/sci/ - Science & Math

>>15486071
numbers larger than 10 are a fiction

>>15486077
>10
Funny way of saying 1 Ten.

I love pure but I'm better at applied.

>>15486099
The one question applied folks can't answer:
>Is there anything, in the entirety of mathematics, that you are excited about?

>>15486071
anyone know anything compact sheaf symplectic morphisms?

>>15486103
i like drawing triangles

>>15486118
>drawing
Construct, youre basically an Architect.

>>15486110
should have brushed your teeth retard

It's a good thing that this is /mg/ because you'd need a goddamn math degree to count all of the splinter threads we're seeing today

>mathfags are so retarded it takes them three tries to start their own general correctly

doing a statistics course online over the summer, and i don't really understand how to do this thing and i don't really know how to put it words either so i'll just give an example.

i've got this probability distribution table of discrete values, 1 to 5, pic related.
i understand how to find the mean of it; multiply the x to the corresponding probability, and add them up over the entire set.
that's the effective value, or mu. i can comprehend what is going on under the hood and i can believe that it represents "if you did this thing a million times, all averaged out your result would be this value"

and i get how to do the variance of it, in terms of i know the equation to use and steps to take, but i don't understand what is going on under the hood or what that really represents.

and also i can get the square root of that value, and that's the standard deviation, and i get that the STDEV relates to how far each value is, on average, from the mean. that makes sense, i get that. it would be like "if you did this thing a million times, you'd usually get this outcome, and the typical outcome would be within some distance Y of the mean"

but additionally, i don't get what it means to calculate the expected value of some function, relative to that set of random values. or the variance of some function relative to that set of random values.

like, i'll have a problem where i have pic related table, and "calculate E[3x+(2x^2 + 3)] "
or "calculate Var[3/(x^-2)] "
like what the fuck does that even mean?
i can see the steps to take in the textbook or whatever but i don't know what the fuck is going on.

my working model is that i basically add a new column, and plug in values of x for that equation. so like, instead of [1], my first value would be (3*1 + 2*1^2 +3) and it just has the same fuckin .5 probability as the normal x.

but what does that mean? like why would i do that? what's the purpose?

>>15486110
Should have been born in Europe or the Common Wealth.

>>15486520
>and i get how to do the variance of it, in terms of i know the equation to use and steps to take, but i don't understand what is going on under the hood or what that really represents.
Variance, squared deviation from the mean, is the distances from the mean average for all your data.
https://en.wikipedia.org/wiki/Statistical_dispersion
You could also think of it like how spread your data is in raw terms, rather than standardized terms.
>but additionally, i don't get what it means to calculate the expected value of some function, relative to that set of random values. or the variance of some function relative to that set of random values.
The expected value conceptually changes a little based on what you are doing. Simplest is random variables like die throws. https://en.wikipedia.org/wiki/Expected_value#Random_variables_with_finitely_many_outcomes
Though you may benefit most from this very short example of the fair die as it explains both expected value and variance and how they're related, https://en.wikipedia.org/wiki/Variance#Fair_die
>but what does that mean? like why would i do that? what's the purpose?
The purpose is to know what you should expect and by what range you should expect it. In other terms, to have a much more accurate and more precise guess. Examples are limitless but, say, being able to determine whether in some given range of throws of a die whether the die is fair or not and to what degree it may not be. You want to know what is or isn't "extremely improbable" for example.

Hi, I'm a retard who doesn't know algebra. Or remember fractions, calculating cirumferences, any of that. I wanna learn math to a competent enough level to take a crack at SB5 (Stanford Binet 5).
Any suggestions on how to go about teaching myself?

>>15486666
khan academy

>>15486666
>The Stanford–Binet Intelligence Scale is now in its fifth edition (SB5), which was released in 2003. It is a cognitive-ability and intelligence test that is used to diagnose developmental or intellectual deficiencies in young children.
Good joke on /mg/, a rare sight. Anyway, I've been shilling Openstax since 2018, so give them a try if you're actually serious. They are easy for midwits to download and have colorful pictures.

