curiosity on parallel lines and spheres

so…first: I’m not a math expert, second: I think this topic is moar like geometry but anyway…

if we have 2 parallel lines on a sphere they meet at the poles right? like the Longitudes of the earth. said this if we start “squashing” the sphere the lines keep on touching at the poles. if we keep doing this until we have something so similar to a plane that it is indistinguishable from it, the lines still meet at the pole… so is it right to say that parallel lines actually do meet…? and why doesn’t this happen for the latitude of the earth? because of technically speaking they to have the same proprieties of the longitudes.

submitted by Nevin Manimala Nevin Manimala /u/TheCubeMan4
[link] [comments]

Should the Control of a Dynamical System Depend on the State?

Hey, all.

I am working with a dynamical system of the form [; begin{align*}dot{x}_i(t) &= u_i(t)\ &= d_isum_{jinmathcal N_i}w_{ij}(x_j-x_i)end{align*} ;], where [; d_i ;] are control parameters, and [; w_{ij} ;] are predetermined weights specified arbitrarily (randomly).

I want to find optimal control of the system, but I have not ever worked with a dynamical system where the control depends on the state itself. In particular, the dynamical system can be written in the form [; dot{bf x} = -DL{bf x} ;], where [; D ;] is an invertible diagonal matrix containing the [; d_i ;]‘s created to stabilize the system by sending [; dot {bf x} ;] to 0, [; L ;] is a matrix containing the weights (Laplacian), and [; {bf x} ;] is the state vector.

How does setting up a performance index differ in this case? In particular, what I want to do is have [; dot{bf x} approx {bf 0};] in a given time [; t_f ;], and the way I can check this is with determining if [; |dot{bf x}|_2^2 approx 0 ;]. Thus, how do I need to set up my performance index? I think I need to set it up like so:

[; J = int_0^{t_f} lambda^top(left[|dot x|_2^2 - 0) + mu^top(-DL{bf x} - dot{bf x})right],dt ;]

But I’m not sure if this is appropriate setup. I need to incorporate the dynamics of the system with a Lagrange multiplier, given here by [; mu ;], right? Do I also need the Lagrange multiplier [; lambda ;] on [; |dot{bf x}| ;]?

I apologize for the nature of my questions. I just unfortunately haven’t seen a system like this before in control theory, and I’m not sure how it affects finding the Euler-Lagrange equations.

submitted by Nevin Manimala Nevin Manimala /u/Lafojwolf
[link] [comments]

Need help finding a particular website

There was a webpage I used to visit a long time ago that listed every positive integer from 1 to 1000 (or it might have been 10,000), and beside each number it gave a mathematical fact that uniquely characterized that number (i.e. 137 is the first such and such to have the property such and such).

Does anyone remember this site? If so, I’d really like some help finding it. Thank you in advance.

Edit: Has been found

submitted by Nevin Manimala Nevin Manimala /u/American_Gambit
[link] [comments]

Algebra practice for college undergrad level CMS students?

I am a first year undergrad student studying computer science, and I am absolutely garbage at math. In high school, I barely passed the first 3 levels of maths, and then aced AF12 and Calculus+Vectors; which is how I got into uni (CGPA doesn’t matter as much in Canada). But the only reason why I was able to do this was because we were tested every chapter, and the expectation on us to retain everything by the end wasn’t as high. Now that I’m doing college level math, I often get stuck on trivial algebra in the middle of much more complex proofs and evaluations. It takes forever for me to process a few equations because I worry about making mistakes. Also, a lot of definitions and examples apply algebra without explanation, leaving me confused. Is there a resource or method for practicing algebra so that I don’t have to comb through 10 websites to figure out what happened? Thanks.

submitted by Nevin Manimala Nevin Manimala /u/Aqkchua
[link] [comments]

Non-Newtonian Calculi

Recently I was thinking about what a multiplicative derivative and integral would be. I came up with:
rfdx = limh->0 (f(x+h)/f(x))1/h
which is the geometric derivative. Of course after googling it I found out about the geometric and bigeometric derivatives.

Next I tried using a different operation. Analogous to the harmonic mean, I defined harmonic addition which is
a⊕b = [a-1 + b-1]-1 and the harmonic difference a⊖b = a⊕(-b)=[a-1 – b-1]-1 And the harmonic derivative
hf dx = = limh->0 h [f(x+h) ⊖ f(x)]

h is multiplied by the difference because of the natural repetition operator.
x = x
x ⊕ x = x / 2 x ⊕ x + … + x ⊕ x = x / n

Repeated addition is multiplication, so the standard derivative multiplies the function difference by 1/h. Repeated multiplication is exponentiation, so the geometric derivative takes the function ratio to the power 1/h. Repeated harmonic addition is division, so the harmonic derivative uses 1/(1/h) * a = h * a. It’s probably skipping over a bunch of details though. Obviously after doing this I found the derivatives of the usual common functions
h[x-2] dx =1/(2x) and found how each was related to the normal derivative
rfdx = ef’/f

I haven’t been able to find anything talking about this harmonic derivative. I found sources talking about infinite families of Non-Newtonian calculi but they only ever talked about geometric and bigeometric. One site did mention quadratic and harmonic derivatives, but only in passing. Also there were a lot of papers that seemed beyond my level (calc 1 and 2, lin algebra, and general self study)

Basically I’m looking for the correct terminology and resources to look into this more. As well I’m wondering if there are any other notable generalized derivatives. Perhaps some that aren’t just

EDIT: Sorry if the formatting is terrible. I composed this on mobile during lecture lol
Tried fixing the formatting. I may have forgotten an negative sign. I’m trying to figure out where specifically I got my formula for it from. It doesn’t change the idea of the post though.

The article mentioning the quad and harm.

I think I didn’t give this article a fair chance at first. It’s seems to answer my questions.

submitted by Nevin Manimala Nevin Manimala /u/SetOfAllSubsets
[link] [comments]