2b:null 2c:["$","title",null,{"children":"Klein Four Dichotomy Correlation | Lao | Observable"}] 31:T168a,dichotomyCorrelation = { const w = width; const h = w/4; const padding = 4 let columns = 4 let rows = 1 let xTrans = w/columns let yTrans = h/(rows) let domain = [ -1, 1] let range = [0,4] let offset = { x: 0, y: 0 }; let xScale = d3.scaleLinear(domain, [padding, xTrans-padding ]); let yScale = d3.scaleLinear(range, [yTrans-15, 0+15]); let handles = [ {curve: "X", x:xScale(0), y:yScale(.5), a:1.001, b:1.001, xTrans:0, yTrans:0}, {curve: "Y", x:xScale(0), y:yScale(.5), a:1.001, b:1.001, xTrans:1, yTrans:0}, {curve: "Z", x:xScale(0), y:yScale(.5), a:1.001, b:1.001, xTrans:2, yTrans:0}, ] let X = new Dichotomy(handles[0].a, handles[0].b); let Y = new Dichotomy(handles[1].a, handles[1].b); let Z = new Dichotomy(handles[2].a, handles[2].b); let XYZ = intersect(X,intersect(Y,Z)) let X0 = mix(X); let Y0 = mix(Y); let Z0 = mix(Z); let X0Y0Z0 = intersect(X0, intersect(Y0, Z0)) let XYZmin = minimum(XYZ, X0Y0Z0) let t let plot = (U) =>{ let points = [] let step = Math.floor(w/columns-2*padding) for(let i=0; i xScale(d[0])) .y((d) => yScale(d[1])) const clip = svg.append("clipPath") .attr("id", "rect-clip") // to use: .attr("clip-path", "url(#rect-clip)") .append("rect") .attr("x", padding) .attr("y", padding+15) .attr("height", yTrans+15) .attr("width", xTrans-2*padding); let curves = [ {variable: 'X01', color:'#1EDE89', op:.1, x:0, y:0, data: X }, {variable: 'X10', color:'#1EDE89', op:.1, x:1, y:0, data: Y }, {variable: 'X11', color:'#1EDE89', op:.1, x:2, y:0, data: Z }, {variable: 'H_1', color:'#1EDE89', op: 1, x:3, y:0, data: XYZ }, {variable: 'S01', color:'#9022DD', op:.1, x:0, y:0, data: X0 }, {variable: 'S10', color:'#9022DD', op:.1, x:1, y:0, data: Y0 }, {variable: 'S11', color:'#9022DD', op:.1, x:2, y:0, data: Z0 }, {variable: 'H_0', color:'#FF3131', op: 1, x:3, y:0, data: X0Y0Z0 }, {variable: 'min', color:'#000000', op: 1, x:3, y:0, data: XYZmin } ] const path = svg.selectAll("g") .data(curves) .enter() .append('path') .attr("clip-path", "url(#rect-clip)") .attr("transform", (d) => "translate(" + d.x * xTrans + "," + d.y * yTrans + ")") .attr('fill', (d) => d.color) .attr("fill-opacity", (d) => d.op) .attr('stroke', (d) => d.color) .attr('d', (d) => line(plot(d.data))); let textData = [ {text: "X", x:xScale(0), y:15, xTrans:0, yTrans:0, weight:700}, {text: "Y", x:xScale(0), y:15, xTrans:1, yTrans:0, weight:700}, {text: "Z", x:xScale(0), y:15, xTrans:2, yTrans:0, weight:700}, {text: "X⋅Y⋅Z", x:xScale(0), y:15, xTrans:3, yTrans:0, weight:700}, {text: "−", x:xScale(-.85), y:h, xTrans:0, yTrans:0, weight:400}, {text: "0", x:xScale(0), y:h, xTrans:0, yTrans:0, weight:400}, {text: "+", x:xScale(0.85), y:h, xTrans:0, yTrans:0, weight:400}, {text: "−", x:xScale(-.85), y:h, xTrans:1, yTrans:0, weight:400}, {text: "0", x:xScale(0), y:h, xTrans:1, yTrans:0, weight:400}, {text: "+", x:xScale(0.85), y:h, xTrans:1, yTrans:0, weight:400}, {text: "−", x:xScale(-.85), y:h, xTrans:2, yTrans:0, weight:400}, {text: "0", x:xScale(0), y:h, xTrans:2, yTrans:0, weight:400}, {text: "+", x:xScale(0.85), y:h, xTrans:2, yTrans:0, weight:400}, {text: "anti", x:xScale(-.72), y:h, xTrans:3, yTrans:0, weight:400}, {text: "incoherent", x:xScale(0), y:h, xTrans:3, yTrans:0, weight:400}, {text: "coherent", x:xScale(0.72), y:h, xTrans:3, yTrans:0, weight:400}, ] const text = svg.selectAll("g") .data(textData) .enter() .append('text') .attr("transform", (d) => "translate(" + d.xTrans * xTrans + "," + d.yTrans * yTrans + ")") .attr("x", (d) => d.x ) .attr("y", (d) => d.y ) .attr("text-anchor", "middle") .style("font-size", "14px") .attr("font-family", "Arial") .attr("font-weight", (d) => d.weight) .text((d) => d.text); // Mouse