>>15486666
if you are serious, domain knowledge wont help with an iq test
as much of a meme as it is, the free brilliant wiki is pretty good
eg https://brilliant.org/wiki/pythagorean-identities/

Why did the subject of Analytic Geometry become compressed into a few chapters of a Calculus textbook? I was looking for books on the subject matter and you're either looking at textbooks prior to the 1920s, or some very fantastic Soviet Era books from the '80s. Pic rel is freely available as a Mir publication which is more than suitable for me, but I do wonder why the fashion changed.

https://mirtitles.org/2021/11/06/lectures-in-geometry-semester-1-analytic-geometry-postnikov/

>>15486725
Because there's very little material left once you take away the linear transformations to put them in a different class, in an algebra setting.

>>15486709
if all you don't know is the math section then it'll help you take the rest of the test actually
won't make you smarter but it'll let you take the subtest, the math stuff is usually just PSI i think
a lot of the better iq tests have math subsections

>>15486666
>Stanford Binet 5).
did I do well?

base 12 >> base 10

>>15486110
>compact sheaf symplectic morphisms
word salad

>>15486884
>if all you don't know is the math section then it'll help you take the rest of the test actually
i can not imagine so. the math is more intuition and patterns rather than actual equations

Sphere - can be integrated
Sphere/Sphere intersection - can be integrated
Torus - can be integrated
Torus/Sphere intersection - can be integrated
Torus/Torus intersection - no methods are known to exist
Why?

>>15486750
And then we have students that cannot even perform basic equations with an ellipse or any sort of conic.

there is a connection between euler's elliptic addition theorom and the group law on elliptic curves, and i am desperately trying to understand it. heres what ive managed to gather so far:

elliptic addition theorem
>fagnano (lol) managed to find the theorem for doubling the arclength of a lemniscate
>euler later generalized it into the addition theorem for lemniscate arclengths, pic rel.

group law on elliptic curves
>chord and tangent method
>???

connection
>inverse of [math]\displaystyle \int_{0}^{a}\frac{dx}{\sqrt{P(x)}}[/math] can be used to parameterized elliptic curves (why?)
>the addition theorem for the above integral (pic rel?) is identical to the chord and tangent method (why????)

any help is appreciated.

The limit above is equivalent to

[math]\lim\limits_{x\to a}\dfrac{f(x) - f(a) - Df(a)(x-a)}{||x-a||} = (0, 0, \dots, 0) \in \mathbb{R}^{p}[/math]

right?

>>15486666
Cannot tell if bait, but your quads are nice and checked

>>15488581
Yes. One direction is one line. The other direction involves knowing properties about limits and norms but is also easy.

what's the purpose of free groups and torsion supposed to be?

>>15488936
Free bits wander off infinitely. Torsion bits loop back on themselves. In the case of finitely generated abelian groups, a group is made up of *only* free bits and torsion bits; no other weirder bits.

