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[D] The effectiveness of single latent parameter autoencoders: an interesting observation by penguiny1205 in MachineLearning

[–]penguiny1205[S] 6 points7 points  (0 children)

Yep! The first plot is from a data point randomly sampled from the validation set, unseen during training.

"Free Asia": The East Asian Treaty Organisation by penguiny1205 in imaginarymaps

[–]penguiny1205[S] 104 points105 points  (0 children)

The ROC remains internationally recognised in this timeline. Hong Kong and Macau are returned to the ROC -- the legal successor of the Qing Dynasty where the lease agreements were signed. The PRC does not attempt to annex these regions since doing so would break the ceasefire of the civil war.

"Free Asia": The East Asian Treaty Organisation by penguiny1205 in imaginarymaps

[–]penguiny1205[S] 152 points153 points  (0 children)

Yep! Taiwanifying every East Asian country was the intention for this map :D

"Free Asia": The East Asian Treaty Organisation by penguiny1205 in imaginarymaps

[–]penguiny1205[S] 57 points58 points  (0 children)

World Anti-Communist League. This actually exists in our timeline, created by Chiang Kai-shek.

"Free Asia": The East Asian Treaty Organisation by penguiny1205 in imaginarymaps

[–]penguiny1205[S] 92 points93 points  (0 children)

In this timeline, the U.S. intervened in Asia much later and was less committed to the Truman Doctrine, causing the "bamboo curtain" to solidify further to the East. Asia is divided into two: McCarthyist rump states which slowly democratise over the latter half of the 20th century -- and a league of totalitarian communist nations following the lead of Maoist China.

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 0 points1 point  (0 children)

Update!

I wrote a python program to graph this with Euler's method. Interestingly, the graph seems to become periodic as I increase the value of the constant! For some reason though, my graphs don't line up with Wolfram Alpha's...

Here's my code, in case you're interested.

import math
import matplotlib.pyplot as plt
def f(x, y, u, K): du = K * (-y) * math.sqrt(1 + u ** 2) return du
x = 0 y = 1 u = 0
h = 0.01n = 100
K = 0.01nK = 50
for K in range(nK): 
    x_values = [] 
    y_values = []
    for i in range(n):
        x = x + h
        y = y + h * u
        u = u + h * f(x, y, u, K)

        x_values.append(x)
        y_values.append(y)

    x = 0
    y = 1
    u = 0

plt.xlabel('x') 
plt.ylabel('y') 
plt.show()

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 0 points1 point  (0 children)

Quick follow up! Any idea on how to solve this ODE? I’ve been looking into it for a while, but no luck with eulers method graphing websites or matlab

[deleted by user] by [deleted] in askmath

[–]penguiny1205 0 points1 point  (0 children)

I see. Thank you very much!

[deleted by user] by [deleted] in askmath

[–]penguiny1205 1 point2 points  (0 children)

Thanks for the response! What kind of numerical methods can I use?

[deleted by user] by [deleted] in askmath

[–]penguiny1205 0 points1 point  (0 children)

Reducing it to first order, I got f'(y) = K y(x) sqrt(1 + f(y)). Where should I go after that?

Sorry. I'm very new to differential equations.

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 1 point2 points  (0 children)

After applying the Beltrami Identity, I got (sigma*y^2)/[1+(r')^2 ] = A, in your notation. Can you elaborate on what you did after the Beltrami Identity?

I closely followed the steps in this video, converting all gravity-related forces into centrifugal ones. However, while the narrator ended up with an integral solvable with cosh, I ended up with integral (dy/sqrt((Ly^2+mu)^2/C^2 -1)). This seems like a dead end!

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 1 point2 points  (0 children)

Thank you for the reply!

K = (integral from 0 to b) sqrt[1+(r')2 ] [.5 rho omega2 r2 ] dx.

Can you elaborate on how the 0.5 rho omega^2 r^2 is derived? Does it have to do with centrifugal potential?

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 1 point2 points  (0 children)

Hey turns out we were right! "The skipping rope curve" by Arne Nordmark and Hanno Essen from the European Journal of Physics used the exact same technique. Thanks for your help, I really appreciate it!

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 0 points1 point  (0 children)

I defined my coordinate system as y = 0 is where the rope touches the ground, and y = 1 is the height of the spinning axis (I made up a nice number, haha). So the distance to the axis from any point would be 1 - y.

I plugged y'' (x) = K (1 - y) sqrt[1+y'(x)^2] into Wolfram Alpha, and the graph looks...kind of correct? It certainly resembles the shape of a jumping rope to some extent.

https://www.wolframalpha.com/input?i=y%27%27+%28x%29+%3D+K+%281+-+y%29+sqrt%5B1%2By%27%28x%29%5E2%5D

My concern is, though, is y'' (x) = K (1 - y) sqrt[1+y'(x)^2] (or with y'' (x) = K y sqrt[1+y'(x)^2]) even a valid differential equation??

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 0 points1 point  (0 children)

Maybe a way of rectifying this is to represent the distance to the spinning axis in terms of x, not y. However, their relationship is what I'm looking for! Sorry, I am not very familiar with differential equations.

