/sci/ - Science & Math

>>15192912
Most of them can't even be compared with one another. Also, Newton, Fermat and Lagrange should not be on this list.

Why do people who study mathetmatics always talk down to everyone else? Then you ask them what they do other than math and they just give you this blank stare. They never actually end up doing anything productive with all the math they know. Its just mental masturbation.

>>15192912
>>15177122
Okay I have a question about what a statistical model is. Here is the wikipedia
https://en.wikipedia.org/wiki/Statistical_model
It defines a statistical model as a pair [math] (S,\mathcal{P})[/math] where [math]S[/math] is the set of possible observations and [math]\mathcal{P}[/math] is a set of probability distributions on [math]S[/math]. So if one were to apply this to a simple linear model, what does [math]\mathcal{P}[/math] consist of? Suppose we have two random variables [math]X,Y[/math] (independent variable, dependent variable respectively). Is each [math]P_{\theta}\in\mathcal{P}[/math] of the form [math]P_{\theta}\sim N(\beta_0 +
\beta_1X,\sigma^{2})[/math]?

>>15192938
My bad, I'm phonefagging and can't delete. That should read this
>>15177122
Okay I have a question about what a statistical model is. Here is the wikipedia
https://en.wikipedia.org/wiki/Statistical_model
So it defines a statistical model as a pair [math] (S,\mathcal{P})[/math] where [math]S[/math] is the set of possible observations and [math]\mathcal{P}[/math] is a set of probability distributions on [math]S[/math]. So if one were to apply this to a simple linear model, what does [math]\mathcal{P}[/math] consist of? So say we have two random variables [math]X,Y[/math] (independent variable, dependent variable respectively). Is each [math]P_{\theta}\in\mathcal{P}[/math] of the form [math]N(\hat{\beta_0} +
\hat{\beta_1}X,\sigma^{2})[/math]?

Bumpin' >>15187298

>>15192912
>The Mathematician Greatest Mathematicians of Mathematician All Time

If [math] \Omega [/math] is the curvature of a connection on a smooth principal bundle [math] \pi : P \rightarrow M [/math] for some gauge group [math] G [/math] , and [math] v \in \mathrm{ker}(\pi_*) \subset TP [/math] , then do we have [math] \iota_v \Omega = 0 [/math] ?

Here [math] \iota_v \Omega [/math] denotes the contraction of [math] v [/math] into [math] \Omega [/math]

>>15192912
>no Khwarizmi, the father of algebra who was Persian therefore Aryan
westoid chud thread

>>15193064

>>15192934
why do you resent mathematicians?

>>15192934
lol what? I think you're just upset that you couldn't grasp a basic concept.

>>15192934
https://boards.fireden.net/sci/last/50/13576595/#q13582189

>>15193082
>1400 to 1950
now show me a pre-1400 chart where western europeans lived on top of trees and ate their own feces and didn’t have the capability yet to turn the rest of the world into a shithole like savage mongoloids
also
>implying the scientific community in the west wasn’t and isn’t filled with non-westerners

>Euclid
WE WUZ GREEKS N SHEEIIT

James Stewart or Spivak? I read and skimmed the exercises on Stewart already, but I'd like to thoroughly 'learn the concepts intuitively' as he outlines in the preface 7th edition Early Transcendentals. I also possess a copy of Spivak Calculus 4th edition, but looking through it there's more new information I'd have to learn, probably, which makes it difficult as doing a thorough rundown of all the exercises in Stewart.

>>15193230
The pre 1400 chart is visible on the right page.

>>15193236
sorry for nine9000 posting, but nobody answered the question. it's a lateral interest of mine, and i still don't know how to ask it even.

>>15193064
to be fair nothing the arabs ever did can compare to what was accomplished by the greeks and later the europeans. we just took the indian numerals, added decimals, did some shit with astronomy and trigonometry, then handed it off to the europeans.

t.arab

>>15193259
>we wuz persians

>>15192912
Hello.
I've had thoughts again.

Isn't the high school math curriculum just bad in all aspects?
It's just numbers, all you deal with is numbers in high school math.

Why? Lack of education among teachers? Are kids not capable of understanding anything else?

In my opinion, proofs should be introduced early, and critical thinking in mathematical concepts.
Teaching axioms to kids and start working on simple properties of numbers, and what we can find out about it.
Work through logic and critical thinking, through proofs and the likes.

Very few kids will ever use the math they learn in school. So why not reduce the "learning outcome" and instead introduce proofs?
That'd give kids fundamental logical abilities and help with critical thinking.

>>15193242
I said a chart including only the pre-1400 achievements
Of course everything becomes insignificant compared to the internet and television if you wanna be biased about it. It’s exponential, even though none of modern achievements would exist if it weren’t for pre 1400 ones. (especially in mathematics)

>>15192921
Removing Newton and Lagrange but keeping Hilbert is a good kek

>>15193259
>persians are arabs

>>15193270
If you're going to introduce proofs to kids, you might as well introduce leetcode, which would be much easier to justify to educators anyway.

>>15193230
>implying the scientific community in the west wasn’t and isn’t filled with non-westerners
It was not in the periods in question. Name the greatest x in any scientific field -- they are all Europeans in virtually every case. And when they're not they're Ashkenazi Jews i.e. Jews that stole some European genetics.
Like the other guy said, the 800 BC to 1400 chart is on the right. Note Europe is being compared to EVERYWHERE ELSE and by my count the area under the graph was also greater prior to 1400.
Btw this would also apply to philosophy, music, art, architecture etc., Example with music the world adopted European notation and the most played pieces are European (Beethoven, Bach, Mozart etc.).
This is math related because this trivially applies to the field of math.

>>15193270
Nah, proofs are kind of useless and the kids would complain a lot. It's actually better to just actually teach them applicable math. Teach them some computational math that can be done on spreadsheets then, teach them all the basic functions of excel and the math it's doing. Literally, the majority of those kids will touch a spreadsheet at some point in their lives and it's useful for them to understand what the functions are doing mathematically. Integrate this with what they're doing in algebra and do not punish them for coming up with computational solutions.

From there move them on to learning some basic leetcoding with a focus on again computational math and statistics. Again, keep it focused on the use cases. Integrate it with learning calculus and you will be providing real value to education. Proofs are fine with where they are taught, they're only ever going to be appreciated by mathematicians. Those other skills though are appreciated by mathematicians, scientists, engineers, corporations, small businesses and basically people of all stripes and colors in every single field.

I failed another calculus test again, I need to form study habits. Proofs are already hard to comprehend let alone write. I am going to get below a 70 at this rate.
I've failed you Kurisu, please forgive me.

>the set of continuous functions of bounded variation is of measure zero
Is there some kind of proof that all of the interesting subsets are countable?

>>15192912
>aryan
Did the (((Georg Landsberg))) thread upset you little baby?

>>15193358
There is at least one set of interesting, undefinable, continuous functions of bounded variation: polynomials where at least one coefficient is Chaitin's constant.

>>15193270
I wonder if there'll ever be a better way to manage teaching math. Sure it's nice that we push for more STEM and all, but since a society can't be ran by STEM alone. Why can't there just be a burger flipper track? Or a Walmart track? Or an artist track? Maybe the idea of math teaching critical thinking is just a flawed idea. That's not say that it doesn't work. Really, how else are you supposed to teach critical thinking anyways? The Darwin way? Math is a suitable replacement if you assume the benefits outfit the costs of the Darwin way. Survival isn't simple in a society of this scale, but mathematics allows one to thrive despite that.
Therefore, proofs or no proofs, I think the most important part is still logic and critical thinking, but if the best way to teach those is experience. Then the second best way might as well be math even if people seem to hate it or forget it. The alternative is too grim for them to even imagine.

>>15193347
Shut the fuck up worthless talentless schizo

If you drop the axiom of infinity but keep the axiom of choice do you still get well-ordering?

>>15192921
>t. Retard

Does [math]\frac {\mathbb{Q}} {\mathbb{N}} =3[/math]?

>>15193325
i just don't think this chart is that accurate pre 1400
but still, i was talking about western europeans. during 400bc europe is mostly referring to greece. I wouldn't put ancient greece in the western europe category but if you wanna we wuz go ahead. Its also weird how there's no trace of ancient persia when they had a bigger and more successful civilization than greeks and romans at the time. is it being counted as europe cuz it stretched to eastern europe at some point?
Also, westerners like to say the islamic golden age is a meme when they don't even learn it in schools, but art, poetry, chemistry, mathematics, etc. really flourished there from 800 to 1300ad, even if some of it could've been borrowed or translated from older greek and indian texts.

>>15193613

>>15192921
why do brainlets always underrate Newton so badly?

>>15193613
Amazing amount of cope

>>15193293
Persians did get arabized and browned from their original aryan selves, OBVIOUSLY. There was a conquest of the area and conversion of religion to shitslam as well from the aryan religion. This is well they got browned. There was probably already a browning from mixing of the white aryans with the pre-aryan peoples. The people there now look basically like arabs except some of the higher caste looks a little lighter than arabs.

>>15193700
>i like to make shit up

>How come the winners of the great divergence didn't instawin at an earlier point in time and how come the losers of the great divergence didn't instalose at an earlier point in time
Not science or math
>>>/his/
>>>/lit/
>>>/faggotry/

>>15193677
why are *nglos so desperate for a mathematical figure that they pretend newton is great?

>omega_0 is the closed set containing every natural number
How can this be done? There is no limit to the natural numbers?

>>15193711
Lol
>although the population is made up of rape babies, the genes are actually [related to proto humans that passed through the region before they evovled in other places]

>>15193259
Khwarizmi actually contributed to the story by organising the knowledge of the Greeks and he was cited by Europeans. Besides getting the font for your number system from Indians you didn't get anything from them that wasn't already present in MENA for 2000+ years before.