>>15488466
Consider the elliptic curve [math]E: y^2=g(x) [/math], then you're integrating [math]\omega=\mathrm{d}x/y [/math], which is called an invariant differential, over a curve inside [math]E [/math] (which is a torus, we're looking at the complex solutions). Then, pick [math] \gamma_1,\gamma_2[/math] two nontrivial loops starting from the origin of [math]E [/math], such that they generate [math]\pi_1(E)=\mathbb{Z}^2 [/math].
At this point you have the Abel-Jacobi map (https://en.wikipedia.org/wiki/Abel%E2%80%93Jacobi_map)) that takes [math]P\in E [/math] and sends it to the complex number [math]\int_O^P \omega [/math]. This depends on the path [math]\vec{OP} [/math] you've taken, but by our choice of [math]\gamma_1,\gamma_2 [/math] two different paths [math]c_1,c_2: \vec{OP} [/math] are hopotopy equivalent up to some combination [math]m_1\gamma_1+m_2\gamma_2 [/math]. Call [math]I_1=\int_{\gamma_1}\omega [/math], [math]I_2= \int_{\gamma_2}\omega[/math].
This means that the Abel-Jacobi map is well defined modulo [math]I_1\mathbb{Z}\oplus I_2\mathbb{Z} [/math]. As it happens, this map [math]J:E\to \mathbb{C}/I_1\mathbb{Z}\oplus I_2\mathbb{Z} [/math] is an isomorphism. In particular, it preserves the sum, which means that [eqn]\int_O^{P+Q}\omega=\int_O^P\omega+\int_O^Q\omega \quad\mod I_1\mathbb{Z}\oplus I_2\mathbb{Z} [/eqn]
And okay, (a) the origin is not the same as the point [math](0:1:1) [/math], and (b) we have some [math]I_1,I_2 [/math] to deal with. For point (a), we could have done everything starting from [math](0:1:1) [/math] instead, it's not like the starting point bears any particular relevance. For point (b), this is a continuity argument: fix [math]P [/math] and let [math]Q [/math] vary continuously, then the integers [math]m_1I_1+m_2I_2 [/math] which appear have a continuous dependence on [math]P [/math], but since they are integers they must be constant. Now, pick [math]P=O [/math], and conclude [math]m_1=m_2=0. [/math]

>>15489323
Note that this is not a complete proof: the opposite is true, I simply hid all the difficulty behind the fact that [math]J [/math] is an isomorphism of groups. But this point of view clarifies the situation a bit, I think

>>15486071
I have to pick a topic for an algebra presentation. Module theory, field theory or Galois theory. I had chosen the structure theorem over PIDs but realized the proof's pretty boring, and then Galois theorem and Abel-Ruffini but the professor said we were going to cover that in class. So any ideas for an interesting topic?

>>15486608
coming back to this a couple days later, i more concretely understand it.

the average, or expected value of a function, is represented by the integral of x*f(x) over some given range.

f(x) is the probability of the thing happening, so you're just multiplying x (the value of the thing) times the probability of the thing, and adding up all those over the entire range.
which of course gives you the average value of the function.

which means that finding the average of some arbitrary function is done the same way, you just multiply the arbitrary function by the function that gives you a probability, and then integrate over the range you care about.

this is obvious but before i was just plugging the numbers in because "well that's how you get the average" but understanding why it gives you the average makes the entire thing more intuitive.

>>15489657
idk what the curriculum is like, but one topic I wish I had studied early on are integral extensions, or even faithfully flat extensions. Otherwise idk, something about transendence degree, usually these kind of courses want to get to Galois theory as soon as possible so they ignore transendental extensions

>>15489726
I think trascendental extensions are taken, but I'm going to ask. What could be the objective of a short presentation on integral extensions? (besides just explaining what they are). It'll be on a blackboard and I don't have a lot of time, so maybe an specific theorem/application.

>>15489808
>What could be the objective of a short presentation on integral extensions?
lying-over and going-up are the standard applications, you can look it up on Atiyah-Macdonald

I couldn't find much on Google for simplifying cos(1/2 * arccos(x)), so I farted around and got this. I'm not a mathematician so I'm not sure if I violated domains or something, or if there was a really easy way to simplify that I didn't see. Does this look right?

>>15489859

>>15489859
>>15489861
Have you ever heard of the half-angle formula?

>>15489873
Well shit. I looked at it at first and somehow convinced myself that the square root was gonna mess things up. At least the answers come out the same.

>>15489322
thanks

Non-math bro here. Can you help me come up with an equation for this?

Knowledge of knowledge of knowledge of knowledge as infinitum

Like I know that you know that I know that you know, on and on to infinity.

Would it be X^X? X being knowledge.

Thanks. Pic unrelated.

>>15490441

Ad*

Any PhD student here who hates both the industry and academia?
I don't like a 9-5 job even though the salary can be appealing.
For academia, I absolutely don't want to spend 2-4 years of post doc just to land a shitty tenure-track job at some low-paying school, and grind for 5-9 more years to get tenured.
At this point I might actually become a NEET once I defend in december.