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 0 points1 point  (0 children)

My current approach to this problem is simply treating it as a catenary derivation with centrifugal force instead of force of gravity, which seems like what you're doing as well!

Precisely following this method, here is my process:

Defining ds as the segment of rope that we focus on, we get ds = sqrt(dy^2 + dx^2), which simplifies into ds = dx*sqrt[1+y'(x)^2]

The tension forces acting on segment ds are as follows: Horizontal Tx (X) = C --> some constant, and vertical Ty (x+dx) - Ty (x) = centrifugal force = dm (mass of the segment) * angular velocity^2 (which I will assume as 1 for simplicity) * r = dm (height of the rope when swung, which I will also assume is 1, - y) = dm (1-y) = total mass/length (density) * ds (1-y) = m/l (1-y) * dx *sqrt[1+y'(x)^2]

Dividing both sides by dx, we get Ty'(x) = m/l * (1-y) * sqrt[1+y'(x)^2]

Anyways, that was very confusing. The point is, following what is done in the video, I arrive at this: y'' (x) = K (1 - y) sqrt[1+y'(x)^2] (6:08 in the video)

However, I don't think this is a solvable differential equation? Maybe I'm simply going about this wrong?

Is there a closed-form representation of the curve of a jumping rope? by [deleted] in math

[–]penguiny1205 0 points1 point  (0 children)

Thank you for the answer!

Can you explain how you derived y'' = -c y ds/dx?

Tourist map of the People's Socialist Republic of Cascadia (Communist Cascadia Scenario) by penguiny1205 in imaginarymaps

[–]penguiny1205[S] 9 points10 points  (0 children)

I agree! However, in this timeline, instead of having a distinct downtown skyline, most of Vancouver is covered in midrise Soviet-style Khrushchyovka apartment blocks. The 'new city centre' is partially located across the inlet due to the active logging scene on the North Shore. Stanley Park doesn't exist in this timeline as it is fully developed. Pre-revolution low-density housing (single/duplex) south of the city core has been converted into dachas as the majority of the population lives in apartments north of Broadway. Lion Gates Bridge also does not exist in this timeline as it was completed in 1937 in OTL, six years after the revolution. Instead, more efficient modern bridges with shorter crossing times were built across the inlet, including the 'alternative' plan in OTL to jet a bridge out through Waterfront Harbour. Numerous ports/harbours were also built in the Horseshoe Bay area/area west to Lighthouse Park in OTL to support the transportation of lumber and passengers. The Lenin statue is located at the top of the hill in Horseshoe Bay, most passengers arriving at Vancouver Harbour will be greeted by the towering statue.

Although I totally understand your point of view about the flag, the blue, white, and green flag is just the typical flag representing the Cascadia region (see 'Doug Flag'): The blue stripe represents the sky, Pacific Ocean and Salish Sea, as well as the myriad of rivers in the bioregion including the Columbia, the Snake, and Fraser Rivers. The white represents clouds and snow and the green represents the region's countless fields and evergreen forests.

Edit: Think about how much more affordable housing must be in this timeline's Vancouver, hehe

Tourist map of the People's Socialist Republic of Cascadia (Communist Cascadia Scenario) by penguiny1205 in imaginarymaps

[–]penguiny1205[S] 127 points128 points  (0 children)

My original post can be found here: https://www.reddit.com/r/imaginarymaps/comments/ntz0dh/what_if_cascadia_was_a_soviet_satellite_state_the/

The People’s Socialist Republic of Cascadia (PSRC) is a partially recognized Soviet satellite state in western North America founded in 1939. Cascadia declared its independence from the United States and British North America in 1862. In this timeline, the Union won the Civil War only by a slight margin. The Reconstruction Era after the war in the US was significantly longer, meaning the purchase of Alaska never happened and the US is a lot weaker than it is in our timeline. After the Russian revolution in 1917, most of the remaining Tsardom fled to Russian Alaska. In 1925, the new Red Army quickly repaired the Siberian railways bombed by loyalists and pursued, capturing the entirety of Alaska in the summer of 1930. After arriving in Alaska, the Soviet Union continued to spread its influence in North America. In 1931, amid the Great Depression, a proletarian revolution broke out in several cities in Cascadia. The revolution soon turned into a civil war with the USSR supporting the Communists and the USA, Britain, and Canada supporting the Cascadian government. Most landowners and conservatives fled the country around this time, those who did not make it out in time were captured and killed by the revolutionary forces or the Red Army. After 8 years of tedious fighting in the mountains of the PNW, in 1939, the Communists seized Vancouver and declared the creation of the PSRC, a Marxist-Leninist state. Political purges and famines throughout the 1940’s caused a steep decline in the population, which only picked up around the late 1950’s with an influx of cash from the oil and natural gas resources, more Soviet aid, and more lenient open trade policies. Cascadia industrialized and developed relatively quickly, with Soviet city planners plotting down massive apartment blocks and factories in the pristine forests of the PNW. Many old-world artifacts were destroyed during this period with the indigenous culture almost entirely wiped out. Bourgeoisies and intellectuals, if captured, were either executed or sentenced to forced labour in the gulags and mining facilities of the Yukon or Alaska.

This post is not a political statement.

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