/sci/ - Science & Math

thread theme:
https://youtu.be/_9CPKtDy3fM

>>15159206
Math is for niggers

>>15159206
Sigur Ros is literally gay

So do any of you nerds have the psionic ability to calculate an answer to the following admittedly absurd question.

Hypothetically... if we were able to contain all the solar output from the sun and focus it in a single direction... How fast could we go?

Then sun is heavy but it sprays out a lot of energy. Would it even be enough to move our star if it could all be focused in a single direction?

>>15159206
Ay I like that music

The previous thread hasn't even hit the bump limit. But I guess you just have to be quick to make sure you get to advertise your epic taste in music to everyone, huh?

>>15159250
I know one nigger who was good at math - Blackwell. Any others?

>>15159651
It was but some janny deleted posts.

t.not op

>>15159530
>Then sun is heavy
That means nothing. It's in space and if you give it a hard shove then it'll start moving in that direction indefinitely.

I suppose the sun stays in the same place because the energy goes radially in every direction.

That's my contribution. I dunno how fast the sun would be cheesing it in one direction if it all got concentrated.

>>15159530
The radiation pressure of photons is given by E / c, and the energy produced by the sun is 3.8 x 10^26 Joules per second. So this gives you a force of 1.267 x 10^18 Newtons. The mass of the sun is 2 x 10^30 kg so if you focused all the photons in a single direction you would get an acceleration of 6.34 x 10^-13 m s^-2. So not very fast at all. To give you an idea of how weak it would take 2 million years to accelerate to a speed of 1 metre per second.

>>15159901
Whoa. Big brain answer. Thanks.

Next project: how to make the sun more excited.

what's the last cool theorem you've learned the proof of? for me, it was Artin-Schreier yesterday: [math]C[/math] alg. closed, [math]1<|C:K|<\infty\implies|C:K|=2,\,\operatorname{char}(C)=0,\,C=K(i)[/math] for [math]i^2=-1[/math], i.e. if the alg. closure of a field is a finite extension of it, then it looks like [math]\mathbb R\subset\mathbb C[/math]

How the hell do you find the maximum for picrel? I know that you need to take the gradient and look where it equals 0, but gradient of f can't equal zero..

>>15160897
maximum on what domain? on all of R^2 it doesn't have a maximum, because f(x,0)=e^x->infty

>>15159530
I'm too lazy to write the procedure here so I will just tell you that knowing the sun has a mass of 2*10^30 and outputs 3.8*10^26 joules per second it would take 44 minutes to accelerate it to 1m/s. Disappointing, innit?

>>15159901
Oops, I'm >>15160977, just realized my answer is wrong!

>>15160981
Sad face. My only two hopes now for turning the solar system into a spaceship is that the calculation does not account for all possible sources of thrust emitted by the sun or;

Squeezing the sun hard enough using magnets or something to produce additional thrust.

What could go wrong.

>>15161181
Your idea is fated to fail. The only other particle the sun generates during fusion apart from photons are neutrinos. However these produce no thrust at all since they essentially don't interact. The only time this would not be true would be during a supernova but that's a single one-off event.

>>15161287
Yea but if the plasma surrounding the core can be manipulated into shedding in a single direction in a controlled way the exhaust would consist of more than photons and neutrinos.

>>15159206
Scientifically, how accurate are these videos on the metaphysics of mathematics?

https://www.youtube.com/watch?v=mhIkyqLDl9M
https://www.youtube.com/watch?v=ZQOwG-hcd_k

Need help from smart math anons.

My CS degree has a linear algebra module and a Linear Algebra and Statistics for Engineers module. There is no statistics module, so if I want to learn statistics, I have to take the latter.
I want to do an AI/ ML graduate degree, so it'd be nice to do the module that covers statistics. I'm scared to choose it because it's designed for engineers and engineers have hard math and my uni is known for its engineering degrees.

Can you look at the descriptions of the modules and tell me what I can expect?

Linear Algebra 1:
This unit will consider problems arising from science, engineering, and business-related fields. Students will learn the necessary skills to model and solve such problems through the introduction of mathematical techniques of linear algebra as well as complex analysis. Students will be introduced to the idea of a complex number together with their applications and use in solutions to polynomial equations. This unit will cover vectors, lines, and planes and their extension into n-dimension space. This unit also covers matrices and their use for solving systems of linear equations through a study of a number of different types of solution methods. Eigenvalues and eigenvectors will be considered.

Linear Algebra and Statistics for Engineers:
This unit will consider problems arising from engineering-related fields. Students will learn the necessary skills to model and solve such problems through the introduction of mathematical techniques of linear algebra, data analysis as well as statistical inference. This unit will cover vectors, lines, and planes and their extension into n-dimension space. This unit also covers matrices and their use for solving systems of linear equations through a study of a number of different types of solution methods. Students will be introduced to the world of statistics by looking at the concepts of descriptive statistics and inferential statistics.

>>15161555
I responded to you in the /sqt/ as well, got here because I followed your link, but to be clear I wouldn’t take my own advice irl, but maybe I should. My degrees were in cs and pure math. I wish I’d just done pure statistics, so maybe I’ve got a boiled onion to toss in here or so. If you want to go to grad school for AI/ML, the canonical advice is to not take undergrad stats eng classes, and instead focus on theoretical stuff. But it’s skewed by 4chan elitism. It sounds like you’re a sophomore, or so, so I’d stick to the classes that underpin the field, namely pure stats and the ability to code up what you want expressed.

If I can say one thing, don’t get me wrong in thinking stats, ai and ml is all just stats. ML just has more code monkeying than insurance companies do.

Another thread, another problem. Last one was quite fun. I hope this is as well. Good luck to everyone who attempts to solve it. I appreciate anyone who does. I don't think it's that difficult but I don't actually know, I haven't attempted it yet. It might be very elementary.

>>15161811
Is [5] straightforward or does it mean some other set?

>>15161811
Define your stupid fucking notations and italicise the variables.

How many functions [math] f : \{ 1, 2, \dots, 5 \} \to \{ 1, 2, \dots, 5 \} [/math] are there such that:
[math] \forall y \in \mathbb N \operatorname{card} \{ x : f(x) = y \} \leq 2 [/math]

>>15162303
>>15162350
Yes...yes...thank you!

#MathEducationNeedsToChange. It's becoming untenable to claim students profit from being exposed to calculus.
Calculus is sitting like a fat walrus on the harbor pier, blocking access to your speedboat.
Before I knew about calc, I wondered to myself "how does one calculate something with variable speed?". So I'm already like <10% of the pop. But I didn't ever have interest in learning it, or need it.
>they're a solid foundation for further STEM/accounting career
Pre-algebra, perhaps. But, actually, my thesis is the opposite: all this needless crud takes time away from attempting to instill a deep understanding of uncontroversially essential math.
>so how would school math look like?
Just arithmetic, intro to functions, and 80% of high school math would be focused on devising actual solutions to personal finance and some other "productive normie" issues. Not merely as text exercises, it could be used for genuinely their individual (unrealistic) or representative elements of young adult budgeting.
Most of geometry is kill. Trig is kill. Calculus is kill. Permutations are kill (useless for "apples" situations without binomial coeffs). That clownishly tiny vectors bit is kill.
Geometry past volumes is the most egregious holdout of the ancient Greek ideal that we should do math for inherent beauty of it. It's a beautiful philosophy. Form your uni lectures around it.
You have literally never used triangles in your life. The ratio of "usefulness of rectangle geo: triangle geo" is comical.
You have used angles, but only as a pre-masticated reference points. You can just teach the students the values and what they stand for. Now THIS is an area where rote memorization actually makes sense! Because these numbers are fundamentally arbitrary, while it's never useful to calcuate angles for 99+% of lives. I know you. I have watched you. You have not calculated an angle ONCE outside the scope of your STEMlord education or wage job.

School math would be "Arithmetic+".

(1/2)

all rectangle geometry is just triangle geometry, dipshit

>>15162468
(2/2)
FAQ:
>Waah waah just want to make office-skulking accountant NPC husks
Shut your whore mouth. Remember the rote anecdote? All this arithmetic would be spent on actually trying to marshal every single last IQ point of avg student so he can now multiply hundreds in his head, understand fractions intuitively, all that good shit.
If a 100 IQ normie doesn't leave school having an intuitive understanding why distributivity is important and why it makes mental or paper math trivial -- Artihmetic+ has failed. While this is a project that tries to face the dimwit/midwit
It would take time. Which is why the entire curriiculum would be stretched. Drake good: deep understanding of a limited toolset, like an artisan or the menu card of class restauraunt. Drake disgusted: rushing a million overviews through, instantly being deleted from your brain when the next deadline drops.
>so common core math?
I don't know about American issues. But from what I gather, it's aimed at elementary school students and uses ambiguous language as fuck to foster "creativity". Yeah no. Just use adult, unambiguous language with middle schoolers to try and continue to teach such principles.
>wtf I am bored now
Yes. We are just <5% of the populace. Why should us not having to sit bored have a precedence over most of the population literally gaining nothing from math, except regarding it as "that school subject. rather than a field of FUNCTIONAL human endeavor like any other? They should regard mathlike a non-corporate-golem regards cooking: absolutely vital for daily tasks, and perhaps a little delight whose art and beauty can easily be further explored without needing to be an aspiring Michelin-tier chef and centering your entire adulthood around it.
>they'd still post #IhateMath on their instas and spout such kinda anti-humblebrag IRL
But the sentiment would be less; and also less meaningful.

