/sci/ - Science & Math

I can't and I'm not pretending

>rational
>literally using imagination numbers
cringeberger

>>15969522
>"These little tic-tacs that are flying around in our neighborhood all these- these little aliens perhaps that are living under the water or whatever, you know, that we're soon probably gonna be chatting with, you know, in the next few decades..."
What did he mean by this?

>>15969522
>Don’t pretend that you can do something that you can’t
The problem is people think they can do something and it's not until they can actually do the thing that they realise that earlier on when they thought they could do something they actually couldn't. Dunning Kruger etc. Then one may ask is that pretending, and yeah I think it is, it's subliminal pretending

>>15969619
Yes.

On some days I doubt whether I really can do the things I normally can do.

How do you personally study textbooks on your own? Do you take notes on your computer? Paper? What do these notes consist of? A more or less re-writing of the textbook in your own words? Just definitions, theorems, and proofs?

>>15969764
Rewriting shit from the book is pointless. Instead focus more on solving many exercises.

>>15969853
I see. I ask because my notes look like pic rel from Obsidian. It's so very appealing but does involve writing out lots. However, the self-referential nature of math with definitions, theorems etc. makes it appeal to me in theory.

>>15969942
This always seemed more effort than it's worth, to me. The definitions you really should just learn by heart, and if you forget something you can just look it up on wikipedia, these days.
What I do often do, however, is writing out a big proof in tex, if there's some difficult step I don't expect to remember. Then if I want to look it up later I can just grep the files and find it pretty quickly.
As for >>15969764 I just try to prove the theorems and make the exercises. No notes, generally. It has done me well thus far.

Should I try to make a relationship with the math department for the grad schools I applied to?
I only applied to 3 because if I don't get in I'll just do an accelerated masters and go be a wagie retard like everyone else.

>>15969977
Don't try to force a contact if you have nothing meaningful to say. Tenured profs are better politickers than you, it's only going to annoy people
If you're applying to a school for a very specific reason related to 1-2 profs it's a good idea to contact them. In most American schools they can't really pull strings to get you in but it's still prudent to scout somebody if you want to work for them for 3-5 years. Make sure they're taking students, have a personality/work style you like, aren't a psychopath, actually still care about what you think they do, that kind of stuff

>>15970008
>In most American schools they can't really pull strings to get you in
i will say, you never really know which professors are sitting on the committee accepting incoming students. i emailed a prof i was very interested in working in and he was on the committee, i (think) it helped get me into grad school

I'm taking algebraic topology in a few but have never (formally) studied general topology. What concepts from topology should I repeat/learn before taking on alg. top?

>>15970024
Do a quick review on non-Hausdorff spaces

im young, how to olympiad?

>>15969567
I don't see anything imaginary in the part titled "rational parametrization".

If I have a family of sets [math] A_{ \lambda} [/math], and I want a set [math] A [/math] such that one and only one element of [math] A_{ \lambda} [/math] is in [math] A [/math] for every [math] \lambda [/math], do I need Axiom of Choice. Note that the choice is "unordered".

>>15970319
Do you know anything about the sets [math]A_\lambda[/math]?

>>15970323
No, completely general.

How is recursion theorem used to, say, the definition of factorials is valid.

>>15970382
Take
X = N^2
a = (0,1)
f(a,b) = (a+1, (a+1)b)
The theorem will tell you that a F exists with
F(0) = a
F(n+1) = f(F(n))
Then define the factorial as the second component of F(n).

>>15970294
you clearly dont understand
please review my 3,000 videos before speaking again

Dear Norman, you didn't take my music dodecahedron seriously enough, sincerely, that guy

Sps [math]x(t)[/math] is a polynomial of degree [math]n[/math] and [math]y(t)[/math] is a polynomial of degree [math]m[/math]. Then [math]
x(t)y(t) [/math] is a polynomial of degree [math]
n+m [/math].
Can I define a product [math] \cdot_{\mathcal{P}}
[/math] s.t. given two vectors [math] \mathbf{v}\in\mathbb{R}^n
[/math] and [math] \mathbf{u}\in\mathbb{R}^m
[/math] I'd get a vector [math] \mathbf{v}\cdot_{\mathcal{P}}\mathbf{u}
\in\mathbb{R}^{n+m-1}[/math]?
Being more specific, if [math] \mathbf{v}
[/math] and [math] \mathbf{u}
[/math] are isomorphic to [math] x(t)
[/math] and [math] y(t)
[/math] respectively, then this new product between these vectors would wield a vector isomorphic to [math] x(t)y(t)
[/math].
Is this possible? Am I allowed to do this?

>>15970804
yes but why is it R^n+m-1?

>>15970841
Just take, for example, [math] t
[/math] and [math] t^2
[/math]. If you take one isomorphic vector to each of them, you'll have one vector in [math] \mathbb{R}^2
[/math] and the other in [math] \mathbb{R}^3
[/math].
The product of the polynomials I presented here is a polynomial of degree 3, which is isomorphic to a vector in [math] \mathbb{R}^4
[/math], and [math] 4 = 5 - 1 = 3 + 2 - 1
[/math].

>>15970852
Now I'm just here querying chat but here's what I got
https://tches.iacr.org/index.php/TCHES/article/view/9305

>To create this product operation, you could define it element-wise. For instance, if ( v = [v_0, v_1, ..., v_{n-1}] ) and ( u = [u_0, u_1, ..., u_{m-1}] ), then each element of ( v \cdot_P u ) could be defined as a sum of products of the elements of ( v ) and ( u ), akin to how polynomial coefficients are multiplied and summed:[ (v \cdot_P u)k = \sum{i+j=k} v_i \cdot u_j ]Here, ( i ) and ( j ) are indices for vectors ( v ) and ( u ), respectively, and ( k ) ranges from ( 0 ) to ( n + m - 2 ), which ensures the resulting vector is in ( \mathbb{R}^{n+m-1} ).

>>15970883
Thanks for sharing that, anon.
I was working with this idea a few days ago and the formula I got was
[eqn]
\mathbf{v}\cdot_{\mathcal{P}}\mathbf{u} = \sum_{i=0}^{n}\sum_{j=0}^{m}\xi_i\zeta_j\mathbf{e}_{i+j+1}
[/eqn]

>>15970913
Oh, forgot to mention:
the xi's and zeta's are the components of v and u respectively.

>>15970804
That's just convolution
If you embed your polynomials in an infinite discrete vector space (for example, the ring of Laurent polynomials), you'll get closure, otherwise you have to worry about specifying the degree of the space the product occupies (i.e., the sum of degrees of the spaces of the multiplicands, less 1)

Incidentally, the familiar base ten way of multiplying that you learn in school is *also* just convolution. The only difference between numbers and polynomials is that you do carries with the former

>>15970974
>otherwise you have to worry about specifying the degree of the space the product occupies (i.e., the sum of degrees of the spaces of the multiplicands, less 1)
Granted I do that, can I define such product?

>>15969522
Starting 2 grad level math courses as an EE PhD student on Monday. Measure Theory and Convex Analysis (think nonlinear programming but more of a proofs orientation than algorithms focused). I'm generally competent and did alright in my real analysis prereq, but I can't help feeling like I'm a bit fucked. Any recommendations to keep myself sane?

>>15970980
Of course you can, but I suspect what you actually mean is whether you could do it with matrix operations. Even though I'm fairly sure I've seen it recently, the best I could find in the first 10 seconds is pic related and page 16 of this:
https://icarus.csd.auth.gr/wp-content/uploads/2020/08/13-Fast-1D-convolution-algorithms.pdf

Is it true that there are a near infinite number of unprovable and unfalsifiable conjectures that are numerically verifiable but ultimately a huge waste of time?

Just found out I need to become the man of the house within 1 year because no one fucking tells me anything and now I have to deal with all the shit.
How do I go from a bs in math to a guy that can save his family from the the poor house.
I've been repeatedly told 300k starting for mathematics, but I think that's only for quant researchers with PhDs, so I need something faster.

