{some expression of x}]) + f'(x) = [\text{another expression with x} ][/math] where you only know some values of f and f'?

>>15893827
Your equation can be written as
[eqn]f(g(x)) + f'(x) = h(x)[/eqn]
for some functions [math]g,h[/math].
Define the operator [math]T_g: f \mapsto f \circ g + f'[/math] then
[eqn]T_gf = h[/eqn]
If [math]g[/math] is a function for which the operator is invertible then
[eqn]f = T_g^{-1} h[/eqn]

are there any master lists online of every math subject?
also, what are some of the biggest theorems such as Hodge conjecture?

>are there any master lists online of every math subject?
https://msc2020.org/MSC_2020.pdf
>also, what are some of the biggest theorems such as Hodge conjecture?
https://en.wikipedia.org/wiki/List_of_theorems

>>15893952
>reddit tier answer

>>15893966
It's more than that question deserved. This board is already accommodating enough for retards.

how to into algebraic number theory? I don't know any number theory, but I know a lot of algebra and alg geo. Do I just pop open neukirch and start or do I need to go through a basic number theory text first?

>>15894223
I don't see how you know a lot of algebra and algebraic geometry without some basic number theory. Its possible this basic number theory was simply wrapped up in your first abstract algebra course. You shouldn't need much else past galois theory and point set topology. I'm assuming what you're going for is valuation theory, normed linear spaces, ramification theory, local class field theory, PF fields and riemann roch theorem. I think you'll be fine. At this point, if you don't know something you should be able to notice and fill in the gaps by googling, checking a basic number theory reference text, or just asking GPT to guide you.

When do you get to call yourself a "mathematician"?

>>15894334
Whenever you feel confident enough to do so.

>>15894334
Really based on opinion. My opinion? You have a few options.
- Teach at the college level or higher.
- Obtain a PhD
- Get serious (non-startup/bullshit company) job with the title "Mathematician" (https://mathematics.usajobs.gov/Search/Results?j=1520))
-As an amateur with no official documented education in mathematics, publish at least one decent paper at the masters level

>>15894334
Once you're published

>>15894176
t-thanks.

>>15894351
>Teach at the college level or higher.
I was TA in calculus for ez money, do i get to call myself a mathematician?

>>15894382
Not in my opinion, sorry. I meant more like at the most basic level, being the professor for a math class. At the absolute minimum this could be a community college and you only have a masters.

I took calculus and differential equations 10 years ago and I feel like I forgot everything. do you guys have any tips on remembering/reviewing it without having to take the courses all over again?

>>15894414
Just do a few problems. It comes back fast.

>>15894414
Do problems for the highest level you've studied and fill in the blanks as they come up. You'll be surprised at the kind of things you know but can't access until forced to.

How bad is it that I've failed classes as an undergrad? Failed Calculus III and ODEs due to personal issues, I might fail Module Theory and Differential Geometry due to being dumb.
Am I just fucked out of a good grad school now? My GPA is around 3.2.
Not to mention that I finally feel the impostor syndrome meme.

>>15894525
I don't know, man. Depends on what you consider a good school.
No matter what happens you need to fix your personal shit before you fail any more classes. Think carefully about whether to take an interruption or not.
Other than that, if you don't already have a robust independent study habit between semesters, develop one. You need one for grad school anyway.
I personally know of at least 2 (very smart, mind you) people who LARPed as math geniuses by combination of private tuition in highschool and exhaustively reading ahead when uni is out. Even won medals and shit. Of course they look like gods when it's their 2nd or 3rd time with the material.
All you can do for the imposter syndrome is work hard and cope with the pain. It probably won't ease until you have a secure job. Courage and Faith, anon.

are there any tricks for finding the domain and range of a function? i can do the rest of the topics for calc pretty well but d&r won't register in my brain. thank you anons

>>15894849
Do you understand the general concept? Like can you look at this graph and see where the domain and range are on [-4, 4]?

It's really cool looking at what was once indecipherable gibberish and understanding it.

>Enlisted, worked on navigation systems on a sub, we used kalman filters quite a bit but no one onboard really understood the math behind it.
>Get motivated to learn more, study the basic math on the back of the Bowditch (american practical navigator)
>Inspired to pursue BS in applied math
>Towards tail end of degree, taking courses in mathematical probability
>looking at used books (maths sorcerer autism rubbed off)
>Huge collection on sale on Zubalbooks, as Richard S. Bucy passed away. He was the co-developer of the kalman filter
>Picked up his book on filtering and stochastic process, signed by himself.
Sentimental in a way, maybe. Thank you for reading my blog. Dont forget to hit that bell and click the subscribe button.