>>15162474
>While this is a project that tries to face the dimwit/midwit
"This is a project that tries to face the realities of dimwit/midwit limitations."

>>15162303
>>15162350
>>15162404
Guys, guys, I apologize. I thought this was standard notation. I've seen friends use it without any explanation so I figured most people are familiar with it.The problem is straight from a book I haven't read and I understood what the notation meant so again, I thought it was clear but apparently not. I'll be more careful next time, explain my notation and italicize the variables.

>>15162678
It is completely standard notation. I've never read Stanley's book and I was thoroughly familiar with the notation.

>>15162704
Sometimes, notation written like '[a]' may mean that
it could be the greatest integer function in some texts
or maybe it's a real number interval where either
[a...a] or {a} would make more sense. So, in case
of unfamiliarity a brief definition or equivalent notation
would be helpful.

Does every simply-connected open subset of the plane have connected complement?

Note this is untrue if we strengthen connected to path-connected; consider e.g. the complement of the (closed) topologist's sine curve

>>15161811
>it might be very elementary
It's very elementary if you only try to do 5, because 5 is too small compared to 2. You only have enough stuff to screw up one preimage. So it's easy to count the bad ones, you just split into cases based on which preimage is the one that has too many elements

If you want to replace [5] with [N] then it's a little bit less elementary, you can extract it as [math]N![x^Ny^N]\frac{1}{1-y(1+x+x^2/2)}[/math] , I think. I'm sure that fiddles down to some sum of binomials

>>15159206
I accidentally posted in the old thread.

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

>>15162853
No.
Consider [math]\{ (x, y) \in \mathbb{R}^2 : 0 < x < 1\}[/math]

>>15162935
Oops I'm retarded lmao

>>15162935
Follow-up question:
If a compact subset K of the plane has simply-connected complement, then is K connected?

>>15159530
>How fast could we go
See the no communication theorem with regard to constraints on sending classical information. There is a reason this theorem holds by the way, it has to do with with the fact that the physical world is virtual. The refresh rate of physical reality, see pic.

>>15162961
Sorry that question is dumb too.

Here's a better question:

If a closed subset K of the 2-sphere has simply-connected complement, then is K connected?

>>15159530
The second part of this
>>15162973
that should be mentioned is that bell type correlations are in fact FTL but no classical info can be sent, which goes back to the no communication theorem. The reason that these non-local FTL correlations can exist is that all points in the virtual spacetime are equadistant to the processor. See pic. specifically
>notions of locality and distance defined within the simulation do not constrain the action space of the system performing the simulation (i.e. from the perspective of the system performing the simulation, changing the values of variables of spins/particles separated by 1 meter or 1 light year has the same complexity).

Have Gödel's incompleteness theorems caused work in the foundations of mathematics to stagnate?

>>15162989
no

>>15159206
I'm tired

>>15159530
One more note with regard to these
>>15162973
>>15162981
These are in regards to quantum info no go theorems, but the speed limit holds for anything due to the processor specs of the rendering entity rendering the physical world. This speed limit can and has changed by the way, as the players immersed in the reality came up with more precise ways of measuring the speed of light to more precise degrees and decimal places, this made the need for processor upgrades.
see vid
https://www.youtube.com/watch?v=Z1axh6ki0oc

>>15162350
Replacing 3, 4 with ellipses in your sets seems silly, they have the same number of characters

does a cubic count as an elliptic curve?

Should i take a class on axiomatic set theory? I had introductory set theory first year and i think it was very useful

>>15162975
True but I don't have a slick proof.
>inb4 why is it true
Closed subsets of S^2 are compact, so we immediately have that they have a finite number of connected components. These connected components are also all at a positive distance from each other, so we can put them in very tiny balls within the sphere. Then, we choose two connected components and argue from the cohomology group of the twice-punctured sphere.

>>15163765
Go sit in a few lectures and see what you think of it.
Foundational stuff is either something you love or something you think is inane autism. You'll have to check it out for yourself to decide

It's not "useful" in the same sense that introductory set theory is. It's a specialized topic that even most mathematicians don't really know much about.

>>15163532
If it's smooth and you regard it projectively, then of course

>>15164084
>if you regard it projectively
>you will never be a surface

>>15164084
Oh and I forgot, you also need to pick a point on it, which over a non-algebraically closed field might not be possible
FYI I couldn't come up with an easy example so I looked it up and there's [math]E:\quad 3x^3+4y^3+5z^3=0[/math] with no [math]\mathbb{Q}[/math]-point, but I'm too much of a brainlet to show it

I’m not the person you’re replying to but

>>15164122
Fermat’s last theorem would imply this, no?

>>15164084
Could you remind me what it means for a curve to be projective?

>>15164211
>Fermat’s last theorem would imply this, no?
I don't see how
>Could you remind me what it means for a curve to be projective?
There are some subleties (especially over a non-algebraically closed field [math]k[/math]), but the gist of it is that a projective variety over a field [math]k[/math] is the subset of [math]\mathbb{P}^n_k[/math] given by the vanishing of some homogeneous polynomials [math]f_1(x_0,\dots,x_n),\dots, f_m(x_0,\dots,x_n)[/math]. Remember that it makes sense for a homogeneous polynomial [math]f[/math] to say that [math]f[/math] vanishes at a point [math]P[/math] of [math]\mathbb{P}^n_k[/math]. Then, a projective curve is a curve which is projective.

Now in the case at hand the context was
>does a cubic count as an elliptic curve?
So presumably one is given a cubic polynomial in two variables, say [math]f(x,y)=3x^3+4y^3+5=0[/math], and considers its solutions in the affine plane [math]\mathbb{A}^2[/math].
What I meant to say is that we first need to "make this projective" by "adding points at infinity", by which I mean to perform the standard operation of taking [math]f(x,y)[/math], making it homogeneous ( [math]F(x,y,z)=3x^3+4y^3+5z^3[/math] ), and considering this as a projective curve in [math]\mathbb{P}^2[/math].
Note that whenever [math]z\neq 0[/math], we can normalise [math](x:y:z)=(x/z:y/z:1)[/math], and we are reduced to the "affine case". When [math]z=0[/math], we get the homogeneous equation [math]f(x,y,0)=3x^3+4y^3=0[/math], which consists of a finite number of points

>>15162704
Thank you anon, I'm glad you agree with me at least.
>>15162857
Hmm hmm anon. Well I didn't count the bad ones, I counted the good ones and got the correct answer. I don't know if it was the easiest way to go about it.

So is a generalization possible then?
I didn't really understand the last part of what you wrote with the y and things.
>>15161811
And yeah, basically the problem is very easy for 5 and everyone here should be able to solve this. I'm referring to the people who asked about the notation! Solve it!!!

Two weeks into linear algebra. Its extremely boring. Learned some near tricks, but nothing that requires intense focus like calculus 3 line integrals through parametric vector fields or stokes theorem. I'm missing excitement in my life. Does it get better at abstract algebra?

Am I in out of my head in going for an applied math postgrad after getting an undergrad in civil engineering? I've already studied 3 sems of calculus, linear algebra, differential equations, I'm just missing real analysis for which I can take a remedial course. Is there any other prerequisite I'm missing? Will I be able to hack it coming from an engineering background?

Keep in mind I'm just doing applied math, not pure.

>>15164875
What is an applied math postdoc? Isn't engineering just applied math?

>>15165031
I meant postgrad as in an msc degree not postdoc.
>Isn't engineering just applied math?
That's basically my question. I know I have to learn basic proofs and real analysis but I wonder if there's anything else I'm missing that I didn't learn during engineering undergrad.

>>15165055
I think you should go for a real applied math degree like Statistics, Physics, or Engineering, instead of meme shit like le "applied math." What the fuck does that even mean? I think it's just one of those scam degrees colleges offer as a money making machine.

>>15160897
They don't address the fundamental issue with why modern set theory is invalid. First and foremost, the law of logical honesty states that you cannot claim you can do something you cannot do. This means that you cannot claim to do any of these operations with these so-called "infinite sets" because the idea of an set is that you collected some number of objects. You cannot collect an infinite number of things.