>>15971084
Automate accounting for small businesses

>>15971091
and my knowledge of complex analysis will be useful for this?

>>15971093
Rigor provides reliable accurate code
It's just one idea

You could go further into data analysis but even just accounting services for general contractors will net you a nice check
If you went further you could provide such services to progressively larger companies where analysis becomes even more critical for their margin and that is a math heavy intersection. Not necessarily accounting but examining the books at minimum and rather speculative business specific stuff at the far end

Outside of coding for analysis idk, if you wanted you could look into being an actuary they make bank but it's a lot of certs that don't come overnight
I have no idea what I'm talking about but these are the possibilities I'm aware of

>>15970319
>>15970378
You do need it, since this allows proving AC. Suppose we have an arbitrary family of sets [math]U_\lambda[/math] which we want to make a choice function for. First ensure all the [math]U_\lambda[/math] are disjoint with some tagging scheme, then define [math]f(U_\lambda)[/math] to be the unique element in the intersection of [math]U_\lambda[/math] and the set [math]U[/math] given by your proposition.

>>15971084
If you're in the US you could get a teaching certification and become a HS teacher. Shit pay and stressful but you're basically guaranteed a job. Might take a while to get a certificate depending on where you live tho.

I heard college tutoring pays well, plenty of rich kids are trying to pass calculus, but IDK how you would break into that market. Also less stable than other options

>>15971192
Hmmm I don't really know what high school is like since I never went, but I've heard it's awful. So the tutoring might be more my speed.
I've never done it before but I assume it's just like baby shit like calculus, trig, and maybe linear algebra.
But you're right, I have no idea how to become a tutor. I will see if chatgpt knows.

>>15971007
Thanks anon

is there a word that sounds like "fourier" that means something like "a brief introduction [into a subject]" or did i totally imagine it? i dont think its "foyer" because none of the definitions i found really talked about it being used like that.

>>15971355
foray?

>>15971356
that could very well it be, thanks anon

>>15970393>>15971130
Thanks.

How do I prove that if a x b + b x c + c x a =0, then a,b,c are complanar.

does anyone know what work the highlighted portion here is referring it? theres no source given. really im just looking for an explanation of this "link" it refers to, not necessarily the original work.

wildberger's ideas will become mainstream before the end of the century.

>>15972219
https://en.wikipedia.org/wiki/Weierstrass_elliptic_function#Relation_to_elliptic_curves
https://en.wikipedia.org/wiki/Weierstrass_elliptic_function#Addition_theorems

>>15972321
thanks, fren

>>15970221
dont bother

focus on school and understanding undergrad-level material

why can't you quantify over relations in first order logic?

Can someone elaborate one this? I'm not quite understanding. If for a closed set, x + h is outside the domain, why not just use x - h instead? And if that's truly a problem, why not just take the derivative on the open set without the last bit that makes it closed? Or is this really only a problem with sets that are neither open nor closed?

>>15972893
If X is a closed set, then you can have a derivative that is only defined in particular directions, because f(x) will be defined at limit points that are not solely interior points.

If the set X that f is defined on is open, then if f(x) exists, you can be certain that x is an interior point to the set, and thus can be approached for the limit in any particular direction. This is an idea that matters more when you are talking about functions of general metric spaces or R^n for n> 1, but it would be like if you were talking the derivative of a function in which the limit existed only from the left and not from the right.

>>15972926
that makes sense. Is it really a problem, or is it just for easy definitions? If one sided limits are different, I don't understand why that's a problem and not just indicative of discontinuous behavior.

>>15972957
Recall that the derivative of a function at a point is more-or-less the slope of the tangent line at that point.
How do you describe the tangent line at a discontinuous point?

>>15972962
if it's discontinuous at the point you want to take the derivative, you can't since that's a requirement for the derivative. but if it's discontinuous next to the point, for example, defined as f(x) = 0 on (0, 1] and f(x) = x on (1, 2), why can't you just say the derivative is 0 at f(1) since you can arrive at that by taking the left-side limit for the derivative where the point is defined? Why is a closed set any more of a problem than a discontinuity in the middle of an open set?

Yeah why do you need continuity on both sides for the derivative …… just take the slope on the side of the border that is continuous

>>15973011
thats what im saying my nigga. why a set gotta be open. close that shit nigga

>>15972999
It's not discontinuous itself, but as an example of why the derivative needs to be two-sided, let's consider the function [math]f(x)=|x|[/math].
Clearly, for all strictly positive values of x, [math]f(x)=x[/math] and so [math]f'(x)=1[/math], and for all strictly negative values of x, [math]f(x)=-x[/math] and so [math]f'(x)=-1[/math].
But what of the point at x=0? You could close the interval on either side, so if we allow for one-sidedness in derivatives, how do we interpret this? Does it have two derivatives at x=0? Should one of those derivatives take priority - and if so, why?

>>15973011
Because if the derivative isn't the same from all limit approaches, then there isn't one single derivative. As an example, look at the f(x) = abs(x) example that anon posted >>15973025

If you take the derivative at 0 approaching from the left it would be negative 1. If you take the derivative approaching from the right it would be positive 1.

If you allowed a defined derivative to exist where you have some directions of approach which produce a different limit, then you have multiple possible derivative values for a single evaluation point (i.e., it isn't a proper function).

>>15973025
ok thx. so you need two sidedness in case there is no clear priority, you need it to be defined at the point obviously, and you need it to be open to resolve one-sidedness

>>15972893
when x is interior then you can get both sided limits
when you have a boundary point you can only go from one direction
when you have an isolated point you have no space to compute a limit in the domain.

>>15972957
>and not just indicative of discontinuous behavior.
Because derivatives don't exist at points of discontinuity, a derivative is information about the function a point.

>>15972999
the domain isn't discontinuous, the function is

So you're telling me
17/2 = 8*2+1 = 5*3+2 = 17/3

>>15971192
Any state that pays well for teaching requires a masters. There are some programs in top states like Mass or Connecticut that will pay 100% of your tuition and let you work on your masters while you teach, but you have to teach at the worst schools in the state for that. So yeah you can make good money, but all the states that pay good money need more than a BS.

>>15970221
just do problems and git gud

>>15969522
Idk where else to ask about this so I'll ask here.
I've thought of a problem while solving a puzzle in a game. It should be a very famous problem and it 100% has an optimal solution, but idk how to google it. So we have 2n cells where n>=4. All cells contain either 0 or 1 inside. You can change the value to the opposite, but it changes the values of adjecent cells too. You need to create an algorithm which will change all cells' value to 1 in a finite number of steps(~O(n)). I've tried some solutions myself but I'm too lazy to code this solution and check if it works for all n and all starting conditions. Do you guys know how this problem is called and what's the solution?

>>15972676
Because the language of first order logic is not expressive enough. For quantification over relations you go to second order logic

>>15974386
You essentially just have to solve a linear system
[eqn] A x = b[/eqn]
in the vector space [math]\mathbb{F}_2^{2n} [/math]. The matrix A is a tridiagonal matrix with 1 on the diagonal and the two subdiagonals, the vector b is the initial configuration reversed (0's where 1's are and 1's where 0's are), x tells you which cells you have to press.

Such a system can be solved in O(n) with Thomas algorithm.

>>15974386
This is very easy. This is equivalent to expressing the vector (1,1,..1) as a linear combination of vectors (0,..,0,1,1,1,.0,..) in the Z/2- vector space (Z/2)^(2n).
To solve the puzzle, just invert the matrix (e1, e2, ..., e_2n)^T.
As for a relatively fast algorithm, consider the following.
In order for all cells to be on, the first cell will either be pressed or not. In either case, there is a unique solution whether to toggle the succeeding cells or not, determined by the leftmost neighbor of each cell.
For example, take
0 0 0 0
Try turning the first cell
1 1 0 0
the first cell is already on, so do nothing for second cell. The second cell is already on, so do nothing for third cell. The third cell is off, so turn on the last cell
1 1 1 1. Voila.
In a case starting from 1 0 0 0, you first try switching the first cell, realize this doesn't work after reaching the end, and try again now switching the first cell.
If a solution exists, this algorithm has either 2n or 4n steps.