Interaction let par function dragged(event, d) { d3.select(this) .attr("cx", () => { d.x = Math.max(xScale(domain[0]), Math.min(xScale(domain[1]), event.x)) return d.x }) .attr("cy", () => { d.y = Math.min(yScale(range[0]), Math.max(yScale(range[1]), event.y)) par = map(xScale.invert(d.x)*.5+.5, yScale.invert(d.y)*2) d.a = par[0] d.b = par[1] return d.y }) .raise(); //brings point to front for( let i=0; i<3; i++){ curves[i].data.set( handles[i].a , handles[i].b ) } XYZ = intersect(X,intersect(Y,Z)) X0Y0Z0 = intersect(X0, intersect(Y0, Z0)) XYZmin = minimum(XYZ, X0Y0Z0) curves[3].data = XYZ curves[7].data = X0Y0Z0 curves[8].data = XYZmin path.attr('d', (d) => line(plot(d.data))); } const dots = svg.selectAll("g") .data(handles) .enter() .append('circle') .attr("transform", (d) => "translate(" + d.xTrans * xTrans + "," + d.yTrans * yTrans + ")") .attr("cx", (d) => d.x) .attr("cy", (d) => d.y) .attr('r', 5) .attr('fill', (d) => '#1EDE89') .attr("fill-opacity", (d) => d.op) .attr('stroke', 'none') .on('mouseover', function() { d3.select(this).attr('r', 7) }) .on('mouseout', function() { d3.select(this).attr('r', 5) }) .call(d3.drag().on("drag", dragged)) // print return svg.node(); }32:T40b,md` Random variables, ${tex`X, Y, Z`} theoretically have a Klein four relationship. To test how well this fits the data, each variable can be split into the observed data ${tex`X_{1}, Y_{1}, Z_{1}`} in green, and the variable if the polarity was random ${tex`X_{0}, Y_{0}, Z_{0}`} in purple. The product of these triads make the null hypothesis in red and alternative hypothesis in green: > ${tex` X_{1} \cdot Y_{1} \cdot Z_{1} = H_1 \\ X_{0} \cdot Y_{0} \cdot Z_{0} = H_0 `} The black region is the overlap of ${tex`H_0`} and ${tex`H_1`}. Since each pair of dichotomies is independent, but the set of three is dependent, all three random variable have to have an average polarity for the null to split from the alternative. If it does, only the splitting of the green to the right is coherent with the model. In practice, it is very unlikely to have a metric that is consistently incoherent, unless something got flipped in the calculation, so the real comparison is between the case of ${tex`H_0 \geq H_1`} and ${tex`H_0 < H_1`}. `33:T1b8d,unitCircle = { const w = Math.min(width, 450); const h = w; const padding = 0; const pi =Math.PI let theta = pi/6 let color = ['#B93CC9', '#2CB1B3', '#55D15D'] // scales let xScale = d3.scaleLinear([-1.5, 1.5], [padding, w - padding]); let yScale = d3.scaleLinear([-1.5, 1.5], [h - padding, padding]); // canvas let svg = d3 .select(DOM.svg(w, h)) let line = d3 .line() .x((d) => xScale(d[0])) .y((d) => yScale(d[1])) let makeCircle = (radius, angle) => { let list = [] let step = Math.round(angle*100) for(let i=0; i [ [[-3,0],[3,0]] , [[0,-1.5],[0,1.5]] ][i] ) .attr('d', line) .attr('fill', 'none') .attr('stroke', '#DDDDDD') let angle = svg.append('path') .datum( makeCircle(.23, theta) ) .attr('d', line) .attr('fill', 'none') .attr('stroke', 'black') let hypotenuse = svg.append('path') .datum( [ [0,0], [Math.cos(theta), Math.sin(theta)] ] ) .attr('d', line) .attr('fill', 'none') .attr('stroke', '#000000') let base= svg.append('path') .datum( [ [0,0], [Math.cos(theta), 0] ] ) .attr('d', line) .attr('fill', 'none') .attr('stroke', color[0]) let height = svg.append('path') .datum( [ [Math.cos(theta),0], [Math.cos(theta), Math.sin(theta)] ] ) .attr('d', line) .attr('fill', 'none') .attr('stroke', color[1]) let tangent = svg.append('path') .datum( [ [Math.cos(theta), Math.sin(theta)] , [1/Math.cos(theta), 0 ] ] ) .attr('d', line) .attr('fill', 'none') .attr('stroke', color[2]) // text let label = [0,0,0] let textData = (a) => { if(Math.cos(a)>=0){label[0] = 0}else{label[0] = 1} if(Math.sin(a)>=0){label[1] = 