{"version":9,"randomSeed":"55503bdde1bc71d8155fecee6e7afbac","graph":{"viewport":{"xmin":-10.000000000000002,"ymin":-10,"xmax":9.999999999999998,"ymax":10}},"expressions":{"list":[{"type":"text","id":"12","text":"inspired by \"Wavelets: a mathematical microscope\" by Artem Kirsanov (2022) https://www.youtube.com/watch?v=jnxqHcObNK4"},{"type":"expression","id":"1","color":"#c74440","latex":"f\\left(x\\right)=1"},{"type":"expression","id":"2","color":"#2d70b3","latex":"g\\left(x\\right)=x^{2}"},{"type":"expression","id":"13","color":"#2d70b3","latex":"A_{0}=12","labelSize":"medium","slider":{"hardMax":true,"max":"100"}},{"type":"expression","id":"14","color":"#388c46","latex":"A=10^{A_{0}}"},{"type":"expression","id":"3","color":"#388c46","latex":"C=\\frac{\\int_{-A}^{A}f\\left(x\\right)g\\left(x\\right)dx}{\\sqrt{\\int_{-A}^{A}f\\left(x\\right)^{2}dx}\\cdot\\sqrt{\\int_{-A}^{A}g\\left(x\\right)^{2}dx}}"},{"type":"text","id":"18","text":"u . v = |u| |v| cos θ, so cos θ = (u . v) / (|u| |v|) = (u . v) / (sqrt(u . u) * sqrt(v . v))"},{"type":"text","id":"24","text":"when f(x) = x and g(x) = x^x_1, C = y_1 = sqrt(z_1) / w_1"},{"id":"28","type":"table","columns":[{"values":["1","3","5","7","9","11"],"hidden":true,"id":"26","color":"#388c46","latex":"x_{1}"},{"values":["1","0.916515","0.8206518","0.74535599","0.686348585","0.638971"],"hidden":true,"id":"27","color":"#6042a6","latex":"y_{1}"},{"values":["1","21","33","5","57","69"],"hidden":true,"id":"29","color":"#000000","latex":"z_{1}"},{"values":["1","5","7","3","11","13"],"hidden":true,"id":"30","color":"#c74440","latex":"w_{1}"}]},{"type":"text","id":"34","text":"when f(x) = 1 and g(x) = x^x_2, C = y_2 = sqrt(z_2) / w_2 = cos(u_2), sin(u_2) = p_2 / q_2"},{"id":"38","type":"table","columns":[{"values":["0","2","4","6","8","10"],"hidden":true,"id":"36","color":"#c74440","latex":"x_{2}"},{"values":["1","0.74535599","0.6","0.51507875","0.4581228","0.41659779"],"hidden":true,"id":"37","color":"#2d70b3","latex":"y_{2}"},{"values":["1","5","9","13","17","21"],"hidden":true,"id":"39","color":"#388c46","latex":"z_{2}"},{"values":["1","3","5","7","9","11"],"hidden":true,"id":"40","color":"#6042a6","latex":"w_{2}"},{"values":["0","0.729728","0.927295","1.029697","1.094914","1.141097"],"hidden":true,"id":"42","color":"#2d70b3","latex":"u_{2}"},{"values":["0","2","4","6","8","10"],"hidden":true,"id":"125","color":"#2d70b3","latex":"p_{2}"},{"values":["1","3","5","7","9","11"],"hidden":true,"id":"126","color":"#388c46","latex":"q_{2}"}]},{"type":"expression","id":"44","color":"#6042a6","latex":"\\left(\\cos u_{2},\\sin u_{2}\\right)\\left[6\\right].y","labelSize":"medium"},{"type":"text","id":"46","text":"sin(u_2) is a simple fraction?"},{"type":"text","id":"130","text":"conjecture: the sine of the functional angle between x ↦ 1 and x ↦ x^(2n) is (2n) / (2n + 1)"}]}}