>>15486071
Makes you think

>>15490599
>wah, I don't like industry because there's low freedom and they make you work
>wah, I don't like academia because there's low pay and options and they make you work (sort of)

>>15490605
Is there an escape?

>>15490609
there's always an escape

>>15490605
>and they make you work (sort of)
Do you know how stressful it can be if you are a professor at an R1 school?
I think I should be fine with being a teaching professor (+ doing math on my free time) for now. I just can't deal with the stress of fighting for tenure.

>>15489323
>>15489324
Ok I'll be honest, my reply is off from the start, since [math]y^2=g(x) [/math] is not an elliptic curve. I simply assumed, without taking a look at the equation, that [math]g(x) [/math] was a cubic polynomial (because you were talking about elliptic curves, so only a cubic would do the job).

At this point I think the link between your pic related and elliptic curves is more historical than mathematical, and goes as such: people studied elliptic integrals (i.e. integrals computing the arclength of ellipses), then went on to study lemniscate integrals and such. They found clever addition theorems (probably using ugly integral substitutions). Weierstrass comes along with his own special kind of "elliptic" integral, which turns out to be related to elliptic curves (i.e. smooth plane cubics) by the argument I gave.
This isn't the whole story, for example I have no idea where https://en.wikipedia.org/wiki/Weierstrass_elliptic_function#Relation_to_Jacobi's_elliptic_functions comes from, but it's not easy for me to find any more concrete info online. I also have no idea whether one can find a "conceptual" proof of Euler's addition theorem coming from geometry (akin to the one I gave), since the geometry of [math]y^2=g(x) [/math] is pretty dull (a singular rational curve, ugh)

>>15490609
Yes. Be good. If you make ground breaking research you will easily get tenure in an Ivy League, but of course most Ph.D.s are worthless.

>>15490599
When the choice is between unhappy and poor, and unhappy and rich the answer is kind of obvious.

>>15490705
Complete lie. Being good is certainly not enough. There are all sorts of cases of people who are great but step on the wrong toes and have their careers destroyed. A few years ago even Serre pretty much bullied someone into giving him credit for a result that the other guy proved but Serre wanted the problem as his own.

If [math] T [/math] is an asymptotically unbiased estimator of [math] \theta [/math]. What is the condition for [math] f ( T) [/math] to be an unbiased estimator of [math] f ( \theta) [/math]?

>>15489657
squareable lunes

In software, it is possible to use a high level language to implement the same functionality of a low level language, at a huge cost in complexity (the amount of code required is large and complicated, and takes way more CPU).

From my limited perspective, it seems like math is a high level language that when being used for particle physics, becomes highly complicated. In the world of software engineering, this is an indication that we've taken a wrong approach, that we should use a lower-level language.

Is it possible that this is happening with math? And that there exists some more fundamental (simpler, less abstracted) language that can describe the universe? If so, is there any speculation about what this language might look like?

In how many ways can the set [math]\{1,2,3,4,5,6,7,8,9,10,11,12\}[/math] be partitioned into three subsets such that the numbers in each subset sum to [math]26[/math]?

>>15490787
> A few years ago even Serre pretty much bullied someone into giving him credit for a result that the other guy proved but Serre wanted the problem as his own.
Can you elaborate more on this?

>>15491222
Is there a name for the general form of this problem?

>>15491222
The Stirling partition number S(12,3) gives 86526 possible ways to partition a 12 element set into 3 sub-sets. So I think the only way to work that out is by brute force. Should be fairly straight forward in the programming / scripting language of your choice.

>>15486071
>pic
learn long division and what 1/3 is in base twelve

Are there some good modern texts on lattice theory? Surely Birkhoff must be at least a little dated nowadays or is that still the standard reference?

As an aside, why does math talk on this board (specifically outside of /mg/) boil down to people arguing over the same handful of topics (set theory, infinity, ...) again and again? Is this board full of cranks?