>>15193613
>I would not say that Europeans are European
Fortunately for us we don't need your input on how we identify ourselves. As a Greek I identify as European and maybe to a lesser extent with other IE peoples who shared our culture in the Hellelenic era, but definitely not any of the current We Wuzing occupiers of MENA shitholes obsessed with trying to destroy EU coherence.

>>15193613
>Also, westerners like to say the islamic golden age is a meme when they don't even learn it in schools, but art, poetry, chemistry, mathematics, etc. really flourished there from 800 to 1300ad, even if some of it could've been borrowed or translated from older greek and indian
Yes. Those fields really flourished _in Europe_, where the first true modern universities, hospitals etc. were being built, manuscript output was at record levels. New techniques in civil engineering (first buildings in human history taller than the pyramids), mathematics, automisation, fractional distillation, metallurgy (first nanoscale materials), medicine etc.

Throughout this time, contrary to popular belief, the most important ancient Grecian texts we still taught in Trivium and Quadrivium curriculum even in Western Europe.

>>15193762
Even Leibniz thought he was a great mathematician. It's because you faggots read Principia in isolation that you don't know how quickly the field and notation progressed after that which in itself was largely due to widespread dissemination of Newton's work.

>>15193711
>mostly
Yeah, only the elite of the society were ever aryan. That's the way it always worked when the aryans conquered a brown people. The same was with the indian people. The aryans conquered, the ran shit and built the civilization and created all of the notable things in the arts and sciences and built the great structures ect, then the aryan genetics get slowly submerged in the more numerous pre-existing brown inhabitants. Look at a city like new york. In a hundred years the population will likely be 99% poopie people. That doesn't mean that the poopie colored people founded the city though. Poopie people can't now nor never could built great civilizations.

>>15193762
literally every contemporary of his (except Hooke, probably) considered Newton a genius
And Britain is hardly short of great mathematicians, kek. Anyone dumb enough to not know the many great British mathematicians is just parading his ignorance.

>>15192912
Riemann should be n.1 and Gauss shouldn't be on there.

>>15193700
Where do you find this shit? The Aryans were the ruling caste. The root "arya" is pretty much identical to the word "noble" (which is "aristos" in Greek, again from the same root).
The Persians were not Arabs and they aren't now either. Literally nobody with a brain calls Persians Arabs.

>>15193354

>>15193830
>can't now nor never could
You talk a lot of shit for someone who sounds like a nigger

>>15193900
Dumbest post of the year

>>15192943
I'd follow the reference given in the section first, you know.
To be explicit:

[3] McCullagh, P. (2002), "What is a statistical model?" (PDF), Annals of Statistics, 30 (5): 1225–1310, doi:10.1214/aos/1035844977.
http://www.stat.uchicago.edu/~pmcc/pubs/AOS023.pdf

>>15193082
Why did they not normalize by population size?
This pic tells us nothing about Switzerland, for example. Of course the numbers wouldn't match those of France or Germany, which each are >10 times as big

>>15193259
>In the 12th century, Latin translations of his textbook on arithmetic (Algorithmo de Numero Indorum) which codified the various Indian numerals, introduced the decimal positional number system to the Western world. The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical text-book of European universities.

>In addition to his best-known works, he revised Ptolemy's Geography, listing the longitudes and latitudes of various cities and localities. He further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial. He also made important contributions to trigonometry, producing accurate sine and cosine tables, and the first table of tangents.

>Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before.

The fact he wrote these methods of algebra in the 820 CE, is simply astounding, and anyone foolish enough to see how simple it is today, can't grasph how inconcievable his methods may have been then.

>>15193813
>>15193899
not an argument

somebody needs to make the 4th volume of principia mathematica, and that person is not me (because I'm busy)

Among the square matrices of dimension n times n with rational entries and non-zero determinants, up to which n do nice formulas for the matrix inversion exist?
I expect them to also always be rational matrices.

>>15194014
All of them
[eqn]A^{-1} = \det(A)^{-1} \text{adj}(A)[/eqn]
where [math]\text{adj}(A)[/math] is the transpose of the cofactor matrix of [math]A[/math].
This is a nice formula.

>>15193613
>i just don't think this chart is that accurate pre 1400
but still, i was talking about western europeans. during 400bc europe is mostly referring to greece. I wouldn't put ancient greece in the western europe
The chart is about Europe, not western Europe. Greece is part of Europe.
>ancient persia when they had a bigger and more successful civilization than greeks and romans at the time
I don't know what metrics you are using, we the chart is about innovations, not size of the land or population.
>westerners like to say the islamic golden age is a meme
We do talk about Islamic world too, at least in Europe. I believe it is part of the curriculum in most of the history classes.
>art, poetry, chemistry, mathematics, etc. really flourished there from 800 to 1300ad, even if some of it could've been borrowed or translated from older greek and indian texts
On the contrary to the popular belief, not everything was just forgotten during the middle age in Europe. Latin texts were studied during that time, not only arabic.

>>15193961
nobody is arguing with you, you're just being informed that you're a retard whose opinion of Newton doesn't matter

>>15194045
>We do talk about Islamic world too, at least in Europe.
Yeah I bet you do nowadays.

>>15194062
newton as a whole was genius tier, but his math contribution was mid and replacable

>>15194126
Each post from you has been a disaster.

non-axiomatic systems are the future

i was thinking about axioms and primitive notions again. axiomatic systems still have primitive notions. if you could have a system with no axioms, and only primitive notions, i wonder what can be done with it.

>>15192934
some people might think maths is the pinnacle of academic purity, the essence of all truths and meaning. Of course, if somebody thought that, they would be wrong, because you can purity spiral more to get from mathematical logic to philosophical logic, and semantics of natural language, which are outside the reach of mathematics.

>basis, dimension, rank of systems of linear equations

filtered

>>15194032
thx

Hey ya!
I'm back with another problem. Looks veeeery interesting hmm hmm don't you think so?
I haven't attempted yet, no idea about difficulty but hopefully it's somewhere in the middle.
I appreciate anyone taking the time to attempt it. I can of course try my best to clarify things about the problem if there's anything unclear.

>>15194261
Math is philosophy, but a logical and pure form.
Normal philosophy works on natural languages and cannot form real logic, nor can it actually really explain much. It's an "idea guy" type of thing.

I'd argue normal philosophy is probably above sociology when it comes to "purity".

>>15193270
Highschool education in general was watered down after world war 2, prior to that not everyone graduated and even fewer people went to university. Its why a HS diploma used to be able to get you decent positions working at banks .

>>15194308
If he is a real jew the probability is 0 and even if the turk did get the keys the money would be long gone.

Why do the majority of modern texts present Bernays-Gödel set theory as a single sorted theory (so basically a pure "class theory"), instead of formulating it as a two-sorted theory (sets and classes)

>>15194391
Donno, but maybe to fit more with the dominant theory.
It's all fringe anyway, why overcomplicate it.

>>15193677
He was a glorified librarian. I'd sooner worship the street-shitter and suicide bomber mathematicians of the 10-13th centuries than Newton or Leibniz.

>>15194408
I bet you would.

>>15194407
Maybe the single-sorted approach makes the metatheory easier, but in terms of actually working inside the theory I'd argue a two-sorted presentation feels much more natural.

/sci/ is playing in the 4cc today. Shinichi Mochizuki, Norman Wildberger, and Yukariposter are on the roster. Game starts in about 40 mins>>15194318

>oral exam
>lecturer says to write him email for an appointment
>do so, receive no reply
>wait a week and talk to him after lecture
>says he received it but he's still collecting appointment dates so he can bundle them
>wait another two weeks, the week I suggested to him is coming
>ask him again per mail for an appointment
>no reply
>lectures are over so I cant pester him easily
epic, simply epic

>>15194875
Just go one level up? How is this even a problem?

>>15194308
I can't find any nice or easily computable formula, i surrender.

care to share solution/source?

r8 my shitty meme

>>15194308
>>15195046
Actually I think I'm just dumb and this problem is easy.
If I'm not wrong about the above statement then the answer is [math]\frac{k}{n}[/math].

>professor says he'll send email about some additional sources to better absorb the new material
>He sends out an email
>It's two links to libgen of the two books
Alright. He's actually by far the best professor I've ever had.

>>15195046
>>15195530
Hi anon. First of all thanks for your effort and time solving the problem. Unfortunately I have just arrived in the library and will begin attempting it now so I can't check your answer or look at the solution without spoilering myself. Sorry about this. The book is called "Combinatorial problems and exercises" by Lazslo Lovasz. The problem is on page 30 and it is the 6th problem of the chapter. The solution is on the back of the book. You can find the book in library genesis. Again, I'm really sorry about this. But I'm sure you got it correct! And thanks a lot again! I appreciate your time and effort solving it.

>>15196232
>>15195530
Alright, I solved it as well and checked the solution from the book. Nice job anon, your answer is correct! Congratulations.
Again, thank you a lot for your time and effort solving this problem. I think it's a pretty dope problem. I liked it.

[math]\to[/math]
should be used to denote logical implications.

[math]\implies[/math]
should be used to denote metamathematical or informal implications.

[math]T\vdash P[/math]
or just [math]\vdash P[/math], if the context is clear or left up in the air, should be used for formal provability claims.

[math]\frac{P}{Q}[/math]
should be used to denote imposed inference rules and rewriting principles.
This includes principles like Bayes rule when adopted as a principle. (The latter is not to be confused with Bayes law proven as just a basic factoring result in measure theory, and so for essentially combinatorical situations just as applicable in frequentist applications)

Those should not ever be mixed.