>>15164875
post grad? you mean a graduate degree?
Most applied math masters programs are heavily into PDEs, numerical and functional analysis.
If you haven't real/complex analysis its going to be a prereq.

>>15165161
applied math degrees have existed for decades. Money making machine these days is data science.

https://youtu.be/Vp570S6Plt8

>>15165210
>You cannot collect an infinite number of thi-

Discuss pic related

>>15165480
Writing down a symbol as shorthand for a type of object is not the same as collecting objects of that type.

Trying to prove that [math]({\bigcap_{\alpha \in I} A_\alpha}) \cap ({\bigcap_{\alpha \in J} A_\alpha}) = {\bigcap_{\alpha \in {I\cup J}} A_\alpha} [/math].

I need to justify the following step
[eqn](\forall \alpha \in I, x\in A_\alpha) \land (\forall \alpha \in J, x\in A_\alpha) \iff\\
\forall \alpha[(\alpha \in I \land x\in A_\alpha) \lor (\alpha \in J \land x\in A_\alpha)]
[/eqn].

How?

>>15165537
What do you think we do in math, walk around plucking numbers from bushes and putting them into baskets?

>>15165580
what you wrote as your step is invalid; for example if both I and J are empty

>>15165634
oops sorry, assuming I and J are non-empty

>>15165644
what you wrote is still invalid -- it implies either every alpha is in I or every alpha is in J, which is false if there are objects not in I and (other) objects not in J

>>15165653
^correction; it implies that for every alpha, either alpha is in I or alpha is in J, which is not true if there is something not in I union J

>>15164550
Yes, abstract algebra will blow your cock off

>>15165657
Ok, what about this?

Let [math]I[/math] and [math]J[/math] be non-empty sets.
[eqn](x\in A_\alpha \quad \forall \alpha \in I) \land (x\in A_\beta \quad \forall \beta \in J) \iff\\
\forall \gamma[(x\in A_\gamma \land \gamma \in I) \lor (x\in A_\gamma \land \gamma \in J)]
[/eqn]

>>15165680
now you really are implying what I said in >>15165653

>>15165680
>>15165783
lol I'm retarded today, no you aren't implying that, but you didn't change anything from >>15165580 except turning alpha into gamma

For every real algebraic x with 0<x<1 , is there some rational q such that cos(qπ) = x ?

>>15165784

The point is that the alphas in the first line of >>15165580 (You) have nothing to do with each other since they belong to separate for-all quantifiers. So what you said in >>15165653 does not apply.

For 0<x<1 , if x is algebraic (over the rationals) , then is there some rational q such that sin(qπ) = x ?

>>15165867
consider x=1/3

I found this formula to ennumarate amonium compounds
x = cycle_index(S4, A)
Where S4 is probably symmetric group and A is a random value for ex: 210
How to find x?

>>15165865
what you wrote in >>15165680 is "for every gamma, either x is in A-gamma and gamma is in I, or x is in A-gamma and gamma is in J". What if gamma is not in I or J?

Prompt: Generate a textbook reading list for a strong foundation in applied/pure mathematics that will leave the student prepared for graduate studies.

Applied.
Numerical Analysis by Richard L. Burden and J. Douglas Faires
Calculus of Several Variables by Serge Lang
Differential Equations: A Modeling Approach by Reinhold Remmert
Linear Algebra: A Modern Introduction by David Poole
Convex Optimization by Stephen Boyd and Lieven Vandenberghe
Applied Partial Differential Equations by J. David Logan
Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers by Roy D. Yates and David J. Goodman
Numerical Methods for Scientists and Engineers by Richard Hamming
The Finite Element Method: Its Basis and Fundamentals by Olek C. Zienkiewicz and Robert L. Taylor
Optimization: Algorithms and Applications by Rajesh Sundaram

Pure.
Principles of Mathematical Analysis by Walter Rudin
Linear Algebra: A Modern Introduction by David Poole
Abstract Algebra: A First Course by Fraleigh and Beauregard
Real Analysis: A Long-Form Mathematics Textbook by J. Donald Augustine
Topology: A First Course by James R. Munkres
Algebraic Topology by Allen Hatcher
Combinatorics: A Guided Tour by Miklós Bóna
Number Theory: An Introduction to Mathematics by Paulo Ribenboim
Geometry: A Comprehensive Course by Dan Pedoe
Group Theory: An Introduction by John F. Humphreys

Is ChatGTP doing a good job here?

>>15159206
Decided to go back to school to pursue a CS degree, and calculus is one of the prereqs for enrolling in the program. I've taken college algebra and trig, but that was ~5-6 years ago, so I'd like to brush up on Khan Academy. What would be the best set of classes to take? It looks like they offer college algebra, trig, precalc, and a pre-precalc. Would it be sensible to jump straight into precalc?

>>15166203
Just do precalc. If you can't get up to speed on precalc you wont make it through calculus I.

Its what I did in a similar situation. Precalc has been my lowest grade so far with a B. Have had an A on every other math course, and I have two semesters left now.
If you take college algebra and trig separately you will be wasting your time and bored.

Has anyone read Vladimir Zorich's Analysis textbook? Is it worth reading for somebody at the graduate level who needs to shore up their analysis study or are there other texts that would be more robust

>>15166133
good point...

so then how do I justify step (3) to (4)?

>>15166208
Thanks for the advice.

>If you take college algebra and trig separately you will be wasting your time and bored.
I just wasn't sure if I should be super autistic and do a thorough review of everything just to make sure there aren't any gaps in my knowledge.

>>15166219
NTA you replied to but this looks fine so far.

The equivalence between (3) and (4) boils down to the fact that [math](\forall x(\phi_1\to\psi))\land\forall x(\phi_2\to\psi)[/math] and [math]\forall x((\phi_1\lor\phi_2)\to\psi)[/math] are equivalent.

If you expand your bounded universal quantifiers, remembering that [math]\forall x\in X\,\,\phi[/math] is short for [math]\forall x(x\in X\to\varphi)[/math], you will recognize that this is an instance of the aforementioned equivalence (you'll also need to appeal to the definition of [math]\cup[/math] once to get there).

As for actually proving the equivalence mentioned earlier, it's a combination of the fact that universal quantification distributes over conjunction, [math]\forall x(\alpha\land\beta)\equiv(\forall x\,\,\alpha)\land\forall x\,\,\beta[/math], and also that [math](\phi_1\lor\phi_2)\to\psi\equiv(\phi_1\to\psi)\land(\phi_2\to\psi)[/math], which basically states that disjunction is the least upper bound w.r.t. implication.

You just prove them (informally) the way you prove everyday mathematicial statements.

To prove that [math](\phi_1\lor\phi_2)\to\psi[/math] implies [math](\phi_1\to\psi)\land(\phi_2\to\psi)[/math], you suppose [math](\phi_1\lor\phi_2)\to\psi[/math] and prove each of the conjuncts in the conclusion separately.
For [math]\phi_1\to\psi[/math] you continue by assuming [math]\phi_1[/math] and trying to derive [math]\psi[/math].
However if [math]\phi_1[/math] holds, then so does [math]\phi_1\lor\phi_2[/math].
But we still have that [math](\phi_1\lor\phi_2)\to\psi[/math] and so [math]\psi[/math] follows by an application of modus ponens.

The converse, i.e. that [math](\phi_1\to\psi)\land(\phi_2\to\psi)[/math] implies [math](\phi_1\lor\phi_2)\to\psi[/math], follows similarly but requires the usage of some other rules of inference (proof by cases, since the proof will inevitably assume a disjunction, namely [math]\phi_1\lor\phi_2[/math]).

Thoughts on this text?

>>15165210
>First and foremost, the law of logical honesty states that you cannot claim you can do something you cannot do.
Do you need to be able to make a car from scratch when you talk about cars? When we talk about an infinite set we just refer to the existence of such a set. You can of course choose not to believe an infinite set exists but your mathematical model would be useless to anything more than simple arithmatics.

Since there are so many finitism schizos in this thread: what's a possible foundation for finite mathematics? Can you just take your favorite set theory and remove whatever axiom gives you an infinite set? So like ZF/NBG/whatever minus infinity or ETCS minus NNO?

>>15166308
At least they don't have to argue whether the choice function exists kek

>>15166479
Is that because choice is derivable for finite sets?

Are measurement devices energetically weaker/inefficient in the grand scheme? An ammeter converts current to a mass spring, then marks it with a single dimensionless unit rather than comparing springs more generally. Its effectively 0.5 dimensional. Which is quantum precise, but in n+1 spacetime probably loses out.

