/sci/ - Science & Math

How do I biject a countable infinite cartesian product of an uncountable set X, with X

>>12505951
that might be impossible though
for instance, assume that [math]\aleph_0 < \aleph_1 < 2^{\aleph_0} [/math] then [math]|\aleph_1 ^ {\aleph_0}| \geq |2^{\aleph_0}| > \aleph_1[/math]

>>12505951
In general that's impossible.
[math]X= \omega_\omega [/math] doesn't work.
Konig's theorem says that if the cofinality of |X| is [math]\omega[/math], then [math]|X|^\omega > |X|[/math].
In general cardinal exponentiation is very undetermined without assuming additional axioms. However, under GCH (Generalized Continuum Hypothesis), things become much simpler:
If [math]\lambda, \mu[/math] are cardinals, then
[math] \lambda \leq \mu \implies \lambda^\mu = \mu^+ [/math]
[math] \lambda \geq \mu \geq cf(\lambda) \implies \lambda^\mu = \lambda^+ [/math]
otherwise [math] \lambda^\mu = \lambda [/math]

>>12505984
>>12505994
What if the set X is {0,1}^w
The munkres asks me to prove that the set of all countable subsets of {0,1}^w is bijectable with X. I take that as a cartesian product with countably infinite slots where all combinations of elements of X are represented as in cartesian products.

>>12506032
I guess subsets lack order so it has fewer possibilities than cartesian products. And since it's a product of infinite that might be relevant

Either way I'm stumped on the problem

>>12506038
Well the set of countable subsets, call it Y, is at least as big as X, by an argument that depends on your definition of "countable".
So it suffices to show Y<=X. But for that, we know Y<= X^N (the functions from N to X) and X^N = (2^N)^N= 2^(NxN)= 2^N (using NxN=N, by a zig-zag argument).

>>12506054
Woah... You turned an uncountable x countable into a 2^ countablexcountable. Thank you anon. So this method works for cartesian products right?

Does IUT prove the abc-conjecture?

>>12506093
No. 3.12 cлeдcтвиe cocaтб.

Say we have [math]f : H \to \mathbb{C}[/math] meromorphic on the upper half complex plane [math]H[/math] and we want to construct a meromorphic function [math]g : \dot{D}\to \mathbb{C}[/math] defined on the punctured disc [math]\dot D[/math] satisfying [math]f(z) = g^{2\pi i z}, \ \forall z \in H[/math].
What is the best way to explain how [math]g[/math] is constructed?

I thought of one by taking two different branches of the complex logarithm (say, the principal and the one undefined on the negative part of the imaginary axis), defining [math]g_1, g_2[/math] each one with each branch and then using then to get a meromorphic extension (of both) to the whole punctured disk.
I remember the proof of these log-functions (say log of h) is by defining them as an integral of h'/h, which is then a primitive of it. So I guess I could do something like that here too.

Is there a more instructive way? I wonder if I'm overcomplicating things.

Say we have [math]f : H \to \mathbb{C}[/math] meromorphic on the upper half complex plane [math]H[/math] and we want to construct a meromorphic function [math]g : \dot{D}\to \mathbb{C}[/math] defined on the punctured disc [math]\dot D[/math] satisfying [math]f(z) = g(e^{2\pi i z}), \ \forall z \in H[/math]. What is the best way to explain how [math]g[/math] is constructed? I thought of one by taking two different branches of the complex logarithm (say, the principal and the one undefined on the negative part of the imaginary axis), defining [math]g_1, g_2[/math] each one with each branch and then using then to get a meromorphic extension (of both) to the whole punctured disk. I remember the proof of these log-functions (say log of h) is by defining them as an integral of h'/h, which is then a primitive of it. So I guess I could do something like that here too. Is there a more instructive way? I wonder if I'm overcomplicating things.

Say we have [math]f : H \to \mathbb{C}[/math] meromorphic on the upper half complex plane [math]H[/math] and we want to construct a meromorphic function [math]g : \dot{D}\to \mathbb{C}[/math] defined on the punctured disc [math]\dot D[/math] satisfying [math]f(z) = g(e^{2\pi i z}), \ \forall z \in H[/math].
What is the best way to explain how [math]g[/math] is constructed?

I thought of one by taking two different branches of the complex logarithm (say, the principal and the one undefined on the negative part of the imaginary axis), defining [math]g_1, g_2[/math] each one with each branch and then using then to get a meromorphic extension (of both) to the whole punctured disk.
I remember the proof of these log-functions (say log of h) is by defining them as an integral of h'/h, which is then a primitive of it. So I guess I could do something like that here too.

Is there a more instructive way? I wonder if I'm overcomplicating things.

>>12506356
You need f to be periodic.

>>12506356
>>12506369
When you have f periodic, you don't need to consider which branch of the logarithm you take, you can take them all together at once, since f(z + 1)=f(z) so the resulting map is well-defined.

>>12506369
Ah yes, f is [math]\mathbb{Z}[/math]-periodic.

>>12506377
So, basically, the argument defect of a branch of log is fixed when I consider the composition [math]w\mapsto f(\log(w)/(2\pi i))[/math] due to the [math]\mathbb{Z}[/math]-periodicity of the function? Cool.

>>12505939
I'm starting over learning maths from high school level. I've done some Khan Academy lessons, but I want something more throughout, especially for (analytic) geometry. What's a good book?

>>12506406
Exactly. This idea is extremely common in all of mathematics.
When the function has some symmetry, i.e. when you can alter the argument of the function in some way without changing its value, what you often do is "take the quotient" and create a new function which takes groups of arguments instead of one argument as the argument.
This is the whole idea behind the first isomorphism theorem.

Merry Christmas /mg/. Does the usual way of rotating a function around the origin rotate it's Riemann surface around the origin?
Also, what's the simplest function takes the real axis to both branches of sqrt(|x|) and the imaginary axis to [math]\pm[/math]i*sqrt(|Im(x)|)?

The symbol
[math]:=[/math]
isn't part of the primitive set theory language.

So if I write
[math]f(x):=7\sin(3x)[/math]
how does this actually translate to a set theory set?

>>12507501
{(x,y) in RxR | y=7sin(3x) }

>>12507501
In terms of set theory, := and = aren’t different. It’s just notation humans use to clarify to other humans that this equality is definitional, and not the result of something established earlier

>>12507501
https://en.wikipedia.org/wiki/Skolem_normal_form

>>12507340
For your second question, do you mean to create a surface that includes two real parabola (sqrtx mirrored at zero) and two purely imaginary parabola of the same proportion, but at 90°?

>>12507552
Completely unrelated to what anon was asking for.

Repill me on Vertex Operator Algebras and product expansions.

Anybody here like them?

>>12507565
Michael Penn does.

>>12507567
lol, I've seen his channel before but I didn't know he does this kind of stuff too. Must be very weird jumping in topics and level so strongly

is Nikolaj-K an /ourguy/?

>>12507340
Geometric intuition ahould not be applied to calc 2

>>12507653
Worst take I've ever seen on this general

>>12507655
Why? People doing calc 2 (and 1, and most of 3) need to focus on memorizing the algebraic methods to not fuck everything up. Don't run before you know how to walk.