[math]P\to Q[/math]

[math]P\leftrightarrow Q[/math]

[math]P\implies Q[/math]

[math]P\iff Q[/math]

[math]\vdash Q[/math]

[math]\frac{P}{Q}[/math]

[math]\dfrac{x\in{\mathbb N}}{x+1\in{\mathbb N}}[/math]

[math](P\leftrightarrow Q)\ \vdash\ (P\to Q)\ \ \implies\ \ \big((x+1=7)\leftrightarrow (x+2=8)\big)\ \vdash\ \big((x+1=7)\to (x+2=8)\big)[/math]

[math]P\supset Q[/math]

Thoughts?

i hate computer scientists
that is all

>>15194032
I want to add that the identity
[eqn]\det(A)=A~\mathrm{adj}(A)=\matrhm{adj}(A)A [/eqn]
holds over any commutative ring. One uses this to prove Cayley-Hamilton, for example

what's an intuitive explanation for why the definition of subobject classifier involves a pullback diagram

take a system with no axioms, only primitive notions, it can be both consistent and complete can't it? why isn't more work done on this topic.

>>15196964
the empty theory is certainly consistent since any structure models it (vacuously, as the theory is empty)
it's most definitely not complete though. take for example the language with one binary relation symbol only; we can neither prove nor disprove that the relation is reflexive as we can come up with easy countermodels in each case (one where the interpretation of that relation symbol is reflexive and one where it isn't). again, those are actual models of our theory because they vacuously satisfy it

Why did Grothendick contribute of such an importance as the others in that list?

He just did abstract nonsense

>>15197003
Grothendick [sic] was actually well known for his functional analysis work before he did algebra stuff

>>15196881
Intuitively, the pullback diagram allows us to determine whether a given morphism "lands inside" a subobject S by comparing the image of the morphism under the subobject classifier with the terminal object. If it does, we say that the subobject S "classifies" the morphism f. The definition of subobject classifier involving a pullback diagram thus provides a convenient and natural way to describe subobjects and their relationships with morphisms

>>15193064
>>15193082
>>15193230
It's by charles murray so it's obviously biased.
I still think that "the west" has made most contributions by both volume and impact but he is very biased. he will consider every single british invention or book an accomplishment while only considering big accomplishments like gun powder an accomplishment for "the east".
yeah because the patented hidden gun dresser or picrel are equal to being the first to do combinatorics

>>15197055
Thank you anon. I think I'll still need some time to understand the motivation behind the definition.
I get most parts of the definition, i.e. a subobject classifier is an object (thought of as containing truth values) together with a designated element (i.e. arrow out of the terminal object), which we think of as being the truth value "true".
And they satisfy that for every object B and subobject (A, a) of B (where a is a mono from A to B), there exists a unique arrow from B to the subobject classifier (thought of as the characteristic function of the subobject) such that some kind of property holds.
And that kind of property is supposed to encapsulate that every element in the subobject A is sent to "true" and everyone else isn't.
I just kinda don't know how saying "such that this diagram is a pullback diagram" manages to capture exactly this property.

I'll think about this some more tomorrow, I can't concentrate anymore

>>15197003
On the contrary, Grothendieck's work was always aimed at the concrete, even if it used abstract methods.

>>15197119
This is just the universal property of pullbacks. If the diagram is a pullback, then A is simply the inverse image of the truth value.

My college requires you to take "introduction to complex analysis" before taking "complex analysis". is it really that hard?

>>15196964
Assuming you treat equality as a logical symbol, a theory, even in an empty language, will not be complete. You cannot prove or disprove, for example that “for all x,y, x=y”. There are many other theories that are both consistent and complete, like the theory of real closed fields, euclidean geometry, and the theory of directed graphs containing exactly one n-cycle for each n>1. They just are not powerful enough to encode arithmetic.

>>15197140
If you think you can skip it, talk to your advisor. Too many engineer and CS tards enrolling in analysis courses and crying about bad grades.

>>15197145
>a theory, even in an empty language, will not be complete

An empty theory*

>>15197140
it feels like linear algebra but its analysis instead

im studying math (undergrad 2nd year) and im realizing that i dont really like it i just dislike it the least out of all other curricula

>>15197181
Take physics and you'll learn to hate something even more.

>>15197181
If I were you I would find the thing I like the least and study that.

I'm doing calc 3 right now and I'm vaguely interested in model theory. What's my path to get there? I assume it looks something like this:
>Real Analysis
>Set Theory
>Abstract Algebra
>???
>Model Theory

For example, would I need any graph theory or topology? What's the usual way people get into model theory?

>>15197290
You need a mathematical logic course, as well as a course in algebra and topology. Set theory would be very helpful, as well as graph theory, but you can probably pick that up on the fly. The best thing you can do is learn as much as you can while learning the prereqs.

>>15197181
congratulations you've discovered the reality of life, everything sucks so you do the thing that sucks the least in order survive and not go insane

I'm in the middle of a differential equations class (ordinary differential equations), does it get better? The whole course is basically a bag of tricks to solve differential equations, with proofs scattered in to justify the techniques. But actually all of them outside of a few special cases are unsolvable so we use numerical techniques to "solve" them instead.

>>15197487
have you tried not being american?

Proofs are filtering me. Not just for particular theorems or problems either. I get stuck at the most basic proof problems. I can't figure out how to "manipulate" the assumption so it becomes the conclusion.
Analysis has turned me off of Calculus so hard, but I feel like I need to learn this just to improve my thinking

>>15197487
I think differential equations is not a topic that's bad in this regard.
I.e. there's certainly math subject with "less systematic theory" than differential equations.
Just gotta learn Hilbert space theory, integral transforms and how the interact with the derivative, operator theory, special functions theory. It's pretty rich and fairly systematic, really.

>>15197610
Learning formal logic helped my quite a bit, but it's a somewhat long path if you're not motivated by the structure of logic itself.
Another silver lining is that it will inevitably make you opinionated on debatable topics that you can then shitpost about on /sci/

>>15193259
>thinking persians are arab
Typical Arab niggerbrain

Is there a holomorphic function on the complex open unit disk whose limit does not exist (or infinite) at every point on the disk boundary?

I know examples which have no limit at any root of unity, for example [math] f(z) = \sum_{n=1}^{\infty} z^{n!} [/math] , but I'm wondering whether it's possible for the limit not to exist at *all* points on the unit circle.

Does the complex power series [math] f(z) = \sum_{n=1}^{\infty} z^{n!} [/math]
converge for any z in the unit circle |z|=1 ?

Note that it diverges for z any root of unity

>>15197713
Why wouldn't it?
The n'th term is a finite product of z's that's smaller than z^n.

>>15197740
Sorry, I misunderstood you asking for inside the disc bound by |z|.

A simpler question might be first to ask if the points are uniformly distributed in the limit.

>>15193082
book?

Might also be that one can argue in terms of a passive coordinate transform of the frame that all situations are equivalent (and thus all diverge).

>>15197744
>first to ask if the points are uniformly distributed in the limit
Lel I have no idea, should they be?

>>15197751
Hmm but other series can converge at some points and not others.
Consider for example the series [math] \sum_{n=1}^{\infty} \frac{1}{n} z^{v(n)} [/math] where v(2k+1) = 1 * 3 * 5 * 7 * ... * (2k+1) and v(2k) = 2 * 4 * 6 * 8 * ... * (2k) . This should converge at z= -1 but diverge at z=+1 , giving the alternating and non-alternating harmonic series.
(I'm assuming you're replying to >>15197713)

>>15197610
Don't be discouraged by proofs. You're learning a new way how to think while simultaneously undoing how you've been taught to think about math before proofs. It's a hard transition, it was for me to.

Also analysis is definitely rough, you're having to juggle a million quantifiers (for all, there exists) in your mind, while doing many many inequalities over and over again.

>>15197635
Yeah I'm hoping it gets better, I've just been turned off by it so far. I think it'll get better once we get into "whats actually going on here" rather than the bag of tricks computations.

One thing I'm led to is that if [math]f\colon{\mathbb N}\to{\mathbb N}[/math], then

[math]\sum_{n=0}^\infty {\mathrm e}^{2\pi\, x\, f(n)} = \sum_{k=0}^\infty \chi_f(k)\,{\mathrm e}^{2\pi\, x\, k} [/math]

where chi is 1 iff k is of the form f(m) for some m.
I.e. a Fourier series representaiton.

forgot my i

>>15197862
>>15197871
Ya the real and imaginary parts will give Fourier series when you plug in z=e^{it}

>>15195446
10^99 and 11*11 made me kek
i like it anon

>>15197713
It divergers for every [math]z[/math] with [math]|z|=1[/math]. Just consider the sequence of partial sums
[eqn]s_k = \sum_{n=1}^k z^{n!}[/eqn]
for a given [math]z[/math]. It can't be a Cauchy sequence since
[eqn]|s_{k+1} - s_k| = 1[/eqn]
for all [math]k[/math].

>>15197898
Ah right, thanks a lot anon

>>15197898
Wait a minute, this shows the complex series [math] \sum_n \exp(2\pi i n! t) [/math] is nonconvergent, but how do we know its real part can't converge to 0?

>>15197898
Wait a minute, this shows the complex series [math]\sum_n \exp(2\pi i t (n!)) [/math] doesn't converge in the complex plane;
but how do we know its real part also doesn't converge?

Let [math] C = \{z \in \mathbb{C} : |z| = 1 \} [/math] be the complex unit circle.

For any given subset [math] A \subset C [/math] is there a(n origin-centered) complex power series [math] \sum_{n=1}^{\infty} a_n z^n [/math] whose convergence set, when intersected with C, gives A ?

>>15192912
Anyone have experiences with excel and can help me with this?
I've made a little excel sheet for functions. The upper tables are for defining the variables of the functions. The lower ones then calculate the y values and its derivatives. The problem I have is that Excel returns an error for any 0^0. For example when I just have a constant 10 (x^0) then the 10 in 0 would return an error. I've now added #NV to hide the result in the graph but that isn't really a solution. I can't see the values of most functions for the y-axis in the table and it leads to wrong depictions of other functions. Anyone got an idea how to solve this?

I got a question for you math people. Say you have a population of data points with a certain distribution. Obviously, the closer you get to sampling the entire population, the closer the sampling distribution's shape will resemble the real distribution. Is there a way to calculate the minimum sampling size required for the sampling distribution to closely resemble the real distribution? Or should I just simulate this in R or something?

