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Discuss maths
Previously: >>15107772
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Well, /mg/?
>>
thread theme: https://youtu.be/FrHW6_aJSKc
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>>15140412
a) this is clearly a linear map and it has trivial kernel: L(0, dot) = 0 iff x = 0.
b) by definition X and Y are isomorphic by nondegeneracy of L. propagate y1 thru yk in Y back through this isomorphism to vectors x1 thru xk in X, then take their duals; these are the "vectors" you're looking for.
c) part (a) yielded an injection X -> Y*, blah blah blah you get the point
>>
I'm a grad student coming across hypergeometric functions for the first time and I gotta ask, are there any applications outside of DE solving?
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https://www.youtube.com/watch?v=mhIkyqLDl9M
https://www.youtube.com/watch?v=ZQOwG-hcd_k
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Scientifically, how plausible is it that the Mathematical Universe Hypothesis is the correct theory of everything?

https://alwaysasking.com/why-does-anything-exist/
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>>15140493
P=0.1
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>>15140491
>>15140493
fuck off
>>
Asking again.
Are linear spaces over finite fields ever useful?
I've only used them to construct some "interesting" groups, but have never seen them come up in anything more interesting ie. algebraic topology / geometry or such.
>>
>>15140702
I'm a geometer so IDK but probably some interesting cryptography applications that are, unfortunately, less powerful than elliptic curve crypto. I wonder if you can show correspondences to any interesting families of matrices.
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What is a set?
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>>15140717
it's left as an undefined primitive just like members of a set.
>>
I used to enjoy math in school, but now that I'm doing it in college I'm not having fun;
Algebra and discrete math are enjoyable, but most of my other courses are either calculus or oriented towards calculus.
And I can't differentiate or integrate for shit.
>>
>>15140729
Why do mathematicians accept systems that have more than one undefined primitive?
>>
>>15140403
whats with the basic /mu/core as op image
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>>15140712
>some interesting cryptography applications
True. I mean, the only I know of are uses of [math]\mathbb{Z}_2^n[\math] which is not quite usage of interesting LA theory, but maybe there are others.
To be fair, finite projective planes are somewhat interesting combinatorial objects and one would probably go constructing one with a linspace over a finite field.
Btw, how are you enjoying being a modern geometer? Are you in Rieman geom., or some fancy axiomatic geometry stuff, or something else? I'm not up to date with modern geom. research.
>>
>>15140717
>>15140729
>undefined primitive
wdym? They are the object of set theory, i.e. first-order logic with equivalence + the [math]\in[/math] with several axioms and axiom schemes.
>>
>>15140901
>the object
*the objects
>>
>>15140901
Is the following true?
>Sets are the objects studied in set theories, where a set theory is any extension of first order logic to incorporate membership and equality relations.
>>
>>15140956
>Is the following true?
Yes, I believe so, although it's not the unique definition.
I guess it's true that sets may be understood as "undefined primitives" and basically function on an intuitive level without a rigorous logical backing.
It seems to be the case in math though that any set that we talk about on an intuitive level may be formalized using a set in the set of i.e. ZFC or presumably other axiomatizations (I don't know anything about the other ones).
>>
>>15140478
Hypergeometric series show up in combinatorics. There's some theory of stuff called Wilf-Zeilberger pairs involving hypergeometric series you can use (with a computer) to solve a big family of combinatorial-looking summations algorithmically
>>15140702
Linear algebra over F_2 is useful a lot. There's a famous elementary folklore problem called eventown and oddtown with a neat proof using linear algebra over F_2.
There's also a method in crypto you can use for like randomized factoring and stuff that uses F_2 linear algebra called Dixon's algorithm. F_2 is useful all the time.
I don't think I know any problems that use bigger fields, 2 is kind of special. I guess there's Galois theory?
>>
>>15140403
Shit guys.

I passed real analysis, and I promised I wouldn't do any more pure math.
But now I'm doing measure theory, and teacher said the course is very hard. Did I fuck up? I guess I did... But also it feels like a very necessary course to understand anything about math.
>>
>>15140717
What does chatGPT say?
>>
Retard here
Does [math] x^2+y^2=/=0 [/math] imply that both x and y are not zero?
Why?
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>>15141528
no, just implies one is not 0.
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>>15141528
No. Consider x=1+0i, y=0+1i.
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>>15141538
My bad, should have also mentioned both are rational
>>
Back in 2021 /mg/ use to have 100+ post derails about what a set was or wasn't. Why doesn't that happen anymore?
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>>15140702
See matroids.
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>>15141561
contaminated water.
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>>15140702
Enumerative combinatorics
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Rationalize developing an AI that
- understands the major points of this short conversation
- correctly assess the value of the comment
- explains the value of the comment* to an interested party such as a chat-interacting user of the AI
*The comment is correct, the typeset page needs to be rewritten to be up to the standard of what mathematicians consider genuine, true, authentic, clear, precise, and informal as far as style is concerned.
However, essentially nobody except mathematicians are exposed to this austere and refined method of communication, and there is absolutely no reason to develop computers capable of handling the finer points of subtle communication between a handful of people unless there is a demonstrated market, and there isn't because mathematicians want you to hand over the blueprints, the formula, the secret sauce, and they will just rip you to shreds if you put up any resistance.
So, rationalize developing "AI Jesus" and walking into the lion/mathematician's den with a printout of the source code...
>>
>>15142225
I am impressed that you are capable of understanding what a monoid is and writing tex given your severe mental health issues. I hope you can continue to learn and enjoy mathematics without being distracted by those delusional thoughts.
>>
How abstract are higher pure math topics such as measure theory and topology compared to calculus I-III?. I found most calculus I-III very intuitive and "obvious". Does that mean I can ace something like topology?
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>>15142831
Topology is considerably more abstract.
But a significant amount of point-set topology, the usual introduction, just serves as a generalisation of concepts you'd find in real analysis, and that's more-or-less the theory behind calculus.
So if you haven't already done it, real analysis is a good stepping stone, either sequentially or simultaneously.
>>
>>15142873
Suppose that I'm done studying calculus I-III, would I be able to just jump into real analysis aftee studying the velleman's how to prove it book first. Or do i need other prerequisites.

I'm going to enter uni as a civil engineering major next 2 and half years but i just want to further my math education by self study
>>
>>15140717
a set is an object in the category SET of course
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we define the square root of negative one to be an imaginary number. and this concept has opened up a ton of real world math and physics and engineering.

wait until the first dumb human nigger figures out that there is another operation on negative one thats even more valuable and it will be defined like an imaginary number. and will be even more useful
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>>15142944
Real Analysis, Topology and Algebra have no prerequisites aside from high iq.
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>>15143234
Wouldn't a solid proof writing command make real analysis easier?
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>>15141049
See if you can make it past the drop date
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>>15143236
That's called high iq.
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>>15143237
I could fail both of my courses now, as long as I pass my thesis I'm fine, I have 5 extra courses already.
So, I'm not dropping it, but I might need all the 3 attempts at the final exam.
I'll have extra time because I'm taking a "grad level" CS course that's dumbed down enough to enable "cognitive science" students to take it (only requires a statistics course, which only requires single variable calculus and potentially linear algebra). And I've already gone through the maths behind PCA, SVM, poly/linear regression, feed-forward neural networks and CNN.. Which is probably like 3/4th of the course.
>>
>>15141561
Because those retards left the general.
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>>15143368
Shame, it was fun to spectate
>>
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>>15140717
>he/him
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this filters me
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>>15143628
Isn't it just a basics mathematical statistics course?
brosef... I guess this is why you chose comp sci?
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>>15143639
>Isn't it just a basics mathematical statistics course?
yeah, it is, with probabilities and combinatorics as well (at my uni)
i always hated those in high school so this is why i hate this as well
i actually liked calc 1 and 2 and diff eq
>>
I'm liking the algebra class I'm taking but the TA mentioned that in a branch of math called type theory the main objects that gets manipulated and transformed are proofs and spaces of proofs.
A - this sounds crazy.
B - is he correct?
C - if he is correct, how do I get into type theory?
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>>15143657
>how do I get into type theory?
Transition.
>>
>>15143657
Regardless of the jokes about what kinds of people bother with metamathematics, the TA is wrong. There are no branches of mathematics that manipulate proofs or "spaces of proofs", whatever that bizarre lunacy might be a mistranslation of, citizen. The whole idea is impossible. You cannot do binary operations on proofs, you cannot establish equivalence relations on proofs and you cannot represent a proof as a set.
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>>15143688
Isn't there a branch in CS that tries to basically automate theorems? Basically branch them and check so all theorems are correct, and then just I guess let it rip until it finds some useful theorem.