>>12507704
Holy shit man I hope this is higher bait. Read
uni-muenster.de SLASH Physik.TP SLASH ~munsteg SLASH arnold.html
and think about that again. Also fuck Hiro for making me reformat.

>>12507653
Calc 2 is complex variables? Huh.

I sort of hate differential geometry, but I think geometric intuition is always a useful tool to exploit. Tool, though, shouldn't be more than that unless it's actually a course explicitly in geometry or a high level course with big overlap with the area.

>>12507725
>>12507655
Geometric thinking is only required where creativity is needed. Calc 2 doesnt need creativity.

>>12507775
stop

I know he is well-respected here on /mg/ but why do you think mathematical plebs are so repulsed by his ideas, constantly misinterpreting him and calling him mean names without any arguments? You would think the plebs would be attracted to a finitist approach to mathematics.

I know he is well-respected here on /mg/ but why do you think mathematical plebs are so repulsed by his ideas, constantly misinterpreting him and calling him mean names without any arguments? You would think the plebs would be attracted to a finitist approach to mathematics.

>>12507340
You're looking for the family of curves that takes all rays extending from the origin rotated from the positive reals by 2pi/theta to ±sqrt|x| for all integer theta, can't remember the name but the extension to rational theta was a fun introductory problem about a year ahead of where you currently are.

>>12507816
if you like philosophical discussions on aesthetics then go watch wildberger
but if you actually want to do math then his videos are peak boredom

>>12507838
>discussions on aesthetics
I think you mistook him for someone else.

Is the cardinality of the continuum undecidable because you can choose any version of ZFC where the solution is a particular aleph, and that aleph_1 and aleph_23,401,219 are just as equally viable as any other aleph?
If no aleph solution has special significance, are there even any problems from other topics that depend on the choice of aleph?
Also, how the fuck does that even work, anyway? Can you arbitrarily make the irrationals denser in any way you like? Is the answer actually aleph_infinity?

>>12507851
The answer is that ZF/ZFC is inadequate.

>>12507851
All the alephs you mentioned are possible. For more generality, look up Easton's theorem.
Lots of topics and lots of other problems in set theory depend on cardinal exponentiation, so the choice does matter.
>Is the answer actually aleph_infinity
If by infinity you mean omega, then that is impossible by Konigs theorem, since its cofinality is omega.
>Also, how the fuck does that even work, anyway? Can you arbitrarily make the irrationals denser in any way you like?
None of this actually works or means anything. It's a purely formal symbol manipulation that has absolutely nothing to do with geometrical intuition about the real line. The concept of irrationals dont even come in the proofs, since youre dealing with 2^aleph_0 rather than standard definitions of the reals. The proof in one direction is not too hard and uses constructible universe model, while the other direction uses forcing which is absolutely ridiculous and depends on a lot of syntactic fuckery.

>>12507884
Inadequate for what? The infinitist schizos seem to be doing just fine with ZFC.

>>12507937
Inadequate for solving the continuum hypothesis. Despite all the arguments about forcing, the real world has only one solution to the problem which we have yet to discover.

>>12507949
How much more will it take to convince you schizos that the question is meaningless? Does the opinion of seasoned set theorists like Feferman mean nothing to you? It's literally all just a delusion. Like asking whether the number of breadcrumbs that Greten left on her way is even or odd. The answer is that it's a fairy tale, so who cares?

>>12507978
It's not schizo to have the axioms:
-> Non-Algebraic Irrationals exist
-> There is only one correct answer to any question of absolute or relative size in mathematics and logic

>>12505939
test

>>12507851
I think it will help to think about it if you bring ordinals into the picture, not just talk about aleph_x so abstractly.

>>12505939
How do I find meaning in mathematics? Or even take an interest in it?
I can memorize formulae and crunch numbers 'till the cows come home, but I never feel like I actually learned anything worthwhile.
I guess I kind of enjoyed my statistics classes, but only because when you're working with running ANCOVA on data you've collected there's a clear purpose to what you're doing.
I never took calculus or pre-calculus, never did trigonometry, just basic algebra, statistics, and symbolic logic needed for my degree. Now whenever I open up the calculus or pre-calc textbooks I own I just get bored senseless and can't bring myself to turn the page or even do the sample problems. I'd like to understand the physics books I own a little better and that really does require some basic knowledge in calculus, but even with a clear goal in mind I just find mathematics so tedious during the learning phase that I find myself continually closing the books and reading more entertaining topics like philosophy of mind, metametaphysics, etc.

>>12508201
Mathematics is just a toolset. All you're doing is learning how the tools work and then using it to understand what's going on when someone describes an object with a mathematical model.

>>12506113
чтo?

>>12508201
>>12508259
There are two types of people:
Applied maths and pure maths people
You guys are applied maths people. Stay in your lane. Find enjoyment in your own pursuits and use math when it's useful.
But you'll never know the true joy of pure maths.

>>12508265
When I study pure math I feel like I'm aimlessly memorizing random facts. Is there a way not to feel this way?

>>12508265
Mathematics exists solely for the purpose of describing idealized objects so that one can better understand the world we live in. There is nothing wrong with trying to develop new tools with which one can understand the world. The point of math is to develop and teach how to use computational tools with which one could potentially use to better understand the world. Trying to divorce yourself from the real world and studying some abstract category you imagined in your head for no reason other than you like some abstraction is just masturbation.

>>12508274
And masturbation is orgasmic
And pure maths is above that

Bros... I think I just created a formal definition for formality:
>Lemma Definition: A mechanical element of a topology is an element which maps to other elements only in response to which elements are filled with uniformally distributed energy (such as heat, rotation, linear motion)
>Formality = An ordered pair containing symbols for the cardinality of mechanical elements and the cardinality of total elements

This can be a gear system, digital system, or mentally it is different symbols being neurally activated, and logical connectors being activated, which is mapped to a new state in the system. Intelligence uses memory to store a history of states to guide the energy distribution, to favor maps/implications that lead to desired concept.

>>1250829
I see. I shouldn't defame masturbation by comparing it to the way pure math is practiced these days. Masturbation is an attempt to satisfy the natural need to reproduce when one isn't able to satisfy it with the opposite sex. Pure math is just a few autists in a room making up abstractions because they don't have anything better to do.

>>12508303
Also each element can have distinct energy levels but the energy is uniform. One gate can be hotter or one lever be opened wider, to cause more implications to be discovered in a given direction.
I say the energy must be uniform because I assume that non uniformity would differentiate the elements and cause them to not be truly atomic. Gradient formal systems like russian water based diff eq calculators require continuum methods.

>>12505939
Has anyone here worked on genetics?
Also to motivate myself I'll post the page number I am on my current topology text book every time I post.
17.

All systems of elements are formal because all consciousness and matter evolves according to recursive principles. The matter is how to select a formal system out of a group of particles. This requires connecting them in sets. The sets are linkage neurons or co-perceptions. These are mechanical elements of a formal system which is topologically equivalent to the material system being perceived.

The pen and paper used in math is a partial projection of the whole system contained mentally. Sometimes ink/muscle memory can perform some acts without neurons and then feed back to neurons, however this whole cycle is mechanical. It is mirrored by the topologically equivalent sentience domain system, only where neurons are involved. The non neuron aspects from muscle memory are perceived by the nervous system rather than mind. A being that perceives both nerves and neurons would perceive the entire formal circuit while the sub-beings perceive holes. For all sentiences there is a sentiences for their union and their intersection.

