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286 lines
7.3 KiB
Plaintext
286 lines
7.3 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "recorded-albany",
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"metadata": {},
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"outputs": [],
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"source": [
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"import torch\n",
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"from torch.autograd.functional import jacobian"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "senior-characterization",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([1.0000, 1.0000, 1.0000, 1.0000, 0.5000], requires_grad=True)"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"#set states\n",
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"stocks = torch.ones(5)\n",
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"#Last one is different\n",
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"stocks[-1] = 0.5\n",
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"#now add the tracking requirement in place\n",
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"stocks.requires_grad_()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "sublime-trance",
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"metadata": {},
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"outputs": [],
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"source": [
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"launch = torch.ones(5, requires_grad=True)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "double-climb",
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"debris = torch.tensor([2.2],requires_grad=True)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "blind-reunion",
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"metadata": {},
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"outputs": [],
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"source": [
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"scaling = torch.ones(5)\n",
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"delta = 0.9 \n",
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"launch_debris_rate = 0.05\n",
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"collision_debris_rate = 0.07"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "worldwide-winning",
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"metadata": {},
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"outputs": [],
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"source": [
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"def survival(stock, debris):\n",
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" #Gompertz distribution for simplicity\n",
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" #commonly used with saturation\n",
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" #TODO: ACTUALLY DERIVE A SURVIVAL FUNCTION. THIS IS JUST A PLACEHOLDER\n",
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" eta = 1.0/(scaling@stock)\n",
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" b = 1/debris\n",
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" \n",
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" return 1 - ( b*eta*torch.exp(eta+b*stock-eta*torch.exp(b*stock)))\n",
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"\n",
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"\n",
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"def g(stock, debris, launches):\n",
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" new_stock = stock*survival(stock,debris) + launches\n",
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" \n",
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" #Ignoring autocatalysis\n",
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" new_debris = (1-delta)*debris + launch_debris_rate * launches.sum() + collision_debris_rate*(1-survival(stock,debris)) @ stock\n",
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" \n",
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" return (new_stock, new_debris)\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "naughty-transport",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([0.8600, 0.8600, 0.8600, 0.8600, 0.8802], grad_fn=<RsubBackward1>)"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"survival(stocks,debris)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "large-trigger",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(tensor([[-0.0142, 0.0271, 0.0271, 0.0271, 0.0271],\n",
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" [ 0.0271, -0.0142, 0.0271, 0.0271, 0.0271],\n",
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" [ 0.0271, 0.0271, -0.0142, 0.0271, 0.0271],\n",
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" [ 0.0271, 0.0271, 0.0271, -0.0142, 0.0271],\n",
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" [ 0.0251, 0.0251, 0.0251, 0.0251, -0.0142]]),\n",
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" tensor([[0.0825],\n",
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" [0.0825],\n",
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" [0.0825],\n",
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" [0.0825],\n",
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" [0.0634]]))"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"#Get the derivatives seperately\n",
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"jacobian(survival, (stocks,debris))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"id": "prime-projector",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([[-0.0142, 0.0271, 0.0271, 0.0271, 0.0271, 0.0825],\n",
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" [ 0.0271, -0.0142, 0.0271, 0.0271, 0.0271, 0.0825],\n",
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" [ 0.0271, 0.0271, -0.0142, 0.0271, 0.0271, 0.0825],\n",
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" [ 0.0271, 0.0271, 0.0271, -0.0142, 0.0271, 0.0825],\n",
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" [ 0.0251, 0.0251, 0.0251, 0.0251, -0.0142, 0.0634]])"
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]
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},
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"#Get the derivatives as a single result\n",
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"torch.cat(jacobian(survival, (stocks,debris)), axis=1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "urban-decision",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(tensor([1.8600, 1.8600, 1.8600, 1.8600, 1.4401], grad_fn=<AddBackward0>),\n",
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" tensor([0.5134], grad_fn=<AddBackward0>))"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"#Testing state updates\n",
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"g(stocks, debris, launch)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "congressional-kelly",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"((tensor([[0.8457, 0.0271, 0.0271, 0.0271, 0.0271],\n",
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" [0.0271, 0.8457, 0.0271, 0.0271, 0.0271],\n",
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" [0.0271, 0.0271, 0.8457, 0.0271, 0.0271],\n",
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" [0.0271, 0.0271, 0.0271, 0.8457, 0.0271],\n",
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" [0.0126, 0.0126, 0.0126, 0.0126, 0.8731]]),\n",
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" tensor([[0.0825],\n",
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" [0.0825],\n",
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" [0.0825],\n",
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" [0.0825],\n",
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" [0.0317]]),\n",
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" tensor([[1., 0., 0., 0., 0.],\n",
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" [0., 1., 0., 0., 0.],\n",
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" [0., 0., 1., 0., 0.],\n",
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" [0., 0., 0., 1., 0.],\n",
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" [0., 0., 0., 0., 1.]])),\n",
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" (tensor([[0.0042, 0.0042, 0.0042, 0.0042, 0.0013]]),\n",
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" tensor([[0.0747]]),\n",
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" tensor([[0.0500, 0.0500, 0.0500, 0.0500, 0.0500]])))"
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]
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},
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"#Note the two tuples of jacobians: the first is for stock evolution, the second is for debris evolution\n",
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"jacobian(g, (stocks,debris,launch))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "identified-insertion",
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"metadata": {},
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"source": [
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"## Next step: Construct the intertemporal-transition function(s)\n",
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" - Note: There are a couple of different ways to do this\n",
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" - Just a single period transition function, manually iterated\n",
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" - A recursive function that creates a $p$ period iterated function\n",
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" - A recursive function that returns a list of functions iterated from 1 to $p$ periods\n",
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"\n",
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"I am planning on doing the latter, as each version is needed."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "utility-browse",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.8"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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