>>15143643
There's a lot of names and stuff, not much fun no.
For combinatorics you just need to understand permutations. To remove possibilities you divide them. If you want to remove equal sets that just come in different order, then you gotta check how many ways those can be combined and remove those too.
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>>15143688
A proof is just a tree. Of course you can apply operations to trees.
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>>15143688
You're wrong.
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>>15140702
As an algebraic geometer, I'm legally obliged to mention the Weil conjectures
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>>15143702
Proofs are not graphs of any kind.
>>15143716
Evidence..?
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>>15143799
>for all, blabla bla, implies, bla bla
That's a graph.
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>>15143976
That is not a graph
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>>15143657
>let A be the set of men
>let B be the set of mortals
>let socrates be an element of the domain of discourse
>[axiom 1] x:A, where x=socrates
because typing [math]x \in A[/math] on my phone keyboard is a pain
>[axiom 2] f:A -> B, where f=proof that all men are mortal
>[theorem] f(x):B
apply f to x (who can be interpreted as a proof that A is nonempty)
conclude that socrates is mortal

disclaimer: i don't actually know type theory and this example is completely made up
>also inb4 someone claims that A->B should be a dependent type, let's just say that f=hemlock, which works the same way for all men
>>
>>15143657
>proofs and spaces of proofs
There is a particular style of looking at types called Brouwer-Heyting-Kolmogorov (BHK) semantics that works a bit like that. In this style, saying “t is of type T” really mean “I have a proof that t is of type T”. Hence in this style to construct, compare, etc. types and their elements you inevitably are constructing and comparing those proofs. This “constructive” or “intuitionistic” is point of view is explicit in systems such as Coq. So this might be what your TA was talking about.
>>15143688
>There are no branches of mathematics that manipulate proofs or "spaces of proofs",
This actually was one of the claims being made to sell Homo/tranny Type Theory to normies , but mercifully, HoTT is dead.
>>
>>15143688
>There are no branches of mathematics that manipulate proofs
This is true. The relation of proof to math is quite strange, and doesn't go back all that long. The idea that proof and math are intimately related is likely a modernism.
Generally speaking, raw informal proofs are not considered to be "digested" enough into easily manipulatable formal parts, and students aren't taught formal proof styles like Fitch, so they have no intuition as to how formal proofs might be manipulated.
And students aren't exposed to formal inference rules, what would have to be taken as the definitions of operations that algebraically manipulate proof fragments.
As a matter of practice, no mathematician ever touches a formal proof. That work is left to logicians and computer scientists.
You could say that manipulating proofs is what mathematicians do, and it is left to logicians, philosophers, and computer scientists to contemplate what it is the mathematicians do. Although everybody agrees that mathematicians demand their work be treated as proof candidates and subjected to a review process that authenticates their work as genuine proof. But you have to do more, you have to demonstrate goodwill when it comes to the organization and presentation of your work; you can't use your work and the buzz it generates to abuse your enemies or launch a political campaign. So proof review is as much about defending the integrity of the perception of proof as it is about the formal correctness of the argument.
You can't expect me to read your proof where you define a subset of the natural numbers to be niggers.
>>
>>15142369
Alright, take the market to be "capable of understanding what a monoid is."
Think about it. What does this mean? We're already in trouble (and we're pretty sure that this is a small market) because
- the Wikipedia page for monoids is actually pretty great
- there are lots of insightful details on the Wikipedia page for monoids, unlike most Wikipedia pages
So as a result, our small (perhaps tiny) market is actually super stratified, with many layers of subtleties of understanding, deeper understanding, subtle and relatively well-established connections to category theory, universal algebra, and so on.
Once you understand monoids, suddenly the juxtaposition of words manifesting language is
>mere multiplication in a monoid.
So our "market" is actually our developer pool. We've confused our workers with our customers.
The people we're trying to message are so smart that we don't have solid ground to stand on to understand how we can support them in achieving their goals. We aren't twice as smart as them, so we can't observe them and "debug" the problems or issues as they go about their activities. That denies the math AI developer a potentially rich source of "value propositions" or reasons why the developer might have something of interest for the market.
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>>15142369
There's nothing delusional about confronting bullies like Bill Gates. Bill Gates is on the side of organizing denial of access to source code, i.e. Gates is an anti-academic and an anti-mathematician.
Ordinary people could get hurt (easily) by confronting bullies like Gates. Gates went out of his way to hurt people he perceived to be his enemies.
However, it's simply cowardly to rationalize patting yourself on the back for refusing to stand up to bullies like Gates.
You shouldn't take the side of bullies like Gates and suggest that only crazy people are strong enough to stand up to bullies like Gates.
>>
>>15142225
>hand over the blueprints, the formula, the secret sauce
>>
>>15144329
>>
>>15144343
Bill Gates is saying,
>"There will be no good software unless software is anti-mathematical, anti-academic, anti-scientific, and secret."
>>
>>15144344
cowardly
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>>15144352
Right. Cowards think standing up to bullies is delusional behavior because they have a mental block. Their cowardice actually clouds their judgement, convincing them that they will never develop courage and actually convincing them that it's reasonable to neglect to develop courage.
Is it?
is it reasonable to neglect to develop courage?
When bullies like Gates are at large?
Is that reasonable?
Maybe if you're weak af.
>>
what is it about math that attracts so many schizos?
>>
/sqt/ didn't answer, so I'm asking here. Is this correct?
>>
>>15144366
>"I'm a coward, I don't stand up to bullies, and it makes me feel uncomfortable when people stand up to bullies, so I call them names."
Fuck off, psycho. Gates took anti-West positions, and that's a problem for the West.
You are doing the dirty work of anti-Western psychos like Putin.
>>
>>15144366
Anti-Westerners attack Western mathematicians and brag about it.
That's why Western mathematicians want Putin murdered.
It's a reasonable response to a thread.
It isn't crazy.
>>
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>>15144383
Western mathematicians want to crank as hard and as fast on the murder crank as possible in order to roll out the red carpet for
- Western liberal democracy
- Market economy
- Western values
And then stick those dead motherfuckers in a mass grave, pave over it, install a sound system, hire a dj, hire a vj, install a light system, and throw a rave
And I want those kills.
I want credit for murdering the enemy.
http://mp3.hardnrg.com/ChrisC-Moon_Bass_Delta.mp3
This is about putting Russians & Ukrainians on kill lists and carving out exceptions to equal protection and other Constitutional Amendments in order to wage war on the enemy.
Western mathematicians want jackpot mass murder to defend the West and Western values, and they want it right.fucking.now.
>>
>>15144369
don't bother, it's solved (f_n and f need to have the same domain)
>>
Can someone tell me how to get Wolfram alpha to solve something like this

Graph f and locate all points of discontinuity
f(x)=
{ 8+x if x<1
{-1-x if x>=1

I'm a complete idiot I know
>>
Should I get a masters in math while I work full time? I’m finishing a cybersecurity masters while I work full time. I’m bored.