Why is mathematics filled with inane sophistry like imaginary numbers and uncountable sets (actually the real problem here is diagonal arguments)?

>>15974485
Where the tridiagonal matrix thing came from? This does make sense, but I'm interested why exactly this operator works like that. I've kinda skipped my linear alebra course and just learned things that I needed for work afterwards.
>>15974522
Well I do understand that for any starting condition I can transform the vector into (1, 1...1) or (1, 1...0). So the problem essentialy is just to turn that last 0 into 1. Also I do understand these words in a vacuum:
>This is equivalent to expressing the vector (1,1,..1) as a linear combination of vectors (0,..,0,1,1,1,.0,..) in the Z/2- vector space (Z/2)^(2n). To solve the puzzle, just invert the matrix (e1, e2, ..., e_2n)^T.
but can't grasp how exactly it is related to this problem. What's the Z here? And what basis vectors are you talking about?

>>15974671
>What's the Z here?
Z means that your elements are integers.
Not sure if Z/2- is supposed to be Z2 or Z/2Z, but they mean basically the same thing with different notation in either case: You're working with rules where 1+0=0+1=1 and 1+1=0+0=0 (basically, the even/odd parity of the sum is what matters, corresponding to the on/off parity of the cells)

>>15974724
meant to reply to >>15974709

Not sure if this is a stupid question, but I'm not sure how to prove it, nor find a counter-example:

Let [math] U,V \subset \mathbb{R}^2 [/math] be open subsets with compact closure.

Suppose [math] \partial U = \partial V [/math].

Then does [math] U = V [/math] ?

>>15974485
Ok, I've coded it and this thing actually works, but I don't know why exactly and would love to know how you reduced this problem to just a linear system and how exactly you thought of that operators' form.
Also, I've noticed that in cases of odd vector lengths the problem sometimes doesn't have a solution and it would be interesting to find the criterion of solution existing. For example, solutions for 00001 and 10101 doesnt exist.

>>15974743
Consioder [0,1) and (0,1) in R as the x axis in R^2.

>>15974769
>[0,1)
not an open subset

>>15974751
My bad.
Consider the space R^2 - {(0,0}}. It's open and homeomorphic to D^2 - {(0,0)}, which has compact closure. Therefore it's enough to find such a counterexample on the cylinder R x S^1.
Consider the open sets U = union over Z of (2n, 2n+1) x S^1 and V = union over Z of (2n+1, 2n+2) x S^1. Both sets have the same boundary: union of {n} x S^1. Both have compact closure in R^2, yet are different sets.

>>15974804
Meant for
>>15974777

>>15974804
>>15974808
this is basically what it looks like. The circles get infinitely close together near zero and near the unit circle.

Is there a difference between the Axiom of Choice for Finite Sets and the Axiom of Finite Choice? I ask because I've seen it said many times that the existence of Choice functions for finite sets is provable in ZF alone, but I've also seen it said that the Ultrafilter Lemma implies the Axiom of Choice for Finite Sets.
If they are different, is it that the one provable from ZF only applies for finite families of finite sets, while the one that follows from the Ultrafilter Lemma also applies to infinite families of finite sets?

>>15974882
Finite axiom of choice says that a finite family of sets has a choice function. It's provable from ZF by a simple induction.
Axiom of finite choice says that a family of finite sets has a choice function. It's not provable from ZF.

>>15974804
>Consider the space R^2 - {(0,0}}. It's open and homeomorphic to D^2 - {(0,0)}, which has compact closure.
However R^2 - origin has closure R^2 , which is non-compact.

>>15974813
This makes sense though. So you mean, U and V are alternating open annuli in the unit disk, getting thinner as you approach the origin, or the unit circle. I think this works, thanks.

>>15974743
>>15974813
>>15975044
Then a folllow-up question:

Let [math] U,V \subset \mathbb{R}^n [/math] be *connected*, bounded, open subsets.

If [math] \partial U = \partial V [/math], then does [math] U = V [/math] ?

How can scientists know if a number has an infinite number of digits? They can't check, it's impossible to know if even ONE (1) such number exists because it would take more computational power to check it than exists in the universe

>>15975299
we can't directly prove that a number has an infinite number of digits, no, but in many cases we can prove that it cannot be rational (and thus cannot have a finite number of digits, necessitating an infinite number)

>>15975299
How can scientists know there an infinite amount of numbers??

>>15975333
They can't but they've used computers to compute most of them

>>15975344
*"a LOT of them" I mean, obviously

>>15975058
Got the answer from Reddit:

The Lakes of Wada give a counterexample.

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

>>15972893
are you reading https://pi.math.cornell.edu/~hubbard/vectorcalculus.html?
i really want to read something that mixes multivariable calculus and linear algebra to truly give a better view of the big picture, would you recommend it?

>>15975058
You can prove though that in this case either U=V or U and V are disjoint

where's my fields mettle

>>15975780
In English?

>>15975333
>Let S be a set of all integers

>>15975333
Are you memeing or are you serious? The way they know is pretty straightforward. Test the hypothesis that there are a finite number of numbers.

If this hypothesis turns out to require contradictions, it cannot be true.

>>15976275
I'm always curious why they don't include set theoretic constructions of integers and the standard operations, and instead choose to just "assume" everyone "knows" about "that" and proceed axiomatically

>>15976275
How do you test it?

>>15976288
Consider the idea that there are finitely many positive integers.

What happens when you reach "the last" integer that prevents you from adding another one?

>>15976294
You die?

>>15976297
Here I'll generate all the infinities for you
Start with 1
Add it to itself we get 1+1
Repeat

>>15976297
Okay, so you die and the next person picks up where you've started off.

What happens when they take this "last" integer, and add it to itself. You've now doubled the number of finite integers in the set of integers.

This next person who has continued the counting now repeats, but instead of counting one by one, they double the size of the set, knowing that the integers between the ones they produce "can be counted" by some person or some group of people sharing the task.

When does it stop?

>>15976301
For how long?

>>15976309
xy is finite for any finite x and y

>>15976315
Yes, that is true that if you stop at any particular finite (x,y) tuple, x*y will be finite.

The question is, what is preventing xy from growing arbitrarily large? To claim that there are a finite number of integers is to claim that there is a maximum possible pair xy somewhere that no other xy can be greater than.

>>15976310
Oh you're like you can't generate something from nothing or
You can't generate infinity from finitude as in if there's finite time no matter how fast you continuously add 1 you'll never get to infinity

So let's engage a fun thought experiment about self replicating nanobots who can derive their resources from their environment regardless of their position in time or space
How long does it take the nanobots to consume the entire universe

>>15976324
Time? The hypothesis you're testing is that there are arbitrarily many numbers if there is an arbitrarily long time. How do you "test" whether time goes on forever or stops?

>>15976325
How large is the universe?

>>15976347
We don't even need infinite universe theories. All we need is to examine the acceleration of the nanobots expansion, and compare the expansion of the universe. If the universe grows faster than the nanobots growth, the nanobots can't consume the entire universe

If this was the case, what is possibly capable of continuously out expanding such nanobots? The answer is only other nanobots that got started before the nanobots, because these nanobots make themselves. Define a possible thing that doesn't do this that also expands faster than this. In this context nanobots might as well just be self replicating energy

So you don't actually need infinity you just need an endlessly accelerating acceleration and enough time to sufficiently approach infinity or enough acceleration that the nanobots within the nanobots cannot expand beyond the outer or older nanobots while both spheres of nanobots are essentially infinite while also being, essentially, finite

>>15976368
>enough time to sufficiently approach infinity
You don't get closer to infinity.