>>15198284
not an expert for this, but is there a set of possible distributions you are looking at to begin with?

What does [math]R^{d\times n}[\math] mean? What happens to the r and d? do they just mean multiplied? I mean they're both one dimensional, so wtf is the cross product? Just time each other???

>>15198590
Also what is the {+/- 1}^n ?
Is it a set of neg/pos 1's of n dimension???

>>15197181
>pic
go back

>>15198476
Yeah I don't know the name of what it would be, but it would look really similar to a negative binomial distribution. High probability of low values and low probability of high values, probabilities drop exponentially you move right on the graph

>>15198590
X is a real-valued matrix with n rows and d columns and y is a vector with n entries that are all either 1 or -1.

>>15192912
What is a good book for learning Modular Arithmetic? Preferably an old one so it isn't hard to get.

I like Modular Arithmetic because it goes in a circle, but I have a hard time remembering how the rules work. When I stop looking at the book I forget what the book told me.

Also, should I care about what a lattice is? I like how this word sounds and I see nice drawings of lattice.

>>15198745
>book for learning Modular Arithmetic
I mean, there is not much to learn anyways, just watch a good youtube video: https://www.youtube.com/watch?v=5OjZWSdxlU0 (Haven't watched that one but should do the work).
A book would be useful if you still wants to learn more after. If you want to solve fun problems try something like: Art and Craft of Problem Solving by Paul Zeitz. If you want to learn new ideas try something like Concepts of Modern Mathematics by Ian Stewart. Use libgen to get free all books.

A lattice is not something very interesting actually, you can think of a lattice as the set of vectors with integer coefficients inside a vector space. They are interesting but you need to focus in other things for the moment. Try to understand what is a vector space for example, that is more important. Try to think about what "linear algebra over the field of integers mod p" is for example.

Any probability anons have an idea for this problem?

Let [math]X,Y \in \{0,1,2,\ldots,n\}[/math] be discrete random variables and [math]Z \in [0,1][/math] a continuous random variable. What assumptions are needed about the joint distribution of [math](X,Y)[/math] such that
[eqn]
f_{X|Y}(x|y) = \int_0^1 f_{X|Z}(x|z) f_{Z|Y}(z|y) \; dz?
[/eqn]

>>15198284
>Obviously, the closer you get to sampling the entire population, the closer the sampling distribution's shape will resemble the real distribution
It turns out that this is quite far from the truth. Read up on the central limit theorem.

>>15198947
I'm not sure I follow. CLT states that the distribution of descriptive statistics of the sampling distributions will tend to be normally distributed, but this doesn't say anything about the shape of the sampling distribution itself, right?

>>15195530
>>15194308
I am not convinced.
Let P(rest | k, n) be the probability of being able to open the rest of the n boxes given you have broken open k.
Suppose k < n/2. Then P(rest | k, n) = 0.
Let's consider the smallest case, n=2. P(rest | 1, 2) = 1.
For n=3, we have P(rest | 2, 3) = 1.

>>15198981
Anon I believe there has been a misunderstanding. In my opinion what you said about P(rest | k, n) = 0 when k<n/2 is simply not true. I think I understand how you thought this was the case however notice that when you open a box with your key, that box also has a key inside it that could very well open an unopened box. I think you didn't take this into consideration hence your objection to the answer. The rest of the examples you've given for n=2 and n=3 also looks mistaken to me.
For n=2 you claim P(rest | 1, 2) = 1.
In other words, when we have two boxes and the turk breaks open 1, he is guaranteed to be able to open the other(with the help of the key he found in the first box). Again, I believe this is completely wrong. The first box he opens could have the key to the box it was contained in, hence he couldn't open the second box. A similar thing is true for what you have said about n=3. Please let me know if this misunderstanding is cleared or if you still disagree and think the answer is wrong.

>>15197139
ah, thanks so much. I think this made me understand.
In [math]\mathbf{Set}[/math] the property of the characteristic function can be stated as [eqn]\forall b \in B \;(b \in A \Leftrightarrow \chi(b) = \text{true})[/eqn]
Using the preimage, this can be more compactly stated like this
[eqn]A=\chi^{-1}(\text{true})[/eqn]
So the property really translates to "A is the preimage of true under the characteristic function".
I'll be honest that I already forgot what the text taught me about preimages in general categories, but I believe it was related to pullbacks in some way.

Thanks for pointing me in the right direction.

>>15199157
Thanks for the response, anon. I did indeed misunderstand the problem. I did not consider that the unlocked boxes could help open others, and I for some reason assumed that the Jew was generating derangements instead of all permutations.

>>15199638
Thank you as well anon for raising your objection. I'm glad we cleared up the misunderstanding. I'm curious though how the problem would have looked if it was as you understood it at first. A problem like "Given a random permutation f of [n], how many of f(1),f(2),...,f(k) are greater than k on average.". A problem like this maybe. I don't know, maybe it's an interesting problem, maybe it's not. I'm not sure what the solution would look like.

>>15192921
Leibniz > Newton.

>>15192934
Math is one of the few things that's actually real.
In that your entire life could be a dream or simulation, and all the math you know would apply when you wake up.
Or in any alien civilization in any galaxy.
Might have to change the symbols and names, but the logic and values would be the same.
And that applies for the rest of eternity.
I dunno, kind of a big deal, I think.
That's why the secret culture of the elite revolves around it.

>>15194330
>I'd argue normal philosophy is probably above sociology when it comes to "purity".
For example, nominalism.
This is the belief that when the trillionth set of ayylmaos discover pi, it's still just a coincidence.
Of course, that's false, but modern culture encourages the population to believe that everything is up for debate, and so nominalism is kept around for that reason.

>>15197747
Human Accomplishment by Charles Murray
>>15197097
It was a statistical exercise. Essentially, what he did was the following. Pick a subject -- in our case say math. Then grab various books on the topic (say history of math book) from various sources including European, American, Indian, Chinese, Japanese, Arab etc.. Then analyze which names and "events" were mentioned the most AND took up the most space. Those mentioned in over 50% of sources are then considered significant. Then he standardized for people who was "mentioned" most AND took up the most spaced, and you get lists like pic rel -- rather accurate if you ask me.

>>15199978
Here are a few more

>>15199985
technology

I don't understand the first answer in this problem:
https://math.stackexchange.com/questions/3661870/bayesian-problem-what-is-the-expected-number-of-mattresses-that-the-princess-sl

where does the 0.5 come from? and the generalisation of p(slept well)?

Guys, whats the difference between Analysis, Mathematical Analysis, Real Analysis and Calculus?

>>15200213
Calculus [math]\subset[/math] Real Analysis [math]\subset[/math] Mathematical Analysis = Analysis

>>15200229
Thanks anon. I wasn't sure thats why i asked. I already got books on calculus, real analysis and recently got a mathematical analysis book (the one from Apostol). Should i get an additional book on "Analysis"? shouldnt it be covered by Calc, real analysis and mathematical analysis, right?

>>15200264
Analysis and Mathematical Analysis really mean the same thing (depends on the context,but generally).
The Apostol book -- I just checked the table of contents -- covers real analysis, some calculus, and complex analysis.
I will be more through. Analysis can be thought of as studying the properties of topological spaces (basically a space with a fundamental structure; we only care about the properties preserved under "continuous" deformations like "stretching" and "bending" but without ripping it, or making any holes) with a notion of "distance" (metric), and especially the functions between such spaces. Within here there will include properties that one can study such as the "volume" (measure) of sets -- sets that are uglier and more complex than one might be accustomed to. Note I am being abstract here. Real Analysis and Complex analysis, deal with the spaces [math]\mathbb{R}^n[/math] and [math]\mathbb{C}^n[/math] respectively.
Calculus is then a set of tools derived from these fields, which are then applicable to real world problems. Here we have derivatives, integrals, how to solve them in various dimensions etc.
As for what books, it depends on your level of maturity. If you are familiar with the ideas of calculus and have a HS level understanding of mathematics, then you can probably move on to a Real Analysis book. I personally like Tao's and N.L. Carother's books on Real Analysis. Though perhaps brush up in set theory and the language and foundations of mathematics e.g. set theory, how to prove things, logic etc. a good book on this is called "How to Prove it".

>>15200318
Thanks anon. I wasn't really sure what "Analysis" is about since i saw some books like Rosenlichts Introduction to Analysis which seems to be a Real Analysis book judging by its table of contents. I als found Eschers and Amanns series "Analysis" which seems to cover Real and Complex Analysis, Manifolds, Differntialforms and Measure theory. Have you read the series and would you recommend it as additional literature?

>>15200343
I have not. Don't get ahead of yourself; first really do commit and learn real analysis THEN you can learn about manifolds and differential forms etc. Whatever book you choose, make sure you can find solutions to the exercises (be it official, or some blog). And DO the exercises. Not necessarily all of them. If the author stars the problems DO THEM. Otherwise I sometimes followed a rule of doing the prime numbered problems in each section (assuming they start from 1). Good luck!

where can i learn about second-order propositional logic?

>spend entire day working through a single section of a text
>stuck at one exercise for hours on end
>3 am
yep im retarded and hate myself
surely tomorrow it will all work out
goodnight /mg/

Keep fucking up a problem even though I know how to do it and I ended up punching myself in the face repeatedly. I'm gonna call it quits for tonight. I didn't get anything done. I hate myself.

>>15201285
>>15201472
get some good sleep anon been there

Exercise: show for a positive integer n, the number of ways n can be expressed as the difference of two triangular numbers , is equal to the number of odd positive integer factors of n .

As usual, the triangular numbers are 0, 1, 1+2, 1+2+3, 1+2+3+4 ... Note we are allowing 0 as a triangular number.

So for example, 3 = 3-0 = 6-3 , and 3 has two odd factors, 1 and 3 .