I dated a cute girl in high school calculus class, so I never really learned any of that despite getting the college credit. I have a BS in CS but I still perceive a gap in my math knowledge especially since a lot of math comes up with the data science techniques I work with. Should I do a free online course instead? Should I quit working and return to a CS PhD program to research AI applications for cybersecurity?
>>
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>>15142225
A quote from scripture, something about those who do not see the light, who cannot receive the love of our Lord Jesus Christ.
In short, if you've already decided to have FAITH in the people who develop artificial intelligence to assist math students in
- organizing their work
- participating in fraternal orders
- polishing their resumes
- tidying up their social network
- proactively developing opportunities to foster growth and engage the community instead of passively standing back, going into a cave, never coming out, and defending attacks with appeals to personality or introversion
>"It's just the way I am."
>"Go away."
>"God made me this way, there isn't anything I can do about it."
Faith begins when you refuse to put God in that box you made.
That FAITH is GOODWILL.
It's Field of Dreams.
If you build it, they will come.
Capitalism.
Dreams.
>>
>>15144619
Do you only have math up to calculus? How do you intend to take a masters with that? You wouldn't have the prereqs to almost any grad level math course.
>>
>>15144329
Question: do you try to get the market to self-identify? In other words, do you try to overcome the challenges
- small potential market consisting of individuals capable of understanding what a monoid is
- market is fractured and dazzled by the fantastic magic tricks on the Wikipedia page for monoid
by getting the market to think of itself as actually on some kind of "secret monoid club" where there are levels of understanding just like secret societies or management of large organizations, entire educational philosophies developed for the specific purpose of management or secrecy?
Is there an effort to convince the potential market that they're part of some social organization that tracks their monoid understanding capability?
And how do such efforts manage to convey a message that isn't just
>"Oh hi. We're the latest and greatest secret society. Sexy babes love us. Strong men with bombs want to help us. Join us as we dominate the world!"
It's a bit of a bland message. Did you watch too many Bond films?
>>
>>15144711
I have Calculus from high school, linear algebra from college, stats from college, and discrete math from college. I may be forgetting some
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>>15144778
>set with associative operation and identity element
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>>15144816
Anon, if it was going to be easy, I wouldn't have come to you.
You're going to be operating behind enemy lines.
I'm sending you to infiltrate dangerous organizations.
These organizations are known for powerful brainwashing techniques.
They're going to try to get you to associate with them.
They're going to try to get you to identify with them.
I'm counting on you, Anon.
I know you're reliable.
I can trust you.
>>
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>>15144825
Monoids are powerful magical objects that corrupt the hearts mortal men. I cannot bear such a burden. You will have to find another monoid-bearer for you quest, O stranger.
>>
>>15144731
I don't think you have any chance at a masters in math, and especially not if you work full time. I don't think anyone would even admit you to a grad level math course, you just don't have prereqs to really any grad level math course.
>>
>>15140416
this is really good. thanks for posting my dude.
>>
what did she mean by this?
>>
>>15145049
nothing, they only output a string that looks the most like it could be an answer
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>>15145055
dont say that about my wife
>>
>>15144619
MIT OCW
Unless you have a firm idea within 'AI applications for cybersecurity' I doubt many schools will admit you for a PhD.
>>
/mg/ is dead and its corpse defiled
>>
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I cannot into proofs
I'm too fucking dumb, I don't get the logic behind any of this
I look at the assumption, then the conclusion, but I can't tell what the fuck is going on in-between.
I'm supposed to be able to prove integrals with partitions right now but I can't even prove basic stuff
I wish Prof. Leonard had videos about these...
>>
Why are there topological proofs of fundamental theorem of algebra? What a waste of time.
>>
what's a dr. habil.? my lecturer makes us call him that instead of sir or mr
>>
I am cramming for an exam for the 7th time. It got me thinking. I do all of this in 3 days, where the first two days are idle, rereading, noticing patterns, writing out techniques and thoughts how to approach problems and then I cram theory and solve practice exams until I feel confident enough to stop. It's usually always an all-nighter the third day.

This gives me above average results. Scoring in the high 70s and mid 80s. Sometimes I even get to 90. My grades aren't half as bad.

But then, other students study for weeks and most get similar results, while the rest usually gets higher grades.

I never learn from my mistakes and always leave everything for the last minute.

Uni is a joke. This is coming from someone attending a masters programme in maths. Just last semester I learned and passed two algebraic topology exams, one dealing with cardinals and one in commutative algebra in under 10 days. What gives? Is the bar that low that even a retard like me can manage this?
>>
If you are struggling with a question should you go straight to looking up the solution or struggle for hours on end until you figure it out yourself?
>>
>>15140416
tropical island >>>>
>>
>>15140403
I feel like.

>Applied math
Is hard, boring, and frustrating

>Pure math
Still hard but curious, might not quite be "fun" but it's at least a bit closer to being a game.

I think I'll probably stick with pure math, and then once I have done "all of it", maybe applied math will be easier to understand.
>>
Just solved a fun small problem, if any of you want to think about it:
Given a finite group G og matrices in M_n(C) and the usual multiplication of matrices, prove that every matrix in G is diagonalizable
>>
>>15147777
you can do this with character theory btw
>>
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I'm gonna try calculus 1 tomorrow but I already know I'm gonna have to re-take it cause I know too little about integrals, limits and differential equations. Honestly I wonder if I'll even pass the barrier test. these are the moments when you regret not choosing a more scientific-focused highschool
>>
>>15148191
>differential equations
are absolutely not coming up in the context of an introductory calculus course
>>
>>15145954
The fundamental "theorem" of "algebra" is about analysis and topology, and not about algebra. The real numbers are a topological object, not an algebraic one.
>>
>>15148203
The prof added them to the exam this year, first order linear and separable variables(?), so you either get one of those or an integral. Basically he said the physics prof asked him to. I'm in Europe so I don't know if the curricula is the same as in america
>>
In the past I’ve seen people on /sci/ post a list of books that should be used to remediate one’s math knowledge starting all the way from basic algebra.

Do any of you have such a list and if so can you please post it?
>>
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>>15148203
let n be the order of an arbitrary matrix M in G, then M = M*M^(n+1)M^(-1)
>>
>>15148784
>>15147777
>>15148021
oops wrong (you)
>>
Hang on, which is the more appropriate notation? Is it
[math]x_1=\dots=x_n[/math]
or
[math]x_1=\ldots=x_n[/math]
??????
The former uses \dots whereas the latter uses \ldots.
>>
>>15140416
faggot
>>
>>15148869
\dots is context sensitive and automatically uses the standard alignment given its surroundings.
>>
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>>15148973
Really? Woah. Somehow I never knew that despite having used LaTeX for a decade, whoops. Thanks a lot anon. I appreciate this.
>>
What happens to the plotted line in the fourth graph of Anscombe's quartet if you remove the odd dot?
>>
Asking here because I didn't get an answer in sqt:
Any good books that focus on the counting aspects of combinatorics and probability. I picked up Casella and Berger to refresh myself of probability and got filtered by some questions in the first chapter itself.
Ex from Casella
My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get at least one call each day?
>>
>>15149160
this can be solved by finding how many ways you can distribute the calls so that every day has at least one, divided by how many total ways you can distribute the calls without restriction. I would use the stars and bars type logic to figure it out. Most any book on probability/statistics should have this, I used Solomon's Probability and Stochastic Processes.
>>
I hate linear algebra and I hate uni math education (sic how I phrased that).
I got a load of old exams from my uni. They all ask for shit like "calculate the determinant" (usually as part of an overarching exercise).
I have ADHD, but it doesn't matter much for the point I am trying to make, which is: this is literally the old "why do we need to study math if we have calculators now" adage, but this time unironically. Calculating matrices is error prone brainless, mindless busywork.
If I fuck up one -+ sign, you can chuck the entire exercise into the bin. This is dysfunctional. This is shit. Why should they not focus on conceptual big picture questions basically exclusively? Stuff like "if xyz is a linear map from blah lah then what are the criteria for ..." or "under what circumstances does abc satisfy a vector space for..." or "define the direct sum".

It's not like these exams don't have these conceptual exercises -- but getting all these right account for maybe 33% or less of the minimum points needed. What I am saying is that conceptual questions should account for almost all points needed.
This kind of small picture calculator work is just useless. It requires a certain type of person to find it "unproblematic".
But because I have to exercise and exercise and exercise these calculation problems I just have literally not enough minutes in my productive day where I can think about the actual conceptual foundations of linear algebra (and some other math fields). I just have focus almost entirely on not fucking up calculating determinants and matrix multiplications.
>>
>>15149255
you are simply lazy, you should probably quit math now as you will forget everything as soon as you leave the class
>>
Can someone explain this proof of why determinant is zero when two rows are equal in a field of characteristic 2.
>>
>>15149255
>If I fuck up one -+ sign, you can chuck the entire exercise into the bin. This is dysfunctional. This is shit.
What's dysfunctional about "you fucked up so it doesn't work?" Everything is like that.
You think your proof is going to come out okay if you drop a sign in a "conceptual" question?
>>
>>15149255
REAL math requires a lot of computations. If you ever research math you're not gonna be doing cherry picked problems carefully crafted to minimise computation. No, you're gonna be making 100s of conjectures. Also, it may that you are retarded and the question requires you to solve the computation a clever way, and not just brute force it.
>>
>>15149255
You don't lose all the points for making a small mistake.
Conceptual stuff is usually just for intuition, and not actually correct.