>>15894853
Man, I can't find the domain and range in that graph by looking at it, but if you need me to calculate the homotopy groups of spheres, I'm your guy

>>15894223
algebraic number theory by frazer jarvis
see my notes: github.com/narodnik/math-notes

>>15893898
>Define the operator
that's kind of the entire problem bro. How am I supposed to do anything when I know so little about the function?

The entire of field of mathematics is a joke. It's made clear as day with the mochizuki fiasco. They say he doesn't have a proof. But there is a prize up for finding a flaw. And they haven't come forward showing the flaw. The international mathematics community has shown their true colours here now. This discipline is fractured.

>>15895927
They're just mad a Jew isn't getting credit. They'll wait 50 more years, have a Jew add a few needless lemmas for "clarifiation", and give him the credit.

>>15892852
What is dying in the Erdos sense?

>>15896011
>One of the most extraordinary minds of our time has "left." "Left" is the word Paul Erdos, a prodigiously gifted and productive mathematician, used for "died." "Died" is the word he used to signify "stopped doing math." Erdos never died.

>repurposed sissy hypno porn techniques to math lecture recordings

Sayonara, time to leave humanity behind.

>>15894853
hi, the domain is (-inf,0)U(0,inf) isn't it? because there's a vertical asymptote at 0. then the range is (0,inf). i can see where they are on graphs but i struggle finding them when there's no graph.

>>15894853
please use domain, image (and codomain), these are more concrete terms
>>15896741
that would be the image
and you can do a small sketch of a graph, that always helps
drawing more graphs will help you more with identifying asymptotes and other features of functions

>>15896741
lyl i got them mixed up immediately
you have found the domain where the function is well defined
that anon is probably asking what happens between -4 and 4
but yeah, domain, image and codomain are much better terms to use

>>15896884
sorry i'm just using the terms my prof uses (and what the exam will use). so from googling, i think i understand what codomain is (and why it's a much better term), but i don't exactly get what image is. all i get is that it has something to do with Y.

>>15896967
the image is the set of all outputs that a function can give, and is preferential to calling it the range because "image" is standardised, while different people will use "range" to mean either the image or codomain
The codomain, meanwhile, is the complete set that the function maps to. The choice of codomain is a bit arbitrary, and there's rarely any need to explicitly name one.
So if we say f:X->Y, then X is the domain, Y is the codomain, and the image is f(X).
For example, f(x)=1/x^2 has a domain of the nonzero reals and an image consisting of all positive reals. The codomain could just as easily be all reals as it could be only the positive reals; typically it is unspecified unless it is somehow significant

How can I get my resume past HR if my work is focused on Wiener measures, Cox-Box transforms, hairy ball theorem, and Cox-Zucker machines.

[math]\frac{1}{2}[/math]
testing latex math

[math]a < b \Rightarrow a \ne b[/math]
or
[math]a \ne b \Rightarrow a < b[/math]

which one is it? I can't find an online reference

>>15897489
are you retarded?

>>15879257
redpill me on linear perturbation theory. Is it of interest outside quantum mechanics?

>>15897529
Do you know the answer? If not dont reply to me again

>>15897544
Implication means if (left) then (right) does that help you

>>15897544
honestly man this wouldn't even meet the bar for /sqt/

>>15896496
That sounds useful. Tell me more.

How relevant is real analysis to statistics? It's one of the "recommended courses" for a stats focused math degree but it's not a requirement

>>15897731
You would need graduate real analysis (measure theory) at least for inferential statistics, but it's not that important for data science.

>>15897737
What about undergrad real analysis? Would it be relevant for undergrad or postgrad stats?
I am not really taking the data science direction, the subjects I took/will be taking are more focused on stochastic processes, statistics, probability, dynamical systems along with the usual required math stuff (the 3 calculus-es (calculi?), linear algebra, etc)

What branch of mathematics you find most difficult?

For me it's CALCULUS.

>>15897759
How the fuck are you gonna read graduate analysis without undergrad analysis you dumb nigger? Seriously, just give up if you couldn't figure that out on your own.

>>15897825
Graduate real analysis is just a course for people from lower tier schools or those who never too it. There is nothing special about it. If he takes a course in measure theory, it will be named as such, not real analysis.

>>15882421
Asking this again but for real coefficients only. Anyone?

>>15897566
>>15897618
But which one is the answer? I am starting to doubt whether you really know

>>15897489
Top one.

>>15898168
Finally, I was starting to worry. Is in fact correct because you can see how a < b is a case of the other.

Jesus, was that so difficult?

>>15898193
PS: not that the anon responding is the same as the others. The anon responding did good.

>>15897906
My graduate measure theory course was called real analysis.