>>15492119
All boards have a bunch of threads that repeat over and over, I don't know what causes this phenomenon but here are two conjectures
>Some (you) addict just wants to farm for replies but isn't feeling creative enough for fresh OP so just goes with something that's worked before
>Bots (or sad Discord users) wish to degrade the quality of this site for some reason, scan for threads that generate particularly retarded discussions and continually repost with slight variations

>>15492499
>>15492119
That said, be the change you wish to see in the world and just make some good math threads

>>15492119
>Is this board full of cranks?
Since they aren't banned in spite of chronic rule breaking, yes, the board has very high crank content.

bvump

Why is tetration never covered in any math curriculum? What area of math does it even belong in?

>>15493122
You can always add more stuff, you can always iterate one level more

As it happens, in the universe, most of the good interesting stuff is clustered at the ... center
Tetration is just adding one more of something interesting. But the more you do that, paradoxically, the less interesting it becomes.
Until you add a meta level. And then iterate on that. But it also stops being interesting pretty fast.

Find new concepts instead of adding one level of iteration. Tetration belongs in the branch of math where people like to do the same thing obsessively every day instead of branching out

>>15493122
Because it's not useful at all. I have never noticed it being used in any field, pure or applied. I can't think of a single case where you would be powering from the top.

>>15491222
[math]\Prod_{i=1}^12(x^i+y^i+z^i)[/math] find the [math]x^{26}y^{26}z^{26}[/math] coefficient

>>15491222
[math]\prod_{i=1}^{12}(x^i+y^i+z^i)[/math] find the [math]x^{26}y^{26}z^{26}[/math] coefficient

>>15493280
Do you not have to divide it by 6?

We lost one lads

>>15493154
Well it's needed to define Graham's number, which all the pop soi youtubers are obsessing about.

It do be like that

>>15493868
As I said, not useful.

>>15493888
Like original Shakespeare and the translated pages.

>>15492548
>has very high crank content
Not cranky enough, I say.

Hello friends looks like the math reddit has shut down so now Im here
What areas of math do you guys all like?
I myself am partial to algebraic geometry and category theory

I've heard that Con(ZF) implies Con(ZFC). Is it true that the argument for this goes like
>assume ZF is consistent
>then ZF has a model
>within this model is something called the "constructible universe"
>this constructible universe is also a model of ZF
>choice is true in the constructible universe
>so it is a model of ZFC
>therefore ZFC is consistent
?

>>15493858
Uncle Ted should be the next /mg/ OP image.

>>15495680
Which way, Crank-Anons?

>>15493858
BASED TRANNY

>>15495680
>Uncle Ted should be the next /mg/ OP image.
father ted*

>>15495680
pic related

>>15495888
I too would be depressed if this were my life’s work

>>15495642
Judging by the folks here, we love babby real analysis, babby complex analysis, babby model theory, babby probability and arithmetic geometry.

>>15495642
We like discussing undergraduate calculus books.

>>15495642
real numbers
man it sucks now the math subreddit is gone, MJR is gone, guess only math discords and here left now to shitpost on

>>15496513
Check out fractalforums and mathchan, both slow, both of them like diff geo

>>15486520
unironically just look this shit up on youtube, there are a bunch of intuitive explanation videos on all sorts of mathematical topics including statistics

>>15496513
>>15495642
It's temporary. It's just some dumb protest they're doing.

>>15486520
Every transformation of a random variable is obviously a random variable in itself, so its expectation just gives the expectation of that. What is so hard to understand about that? For example, we can consider the function that transforms even numbers to 1 and odd numbers to 0. Applying this transformation to the random variable of the number on some dice gives the indicator variable of the dice rolling an even number. Its expectation will roughly give a measure of how often will a dice roll even. It is no coincidence it equals 0.5 i.e. the probability of getting an even roll

>working in my thesis
>assessor: you should put y here and remove x, sweaty
>few days later
>assessor: you know, I think x would be more fitting instead of y
>few days later
>assessor: I don't see how x is relevant, delete it and add z
why are doctors like this