>>15159530
I suspect this is fundamentally impossible, by conservation of momentum.
Like trying to move a boat by blowing on the sail.
You need outside momentum coming in instead somehow.
Use the sun for energy but use something else for "leverage" if it make sense.
Maybe magical infinite thrusters on your redirection units idk.

nihilists: accept categorical language or we cut your johnstone!
me: (accepts categorical language)

>>15159206
Does pic-related is always decreasing function or I am retarded and this has a local minimum?

holy fuck i hate the entire field of analysis

Can you apply the divergence theorem when the field has a singularity at the surface you're integrating in/over? Can you even use the surface integral directly?
I know about integrating F=1/r^2 over the unit sphere etc, what I'm interested in is integrating a surface that contains (0,0,0). In general, my field doesn't have divF=0.

Does math help your memory/brain?
Like I finished CS degree 3 years ago and been a web developer ever since, I forgot a lot of the math stuff I learned. I feel like I'm getting dumber and my memory is becoming worse.
Would practicing math by myself help memory/"exercising" the brain or is that a bad reason?
Is there a math guide/chart to follow when self learning? Maybe Ill read some book by myself slowly or pirate a course and follow along.

Is there a difference between intuitionism and constructivism in logic/foundations or are these synonymous?

>>15166800
logarithms are strictly increasing on the positive reals, so your function here is strictly decreasing, yes
>>15166853
They're synonymous

I need help with a recurrence relation/ growth rate problem?
I posted a link that shows my question I’m trying to solve and the part I’m having trouble with here. If any kind knowledgeable souls can help me please do
>>15166797
>>15166993

>>15166817
Find a university that you respect.
Find their math program syllabus.
Copy textbooks, study in that order.
Whalla, you now have the guide you need and like.

>>15166671
Yep

holy fuck i love the entire field of analysis

>>15165876
How do you show arcsin(1/3)/π is irrational?

>analysis
SOVL

>Algebra
Soulless.

>>15167071
look into Niven's theorem

>>15167093
most soul to least soul:
>combinatorics
>analysis
>number theory
>topology
>algebra

For a bachelors thesis, would I be in over my head by attempting something relating to kalman filters?

>>15167149
You're out of your mind though I don't necessarily disagree with combinatorics being on top

Springer TAM series has just about all of my favorite textbooks, its no question. Whoever is curating that is really on point.

[math]ZF\neg C[/math]
Anything more retarded and pretentious than this?

>>15167353
ZFC

>>15167353
for me it's ZF + DC + AD

>>15167353
it's true though, sometimes the sets just get too big for any meaningful choice to occur

>>15167353
>>15167358
>>15167365
For me it's ZFC + three angels can fit on a pin head + 2^(aleph_0) = aleph_42

>>15167392
I know you are shit posting but any axiomization of set theory where the continuum hyphotesis is false is absolute fucking bullshit

>>15167395
Any axiomization of set theory is absolute fucking bullshit

>>15167395
[math]\mathfrak c = \aleph_2[/math]
I'm not even shitposting here

>>15167403
why
>>15167405
support your claims

When I'm factoring polynomial functions, I sometimes find a '0' that is before the last coeficient (using Ruffini), and it seems to lead to mistakes.
What's happening, and whats the problem?

Say function is:
2x^2 - 2x^2 - 10x - 6=f(x)
First root could be 3.
What we have left is
(x-3) (2x^2+4x+2)
Solving 2x^2+4x+2=f(x), root is -1.

Though if we apply Ruffini with -2 it leads to:
| 2 4 2
-2| -4
|------------------
2 0

With a 0 amidst it, leaving c at the right.
What is to be thought of it?

Thanks beforehand

>>15167410
PFA

>>15167410
Because continuum hypothesis is undecidable. If you want to do maths that's not bullshit, you have to stop at countable sets (subsets of N/ SAr) at most.

>>15167418
how would you make the reals countable

wtf this general has no opening post with resources? even fucking pony generals have one

>>15167413
>2x^2 - 2x^2 - 10x - 6=f(x)
Assuming you meant [math]2x^3 - 2x^2 - 10x - 6=f(x)[/math]
Applying Ruffini's rule with -2 means dividing [math]2x^2+4x+2[/math] with [math]x+2[/math], so you get [math](2x+0)(2x+4)[/math] and remainder +2.

I don't see a problem or a mistake here. Could you explain on that a bit more?

>>15167455
Resources for what? You need to at least be well into your bachelors to understand anything going on here. By that point you should already know what resources you need and use.

>>15167455
check the stickies

>>15167464
>(2x+0)(2x+4)
the latter term should be [math](x+2)[\math].
Sorry.

>>15167414
Forcing axioms are where set theory gets really fun. Jechs Multiple Forcing book is goat.

>>15167423
The reals are already countable, when considered as subsets of the natural numbers, or the rational numbers. Only in Cauchy's construction of the reals are they uncountable.

>>15167464
Yes, I meant
[math]2x^3-2x^2-10x-6[/math]

Had I to be honest, it seems I possibly haven't studied or learned Ruffini's rule carefully enough. I guess I might have to go through it again.

I'm looking at what I tried to do for (2x^2+4x+2)/(x+2) and had as a result of ruffini's operations:

2 0 2

But I'd have thought this would be construed as:
[math]2x^2+0x+2[/math]
Though I guess I wouldn't make sense.
Would you mind showing how that one is solved?

>>15167578
>But I'd have thought this would be construed as:
>2x^2+0x+2
That is incorrect. Your result
>2 0 2
should be interpreted as [math]2x +1[/math] with remainder [math]+2[/math] so when you multiply the result and add the remainder you get the original polynomial:
[math](2x)(x+2)+2 = 2x^2+4x+2[/math].
See https://mathworld.wolfram.com/RuffinisRule.html for another example.

>>15159210
https://ia803401.us.archive.org/8/items/heart_of_darkness/heart_of_darkness_2a_conrad.mp3

I recently gave the second stage of Mathematics Olympiad (INMO) in my country, quite confident that I will clear it.

Thoughts on note taking apps such as Obidian? Here is what a PDF looks like (stats class, as a math major I try to abatract when possible or be more general). I am finding the rewriting drills it into my brain, that and of course you get used to using LaTeX. (Note I was temporarily disabled in the question block).

>>15167690
That should read obsidian. Here is an example of a subgraph right now in it (also use trying to use it for non-university stuff). Red means there is no existing file named that yet but there are files linking to a would-be file.

>>15167698
Wrong image. There should only be one "1 - Distributions" file.

>>15167690
no computers in math class
it's about the sensuality of chalk
the subtle movement of pencil
across the paper pulp page

This question is regarding the logistic map for x_n+1 = r x_n(1-x_n)
So the map was made as y=rx(1-x)
I got fixed points/steady states at x=0, and x=(r-1)/r
Like the question says at r~3.1, there is a period 2 cycle and the steady state/slope is ~ -1
How do I write this as a function of r??

>>15167690
im interested in this and asked about it previously in /mg/ but no one chimed in. say more what does your workflow look like and how do you fit it into your studies

>>15167826
I pretty much just started using this.
I saw others such as https://github.com/zhaoshenzhai/MathWiki
atomize everything. I think there is certainty value in doing so, but I also think some structure is useful. So I am atomizing (new fies) for every definition or important theorem. Which are then referenced or linked in more structured notes such as that PDF I posted.
Most of the value itself imo comes from re-writing the notes from class (I am not yet quick enough to keep up in class, though that is likely to change soon), as that — like I said — drills it into my head.
In essence I am currently using it as one would normally take notes (structured folders e.g. School/Semester/CLASS400/Notes), and just taking advantage of the linking and slight extra motivation provided by the graph.

>>15167891
I am also testing it as a book cataloguer. Though this is arguably redundant because goodreads exists.

>>15167898
Maybe for learning Japanese too. Though again this would really be more for aesthetics as anki or wanikani exist. I know there is actually an anki-like plugin for this.

Read through a College Algebra textbook's section on Real Numbers. I have a few questions.

1. Irrational Number's relationship towards the Rationals is that they cover the elements which the Rationals do not, with the Real Numbers acting as the Universal set. Therefore, is it appropriate to say that the Irrationals are the complement of the Rationals?

2. Is the fact that multiplication is repeated addition sufficient to say that the commutative property applies to Multiplication as well?

3. Is it an appropriate analogy to say that the commutative property deals with distances and the associative property deals with rearranging the distances?

4. Reason for how the formula [math]LCM(A,B)=\frac{AB}{GCD(A,B)}[/math] works?

>>15167353
Would be weird if you actually adopt negations as axioms, but there's little problem with this. It's stronger than ZF and e.g. weaker than assuming every set is measurable.

>>15166853
Brouwers intuitionism was essentially an analysis with some non-classical axioms.
Meanwhile "intuitionistic logic" denotes a logic without LEM (but you can also be more conservative than that)

I don't think many people would use the term "intuitionism" for foundations and just means a base constructivism, and so I have to disagree with >>15167006

fourier transform chads, did we get too cocky?