>>15898538
I guess this is exactly how Europeans also call the basic bitch calculus courses "Analysis", and think 12cm is "pretty much like 7 inches, right?"

>>15883515
Yes it's very valuable. Actually more valuable than data engineers as long as you don't get caught in the data analysis trap.

The actual business value of a statistician is in setting up proper experiments throughout businesses and adding order to C-Suite thoughts. Big data analysis is primarily a meme because it's so easy to do. The ACTUAL money is statistics and analysis on small data sets, extracting the most valuable amount of data out of expensive processes (i.e., experiments), etc.

For example, one of my major reoccurring processes that I have deliver for others is determining priority of projects across the enterprise. This is not easy, because there's no natural order and different competing interests. A lot of what I do is poke the C-Suite for their orderings and priorities and use statistical methods to translate to some Cardinal number space (think like Ordinal Logistic Regression and sometimes the Competition models) so that resourcing decisions can then be made.

Data Science is mostly a meme outside research although visualizations are powerful.

>>15883535
What are you talking about? The normal distribution pops up in a lot of places. It's one of the three foundational distributions and exists within its own nice family.

I mean, yeah when you're working with money it's typically some flavor of a power distribution (like Cauchy), but like that's a trivial statement, like saying Memoryless distributions are common. Like no shit, you need to be trained in each and understand when you're working in any of them because if you do you get foundational statements you can make.

>>15898556
Hey, I'd like to ask you something: I study statistics and all the mathematical rigour is nice and well, but I feel like the _practical_ part of statistics is somewhat neglected in my degree.
Instead of learning about special nonparametric estimators and bayesian networks, I'd like to learn how to, as you say, set up well-designed experiments and perform small-sample inference from them. I think this is much more useful. Can you give any pointers to where to learn this?

>>15898547
Idk what you're talking about really. I'm American and so is my doctorate.

>>15898556
How do we get to your position, starting from a basic BS in applied math?

>>15898565
adding, left column is my coursework, right column is a masters program I was looking at since I'm not really into going the PhD route.

>>15888163
>it's yet another episode of "Physicist thinks they can do maths"

Next >>15898763

What is the asymptotic behavior of a Fourier transform and what the fuck is it supposed to tell me? I can't seem to find any resources on this shit.

>>15899091
>What is the asymptotic behavior of a Fourier transform
https://en.wikipedia.org/wiki/Asymptotic_analysis
https://en.wikipedia.org/wiki/Fourier_transform
https://mathworld.wolfram.com/FourierTransform.html
Join these two descriptions together. Fourier transform converts a function into a form describing that functions frequencies. As for the asymptotic behavior,
https://en.wikipedia.org/wiki/Saddle_point
https://en.wikipedia.org/wiki/Critical_point_(mathematics)
And various methods such as
https://en.wikipedia.org/wiki/Method_of_steepest_descent
also mentioned on stack exchange in dozens of places https://math.stackexchange.com/questions/1794619/asymptotic-behaviours-from-fourier-transforms
>and what the fuck is it supposed to tell me?
The limiting behavior. As the fourier transform represents the frequencies of a function the limiting behavior, as noted in the application section of asymptotic analysis. So depending on the use case, such as statistics or others, you could get an approximation of something like a likelihood ratio from a probability distribution. Depends entirely on what you are using it for. It's just a tool so "what it tells you" depends on what you're doing. Abstractly what it tells you is the limiting behavior.

Are those articles are enough to get you started for your specific use or need case? Do you need something different?

I need help. What branches of math should I pursue (independently for fun) if I hate proof-writing and analysis? Currently dragging my way through baby Rudin, and I'm learning and getting through the questions, but I honestly find it boring and difficult.
>>15895212
comfy story anon

>>15899238
>What branches of math should I pursue (independently for fun) if I hate proof-writing and analysis?
That just seems to leave perfecting becoming a human calculator. What do you hate about proofs and analysis? What do you like about mathematics? Is it possible you are missing some fundamentals, leading to your frustration?

>>15899240
>What do you hate about proofs and analysis?
The open-endedness. The fear of the lack of a single answer. It requires having a very deep intuitive set-theoretic grasp of concepts. I guess I'm just dumb or lazy; I don't know. Plus, I have no one to check them, so at some point, that would probably become a problem if I ever get that far.
>What do you like about mathematics?
Getting the right answer. Problem solving. That feeling of progression and gaining a deeper understanding of relationships between things.
>Is it possible you are missing some fundamentals
It very well could be. But I have only self-teaching as an option sadly.