>>12508274
>implying that art is useless.masturbation

>>12508346
Note: Neurons are material but are distinct from the copied topology in the way that sleep cycles copy sun cycles yet are distinct

>>12508349
Yes. It's like pure math. Lower than masturbation.

Pure math is either structural philosophy or alien-world finding. A lot of alien worlds seem defective like universes with improper force ratios, and there should be discretion used as to whether or not it's worth exploring

Assuming the letter T is two intersecting one dimensional lines, does the point at the intersection container a neighborhood which is homeomorphic with an open subset of R?

>>12508498
certainly not. if you consider such a neighborhood, consider a connected subset of it containing the intersection point. well removing the intersection point from the connected neighborhood gives 3 connected components. but removing any point from any connected neighborhood in R gives 2 connected components. this leads to a proof that the neighborhoods cannot be homeomorphic.

>>12507565
>Anybody here like them?
Sure.

>>12508510
So is this a shape that is neither a 1 manifold nor 2 manifold? What do we call it?

>>12508545
I mean... we call it a topological space? You could think of it as a quotient space of a manifold I guess. There's no special name for it.

Can you guys help me with something here.
Let f(x) be x---> Z
let K be cross product of Z
find elements of Z that are not in K
What is the method here ?

>>12508545
I think you might call it a stratified space. Specifically a stratification of 1-manifolds and 0-manifolds.

>>12505939

>>12508903
god i want a hot tranny to shove a cock up my ass

>>12508545
CW complex

>>12509116
What's the point? They can't get hard.

>>12508267
Your aim should be the ability to post the most pretentious shit possible on sci.
Either that or embrace traditional mathematics and rid the world of the tranny and jew.

Is the following legit?

"""
f: R --> R such that:
for all x,y in R: f(x + y) = f(x) f(y)
f does not map everything to 0
f is differentiable at 0 with f'(0) = 1

The above seem to be enough to define the exponential.

f(0) = f(0 + 0) = f(0) f(0) ==> f(0) = 0 or f(0) = 1
If f(0) = 0, then for all x in R: f(x) = f(x + 0) = f(x) f(0) = 0, which is not allowed in the construction.
Thus it has to be f(0) = 1.

Notice that
(f(x+h) - f(x)) / h = f(x) (f(h) - 1) / h

We have 1 = f'(0) = f(0) lim (f(h) - 1) / h
therefore lim (f(h) - 1) / h = 1

This means that f is differentiable for all x in R with f'(x) = f(x) lim (f(h) - 1) / h = f(x).

We can now proceed as usual with the derivative definition.
"""

>>12509298
Also, is there a way to make the conditions even simpler?

>>12507653
developing geometric intuition wherever you can is excellent practice

>>12509383
What about topological intuition instead

>>12509326
Yes.
>f does not map everything to 0
This is redundant because f'(0)=1 implies f is not constant.

Thinking about smart ways of checking the Jacobi identity for the cross product in R^3.
One way is to consider the trilinear form
f(a,b,c)=(axb)xc + (bxc)xa + (cxa)xb
and check on the tensor basis. Notice that when a=b=c then f=0.
When a,b,c are distinct basis vectors f is also 0. When two of them are equal and the other is different it's also easy to check that f=0, so by linearity it follows that f is 0.
But this doesn't look very elegant. I expect there to be a simpler way to check it.
At the end of the day the Jacobi formula states something like the cross product is distributive. If ad_a is a map s.t. ad_a(b)=axb then Jacobi formula states
ad_a( bxc ) = (ad_a(b))xc + bx(ad_a(c)) which looks more motivated.
Maybe there is some more general property about the cross product that's easy to check from the definition (no casework) that immediately implies it's ad-distributive?

>>12509298
Sounds fine.
One note, if f'(0) = 1, then f is already not the zero map (which has f'(0)=0).

>>12508540
What do you like about them.

Bros wtf is a hyperplane? My prof gave this as a motivation for differentiability. We say if a function is diffable at a point then it must have a tangent hyperplane at the point. I'm lost bros, I know hyperplanes as a vetcor space vaguely but I don't see why differentiability implies such a vector space's existence.

>>12509634
Maybe he meant for the given dimension? Like a 1d function has a 1-plane (line) tangent, 2d space has a plane tangent, 3d has a cube tangent and same in 4+

Just means locally euclidean

>>12509634
a hyperplane in an n-dimensional space is simply an (n-1)-dimensional affine subspace
hyperplane in 2D space is a line
hyperplane in 3D space is an ordinary plane
hyperplane in 4D space is something 3D
etc.

a graph of a function of n variables is an n-dimensional surface living in an (n+1)-dimensional space. it makes sense to consider its tangent space which will be also n-dimensional, therefore a hyperplane in (n+1)-dim space. the motivation of your prof was that a function is differentiable at a point if its graph is smooth enough so that there exists a well-defined tangent space at this point. it will mean that the function can be approximated by a linear expression (so-called differential).
function of 1 variable has a graph which is a 1D curve living in 2D space. its tangent space is the usual tangent line, therefore a hyperplane.
function of 2 variables has a graph which is a 2D surface living in 3D space. its tangent space is 2D, therefore a hyperplane.
etc.

Post interesting facts about knots I need help for my philosophy thesis. Something about how they form bonds without any overtly interacting components. Or how they have structure that the inside can't perceive (it just thinks it's the circle group).

>Joshua King came to Cambridge from Hawkshead Grammar School. It was soon evident that the school had produced someone of importance. He became Senior Wrangler, and his reputation in Cambridge was immense. It was believed that nothing less than a second Newton had appeared. They expected his work as a mathematician to make an epoch in the science. At an early age he became president of Queens’; later, he was Lucasian Professor. He published nothing; in fact, he did no mathematical work. But as long as he kept his health, he was an active and prominent figure in Cambridge, and he maintained his enormous reputation. When he died, it was felt that the memory of such an extraordinary man should not be permitted to die out, and his papers should be published. So his papers were examined, and nothing whatever worth publishing was found.

>>12509736
Based.

>>12509736
lol, nice find

Thread theme
https://youtu.be/7NyxyKQ98ko

>>12509647
>>12509667

Thank you guys. Now I understand it.

>>12509736
How does this happen? Pure social circuits? This is why I say the OP pic should more often be math itself than mathematicians. I'm making the next OP and it's gonna be the cover of a book on elasticity from some soviet scientist in the 6/70s

Gonna be called ziel edition

>>12509736
Proof that mathematics is just autistic masturbation

>>12509456
>What do you like about them.
The fact that their representation theory is extremely rich and forms the basis of CFT and ST

>>12509736
based, i love how atheists got scammed

>>12509736

this is what IQ masturbation does

>>12507635
Yeah. He's doing a small series on basic proofs now

>>12509736
Literally how...

>>12510270
Nepotism, cult of personality.

>>12509127
False, if they keep using it they can always get hard.

>>12510296
WTF I love cults of personality now... I'll make myself the goddess of algebra, similarly, without any merits. Wait and see, my friends, wait and see.

>>12509780
yeah okay but do you like something inherent to them, something mathematical

>>12509776
Sorry, but next edition is gonna be Kummer to complete the K3 gang.