>>15192912
I just finished self-studying multivariable calculus with Fleming's book and I want to start with Complex Analysis. I've thought about VCA by Needham but it seems too long and I don't want to spend the next 8 months studying Complex Analysis (I do math 2-3 hours a day). Any suggestions? Thanks guys

>>15201778
Oh anon, this is a really cool problem. I don't know if I shared it here in 4chan however I shared it in a couple other places. Another way to phrase is to ask how many ways you can write a positive number n as sum of consecutive positive integers. Same problem, but this is how it was phrased in the book "Concrete Mathematics" where I first encountered the problem.

>>15192912
Why are they all fucking white

>>15201929
I consider this book idiot proof.

Hello /mg/. Computer science fag here.
I'm studying for my exam in "Foundations of Computer Science" by going through some of my profs past exams.
One question is (rough translation): Let [math]\mathbb{P}[/math] be the set of all well-formed formulas that are built up out of a single propositional variable [math]p[/math] and by repeated application of the connective [math]\rightarrow[/math].
Show that for any [math]\alpha\in\mathbb{P}[/math], if [math]\alpha[/math] is a tautology of classical logic, then it is also provable in intuitionistic logic (our deduction calculus has that usual modus ponens and introducing [math]\rightarrow[/math] via conditional proof rules, I don't think the rules for the other connectives should matter?).
I have tried using induction: Prove the claim first for the single prop variable [math]p[/math] (which works, since [math]p[/math] isnt a tautology), then try to prove that if it holds for wff [math]\alpha,\beta[/math], then it also holds for [math]\alpha\rightarrow\beta[/math].
But the induction step doesn't seem to get me anywhere? Like the assumptions in the inductive step don't really help.
Is induction the wrong approach? We did a lot of these types of inductions in class so I just assumed it would work.
Or is induction the right way but I just don't see how to prove the inductive step? Maybe I need some intermediate steps?

I'm not asking for a solution btw. Just a hint so I don't bash my head against the wall if my approach is wrong anyway.

>>15202127
Lol, you actually think /sci/ can help? How retarded are you? We just post memes and namedrop books we've never read like >>15201949.

>>15201938
Who is that girl at the top? She cute as hell.

Bros, I need help with maths. How do I get good at maths? I'll be honest, currently I absolutely suck at it. I barely passed math in my school life. I hated how it was taught and I hated maths.
Now in uni, I had a teacher who absolutely made me love maths. It made me realize how much I missed out. That faculty's course was the first time in my life I was one of the top scorers in my maths class. Although, this might have been just lucky since the next semester I got a bad teacher and I barely passed that class. I kind of find it hard to motivate myself without a great teacher.
Now, I'm trying to learn maths myself so I can catch up. But, I don't know how. How exactly do you get good at math? How do you cope with workload? How many hours a day do you have to study math? I need to know these since I need to juggle my math with my usual university courses which are also quite difficult.

>>15202313
What are some basic maths books that even an absolute brainlet like me can handle? It doesn't even have to be a book. An online course will do too.

>>15192912
Where Ramanujan?

>>15202336
In his shitting street

>>15202279
The one on top? Katyusha from Girls und Panzer. She's a soviet larper.
The girl she's standing on top of is Erika, german larper.
Solve this >>15194308 problem of mine and as a gift I can edit a picture of Erika holding a math book of your choice.
If you attempt the problem but can't solve it I'll still do it for you.

>>15202263
I've received help from some knowledgeable anons here before!

>>15202127
You want to prove that [math]\alpha \to \beta[/math] is intuitionistically provable, given that it's a classical tautology. This can happen in one of two ways:
>[math]\beta[/math] is a classical tautology
Then by the induction hypothesis you have a proof of [math]\beta[/math]. Look among your intuitionistic inference rules for something of the form [math] \beta \to (\alpha \to \beta)[/math] (it appears in the proof of the deduction theorem, if you've covered that) and apply modus ponens.
>[math]\beta[/math] is not a classical tautology, but [math]\alpha[/math] is a classical contradiction
Then [math]\alpha \to \perp[/math] is a classical tautology (by completeness of classical propositional logic), so by the induction hypothesis you can derive a proof for it. Intuitionistic logic should also have an axiom (= one-line proof) for [math]\perp \to \beta[/math], so you just need to look for (or use induction on proofs to derive) an inference rule that lets you smash the two together into [math]\alpha \to \beta[/math].

Also, ask on >>>/sci/sqt/ next time kthx

>>15202127
>>15202418
Errata: forgot that you can have tautologies with overlapping variables, like [math]\alpha \to \alpha[/math]. But you should be able to cover that case by using Craig's interpolation theorem to split it up into two smaller (classical) tautologies, and induct on the number of variables until it reduces to the base case [math]\alpha \to \alpha[/math].

>>15202313
>catch up
2bh school math isn't a lot of work and not that important. Barely passing it has roughly the same value (knowledge wise) as graduating top of your class. Just study new math, you can fill in the gaps as you need to.
>How exactly do you get good at math?
by doing math
>How do you cope with workload?
by doing an appropriate amount of math and working hard
>How many hours a day do you have to study math?
you don't /have/ to study math. If it's not for your classes, then treat it for what it is - a hobby - and assign time accordingly.
What are you studying? What uni math classes did you take? If you're studying something science related, then you should consider taking some math classes on the side.

>>15202313
As someone that has been self-studying math for more than a year (ik, not much but still some time) I can say that math is hard to love, even if you like it. I spend 3 hours a day studying it and it's not something I enjoy everyday but sometimes you have to do it to keep up with the habit, you will feel that it's very fun sometimes and very boring at others but it's very important to keep up and to not let the inertia of the joy it brings you be wore off by the days that you find it boring, it is a very recompensating hobby if you do it for long enough.
>How exactly do you get good at math?
Read books, do excercises (this is very important as it's what solidifies your knowledge. Don't give up after reading the excercise, try to work it out for at least 20 min)
>How do you cope with workload?
You don't, you just get what you don't like out of the way as fast as you can.
>How many hours a day do you have to study math?
Up to you, I started doing 2 hours a day, then I did 5 hours a day in Summer and back to 3 hours a day when september started.

50% into my first linear algebra course. This just keeps going into a deeper and deeper rabbit hole. I'm getting kind of nervous. Vector spaces are becoming too abstract for me, and I'm only at coordinate representation relative to the basis of a vector space.
Its like riding on a monocycle and the road I'm on keeps getting thinner and thinner, I can't keep this up at this pace for the coming exams and keep an A, but I desperately want an A. The course covers the entirety of Hoffman and Kunze.

>>15203014
>Vector spaces are becoming too abstract for me
you are going to have a heart attack when you reach Galois Theory

>>15203019
I'm considering switching to applied maths for that reason. I want to actually be good at it, not just barely scrape by with little understanding because I'm a good test taker.

I did up to calculus 3/linear algebra in college and am interested in going back to school for applied mathematics. I'd like to try and get the ball rolling myself again but I'm not sure how much review is needed. I've heard real analysis is the first major roadblock people run into so I'd like to start studying that myself, how much review from calculus would be required before I can do that? Do I need to finish a basic proofs book beforehand?

>>15203137
You should have already done a basic proofs class before linear algebra and calculus 3.

>>15200368
For arithmetic, the 2000's Reverse mathematics book

is there a natural number [math]a[/math] such that [math]\frac{3^{k}}{2^{k}}a[/math] would be a natural number for all values of [math]k[/math], where [math]k[/math] is a natural number.

>>15203326
[math]a=0[/math]

>>15203328
is 0 the only one

>>15203026
>switching to applied maths for that reason
You do realize the applied math students take the same linear algebra course, right? Unless it's a joke university.

>>15203332
nevermind, it's a dumb question. i'm dumb. so dumb.

>>15202369
>The one on top? Katyusha from Girls und Panzer. She's a soviet larper.
That's the one, thanks. She appears in need of headpats and/or ice cream.

>Solve this >>15194308 problem of mine and as a gift I can edit a picture of Erika holding a math book of your choice.
I have been looking into it, but I'm not making much progress.

>>15192912
I am using my computer to make Maid Space Operators out of Successor Function and Unsuccessor Function. So Maid Space can go up or down and have operators.

I made a radix changer where you take a Maid Space you want to change and Zero in a Maid Space with the radix you want.

If the first Maid Space is Zero then return the second. If it isn't, use Unsuccessor on it and Successor on the second one. Keep doing this until the first one is zero.

This works but probably there is a better way to do it? When I try to search for a radix change algorithm, all I find is something unrelated about radix sort.

The radix getting changes is important.

>>15203137
proofs are more important

>>15203137
Applied math is pure math + applications. Unless you go to a very shit university. You'll take the same core classes with the pure math people. Fair warning.
Are you planning to do a masters? Or a second bachelor?
Any applied math master program that is willing to accept someone with your background is probably a joke.
A second bachelor will probably do things from the ground up. So no need to do special preparation, unless you think it will improve your admission chance.

I don't think (real) analysis has any real prerequisite to it. I suggest going through a gentle problem-based intro book like picrel.

>>15203340
Yes, but like the other anon said, I will take higher up classes like galois theory that are frightening.
Linear algebra is the first class that is making me feel like getting an A isnt a certainty and I'm studying 2 to 3 hours every single day for it.
Maybe I'm not cut out for it, and maybe I should switch to a program not geared towards grad school or bust.

>>15202605
>>15202605
I can't cover every branch of math since its huge. I have interest in robotics and AI/ML, so I want to do maths that deal with those fields. But I'm not sure what path I should tackle everything for that. I asked ChatGPT for a roadmap but it isn't being helpful.

>>15203838
>But I'm not sure what path I should tackle everything for that.
Apply for a job, and learn in the job? https://search.therobotreport.com/listings/
Wnat you're doing now is basically procrastination.

>>15203884
I'm still in dergrad, I can't aply for a job. And how will I get a job if I don't know any math? You're right about the procrastination part, hence why I'm gonna start soon once I know where to start

>>15203494
>I have been looking into it, but I'm not making much progress
That's already enough then. Tell me what math book you want edited.