0}else{label[1] = 1} if(Math.cos(a)*Math.sin(a)>=0){label[2] = 0}else{label[2] = 1} return [ {text:"θ = "+Math.round(a*180/pi)+"°", x:h/2+1, y:w/2+4, color:'#000000', size: '14px', weight: 700, anchor:"middle", opacity:1}, {text:'+', x:xScale(Math.cos(a)), y:yScale(Math.sin(a))+5, color: '#000000', size:'20px', weight: 200, anchor:"middle", opacity:1}, {text:['+','−'][label[0]], x:xScale(.6*Math.cos(a)), y:yScale(.1*Math.sin(a))+5, color: color[0], size:'20px', weight: 200, anchor:"middle", opacity:1}, {text:['+','−'][label[1]], x:xScale(.9*Math.cos(a)), y:yScale(.5*Math.sin(a))+5, color: color[1], size:'20px', weight: 200, anchor:"middle", opacity:1}, {text:['+','−'][label[2]], x:xScale(Math.cos(a)+.1/Math.cos(a)), y:yScale(.98*(1-Math.cos(a)**20)*(Math.sin(a)-.1/(Math.cbrt(Math.sin(a)))) )+5 , color: color[2], size: '20px', weight: 200, anchor:"middle", opacity:1 }, //quadrant I {text:'hyp +', x:w, y: 15, color: '#000000', size: '14px', weight: 700, anchor:"end", opacity:1 }, {text:'cos +', x:w, y: 15+35, color: color[0], size: '14px', weight: 700, anchor:"end", opacity: 1-label[0] }, {text:'sin +', x:w-45, y: 15, color: color[1], size: '14px', weight: 700, anchor:"end", opacity: 1-label[1] }, {text:'tan +', x:w-45, y: 15+35, color: color[2], size: '14px', weight: 700, anchor:"end", opacity: 1-label[2] }, //quadrant II {text:'hyp +', x:0, y: 15, color: '#000000', size: '14px', weight: 700, anchor:"start", opacity:1 }, {text:'cos −', x:0, y: 15+35, color: color[0], size: '14px', weight: 700, anchor:"start", opacity: label[0] }, {text:'sin +', x:45, y: 15, color: color[1], size: '14px', weight: 700, anchor:"start", opacity: 1-label[1] }, {text:'tan −', x:45, y: 15+35, color: color[2], size: '14px', weight: 700, anchor:"start", opacity: label[2] }, //quadrant III {text:'hyp +', x:0, y: h-3, color: '#000000', size: '14px', weight: 700, anchor:"start", opacity:1 }, {text:'cos −', x:0, y: h-3-35, color: color[0], size: '14px', weight: 700, anchor:"start", opacity: label[0] }, {text:'sin −', x:45, y: h-3, color: color[1], size: '14px', weight: 700, anchor:"start", opacity: label[1] }, {text:'tan +', x:45, y: h-3-35, color: color[2], size: '14px', weight: 700, anchor:"start", opacity: 1-label[2] }, //quadrant IV {text:'hyp +', x:w, y: h-3, color: '#000000', size: '14px', weight: 700, anchor:"end", opacity:1 }, {text:'cos +', x:w, y: h-3-35, color: color[0], size: '14px', weight: 700, anchor:"end", opacity: 1-label[0] }, {text:'sin −', x:w-45, y: h-3, color: color[1], size: '14px', weight: 700, anchor:"end", opacity: label[1] }, {text:'tan −', x:w-45, y: h-3-35, color: color[2], size: '14px', weight: 700, anchor:"end", opacity: label[2] } ] } let dynamicText = svg .selectAll("g") .data( textData(theta) ) .enter() .append('text') .attr("x", (d,i) => d.x ) .attr("y", (d,i) => d.y ) .text((d,i) => d.text ) .attr("text-anchor", (d) => d.anchor) .style("font-size", (d) => d.size) .attr("font-family", "Arial") .attr("font-weight",700) .attr('fill', (d) => d.color) .attr("opacity", (d) => d.opacity) .attr("stroke", 'white') .attr("stroke-width", 5) .attr("stroke-linejoin", "round") .attr("paint-order", "stroke"); let x,y let area = svg .append('rect') .attr('width', w) .attr('height', h) .attr("opacity",0) .on('mousemove', function(event) { x = d3.pointer(event)[0]-w/2 y = -d3.pointer(event)[1]+h/2 if( y>=0){ theta = Math.atan2(y, x) } else{ theta = Math.atan2(-y, -x) + pi } angle .datum( makeCircle(.23, theta) ) .attr('d', line) hypotenuse .datum( [ [0,0], [Math.cos(theta), Math.sin(theta)] ] ) .attr('d', line) base .datum( [ [0,0], [Math.cos(theta), 0] ] ) .attr('d', line) .style("stroke-dasharray", () => { if(x>=0){return "0"}else{return "3"} }) height .datum( [ [Math.cos(theta),0], [Math.cos(theta), Math.sin(theta)] ] ) .attr('d', line) .style("stroke-dasharray", () => { if(y>=0){return "0"}else{return "3"} }) tangent .datum( [ [Math.cos(theta), Math.sin(theta)] , [1/Math.cos(theta), 0 ] ] ) .attr('d', line) .style("stroke-dasharray", () => { if(x*y>=0){return "0"}else{return "3"} }) dynamicText .data( textData(theta) ) .text((d,i) => d.text ) .attr("x", (d,i) => d.x ) .attr("y", (d,i) => d.y ) .transition().duration(100).ease(d3.easeLinear) .attr("opacity", (d) => d.opacity) }) // print return svg.node(); }34:Tb22,gaussIntersection = { let cInterval = 2 // number of intervals over the range of c let cn = 40 // number of hyperbolic paths in each product interval let hn = 20 // samples per hyperbolic interval let ln = 101 // number of [s(x),x] points for cubic look up let m = 10 // number arc length gaussian samples per interval let arcInterval = 2 // intervals per hyperbolic path // hyperbolic segment information let array = [] let list, x0, x1, s, interval, c, spline, arcLength, x, y let f = (x,c) => { return Math.sqrt(1 + ( c * x**(-2) )**2 ) } for( let i=0; i 0; }); // index of first 0 d[0]), list.map((d) => d[1]) ) for( let i=0; i d[0]), this.data.map((d) => d[1]) ); this.linear = (x) => { if(x <= this.data[0][0]){ // linear exterpolation to the left of sample points return this.data[0][1] + (x-this.data[0][0]) * (this.data[1][1]-this.data[0][1])/(this.data[1][0]-this.data[0][0]) }else if(this.data[this.data.length-1][0] <= x){ // linear exterpolation to the right of sample points return Math.max(0, this.data[this.data.length-2][1] + (x-this.data[this.data.length-2][0]) * (this.data[this.data.length-1][1]-this.data[0][1])/(this.data[this.data.length-1][0]-this.data[this.data.length-2][0]) ) }else{ // regular linear interpolation index=this.data.findIndex(function(n) { return n[0] > x; }); return Math.max(0, this.data[index-1][1] + (x-this.data[index-1][0]) * (this.data[index][1]-this.data[index-1][1])/(this.data[index][0]-this.data[index-1][0]) ) } } this.f = (x) => { if(x <= this.support[0] || this.support[1] <= x){ return 0 }else if( -.999 < x && x < -.01 || .01 < x && x < .999){ return this.spline.at(x); }else{ return this.linear(x) } }; }; return new intersected() }36:[{"id":"e324e077ecd6f0913a50642ee7f56070e919478ff0e353ef7de4d87e1df311b21827f4243b390814e0f5ea661074984a17ec9c103ed5abe0b02cc799583cc3f8","href":"https://static.observableusercontent.com/files/e324e077ecd6f0913a50642ee7f56070e919478ff0e353ef7de4d87e1df311b21827f4243b390814e0f5ea661074984a17ec9c103ed5abe0b02cc799583cc3f8","name":"gaussNodesWeightsTo100.json","type":"application/json","lastModified":1644102102682.257,"size":198616}] 2f:{"id":"741d2d26d203ca60","slug":"klein-four-dichotomy-correlation","title":"Klein Four Dichotomy Correlation","thumbnail":"cba1430535793a89e104faad0d97a8c5a7b2a2a5ff068795a83372d4cd694124","access_level":"public","create_time":"$D2022-02-04T06:40:02.384Z","update_time":"$D2022-07-01T12:00:55.249Z","view_time":null,"first_public_time":"$D2022-02-19T13:39:01.868Z","fork":false,"stars":12,"starred":false,"owner":{"type":"team","id":"cf45be0120eccaad","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/aeddc181df6ca80a347940ba62e17601ac6395e9d2c21142887eca5fd277326a","tier":"starter_2024"},"version":2046,"publish_version":2046,"role":null,"authored":false,"authors":[{"type":"individual","id":"5da1d0646b8496ac","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/9c14da488e1a68ca1868ed025eb7edcce3d300c8a9f3fbcdae66a3daa9ccca63","tier":null}],"members":[{"type":"individual","id":"5da1d0646b8496ac","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/9c14da488e1a68ca1868ed025eb7edcce3d300c8a9f3fbcdae66a3daa9ccca63","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Klein Four Dichotomy