>>12509403
>>12509456
oooh right!
thanks

>>12510323
Yes of course, I just told you.

bros is there any hope for a simpler closed form of this bad boy?

>>12510922
for context, [math]a,\sigma^2,t >0[/math]

>>12510922
and [math]0<x_0<a[/math] too

Does anyone have the probability theory textbook chart?

>>12510945
Probably.

>>12510922
replace the ((-1)^n - 1) with summation over odd terms
use the formula cosx = (e^ix + e^-ix)/2
set a new variable z=pi / a
call this shit f(z)
differentiate this shit once or twice and you may get something nice like f'(z)=geometric series

>>12510968
thanks I'll try

>>12510959
cute

>>12505939
bet you geniuses can't tell me what this function is?

>>12510987
Based wolfram alpha shitting on intuitionlets

>>12510987
Is this one of those uncomputable functions?

>>12510998
problem is what you see there is the most computation you are going to get from wolfram alpha, unless u pay for mathematical but even then only 5 minutes of computation

>>12511009
*mathematica

>>12511009
Which is exactly why you shouldn't rely on graphing programs to visualize functions

>>12510517
Not really tho

>>12510987
function is (a + b*i) ^ i mod a

>>12509116
Scrolller has what you need

The square root maps the ray 0 to ±sqrt|x| and the ray pi to ±isqrt|x|, alternating for each pi rotation.
What is the analogous function that alternates like the above for each pi/3 rotation, mapping the rays 0, 2pi/3 and 4pi/3 to ±sqrt|x| and the rays pi/3, pi and 5pi/3 to ±isqrt|x|? If we say the square root is F(1) and the function just described is F(3), can we have F(Any Q)? - (think of star polygons)

Of all the situations where |a-b| is useful in a different way to a-b, are any of them interesting in any way?

>>12511102
Isn't this >>12507340?

People stop shitting /mg/ with kids questions

>>12510922
Appears to be a theta function
Got banned so posting from my phone and can't do much here but look up the series for the Jacobi theta functions and you will see yours is identical except for additive constant, scale factor, and the 1/n^2 which indicates your series is probably simply related to them

>>12511292
Yes, I am aware of this, but this shit is very difficult to manipulate effectively. I will probably end up with some theta function which is not very satisfying but it's like that sometimes. Thanks

>>12511292
I'll state my issue in full: basically I'm trying to evaluate this integral. I know for sure that it should be a function of both [math]x_0[/math] and [math]t[/math].

>>12511392
I tried fucking around with wrapped distributions but I'm not getting anywhere near the answer (that I know numerically)

>>12511392
less blurred here

What's the expected determinant of a random matrix in M_2x2(C) who's four entries are sampled from U(1)?

>>12511438
[math]\int_{T^4} (ad-bc) d\mu[/math]

Can we even define the expectation value on a circle without faking a base point?
If I'm selecting a random number in a compact interval [0, 2pi] with endpoints identified, it seems that the expected value is not defined at all. As it's not like measuring from starting point 0 should be different than any other starting point.

>>12511438
Are the entries IID? In this case shouldn't it be 0?

>>12511392
sorry, it's easy to get misled by the large amount of low level questions into thinking there are people on /mg/ who don't know what theta functions are
Personally I think the residue theorem is the way to attack integrals with theta functions or theta-like functions, but typically that will give you a series as your answer.

>>12511457
I have a hard time justifying this on a space that's not totally ordered.

Maybe in exp(iX)-exp(iY), indeed I can say that the chance of the first exp(ix) calling on the upper hemisphere is as big as the exp(iy) falling on the bottom one, so all chances cancel out.

But for the second question, if I should give a best guess on which exp(ix) would land on the circle, there should be no reason that it should be e.g. at 1 or -1. What's the math to do that? What if compact spaces are involved but the situation isn't a symmetric one as in f(X)-f(Y) above?

>>12511453
It works as long as your space has a probability measure on it. The circle has an obvious measure.

>>12511109
I personally think of |a - b| as a distance that is not a part of the line where a, b or a-b lie, but a thing of its own.
I mean, I view it as a special case of ||u - v|| for u, v in some normed vector space. u - v is not the same as ||u-v||.

>>12511506
I'm fine with formally parametrizing the situation and computing the expectation - but this doesn't appear to capture the probability scenario.

Say I visit a friend in their hometown I don't know, I don't know where their apartment is and we decide to meet "at the circular lake close to the center." I have no knowledge of the geography and so I can't have any best guess as to where on the lake, w.r.t. nord-east-south-west, he's likely to show up.

For any [math]d\in [0,2\pi)[/math], I can parametrize the boundary of the lake via [math]z_d(\varphi){\mathrm e}^{\phi+d}[/math] and equip the circle with the uniform measure alla
https://en.wikipedia.org/wiki/Circular_uniform_distribution
(there done for [math]z_0[/math])
but then [math]\langle \varphi \rangle[/math] will depend on [math]z_d[/math].
Given that I indeed should not be able to compute a best guess "expectation", and for compact spaces the answers seem to inconveniently depend on my chart, there doesn't appear to (and arguably should not) exist a sensible model of probability for this situation.

Corrections:
[math]z_d(\varphi):={\mathrm e}^{\varphi + d}[/math]
and computing
[math]\langle z_d \rangle[/math]

>>12511518
Found an account given in
https://www.amazon.com/Topics-Circular-Statistics-Rao-Jammalamadaka/dp/9810237782

The Descriptive Statistics case is handled via "resultant" vectors R as in the Wikipedia article

>>12511039
Oh, ok. Whatever you say.

>>12511392
Also, with your first series, is there a problem with considering it as a derivative of a Jacobi theta function with respect to tau/t? Because this is where the factor n pi/a comes from, just out of the exponential/nome being differentiated.
Or is your question how to relate your series to your integral?

>>12512125
Sorry I mean integral/integrated. Hate posting from a smaller screen

I like counting to 1 while I masturbate

>>12511438
you're asking for E(ad-bc) = E(ad) - E(bc)
but in this case, clearly E(ad) = E(bc) because it's both "choose two random numbers from the same distribution and multiply"
so the answer is zero

>>12512125
The first series I posted was an attempt to obtain a solution to the integral. The squared term appeared from integrating x (if I didnt do any mistake).

>>12512670
As far as I know, integrals of theta functions have no closed form for general values of z, or x in your notation. For x=0 you can express that in terms of hyperbolic functions but in general it's just complicated.
So I wouldn't be too hopeful about a closed form for this in terms of common functions.

>>12512716
A sad relief but a relief indeed. I will focus on other aspects then. May God bless you anon

Which one of these three for 1st formal book cover to cover in algebra
> mines - a course in constructive algebra
> mac lane - algebra
> brešar - undergraduate algebra
for preschool type theorist foundational mathematician (no categories yet, unfortunately)

>>12512777
you definitely shouldn't do nonstantard shit like "constructive algebra" if it's your first book in algebra
the other two are probably ok

So how do we know, for real, that the Finnish algebra tranny is a tranny? Are there pictures?

>>12505939
If two square matrices A and B share a common eigenbasis, then AB=BA. That's a sufficient condition, but is it necessary? What about defective matrices?