As for the problem, are you familiar with writing permutations as its cycles?
What the problem is asking could be rephrased as, what is the probability cycles containing 1,2,....,k cover all cycles of the permutation. There's a certain way you could write these cycles such that looking at the last element makes it clear whether or not every cycle has been cleared or not. I won't say more for now because I think this might have given the push you needed to carry the problem to the end, however if that's not the case there is nothing wrong with that. I can tell you the whole solution or give some more hints.


>She appears in need of headpats and/or ice cream.

Canonically speaking she's a horrible human being responsible for executing and torturing POWs and many more crimes I dare not describe.

>>15203587
A masters would be more applicable to what I actually want to do which is quantitative analysis, but I'm willing to do as many prerequsities as I need to in order to get there. I've also considered doing statistics instead if that would be a little bit easier in terms of what I need to know beforehand.

What math fields have high internet popularity and low IRL popularity besides metalogic and intuitionistic type theories? Conversely, what math fields besides computational linear algebra have no internet popularity despite high IRL popularity?

>>15201929
Help, /m(a)g/gots

>>15203961
I'd add
>Mochizuki's IUT (arguable not real math)
>category theory for the sake of category theory (topos theory and categorical logic goes here, some parts of higher categories are irl popular though because of Jacob Lurie and the Langlands program)
>things that sound cool to undergrads like fractional calculus, infinite dimensional calculus, hard computations of integral, series, inequalities,... math olympiad style of problems also goes here
>everything related to philosophy of mathematics
>old dying topics like logic (except first order logic maybe but still pretty niche), set theory, general topology, universal algebra,...
If you actually want to be a mathematician please avoid spending extra time in ANY of these topics and ask any of the successful researchers in your department for an advise in what to study.

anyone familiar with sympy?
I was to get the derivative of a series, but it seems that series is reserved only for taylor series

>>15198046
A precise characterization is open.
https://mathoverflow.net/questions/49395/behaviour-of-power-series-on-their-circle-of-convergence/49411

Ted Kaczynski (yes, that Ted Kaczynski) solved an easier variation of this problem in his PhD thesis:
On a Boundary Property of Continuous Functions
https://projecteuclid.org/journals/michigan-mathematical-journal/volume-13/issue-3/On-a-boundary-property-of-continuous-functions/10.1307/mmj/1031732782.full

>>15203986
2-3 hours a day should be easy for VCA. Skip starred sections, only go through the end of Chapter 9. It's probably the easiest text for complex analysis on the market that's actually decent too.
If you really want to speedrun (not recommended for learning math properly) then just read faster, and mark down but skip places you get stuck for at least 5 minutes. It'll probably get clarified later, or you'll retread some ground in later courses. But it's always better to figure things out yourself.

English is looking for a replacement for the word "field" due to the racist connotations of it. Personally, as a German I don't understand it. but you Anglos do you I guess.
In German, we use "Körper", which means "body" (etymologically related to "corporeal"). This could also serve as a replacement for "field", as "body" isn't really used in math-math, just physics. What's the term for field in your language?

>>15204231
Abelian division ring

>>15203900
>As for the problem, are you familiar with writing permutations as its cycles?
That is exactly the direction I was thinking in myself, pretty much to the letter; trying to find a representation of the cycles of a permutation in a form that provides some insight in the probability of the problem. But life got in the way and interrupted be; I will give it another go later.

>Canonically speaking she's a horrible human being responsible for executing and torturing POWs and many more crimes I dare not describe.
Well, don't meet your heroes I guess. She still looks cute in the pic.

>That's already enough then. Tell me what math book you want edited.
I'm good, thanks.

>>15193961
you were provided several arguments
your entire existence is an embarrassment

>>15204272
You are quite right, and my original response was wrong. Thank you for spotting my error.

I withdraw my claim.

Imagine if AI solves one the millennium problems, wew lad.

>>15204673
>Imagine if AI solves one the millennium problems, wew lad.
hasn't this already happened with protein structure prediction or whatever that's called

>>15201938
12

>>15193541
[math]\frac{\mathbb{Q}}{\mathbb{N}} = \mathbb{Q}[/math]

Without the axiom of infinity, does [math]\omega[/math] become a class or is it not definable at all?
Like does there exist a least class containing [math]\varnothing[/math] and closed under [math]x\mapsto x\cup\{x\}[/math]

>>15204696
I dunno about that, but I wonder if something similar to "you can't use math to prove math" applies here.

>>15204743
>I wonder if something similar to "you can't use math to prove math" applies here.
How would it apply

>>15203961
Intuitionistic type theory has become quite an active field of research in recent years.

>>15203988
Higher category theory is used in much more areas than the Langlands program. It's pretty much omnipresent in (algebraic) topology and (to a slightly lesser extent) algebraic geometry. Why don't you shut up if you clearly have no idea about modern mathematics?

>>15204823
Uh... Let me think about it.

is there a way to tell if a proof is the most efficient proof possible for a problem?

Have 3 chapters of homework to do tonight because I put stuff off. I'm not usually like this. I have all A's, not a dumbass. Anyway yeah. Please pray for me.

Also, I'm trying to impress this girl in my calc 1 class but I fucked up on the whiteboard today. The embarrassment made me so mad I bent my pen.

>>15205057
Technically it's all due tomorrow at midnight but I want to do it tonight to get ahead again.

I'm getting kind of hypnotized messing around with these attractive fixed point equations
x = 0.8*x+0.6
x = cos(x)
x = sqrt(x)
there aren't many practical resources that I can find regarding them in general, anyone know if there is a list of more similar ones somewhere? without getting too far off the deep in with theory, etc? I'm just more interesting in seeing how they converge from both sides towards a single value. If anyone knows, thanks.

>>15202313
>How many hours a day do you have to study math?
Taking 3 math courses at the moment, so about 4-8 hours a day during the week (including lectures), and typically 0-6 hours on weekends.

>How do you cope with workload?
Stockholm syndrome

There is no stem career general up so I'll ask here. I'm going to a very christian school, and when looking it up on reddit I see constant comments that they intentionally throw out applications that came from my school because its christian, white supremacist, anti-lgbtq, etc. Is this true? So even with undergraduate research, fantastic GRE scores, letters from active research professors, I am going to have my life thrown apart because the uni I went to was founded by christian people?
Has anyone here in academia heard of this, or is it just redditors larping?

>>15205057
>I'm trying to impress this girl in my calc 1 class but I fucked up on the whiteboard today.
Don't think about it too much. Just retain the confidence. I once did the same and failed horribly but through my sheer will I was able to suppressed my sperg self and brought out the inner chad in me to look confident. I got together with that girl in the end. gl

>>15193937
The link didn't work. Searched up the article else where. It seems that each [math]P_{\theta}\in \mathcal{P}[/math] is of the form [math]\mathcal{N}\left(\beta X, \sigma^2 I\right)[/math]. In other words a multivariate normal distribution.
As a math schizo I like when everything is well defined and with zero ambiguities, so non-math classes that often gloss over this information make me sperg out.

Fuck chatgpt
>very unsure about a question on an assignment for category theory class
>ask chatgpt
>tells me my answer is wrong
>change my answer to chatgpt's at last moment
>turns out my answer was right
I guess it's not over (yet)

>>15205322
Chat GTP is abhorrent for math. I also expected it to serve as a fantastic study partner, but it just gives flat out wrong answers.

>>15205325
Chat GPT is a good bullshiter and great at doing bullshitter's job like writing an essay. But when it comes to math, bullshitting doesn't work since it requires logic and actual understanding

>>15204735
I think the answer is yes, found some lecture notes on how to define classes inductively so I guess that would work

>>15205325
>>15205342
I have found it useful. For example I asked it why does
[math] \frac{\bar{X}-\mu}{S/\sqrt{n}}[/math]
have a t-distribution. And it correctly answered:

When we replace the population variance [math]\sigma^2[/math] with the sample variance [math]S^2[/math], the resulting statistic becomes

[math]T=\frac{\bar{X}-\mu}{\frac{S}{\sqrt{n}}}[/math]

where [math]S[/math] is the sample standard deviation, defined as [math]S=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2}[/math]. Note that [math]S^2[/math] is an unbiased estimator of [math]\sigma^2[/math], but it has a slightly different distribution than [math]\sigma^2[/math].
To derive the distribution of [math]T[/math], we can use a similar approach as before, but now we need to take into account the fact that [math]S[/math] is a random variable, and that it is not independent of [math]\bar{X}[/math]. Specifically,

[math]\frac{\bar{X}-\mu}{\frac{S}{\sqrt{n}}} = \frac{\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}}{\sqrt{\frac{S^2}{\sigma^2}}}[/math]

The numerator is a standard normal random variable, as we showed before, and the denominator is a scaled chi-squared random variable with [math]n-1[/math] degrees of freedom, since [math]S^2[/math] is an unbiased estimator of [math]\sigma^2[/math] and follows a chi-squared distribution with [math]n-1[/math] degrees of freedom. The ratio of these two random variables follows a Student's t-distribution with [math]n-1[/math] degrees of freedom, which is a distribution that accounts for the variability introduced by using [math]S^2[/math] instead of [math]\sigma^2[/math].
Therefore, we have shown that
[math]T=\frac{\bar{X}-\mu}{\frac{S}{\sqrt{n}}}[/math]
follows a t-distribution with [math]n-1[/math] degrees of freedom.
This distribution is used in hypothesis testing and confidence interval estimation for the population mean [math]\mu[/math], when the population standard deviation [math]\sigma[/math] is unknown and has to be estimated from the sample.