The idea is to use the intuition to do more effective number crunching. If you can't do the number crunching correctly you probably don't know how to do the number crunching.
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>>15149314
Since a_k = a_l, every term in the summation appears twice; once for a permutation pi, and again for the permutation pi' which is equal to pi except with the kth and lth position swapped
adding a number to itself in a field of characteristic 2 kills it, so each pair adds to zero
>>
>>15149314
Determinant is unchanged by row operations

And a row operation can make one of the rows zero, if there are linearly dependent rows

And a matrix with a row all zero has zero determinant

(You can also replace row with column above)
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>>15149355
Should add, this is true regardless of the field being used
>>
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What the fuck is this construction? At first I thought it was union(X*[0,1]),(Y*[0,1]) but it's actually breaking up the lines into points. However, if you break the lines up into points and place them wherever you like, even if you have uncountably many of them you only have countably many ways of manipulating them.
>>
>>15148207
You make it sound like analysis is just a dense part of topology
>>
>>15149325
Not how proofs work, undergrad.
>>
>>15149314
terrible proof
>>
Is there 5 Simple Math Problems No One Can Solve (Easy to understand, supremely difficult to prove)?
>>
>>15149341
Arigato gozaimashita anon-kun!
>>15149355
>Determinant is unchanged by row operations
Not true for all row operations anon-kun, but I see your point. Though the book uses this theorem to prove your claim, so I cannot use that.
>>
>>15149415
What proof would you prefer anon-kun?
>>
>>15140786
It also happened to me, you just have to practice a lot, like an engineer does, then you can start to understand what's happening, and it'll look trivial.
>>
>>15149423
>>15149427
You'll never be Japanese.
>>
>>15149429
I just want to learn Linear Algebra. Why do you hurt me so?
>>
>>15149429
>He said
>While being on a image board for japanese manga
>>
The Mochizuki ABC conjecture manga releases in eight days.
>>
>>15149399
This is literally from r*ddit
I would not bother
>>
How many ways to visit 6 cities twice without visiting the same city twice in a row?
A solution on Stackoverflow gives img related. I understand using inclusion/exclusion, but I don't understand this part: The solution also says There are 2* k^−6 * (12−k)! arrangements in which k cities are visited twice in the row, I'm confused on how they got this.
>>
>>15149435
>>15149446
Just act like a normal human being.
>>
test

[eqn] begin{matrix} \; & \; & \cdot & \; \\ H & -- & O & \cdot \\ \; & \; & | & \; \\ \; & \; & H & \; \\ \end{matrix}[/eqn]
>>
>>15142831
Much more abstract. You may like it or you may not. Personally I liked it even though they were very different from my applied courses, especially abstract algebra. You may find that you ace them, you may also find that you struggle with them. I hated the exams for the abstract courses, forcing out a proof is really not fun and so studying for exams mostly comes down to memorizing a bunch of bullshit compared to applied courses where you mostly just flow and do the problems on exams.

Still, the value of them is somewhat questionable unless you're really interested in academia. Pretty much the only thing I got out of them was a much cleaner and clearer mathematical writing style and a much stronger understanding and technique for proofs (which does come in handy in higher applied courses because proofs do show up).
>>
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Albert Einstein absolutely blows my mind with his genius, so maybe someone here can give an example of zeta regularization in physics for a simple minded fellow like me.
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>>15149482
>There are 2* k^−6 * (12−k)! arrangements in which k cities are visited twice in the row
I think you are misreading the answer, as well as misunderstanding how inclusion-exclusion works. This is the number of solutions in which _a specific set of k cities_ is visited twice in a row. There could be more loops than just those.
>>
applications of mathematics
>>
>>15150034
you're a homosexual pedophile and you jack off to children's cartoons, that has nothing to do with math. why do you feel so driven to spam the math and science board with your pedophillic cartoon masturbation material? 4chan already has dozens of other boards for that material, why do you feel the need to force your porno on /sci/ instead of posting it in the appropriate venue?
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>>15148191
so here I am again /mg/, I did what I could which was almost nothing. I think I got a limit right thanks to my class notes but other than that I improvised. Realised I don't know shit about exponential functions.
Can someone pls link a good repository with calculus 1 materials (study of functions, limits, integrals) that go in detail and explain each step?
>>
>>15142831
I don't know specifically topology, but "pure math" is quite different.

Real analysis is incredibly difficult if you've never done any proofs before. You also have to write proofs or "show" that a statement is true. That means remembering about 100 theorems, and know which one to use to solve a question (or just restate the entire proof).
It takes a while to understand that you take some assumptions, and suppose they hold, then you that you use them to show that something else is fulfilled. That's a very different concept from ordinary calculus.

Now doing measure theory, and if I pass this then topology is supposedly a "kindergarten course".
Some things are just absolutely obvious, but proving it is a while different thing.

But ye, it's completely different type of math. Kinda nicer, but also a lot more work.
>>
>>15150034
God I fucking love these autistic Japanese diagrams
>>
I was reading my profs notes on free algebras and he motivates them by saying that [math](N,s,0)[/math] with [math]s:N\to N[/math] and [math]0\in N[/math] and satisfying the three peano axioms ([math]0\notin s(N)[/math], [math]s[/math] injective, [math]N\subseteq M[/math] for any [math]M\subseteq N[/math] with [math]0\in M[/math] and closed under [math]s[/math]) is an example of an (absolutely) free algebra (of type (1,0) and on empty generators).
Now a footnote says that one can check that the structures above (he calls them Dedekind-structures) are exactly the ones satisfying the property that for any other triple there exists a unique homomorphism from that structure to that triple (this is also how we define freeness later on).
As an exercise I tried to prove this equivalence. I'm familiar with the proof of Dedekind-structure => freeness (i.e. that unique existence of homomorphism property), since uniqueness is just an induction argument and I'm familiar with the existence argument from set theory texts (the recursion theorem).
I also managed to prove that freenes implies axioms 1 and 2 (0 not in image, successor injective) by using the existence part of the freeness property.
However, I can't seem to prove freeness => induction. Surely I'll need to use the uniqueness part of freeness (since I didn't need it for the other 2 axioms). But I can't even come up with two maps that would help me conclude if i knew they were equal... Do I maybe need to use the full power of freeness and use both existence and uniqueness?
>>
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We actually have four ways to enter math
- ASCII pidgin
- square bracket math /math
- pic related
- PDF file
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>>15149150
Trying to calculate the regression line gives a division by 0
(10*551.2-64*68.9)/(10*640-6400)=∞
so it wouldn't exist
>>
>Every open subgroup of a topological group is closed.
Math was a mistake.
>>
I'm taking my first ever applied math course (it's a first year graduate level course) and I have no engineering background, just pure math. Maybe this is a retarded question but like I said I have no knowledge of how the physical world works except for that which I've gleaned from studying differential equations (which I'm quite fond of).

What physical situation could this differential equation be describing?
[math] x'' -\frac{k}{m}x + \frac{\alpha}{m}x^3 = 0[/math]

I don't want help with the actual problem he's given (he wants us to do something specific, and I'm already done). But like... why would an autonomous system look like this? The force acting on an object is inversely proportional to the CUBE of its position, with constant proportion... What? Anybody know of any examples of something like this?
>>
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https://archive.org/details/quadratox-the_reprogrammation_loops/+-+The+Reprogrammation+L∞ps+-+01+Redirection.flac
>AWWWWWW
YISSSSSSSS
>>
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>>15150895
trying to think of an example of a group embeddable monoid that has the property of the second exercise...
...or do we think B <= C iff B << C when A is group-embeddable...
>>
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A little meta-observation, please discuss:
(My ultimate goal is TCS, but I'm studying math.)
Certainly, math is work. It's not (just) about fun, it's mostly hard work and research-wise a lot of frustration, but anyway, I see two main end goals in going to study mathematics:
1. To learn enough to be able to participate in the creative art that is pure math and enjoy the ultimate intellectual esthetic.
2. To learn the craft for applied maths, i.e. understand e.g. PDEs and numerical maths so well you can really shine in physics research or such (or even engineering, finance, you name it).