Correlation","cells":[{"id":3,"value":"md` # Klein Four Dichotomy Correlation `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1642,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1980,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":863,"value":"md` ## Klein Four Group`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"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 ${tex`C_2 \\times C_2`}. Implicit in the Klein four group is three dichotomy subgroups - the first two are the dichotomies that generated the Klein four group, and the third is a new dichotomy.\n\nAn example of the Klein four group from basic mathimatics is the charge of sine, cosine, and tangent in the 4 quadrants of the XY plane.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":64,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":318,"value":"md` In this geometric example, the original 2 dichotomies come from the plus and minus directions of the x and y axes. A point on the unit circle can be measure in the x direction by cosine and in the y direction by sine. Tangent is the ratio of sine divided by cosine, and it inherits its charge by properties of division:\n\n${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","author":"$undefined","summary":"$undefined"},{"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","author":"$undefined","summary":"$undefined"},{"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`+`} | ${tex`-`} |\n`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2003,"value":"md` ## Related Notebooks `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2007,"value":"md` Reparameterizing the beta distribution to have a nice interactive handles
\nhttps://observablehq.com/@laotzunami/beta-distribution-ui`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1880,"value":"function map(mean, gamma) {\n function betaPDF(x, alpha, beta, B){\n if(x==0){x=.00000001}\n if(x==1){x=.99999999}\n if(0\nhttps://observablehq.com/@laotzunami/product-of-two-random-variables`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":466,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":736,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":871,"value":"md` ** Other Functions** `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":553,"value":"Spline = (await import(\"https://cdn.skypack.dev/cubic-spline@3.0.3?min\"))\n .default","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":396,"value":"function Dichotomy(a,b) { // beta distribution transformed from [0,1] to [-1,1]\n this.class = \"dichotomy\";\n this.alpha = Math.max(0, a);\n this.beta = Math.max(0, b);\n this.B = cephes.beta(this.alpha,this.beta); \n if(1 < this.alpha && 1 < this.beta){this.support = [-1, 1];}\n else{this.support = [-.9999999999, .9999999999];}\n this.set = (a,b) => {\n this.alpha = Math.max(.01, a);\n this.beta = Math.max(.01, b);\n this.B = cephes.beta(this.alpha,this.beta);\n if(1 <= this.alpha && 1 <= this.beta){this.support = [-1, 1];}\n else{this.support = [-.9999999999, .9999999999];}\n };\n this.f = (x) =>{\n if( x< this.support[0] || this.support[1] {\n return (X.f(x) + X.f(-x))/2\n }\n }\n return new eq()\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1779,"value":"function minimum( X, Y ){\n function min(){\n this.class = \"minimum\"\n this.support = X.support\n this.f = (x) => {\n return Math.min( X.f(x), Y.f(x) )\n }\n }\n return new min()\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":747,"value":"md` **Libraries** `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":398,"value":"cephes = {\n const c = await require(\"https://bundle.run/cephes@1.2.0\");\n await c.compiled;\n return c;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W36","stdlib":"1","titling":"auto"}} 30:"$2f:body"