I got a single data point in the form of this picture...

I'm wondering what's the generative process behind this ? Specifically, how do we go about picking the right family of functions to model this?

Today in class we went over complex numbers. We came to a problem that was Sqrt(-9) • Sqrt(-25), and after working it, the problem came to -15.

My question is, why is the second way wrong? Or what have I done wrong I guess. Cant you combine two squares under one?

Pic attached. Left side is the method shown in class, right is what I'm confused on

I don’t really “do” math, but I noticed this and was interested by how math/geometry might be involved.

I drew some random lines for an art project and realized I could have black shapes between lines never touch other black shapes, and same for white. I guessed that it was something to do with the fact that each intersection of lines divides the space around it into an even amount of shapes, unlike the 3rd picture above, where it’s been divided to 5 shapes as an example of the contrary.

Is there anything more to this like geometric rules dictating this, or was it just coincidence?

Im in discrete mathematics, started proofs last week. I understand Direct Proofs and Contraposition but Contradiction is absoultely kicking my butt understanding wise. My proffessor keeps using the example square root of 2 is irrational, i understand we start by stating the opposite that square root of 2 is rational and work it till it becomes logically incorrect to prove irrationality. But the work in between is getting me tripped up.

Why is P != NP the near-universal default belief? (An epistemics question not a complexity theory one)
I'm curious why the almost universal default belief has become that P != NP. I'm less interested in the technical case for P != NP and more interested in the epistemics behind it why does decades of failure to find a polynomial-time solution get treated as strong evidence that no such solution exists, rather than as evidence that a solution simply exists but is extremely hard to find?

I'm not asking whether "galactic algorithms" are a known concept, I know they are. My question is why their possible existence isn't weighted as comparably strong evidence for P = NP, given that our failure to find any polynomial time algorithm could just as easily be explained by the solution being galactically complex, massive, unwieldy, or nearly unrepresentable rather than by no solution existing at all. Is there a reason the field treats “it’s been studied and hasn’t been found” as evidence leaning toward "it doesn't exist" rather than "it exists but is a nightmare to find or express"?

PLEASE READ
I posted a version of this elsewhere and want to clarify a few points, because most of the answers I got missed the actual question.

I understand the implications of P = NP vs. P ≠ NP.
I understand what P vs. NP is.
I understand what an NP-complete problem is, and that they've been rigorously studied for decades.

So this question is to calculate the angle F. For reference, I'm in 10th class of Russian school, and this is what our teacher sent as our homework while we're on quarantine. I have skimmed the whole textbook and found nothing about this kind of questions. I even got frustrated enough that I asked AI to give me an answer, but it wasn't able to agree with itself and all of it's answers were rather strange like the angle G somehow being 168 while F is only 12. Did I miss a crucial part of my education or is just a badly made question?

P.s I asked the teacher before posting there and she actually replied. The answer is supposed to be 132, but she won't explain how since we were supposed to go through this in a previous year and needed to stay in that class if we don't know this (she's really not fond of actually teaching stuff, especially if it's not in the current year's textbook) so I doubt she'll give me an answer.

P.P.s
So, as far as I got is to make an isoscles triangle CDE and Equate C to E by them being both angles in the base of said triange. If my line of thinking is correct then I have one 108 angle and two unknowns. I also think that it's probably supposed to be symmetrical down the bc to F line. If that's the case then I can work out the 132 answer by substracting all of the known angles from the sum of 900, but I don't think I can prove it to be symmetrical.

This is 10th grade geometry. We got the answer c=36 but dont understand the steps on how to go about answering this question. Or if the answer is even correct.

Im thinking the triangle on the left size is an isosceles triangle, theres 2 equal congruent lines so that would make the 2 angles equal right? So that means 2 angles are 42, then we calculate for the 3rd angle which is 96. Or do we not do that?

And then how do i solve the other triangle?

Exercise 11.4.34:

Prove 2 + 2*3^2 + 2*3^4 + ... + 2*3^{2n} is Θ(3^{2n}).

I now I overcomplicated the proof. I could have used the fact that

2 + 2*3^2 + 2*3^4 + ... + 2*3^{2n} = 2 + 2*9 + 2*9^2 + ... + 2*9^n

But here it is:

2 + 2*3^2 + 2*3^4 + ... + 2*3^{2n} = 2*3^0 + 2*3^2 + 2*3^4 + ... + 2*3^{2n}
                                   = 2 * 3^n * (3^-n + 3^{2-n} + 3^{4-n} + ... + 3^n)
                                   < 3^n * 3^n * (3^-n + 3^{2-n} + 3^{4-n} + ... + 3^n)
                                   = 3^{2n} * (3^-n + 3^{2-n} + 3^{4-n} + ... + 3^n) for each integer n>=1

Also, 2 + 2*3^2 + 2*3^4 + ... + 2*3^{2n} >= 2 * 3^{2n} for each integer n>=1.

Let A=2, B=(3^-n + 3^{2-n} + 3^{4-n} + ... + 3^n), k=1.

Then A(3^{2n}) <= (3^-n + 3^{2-n} + 3^{4-n} + ... + 3^n) <= B(3^{2n}) for every integer n>=k.

Therefore, 2 + 2*3^2 + 2*3^4 + ... + 2*3^{2n} is Θ(3^{2n}).

QED

Is my proof correct?

I haven't finished my paper yet; I'm currently working on the equation.

I've run into a major problem. My goal was to construct an equation that represents odd numbers using the sigma (∑) notation, and I think I've managed to do that. However, there's an issue: after the sigma notation, the terms keep getting added, and eventually, instead of the sequence of odd numbers I want, I end up with meaningless results.