That said, most courses I take have a flaw in these regards: they mostly make you learn theorem-proof, theorem-proof, theorem-proof ad nauseam + the additional drill of solving written-test exercises by hand. I can't help but look at my schoolmates and just see how most of them are mechanical theorem-monkeys. They are quite good at applying the admirable stack of theorems they have memorized for "theoretical applications", but there is no deeper understanding and frankly, no real skill. It's just being a nerd who does his job right. You can just tell they aren't gonna be the next Hilbert.
Anyway, it just seems like the logical conclusion of the way the courses are taught. For the first end-goal I stated, we are too focused on the definition-theorem-proof carousel that the bird-eye view of the theory is lost and mathematical creativity is stumped to the ground. For the second one, we get lost again, because how do we react when we are presented a real-life model? Will we remember Theorem #378546 along with Lemma #648364 can be applied? I doubt it.

My question is, do universities generally develop their math courses in this way as well, or is any one of you lucky enough to study in a way that gives you more than the (memorize a lot)->(pass exam)->(probably forget in five years) brainslaughter?

>inb4 I'm dumb CS brainlet getting filtered
I've got perfect marks so far.
>>
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>F. William Lawvere (February 9, 1937 – January 23, 2023) was an influential category theorist.
>>
>>15141528
Yes
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>>15151077
>influential category theorist.
Lost
>>
>>15151072
actually intelligent people get terrible grades because they're constantly dancing, doing drugs, and threatening God if impossibly terrible study habits aren't rewarded with super-intelligence
but in all seriousness...this is completely moronic
you have to do two things
1. read what you wrote out loud, and get a copy of Audacity and record it and listen to it, over at audacityteam.wtf or teamaudacity.ass or whatever
2. go to /lit/ or /his/ and bring these subjects up with people who actually have a background capable of giving you intelligent responses
3. it's super easy to reinforce 1 if you get your computer to read what you wrote out loud for you
also
>muh supermemo/spaced repetition
>muh dharma
>>
>>15151077
Category theory is C++
Important for historical reasons
Avoid it like the plague
>>
>>15151191
C++ is great. C++ is a proper programming language with rigorous design. it's the java, javascript, html, webfaggotries... that should be avoided.
>>
>>15151072
In all fucking seriousness, just fucking write down every fucking theorem you've heard anyone mention in the last 18 months, as much as you can from memory, and spend some time figuring out which theorems, on which pages, in what books, &c.
Fuck the unknown theorem.
Do not give me formal theorem string rewriting crap.
You're not a fucking machine.
Real people don't give a shit about formal shit.
Real mathematicians force all actually effective formal-cognitive contradictions through cryptolects, to compress the "business logic" into the same socio-linguistic modules used to...nevermind...and those cryptolects are organized by model theory, see pic related
>give it up
http://mp3.hardnrg.com/morgan/Morgan-Sentinel.mp3
>>
>>15151187
Are you sure you replied to the right post? Either that or you might have forgot to take your meds today, but maybe I'm just misunderstanding you.
>>
>>15151195
Ah, you haven't read other people's code. Why bait the data harvesters?
t. quadradius-playing cold reading necromancer
>>
>>15151200
meds
now
>>
>>15151200
I'm ordering you to speak because you sound like you have a confidence issue, and I want to spam your brain's speech centers.
Should make you more confident.
>>
>>15151218
This doesn't remove my concern. I never mentioned confidence, spaced repetition or dharma.
I didn't want a conversation about myself, but about mathematics education.
>>
>>15151223
you don't know any education philosophy or history
how are you going to have a conversation
you don't know how to have a conversation about education philosophy or math education without injecting yourself into the conversation
I assume you think it's less of a slog to get through all of these /lit/ and /his/ topics
Who knows...if you don't try, obviously you aren't putting the strength you used to crack math into those subjects
those are terribly dull subjects
lots of politics, lots of money
academic politics
what a mess
>>
>>15151223
ZFC destroyed Europe
We should call WW I and WW II by their real names, ZFC-1 and ZFC-2
>>
>>15150260
symbolab and khan academy
>>
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>>15140403
what would you answer for pic? the given definitions are

>a ratio between two quantities is A : B if there is a unit so that the first quantity measures A units and the second quantity measures B units.

>a fraction a/b is defined as the number formed by taking a parts of size 1/b. 1/b is defined as the number formed by 1 part when a whole is partitioned in b equal parts.
>>
Is a Tetrahedron a 3D rendition of a triangle? And if so, then what is a 4D rendition? And so on?
>>
>>15151644
I'm waiting with my dick out, stroking btw
>>
>>15151644
"Rendition" is the wrong word, but it is the "equivalent" to it, yes. The 4D equivalent is called the pentachoron, or the 5-cell, or the pentahedroid, or any other number of names.
In general, though, the n-dimensional equivalent of a triangle is called the n-simplex (so the triangle is a 2-simplex, as an example)
>>
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I come again, with another cool looking problem that I myself haven't attempted yet but it looked very cool. I do hope it's very cool indeed. Also a bit unrelated but one side of my family is from Erzurum. It's an interesting place. Has soviet style buildings everywhere. The people are certainly interesting as well, a bit angry perhaps.
I'll be attempting the problem myself when on breaks, I hope you to might find it interesting enough to attempt. I appreciate anyone replying.


>>15147243
Anon, I'll only parrot what I heard recommended from some book, I believe the book was about Math Olympiads.
You should NOT immediately straight to the solution. You should struggle and take breaks and try different things. However if you feel that you are not making any progress and that it's a bit useless at that point, you should read the solution and figure out what you missed. If your inability to solve the problem was due to not understanding the material, you should go back and review.

As a start, why don't you attempt this problem?
The solution is in the book and I'll post it eventually but if I'm late, you can ask for it earlier.
>>
>>15151655
How many sides does a 5-cell have, and why is Wikipedia calling it a solution to a hypothetical maths problem instead of a 4D triangle? Should it not follow some pattern, they are I think claiming to sides to go from triangle (1), tetrahedron (4) and pentachoron (10) when intuitively I came to conclusions of 6, 7 or 8 but in no way did I fathom it could have been 10.
>>
>>15151655
Sorry by sides I mean the amount of symmetrical triangles (faces)
>>
>>15151658
>>15151659
For an arbitrary n-simplex, the number of triangular faces it has is the (n-1)th tetrahedral number, which you can just view as the sum of the first (n-1) triangular numbers
>>
>>15151702
Give me a number of faces please in laymen terms, I didn't finish high school :'( please don't be like this
>>
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Is galois theory the most beautiful theory in math, bros?
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>>15151707
the nth triangular number is just the sum of all numbers from 1 to n
the nth tetrahedral number is the sum of the first n triangular numbers
so for the 4-simplex, we're looking at the 3rd tetrahedral number, which is the sum of the first 3 triangular numbers, which are 1, 3, and 6. hence 1+3+6=10
>>
>>15151712
I don't understand the pattern. Which words are you using to reference the 1 face and the 4 faces in the previous two shapes, before getting into the (hypothetical) fourth dimension.

I think what I'm actually getting at is I don't think this pentachoron theory is accurate, would you be open to this or is it in your mind quite solidified? Who invented this theory?

So you're telling me, that because a tetrahedron has the faces of a regular triangle that a pentachoron is quite literally made up of faces consisting of a 3D object? Ah. Again I'm not understanding how this would result in 10 but I see how complicated this is now. Thanks for humouring me. <3
>>
>>15151733
All the numbers I've been using have been strictly in reference to the triangular faces.
The triangle is the 2-simplex, so we can count the number of triangular faces it has by looking at the first tetrahedral number. This is the sum of the first triangular number. The first triangular number is the sum of the first integer, which is 1. Hence the triangle has 1 triangular face, which is obvious.
The tetrahedron is the 3-simplex, so we look at the second tetrahedral number. This is the sum of the first two triangular numbers. As mentioned, the first triangular number is 1. The second triangular number is the sum of the first two integers, 1+2=3. Hence the second tetrahedral number is 1+3=4, so a tetrahedron has 4 triangular faces.