>>12510517
Let's analyse your post for a second.
>The fact that their representation theory is extremely rich and forms the basis of CFT and ST
Rep theory is physics (by your own admission, see >>/sci/thread/S11539096#p11548902)), CFT is physics and ST is physics. I'm not seeing where you answered his question, since there's no maths in your post.

>>12513310
can you state the precise claim that you want to prove/disprove?

>>12513319
Essentially I'm asking whether there's a concise description of when matrix multiplication is commutative.

>>12513510
https://en.wikipedia.org/wiki/Commuting_matrices

Is the IVT part of Lean yet? What about the Cayley-Dickinson construction?

Is it a good idea to mention specific profs on a statement of purpose for a grad school application? Also I started writing a paragraph on why I want/need a PhD for intellectual+carreer reasons, but not sure if this is a good thing to include. Physics

>>12513139
yes we have pics of him on /mg/ discord

>>12513917

>him

someone is getting reported to the department's gender equality and minorites council

I have a doubt but I don’t want to bother my profesor assistants on vacation about an old topic in my analysis class.
We proved that the following are equivalent.
1) X is compact
2) Every sequence in X has a convergent subsequence.
3) X is complete and completely bounded. i.e. for all epsilon greater than 0 there is a finite cover of balls with center an element of X and radius epsilon.
This implies that every compact set intersected with the rationals is not compact due that the rationals aren’t complete.
My doubt is that if we can explicitly find an open cover of the interval [0,1] intersection Q that has no finite sub cover prooving more directly an special case of that strong affirmation

>>12514047
*Compact set in the reals

>>12514047
>This implies that every compact set intersected with the rationals is not compact due that the rationals aren’t complete.
Wrong. {0} intersected with the rationals is {0}, which is compact.

>>12514047
take some irrational number x lying in [0,1]. then take some sequence of open sets approaching [0,x) and (x,1], something like [0,x-1/n) and similar, and intersect it with Q. this will be an open cover of [0,1] intersection Q without finite subcover.

>>12508267
Yes, study Euclid's Elements first.

>>12514068
Just set [math]A = \{B(q, |x - q|/2) : q \in \mathbb{Q} \cap [0, 1] \}[/math] for the irrational [math]x[/math].

>>12507639
Why yes I believe Nikolaj-K is our guy. What gave it away?

>>12513917
What is the "official" /mg/ discord nowadays, btw?

Is there a non boring way to learn? I just feel tired and dead inside when I have to study. I have to learn calc 1 to not be fucked 30 days from now. I am intelligent, but this inability to focus is killing me

>>12514156

microdose aderall or modafinil

>>12514123
There isn't one.
If there is one, it's never actually posted, at which point I'd argue it's not an /mg/ discord, it's an unrelated discord whose users happen to browse /mg/.
There's the /sci/ discord linked in the wiki tho.
Also the two reading groups.

Would you participate in a math imageboard that gatekeeped users with a random problem from a book? Asking for a friend

>>12514172
I don't ask for help online because people always gatekeep. I tried the stackexchange forums a few times and never got answers, only a bunch of posts picking on things that were obvious but I didn't ask perfectly.

>>12514169
I see. I used to waste my time on several, but then I got banned for spamming yaoi or something.

>>12514172
No because I most likely wouldn't be able to enter it.

>>12514172
Yes, but because I have to solve stupid shit to post I'd be strongly inclined to put less effort into my posts.

>>12514176
Oh dear, Stackexchange is probably the most triggering maths site online. Overflow is OK because there you will not encounter the useless nitpickers, but Stackexchange, oh my oh my... I remember nuking my user when someone edited my and the OP's posts because they didn't like the notation and terminology. The OP used something like n(A) for the number of elements in a finite set A, and both the notation and calling the cardinality of A the number of elements in the set, those triggered this mouthbreathing autist so bad, they edited both posts and started sperging in the comments. I hate that site. I hate all of you, too, but I also love you all.

>>12514172
I don't think "users with a random problem from a book" are the problem - this can lead to good discussion.
But I barely have hope in a separate image board, since Discords usually don't work either, and those are not really so far from it. In fact, usernames usually help for such niche thing.
You can also barely keep the racists and misogynist out if you recruit from 4chan

how do you cope?

Bros... I think I nailed it...

>>12514269
Can you share your proof? I still haven't been able to prove this. At least I managed to do this: If A is a non-empty set and B is a subset of A, then B is non-empty.

>>12514078
Who is this new girl you are avatarfagging as now?

>>12514176
It obviously would not be the perfect online community but I like /mg/ enough that I think it'd be nice to have a replacement if 4chan gets nuked.
>>12514197
It would probably just be high school math (Calculus). Dunno how to make it so people can't simply use a calculator, though.
>>12514211
It'd be a one time thing, like exhentai.
>>12514227
I browse wizchan and lainchan and I think there are too many dead boards, so I'd just make a single one, maybe it'd thrive that way.
Recruiting is also a problem. I thought about asking users to put posters in their unis, but that'd probably attract too many normies.

>>12514172
Just go to the math stackexchange if you want to talk to people who actually know math. I only come here to shitpost.

>>12514284
No. You can read it in the Acta eventually.

>>12514320
>Who is this new girl you are avatarfagging as now?
You're misunderstanding. I'm rereading Helck, so I'm occasionally cropping Vamirio's faces and posting them.
Feel free to download what I've already cropped in https://mega.nz/folder/MtsCHDSJ#BRbVnZxOk0h5zZ6G03qhlA and then post it in my place.

>>12514353
What's wizchan?

I don't believe in normies

>>12514353
>It would probably just be high school math (Calculus). Dunno how to make it so people can't simply use a calculator, though.
Sounds reasonable.
>I thought about asking users to put posters in their unis, but that'd probably attract too many normies.
I can do that in my uni when I get back there. The dept is so small.

>>12514362
It's better be good then! I'll prove it using Zorn's lemma by showing that any ascending chain of non-subgroup subsets would have an upper bound, and this would then give me a maximal non-subgroup subset, but I haven't figured out how to derive the contradiction out of that yet.

recommend a book on motives

>>12513765
Yes, no

I have 29 non intersecting circles in the complex plane of arbitrary center and radius. An infinitely continuous function takes the boundaries of at least one of them to 1, at least one of them to i, at least one of them to -1 and at least one of them to -i.
I'm allowed to pick any centers, radii and which circles get the rule about the output. What method do I then take to construct the function that satisfies my choice? It's been a long time since I've looked at anything besides combinatorics.

>>12514556
This question is absolutely disgusting. It makes me want to vomit my lungs out.

>>12514531
No CDC? Meme language

>>12514556
I forgot: the number of circles in all of [math]\mathbb{C}[/math] that have their boundaries mapped to any of those is capped at 29. Specifically, because we choose the 4th roots of unity, we have to pick the largest prime below (4^2-4)/2 which is 29.

>>12514580
Where does this problem come from?

Can /mg/ explain what is a set? /sqt/ never answers.

>>12514580
>the largest prime below (4^2-4)/2 which is 29
What?

>>12514586
A set is one of the two objects studied in set theory.

>>12514588
Sorry, I meant (4^(4-1)-4)/2

>>12514589
And what is that object? Do you have a definition or are mathematicians supposed to guess what it means?

>>12514395
Nice collection. I’m not sure I understand your naming system though.