>>15976378
Unless you're already there exactly
The sufficient condition is as the new inner sphere of nanobots expands we would expect as it accelerated it would approach the outer sphere or the edge of the universe
But if this edge either doesn't exist or is expanding at a proportionally greater rate, the inner sphere, from the perspective of the outer, never grows
So from my perspective, this continuous recursion of nanobots expanding, at an accelerating rate, into a sphere of nanobots, also expanding at an accelerating rate, this nesting, can continue to infinity, as can the distance between each nested sphere, can be infinity
Because they will never meet, but you can nest them endlessly, they all accelerate, but the ratios between them is infinite because they will never meet
You need to imagine not infinite time for this, but infinite relative expansions or accelerations of expansions. If one is so large the other will never expand within it, then this is possible to occur recursively

In this context, infinity is much closer to 0, or the amount of space the inner nanobot mash is capable of occupying within the outer sphere
Both expand, but the rate of the outer is infinitely larger than the inner, because no matter how much time passes, no matter how many nanobots are made, within the outer ring, the inner ring never occupies a proportion of space greater than 0

I'm interested in these very banal equalities
I'm wondering, is the following only true for exactly one set of values

a^b + (a-1) = b^a

This is true for a,b 2,3 where it's 9

>>15976449
(a,b) = (1,1) is another solution

>>15976415
You also can't get to 0 from any number
So for example, define the smallest number greater than 0
You can't, you can only approach 0 and get infinitely close to it
Also the largest number less than infinity is equally undefinable

A fun example is division by 0 is infinity because nothing can go into any value of something infinitely many times

>>15976449
I assume you mean only allowing integer values for a,b ? Otherwise there are many real solutions

Let [math] U \subset \mathbb{R}^n [/math] be a simply-connected open subset.

Suppose the intersection of U with any open n-disk is connected.

Then must U be convex?

>>15976469
Consider the open ball in R^3 with the center removed.

>>15976548
Oh of course, thanks.
What about just in [math] \mathbb{R}^2 [/math] then? In full:

Let [math] U \subset \mathbb{R}^2 [/math] be a simply-connected open subset.
Suppose the intersection of U with any open n-disk is connected.
Then must U be convex?

>>15976573
>n-disk
Meant to say 2-disk

>>15976577
The answer is yes. I have a proof but its a bit tricky. Heres a sketch: take two points with their midpoint not in the open set. Consider a big open ball whose boundary passes through those two points. For large radiuses, there are exactly two such balls, corresponding to the sides of the line through them. Using the connectedness of these balls and openness of the set, form a loop half of which is on one side of the line and the other half is on the other. The loop wraps around the missing point once. Therefore its not contractible in the open set, contradicting simple connectedness

>>15976342
Why is time involved in your thought process? You can start counting (x,y) from anywhere with the full knowledge that between any two pairs of integer tuples (x,y) are all of the intermediate values.

Is this a mistake? If f(x) = 0 on [-1, 0) U (0, 1] and f(x) = 1 for x=0, why would lim x ->0 not exist?

I assume this is a consequence of the books limit definition, which says x can equal x0. But doesn't this mean limits are only defined for points in the domain if the function is continuous at that point?

>>15976261
дe мoя фiльдз мeдaл

>>15976673
It's not continuous I think it's the open interval at the edges is a discontinuity

>>15976777
It's a jump discontinuity at 0 by definition of the piecewise function

>>15976657
Hm I think I see what you mean. Thanks anon

WHAT THE FUCK
I HATE BIBTEX & BIBLATEX
This shit is so esoteric. It was working a while ago and now I run it and the References just won't show up. I've tried deleting the aux and bbl files and it still won't show up. I typeset pdftexmk followed by XelateX and it still won't show up.
AAAAAAAAAAAAAAA

>>15976449
If you assume a is a prime number, then some stuff happens. Fermat’s little theorum gives (b mod a) = (a-1).

If b is a multiple of a, then (a-1)=0, which results into the 1,1 solution.

Otherwise, if b is smaller than a, then bmoda would be b, so (a-1)=b. Replacing a with X and b with X-1 gives the only solution, X=3.557, which paradoxically isnt a prime.

Now, if b is larger than a, then b is larger than a by something that is a multiple of (a-1). That gives:

a^(a+(a-1)N) + a-1 = (a+(a-1)N)^a

Replacing N with X, and a with a prime number, and graphing it in logarithmic mode it for various a, only the prime a=2 creates two functions that intersect when N is not 0.

I’ll try to see if anything chanhes when b is negative.

>>15976315
Not rigorous.

>>15977115
I made an error in the modular arithematic during the “b larger than a” part. But, I have this more general way now.

~~~~~

If a and b are positive whole numbers, and a is prime, then Fermat’s Little Theorum means (a-1) = bmoda, which has three possibilities:

>(a-1)=0, for b=aN
>(a-1)=b, for b<a
>b = aX + (a-1), for b>a and X=positive integer.

In all cases, you can substitute b, and find the solutions (1,1), (a paradox), (1,1) and (2,3). Graphing the last one for various primes quickly shows that X is never an integer unless a is 1 or 2, X keeps going closer to zero as the chosen prime a grows.

~~~~~

If b is a negative number, then (a-1)={ (a-1)(bmoda) }moda.

>(a-1)= 0, for b=aN, ends with paradox
>b = aN + 1, for b<a, is a paradox
>bmoda= aN + 1, for b>a, which is a paradox since #moda will never be greater than a. Unless bmoda = 1, b = aM +1, which again results in a is 1 or 2

And of course -b means a^(-b)=some fraction will somehow equal some integer, a paradox.

I dont know of any method to tackle this if a is not considered a prime, or if a is negative. Good luck to you.

>>15977113
Skill issue.

>>15977325
Funnily enough, an hour after I posted that I managed to get it to work.
Still fucking hate the arcane nature of it though.

can anyone tell me how the part in the green tick answer the part starting
a^4b^4=.... is true?
https://math.stackexchange.com/questions/1073870/for-an-ellipse-with-minor-radius-b-show-that-the-product-of-distances-from-th

>>15969764
i do prefer to take (latex) notes, same as when studying a textbook of any subject. however, there is on value to simply rewriting the textbook. my general approach is to have my notes separated by topic, rather than being tied to a specific textbook. so e.g. I have topology notes, model theory notes, etc. then, when reading a textbook in a specific area, I append my notes with any new definitions and theorem statements of interest

>>15970024
make sure you know the basic definitions, examples, and some compactness. you can skip metrization and all that, since that's more of interest to analysists.

I feel like intuition can sometimes be to the detriment of the understanding, not to the benefit.

>>15976662
Time has to be involved if counting is. It doesn't matter where or when you start—you can't accumulate infinite numbers unless you have infinite time. To properly "test" for infinity, you'd need some noninductive measure.

>>15977226
How so?

>>15977724
Interesting. So I don't think you understand what "infinite" or "infinity" as a concept means. Infinite means that there is no upper bound, or that there is no boundary which prevents you from continuing. You don't need to "count" the integers to know that there are an infinite number of them. There are an infinite number of them simply by virtue of there not being an upper bound on their number. Any particular integer you specify at any point in the process, there will always be a integer lower or an integer higher. That's what it means for them to be infinite.

You don't need to have a person sitting there counting one by one. Just by virtue of the fact that there is no upper bound, one can know that there is no possible finite set which contains all of the integers.

>>15977771
At which number does infinity begin? Which number is in the neighborhood of infinity? Which are far away?

>>15977778
Every number is in the local neighborhood of infinity if you let your radius get large enough.

In all seriousness, your response here indicates to me that I was correct. You are very confused about what this concept means. There are no numbers in the "neighborhood of infinity" because infinity isn't a number. "Infinity" doesn't begin or end anywhere, but rather represents a direction of divergence.

>>15977577
He multiplied both sides of the tangent line equation by aabb, and then, since the tangent intersection point (x0,y0) was defined as being on the ellipse, its also a solution to the ellipse, so he set xx0 and yy0 in the tangent line equation into just x0x0 and y0y0, back into an ellipse equation.