In general, we can briefly summarise as follows: The kth triangular number is the sum of all integers from 1 to k. The mth tetrahedral number is the sum of the first m triangular numbers. The number of triangular faces on the n-simplex is equal to the (n-1)th tetrahedral number.
For another example, the fourth triangular number is 1+2+3+4=10, so the fourth tetrahedral number is 1+3+6+10=20. Hence the 5-simplex has 20 triangular faces.
>>
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Feels so comfy reading the proofs of stuff I used to take for granted in high school.
>>
>>15151733
Also, yes, in general a polytope (the name for these things in general) in n dimensions will have polytopes of n-1 dimensions as its faces
>>
>>15151745
You know I don't know what you're saying, you're a horrible person and I think the entire theory is not only wrong it's dangerously wrong.
>>
>>15151745
*speaks a language you don't understand
You're rude, you're dumb, everything you know was taught to you including those precious definitions.
>>
>>15151739
If you can't explain it simply you don't understand it. It's a triangle. Fuck off.
>>
>>15151747
>>15151748
>>15151751
look buddy I'm not sure how to explain it much more simply to you

you sum up the integers from 1 to n and write them down
you sum up THOSE numbers
and you get your value
also not sure what the fuck you mean by "dangerously wrong" but that sounds like your problem and not mine
>>
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I am in an unusual pickle /mg/. I started seriously studying math from advanced calculus (European real analysis) and linear algebra (rigorous). While I am good at all that, my weak base in high school math remains to be a nuisance. I have trouble whenever questions pertain to geometry, for example. Can someone recommend a geometry book for someone like me? I don't want something too easy for high school level kids, since that would be too boring. Or should I just jump straight to modern geometry, would that be practical? After all, I did jump to advanced calculus without doing basic calculus first.
>>
>>15151753
I think you paid a lot of money for a University degree and that's great and I'm proud of you for being so rude about it and repeating other people's theories in such an obnoxious way as well as making a staunch point between the definitions of
Rendition
>an interpretation
and Equivalent
>equal in value

Jesus fucking CHRIST
>>
ah yes, I'm rude for not resulting to three posts in a row to spew meaningless insults
wicked bait bro
>>
>>15151657
Let [math]X_k = 1[/math] if the k-th person breaks the record and let [math]X_k = 0[/math] otherwise.

[eqn]E \left[ \sum_{k=2}^n X_k \right] = \sum_{k=2}^n E[X_k] = \sum_{k=2}^n P(X_k = 1) = \sum_{k=2}^n \frac{1}{k} = H_n - 1[/eqn]
>>
>>15151246
Bloody hell, you are thick.
I asked a simple well stated question and you start accusing me about trying to make world-saving suggestions or something.
>>
>>15152020
The solution looks pretty good anon
and I want to add on to it, if I could.

Since the expectation is some fraction of a whole
number, sending it to the floor ensures for small
groups of skiers that there's a chance no one
could beat the first skier's record at minimum.
Is that fine?
>>
>>15151077
F
>>
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>>15152020
Hello anon, problem poster here. I think you're correct, that is also the answer I found. However the book gives a different answer, namely [math]H_n[/math]. So it doesn't subtract one.
I think the author just counts it as a record broken if it's the first record of the day. But let me know what you think and why the author's answer is different from ours. I don't think it's a big deal.

But anyway! Thanks a lot taking the time to solve this problem! I'm grateful! And nice job also, really slick. I hate to take more of your time but if you could, could you explain the
[eqn]\sum_{k=2}^n P(X_k = 1) = \sum_{k=2}^n \frac{1}{k}[/eqn]
part of your solution? I'm not good at math as a warning.
Also I've attached the book's solution as an image to this reply. Check it out.
>>15152250
Anon I'm terribly sorry but I personally couldn't understand what you're asking. Could you please ask again and elaborate a bit?
>>
How do you guys denote the end of a summation in your personal notes? All the clearest solutions are ugly and crowd the page (underlines, parentheses, etc)

[math]\Sigma_{n=0}^{\infty}f_n(x) + 2[/math]

Is this (the limit of partial sums for [math]f_n[/math]) + 2, or the limit of (partial sums for [math]f_n[/math] + 2)? It should be clearer, no? I usually don't gripe about notation conventions. The whole "we should break hundreds of years of convention so that we can avoid dividing and multiplying by 2 when we do shit with circles" is retarded, but there should be a simple solution to the Sigma problem, no?
>>
>>15152494
To me it reads: [math] )+2[/math] because I don't see a reason to include a non-indexed term inside summation.
[eqn] \sum_{i=0}^{ \infty} f_i(x) + g_i(x) \text{ reads as the whole thing in brackets.}[/eqn]
>>
>>15146966
Improvement over time isn't linear and you're right, you're retarded. If you were half as smart as you thought you'd be cranking out entire topics every week, but you're pissing your time away until the last 3 days before the exam. A university in the modern day is a business, they want a high student population for funding, but that's not an excuse for you to be a lazy faggot. You still have access to all of their resources, you just don't make any use of it, and it was never the point for them to hold your hand through it.
>>
>>15152494
Always rearrange the summand so that it ends with a term that explicitly depends on the index (so [math]\sum_{n=0}^\infty 2+f_n(x)[/math] in this case), then you can treat [math]\sum_{n=0}^\infty f_n(x)+2[/math] as )+2.
If you can't rearrange it for some reason (e.g. [math]\sum_{n=0}^51[/math]) but still want to make the end-of-scope explicit, then you can append something like [math]+0n[/math] as a last resort. I've never had to resort to this, though.
>>
>>15152494
I disagree that a single set of parentheses is ugly
plus they always work and they're always clear
>>
>>15152460
>>15152250
I was writing to the anon with the actual
solution since H_n is a sum of fractions, they do
not add up to a whole number of broken records
for that day. To account for this, just take the
floor of the result...[math] \lfloor{H_n-1}\rfloor [/math]

When the number of skiers n is small, say 3,
there is a possibility that the first skier keeps
their record and the other two couldn't break.
By the solution, the expectation is 5/6 which is
rounded down to zero. This is the minimum
number of record breaks that team can achieve.
As n gets bigger, the floor tracks upward, but
at least the expectation is a whole number.

Ultimately, this is all just my overanalysis...

Note: the sum index given by anon starts at 2
because the first skier can't beat his own record
or even tie with it by the outline of the problem
and that the skier can only ski once a day.
>>
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>>15152530
Anon I don't understand why you would take the floor of the result. It doesn't need to be a whole number. Like if I have 7 bitches and you have none, on average the two of us have 3.5 bitches. Doesn't need to be a who(r)le number.

Or did I misunderstand your complaint?
>>
>>15152538
>>15152530
No, you got it right. Just my over(anal)ysis, that's all.

The number of times the record is broken isn't usually
some fraction, just a whole number for sense. I chose
the floor specifically because it makes sense for
n=2 or 3 skiers to possibly not break a record and
thus zero records are broken...it's the possible
minimum achieved from the usual average.

If n=10, it's still possible the 9 other skiers can't
break a record but the floor of the expectation
just makes the average a simple whole number.

And, no, I don't expect the author to change the
answer because of my over(anal)ysis either. :-)
>>
>>15151767
Lol wtf man. What happened to how chill you were when you aked the original question?
>>
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>>15152579
I actually kind of agree with you anon. I think it would have been nicer if the problem instead asked what's the average number of records broken on a day. I don't think it's wrong the way it is asked, but it sounds a bit strange when you think about it, like you did.

On an unrelated note, I'm glad we didn't suddenly start screaming like this
>>15151767 dude
I think we both deserve a reward for that.

