1c:{"title":"Klein Four Dichotomy Correlation","cells":[{"id":3,"value":"md` # Klein Four Dichotomy Correlation `","pinned":false,"hidden":false,"mode":"ojs"},{"id":1642,"value":"$1e","pinned":false,"hidden":false,"mode":"ojs"},{"id":1980,"value":"$1f","pinned":false,"hidden":false,"mode":"ojs"},{"id":863,"value":"md` ## Klein Four Group`","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"md` Dichotomies have cyclic two symmetry ${tex`C_2`}, which is to say, one pole of a dichotomy is the opposite of the other. A model made of the direct product of two dichotomies has Klein four symmetry ${tex`C_2 \\times C_2`}. Implicit in the Klein four group is three dichotomy subgroups - the first two are the dichotomies that generated the Klein four group, and the third is a new dichotomy.\n\nAn example of the Klein four group from basic mathimatics is the charge of sine, cosine, and tangent in the 4 quadrants of the XY plane.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":64,"value":"$20","pinned":false,"hidden":false,"mode":"ojs"},{"id":318,"value":"md` In this geometric example, the original 2 dichotomies come from the plus and minus directions of the x and y axes. A point on the unit circle can be measure in the x direction by cosine and in the y direction by sine. Tangent is the ratio of sine divided by cosine, and it inherits its charge by properties of division:\n\n${tex`\\dfrac{sin +}{cos +} = tan + \\space\\space\\space\\space \\dfrac{sin -}{cos +} = tan - \\space\\space\\space\\space \\dfrac{sin -}{cos -} = tan + \\space\\space\\space\\space \\dfrac{sin +}{cos -} = tan -`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":385,"value":"md`Said another way, a sine, cosine and tangent value each define 2 points on the unit circle, and the intersection of two measures equals a single location.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":889,"value":"md `\n${md`\n| Function | I | II | III | IV |\n|:---------|:--|:---|:----|:---|\n| Hypotenuse | ${tex`+`} | ${tex`+`} | ${tex`+`} | ${tex`+`} |\n| Sine | ${tex`+`} | ${tex`+`} | ${tex`-`} | ${tex`-`} |\n| Cosine | ${tex`+`} | ${tex`-`} | ${tex`-`} | ${tex`+`} |\n| Tangent | ${tex`+`} | ${tex`-`} | ${tex`+`} | ${tex`-`} |\n`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2003,"value":"md` ## Related Notebooks `","pinned":false,"hidden":false,"mode":"ojs"},{"id":2007,"value":"md` Reparameterizing the beta distribution to have a nice interactive handles
\nhttps://observablehq.com/@laotzunami/beta-distribution-ui`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1880,"value":"function map(mean, gamma) {\n function betaPDF(x, alpha, beta, B){\n if(x==0){x=.00000001}\n if(x==1){x=.99999999}\n if(0\nhttps://observablehq.com/@laotzunami/product-of-two-random-variables`","pinned":false,"hidden":false,"mode":"ojs"},{"id":466,"value":"$21","pinned":false,"hidden":false,"mode":"ojs"},{"id":736,"value":"$22","pinned":false,"hidden":false,"mode":"ojs"},{"id":871,"value":"md` ** Other Functions** `","pinned":false,"hidden":false,"mode":"ojs"},{"id":553,"value":"Spline = (await import(\"https://cdn.skypack.dev/cubic-spline@3.0.3?min\"))\n .default","pinned":false,"hidden":false,"mode":"ojs"},{"id":396,"value":"function Dichotomy(a,b) { // beta distribution transformed from [0,1] to [-1,1]\n this.class = \"dichotomy\";\n this.alpha = Math.max(0, a);\n this.beta = Math.max(0, b);\n this.B = cephes.beta(this.alpha,this.beta); \n if(1 < this.alpha && 1 < this.beta){this.support = [-1, 1];}\n else{this.support = [-.9999999999, .9999999999];}\n this.set = (a,b) => {\n this.alpha = Math.max(.01, a);\n this.beta = Math.max(.01, b);\n this.B = cephes.beta(this.alpha,this.beta);\n if(1 <= this.alpha && 1 <= this.beta){this.support = [-1, 1];}\n else{this.support = [-.9999999999, .9999999999];}\n };\n this.f = (x) =>{\n if( x< this.support[0] || this.support[1] {\n return (X.f(x) + X.f(-x))/2\n }\n }\n return new eq()\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1779,"value":"function minimum( X, Y ){\n function min(){\n this.class = \"minimum\"\n this.support = X.support\n this.f = (x) => {\n return Math.min( X.f(x), Y.f(x) )\n }\n }\n return new min()\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":747,"value":"md` **Libraries** `","pinned":false,"hidden":false,"mode":"ojs"},{"id":398,"value":"cephes = {\n const c = await require(\"https://bundle.run/cephes@1.2.0\");\n await c.compiled;\n return c;\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W23","stdlib":"1"}