>>15977780
That's not me.
>>15977771
You're describing arbitrarily many, not infinitely many. For example, there are arbitrarily many equally-spaced prime numbers but there aren't infinitely many.

I heard that modern mathematics is too algebraic, but before it was taught based on geometry and mathematics was clearer because of this. Maybe you know such textbooks?

>>15977867
anything from 1900 to 2000

I'll be defending my thesis tommorow.

>>15978422
On wut

>>15977795
"Arbitrarily many" and "infinitely many" describe the same thing if there is nothing which provides an upper bound on the size of the set (or the size of any entry in the set). "Infinitely many" is just the limiting case of "arbitrarily many" where arbitrary is able to continue indefinitely with nothing bounding above/below.

>>15978488
Can you explain what you mean in terms of bounding equally-spaced primes?

>>15978533
I don't know enough about the prime number problem to really give a meaningful answer about whether the number of equally spaced primes is unbounded.

I am mostly referring to the concept of a set being infinite in the first place, which either you or someone else disputed. If a set is allowed to grow arbitrarily (for example, there is no point in the process of adding positive integers where any two positive integers added together does not produce another positive integers of greater magnitude) then it is not possible to for the size of that set to be finite.

>>15978535
I courtsy all that, you describe what you're saying very well. I'd like to leave this >>15978533 question open tho

How do I calculate astronomically large numbers like numbers bigger than the number of atoms in the universe
Like 256 tetrated 256

>>15978629
At that point? You don't.
There are some tricks in number theory that can sometimes help tell you some properties of these numbers (e.g. their last few digits or something of the sort), but it's not feasible to wholly calculate a number that large.

>>15978634
Sadness overwhelm

>>15978629
If you want a decimal expansion, you don't. You can really only approximate by having a finite and fairly short length decimal expansion and an appropriate exponential scaling multiplier. Even that doesn't really work particularly well when the numbers get large enough.

>>15978659
At some point it becomes physically unfeasible. Even if you took all the matter in the universe you couldn't create enough bits to store all the digits required let alone perform any calculations.

>open math homework
>look inside
>some retarded fucking problem that requires knowledge that wasn't in the course
As an aside, becoming a math major seems more and more like a meme with every day that passes. Either I learn some calculating tricks and jump ship to a different field minus the actual field-specific knowledge that should have been covered in my undergraduate study or I go all-in on math and end up insane and financially destitute because pure math is the ultimate meme (other than philosophy, lmao).
... now someone call me a zoomer retard.

>>15978795
Does your school offer an applied math major? Applied Math degrees end up being useful in all sorts of fields from engineering to data science to logistical/operations research.

Pure math undergrad degrees do seem to be mostly a pathway to graduate studies unless you are just taking your B.S. and going into some other field and hoping that's good enough.

Is a linear manifold the same thing as a subspace?

>>15978960
https://math.stackexchange.com/questions/1613939/what-is-the-difference-between-linear-manifold-and-linear-vector-subspace

>>15978629
start learning CUDA or AVX

If all math was presented in the form of python code, I would understand it easier.

>>15980117
If only they taught math as sets code and algebra together as one language instead of a trillion fractured splinters of waste your time paper cheese and hyper granular LCD monotony

>>15980117
You should look into proof assistants

Fucking factorials. Fucking summations. Fucking geometry.

If all math was presented in the form of lisp s-expressions, I would understand it easier.

>>15980686
I find Lisp reverence in lieu of Haskell reverence disgusting

>>15970024
Not a lot, Hatcher has an overview of basic topology concepts needed for algtop.

Why do people say maths isn't anki-able?

I've gone from remedial full retard to being able to notice patterns and synthesize previous learnings with new topics just because I've got all of the old information fresh in my mind. Never mind having definitions, formulas, etc... at my fingertips makes solving problems/building on old concepts much smoother.

I went from not having done any maths in like 10 years to speedrunning middle school maths, algebra 1/2, geometry, trig, and Calculus in 4 months while working a full time job.

>>15981168
That's arithmetic-tier material. Try using Anki to learn complex analysis or combinatorics.

>>15981216
I’m just preparing for cs tard tier stuff so that’s fine by me desu. So there’s no problem with anki-ing baby tier shit after all. I’m looking at my lin alg and discrete maths books but they seem plenty ankiable. I might do combinatronics at some point, so I’ll see if I actually fail at that I guess.

>>15981168
Hey, anon. Can u give me book/courses which you use?

>>15981385
>>15981385
Math can't be learned through spaced repetition at the rate that you are taught in college.
You are supposed to cram a shit ton of conceptual information and be fluent with procedures that aren't mechanical but require intuition and understanding to execute.
Anki is more useful for things like anatomy and vocabulary because those aren't so inter-related, they are mostly isolated pieces of data.
When I was about 19 I tried to Anki my way through college and failed horrendously. I think mostly because I was looking for quick hacks and wasn't willing to just put in the hours. But that's really the only thing that works, there is no quick hack.
You say you learned calculus, but that can mean many different things. Simply understanding intuitively the concepts of derivatives and integrals can be done in a few days. On the other hand being able to closed book solve most of the exercises in Spivak's Calculus, or being able to prove the most important theorems on the fly in a rigorous way, will probably take you years of study.

does anyone have a proof for the sum of angles sin formula that uses the integral form of arcsin? in other words, does anyone have a proof of the addition theorem for arcsin?

>>15981673
> On the other hand being able to closed book solve most of the exercises in Spivak's Calculus, or being able to prove the most important theorems on the fly in a rigorous way, will probably take you years of study.

Is it possible to learn this by yourself?

>>15974743
Sometimes working on the contrapositive helps. If [math] U \cap V = \emptyset [/math] then this is trivial so what about when [math] U \cap V \noteq \emptyset [/math] and [math] U\not=V [/math]?

>>15981765
Haven't done it but I don't see why it wouldn't be.

>>15981765
For some people it absolutely is, you just need patience. For others, having an instructor to help "dig you out of holes" when you run into a concept you have a hard time properly understanding is very important.

One thing I've found useful is to have a couple PDFs of alternative books that cover the same material. For example, my introduction to real analysis course used PMA (as most entry level grad courses do) but I found having a PDF of Kolmogorov's real analysis book very useful when I found myself getting stuck on some of the problems in Chapter's 6 and 7 of PMA.

there are given the parametric equations of the lines l1 and l2 [math]\vec{r}=\vec{r}_{1}+t\vec{s}_{1} \text{ and }\vec{r}=\vec{r}_{2}+t\vec{s}_{2}[/math] what is the necessary and sufficient conditon for l1 and l2 to be intersecting lines(so have one intersection point)?

there are given the parametric equations of the lines l1 and l2 [math] \vec{r}=\vec{r}_{1}+t\vec{s}_{1} \text{ and }\vec{r}=\vec{r}_{2}+t\vec{s}_{2} [/math] what is the necessary and sufficient conditon for l1 and l2 to be intersecting lines(so have one intersection point)?

>>15969522
The entire internet does this daily

>>15981787
What about it?

>>15981939
>It's impossible bro
>Trust me

Simple/dumb question, but when discussing any sort of arithmetic progression or sequence of integers, if I want to talk about two terms that are "next" to one another, which terminology is more suitable between these two:
>consecutive
or
>successive
???????
Like if I want to say "[math]d[/math] is the difference between any two consecutive/successive terms".

>>15982434
successive feels a bit more intuitive to me, but realistically either would probably work

>>15982443
Okay thanks.

>The general solution of
>[math]\displaystyle \frac{dx}{\sqrt{1-x^4}} = \frac{dy}{\sqrt{1-y^4}}[/math]
>can also be expressed as
>[math]\displaystyle \int_{0}^{x}\frac{dt}{\sqrt{1-t^4}} + c = \int_{0}^{y}\frac{dt}{\sqrt{1-t^4}}[/math]
can someone explain to me how this works?