>>15205401
But for example if you ask it something more complex such as
Prove that if two measures [math]\mu_1,\mu_2[/math] are equivalent on some pi-system [math]X[/math], then they are equivalent on the sigma algebra [math]\sigma (X)[/math]
It spits out a wrong answer. However it was a very human wrong answer. It assumed one could split find a representation of any [math]U\in\sigma(X)[/math] by [math]U = \bigcup_{i\in \mathbb{N}} U_i [/math], such that [math]U_j\in X[/math] [math]\forall i\in\mathbb{N}[/math].

>>15205217
>Is this true?/liberal boogeyman
most probably not, I might believe it about humanities/social science/etc.
>undergraduate research
has a wide variance, the rec for that will matter more unless you're published in an obviously good way
>fantastic GRE scores
only useful as a filter
>letters from active research professors
this is the important part, letters from people that professors on admissions committees will believe and rank highly. with liberal arts/religious/non-research focused institutions, you might get a rec from someone who's last research was in their phd. you could legitimately be the best student of that professor, but it won't necessarily mean all that much

ayyyyyyyy
yooooooo

mad respect to the math guys!
yall hella smart!
I still don't understand how fractions work
Back to /pol/! Cya guys!

>>15205499
Hey, I appreciate your reply. I'm sorry I don't mean to fall for boogeyman stuff, but this was from comments on academic subreddits where presumably, active professors and researches post. All I care about is maths, I wish people would leave me and others like me out of their religious or political stuff....

>>15205217
If there is discrimination against Christian colleges at all it's likely that people believe they're not as good academically as regular schools. I could see someone seeing a university they don't recognize with a Christian name and assuming it's not any good. Honestly I would probably do that.
But that's not crippling. Plenty of people get into Princeton and Harvard each year from mid schools and some from even legitimately shit ones.

No one is going to trash your application for being a nazi just because you're religious (nor could they; grad applications are reviewed by an entire team of faculty). You're browsing some nutjob subreddits with persecution complexes and no idea how grad admissions work if that's what they think.

>>15205499
>>15205576
>>15205601
Honest question. Why bother responding to /pol/ bait?

>>15205604
Sometimes I take the more level-headed sounding bait because I know that a non-negligible amount of it is young people who spend too much time online and have been legitimately trolled into believing what they're talking about
Wasn't really paying enough attention to notice that a dude who can't speak English and called it "maths" is trying to pretend he's a white American

Saying goodbye soon. Doing a graduate program in statistics instead. I need the money.

>>15205758
Get a job, dumbass. Stop making your mother cry.

>>15204735
Omega is not an element of V_Omega

>>15205604
Seeing the current state of humanities faculties it's likely that something tangential to that could happen. Like, if there's a limited quota and the instute has to chose between him and a minority they may pick the minority. But I doubt any instution would deny an applicant from a religious school if they meet their qualifications.

>>15196464
>>15196466
Good practice in general.
>NEVER be mixed
Maybe an exception now and then.
>>15196474
Perfectly legible, but obsolete.

>>15204702
You're right anon. Though I think you should answer this guy since he asked it first. >>15201778
>>15204258
>But life got in the way and interrupted be; I will give it another go later.
I know what you mean anon. Just a couple days ago I was going to attempt a problem but then I got haunted by demons. It happens. Let me know if you want any hints or something.
>Well, don't meet your heroes I guess. She still looks cute in the pic.

That's how she gets you. You think she looks cute and then boom, loli stalin sends you to gulag.
>I'm good, thanks.

I'm not crying... it's not like I liked you or anything, baka.

>>15205322
>this nigga really trusted
the ai over himself because the bot sounded confident
i can understand not being able to come up with an answer yourself, but how are you unable to verify if some bullshit answer is correct?

This is a topology problem, right?
>>15206060

the more i study, the more retarded it makes me feel
ironic isnt it
does it ever become easier

OK, I passed this bloated course on functional analysis. Now I would like to be able to solve problems of functional optimization, like deducing that the normal function maximizes information entropy for given mean and variance.
Seems like a simple enough question: We have a loss function defined on functions (let's say I'm content with just optimizing for the Schwarz space, or maybe L^2 to have a Hilbert space). What are good books / sources on this topic?
In Hilbert spaces, I can find the closest function in closed convex subsets, which goes in the direction of optimization, but it's far from there.
Honestly, I'd be glad if I only learned just to maximize linear functionals, which information entropy isn't in particular, but it's something at least.

>got a really long math problem right on the first try.

>>15192912
My first exposure to proofs was Rudins Principles of Mathematical Analysis, and the course which had us do it.
I had not done epsilon delta before.


AMA

If
M=[[A,B],[C,D]]
is a matrix decomposed in terms of four other matrices, is there a nice formula for M^n for n a natural?

>>15204735
Definability does not depend on the axioms but only on the language. If you can define it, it's already a class. That predicate also gives you a class in ZFC. (But there this class exists as a set).
You're just asking whether the von Neumann model of the finite ordinals can be defined in set theory language. Can you take it from here?

>>15206185
No not really. Perhaps you are thinking about knot theory, but knot theory is about trying to differentiate two knots. I never learned the practical uses of knots until I needed to learn some knots for climbing, like the figure-8 knot or a stopper knot(can use a trefoil for this). But for those purposes in the post, I don't think you need to learn much about knots.

There's also the practical question of how to grow your hair long enough and making it strong, which is outside the realm of topology.

>doing my hw by hand
>so hard
>takes 2hrs
>whatever it was nothing just work
>using mathematica or sympy to do my hw
>WTF YOU HAVE TO BE A GOD USE THIS
>TAKES THOUSANDS OF YEARS
>I FINALLY DID IT I AM A FUCKING GENIUS
Anyone else feel smarter for being able to use programs for symbolic computations than actually doing it by hand?

>>15207773
No.
There's problems which, to get the answer in time (like multiplying parametrized matrices), you use Mathematica. There's also problems and questions where you want to use the full white paper and toggle data around and experiment in a way where formal syntax would be cumbersome.
It's both part of the modern toolbox.

>>15205396
Got a reference for this inductive definition? Is it by an infinite number of statements and are you happy with that?

What you can do is define the class of all elements which each are contained in all inductive sets - but without knowing any inductive set exists you can't prove that there's anything in it.

>>15207837
>Got a reference for this inductive definition?
Hi anon, in the meantime I realized that I'm not at all satisfied with the prove given in the lectures notes (https://www.cl.cam.ac.uk/~gw104/STfCS2010.pdf).).
Quick summary: on pages 77, 78 the author goes over general inductive definitions for sets and does the usual intersecting all closed sets stuff.
Fast forward to page 106 and he tries to prove that this also works for classes: For every class of rules there is a least class closed under those rules. He argues that by restricting the rule class to a set one may use the previous result that there is least set closed under the restricted rules. He then just unions them all together.
Intuitively, this feels like a smart construction.
And the (ii) part of the proof that I_R is in fact the least R-closed class is just fine.
But the (i) part is very sketchy to me. During the proof he wants to show that there is some set U such that y is in the set inductively defined by the restriction of R to U.
But it looks like he uses some form of choice here to construct that U (choosing a set U_x for every x in X).

>>15207837
>>15207935
I should also mention that it seems one kind of opens pandora's box when googling something like "inductively defined classes".
I get a shit ton of results about inductive types in type theories and their models, constructive set theories and how to deal with inductive definitions (I didn't know constructive set theory doesn't allow unbounded quantification, so you can't even do the "intersect all closed sets" stuff). I even found stuff about inductive definitions in NFU (in a book by Forster), etc...
Literally the only thing I can't find is an elementary treatment about if one can generally define classes inductively. And if so, how that works. Like you said you can't really do the top-down stuff of intersecting everything

Let [math] f : \mathbb{R}^m \rightarrow \mathbb{R}^n [/math] be an analytic immersion.

Is any local inverse of f (defined on an open subset of [math] \mathbb{R}^n [/math]) then also analytic?

I know it should be smooth, and I know e.g. local inverses of holomorphic functions are holomorphic; but I'm wondering about the multivariable real analytic case.

Let [math] f : \mathbb{R}^n \rightarrow \mathbb{R}^n [/math]
be analytic and an immersion (local diffeo).

Is any local inverse of f (defined on an open subset of [math] \mathbb{R}^n [/math]) then also analytic?

I know it should be smooth, and I know e.g. local inverses of complex holomorphic functions are holomorphic; but I'm wondering about the (multivariable) real analytic case.

>>15207935
I don't know what you really want to achieve and why, but if they start to talk about the set of rules, it seems more like metamathematics to me, and this "infinite number of statements" scenario that I mentioned.
The fact that you can ostensibly get a good collection of numbers by always doing +1 is clear, but that's an infinite number of steps to "get them all"
If you do set theory, then this rule based math like it seems to be used in the paper is another world anyhow.
PS constructive set theory does allow unbound quantification, but the more conservative ones restrict separation. But okay McLane also did this for his category theory types set theory. It's makes the theory be closer to type theories.

>>15207150
>I had not done epsilon delta before.
How the fuck not? I was exposed to that in lower level undergrad calculus classes

To expand Apostol's proof of the Fundamental Theorem of Calculus so that A'(x) = f(x) for f continuous in a, b, do you just plug x = a in the proof?

>>15208560

>>15208389
I did one single variable calculus that only showed the "algorithms", but didn't go into any details.

The several variable one had hand waving proos, and mainly just said "if it has these properties we can use it".

Technically I saw "proof by induction" in discrete math, but that was all.

>>15203900
>Canonically speaking she's a horrible human being responsible for executing and torturing POWs and many more crimes I dare not describe.
Based.
>>15204231
>fields are racist
Excellent. Gott straffe dien ewige Angloid.

ever sense letters started looking like this [math]\mathfrak{S}[/math] math stopped making any sense to me

currently in second year of high school. I have always done really well at math, and continue to do so. currently taking discrete mathematics and linear algebra as a part of dual enrollment. I've been having a similar issue in classes whether it was in trig, calculus, or my current classes.

the issue I'm having relates to memory retention, and application of knowledge. I practically never have issues when it comes to understanding concepts taught in class; as I usually am one of the first to finish practice problems during lecture. I am having issues, when it comes to remembering what it is I have learnt even after a single day has gone by. whether I'm doing homework; or prepping for the test, I feel as though I have to 're-learn' all the applicable material that is covered. I usually ace tests, but I actually want to apply what I learn in mathematics to projects related to graphics programming, not just pass the classes.