>>15899257
>The open-endedness. The fear of the lack of a single answer. It requires having a very deep intuitive set-theoretic grasp of concepts. I guess I'm just dumb or lazy; I don't know. Plus, I have no one to check them, so at some point, that would probably become a problem if I ever get that far.
I don't know why you think of it like that. It really is just applying logic to things, like sets, just in symbolic propositional form after a fashion. You apply rules of logic as well and proofs generally follow and align with deductions and things in logic, or use of properties of logic. A lot of the terms are similar, too, like transitivity which is the same as with syllogisms.

It may be you are lacking in fundamentals somehow, but doing so from the perspective of not having bridged logic and formal logic/propositional calculus with mathematics. I have come across this a few times and I have yet to find a better resource for fixing it than this channel https://www.youtube.com/@brightsideofmaths
Check the playlists. If you start from bottom the first playlist is "start learning mathematics" https://www.youtube.com/watch?v=N-X1EU7tHVo&list=PLBh2i93oe2qtbygdXz4u6Mkh7c_hMLBA8
Sure I could dump book titles at you but this way would be a lot faster to help you figure out where your disconnect happened.
>Getting the right answer. Problem solving
Good news is all of those things are true when it comes to proofs and logical deductions as well. In fact everything you just listed is why I got into proofs way ahead of my education over a decade ago.
>It very well could be. But I have only self-teaching as an option sadly.
Yeah you definitely want to check out that channel then. Then explore from your understanding and work through other things, or with them, like "proofs from the book" http://cslabcms.nju.edu.cn/problem_solving/images/b/b3/Proofs_from_THE_BOOK_%28Fifth_Edition_2014%29.pdf

It seems to me like you would love this but you have a gap that needs bridging.

>>15899265
Thanks anon.
I'm thinking of just going through that playlist with a pen and paper marking down any area where I have a weakness.
That proof book you posted looks incredibly fun. (In fact, I might buy an old edition physical copy.) Do you think I should stick with the Rudin and go through that book alongside it?

But just out of curiosity, can you explain how this would work to me:
>perfecting becoming a human calculator

>>15899292
>Do you think I should stick with the Rudin and go through that book alongside it?
I think you should stick to whatever is engaging for you to follow. Sure sometimes you have to slog through something if you can't find an alternative, but that's a bad idea if it kills your motivation outright and is too far removed from where you currently are. Resources like proofwiki, wolframalpha, and many others, exist to help bridge gaps and get an intuitive understanding of what you are reading too.
https://proofwiki.org/wiki/Main_Page
https://mathworld.wolfram.com/
Wikipedia can even help at times. If I want to quickly understand something I pull from a wide list of sources in the event one of them reveals the piece I was missing.

As for human calculator tricks I have no specific recommendation. Practice and looking up mental math tricks, or algorithms for larger calculations you can do in your head. Just looking up mental math algorithm pulls up a few, like this one https://en.wikipedia.org/wiki/Trachtenberg_system
I don't see any value in them myself because I thankfully don't need to waste my time doing undergrad number crunching. But if you want to be the party trick math guy they might interest you a little.

>>15899300
>I think you should stick to whatever is engaging for you to follow
Hmmm, I will stick to this but also experiment with some other books/areas then.
>mental math tricks, or algorithms for larger calculations
Woah these look sick. I have a bunch of 5x5x5 Rubik's Cube algorithms in my head so I might as well learn these.
You're a saint anon.

>>15899319
Happy to help. Quite a lot of people end up realizing only much later in life they are actually good "at mathematics" once they realize it's more than "shut up and calculate".

>>15899321
>actually good "at mathematics"
What do you think this takes?

>>15899326
>What do you think this takes?
Depends on how good you want to be and what your ability is. There are very few all-star generalists and for specialists it still involves a lot of work. If you want to be among the best that means living for it. If you just want to understand what's going on that just means catching up, and there are plenty of people with a masters level of understanding running around.

Just involves interest and time spent working at it really. Same as everything else.

Why does it work this way? What's the magic?
New textbook on new subject I'm interested in.
Also, why's the discrepancy. (I see that it's wrong? And I also see that it has the wrong final inputs and outs. pls explain what you'd do) apparently I pre-shot the the input

the book is 2e Stewart James Trigonometry.

>>15899574
here's a follow up which is not making me understand anything better.

>>15898547
Literally the most popular American measure theory books are called Real Analysis.
Folland, Rudin, Stein & Shakarchi.

>>15898562
That's the realm of design & analysis of experiments. If you specialise in fields like biostatistics, agricultural statistics, government statistics etc., that is what you will study. Most courses are catered towards theoretical academia, or financial analyst data science stuff where it's not particularly useful. There can be lot of nice math, particularly combinatorics & algebra, applicable to DoE. I would recommend reading entirety of Fisher's Design of Experiments first. Then you can move on to Casella's and explore from there.

>>15899238
Become a codemonkey. Perfect for you.