>>15975707
nta but i know 3b1b likes that book
idk if people like him around here

I'm not the one who posted this on Reddit but I thought it was kind of a curious question. Is there generally a well known mathematical method of combining two functions into one equation so that the resulting curve stays identical?

>>15982813
Multiply them assuming they take value in an integral domain.

The solutions of
f(x,y) g(x,y) = 0
are exactly those pairs (x,y) that either solve
f(x,y) = 0
or
g(x,y) = 0

Is there a standard notation for restricting a codomain? I usually just see [math]f\colon X\to Y[/math] restricted to its image written as [math]f\colon X\to f(X)[/math] but now we have two functions called [math]f[/math].

>>15983119
Wikipedia gives the notation [math]f|^B: X \rightarrow B[/math] for the corestriction of [math]f[/math] onto [math]B \supseteq f(X)[/math].

I dropped out after calc 2 to work as an electrician and it pays well but I saw a math video thumbnail just now (haven't watched the video) and it has filled me with an urge to study math.
The thumbnail had this equation in it:
Det(e^M) = e^(M^T)
Is there really some kind of correspondence for determinants of matrices like this that lines up with the logarithm rules? Is it exactly log(M) when the matrix is just a scalar? What topic of math does it come from?

anyone here use the mathematics discord?

>>15983650
You have to take the trace rather than the transpose.
[eqn]\det \left(e^M \right) = e^{\text{tr}(M)}[/eqn]
It's a trivial consequence of the Jordan Normal Form which you learn in your first year Linear Algebra class.

I'm considering quitting mathematics. I was planning on going to grad school this autumn, but honestly, I don't know if I have what it takes. My math skills aren't that great, and I feel like my undergraduate education has been very shaky. My university had a very poor math department and as such my curriculum lacked anything on algebra, functional analysis, or geometry. So I had to learn these topics more or less on my own.

>operations research applied course
>Normgroids quite literally make massive excel spreadsheets with 300 different cells, solver tools on the data tab, and other convoluted bullshit to find the area under two straight lines, a problem they could solve if they just took the time to learn some extremely basic calculus
>instead, everyone else is supposed to use these spreadsheets and waste their day so that normgroids can just chit chat all day and use excel

Insanity.

Can the following Riemann-Stieltjes integral be simplified further
[eqn] \int_0^{ \infty} F(x_1 + x_2) \operatorname d F(x_2) [/eqn]
where [math] F [/math] is an absolutely continuous distribution function.

Is Khan Academy enough for someone who's got some gaps in their knowledge with high-school maths generally being hazy? Or is there some other resource like a textbook which would benefit me more in relearning the basics before starting Stewart's Precalculus?
pls help ive been over analysing which resources i need to self study again

moving on from black and white space to coloured space!

>>15983879
Cramming knowledge is no way to learn math.

>>15984241
Khan academy is garbage.
Until you reach the upper graduate level, learning and doing math is purely hard work and determination. Unless you have an actual diagnosed mental disability, no you are not bad at math, you are just lazy. If someone is better at math than you, they are a harder worker than you in 99/100 cases.

>>15969522
Does anyone else here think the hilbert paradox is stupid?

This was so much funnier in my head and description.

are the trigonometric functions the only real-valued, periodic, analytic, nonconstant functions?

>>15984870
Wouldn't the restriction of an elliptic function to the real line be all those things?

>>15984900
dont those have poles?

>>15984900
excuse me, i wanted to say analytic everywhere.

>>15984847
>vanishing neighborhood
Don't worry anon, I got it.

>>15982466
From “A first course in differential equations,” 3rd edition by J David Logan

After taking the integrals, they took the derivative just so they could move the dy/dx infinitesimals around to make a prettier match.

>>15984163
Since you mention a distribution function, I assume you know about random variables and such.
There are two independent random variables, [math]Y[/math] and [math]Y'[/math] which have [math]F[/math] as distribution function.
If I'm not mistaken, you can write your integral simply as [math]\mathbb P(Y-Y' \leq x_1)[/math].
Whether this is simpler, I don't know.

>>15984908
Nope. e^sin(x) isn't a trigonmetric function or a finite sum of them. Every periodic differentiable function is an infinite sum of them, since the fourier series for it will converge.

How to prove that if S is an infinite set, there is an injection from IN to S?

>>15985754
Since S is infinite it has a countable subset. Fix an indexing {s_i} for it and then map i to s_i. If some set theory autist complains I'm implicitly using AC here or something I don't care, this is how mathematicians do mathematics, go be a logician elsewhere.

>>15984268
True, but to learn stuff I'm actually interested in there tend to be a lot of pre-recs, and I generally don't like taking theorems on faith without at least having good intuition on the topic. I also find myself losing the ability to prove some theorems I should understand, which is very frustrating.

>>15985763
You can cut out the need for AC (or rather, the much weaker Countable Choice) by noting that by "Infinite" you're using the Dedekind notion of finiteness, and that in ZF alone it might not be true by other definitions of finite/infinite.

>>15985763
>Since S is infinite it has a countable subset.
Why?

>>15985937
Choice implies that every pair of cardinals is comparable (that is, at least one injects into the other). Since every strictly smaller subset of a countably infinite set is finite, it follows that a countably infinite set must inject into every infinite cardinal.
If Countable Choice fails, there may be sets that are incomparable with any countable set, and thus this argument would not hold.

>>15985987
How could any infinite set NOT have a countably infinite subset?

>>15986000
No Dedekind-finite infinite set has a countably infinite subset. Amorphous sets (a specific kind of dedekind-finite infinite set which can't be linearly ordered) were known even to Fraenkel.

>>15969522
bigot

>>15986011
?

Is there a finitely generated group that acts arc-transitively on the rado graph?

>>15980773
Haskell is neat but agda is better

>>15986005
So, if axiom of choice is not assumed, amorphous set would serve as a counter-example to my question?

>>15970986
kreyszig's book

>>15971084
learn python well, apply for data sci/ml/webdev jobs
I've been on the hiring side and if you bring a good, solid GitHub portfolio (all I care about tbqh, can be any git hosting provider), then you have a high paying job.
Math is a strength bc of the rigor and the ability to reason about and use complex algorithms and data structures. A lot of stuff in webdev can be done rather poorly using dumber methods or robustly using more complex data structures and algorithms - this gets noticed by most managers and will to raises and such.
NOTE: it's an art to learn *when* to stand out, it's best to display your abilities when saving a dumpster fire of a problem

>>15986981
Yes

Is there a reasonable metric to compare which university/institute has the largest math department?
-----
I'm planning my retirement. And auditing courses until I die at the local universities sounds appealing. So I'm looking for the university with the best course offering, in terms of number and variety, that also admits guest/auditing students. Preferable in EU. Language, I can start learning from now.
Some possible answer I found:
1. Bonn-Cologne, and possibly other larger/group of universites
2. ETH or EPFL
3. Institutions in Paris, but doesn't look like guest student is a thing over there.

Who’s fuller: UK or Germany

>>15987055
I mean, I plan to stay around those universities during my twilight years. Visa is not an issue.
So the main criteria are: I can audit classes officially, it has decent library, decent selection of course topics
I think the largest math department in EU would be the best according to those criteria.

>>15984912
Thanks.

>>15987055
If you're in Germany, Gottingen is a no brainer.

I'm reading some notes and the author said, a principal bundle over a simply-connected group is trivial.

Does anyone know why this is, or where I can find a proof of this?

hello my friends i have no idea how to do this problem, any help is appreciated

>>15987435
That doesn't make sense. I'm not Oppenheimer. And this is not pre-WW2.

Certified brainlet coming in to ask whether you guys remember the point that math really became a love or otherwise enjoyable thing for you guys. Was there something about it that fascinated you early on, or did you stick with it because you were good at it?