I've noticed that any given concept related to programming that I've learnt is often solidified through repetition. I wish to do the same for math, but was hoping that people here would have a suggested means of practicing, whether it be through an app, an online course, videos, specific books; or any particular combinations of what I've listed. I know that I'm willing to put in the work, and that I have the mind for it, I just need some guidance.

>reteaching myself maths from scratch to get better at it
>get stuck on a problem you get in elementary school
>im currently in college freshman year
recommend painless suicide methods

>>15209461
dedicating every waking second to doing mathematics until you die of exhaustion

>>15209468
I have very limited time as it is while dealing with my daily life and college. I might actually have to give up the few shreds of free time I get. But even then, I probably shouldn't bother because as I said above, I'm too retarded for maths I get stuck on an elementary problem.

>>15209452
>hoping that people here would have a suggested means of practicing, whether it be through an app, an online course, videos, specific books
A good suggestion here would depend on exactly what’s being taught in your courses. The most obvious idea would just to continue doing problems similar to your homework problems. For example, any linear system of equations can be solved by the same algorithm row reducing a matrix, so you can just make them up and solve them. Two other ideas
>linear algebra and discrete mathematics
These two subjects sometimes touch, which can be interesting to explore even if it isn’t in your course. For example, adjacency matrices and markov chains.
>computer programming
If that’s your poison, then why not make programs implementing your course material by hand? For example, type in a system of equations in some format, and get a solution or error message. Type in matrices in some format and get their products or inverses or something like that. No cheating of course: represent them as 2d arrays and implement the operations yourself. Just some ideas

>>15209474
appreciate the constructive reply. however, my primary issue stems from remembering even past subjects such as calc 1/2, as well as trig concepts. I would like to know if there are available means of practice online that anyone has found useful. not just for my current courses, but for even older ones such as the aforementioned calc/trig concepts.

https://stats.stackexchange.com/questions/605596/why-does-t2-change-to-f-in-anova
Can anyone who knows statistics answer this question? Moderator who doesn't even understand a basic theorem in Statistics I was referring to closed it down because I ignored his low iq comments.

anyone of you who knows this
>>15209651

>>15192912
>no beards between the ancient times and ~19 century

>>15193270
In California advanced kids are introduced to algebra and variables in 6th grade, and proofs in geometry in 8th grade. Normal kids are introduced to algebra and variables in 8th or 9th grade, and proofs in geometry their sophomore year, or 10th grade.

Mathematical language is very challenging to all middle and high schoolers from my experience, but students struggle much less with the math itself. Kids have a hard time reading certain questions and understanding what they are supposed to do, but when they figure out what they are supposed to do they can do it.

>>15209479
You can probably check out textbooks from those classes from your library, or talk to a teacher in your school and ask for a textbook to borrow. I'm sure there will be plenty of practice problems in there that can help you.

>>15210281
>Mathematical language is very challenging to all middle and high schoolers from my experience, but students struggle much less with the math itself. Kids have a hard time reading certain questions and understanding what they are supposed to do, but when they figure out what they are supposed to do they can do it.
This is common even in the university level, which is what makes online educators like professor Leonard or the Feynman lectures so popular.

degree or no degree, it doesn't matter, only the content of the work is of any importance. but it holds true that many educated in mathematics have produced nothing of note.

>>15192912
I've made a decision.

The idea that students need "mathematical maturity" before taking harder classes is total bullshit.
You don't get "maturity" from doing calc courses and whatever.

Sometimes it's a prerequisit, like it can be good to know what a vector is before you deal with it, but other than that, mathematical maturity is not something that comes to you before you do any harder courses (real analysis).

it's DURING real analysis that you gain maturity, not before it.
They should just let students take longer time with real analysis, while also not demanding as many prerequisite credits. That way students have time to look up things they don't know, and learn what proofs actually are, and deal with the course material, in time.

should i give up?

>>15192912
Hi, math friends. I'd like to expand my world-view, and I'd like to do that with a better knowledge of mathematics. There seems to be a bunch of versions of pic related. Is this the one mentioned in the sticky? All the links are dead.

>>15210817
Wait a minute...Stewart has a metric version of
calculus...? And the many editions before it?

what are some good math history books focused on a single topic/mathematician? I need to read and report on one but it can't be a general math history book

>>15199987
Federico Faggin (lol) and the others who helped invent the microprocessor are more important of a figure and changed the world more than any of those people.

>>15211207
The time period in question was 800 BC to 1950 AD.

>>15211209
then I would put Eckert and Mauchly, the Wright Brothers, Zuse, and others ahead of most of those

>>15210747
no
give it your all
if you're not going to make it, pivot
but never give up

>>15210817
yes

My homework bonus question is killing me.

[math]L_{n} = 2^{L_{n-1}} * p_{ (2^{L_{n-1 }+1} -1)}[/math]
[math]L_{1}=2[/math]
[math]L_{2}=68[/math]
Show that [math]W(W(L_{ n}))[/math] is always smaller than [math]10^{10^{L_ {n-1}}}[/math].

In case anybody is wondering, [math]p_{n}[/math] is the nth prime and [math]W{n}[/math] is the product logarithm.

Can you guys recommend me a book that deals purely with the mathematics side of AI/ML? Most books I see go deep into the programming and hardware aspects, not the math. I wanna learn the math behind AI models.

>>15211462
>analytic number theory is beaut-

>>15211462
No way is that a real homework problem.

>>15211643
I think you're misguided, there's few books which go into "the programming" aspects of AI. E.g. every deep learning book will go on about how regression w.r.t. some cost function is solved with gradient descent (the information being piped through the system in a multi-layer neural network with sigmoids and what not, so that the setup is differentiable and calculus can be applied). Like
http://neuralnetworksanddeeplearning.com/
That's neither programming nor very hardware near.
You can also find obscure "muh latent space, muh relearning" academic papers, but this won't tell you much if you haven't done the exercise of implementing a small network yourself. It can be done in 5 hours in pure python or C++, so just do that first.

If you think of yourself as a mathematician, it's important to not lock yourself into 3 departments at uni which make you believe only algebraic topology is math. Algorithms are mathematics, they came and won't leave the subject, even if it's the part that made innovations in the last decades. Even if we all like to see products S^n x S^m of two or three factors of n-spheres, a list of 10^4 rationals in the unit interval which you iteratively tune to recognize a 8x8 pixel patch of a cat nose, this is also a mathematical structure.
I know that there's an insaturable desire of math students to only study the fields that guarantee they will only read of great achievers of the 1910's and never themselves make something, I think it's worth getting comfortable with mathematics expressed in code blocks, or, god forbid, python.
This is not talk of programming, it's math. Talk of programming is C++ guys discussing the in's and out's of the new OOP class feature they just stole from lisp ideas from 60 years ago.

>>15211197
I think there's many. If the following counts as source, the first one that crosses my mind is Halmos autobiography, "I want to be a Mathematician"

I'd also recommend spending 2 hours here and find someone you like and find a book, of which there certainly are many

https://www.youtube.com/@moderndaymath/videos

>>15211842
Thanks for the detailed answer. I wanted to contribute to the open source AI projects so I wanted to get started learning AI. But, I wasn't sure where to being, the programming or the maths. I thought it'd be better to get my maths up to speed as I'm not exactly a math person.

>>15211904
Interesting, then my polemic was besides the point. Well I'd just go with the short Nelson text and understand e.g. gradient descent.

>>15192912
little help, what’s the integral of e^y^2 dA ? got final on Monday and prof just posted this question is gonna come in the final

>>15211353
>give it your all
Last time I did that I almost lose the capacity of communicating verbally... Well, as long as I can write I guess it doesn't matter.

>A function from a set X to a set Y assigns to each element of X exactly one element of Y.
This is the general definition of functions, but doesn't "exactly one" imply that it's a bijective function?

This is from wikipedia but the definition in my non-English textbook is more or less the same

>>15212016
Nope. Two different X could have the same Y, for example, making the function not injective. Each X is still asigned to just one element of Y.

>>15212040
Thanks, that clears it up

>>15212056
If X and Y are not the same size, then there aren't any bijective functions. Otherwise, and if |X|=n is a natural, then there's still only n! out of n^n functions that are bijective

I hate math so fucking much

>>15212596
why

>48% on physics II exam
>straight A in math courses
I fucking HATE this course, its such a messy pile of bullshit.

>>15192912
guys, can someone tell me something motivating to continue self-studying math? I don't have much motivation at the moment but I don't wanna lose this cool hobby/habut because some puntual moment of sadness. Thanks!

>>15212834
I just reviewed my exam, on a question worth 10 points I had -12, due to an additional -2 for using the word "radiating" to describe the symmetric distribution of charge along the radius of the surface given.
Would have been better off just leaving it blank.

>>15206742
Desperate bumpless bump

I don't get Cantor's proof for the uncountability of real numbers
Once you use the digits from the diagonals to construct a new number, you get a new real number that's different from all the other real numbers but still belongs to the set. I get it up to that point, but now how does that make it uncountable?

>>15213099
Nevermind, I'm retarded

>professor uses [math]z^*[/math] instead of [math]\bar{z}[/math]
on an unreleated note, how the fuck can I write huge expressions in a paper with lines without it looking weird?

next thread

>>15213505

>>15192912
https://mathjobrumors.com/

Is it better to think of a locally compact space as one where the topology has many elements, or rather as one where the topology has big elements?

>>15212930
what are you doing to self-study? I've been wanting to do so myself, but not sure what resources to utilize.

Why does learning (physics and) math feel so inefficient? It's as if it could quickly be explained simply clearly and explicitly, but it isn't.