I asked chatgpt for a general formula for the volume of an n-ball and this is the answer it gave me, is it correct?
>An n-dimensional hypersphere with radius r is defined as the set of points in n-dimensional space that are at a distance of r from the origin. The formula for the volume of an n-dimensional hypersphere with radius r is given by:

>V = (π^(n/2)) / (n/2)! * r^n

>Where (n/2)! is the factorial of n/2. This formula is derived from the formula for the volume of an n-dimensional unit sphere, which is the sphere with radius 1, and then scaling it by the radius r.

>It's worth noting that finding the volume of a hypersphere can become quite complex for large values of n, and finding the exact formula may require advanced mathematical concepts like integral calculus and multivariate distributions.

>>15166308
ZF minus infinity has the hereditarily finite sets as model.
That said, people who are into finitism don't really bother with formal set theory much. There's many conservative frameworks.
Weyl was very conservative, especially when young, and advocated a sort of second order arithmetic without transfinite recursion. (That theory has the infinite subsets of N though, so it's not finitism in a good sense.)

>>15168113
Looks correct at first glance, I've not checked all the factors in that formula, though.
You'll also find the formula on the n-sphere wikipedia page, I assume, although you can argue there's no perfectly canonical way to the ball volume in non-integer dimensions. But this might be a very good angle

>>15166678
Is it not like a rocket where it loses mass by sending out momentum in one direction?

>>15167649
Good luck anon

>>15168103
Continuum hypothesis itself can be written as a negation though (there does not exist a cardinality between N and R)

>>15168155
I think that's technically a fair point.
If I'd have to dwell on my case, I'd say that this is a double negation in disguise: The order of cardinals is defined via injections, namely what's defined is |X|<=|Y| and |X|<|Y| is derived by adding an injection.
The most straight forward definition - and the one you find if you seek category theoretical phrasings of the hypothesis is that
(|N| <= |X| <= |R|) => (|X| = |N| or |X| = |R|)

>>15166308
Weak linear logics give coherent foundations for ultrafinitism.

Weak linear set theories have some really fun properties, like allowing unrestricted comprehension and the set of all sets to exist.

>>15168071
1. Yes
2. Yes, the only thing we know about multiplication is that it is repeated addition. We derive everything from that fact.
3. I am not sure how you mean this. You can think of addition on the reals as travelling distances on the number line, yes.
4. By the fundamental theorem of arithmetic, if you decompose everything into primes it is easy to see that LCM x GCD = AB.

and |X|<|Y| is derived by adding a negation next to the injection clause, I mean.

>>15168166
>>15168172
Had this sort of phrasing in mind
(pic related is the phrasing of the negation of CH, hence there is a negation when talking about the epis)

>>15163834
>It's a specialized topic that even most mathematicians don't really know much about.
>It's all this board talks about

>>15159206
I am trying to prove the following claim:
For [math]a, b, n \in \mathbb{Z}_{\ge 1}, \; a > b[/math]
[eqn] \sum_{i=1}^n \binom{n}{i} b^i {(a-b)}^{n-i} = a^n - {(a-b)}^n. [/eqn]
This claim is derived from two solutions I came up with for the following problem:
How many strings of letters (from the English alphabet) of length 8 with at least 1 vowel (vowels = {A, E, I, O, U}) exist?
The obvious solution to this is [math]26^n - 21^n[/math], but [math] \sum_{i=1}^n \binom{n}{i} 5^i \cdot 21^{n-i}[/math] seems to work as well.

Can anyone help prove my claim? I tried using weak and strong induction but ran in to dead ends (pic related).

>>15168199
https://proofwiki.org/wiki/Binomial_Theorem/Integral_Index
?

>>15168204
Thank you, anon. Rewriting the equivalency as it is in the theorem makes the proof much easier to approach, It's nice to know I not too far off a correct path.

>>15168224
I'm surprised you didn't know the theorem.
You can also check Wikipedia, I think it gives some combinatorical insight. iirc there's nice generatingfunction proofs.

Also works with Gamma functions - might come in handy

>>15168166
Actually this made me think again. If we assume Cantor-Bernstein (which is non-constructive, but okay), then CH can be stated very positively as

[math]\forall X.\ \left( r(A, X)\land r(X, B)\right)\to \left(r(X, A)\lor r(B, X)\right)[/math]

where r(U,V) is the relation that that U injects into V and A=N, B=R.
This sort of \land to \lor properties might have a name, but I don't see it atm.

>>15168253
I do not have much (read: any) background in combinatorics. What program did you send a screenshot of? Looks helpful, based German anon.

>>15167053
How do you find syllabus of universities?

>>15160897
Cos y is bounded by 1 and e^x is unbounded.

Why do categorists seethe about ZFC so much?

>>15168843
Literally go on their site or google it anon

x^x = 81
How do I find x?

I managed to get it into the form 3^4 = x^x = 81 but I'm not sure if that helps
I tried Logarithms and got x logx = 4 log 3
There must be a way to find X but I just can't work it out

I lost sleep over this last night so any help would be greatly appreciated.

>>15169158
There isn't a "satisfying solution" to this sort of problem in general, so to speak. For most values you could substitute 81 with you have to resort to a representation with the Lambert W function

>>15169158
There isn't a "satisfying solution" to this sort of problem in general, so to speak. For most values you could substitute 81 with you have to resort to a representation with the Lambert W function.
Well, unless you want to resort to the schoolyard approach of "just keep subbing in values for x until we get close enough"

>>15169169
I got 3.505 doing it the old fashioned way
I suppose I will settle for now

>>15161811
Non-empty fibers are 1 or 2 element sets.

If all fibers are singletons then
5!
If one fiber is 2-element set then there are 10 choices and we need to pick value at each fiber which gives
10 x 5!/1!
If two fibers are 2-element sets then there are 5x6 choices and we need to pick value at each fiber which gives
5 x 6 x 5!/2!

3120 in total?

Got an exam for my electronics course this Friday. We’re required to get 100% to pass due to some new rule (pass rate was previously 70%). If I somehow get 100% I am dropping out of my apprenticeship and heading straight for electrical engineering instead.

>>15169174
Veeeerryyy close anon. Me, a friend of mine and now you have made the exact same mistake, getting 3120. But alas, that's more than the actual correct answer. I won't say where your error is because I assume you might wanna find it yourself but I'll say it's at the very end, with the 5x6x5!/2!. You're overcounting there somehow. However nice job regardless! I have no doubt you'll find the correct answer.

>>15169196

5!+10*5!+5*3*5!/2! = 2220

Aww, stupid me! Thanks for the entertainment.
Now, what is the solution to the general problem?

Has anyone here ever used Calculus, Exponentials, Combinations, Logarithms or even Trigonometry in their daily work or life?

My Dad has a work bench where he builds stuff out of wood and I know he has used Pythagoras and gradients before.

How would YOU go about being the pedagogue for a modern Gauss? I don't mean giving them a list of reccomended texts, what would you actually give as a day to day course to someone from infanthood to adolescence to produce a prodigy?

>>15169247
Nice job! That is indeed the correct answer.
>Thanks for the entertainment
Thank you a lot more for taking the time to solve it. I'm grateful.
>Now, what is the solution to the general problem?
What a great question.
I apologize but I may not be able to give you a satisfactory answer when it comes to this.
Well, if you mean general as in how many at most two-to-one functions there are from [n] to [n], then I believe it may look something like pic related. My friend found it and I also found something similar. To be clear, I'm not 100% certain this is true but I don't see why it would be wrong.
If you google A012244 you'll find the OEIS page for this sequence. Perhaps that will interest you.
However for i-to-one functions in general, I'm afraid I don't have an answer. Another friend of mine mentioned that they had found it, however they have not shared it yet.

If you do find something that you think is interesting, I'd appreciate you sharing it here. I shared this problem without actually knowing how nice it was but as usual it seems to be more fun than it first seemed. When and if my friend shares the general answer, I'll let you know and put it here hopefully. And in the meantime you could check pic related and see if you think it's correct. Again, I and my friend do but that does not guarantee it. Have a nice day, hope we talk again soon.

oh and also we have a discord if you're interested for problems like this, it's bit dead but meh, let me know if you'd like to join.

I am getting filtered by fucking propositional logic.
does p then q necessarily imply that if not p then not q?
are the the statements "p if q" and "p only if q" equal?
my intuition says the first implies p being false doesn't imply q is false while the latter implies it.
is it just semantics? can both be written as [math]p \to q[/math] ?

>>15169305
>does p then q necessarily imply that if not p then not q?
No. It implies that if P and not Q, then not "P then Q".

>>15169145
Name one t1 university (not from India, China, etc.) which has a detailed syllabus of all their courses including the books.