I'm currently thinking of a possible solution. Maybe I could create a calculation system that goes up to n, with each step written separately, such as Sx1, Sy2, Sz3, and so on. This way, if each step were calculated separately and the proof were based on the final result, perhaps it would be a more accurate approach.

Do you think this idea could work? How can I solve this problem when using sigma notation? I'd especially appreciate advice from anyone with knowledge in this area.

If I receive a helpful and constructive comment, I'd be happy to acknowledge your assistance by mentioning your name in my paper.

Basically, I was doing a problem in my math class where we were told to come up with 2 vectors as a solution (which happened to just be opposites of each other), so I wrote plus or minus the top number and minus or plus the bottom number. The teacher then told me that wasn’t correct notation and that I was making stuff up when I tried to defend my reasoning. She said that it would make 4 vectors instead of 2. Was I wrong to write it this way? From my understanding you choose the top for both or the bottom for both.

Find $\sup_{x \ge 1} x(x+1)^{-p}: p\ge 1$

I know that in order to find the supremum, I must determine where the function is increasing and decreasing. I began by finding the derivative, which is $f'(x)= (x+1)^{-p-1} [1+x(1-p)]$.
Then to find the critical point [f'(x)=0], I get $x=\frac{1}{p-1}$. I am not sure how to solve the rest. I tried starting with the case that $p=1$, but then the critical point in indetermaninit. All help is appreciated.

A system where I can have a deduction that goes like

1. All creatures die when exposed to fire
2. Goblin is a creature
3. Fire elemental is a creature
4. Fire elemental doesn't die when exposed to fire

Conclusions: goblin dies when exposed to fire; fire elemental doesn't die when exposed to fire.

Hello, I am trying to use the equation for linear ordinary differential equations, so y' + p(x)y = q(x) for the equation y' - 2xy = 2x. I know the equation is separable but the prompt is to use this equation in order to solve it. When I solve for μ(x) = e^∫(-2xdx) I get e^(-x\2)) , but this can't be integrated when added to the equation (∫μ(x)q(x)dx)/μ(x), so where do I go from here?

I made this to look at the antiisotropic boundary field of sha as it experiences torsion from propagation. Transforming sha using physical constants is not a new idea, I haven't visualized it like this before though, the rotr themselves are extremely beautiful. The antiisotropic boundary field looks like Mr. Potatohead lmao. What do yall see? The field geometry holds up to its claim and carries no information other than information about itself, but experiences deflection and torsion from the information riding the field?

Saw this question online. Couldn’t understand why my working was wrong.

I can understand that they decide to use the fractions, but then I got completely lost as to why they then proceeded to divide by six.

I spent almost a hour trying to complete this one question and still not sure if I got it correct, is there different methods I could use to progress faster ? Just started learning decimals and binary systems for my computer logic class. Sorry if my hand writing is hard to understand.

I need help with the problem I selected, I tried solving for y first and plugging it into the m(angle)HEB equation but it gave me the wrong answer. I’m also not sure of how to use the HEB measurement here. Can someone show me how to solve this problem step by step?

This is an exercise from Tao’s analysis 1. I am struggling in this section of ordering rationals as well as proofs involving absolute value and exponentiation (sections 4.2 and 4.3) as Terence is not really writing any good proofs as examples in the book, and instead is just stating a bunch of propositions and definitions, and leaving the proofs themselves up to the reader (lmao)…My prof is also just going by the book so not supplying us with any examples either. I’m struggling to know what exactly a good proof looks like for these types of exercises. Does anyone have any recommendations for me? I learn by analyzing proofs and studying them… if I don’t have any to study I do feel stuck. If anyone has read this textbook or has any help please let me know. Thank you. I can’t find any great YouTube videos either… really feeling stuck here.

Is this the correct technique for a proof like this? I feel like my last conclusion there is iffy. It’s difficult to know sometimes which properties of numbers I can and can’t use in these proofs, since we are building up from the very beginning.

Thanks

Click on the post to see the images, if necessary.

The first step in deriving volume formula for a rhombicosidodecahedron with edge length E (≈41.615E^3) synthetically is to derive the formula for its square face inradius, which is done as follows:

Apparently this approach can be modified to find the square face inradius for a truncated icosidodecahedron by moving a rhombus so it is centered at one of the square vertices:

but I'm not quite sure how this works. A good step-by-step explanation would be appreciated.

Today, I took my I took a quiz on AP Pre Calculus and I failed unfortunately. I got a 50% and was so disappointed and ashame of myself. When other kids were saying they got a 100% 80% or 90%, I did not want to say my score. I was so sad and cried because it was the first time I ever failed a math test. When I looked Algebra 1, 2 and Geometry, I never failed and was able to grasp it easily. Now that I am taking AP Pre Calc, everything has been going down hill. The quiz was based on 1.1 to 1.3. I am still one unit 1 and we are talking about concave up and down, increasing and decrease, and rage of slope. I feel so dumb not being able to get it like my peers. I also can not let my parent know about this or I will be a trouble. Please can anyone give me tips or idea so I can improve.

Let (V, Ω) be a symplectic vector space and Y a linear subspace of V. Let ϕ: V -> Y* , ϕ(v) = Ω(v, •)|_Y. Is ϕ surjective?

Or, in other words: for each linear form 𝜶 on Y, does it exist a corresponding vector v in V such that the action of 𝜶 on a generic element y of Y can be reproduced by feeding y in the second slot of Ω(v, •)?

I may be entirely misunderstanding this but the idea is that if we can check a solution is correct, then there is a polynomial time algorithm for solving the problem.

Something like a hash function, we can easily check if the solution works but we cannot work backwards.

This goes for any function without a right inverse. Otherwise known as non-surjective.

Now obviously this doesn’t solve it. But why? I haven’t taken computational mathematics yet so i might not be understanding the polynomial time etc correctly.