>>12514586
It is an object in [math]\textbf{Ens}[/math], of course!

>>12514598
Just so I understand, would the fifth roots of unity case look at between 5 and 310 circles?

>>12514598
Right.
Can it be the same circle for all four points?
If so, you can eenie meenie one circle, translate it's center to the zero point, and then use an expansion to adjust the radius.

>>12514604
What is Ens?

>>12514606
Yes

>>12514609
The category of sets and functions.

>>12514608
There's no rule against just having all the circles centered on the origin with varying radii, and that is in fact the simplest solution.

>>12514617
So we're back at square one aren't we? What are sets and functions?

>>12514625
Functions are an arbitrary construction built on top of the class of all sets and elements

>>12514628
kek

>>12514628
And what is a set and an element?

>>12514601
I downloaded the images from mangadex and cropped them. They all came with xXX.jpg filenames, where XX is the page number.
Once I gathered a bunch, I passed them through pnggauntlet and moved them to a different folder. Sometimes I tried to move pictures into this other folder but windows gave an error because of filename conflicts, except it didn't offer to rename them automatically, so I renamed them to whatever manually.

>>12514634
Can you take your shit rhetoric somewhere else. Even if you have justified issues with some foundations questions, doesn't mean this culture of poking other people who you know won't convince you should be continued. It's utterly annoying and that's not to say that it's not a question you could ask in the right context. But this sci shitposting that's been going on for a few months is completely disingenuous as it's merely there for the entertainment of bored people who like to fight or push some ideology. Please just don't do it. Unless you want to inflict harm on this community. In that case, go on but be aware of what you're doing.

>>12514665
Such a long response to such a simple question.

>>12514634
Sets and elements are the mathematical objects constructed by the elementhood operation, which has a domain of everything

>>12514665
Jc dude, go to bed and stop taking this seriously. Actual work, when it gets done, gets done elsewhere to here

>>12514675
What us the elementhood operation? Why does it exist?

>>12514673
Well given the question has been asked 200 times in the last months, I suppose my rant is still tiny in a compound sense.
I just you or the Wildberger edgelords in general to understand that it's toxic behaviour that makes peoples lives a tad worse. Of course, if you think toxic is a meme progressives word only, then this is all future discussion

>>12514685
This anon is not me. I'm not as much interested about why something exists, because this seems like a bit of a loaded question, and I'm perfectly fine with existence of abstract platonic objects as long as I can understand what they are and know how to do deal with them, e.g. rational numbers. I just want to understand what a set is.

>>12514681
/sci/ and /mg/ can be a place just as nice as any other - there's not many worthwhile places as far as the internet goes

>>12514694
>>12514699
Have I been mean or in any way insulting in asking questions? I thought I was being perfectly polite.

>>12514694
I agree with you that the finitist poster gets the rope but open ended questions of mathematical philosophy are actually pretty good

>>12514705
you're an annoying faggot

>>12514708
n-no u

Can you all, please, stop forgetting to capitalize the first words of sentences or put punctuation characters at the end of your fucking posts? Undergrad topics are welcome here, but undergrad discourse is not.

>>12514729
>Undergrad topics are welcome here
Great! Perhaps you can explain to us the definition of a set in mathematics.

>>12510922
equation?

>>12514698
To save us some time, have you skimmed the Stanford EoP article on basic set theory? Although the question of what a set really is is an interesting one, your inquiry could really be motivated by a different question about sets which the article can help you express.
For example, are you simply applying Munchausen's trilemma to your mental model of set theory?

>>12510922
>>12511392
>>12511407
Which differential equation are you trying to solve?

>>12514738
A set is any member of the class of primitives in any set theory, where set theories are (so far) attempts to rigorously formalise the human notions of quantity and sameness.

>>12514755
>To save us some time, have you skimmed the Stanford EoP article on basic set theory
I haven't, but I read a bit of Jech's book and most of Kunen's book up to Easton forcing on infinite product p.o.s. I have much less of an idea of what a set is than I did before opening the books.
>For example, are you simply applying Munchausen's trilemma to your mental model of set theory?
No. I don't see Munchausen's trilemma as a real problem, since our reasoning is not done in a formal system. I have a conception of what a natural or a rational number is, it is intuitively clear, but I honestly have no idea what a general set is supposed to be. Can you explain what they are in a way that makes it clear that questions in set theory have definite truth values?

>>12514782
>A set is any member of the class of primitives
Can you elaborate? What is this class of primitives?

>>12514807
primitive notions do not have a definition, there is nothing simpler to define them in terms of
a set is the only primitive notion of set theory, the axioms tell you what you can and cannot do with them

>>12514604
>Ens
>not Set
Post legs for punishment.

>>12509736
I just made a tough find

https://incels.wiki/w/Sylvain_Poirier

>>12514820
Did you find it from browsing the incel wiki, a comment by somebody or a search engine?

>>12514815
Does the word "set" have any meaning at all? What is it?

>>12514843
everything, literally everything in math is a set
so no, in that sense it has no meaning at all

just think of it intuitively, its a box containing things

>>12514830
By accident searching for whether Lubso Motl had a slander article on another person and I think this guy wrote about Motl.
And god the searching-for-a-wife page is horrible.

>>12514586
A set is a container. For any selection of objects in the universe of discourse you can construct a set with the axiom: {x}[math]\neq[/math]x.
Sets, therefore, are the objects produced by this construction.

>>12514862
The property {x}≠x is particular to some common set theories but so many standard axioms are compatible with {x}=x that I'd not use it to characterize or describe sets.

>>12514878
How tf is a set theory where {{{{{{empty}}}}{empty}}}=empty useful for anything? And don't get hung up about the empty set, this problem crops up for any choice of elements.

WHAT DO YOU THINK OF UR_ELEMENTS


In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set, but that may be an element of a set. Urelements are sometimes called "atoms" or "individuals".


https://en.wikipedia.org/wiki/Urelement

here are several different but essentially equivalent ways to treat urelements in a first-order theory.

One way is to work in a first-order theory with two sorts, sets and urelements, with a ∈ b only defined when b is a set. In this case, if U is an urelement, it makes no sense to say X ∈ U {\displaystyle X\in U} {\displaystyle X\in U}, although U ∈ X {\displaystyle U\in X} U \in X is perfectly legitimate.

Another way is to work in a one-sorted theory with a unary relation used to distinguish sets and urelements. As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from urelements. Note that in this case, the axiom of extensionality must be formulated to apply only to objects that are not urelements.

This situation is analogous to the treatments of theories of sets and classes. Indeed, urelements are in some sense dual to proper classes: urelements cannot have members whereas proper classes cannot be members. Put differently, urelements are minimal objects while proper classes are maximal objects by the membership relation (which, of course, is not an order relation, so this analogy is not to be taken literally).

The Zermelo set theory of 1908 included urelements, and hence is a version we now call ZFA or ZFCA (i.e. ZFA with axiom of choice).[1] It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements.[2] Thus, standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements. (For an exception, see Suppes.[3]) Axiomatizations of set theory that do invoke urelements include Kripke–Platek set theory with urelements, and the variant of Von Neumann–Bernays–Gödel set theory described by Mendelson.[4] In type theory, an object of type 0 can be called an urelement; hence the name "atom."

Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences. In particular, Jensen proved[5] the consistency of NFU relative to Peano arithmetic; meanwhile, the consistency of NF relative to anything remains an open problem, pending verification of Holmes's proof of its consistency relative to ZF. Moreover, NFU remains relatively consistent when augmented with an axiom of infinity and the axiom of choice. Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem. Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF. Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical universe.[6] In finitist set theory, urelements are mapped to the lowest-level components of the target phenomenon, such as atomic constituents of a physical object or members of an organisation.

Quine atoms

An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particular type of set. Quine atoms (named after Willard Van Orman Quine) are sets that only contain themselves, that is, sets that satisfy the formula x = {x}.[7]

Quine atoms cannot exist in systems of set theory that include the axiom of regularity, but they can exist in non-well-founded set theory. ZF set theory with the axiom of regularity removed cannot prove that any non-well-founded sets exist (or rather, this would mean ZF is inconsistent), but it is compatible with the existence of Quine atoms. Aczel's anti-foundation axiom implies there is a unique Quine atom. Other non-well-founded theories may admit many distinct Quine atoms; at the opposite end of the spectrum lies Boffa's axiom of superuniversality, which implies that the distinct Quine atoms form a proper class.[8]

Quine atoms also appear in Quine's New Foundations, which allows more than one such set to exist.[9]

Quine atoms are the only sets called reflexive sets by Peter Aczel,[8] although other authors, e.g. Jon Barwise and Lawrence Moss use the latter term to denote the larger class of sets with the property x ∈ x.[10]

>>12514889
There's often graph models to interpret the set theory. E.g. the class of Hereditarily finite sets correspond biject exactly with the rooted trees that have no non-trivial automorphism. I'ma post two very different such models (example picture). Sets containing themselves "just" correspond to different directed graph models. I suppose if one is willing to deal with the power set of the powerset of the naturals, one can also deal with sets of infinite depth such as x={x}. Those structures find some applications in logic/CS topics.
https://en.wikipedia.org/wiki/Non-well-founded_set_theory
This was done in large parts by Aczel, which incidentally >>12514951 also mentions

I don't want to get into your guys discussion, but I'd rather characterize sets as reified properties, i.e. properties which can exhibit properties themselves (you can say class instead of property), and such that the always-false property is a set.
Virtually always, the axioms about conjunction, disjunction and a notion of downward flattening of the properties are adopted (Empty Set, Pairing, Separation and Union, introducing algebraic manipulation rules to the game)

>>12514936
>WHAT DO YOU THINK OF UR_ELEMENTS
I think they make things more complicated and I assume they were mostly introduced by people who genuinely took set theory as a good foundation. I never used them.

>>12514956
here's that other model, if you care. Both very classical

I think I'll refrain from posting about finitism for a while because I feel I'm in danger of developing some sort of mental illness.

I have this thread bookmarked but I don't even read what's going on, I only scroll to the last reply.

>>12515018
You should read it, it's lighthearted fun

>>12515072
|confidence|<0

¿The term ultrareal/ultrareals is unused - what are your suggestions besides |x|<0?

>>12513887
>Is it a good idea to mention specific profs on a statement of purpose for a grad school application?
Only if you're genuinely interested in something specific they're doing. They're going to see through you if you're just namedropping and showing off how well you know the faculty.
>Also I started writing a paragraph on why I want/need a PhD for intellectual+carreer reasons
That's pretty much the name of the game, yes.

>>12513887
>>12515232
Also it's obvious but consider asking classmates for opinions if you're comfortable doing so.

I just graduated and have a semester off before work. Which of the following would you self-study?

>Fourier analysis–Stein and Shakarchi
>Partial differential equations–Evans (already worked through a lot of this)
>Analytic number theory–Apostol
>Brownian motion and stochastic calculus–Karatzas and Shreve

Was thinking about working through Hatcher's topology book, but it's been so long since I've done any algebra.

Requesting ñ speakers to improve the discussion in this [math]\textit{lentísimo tablón}[/math] hispachan dot org/c/res/35726 dot html

>>12507653
he's doing complex variables you stupid fucking nigger. it isn't calc 2 at all.

>>12514816
Do I have to?

>>12515303
What did you study the most before graduation? What are you the most interested in? What will your job be like?

>>12515124
Why not call those |x|<0 antireal? They would satisfy a sort of antitriangle inequality: [math]|x + y| \le |x| - |y|[/math], the same way as one would say that a diagram anticommutes if you have [math]i\colon A\to B, d\colon B\to D, k\colon A\to C, c\colon C\to D[/math] giving me [math]di = -ck[/math].

>>12515124
>The term ultrareal/ultrareals is unused -
Friedrich Waismann: Introduction to Mathematical Thinking. The Formation of Concepts in Modern Mathematics. (14. Ultrareal numbers)?

>>12515458
Yes you have to.

>>12515563
Fine. If you want a horrendous Mongol tr*nny to post her legs, you have to do some work. Prove: if we have a connected T4 space in the sense of this article https://en.wikipedia.org/wiki/Separation_axiom#Main_definitions with at least two distinct points, then it contains uncountably many points. You should be able to solve this.

>>12515543
This term reminds me of supervaluationism, which is a philosophical failure

>>12515640
Let x and y be two distinct points in the space. Because it is t4 it is t1 and so {x} and {y} are closed. Because x is not y those are disjoint. Urysohn's lemma gives us a continuous function to [0, 1] that sends x to 0 and y to 1. Because the space is connected its image is connected by continuity so it must be the whole interval. We have no a function onto an uncountable set so its domain must be uncountable. Correct?

>>12515744
I didn't think you'd actually expend your energy on that, but you did your part and I must thus do mine. A nice little exercise that one. Here you go.

>>12515761
Wow, you are disgusting

>>12515824
I know. I did warn you, though, but still the problem used as the condition was solved and I had to keep my word.

>>12515761
thank god that trannies stay in their algebraist ghetto. Yuck!

>>12514820
that led me to https://incels.wiki/w/Scott_Aaronson
so scott aaronson was an incel and even his blog name refers to his incel feelings

Is a hyperbola just a stereographic projection?

>>12515921
Proof (by lampshade)

>>12515761
Oh god, if I'm switching fields I can work with THAT?
I"m becoooming an algebraist

>>12515842
How is the math scenario in Finland? Are there 200 working mathematicians or less? What subjects most of them work on?

>>12515396
Asking about geometric generalizations of basic properties of the square root's riemann surface is just applying calc 2 to pre-existing knowledge about complex numbers.

>>12515843
Sorry.

>>12515941
>consider the diagram

>>12515943
Welcome to the club.

>>12515946
I don't know about the current situation. It was quite bad for my genre 1½ years ago when I left. Probably [math]| \text{now} - \text{then}| < \varepsilon[/math] for some very small epsilon. If you want analysis, logic or inverse problems or some type of mathematical physics, then I guess it's decent(ish). Being a neighbour of Russia implies strong tradition in analysis, and the logic school is pretty big due to Russell -> Wittgenstein -> von Wright. The undergrad & master level stuff is well-taught, though.

>>12515969
Do you consider yourself an incel?