>>15169305
P>Q can be true or false.
I think you're getting tripped up because manipulating propositions isn't the same as evaluating them to ultimately have a value of true or false

>>15169329
Lmao go to their department website, student section, courses, see textbook and lecturer.
What more do you want, a livestream of class? Some even have that

>>15168131
Not since you said you want to use all the sun's radiation (i.e. all directions).
Problem being, it does that in all directions. So to redirect it you'll need thrusters on your mirrors.
So ultimately if you redirect all the radiation back onto the sun, I think you'll be effectively sourcing all of that momentum from the thrusters or whatever else you use to keep your apparatus in place.
You'll end up "holding" the sun by it's radiation pressure, and moving it with the energy from your thrusters.
At least that's what it seems like to me.
Consider a hypothetical giant baloon with equally spaced leaks.
If you put something for it to blow against on one side, it'll move and whatever you put there will move the opposite way.
If you put a sphere around it, it won't move but there will be pressure on the sphere. If you modify the sphere is some such way, the pressure won't equalize and the sphere will try to move like the original obstacle on one side only.
You'll end up using something to keep the sphere in place around the leaky baloon, and the force of whatever you you will be what ultimately moves the system.

The fix:
You can avoid this if you're ok with your redirection system being blown away proportionally, so like you said kinda like a rocket. In other words, don't try to redirect all the radiation, only the part that has vector component opposite the way you want to push. then you wouldn't be using the entire energy of the sun, but you'd be able to move the sun with reflectors if you're willing to let your reflectors be moved away proportionally so no thrusters needed... except to put said reflectors in place in the first place, maybe cover them so you're not pushing as much to get them close to the sun

>>15169079
Trying to seem important. Set theorists are off solving their own problems. Ordinary mathematicians are off solving their own problems. ZFC is such a spook, its just one axiom system out of countless, ordinary mathematicians routinely talk about all groups, or all vector spaces which can't be done in ZFC. Research set theorists either work in subsystems of ZFC, when they are on the more definability side or are assuming forcing axioms or large cardinal axioms. I know many set theorists, I've been to set theory conferences, I can only name two set theorists that care about foundational axiomatics. Until a category theorist, HoTThead, or typetard, can solve an actual problem in ordinary mathematics that a set theorist can't, or the importation of that solution into set theory is untenable, no one is going to rethink foundations.

>>15169471

>>15159206
Is it possible to triangulate a dodecahedron so that each vertex has an even degree?

>>15169504
based

I remember following the discussion on the ncat cafe when that Leinster article was first released and the general consensus seemed to be that this wasn't meant to critique set theory (as its own field of mathematics).
Instead, they believe that set theoretical language/elementary concepts of set theory are too prevalent in undergraduate mathematics and that categorical language or a more structural approach would be beneficial.

Now, maybe I'm too "high on set theoretical language" to see where this is the case in undergraduate courses, because (like you said) ordinary mathematics doesn't concern itself with foundational stuff at all.
My first algebra class started out with defining the naturals up to iso which is clearly quite structural, yet the definition (the usual dedekind axioms) were in a completely membership based language (I guess the categorical way would be to define it as an initial algebra). What's so evil about that?
I do agree that sometimes it might be a good idea to bring up some structural thinking where it makes sense and where it's currently being overlooked. For example, I believe it would be beneficial for students to think of a quotient set X/~ not as the set of equivalence classes but rather (up to iso) as a pair (Q, q : X -> Q) of a set and a surjective function such that x ~ x' iff q(x) = q(x'). Since clearly we sometimes like to use representatives instead of equivalence classes to form our quotient (especially when talking about proper classes).
Speaking of equivalence relations, I could however not imagine trying to completely replace a membership based language for a sets + functions one. Because thinking about a relation as a set whose members are ordered pairs is so much easier to work with than defining it as a pair of jointly monic arrows (I couldn't even write down the definition of transitivity in terms of arrows off the top of my head)

>>15169563
gonna be honest idk where i was going with this rant, im just sick of undergrad (or even high school) category theorists going "ew set theory XD, i like category theory and type theory btw :)" when they don't really know what any of those fields actually deal with and that their undergrad math is almost completely independent from choice of foundation.
it also really invalidates the (very valid) criticism towards how some of the more foundational stuff is taught in undergrad mathematics.
as an side: the same crowd usually also advocates to pay more attention to constructive logic in the sense that you teach classical logic (since that's the default background logic for most ordinary mathematics) but teach it in a more careful way that agrees more with how constructivists work (this way, students won't have to unlearn their habits if they find themselves in a position where they need to work constructively). Examples include simple stuff like explaining the difference between contradiction proofs that prove not p by assuming p and deriving a contradiction (which is constructively valid) vs the classical prove p by assuming not p and deriving a contradiction. I'm by no means constructivist but having to check assignments for early undergrads I'm not convinced that this is a good idea pedagogically as they often end up turning a simple (constructive) definition chasing proof into one using like 3 instances of contradiction + the occasional LEM which wasn't needed at all

>>15169582
>having to check assignments for early undergrads I'm not convinced
ofc I meant that I'm NOW convinced

>>15168379
That's Mathematica and I'm not German

You can also use Wolfram alpha or similar free programs

>>15169504
Minor formal caveat: you can of course talk about all groups. What you can't do is define the set (and thus e.g. the category Grp) of all groups, which you'd need as soon as you'd want to quantify e.g. over all functors F out of Grp

>>15169646
even that isn't a problem (assuming that quantification isn't used within the definition of another class)
the real problem with functors out of proper class sized categories is that they're also proper class sized and so can't be collected into another class, so we're stuck with this size issue when trying to form functor categories
this is assuming plain ZFC as foundation, NBG can at least delay our problem by having conglomerates at the meta-level (similar to how ZF has classes at the meta-level). This is how the ACC book "deals" with size issues but of course nothing beats universes, since even with conglomerates we can't form functor categories of proper conglomerate sized categories (we can now do that with proper class sized ones though), hence why it's just delaying the inevitable

>>15169435
Go on, link to one such link of a t1 university (not from India, China, etc.) that recommends books.

>>15169705
>googles "mit syllabus linear algebra"
>first result
https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/pages/syllabus/
Go on, name one T1 university that has no math syllabi online.

>>15165876
For 0<x<1 with x algebraic (over the rationals) , is arcsin(x)/π algebraic?

For 0<x<1 with x algebraic (over the rationals) , is arcsin(x)/π algebraic?

>>15170228
arcsin(1/4)/pi is transcendental.

>>15169504
>ordinary mathematicians routinely talk about all groups, or all vector spaces which can't be done in ZFC
Except that it can.

>>15170234
Thank you, how can this be shown?

>>15170238
I don't know, it's just an intuition that I have.

>>15170236
Why are you trying to defend something nobody cares about defending?

>>15170242
Not him but math is just - in the end - meaningless entertainment we choose to do for ourselves. Why question participation in the conversation.

>>15170241
What would you say are the error bars on that?

>>15170244
No I mean why defend ZFC, it's basically just a patchwork ad hoc solution to setting a foundation for math, basically thrown together out of necessity

>>15170245
I would bet $20 that it's transcendental (expecting to win $20 if it's not)

>>15170244
>math is just - in the end - meaningless entertainment we choose to do for ourselves
it's hilarious that mathematicians consider this to be an advancement

>>15169513
What counts as a triangulation here? Can the vertices of triangulation lie on original edges?

>>15170244
No I mean why defend ZFC, it's just a patchwork ad hoc solution to the problem of setting up a foundation for math. It was basically thrown together out of necessity.

>>15170314
If so, put a spider like this on every face and you're done.

>>15169513
Please stop associating geometry with anime lolicon filth

is there any function of Dirac Delta comb but on the condition that you can move the teeth of the z comb away without changing its period?

>first week of analysis 2
>get assigned 10 easy problems as repetition of analysis 1
>most classmates can't solve even one
>instructor flat out asks them "did you people accidentally sign up for this class without taking analysis 1? Like you are math students, right?"

>>15170323
The problems included
>argue that f(x,y,z)=x+y^2+z is continuous
and
>assume a given delta satisfies the epsilon-delta definition. Show that delta/10 also satisfies the definition.

>>15170322
Function related

>>15170314
>What counts as a triangulation here?
https://en.wikipedia.org/wiki/Polygon_triangulation
>Can the vertices of triangulation lie on original edges?
No. The original vertices and edges need to remain as they are. What I'm really interested in is what kind of polyhedra have a triangulation that admits a Eulerian trail. I'm sure someone has already figured this out a century or two ago, but I'm too retarded to find the answer.

Does there exist a cardinal assignment function for ZF - Foundation (so we can't use initial ordinals nor Scott's trick)?

https://i.4cdn.org/wsg/1675294309818763.webm

https://i.4cdn.org/wsg/1675295486971780.webm

>>15170322
>>15170333
f(x - z)?