Also not to praise myself but pog problem. I'm glad I found it and shared now.
>>
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>>15140717
Ask norman
>>
>>15140717
>x is a set
x = x
>>
>>15152714
This set does not exist
>>
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>>15140717
>What is a set
Consider multiple objects simultaneously. Objectify this simultaneous consideration into a concept. Extend this concept to include the logical extremes of no objects, or infinite objects. Ta-da. You have a set.
>>
>>15140717
A set of at least no elements.
>>
>>15140717
A set is the mathematical model for a collection of different[1] things;[2][3][4]

I trust this. Wikipedia would never use the word "thing" in the lead section of an article unless they had a really good reason.
>>
Why is counting easier than measure theory?
>>
>keep putting latex off
>get a course that requires latex for assignments
what's the quickest way to learn it???? pls bros if i fail this class i will never become a mathematician earning starting 300k ez
>>
>>15140860
Don't be boring, ayy?
>>
I am starting a masters in industrial engineering but I finished my undergrad a year ago and didnt really use math in my final year. I did pretty okay in my stochastic calculus class but now I remember close to nothing.
I saw in some /sci/ guide that I should start with stewart calculus and do odd chapters only? Is this good advice?
What would be the best path to refresh my mathematical background in a short amount of time?
>>
The moving sofa problem's upper bounds are because we know of sofas that big which don't fit.
>>
An Euler brick is one where the three side lengths and the space diagonal are integers, and no one's found a 2x3x6 yet.
>>
>The male math student who transitioned into being a female math student gets way more respect than the male physics professor who transitioned into being a male math professor
>>
>>15150895
Question: if A is a finitely generated monoid and B_i i=1...n
is a collection of submonoids such that B_i omits B_j when i !=j
and B_i is the free cyclic monoid with a single generator, then
is A the free monoid on n of its elements?
>>
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>>15153283
Question 2: if A is a finitely generated monoid and B_i i=1...n is a collection of submonoids such that the B_i filter pi_{B_i} exists for all B_i and B_i is the free cyclic monoid with a single generator, then is A the free monoid on n of its elements?
>>
Wait, it's all subspaces of [math]\ell^\infty[/math]s?
>>
>>15153236
Correct. Physicists are bad people.
>>
>>15153283
>>15153305
Errata: in both questions, add the extra premise
> A = < U[i=1]^n B_i >
The answer to both questions is no
Take A = Za^2, the integer lattice
And for B,C the axes
>>
>>15153490
sorry, it's Na^2, the upper right quadrant of the integer lattice
>>
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Have you ever solved a problem that wasn't on your homework (or from a textbook)?
>>
>>15153757
any post written in pajeet-tier english can be safely discarded without further consideration
>>
>>15153742
yeah
we were supposed to come up with a problem for galois theory
I came up with one about an infinite tower of field extensions, and you have to prove that the whole tower is a normal extension.
I forget the premises
if I recall, it was something concrete like
Qu[sqrt(2)],Qu[sqrt(sqrt(2))],... although it have been more general...essentially an infinite tower of algebraic extensions or something like that
>>
>>15153403
Always has been...
>>
>>15153742
Once jotted down the wrong problem numbers for homework (was looking at an earlier week's by mistake) and I don't think there was a single problem I did that I was supposed to do.
Thank fuck the professor was lenient enough to just mark it down with "These weren't the problems I assigned" and graded it based upon the ones I ACTUALLY did
>>
>>15153789
Wait, and those are just factor spaces of [math]\ell^1[/math]s?
>>
Last year of my applied math degree. Undergraduate thesis will be the only course I take next spring semester. However, I still have the funds to pay for an additional course to take alongside my thesis course.
I could take a topology class for street cred, or I could take a cryptography course for interest and maybe job prospects?
With topology my degree would be over 50 credits in just math, which is a cool thing to say, I think...
>>
>>15153951
If the crypto course has good quality practicals, I say go for that. Topology is much easier to learn from textbooks whereas a good school can save you plenty of time with a good applied (along with theoretical) crypto course.
>>
>>15153974
I see. I might try and get some more info or just ask someone if they have course notes then.
Its this https://catalog.uconn.edu/course-search/course/CSE/4702/
>>
>>15153985
>Its this https://catalog.uconn.edu/course-search/course/CSE/4702/
Doesn't tell me much. The syllabus itself can be self-taught easier than introductory topology I'd say.
It's also kind of weird to have general topo as the last maths course though. Maybe some actual topologist itt will correct me, but to me general topology opens doors to some interesting generalizations in analysis and also to the rabbit hole of algebraic topology, also leading to algebraic geometry (not that either of them really requires advanced general topology). Taking it without a followup sounds fun only if you are the type to really enjoy the set theory (and topology) branch of abstract maths.
>>
>>15154001
I spent my entire "credit allotment" on stats, pde's, modelling, number theory, machine learning, etc and I did take 3 abstract algebra courses already. I also have a CS minor. I like CS, but only the theoretical side, so I fit all my elective math courses in that nature.

All topology opens up for me is differential geometry, unless I got permission for one grad level course.
>>
>>15153789
>>15153867
I was trying to do a meme response, but actually
the l^1 space is a subset of the l-infinity space.
>>
>>15153951
For job prospects from a course, _with a resume-project_, from here[https://gradcatalog.uconn.edu/course-descriptions/course/CSE/] would be greater than an undergrad cryptography course, unless you're a number theory savant.
>>
>>15140717
From Wolfram Alpha:
>A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored
>>
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Solved a really rough problem by trying things until it worked and now I know how to do it now. Didn't even need to ask for help.

I feel ashamed and angry when I ask for help.
>>
What is a collection?
>>
>>15154050
>actually the l^1 space is a subset of the l-infinity space.
But l-infinity is a factorspace of l^1.
>>
>>15152959
google everything that you need, eventually you'll figure it out. Also use overleaf for a more gentle introduction.
Things to keep in mind at first:
>the characters you'll be using the most are: curly brackets { } that work like the usual parentheses usually do in a programming language; the backslash \ is needed for almost every symbol you'd want to type; dollar signs $ $ delimit a math environment (as opposed to a text environment), write your formulas inside them. Use double dollar signs $$ $$ to display your formulas (they'll be at the center of the page, with a whole line reserved for them)
>some of the more useful stuff: ^ for exponents, _ for indices, \sum for summations, \prod for products, \frac{}{} for fractions, \to for an arrow, \implies for the "imply" arrow (there are lots of arrows but these you'll probably need the most). \gamma for lowercase gamma, \Gamma for uppercase gamma, the same holds for every greek letter (sometimes uppercase looks like a latin letter and isn't used). \mathbb{N}, \mathbb{R} for natural numbers, real numbers (and a bunch of other stuff of course, it's called blackboard bold)
>find a website which covers a bunch of other symbols, you'll learn the ones you need in no time
>>
>>15154621
>>15154050
Yes, you're correct. I had to double check.

Always has been...
>>
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>>15150895
>>
Is y=2x+1 a line in R×R or a sheet in C×C? It can't be both?
>>
>>15155090
The experession "y=2x+1" makes sense in any two-dimensional vector space with chosen coordinates.
>>
>>15154671
>still using tex math environment delimiters in 2023
>>
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>>15155088
requesting two item attachment upload limit so I can put a .jpg and .pdf in the same post
>>
>>15155142
[spoiler]no, let r = ab, s = aa[/spoiler]
[spoiler]then f(rs) = ars and f(r)f(s) = aras[/spoiler]
[spoiler]yet <ab> omits <aa> and vice versa[/spoiler]
>>
I think I've come up with an important math proof that I haven't seen anywhere else yet. What should I do?
>>
>>15155342
Upload it to paste bin and sign it with your own private key so everyone knows that you came up with it first. Then post in on 4chan and send it to your colleagues to evaluate.
>>
>>15155124
I personally use $ $ for in-line and \[ \] for display, idk what kids these days are using. The point was that $ $ and $$ $$ are more intuitive when you start out
>>
How do I tell my calc teacher she is WRONG?
>>
>>15155342
post here
>>
>>15140702
Linear codes
>>
do you guys keep paper you did problems on or just toss it
>>
>>15155370
How is a delimiter which can match on either side and gives less info when there's an error more intuitive?
>>
>>15141049

why is it hard? It's just study function f: X->R, where X is some space.
>>
This is going to sound retarded but is Up(f) the same as U(f, P) in terms of meaning?
>>
Suppose you have two smooth convex surfaces A,B bounding the same amount of volume in R^3 but A is a sphere and B is long and sausage-shaped.
Somebody who doesn't give a damn about triple integrals comes up to you and poses the following question:
> One of A,B is definitely bigger than the other. Which is it?
How do you answer?
>>
>>15156906
That's a retarded way of asking something like the following:
>Which is bigger, a watermelon or a golf club?
>>
>>15156229
thanks for the context
>>
>>15156956
well I thought it was just different notation, I'm talking about Riemann integrals, bounds, partitions, stuff like that.
>>
Mathematically, which is bigger, a watermelon or a golf club?
>>
>>15156962
No one knows what you're talking about until you define what you're talking about.
>>
>>15151761
Just jump into modern geometry. Middle school geometry is not particularly useful for math.
>>
>>15140403
Why is a sigma-algebra called that?
I get that [math]\omega[/math] and [math]\aleph[/math] get different characters to distinguish between the ordinal and cardinal interpretation, but why would you take yet a different letter for a property concerned with countable sets? Why is it not an omega-algebra.
>>
>>15157571
apparently it's for "sum" (well, "Summe", which is just German for sum) in the sense that a union of sets can be considered their sum
>>
>>15156906
I ask them if they mean volume or what. If it's volume, just use water. No need to calculate complex integrals.
>>
If I finish a masters in mathematics from a low-mid tier school, and let’s assume I get a 4.0, what is the best PhD program I could hope to get into afterwards? Will it depend on the kind of research I get involved in?
>>
This makes zero sense.
Prove me wrong.