>>12515969
I see. So you're studying elsewhere, now?
You know, when people ask me where I'd choose to live if I could, Finland always comes to mind. But only if I could be reborn there as a finn. Great basic education, cultural passion for hot drinks (coffee, milk, ...tea?), sauna, nature... Some cool metal bands. (Don't even listen to them anymore, but I have found memories.) Nice humour and friendly attitude towards friends - though the social avoidance with strangers seems really overdone imo. And the cold can get tough I imagine.

>>12515994
Yes and no. I have a bf back home but the travel restrictions, both of our studies and other stuff have pretty much made us incels. Or me for sure, he could get anyone he wants, but I trust him.

>>12516005
>I see. So you're studying elsewhere, now?
Yep.
>You know, when people ask me where I'd choose to live if I could, Finland always comes to mind. But only if I could be reborn there as a finn. Great basic education, cultural passion for hot drinks (coffee, milk, ...tea?), sauna, nature... Some cool metal bands. (Don't even listen to them anymore, but I have found memories.) Nice humour and friendly attitude towards friends - though the social avoidance with strangers seems really overdone imo. And the cold can get tough I imagine.
Have you ever visited the country? The bigger cities aren't too different from (at least) European cities, so there one could live more easily as a foreigner, but then the actual countryside would be like being black and wearing a rainbow shirt in a small Alabaman town. I recommend going for a hike in Lapland some day if you get the chance. There's this one https://www.nationalparks.fi/karhunkierros for example. If you do the whole thing, it's 82 kilometres, you can see reindeer and Aurora Borealis and stuff. And then you can drink the 10 cups of coffee a day like the locals do, as your reward.

Whats the hyperbolic metric? If I have a plane and just make a unit hyperbola above it, do I count the hyperbolas height above it and that's it? Or do I treat each point as the origin of a new hyperbola? The former seems mundane because it'd make objects nearer around the origin and then approach linear distance like euclidean at far away

>>12515914
>My recurring fantasy, through this period, was to have been born a woman, or a gay man, or best of all, completely asexual, so that I could simply devote my life to math, like my hero Paul Erdös did.
I cannot for the life of me understand why someone would have Erdos as hero

>>12516084
yeah it's a bit weird
but erdos seems like a person who very much enjoyed his life and his maths
so he's an enviable person in that sense, especially if you are not very happy with your life

>>12515824
>>12515843
Why are you disgusted? Because she pretended she didn't want to post her legs and proceeded to post a trivial problem? Is it something I'm not seeing?

>>12515761
these are beatiful legs thank you for the photo

>>12515761
ALGEBRAISTS FUCK THAT?!?!

>>12516162
no they just lick the boypussy of the tranny to make it ready for the dick of an engineer

>>12515761
trannies have to stop posting

>>12516116
Because it's a mentally ill man in stockings. Yuck.

>>12516126
Aww, thanks!

>>12516162
Nope.

>>12516182
I agree. The positive thing about a society that allows such freaks to exist is that it will necessarily collapse because of its own rot and decay eventually. It didn't have to come to this, but let us be merciful and accelerate the death of the suffering beast instead of prolonging its dying pain.

>>12516219
>let us be merciful and accelerate the death of the suffering beast instead of prolonging its dying pain
so this is the point of a trannie's life?

>>12516191
Those are kneesocks, you absolute philistine.

>>12516219
What is it with the self-flagellating tendency you trannies tend to have? Is it related to the "feelings of inferiority" Kaczynski wrote about?

>>12516238
No, that is merely a side project. The important part is to construct a certain kind of chain complex. I have two weeks left till my next meeting with my supervisor, and I promised him big things. I hope von Neumann, the pineapple of human intelegense, made his rings sufficiently regular for nice things to happen without any semisimplicity assumptions. Bringing down the cancer infested Leviathan is something we all contribute to.

>>12516240
They are Norlyn 50 den stockings.

>>12516270
Live in an antagonistic relationship with your own body and the social structures around you for like 20 years or something and you will develop a thanatic desire to die and to be punished for your mere existence.

>>12516281
Well... why don't you?

>>12516285
It seems Satan himself is too afraid of my vril levels to let me into his kingdom or something. I have not been able to end my existence thus far, so I will channel that power into a nice dissertation with fancy results and then the universe may have had its use of me.

>>12516308
Your posts indicate that you have a very bad attitude towards things. Perhaps you need to reconsider how you view the world and try to be more positive about things. Maybe you'll even stop being a mentally ill tranny in the process.

>>12516313
Based therapist. I tried to be positive, but the covid test turned out to be negative.

>>12516281
>actually feeling things like "guilt", "shame" and "desire to be punished"
I haven't felt anything other than love and disdain for years now.

What the simplest chaotic string of bools? I want to turn black and white into something other than grey, probably red

>>12516325
Based emotionally [-1, 1] interval touhou poster.

>>12516336
I don't know anything about programming, but what if you do it like having 3 variables for R, G and B, and then you have each of them go from 0 to 1 or to 255 and, then (let's go with 0/1 now) you will have for example R1G1B0 that will give you yellow and so on. Now, to make it more interesting, put the upper bound to 255 and let all three of them vary freely. Then you will get 256^3 (or something like that) different colours determined by those three variables instead of just black and white. No idea if I even understood your question, but you know... #brainstorming #teamwork #togetherwearestrong

>>12516365
Erm that isn't chaotic much
What I'm thinking is a screen that flashes black and white at chaotic intervals and somehow creates the color red in your mind

>>12516379
Oopsie, I somehow replaced chaotic with random in my mind. Sorry.

>>12516272
This guy unironically thinks he's an anime girl

why do we let the peedos, i mean anime posters, flood mathematics threads?

Hello bros I'm the theta function integrals poster, I restated a similar problem and ended up with this another dumb integral. No hope for this one too? What do you think?

>>12516449
for reference, this is the numerical solution

>>12516449
Rewriting the cosign as exp too, the resulting inner integral
[math]\int e^{\pm i(x_0\pm x)k}{\mathrm e}^{-ck^2}{\mathrm d}k[/math]
can maybe be approached with some from this list.

https://en.wikipedia.org/wiki/Common_integrals_in_quantum_field_theory

You might up with a second Gaussian integral that you tackle in the same fashion.

>>12516463
thank you man

>>12515761
Sexy

>>12516468
Hope it helps. tbqh I'm amazed and ashamed I couldn't make up my mind about exp, e and mathrm e there.

Actually there's a bunch of those gem Wikipedia pages out there, such as
https://en.wikipedia.org/wiki/Matrix_calculus
and
https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates
and some others.
And some more far out data bases such as
https://groupprops.subwiki.org/
or
https://www.lmfdb.org/

I should make a github repo or something listing those at one point.

>>12516486
yeah wikipedia math pages are many times nothing short of amazing references

>>12516497
Another comfy one is
https://en.wikipedia.org/wiki/Wallpaper_group

a tranny's butt is like a measurable cardinal
large but inaccessible to me

Daily reminder that while the algebraist fags are going crazy about some tranny legs in stocking, the chad geometers are busy banging 9/10 undergraduate pussy.

New >>12516628

>>12516463
I have found the solution man. It's a sum of gaussian integrals. My wiki savior was this page

https://en.wikipedia.org/wiki/Gaussian_integral

>>12517295
I have to say that >>12516449 can be restated in terms of two Gaussian functions. This is quite crazy but it is true