Have my first calc 1 test on monday. It's an easy A if I can learn the rest of the trig identities by then. All I know right now is sin^2x + cos^2x = 1.

Keep me in your thoughts.

Is there a general consensus on when to use [math]\cong[/math] and when to use [math]\simeq[/math]? My Algebra prof goes back and forth between the two at random but seems to use them interchangeably (usually to denote that two things are isomorphic).

>>15169471
That makes sense, thanks

>>15170767
Anon don't bother learning them, just understand how to pass between complex numbers and trig functions and you can improvise anything. Convert to complex number, simplify and convert back to trig.

who is the landau of math?
for those who don't know landau wrote very terse, well written and extensive books on essential graduate physics

>>15159206
>discover numerical methods
it's an addiction
how do I go back to a pure path?

>>15170884
rudin

>>15170794
The general consensus as far as I'm concerned is that they're interchangeable but the second one looks awful and you should never use it
>>15170884
Lang is probably the closest. I don't know many mathematicians who've written books on a broad array of topics like that, and if people do spam a bunch of textbooks they usually suck.
A lot of Lang's books are pretty shitty too but Algebra is a classic and his analysis books are both pretty great too, so that's 2/3 of the standard graduate trinity there. He never wrote a topology book afaik

>>15170884
Bourbaki

>>15169984
>open courseware
Yes, now try that for an actual graduate level course and not popsci shit they upload on YouTube.
>>15169984
Princeton.

>>15170794
We mathematicians like to abuse notations.

>>15169174
>Non-empty fibers
there is no need for such pretentious language in combinatorics

>>15170578
How do you assign a cardinal to a set without a well ordering?

>>15171090

Yet I have made a mistake.

>>15169303

Thank you anon-san. I was thinking about k-to-1 functions. I am usually too busy for discord.

>>15171214
Anon, my friend has posted his general solution. Before this he posted his proof and what not but that's multiple pages so I won't post those. I'd like to give him a shoutout for his work, he seems to have done a nice job and he's also pretty based.
Anyway, he used this for solving some examples so I'll post it as a second reply to this.
>I am usually too busy for discord.
T-that's okay. It's n-not like I w-wanted to be friends or anything, baka.
>>15171090
I think it's good not to use complicated language especially when it comes to things like this problem but in all fairness, I think it's quite justified sometimes, like this time the way anon used it.

>>15171383
And here's the examples he used to demonstrate his creation.

>>15166149
>rudin pma
Dangerously based

>>15169305
just write out the truth tables and if their final columns are the same, the statements are equivalent.

>>15170884
Bourbaki, obviously, but you also have people like Dieudonné, who wrote 9 volumes on analysis, Lang or Rotman when it comes to graduate Algebra (Gorodentsev for undergrad), Fremlin's 5 volume treatment of measure theory (or Bogachev's 2 volumes for something a bit less extensive but still very comprehensive), Takesaki's 3 volumes on operator algebras, for logic, the 18 volume series "Handbook of Philosophical Logic", Kobayashi and Nomizu for diff geo., etc. The EGA should also be mentioned, here's a partial translation https://fppf.site/ega/

[math] \sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}{6} [/math]
Do I just remember all the formulas for series ^3, k(k+1), etc. or is there a way to "get" to them?

>>15171467
Something like Faulhaber's formula anon?
Is this what you wanted?

https://en.m.wikipedia.org/wiki/Faulhaber%27s_formula

Trying to find the suspension bridge equation in minecraft. I used the catenary equation (https://en.wikipedia.org/wiki/Catenary)) and it turns out something like this (for bridge length 56 and height 5):
f(x)=round(cosh((x)/82.0451)82.04512*2)/2-82
Then I counted the length of the segments manually, but I wonder how to make it a graph. The length of the segments itself steadily decreases with height, but if I try to round x, then it jumps, which does not look beautiful or natural.

Found a good site for charts: https://www.desmos.com/calculator?lang=ru

>>15171503
Yes, thank you

What playlist do you listen to while solving problems?

>>15171542
I used to always listen to noise music (eg. Boris, Watashi no Koko), it really helps with abstract thinking. It lets you cut off from the world, but it is not musical enough for your mind to focus on it. Now I often listen to Bach but I usually end up singing along and it distracts me.

>>15171385

> baka

I understand only the easier formula with only one "bad" fiber possible. The number is correct.

> baka

A based friend indeed.

>>15170665
It doesn't work

>>15172423
Holy Shit didn't want to put the full capture

>>15169079
FTA:
>ZFC has one major flaw: Its use of the word ‘set’ conflicts with how most mathemati- cians use it.
>The root of the problem is that in the framework of ZFC, the elements of a set are always sets too. Thus, given a set X, it always makes sense in ZFC to ask what the elements of the elements of X are. Now, a typical set in ordinary mathematics is R. But ask a randomly-chosen mathematician, ‘what are the elements of π?’, and they will probably assume they misheard you, or tell you that your question makes no sense.
The way this is written is fairly naive. The author is claiming that "ZFC can use the word set" and this suggests that the underlying issue of standardization isn't being manifested. The author isn't dominating the sentences in a way that is directly addressing the relevant issues but rather is imagining things that aren't (that abstract concepts can use words). So we can't have a real conversation about these hallucinations because they aren't real. Also, the elements of pi are simply the Cauchy sequences that converge to pi.
The data abstraction issues are a topic of CS and Abelson and Sussman, not math.
Author hasn't demonstrated a threshold need to drag data abstraction into the conversation, it's a non-sequitur.
Author suggests that readers aren't familiar with data abstraction issues, ignoring relevant CS thought on the topic.
Author is tilting at windmills.
Author is Don Quixote.

>>15171542
I listen to isochronic tones, thoughts?

>>15172441
if you don't want a circular definition of the elements of pi, just say
>sequences of rationals Cauchy equivalent to [redacted closed form series or product expression for pi]
there are lots of idiot mathematicians who go
>OH, YOU KNOW WHAT THE ACTUAL MATH DEFINITIONS ARE
>NEEEEEEEEEERRRRRRRDDDDDD
>YOU'RE SUCH A WEIRDO!!!!!
>OH MY GOD!!!!!!!!
>WHAT THE FUCK IS WRONG WITH YOU?!?!!
anti-intellectual thought is very strong in mathematics
there a lot of bullies in academic math

>>15168071
>2. Is the fact that multiplication is repeated addition sufficient to say that the commutative property applies to Multiplication as well?
Multiplication is repeated addition only in the natural numbers. It's no longer repeated addition in the rationals or the reals.
And even in the naturals, commutativity of multiplication does not obviously follow from the fact that it's repeated addition (why would it?). Exponentiation in the natural numbers is also repeated addition, and it's not commutative.
>. Is it an appropriate analogy to say that the commutative property deals with distances and the associative property deals with rearranging the distances?
No. Commutative property is an algebraic property, not a geometric one. Distances are not involved.

>>15159206
Fa/g/got here, i have a question to ask to those who are better than me at logical proofing(which is probably everyone). I don't know if this is the right thread for it, but it's the most general one i could find, so maybe i guess.

Is a 2 tape nondeterministic turing machine equivalent to a deterministic turing machine? My reasoning is that it is, because a nondeterministic machine can be simulated with a multitape TM, and a multitape TM is equivalent to a deterministic TM, then a multitape TM nondet. is also equivalent, because I can convert the nondeterministic part in a deterministic multitape with 3 tapes. first one contains the input string, second one contains the computation branches (separated by special symbols) where each branch contains the 2 tapes (also separated), and every time a new branch is created, a new subsection containing the 2 updated tapes is created, and the third tape contains the paths.
this machine can then be translated to a deterministic turing machine.

I'm sure the formal proof for this is probably a giant clusterfuck, so a simple yes or no answer is more than enough.

>>15172617
Can you simulate the nondeterministic machine with C#? If so, the answer is they're equivalent.

is he right?

>>15172813
Yes and we need to keep it that way or better yet amplify it

How high level is this general? Can I get help for really dumb shit here or is it inappropiate?
I have neeted for 2 years and now in engineering college I have a lot of basic knowledge missing that I should remember from last year in high school, especially about calculus.
I don't want to ruin actual math discussion by asking trivial shit.

>>15172826
There is a stupid questions thread >>15169850

>>15168895
>>15160897
So in the end f is unbounded by 1

>>15172886
Oh, seems perfect, thanks.

>>15172617
>a nondeterministic machine can be simulated with a multitape TM, and a multitape TM is equivalent to a deterministic TM
I don't remember TMs well, but if you are sure about these two statements, then the answer is yes.

>decide today I will learn homology
>open textbook
>don't understand
>close textbook
I hope your day is going well today, /mg/.

>>15173453
Homology in just algebra, or in algebraic topology?

Any good texts on modal logic not aimed at philosophers?