En and Em are uncountably or countably large sets, with each E_alpha + r_n are a single number.

This means that of course E_n and E_m are not disjoint, because it's possible to create the same number in two ways. e.g 0.1 + 0,2 = 0,3 ... but also 0.05+0.025 = 0.3, so 0.3 will exist in many E_n.
Therefore, they are by obviousness not disjoint, and this whole proof completely falls apart.
>>
>>15158155
Also not using latex, becaue this editor is absolutely horrendus. If you miss the window in your click, or you hold mouse button and move the mouse just a little, all your text disappears and you have to start over again. Completely unusable.
>>
>>15158155
>form a subset with exactly one element from each equivalence class
>hurf but wut if der more den 1 element from the same class ur proof dont work
>>
>>15158175
To be fair, this proof is different from what my teacher did. My teacher had E be the collection of all relations between (0,1) and alpha, which means all real values.
>>
>>15158175
>>15158273
Actually, no, this is the same.
E consists of uncountably infinite elements. It's one element from EACH equivalence class, and there's gonna be uncountably many equivalence classes.

So En is a rational number + all of the uncountably many numbers in E.
This clearly results in duplicates.
>>
>>15158278
>This clearly results in duplicates.
excellent counterexample
>>
>>15158278
E consists of exactly 1 rational element and exactly 1 of each of the ‘different’ irrational numbers
>>
>>15157602
Interesting. Then I have to change my question to "why are sigma-finiteness and sigma-additivity called that".
>>
>>15158278
I like that idea. Let's generalise that logic a bit.
The set of all irrationals is uncountably infinite. If I use strict equality, each element is in an equivalence class with itself exclusively, so there are uncountably many equivalence classes. Therefore, if I take one copy of every irrational number, I will end up with duplicates.
Sound logic.
>>
>>15158415
Some of the irrationals are gonna be space apart by a rational number. lets say 0.1. Now you have at least 9 duplicates for almost each rational number.
>>
>>15158451
which is why we limit ourselves to picking exactly one of each such instance when constructing our set
>>
>>15158415
>>15158451
But we don't even need the irrational numbers, because E also contains ALL the rational numbers.
For some E_alpha there is a r_i which gives E_alpha - r_i = 0
>>
>>15158469
When we construct E we pick one from EACH CLASS, but there are uncountably many, one for each number in (0,1). Then for each E_n we put in each single number in E and for each element in E we add a r_n.
>>
>>15158472
>because E also contains ALL the rational numbers.
dude E contains ONE rational number. just one.
>>
There is an uncountably infinite number of real numbers.
If I take one copy of every real number, then using strict equality, every real number is equal to itself and only to itself. Therefore, I have uncountably infinitely many equivalence classes under strict equality on the reals. But this implies that by taking every real number exactly once, there must be duplicates.
QED.
>>
>>15158472
The difference of any two rationals is rational. They are all in the same equivalence class. You get to pick one (1) representative of this class, like all of the others. You get exactly one (1) rational number to work with in E, of your choice. No more, no fewer.
>>
>>15158486
Except, in the end, you're supposed to add 2 rational numbers, so you have 2 infinite rational numbers.
Each rational number is accompanied by all other rational numbers, for each set.
This somehow creates disjoint sets.

i.e. 0.1 + 0.2 =/= 0.2 + 0.1, apparently.
>>
>>15158529
No, dumbass. You pick exactly one representative from each equivalence class to be in E, including your one rational, and you stick with them throughout. You don't get to suddenly change your set halfway through.
>>
>>15158546
E has both irrational and rational numbers, it's not limited to Q. It just has to have a relation of x-y on (0,1), which includes irrationals.

Now, for each En, each element in E has r_n added to it. Some elements in E will be the same as r_n, E and r_n are not seperate, they have infinitely many duplicates of each other. So then clearly there'll be duplicates in En.
>>
>>15141528
If it is a function [math]\mathbb{R}^2 \to \mathbb{R}^2[/math] than it implies [math]x^2 + y^2 \not = (0,0)[/math] so both [math]x^2 \not = 0 \land y^2 \not = 0[/math]
Otherwise it means that at least one of the components is not equal to [math]0[/math].
>>
>>15144543
Not familiar with Wolfram.
But you'd probably want to create the following set right?
[math]X := \{a \in \mathbb{R} | \lim_\limits{x \uparrow a}f(x) \not = \lim_\limits{x \downarrow a}f(x)\}[/math].
>>
>>15158548
Also, I have never heard of "equivalence class" before this. I looked at video, and I don't know.
>>
>>15158667
>I have never heard of "equivalence class"
How did you end up here?
>>
Is it just me or is the underlying idea that would make everything really simple and easy to understand often shrouded in rigor, and it takes a lot of time to see through the rigor and find the idea?

Case in point: I was reading https://math.stackexchange.com/questions/3672636/show-that-there-exists-a-rectangle-such-that-each-of-its-four-vertices-are-of-sa

I spent a _lot_ of time on the first two paragraphs until i skipped a tiny bit ahead, saw "two line segments L1 and L2 such that they have the same sequence of colouring" and instantly got the solution. I literally could have skipped everything, read "two line segments with the same sequence of colouring" and would've instantly generated the proof myself.
>>
>>15158712
No, most mathematical writing is awful. Sometimes on purpose.
>>
>>15159367
new thread
>>
>>15140403
I'm sorry...but why the fuck are there two concurrent /mg/s?
What's the normal order of/official progression of threads?
>>
>>15159670
> why the fuck are there two concurrent /mg/s?
a troll likes to thread split
>>
I have never once in my life been shown the graph of a tan function. Not in all my years of algebra 1, 2, and 3 + precalculus.

Why does no one prepare you for all the calculus 1 pre-reqs?
>>
>>15141049
weird i was just watching this show too lol
>>
Gotta finish this thread off before jumping to the next.
>>
Next thread:

>>15159206
>>
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You should be able to solve this /mg/
>>
>>15140403
I'm a math idiot but I just played a pokemon card game (online) where I didn't draw a basic pokemon (you need one to start the game) 10 times in a row when my deck is 9/60 basic pokemon cards. For those unfamiliar you draw 7 cards to start the game so you need to have a basic as 1 of those 7.

How unlikely was this to happen?
>>
>>15157960
Theoretically, any schools. Realistically, anything that's not the top 10 schools. Unless you publish something incredible or have an advisor who knows someone who can vouch for you (or some other kind of in)

You're going to want to at least conduct some kind of research and get a publication. I think this will help immensely (and is what I did). I'm waiting to hear back from schools (Ph.D) so I could answer this more affirmatively. Having a good GPA is important but being able to show you can conduct research is pretty much gold. That's what a Ph.D is about.

Also, keep in mind school name brand is less important than your advisor. Obviously if you go to Harvard or whatever that will open doors etc. But you want to pick schools based on research conducted and advisor. Also make sure your advisor is not a massive asshole.
>>
>>15162536
the pythagorean theorem
>>
>>15162536
Infinitesimals (they allow me to accurately measure the size of my dick)
>>
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>>15163350
>>15163657
Wrong
>>
>>15164010
>zero
Thanks for nothing



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