>>15988480
In my undergrad years I was fascinated with logic and set theory. Worst decision in my life. I could've made tons more if I majored in engineering or finance

>>15988194
whats the question

do people actually read math books designed specifically for mathematicians? basically definition => theorem => proof => corollary in loop without even one explanation of what's going on.
is this what makes a difference between an engineer and a mathematician?

>>15988480
I started studying it because it was the only thing I was somewhat decent at, and years after my degree I realized it had turned into a hobby as well.

>>15988194
If you have [math]a[/math] and [math]b[/math] equal to [math]\infty[/math], you get that [math]|f|=\Theta(|g|)[/math] as [math]x\to \pm\infty[/math], so that you can write [math]c|g| \leq |f| \leq C|g|[/math] for constants [math]c,C[/math] and all large enough [math]|x|[/math].
The implications should be easy then.
If [math]L_a=\infty[/math], then you'd get that [math]|f|=\omega(|g|)[/math], so that [math]|f|\geq c|g|[/math] for some constant [math]c[/math] and all large [math]|x|[/math].
The second implication then still holds true.
For finite [math]a,b[/math] the same ideas should work, I think.

>>15988678
I read a lot of papers of that form. Not many books though.

>>15988678
Never experienced a book like this. Pretty sure it's an exaggeration. Some books could do with better exposition though.

>>15988711
i mean papers are not meant to be read by (non) math undergraduates
>>15989162
linear algebra by serge lang or linear algebra done right. two books which are almost always recommended but absolutely suck.
what do you learn from books like that exactly? i can assure that any robotic engineer has a better understaing of linear algebra in the real world better than the authors of those books.

What’s the “funnest” math subject? I enjoyed combinatorics and calc and basically haven’t don’t much more than those lol

>>15976287
There exist set theoretic constructions, why go through them every time you mention integers? That's like defining what vegetation is when you ask the 16 year old employee at the grocery store where they keep their apples

hiya /mg/, resident thread dumbass here. i'm taking complex analysis pretty soon so i wanted to self study a bit before jumping into the course. any book recommendations or tips in general? the one by saff & snider is what i keep seeing but their diff eq book was kind of a mixed bag

real analysis isn't a listed prereq but i'm extremely rusty on it. anything i should touch up on before starting?

>>15989377
>suck & sneeder

>>15989295
depends on what you like. i find algebraic number theory and mathematical statistics both fun. some people find numerical PDEs fun. very personal thing

>>15988701
thanks m8 this totally works,
works for the interval by a change of variables since (a,b) and the whole line are diffeomorphic

Yes I like math and books

But I should probably prioritize Gym more

Another tweet

>>15971078
Yes, you can just enumerate all tautologies in first-order logic and never run out of useless statements and theorems

What was the point of introducing n balls if you're going to pull this shit
Fuck you Apostol

>>15989377
Undergrad complex analysis? My book was Sarason it was pretty good

and I cant wait to learn more to keep up with the people here

>>15990218
What's the problem?

/mg/ exercise:

Let D be a disk of radius R.
If two points in D are chosen randomly, what is the probability that the distance between them is <R ?

>>15990861
>randomly
With which fucking probability distribution, nigger?

>>15990874
The obvious one, tard

>>15990907
>obvious
No such thing exist. We are mathematicans here. We don't assume anything that's not stated in the problem. Fuck off to reddit if you like ambiguously stated problems!

>>15990921
I think you just don't want to admit you can't solve the problem lel

/mg/ exercise:

Let D be a disk of radius R.
If two points in D are chosen randomly, what is the probability that the distance between them is <R ?

>>15990258
yeah it's for undergrad, thx fren

Why is math 55 so hyped? I study at UNICAMP, clearly not the world's best university, and here all of the discipline's scope is obligatory for any math undergrad.

How do I show that if [math] S_1, S_2, \dots, S_n [/math] are orthogonal subspaces, then the projection [math] \hat {\mathbf x} [/math] of [math] \mathbf x [/math] is [math] \hat { \mathbf x_1} + \hat{ \mathbf x_2} + \dots + \hat{ \mathbf x_n} [/math], where [math] \hat{ \mathbf x_i} [/math] is the orthogonal projection of [math] \mathbf x [/math] on [math] S_i [/math]?

>>15990861
(1 - sqrt(27)/4π)

>>15991161
You sound insecure. You're not at Harvard, and that's fine.
I strongly doubt the freshman math sequence in your university cover the same thing as math 55.
Even if that is true, Is it really impossible to believe two courses with the same syllabus may have have different difficulties?
But no need to speculate. Check their old problem sets, can you solve them?
Math 55's OG textbook, Advanced Calculus by Loomis and Sternberg, is one of the most retardedly difficult textbook I've ever read.

>>15989377
>real analysis isn't a listed prereq but i'm extremely rusty on it. anything i should touch up on before starting?
Stromberg is a pretty good book on real analysis. I'm too low IQ for the average shit pushed on this board, but Stromberg's text worked for me.

>>15992262
projection onto what?

>low tier school
>applied math program
>capstone course research project
>Rules: essentially no rules, research what I want, study what I want, just check in once a week and have a paper and presentation ready by the end of the semester

I'm totally lost my dudes. I feel like I already forgot everything. I don't know how broad or how narrow to focus, and they're probably just gonna pass me no matter what but that doesn't mean I don't want to do my best anyways. Its not about the grade its about how I feel. I wish I had a real mentor. I was thinking about something from Kiyosi Ito's selected papers, like Brownian motion in a lie group, stochastic differential equations, stochastic differential equations ina differentiable manifold,well really just one of those three since they're the most approachable of his papers. Another option is to go a more applied route and do a deep dive and dedicated study of sequential (kalman) filtering. The latter has better prospects for inclusion of MATLAB or R, which are the only programming languages I'm competent in.
Super lost, I know I can't be the only one to have ever been in this position.

>>15992272
Correct

By now it should be appallingly obvious that nobody likes autistic people. They do not want to hire us, they do not want to be friends with us they definitely do not want to date us and many of them will not even rent to us! Social exclusion does in fact pose an existential threat to those on the spectrum which is why we have no choice but to form our own exclusive communities.

This is why I am proposing that those with Asperger's syndrome and an above average IQ begin networking with their fellow autists of similar cognitive ability. Despite the fact that many of us excel at STEM fields, many employers refuse to hire us due to our social deficits. The obvious solution to this problem is to start our own companies that only hire above average IQ autistic people. Doing so will solve the problems of employment discrimination and workplace bullying while allowing greater efficiency.

I personally am tired of people like us being the rejected and humiliated laughing stock of the entire internet. This is why it is time for us to separate from the neurotypicals and build our own exclusive communities.

Requirements to join:
1.) ASD diagnosis
2.) 115+ IQ
3.) Fewer than 5 friends irl or online
4.) Y chromosome haver
5.) Special interest is scientific or at least intellectual rather than something insignificant like anime or video games
6.) Must use proper English
7.) Must be actively involved
8.) No spouse or dependents

/gmcuDnZn65

How to learn math from scratch?

Are stupid rich people worth less than middle class professors?

Why do girls hate me?

>>15992821
openstax.org/subjects/math

>>15992832
They're secretly into you but afraid to show it.

>>15992618
kalman filtering is kind of fun. you can do a simple example of a cannonball is fired, you get "observations" from some radar sitting somewhere (so your obs are range/bearing/elevation from the radar location) and add some noise, boom the kalman filter can track the cannon ball

if you want something more mathematically interesting, there's an extension called the unscented kalman filter that uses a neat trick to stay accurate for nonlinear systems

>>15969764
When I finish a chapter I write all the
>definitions, theorems, and proofs
in my own words from memory, using obsidian and excalidraw. I find it to be a lot more engaging and effective than just rewriting everything as is

>>15993233
Thanks. I'm looking at "Applied Stochastic Differential Equations" by Simo Särkkä and Arno Solin, and "Stochastic Integrals" by Henry P. McKean. Not topics I've covered and I figure maybe I can do a good coverage of SDEs with the intention of Kalman filtering coverage, as the text by Särkkä has some coverage of it.
Not sure how ambitious I should be.