>>15173453
iktf, topology is fucking kryptonite to me for some reason
I breezed through my other quals but topology by itself nearly killed me and I still understand jack shit

>>15173523
What non-philosophical reasons do you have to learn modal logic?

>>15173523
Modal Logic by Chagrov and Zakharyaschev

>>15173523
the text mentioned by >>15173635 is the standard one from what I can tell, but the one by Marcus Kracht (Tools and Techniques in Modal Logic) is also worth mentioning

>>15173453
Sections 19-22 of Fraleights A First Course in Abstract Algebra will hold your hand.

>>15173635
>>15173667
thanks anons, I'll check both of them out

>>15173542
curiousity

>>15173184
Yes, I am absolutely sure both of these statements are valid. The proofs are a bit weird but that's irrelevant. Thank you man

>>15173456
Right now in just algebra.
>>15173528
I just want a really gentle introduction to homology. A lot of homology texts kinda breeze through the material, assuming that we're math-savvy enough to fill in the in-between parts that are not explained in detail.
>>15173769
Thanks for the recommendation, that might be useful.

>>15174268
Homological algebra is complicated and it seems to have connections to higher category theory and higher algebra once you get into it. I don't really know the details, but if at any point you're wondering why things are complicated but why they're also connecting a lot to things in topology, just think of higher categories.

>>15174276
>to things in topology
Specifically, in homotopy theory, I should add

>>15159530
This is the idea of the star drive.

The sun emits 10^45 photons a second, each of which has a momentum of p = h\lambda or about 10^-34*10^-9 = 10^-43, or about 100kgm/s every second, ie, 100kgm/s^2.

That's about 10^-28 m/s^2 for the solar system.

The best way to do what you want to do would actually be to make a very very small black hole such that it's Hawking radiation would be a few orders of magnitude over that of the sun, keep it stable, and redirect that in one direction. Then you would actually get enough thrust to move the solar system appreciably.

>>15173453
I hate homos

Could you help me to understand the first question? I can't figure out what exactly the function T is.

What does T:C = R^2 -> R^2 = C mean? Does T map from R^2 to R^2? Why did the author write the equal sign between two distinct fields? They are not equal, but rather isomorphic.

Not exactly math and I apologize for any abusive notation, it has been about a decade since I took qm, but someone may know:

Consider n-particles, each possibly entangled to any other particle. Can these entangled states be considered in superposition?

As an example of what I'm trying to convey, say we have a 3-particle state [a b c] with possible entangled states ("-" denotes entangled) [a-b c], [a b-c], ..., is there some sense in which a wavefunction can capture the probability densities of the system collapsing into any of the possible entangled states? Thanks and I apologize again for not using enough math.

>>15175198
A horrible case of notation abuse. It means [math]T[/math] is a linear map from [math]\mathbb C \rightarrow \mathbb R^2[/math] when we view [math]\mathbb R^2 [/math] as the underlying space of [math]\mathbb R[/math] so the linear map becomes a 2 by 2 matrix and it decouples into [math]Az + B\bar z[/math] by acting on the basis vectors (1,1) and (1,-1).

>>15176162
>as the underlying space of R
I meant [math]\mathbb C[/math]

>>15166299
>reject modernity
>become algebraist

>>15167708
Not sure if this helps, but try organizing distributions in terms of a hierarchy based on how they are derived starting with the Bernoulli distribution.

>>15159206
i'm a tourist from /mu/ and came because Sigur Rós pic. I'm scared of this place.

>>15176197
It's okay fren. We won't bite.

>>15176197
Don't be scared. We'll just whisper some numbers
in your ear and tie you up in pointless debates.

>>15176162
>>15176163
THANK YOU

The silliest thing I continue to try and do is read textbooks on a kindle paper white

does anyone know what happened to the zlib niggas after they got arrested?

>For each of the following functions f(n), determine the largest size n of a problem that can be solved in time t, assuming that the algorithm to solve the problem takes f(n) milliseconds.
I've taken and completed up to calculus 3 and have always done well in math. This is an assignment from my advanced algorithms class (computer science major), wtf is this? Do I solve for n?

How am I suppose to solve this? I am so confused to this problem.... Any hints?

>>15176784
First step is adopting readable handwriting.
I assume the 1-cos(2y) is jointly in a bracket?
Second step is plugging it into WolframAlpha and reverse engineering the solution.
At this point you're done. Third (optional) step is looking at the integral tables on Wikipedia.

>>15176789
This is the original one. I managed to answer most of the problems in the book but not this one given.

How do you solve this one?

[math]3^x+3^(1-x)-4=0[/math]

>>15176917
What's with the misaligned left parenthesis?

>>15176921
anon fucked up his exponents in tex

>>15176926

It seems so.

3^x+3^(1−x)−4=0
Is the equation.

>>15176917
>>15176939
It's just a quadratic equation which you can easily factor.

/mg/ chads I'm clueless about an exercise in Hinman's Mathematical Logic text.
The text doesn't contain solutions or hints to the exercises, so if some logic enjoyer here could just give me a small hint on how to approach (ii), I would be very grateful.

Pic rel shows the exercise.
For context, (i) asks to show that ZF + global choice is conservative over ZF.
Theorem 2.6.23 is basically this stuff https://en.wikipedia.org/wiki/Extension_by_new_constant_and_function_names..
Since [math]\forall x\exists y (x \neq \varnothing \rightarrow y \in x)[/math] is provable in ZF the theorem guarantees that T is conservative over ZF.

Now (ii) argues that "it would seem" that for any set x, one could form the set [math]\{ (z, f(z)) \mid z \in x \}[/math].
This is a choice function for the set x, so any set has a choice function.
And after transferring the result back to ZF (since T is conservative over ZF), one has [math]\mathsf{ZF} \vdash \mathsf{AC}[/math].
We know this is false, so where did this "pseudo-proof" go wrong?

>>15177003
Thanks for replying, anon.
The online solver says more or less the same, also.
I guess I've been away from it all for a while, but I'm not really certain about how to do it. What should be done with the 3^-x?

Thanks again

>>15177039
>one could form the set {(z,f(z))∣z∈x}
Except that you can't.

next thread

>>15177122

Perturbation theory for nonlinear differential equations: based or cringe?

Starting to get the hang of these applied mathematics

>>15177058
It's easier to deal with a variable like y than 3^x. also it's easier to deal with things that are not fractions.

>>15177250
Thanks. I could do it.

>>15177317
Good work!

Is there a general solution for [math]x[/math] in
[eqn]
x \in \mathbb{R}^{n}_{+}, y \in \mathbb{R}^{m}_{+}, a \in \mathbb{R}^{m}_{+} \\
\begin{cases}
x_{1}^{a_1} + \dots + x_{n}^{a_1} = y_1 \\
\ \ \ \ \vdots \\
x_{1}^{a_m} + \dots + x_{n}^{a_m} = y_m
\end{cases}[/eqn]

>>15159206
I have 100 iq and low self-esteem.

If I learn math I will feel really really smart.

Realistically speaking, how much "progress" can a avgwit like me make? I have a degree in Computer Engineering (we are like Electrical Engineers but more retarded) so I have some knowledge of calculus, physics, and so fourth

>>15177344
>Is there a general solution for x
Are you asking if a solution exists, or you're asking for an actual closed-form expression for the general solution?

>>15176361
What's silly about this?

>>15177355
I've seen you post this over and over again. What answers are you hoping to get?

How do I go about pulling tricks out of my ass like this to solve recurrence relations?

>>15178037
By going to the next thread about it
>>>15177122

>>15169513
>>15170404
No, it's impossible.

If I understand your restrictions correctly, then each pentagonal face is triangulated separately, in one of five rotationally equivalent ways.
Each triangulated face "flips" the parity of the two vertices on one of its edges.
When every face is triangulated, we want each vertex to have be flipped either once or thrice.
In total, we have 12 faces to flip 20 vertices, so exactly 4 vertices are flipped thrice.

If any edge is flipped twice, then we've used up 2 edge-flips to accomplish nothing; the remaining 10 edge-flips cannot then share any vertices at all, and we end up with only 2 vertices flipped thrice.
By contradiction, no edge is flipped twice.

Therefore, the only way to flip a vertex thrice is for each of its connecting edges to be flipped once each.
Also, it turns out these thrice-flipped vertices cannot be neighbors: neighboring vertices share two faces but only one edge, so if a vertex is thrice-flipped, then its neighbors can't use its one remaining face to flip both its remaining edges.
In other words, the sets of edge-flips that make up the thrice-flipped vertices don't overlap at all.
Since it takes 3 edge-flips to make a thrice-flipped vertex, it takes all 12 to make all four such vertices, but then we end up only having flipped 16 vertices -- a contradiction.

Quod erat demonstrandum, as the kids say.