>>15992821
start here https://stacks.math.columbia.edu/tag/0001

>>15993668
Actually did not get McKean's book, and opted for Oskendal's book, as I could find an entire syllabus course which utilizes the book from Umea. Should be able to push through it and have time to also get a small Kalman filter project off the ground I think, there is a lot of content available.

Mathematically, how does one solve this mathematical puzzle using maths?

>>15969942
>>15969953
which software is this?

>>15994040
Not that anon, but
>pic rel from Obsidian
I don't know what the other software is

>>15994033
same chance of death anywhere you click, so click (1,4) because if you don't die, the number will give you better odds of not dying next time

>>15994066
Ah thanks
>>pic rel from Obsidian
I am retarded :)
brb, killing myself

>>15994040
>>15994066
The other pic is from https://github.com/org-roam/org-roam-ui

consider the relation 'approximately equal to'
now consider a relation approximately approximately equal to'
now consider 'approximately approximately .... equal to' with unending approximatelys.
now condense that to be 'megapproximately equal to'
Now, it is true that every natural number is megapproximately equal to every other natural number.

What percent of total exercises do you do in a given book you're studying? I think <85% is good enough.

>>15994526
Correction: >85%*. I am terrible.

>>15992821
go find a lecture series on intermediate algebra on youtube or elsewhere, professor leonard has some great introductory math stuff

then watch another on college algebra, he's also got one but find a textbook on libgen to churn out problems to, that's a common theme.after you feel you're competent enough in basic arithmetic, go find a precalc/trigonometry book to learn out of and a lecture series to supplement it wherever needed - the blitzer one is pretty nice, it's what i used in first year HS maths so i at least vaguely remember it. alternatively you could take either one separately, but just know algebra and trig are both rather critical if you want to do any calculus, from there physics or any math involved things

calculus is how stuff changes and accumulates infinitesimally, and you usually start at a semester (or two, depending on where you're learning) course on single variable, then one on multivariable. i'd take some computational linear algebra after singlevar just so you can get used to a few concepts you'll use in multivar and after that ODEs, since either one of those is usually paired with it anyway i feel if you're more prepared the proofs and theorems especially will make WAY more sense, but it covers vectors/matrices/transformations in more detail than you're used to before then. for books i'd go with strang's linear algebra & its applications, vector calculus by colley and literally any calculus book besides spivak for single variable calc, unless you're looking for a challenge to put it lightly you'll have a hard time, so unless you're already competent i'd stay away. ordinary differential equations is usually paired with some basic PDEs, but for a book maybe i'd go for dover or whatever the one on MIT OCW's 18.03 is called

past that you're looking at advanced calculus (complex &real analysis), topology, differential geometry, etc.. by then you'll hopefully be mathematically mature enough to find a book yourself and study out of it

>>15994470
The semantics get rather tenuous with repetitions, but the best I could think of is if you have an exact
number then every "approximately" is a bit off from that number or its approximation (like terms of a sequence). With a certain bound, this can become something like "2 is megaprroximately equal to 3"
or perhaps "1 is megaprroximately equal to 1 million". Simply put, the scale difference of numbers makes
the notion weak, unless it's something basic we can say about real or natural numbers.

>>15994470
define ....

>>15994033
Aren’t (1) and (2) really, really common? That means the centers of the edges are more reliably safe, because they must be (3) or more.

>>15992262
I'm assuming these are linear subspaces since I haven't worked in more general spaces yet. We take [math] S [/math] as the space for which [math] S_{i} [/math] are subspaces, and [math] S_{1}+S_{2}+...+S_{n}=Vect(S_{1}\cup S_{2}\cup...\cup S_{n})=S [/math]. Since the subspaces are orthogonal then [math] S_{1}+S_{2}+...+S_{n}=S_{1}\oplus S_{2}\oplus ...\oplus S_{n} [/math] and so for a given element [math] x\in S [/math], by definition of a direct sum, there exists a unique decomposition [math] x=x_{1}+x_{2}+...+x_{n} [/math] where [math] x_{i}\in S_{i} [/math].

I'm a bit lost with a seemingly simple proof. I want to prove that there is no continuous function f from the 3-sphere to the 2-sphere satisfying f(x) = -x for every x in S^2. My guess is that if we assume such an f exists then we can compose it with a singular 3-simplex to get an induced homomorphism from H_3(S^3) to H_3(S^2), which is trivial. I'm not really sure how to go from here to get a contradiction though

>>15994470
Congrats, you discovered infinity categories.

>>15994470
Every natural number is in the same set as every other natural number.

hi guys, recently my career requires me to do math (linear algebra and calculus) but i'm from a non-math degree. Any recommendations on how to start learning and understanding those?

Law of logs

>>15995072
Bit rusty on this, but have you noticed that f is odd and the trivial homomorphism is even?

>>15995281
Get a different job. You're a stupid nigger out of your element, taking a spot from someone who is actually qualified for it. You are a net drain in your organization and only got where you were based on socializing like a woman.

>>15995150
I don't see a connection.

>>15969522

If A,B are nxn matrices and AB=0, then BA squares to 0.

Does the converse hold?

I.e., if C is an nxn matrix, and C^2 = 0 , then are there nxn matrices A,B such that C=BA and AB=0 ?

>>15995304
Is there such a thing as even and odd isomorphisms?

>>15995072
>function f from the 3-sphere to the 2-sphere satisfying f(x) = -x for every x in S^2
Is this a typo? This doesn't make sense as written.
Do you mean, f(-x) = -f(x) for every x in S^3 ?

>>15995471
Why doesn't it makes sense? Your statement makes sense too but it's not equivalent.

>>15995475
If f is from S^3 to S^2 , then the notation f(x) means x is in S^3 , so f(x) = -x doesn't make sense because f(x) should be in S^2, but -x is in S^3 .

Are you implicitly using some equatorial embedding of S^2 into S^3? If so then you should have clarified this in your original question.

>>15995483
Yes, its an equatorial embedding

>>15995513
Let [math] \iota : S^2 \rightarrow S^3 [/math] be the equatorial embedding
Let [math] \alpha : S^2 \rightarrow S^2 [/math] be the antipodal map

Then [math] f \circ \iota = \mathrm{id}_{S^2} [/math]

So for the induced maps on cohomology we have [math] \iota^* \circ f^* = \mathrm{id}_{H^*(S^2)} [/math]

In particular [math] f^* : H^*(S^2) \rightarrow H^*(S^3) [/math] must be injective

But this is impossible since e.g. [math] H^2(S^2) \neq 0 [/math] but [math] H^2(S^3) =0 [/math]

>>15995548
Oops I had some typos

Let [math] \iota : S^2 \rightarrow S^3 [/math] be the equatorial embedding
Let [math] \alpha : S^2 \rightarrow S^2 [/math] be the antipodal map

Then [math] f \circ \iota = \alpha [/math]

So for the induced maps on cohomology we have [math] \iota^* \circ f^* = \alpha^* [/math] ; note [math] \alpha^* [/math] is an isomorphism because [math] \alpha [/math] is a homeomorphism

It follows that [math] f^* H^*(S^2) \rightarrow H^*(S^3) [/math] must be injective

But this is impossible since e.g. [math] H^2(S^2) \neq 0 [/math] but [math] H^2(S^3) = 0 [/math]

I despise geometry so fucking much.
I'm reading through a differential geometry textbook and I can't stand it, it's driving me nuts.
the thing is, I kinda need to study differential geometry, so I haven't got much choice.

New >>15995883

>>15995764
How about trying “Ordinary Differential Equations” by Vladimir Arnold? That’s the book I’m using, and it seems dandy so far.

What's the strongest form of Determinacy not known to be inconsistent with ZF? I am aware that there are games on sets of reals which are provably indeterminate in ZF alone, so is Real Determinacy the best we can do?